OMTEX CLASSES

Expansion of Ideas: As You Sow, So Shall You Reap | Class 10, 11, 12

aaminomtex@gmail.com (Amin) — Sat, 31 Oct 2026 06:58:31 +0530

Welcome to our comprehensive guide on Expansion of Ideas for Class 10, Class 11, and Class 12 English Grammar. Mastering proverb expansion is a crucial part of the writing skills section in Maharashtra board exams. Below, you will find the exact exam-ready paragraph for the universal proverb: "As You Sow, So Shall You Reap", followed by a detailed breakdown to help you understand the core concept better.

Expansion of Idea: As You Sow, So Shall You Reap (Exam Answer)

(Note: Writing the following paragraph in your exams is sufficient to score full marks.)

The proverb "As you sow, so shall you reap" highlights the universal law of cause and effect. It signifies that our current actions directly determine our future outcomes. Just as a farmer who plants wheat will only harvest wheat and never rice, a person who engages in good deeds will receive goodness, while one who acts with deceit will eventually face misery. In daily life, a student who studies with dedication and honesty will naturally secure excellent marks, whereas one who wastes time will face failure. History shows us that selfless leaders like Florence Nightingale are remembered with reverence, while cruel dictators face ruin. As the parallel saying notes, "What goes around comes around." Thus, we must always sow the seeds of virtue, hard work, and kindness to pluck the golden fruits of success and happiness.

Detailed Breakdown (For Better Understanding)

If you want to understand the concept deeply or expand the paragraph further, here are the core points:

Law of Cause and Effect: Every action we take generates a corresponding consequence in the future.
The Farming Metaphor: You cannot expect a positive outcome if you have planted negative seeds (bad actions).
Personal Responsibility: We are the architects of our own destiny; we cannot blame luck if we did not put in the effort.

💡 Exam Tip: To score maximum marks in this section, always include a parallel proverb in your concluding sentences. For this topic, you can write: "What goes around comes around" or "Every action has an equal and opposite reaction."

Quick Revision Q&A

✅ View Answer: What is the core message of this proverb?

The core message is that our future is entirely dependent on our present actions. Good deeds lead to good results, while bad deeds lead to negative consequences.

✅ View Answer: How does the farming metaphor explain this proverb?

Just as a farmer only harvests the exact type of seed they planted (e.g., wheat brings wheat), a person only receives the results of the specific actions they performed.

👇 Bonus Question for You!

Can you think of a time when your hard work (good seeds) eventually led to a beautiful result (good harvest)? Share your success story in the comments!

Expansion of Ideas: Nothing Is Gained by Weeping Over Past Errors

aaminomtex@gmail.com (Amin) — Sat, 31 Oct 2026 06:57:30 +0530

Welcome to our comprehensive guide on Expansion of Ideas for Class 10, Class 11, and Class 12 English Grammar. Mastering proverb expansion is a crucial part of the writing skills section in Maharashtra board exams. Below, you will find the exact exam-ready paragraph for the proverb: "Nothing Is Gained by Weeping Over Past Errors", followed by a detailed breakdown to help you understand the core concept better.

Expansion of Idea: Nothing Is Gained by Weeping Over Past Errors (Exam Answer)

(Note: Writing the following paragraph in your exams is sufficient to score full marks.)

The proverb "Nothing is gained by weeping over past errors" teaches us the futility of regretting past mistakes. Once an error is made, no amount of sorrow, crying, or guilt can undo it. Dwelling on past failures only drains our current energy and prevents us from focusing on future opportunities. Instead of lamenting, we should analyze our errors and use them as stepping stones to success. For example, a student who fails an exam due to poor preparation will gain nothing by crying over the result. However, if they learn from this mistake and study consistently for the next exam, they can achieve a brilliant score. As the parallel saying goes, "There is no use crying over spilled milk." Therefore, we must always look forward, learn from our past, and work hard to build a better future.

Detailed Breakdown (For Better Understanding)

If you want to understand the concept deeply or expand the paragraph further, here are the core points:

The Futility of Regret: Crying over the past is a waste of time because the past cannot be changed.
Energy Drain: Excessive guilt and sorrow take away the mental energy needed to improve the present.
Learning Opportunity: Mistakes should not be viewed as dead ends, but rather as valuable lessons for future success.

💡 Exam Tip: To score maximum marks in this section, strictly follow the format: Title → Introduction → Example → Concluding Sentence with a parallel quote. You can use "There is no use crying over spilled milk" or "Let bygones be bygones."

Quick Revision Q&A

✅ View Answer: What should be the attitude towards past mistakes according to this proverb?

We should not regret or cry over past mistakes. Instead, we should analyze them, learn valuable lessons, and focus our energy on improving our future.

✅ View Answer: Which parallel proverbs can be used to conclude this expansion?

You can use "There is no use crying over spilled milk" or "Let bygones be bygones" to perfectly conclude your paragraph.

👇 Bonus Question for You!

Have you ever made a mistake that initially made you sad, but later taught you an unforgettable lesson? Share your experience in the comments below!

Expansion of Ideas: Nothing Is Gained by Weeping Over Past Errors

aaminomtex@gmail.com (Amin) — Sat, 31 Oct 2026 06:55:47 +0530

Welcome to our comprehensive guide on Expansion of Ideas for Class 10, Class 11, and Class 12 English Grammar. Mastering proverb expansion is a crucial part of the writing skills section in Maharashtra board exams. Below, you will find the exact exam-ready paragraph for the proverb: "Nothing Is Gained by Weeping Over Past Errors", followed by a detailed breakdown to help you understand the core concept better.

Expansion of Idea: Nothing Is Gained by Weeping Over Past Errors (Exam Answer)

(Note: Writing the following paragraph in your exams is sufficient to score full marks.)

Detailed Breakdown (For Better Understanding)

If you want to understand the concept deeply or expand the paragraph further, here are the core points:

The Futility of Regret: Crying over the past is a waste of time because the past cannot be changed.
Energy Drain: Excessive guilt and sorrow take away the mental energy needed to improve the present.
Learning Opportunity: Mistakes should not be viewed as dead ends, but rather as valuable lessons for future success.

💡 Exam Tip: To score maximum marks in this section, strictly follow the format: Title → Introduction → Example → Concluding Sentence with a parallel quote. You can use "There is no use crying over spilled milk" or "Let bygones be bygones."

Quick Revision Q&A

✅ View Answer: What should be the attitude towards past mistakes according to this proverb?

We should not regret or cry over past mistakes. Instead, we should analyze them, learn valuable lessons, and focus our energy on improving our future.

✅ View Answer: Which parallel proverbs can be used to conclude this expansion?

You can use "There is no use crying over spilled milk" or "Let bygones be bygones" to perfectly conclude your paragraph.

👇 Bonus Question for You!

Have you ever made a mistake that initially made you sad, but later taught you an unforgettable lesson? Share your experience in the comments below!

Expansion of Ideas: The Family Is the Nucleus of Civilization

aaminomtex@gmail.com (Amin) — Sat, 31 Oct 2026 06:42:53 +0530

Welcome to our comprehensive guide on Expansion of Ideas for Class 10, Class 11, and Class 12 English Grammar. Mastering proverb expansion is a crucial part of the writing skills section in Maharashtra board exams. Below, you will find the exact exam-ready paragraph for the statement: "The Family Is the Nucleus of Civilization", followed by a detailed breakdown to help you understand the core concept better.

Expansion of Idea: The Family Is the Nucleus of Civilization (Exam Answer)

(Note: Writing the following paragraph in your exams is sufficient to score full marks.)

The statement "The family is the nucleus of civilization" highlights that the family unit is the most fundamental building block of society. Just as a nucleus controls a biological cell, the moral values, discipline, and culture instilled within a family directly shape the character and progress of the entire human civilization. A child first learns the virtues of honesty, empathy, and respect at home. If a family is broken or lacks moral grounding, the resulting society becomes chaotic, whereas a loving family provides emotional comfort and a sense of belonging. Hence, to build a strong, peaceful, and civilized world, we must first nurture love and good manners within our own homes. As the wise saying goes, "Charity begins at home," reminding us that all societal good originates from a well-nurtured family.

Detailed Breakdown (For Better Understanding)

If you want to understand the concept deeply or expand the paragraph further, here are the core points:

The First School: A child's primary education in morals, ethics, and behavior happens entirely within the family.
The Biological Metaphor: Just as a cell cannot function without a nucleus, a civilization cannot sustain itself without stable families.
Ripple Effect: Peaceful homes create well-mannered individuals, who in turn build a disciplined and harmonious society.

💡 Exam Tip: To score maximum marks, use a parallel proverb in your conclusion. For this topic, you can write: "Charity begins at home" or "The hand that rocks the cradle rules the world."

Quick Revision Q&A

✅ View Answer: Why is the family compared to a "nucleus"?

A nucleus controls the functions of a biological cell. Similarly, the family acts as the central control center for society, instilling the core values that shape how civilization operates.

✅ View Answer: What happens if the family unit lacks moral grounding?

If families fail to teach basic virtues like empathy and respect, the individuals grow up without moral grounding, leading to a chaotic and uncivilized society.

👇 Bonus Question for You!

What is the most important moral value or life lesson you have learned directly from your parents or grandparents? Share your family's wisdom in the comments below!

Expansion of Ideas: A Joy Shared Is a Joy Made Double | Class 10, 11, 12

aaminomtex@gmail.com (Amin) — Sat, 31 Oct 2026 06:41:35 +0530

Welcome to our comprehensive guide on Expansion of Ideas for Class 10, Class 11, and Class 12 English Grammar. Mastering proverb expansion is a crucial part of the writing skills section in Maharashtra board exams. Below, you will find the exact exam-ready paragraph for the proverb: "A Joy Shared Is a Joy Made Double", followed by a detailed breakdown to help you understand the core concept better.

Expansion of Idea: A Joy Shared Is a Joy Made Double (Exam Answer)

(Note: Writing the following paragraph in your exams is sufficient to score full marks.)

The proverb "A joy shared is a joy made double" beautifully captures the essence of human empathy and togetherness. It means that happiness grows exponentially when it is celebrated with others, while keeping joy to oneself restricts its true potential to spread light and positivity. Friends and family are essential because they share our happiness and provide emotional support during our problems. During festivals or personal milestones, celebrating with loved ones instantly amplifies the festive spirit. Conversely, sharing sorrow divides the pain, making it easier to bear. Therefore, we should always share our moments of happiness, as collective joy builds the strongest human bonds. As the Swedish proverb rightly notes, "Shared joy is a double joy; shared sorrow is half a sorrow."

Detailed Breakdown (For Better Understanding)

If you want to understand the concept deeply or expand the paragraph further, here are the core points:

The Multiplier Effect: Happiness is unique because giving it away or sharing it actually increases its overall amount.
Human Connection: We are social beings; our milestones mean little without loved ones to celebrate them with.
The Flip Side (Sorrow): Just as sharing multiplies happiness, sharing grief naturally divides and reduces emotional pain.

💡 Exam Tip: To score maximum marks, provide the beautiful contrast in your conclusion. A perfect parallel quote is: "Happiness is only real when shared" or "Shared joy is a double joy; shared sorrow is half a sorrow."

Quick Revision Q&A

✅ View Answer: What is the central message of this proverb?

The central message is that celebrating your success and happiness with friends and family amplifies the feeling of joy and strengthens your relationships.

✅ View Answer: How does sharing affect sorrow compared to joy?

While sharing joy doubles and multiplies the happiness, sharing sorrow divides the pain and makes it significantly easier to endure.

👇 Bonus Question for You!

Think of your favorite festival or personal achievement. Would it have felt the same if you celebrated it completely alone? Share your thoughts in the comments below!

Expansion of Ideas: Faith Will Move Mountains | Class 10, 11, 12 English

aaminomtex@gmail.com (Amin) — Sat, 31 Oct 2026 06:39:52 +0530

Welcome to our comprehensive guide on Expansion of Ideas for Class 10, Class 11, and Class 12 English Grammar. Mastering proverb expansion is a crucial part of the writing skills section in Maharashtra board exams. Below, you will find the exact exam-ready paragraph for the proverb: "Faith Will Move Mountains", followed by a detailed breakdown to help you understand the core concept better.

Expansion of Idea: Faith Will Move Mountains (Exam Answer)

(Note: Writing the following paragraph in your exams is sufficient to score full marks.)

The proverb "Faith will move mountains" is a profound maxim that emphasizes the incredible power of self-belief and unwavering confidence. It signifies that if a person has absolute trust in their abilities and their purpose, they can overcome even the most monumental and seemingly impossible obstacles in life. When people are beset with problems, they must possess the courage to face them with strong willpower, as confidence turns barriers into bridges. For example, freedom fighters faced the mighty British Empire armed primarily with an unshakable faith in their righteous cause. Believing in oneself is believing in the best, and true faith always paves the way to triumph. As the parallel saying goes, "Where there is a will, there is a way." Therefore, holding onto faith is the ultimate key to achieving the impossible.

Detailed Breakdown (For Better Understanding)

If you want to understand the concept deeply or expand the paragraph further, here are the core points:

Unshakable Self-Belief: Having absolute confidence in your goals gives you the mental strength to endure hardships.
Overcoming Impossible Odds: The "mountain" is a metaphor for massive obstacles that shrink when faced with sheer willpower.
Action Driven by Trust: Faith is not just passive hoping; it is the fuel that drives persistent and courageous action.

💡 Exam Tip: To score maximum marks in this section, conclude with a parallel proverb. For this topic, you can write: "Where there is a will, there is a way" or "Believe you can and you're halfway there."

Quick Revision Q&A

✅ View Answer: What does the word "mountains" signify in this proverb?

In this proverb, "mountains" serve as a metaphor for enormous difficulties, monumental challenges, or seemingly impossible obstacles in one's life.

✅ View Answer: Which parallel proverbs can be used to conclude this expansion?

You can use "Where there is a will, there is a way" or "Believe you can and you're halfway there" to effectively conclude your paragraph.

👇 Bonus Question for You!

Who is someone in history or your personal life who achieved something impossible just because they had unshakable faith? Let us know in the comments!

Expansion of Ideas: To the One Who Tries and Fails | Class 10, 11, 12 English

aaminomtex@gmail.com (Amin) — Sat, 31 Oct 2026 06:38:40 +0530

Welcome to our comprehensive guide on Expansion of Ideas for Class 10, Class 11, and Class 12 English Grammar. Mastering proverb expansion is a crucial part of the writing skills section in Maharashtra board exams. Below, you will find the exact exam-ready paragraph for the phrase: "To the One Who Tries and Fails", followed by a detailed breakdown to help you understand the core concept better.

Expansion of Idea: To the One Who Tries and Fails (Exam Answer)

(Note: Writing the following paragraph in your exams is sufficient to score full marks.)

The phrase "To the one who tries and fails" highlights the immense value of making an honest effort, regardless of the final outcome. It teaches us that failure is not a mark of disgrace, but rather a badge of courage, proving that a person dared to step out of their comfort zone and take a calculated risk. History always remembers those who tried. For example, great scientists failed thousands of times before making revolutionary discoveries, yet every failure served as a lesson for improvement. A student who works hard but scores less is far superior to one who avoids the challenge altogether. True victory lies in the effort itself. As the famous saying goes, "Failure is the stepping stone to success." Therefore, one must always have the courage to try, for an honest attempt is never a true failure.

Detailed Breakdown (For Better Understanding)

If you want to understand the concept deeply or expand the paragraph further, here are the core points:

Valuing Effort Over Outcome: Success isn't guaranteed, but the willingness to try shows genuine character and bravery.
Learning Mechanism: Every failure provides a valuable lesson that pushes you closer to your ultimate goal.
The Real Defeat: True failure is not trying at all due to the fear of judgment or making mistakes.

💡 Exam Tip: To score maximum marks in this section, strictly follow the format: Title → Introduction → Example/Core Idea → Concluding Sentence with a parallel quote. You can use "Failure is the stepping stone to success" or "Ever tried. Ever failed. No matter. Try again. Fail again. Fail better."

Quick Revision Q&A

✅ View Answer: What should be written in the exam for this topic?

Simply write the single paragraph provided under the "Exam Answer" heading above. It contains the meaning, an example of scientists, and a concluding parallel quote.

✅ View Answer: Who is considered superior in this context?

A person who attempts a difficult task and fails is considered far superior to someone who completely avoids the challenge out of fear.

👇 Bonus Question for You!

Can you share a moment in your life where you tried hard at something and failed, but learned a very important lesson from it? Share your experience in the comments below!

Expansion of Ideas: The First Blow Is Half the Battle | Class 10, 11, 12 English

aaminomtex@gmail.com (Amin) — Sat, 31 Oct 2026 06:29:40 +0530

Welcome to our comprehensive guide on Expansion of Ideas for Class 10, Class 11, and Class 12 English Grammar. Mastering proverb expansion is a crucial part of the writing skills section in Maharashtra board exams. Below, you will find the exact exam-ready paragraph for the proverb: "The First Blow Is Half the Battle", followed by a detailed breakdown to help you understand the core concept better.

Expansion of Idea: The First Blow Is Half the Battle (Exam Answer)

(Note: Writing the following paragraph in your exams is sufficient to score full marks.)

The proverb "The first blow is half the battle" implies that taking the initial step toward a difficult task is often the hardest part. Once a person overcomes their initial hesitation and begins with determination, the momentum carries them forward, making the rest of the journey significantly easier to conquer. For instance, a student terrified of a massive syllabus often finds that simply sitting down to study the first chapter removes their anxiety. Taking action eliminates fear. As the parallel saying goes, "Well begun is half done." Therefore, one must always take that courageous first step to guarantee future success.

Detailed Breakdown (For Better Understanding)

If you want to understand the concept deeply or expand the paragraph further in case of higher weightage questions, here are the core points:

Overcoming Hesitation: The most significant barrier to success is often our own fear or laziness to start.
Building Momentum: Taking that very first action generates positive momentum.
Mental Victory: Making the firm decision to start shifts your mindset from "impossible" to "achievable."

💡 Exam Tip: To score maximum marks in this section, strictly follow the format: Title → Introduction of the Proverb → Example/Core Idea → Concluding Sentence with a parallel proverb.

Quick Revision Q&A

✅ View Answer: What should be written in the exam for this topic?

Simply write the single paragraph provided under the "Exam Answer" heading above. It contains the meaning, an example, and a concluding parallel proverb, which is exactly what examiners look for.

✅ View Answer: Which parallel proverbs can be used to conclude this expansion?

You can use "Well begun is half done" or "A journey of a thousand miles begins with a single step" to effectively conclude your paragraph.

👇 Bonus Question for You!

Can you think of a real-life situation where you finally took the "first blow" and successfully completed a task you were initially afraid of? Share your experience in the comments below!

HSC Class 12 Economics June/July 2026 Board Paper Solutions (J-320)

aaminomtex@gmail.com (Amin) — Tue, 27 Oct 2026 22:50:23 +0530

Prepare effectively for your upcoming board examinations with the complete, step-by-step HSC Class 12 Economics June/July 2026 Board Paper Solutions (Subject Code: J-320). Conducted by the Maharashtra State Board (MSBSHSE) on 1st July 2026, this 80-mark Board Question Paper is solved strictly according to the official textbook and board marking scheme, featuring neat comparison tables, point-wise explanations, and exam-scoring tips.

HSC Economics June/July 2026 Board Paper Solutions (Subject Code: J-320)

Subject: Economics (49) | Date: 01/07/2026 | Time: 3 Hours | Total Marks: 80

Q. 1. (A) Complete the following statements : (5 Marks)

$E_d = 0$ in case of Necessary goods.
- (a) Luxury goods
- (b) Normal goods
- (c) Necessary goods
- (d) Durable goods
Answer: (c) Necessary goods
Balance of trade is also referred to as International trade balance.
- (a) National trade balance
- (b) International trade balance
- (c) Balance of payment
- (d) Systematic record of all international economic transactions
Answer: (b) International trade balance
The famous book "General Theory of Employment, Interest and Money" was published by Lord J.M. Keynes.
- (a) Prof. Alfred Marshall
- (b) David Ricardo
- (c) Lord J.M. Keynes
- (d) Adam Smith
Answer: (c) Lord J.M. Keynes
When a government borrows from its citizens, banks, central bank, industries, etc. it is known as Internal Debt.
- (a) Internal Debt
- (b) Personal Debt
- (c) External Debt
- (d) International Debt
Answer: (a) Internal Debt
A period of time in which all factors of production and costs are variable is known as Long period.
- (a) Very short period
- (b) Short period
- (c) Long period
- (d) Very long period
Answer: (c) Long period

Q. 1. (B) Complete the correlation : (5 Marks)

Service utility : Teacher : : Form utility : Toys made of clay
Perfect competition : Large number of sellers : : Oligopoly : Few sellers
Theoretical difficulty : Unpaid services : : Practical difficulty (Statistical difficulty) : Valuation of inventories
Pen and ink : Complementary goods : : Tea and coffee : Substitutes
Trends in exports : Petroleum products : : Trends in imports : Gold

Q. 1. (C) Give economic terms : (5 Marks)

A market for long term funds, both equity and debt raised within and outside the country.

Answer: Capital Market
Demand for a commodity which can be put to several uses.

Answer: Composite Demand
The total quantity of commodity available for sale with a seller at a particular point of time.

Answer: Stock
Index number which is constructed with some specific purpose.

Answer: Special Purpose Index Number
Budget in which the estimated revenue and expenditure of the government are equal.

Answer: Balanced Budget

Q. 1. (D) Find the odd word : (5 Marks)

Scope of macro economics : Theory of income and employment, Theory of product pricing, Theory of general price level and inflation, Macro theory of distribution

Odd word: Theory of product pricing (It falls under the scope of microeconomics.)
Methods of measurement of national income : Output method, Income method, Expenditure method, Ratio method

Odd word: Ratio method (It is a method of measuring price elasticity of demand.)
Reasons for downward sloping demand curve : Law of supply, Income effect, Substitution effect, New consumers

Odd word: Law of supply (It explains the upward-sloping supply curve.)
Exceptions to the law of supply : Agricultural goods, Perishable goods, Rare goods, Giffen's paradox

Odd word: Giffen's paradox (It is an exception to the law of demand.)
Types of utility : Form utility, Total utility, Time utility, Place utility

Odd word: Total utility (It is a concept of measuring utility, not a type of utility.)

Q. 2. (A) Identify and explain the following concepts from the given illustrations (Any THREE) : (6 Marks)

Exam Tip: In Q. 2 (A), the board marking scheme awards 1 mark for correctly identifying the economic concept and 1 mark for writing its crisp textbook explanation. Always write both clearly under separate headings.

Illustration: Arnav sells 80 dozens of mangoes daily at ₹ 500/- per dozen.

Concept: Individual Supply (or Supply)

Explanation: Individual supply refers to the quantity of a commodity that an individual producer or seller is willing and able to offer for sale at a given price during a specific period of time. Here, Arnav's daily sale of 80 dozens of mangoes at ₹ 500/- per dozen represents individual supply.
Illustration: Demand for sugar falls by 10% when the price of sugar rises by 10%.

Concept: Unitary Elastic Demand $(E_d = 1)$

Explanation: When a percentage change in price leads to an exactly equal (proportionate) percentage change in quantity demanded, the demand is said to be unitary elastic. Here, a 10% rise in the price of sugar brings about an equal 10% fall in its demand.
Illustration: Reena deposited a lumpsum amount of ₹ 50,000 in the bank for a period of one year.

Concept: Fixed Deposit (Time Deposit)

Explanation: A fixed deposit is a type of bank deposit in which a saver deposits a lump sum amount for a specified fixed period of time. It carries a higher rate of interest compared to a savings account, and the amount cannot normally be withdrawn before the maturity period without a penalty.
Illustration: Shabana paid wages to workers in her factory and interest on her bank loan.

Concept: Theory of Factor Pricing (or Factor Payment)

Explanation: In microeconomics, the theory of factor pricing explains how factor rewards are determined for the factors of production (land, labour, capital, and entrepreneur) in the form of rent, wages, interest, and profit. Here, Shabana is making factor payments by paying wages to labour and interest on capital.
Illustration: Vrinda receives monthly pension of ₹ 5000/- from the state government.

Concept: Transfer Income (Transfer Payment)

Explanation: Transfer income refers to a one-sided payment received by an individual without rendering any productive service in return during the current period. Grants, scholarships, and old-age or retirement pensions are transfer payments and are excluded from national income.

Q. 2. (B) Distinguish between (Any THREE) : (6 Marks)

Exam Tip: Each distinction carries 2 marks. Always write at least two clear points of distinction (Meaning + 1 specific point or Formula/Example) in a neat two-column table to secure full marks.

(i) Direct Demand and Indirect Demand

Points	Direct Demand	Indirect Demand
1. Meaning	Direct demand is the demand for goods and services by consumers which satisfy their wants directly.	Indirect demand (also known as derived demand) is the demand for goods which are needed for the further production of other goods.
2. Nature of Goods	All consumer goods (consumption goods) have direct demand.	All producer goods (capital goods and factors of production) have indirect demand.
3. Dependence	It depends directly on the utility, habits, and preferences of the consumer.	It is derived from the demand for the final consumption goods that it helps to produce.
4. Example	Demand for food, clothes, books, house, etc.	Demand for workers, machinery, raw materials, land, etc.

(ii) Laspeyre's Price Index Number and Paasche's Price Index Number

Points	Laspeyre's Price Index Number	Paasche's Price Index Number
1. Meaning	Laspeyre's Index Number is a weighted index number formulated by Etienne Laspeyre in 1871 using base year quantities as weights.	Paasche's Index Number is a weighted index number formulated by Hermann Paasche in 1874 using current year quantities as weights.
2. Weights Used	In this method, the base year quantities $(q_0)$ are taken as weights.	In this method, the current year quantities $(q_1)$ are taken as weights.
3. Formula	$P_{01}(L) = \frac{\sum p_1 q_0}{\sum p_0 q_0} \times 100$	$P_{01}(P) = \frac{\sum p_1 q_1}{\sum p_0 q_1} \times 100$

(iii) Direct Tax and Indirect Tax

Points	Direct Tax	Indirect Tax
1. Meaning	Direct tax refers to a tax that is paid directly by the person on whom it is legally imposed.	Indirect tax refers to a tax that is imposed on one person (producer or seller) but paid partly or wholly by another person (consumer).
2. Impact and Incidence	The impact (initial burden) and incidence (ultimate burden) of the tax fall on the same person.	The impact and incidence of the tax fall on different persons.
3. Shifting of Burden	The taxpayer cannot shift the burden of this tax to others.	The tax burden can be shifted by the producer or seller to the final consumer.
4. Example	Personal Income Tax, Wealth Tax, Property Tax.	Goods and Services Tax (GST), Custom Duty.

(iv) Micro Economics and Macro Economics

Points	Micro Economics	Macro Economics
1. Meaning	Micro economics studies the economic behaviour of small individual units such as an individual consumer, firm, or household.	Macro economics studies the behaviour of the economy as a whole, focusing on large aggregates.
2. Method of Study	It uses the slicing method, splitting the whole economy into small individual units for intensive study.	It uses the lumping method, grouping the entire economy into big aggregates for broad analysis.
3. Basic Tools	Individual demand and individual supply are its primary tools.	Aggregate demand and aggregate supply are its primary tools.
4. Alternative Name	It is also known as Price Theory.	It is also known as Income and Employment Theory.

(v) Total Cost and Total Revenue

Points	Total Cost (TC)	Total Revenue (TR)
1. Meaning	Total cost is the total expenditure incurred by a firm on the factors of production required to produce a given level of output.	Total revenue is the total sales proceeds (receipts) earned by a firm from selling a given quantity of a commodity at a given price.
2. Formula	$\text{TC} = \text{TFC} + \text{TVC}$ (Total Fixed Cost + Total Variable Cost)	$\text{TR} = P \times Q$ (Price $\times$ Quantity Sold)
3. Nature of Flow	It represents the outflow of money (expenses) for the producer.	It represents the inflow of money (income) for the producer.

Q. 3. Answer the following (Any THREE) : (12 Marks)

(i) Explain any four features of utility.

Answer: Utility is the capacity of a commodity to satisfy human wants. The main features of utility are as follows:

Relative Concept: Utility is related to time and place. It varies from time to time and place to place. For example, woollen clothes have a greater utility in the winter season or in cold regions like Kashmir than in summer or in Mumbai.
Subjective Concept: Utility is a psychological concept. It differs from person to person due to differences in tastes, preferences, likes, dislikes, natures, and habits. For example, a stethoscope has immense utility to a doctor, but it has no utility to a layman.
Ethically Neutral Concept: The concept of utility has no ethical or moral considerations. It is a morally colourless concept. A commodity may satisfy any want of a person without considering whether it is good or bad. For example, a knife has utility to cut fruits and vegetables for a homemaker, and it can also be used by a criminal to harm someone. In economics, both possess utility.
Utility Differs from Usefulness: Utility is the capacity of a commodity to satisfy a human want, whereas usefulness indicates the benefit or value in use of the commodity. A commodity may possess utility even if it is not useful. For example, liquor or cigarettes have utility to an addict, even though they are injurious to health and have no usefulness.
Utility Differs from Pleasure: A commodity may possess utility, yet it may not give any pleasure to the consumer. For example, a bitter medicine or an injection has utility for a sick patient because it cures the illness, although taking it hardly gives any pleasure.

(ii) Explain any four roles of foreign trade.

Answer: Foreign trade is the trade across the boundaries of a nation, consisting of imports and exports of goods and services. The roles of foreign trade are explained below:

To Earn Foreign Exchange: Foreign trade plays a very important role in earning foreign exchange. The foreign exchange earned through exports can be utilized to import advanced capital goods, raw materials, and technical know-how, which accelerates the pace of economic development.
Encourages Investment: Foreign trade creates an opportunity for producers to reach beyond the domestic markets and sell their products in international markets. This encourages them to produce more goods for export, leading to an increase in total investment in the country.
Division of Labour and Specialization: Foreign trade leads to the division of labour and specialization at the global level. Some countries have abundant natural resources, while others have advanced technology or skilled labour. Foreign trade enables each country to specialize in the production of those goods in which it has a comparative cost advantage.
Optimum Allocation and Utilization of Resources: Due to specialization, resources are channelized into the production of only those goods which yield the highest returns. Thus, there is rational allocation and optimum utilization of scarce resources at both national and international levels.
Stability in Price Level: Foreign trade helps to keep the demand and supply position stable. Whenever there is a shortage of goods in the domestic market, imports can be made to arrest rising prices, and surplus output can be exported to prevent a sharp fall in prices, thereby maintaining stability in the price level.

(iii) Explain any four features of micro economics.

Answer: Micro economics is a branch of economics that studies the economic behaviour of small individual units. Its key features are:

Study of Individual Units: Micro economics is the study of the behaviour of small individual economic units, like an individual firm, an individual price, an individual household, etc.
Price Theory: Micro economics deals with the determination of the prices of goods and services as well as the prices of factors of production (rent, wages, interest, and profit). Hence, it is also known as Price Theory.
Partial Equilibrium: Equilibrium is the balance between two factors. Micro economic analysis is a partial equilibrium analysis as it isolates an individual unit from other forces and studies its equilibrium independently, neglecting the interdependence between economic variables.
Slicing Method: Micro economics uses the slicing method. It splits or divides the whole economy into small individual units and then studies each unit separately in detail, such as studying individual income out of national income.
Based on Certain Assumptions: Micro economics begins with the fundamental assumption of "Other things remaining constant" (Ceteris Paribus), such as full employment, pure capitalism, perfect competition, and laissez-faire policy. These assumptions make the analysis simpler.

(iv) Explain any four factors influencing elasticity of demand.

Answer: Elasticity of demand refers to the responsiveness of quantity demanded to a change in price. The main factors influencing the elasticity of demand are:

Nature of Commodity: By nature, we can classify commodities as necessaries, comforts, and luxuries. Demand for necessaries like foodgrains, salt, and medicines is relatively inelastic because they are essential for survival. On the other hand, demand for comfort and luxury goods like cars, perfumes, and air conditioners is relatively elastic.
Availability of Substitutes: Demand for a commodity will be more elastic if its close substitutes are easily available in the market (for example, lemon juice, sugarcane juice, and cold drinks). If the price of one rises, consumers shift to its substitute. Conversely, commodities having no close substitutes, like salt, have inelastic demand.
Number of Uses: Single-use goods have a less elastic demand. Multi-use (composite) goods have more elastic demand. For example, coal or electricity can be used for cooking, heating, lighting, and industrial purposes. A fall in the price of electricity will lead to a substantial increase in its demand across various uses.
Habits: Habits make the demand for certain goods relatively inelastic. Even if the prices of habitual goods rise, consumers continue to purchase almost the same quantity. For example, a smoker's demand for cigarettes or an addict's demand for liquor will not fall significantly when prices increase.
Durability: The demand for durable goods such as furniture, washing machines, and televisions is relatively elastic, as their purchase can be postponed when prices are high. Conversely, the demand for perishable goods such as milk, vegetables, and fruits is relatively inelastic.

(v) Explain any four types of monopoly.

Answer: Monopoly is a market structure in which there is only a single seller who controls the entire market supply of a product that has no close substitute. The main types of monopoly are:

Private Monopoly: When an individual or a private body controls a monopoly firm, it is known as a private monopoly. The primary motive of a private monopoly is profit maximization. For example, Tata Group or Reliance Industries operating a sole enterprise in a specialized sector.
Public Monopoly: When the production is solely owned, controlled, and operated by the Government, it is known as a public or social monopoly. It is usually welfare-oriented and not profit-oriented. For example, Indian Railways.
Legal Monopoly: This monopoly emerges on account of legal provisions like patents, trademarks, copyrights, etc. The law forbids potential competitors from imitating the design or form of the product registered under a given brand name. For example, Amul's products or patented pharmaceutical drugs.
Natural Monopoly: The monopoly created on the basis of natural conditions like climate, rainfall, specific location, or mineral reserves is known as a natural monopoly. For example, gold mines in South Africa or crude oil reserves in Gulf countries.
Simple and Discriminating Monopoly: In a simple monopoly, the seller or a firm charges a uniform price for its product to all buyers. In a discriminating monopoly, the firm charges different prices to different buyers for the same product or service (for example, a doctor charging different fees to rich and poor patients).

Q. 4. State with reasons whether you agree or disagree with the following statements (Any THREE) : (12 Marks)

Exam Tip: In Q. 4, clearly stating "Yes, I agree with the above statement" or "No, I disagree with the above statement" carries 1 mark, and the supporting reasons carry 3 marks. Never skip the opening agreement/disagreement line!

(i) There are many assumptions to the law of supply.

Answer: Yes, I agree with the above statement.

Reasons: The law of supply states that "Other things being constant, the higher the price of a commodity, more is the quantity supplied; and lower the price of a commodity, less is the quantity supplied." The validity of this law rests upon the following assumptions:

Constant Cost of Production: It is assumed that there is no change in the cost of production. A change in the prices of factors of production would alter the cost and affect supply even at the same price.
Constant Technique of Production: It is assumed that the technique of production does not change. Improved technology increases production and supply at the same price.
No Change in Weather Conditions: It is assumed that natural conditions like rainfall, temperature, floods, or droughts remain normal and unchanged, especially for agricultural commodities.
No Change in Government Policy: It is assumed that government policies regarding taxation, subsidies, and trade regulations remain unchanged.
Constant Transport Cost: It is assumed that there is no change in transport facilities and transport costs. Better transport facilities increase supply at the same price.
Prices of Other Goods Remain Constant: Prices of other related goods are assumed to remain constant so that producers do not shift their resources to other products.

(ii) Product differentiation is the only feature of monopolistic competition.

Answer: No, I disagree with the above statement.

Reasons: While product differentiation (making a product distinct through brand name, trademarks, packaging, quality, design, or colour) is an important feature of monopolistic competition, there are several other essential features of this market structure:

Fairly Large Number of Sellers: In monopolistic competition, the number of sellers is fairly large, though comparatively less than in perfect competition. Hence, no single seller can control a major share of the market supply.
Fairly Large Number of Buyers: There are a fairly large number of buyers in this market. Consequently, no single buyer can influence the price of the product by changing their individual demand.
Close Substitutes: Goods sold under monopolistic competition are not homogeneous, but they are close substitutes for one another. For example, different brands of soaps, toothpastes, or cold drinks.
Free Entry and Exit: Under monopolistic competition, there is freedom of entry and exit. New firms are free to enter the market if there is supernormal profit, and existing firms can leave the market if they incur losses.
Selling Cost: Selling cost is a peculiar feature of monopolistic competition. Firms incur heavy expenditures on advertisements, radio and television broadcasts, hoardings, exhibitions, window displays, and free gifts to create more demand for their product.

(iii) Marginal utility and total utility are the same.

Answer: No, I disagree with the above statement.

Reasons: Marginal Utility (MU) and Total Utility (TU) are two entirely different concepts in cardinal utility analysis:

Difference in Meaning: Total Utility (TU) refers to the aggregate of utilities derived by a consumer from all units of a commodity consumed at a point of time. On the other hand, Marginal Utility (MU) refers to the additional utility derived by a consumer from an additional unit of a commodity consumed.
Formula: Total Utility is the sum of marginal utilities: $\text{TU}_n = \sum \text{MU}$ or $\text{TU}_n = \text{MU}_1 + \text{MU}_2 + \dots + \text{MU}_n$. Marginal Utility is the net addition to Total Utility: $\text{MU}_n = \text{TU}_n - \text{TU}_{n-1}$.
Behaviour During Consumption: As a consumer consumes more and more units of a commodity continuously, Total Utility increases at a diminishing rate, whereas Marginal Utility goes on diminishing continuously.
At the Point of Satiety: When the consumer reaches the point of maximum satisfaction (point of satiety), Total Utility reaches its maximum level and remains constant, whereas Marginal Utility becomes zero $(\text{MU} = 0)$.
Positive and Negative Values: Beyond the point of satiety, if consumption continues, Total Utility begins to decline (though it remains positive), whereas Marginal Utility becomes negative (disutility). Only on the consumption of the very first unit are TU and MU equal.

(iv) Index numbers have limitations.

Answer: Yes, I agree with the above statement.

Reasons: Index numbers are extremely useful statistical tools and economic barometers, but they are not completely reliable and suffer from the following limitations:

Based on Samples: Index numbers are generally based on samples. It is practically impossible to include all items in the construction of an index number. Hence, they are not free from sampling errors.
Bias in the Data: Index numbers are constructed from various types of data collected by individuals or agencies. There may be errors or deliberate bias in the collected data, which affects the accuracy of the results.
Misuse of Index Numbers: Index numbers can be misused. Sometimes a person or institution chooses a base year or commodities that yield a favourable result to prove a predetermined point of view.
Defects in Formulae: There is no perfect formula for the construction of an index number. It is only an average, and so it carries all the limitations of an average.
Changes in the Economy: The habits, tastes, and expectations of the people in a country are always changing over time. All these dynamic changes cannot be accurately included in the estimation of index numbers.
Qualitative Changes: The price or quantity index numbers may ignore the changes in the quality of the product. A higher price may be due to better quality, which is overlooked in simple index calculations.

(v) Public finance is different from private finance.

Answer: Yes, I agree with the above statement.

Reasons: Public finance is the study of the income and expenditure of the government, whereas private finance is the study of the income and expenditure of an individual or a private business. They differ in the following ways:

Objectives: The main objective of public finance is to offer maximum social advantage to society, whereas the primary motive of private finance is to fulfill private interests and maximize individual satisfaction or profit.
Determination of Expenditure: In public finance, the government first determines the volume and different ways of its expenditure and then finds out the ways to raise the necessary revenue. In private finance, an individual considers their income first and then determines the volume of expenditure.
Credit Status: The government enjoys a very high degree of credit in the market, both domestically and internationally. In contrast, the credit of a private individual is limited.
Right to Print Currency: The government has the legal authority to print currency notes through the Reserve Bank of India to meet its financial deficits. A private individual does not enjoy any such right.
Elasticity of Finance: Public finance is more elastic and flexible, as the government can make drastic changes in its revenue and expenditure policies. Private finance is relatively rigid, as an individual cannot easily make major changes in their income.

Q. 5. Study the following table, figure, passage and answer the questions given below it (Any TWO) : (8 Marks)

(i) Observe the following table and answer the questions given below it : (4 Marks)

(a) Complete the above table. (1 Mark)

Commodities	Quantity in 2000 Base Year $(q_0)$	Quantity in 2001 Current Year $(q_1)$
A	20	45
B	40	70
C	60	105
D	20	60
Total	$\sum q_0 = 140$	$\sum q_1 = 280$

(b) Calculate the Quantity Index Number from the above table. (3 Marks)

Given:

Sum of the quantities of the base year $\left(\sum q_0\right) = 20 + 40 + 60 + 20 = 140$
Sum of the quantities of the current year $\left(\sum q_1\right) = 45 + 70 + 105 + 60 = 280$

Formula:

\[Q_{01} = \frac{\sum q_1}{\sum q_0} \times 100\]

Substitution & Calculation:

\[Q_{01} = \frac{280}{140} \times 100\]

\[Q_{01} = 2 \times 100\]

\[Q_{01} = 200\]

Final Answer: The Quantity Index Number $\left(Q_{01}\right)$ is 200.

(ii) Observe the following diagram and answer the questions given below it : (4 Marks)

In the given diagram, $AE$ is the linear demand curve of a commodity plotted with Quantity Demanded on the X-axis and Price on the Y-axis. The points along the demand curve from top to bottom are: point $A$ (on the Y-axis), point $B$ (on the upper half), point $D$ (at the exact midpoint), point $C$ (on the lower half), and point $E$ (on the X-axis). On the basis of the given diagram, state whether the following statements are true or false:

Key Formula (Point/Geometric Method): $\text{Point Elasticity of Demand } (E_d) = \frac{\text{Lower segment of the demand curve below the given point}}{\text{Upper segment of the demand curve above the given point}}$

Demand at point 'C' is relatively elastic demand. (1 Mark)

Answer: False. (Demand at point 'C' is relatively inelastic demand $[E_d < 1]$ because it lies on the lower half of the linear demand curve where the lower segment $CE$ is smaller than the upper segment $CA$.)
Demand at point 'B' is unitary elastic demand. (1 Mark)

Answer: False. (Demand at point 'B' is relatively elastic demand $[E_d > 1]$ because it lies on the upper half of the linear demand curve where the lower segment $BE$ is greater than the upper segment $BA$.)
Demand at point 'D' is perfectly inelastic. (1 Mark)

Answer: False. (Demand at point 'D' is unitary elastic demand $[E_d = 1]$ because it is the mid-point of the linear demand curve where the lower segment $DE$ is equal to the upper segment $DA$.)
Demand at point 'A' is perfectly elastic demand. (1 Mark)

Answer: True. (At point 'A' on the Y-axis, the upper segment is zero, so $E_d = \infty$, which indicates perfectly elastic demand.)

(iii) Read the following passage and answer the questions given below it : (4 Marks)

Passage: The Finance Ministry has proposed a two slab GST structure instead of multiple slabs. This reform will make the tax system simpler, benefiting micro, small and medium enterprises and also making goods cheaper for consumers. Apart from regular tax revenue, the central government earns a large share through Cess and Surcharge.

For the financial year 2025-26, the government expects 5.91 lakh crore from Cess and Surcharge which is a 9.43% increase compared to the previous year.

Recently, GST rates on several items were reduced. This gives relief to consumers, increases the purchasing power and boosts demand in the economy. However, such cuts reduce the government's revenue. According to officials, this loss can be managed through better tax collection and other fiscal measures.

Today, India's public finance system faces major challenges: Revenue shortfall with increasing expenditure. Need to reduce fiscal deficit while promoting investment. To address this, the government must focus on tax reforms, prioritizing expenditure and long term infrastructure and investment.

How much revenue does the government expect from Cess and Surcharge in F.Y. 2025-26? (1 Mark)

Answer: For the financial year 2025-26, the government expects ₹ 5.91 lakh crore from Cess and Surcharge, which is a 9.43% increase compared to the previous year.
Mention the major challenges faced by India's public finance system today. (1 Mark)

Answer: Today, India's public finance system faces two major challenges:
- Revenue shortfall with increasing expenditure.
- Need to reduce the fiscal deficit while promoting investment.
Express your opinion with reference to the given passage. (2 Marks)

Answer: In my opinion, the proposal to simplify the Goods and Services Tax (GST) into a two-slab structure and reduce GST rates on essential items is a highly progressive step. It will ease the compliance burden on Micro, Small, and Medium Enterprises (MSMEs), make goods more affordable for common citizens, increase purchasing power, and boost aggregate demand in the economy. Although tax cuts may cause a temporary dip in government revenue, this shortfall can be effectively balanced by improving tax compliance, widening the tax base, prioritizing productive public expenditure, and focusing on long-term infrastructure investment to ensure sustainable economic growth.

Q. 6. Answer the following questions in detail (Any TWO) : (16 Marks)

(i) State and explain the law of demand with its exceptions. (8 Marks)

1. Introduction:

The Law of Demand is one of the most important fundamental laws of consumption in economics. It was introduced by Prof. Alfred Marshall in his famous book, "Principles of Economics", published in 1890. The law explains the functional (inverse) relationship between the price of a commodity and its quantity demanded.

2. Statement of the Law:

According to Prof. Alfred Marshall, "Other things being equal, higher the price of a commodity, smaller is the quantity demanded and lower the price of a commodity, larger is the quantity demanded."

In other words, other factors remaining constant, if the price of a commodity rises, its demand falls; and if the price of a commodity falls, its demand rises. Thus, there is an inverse relationship between price and quantity demanded.

3. Symbolic Representation:

Symbolically, the functional relationship between demand and price is expressed as:

\[D_x = f(P_x)\]

Where:

$D$ = Demand for a commodity
$x$ = Commodity
$f$ = Function
$P_x$ = Price of a commodity

4. Key Assumptions (Ceteris Paribus):

The Law of Demand is based on the assumption that all other determinants of demand remain constant: constant level of consumer's income, no change in size of population, constant prices of substitute and complementary goods, no change in tastes and habits, and no expectations about future changes in prices.

5. Demand Schedule:

The Law of Demand can be explained with the help of the following individual demand schedule:

Price of Commodity 'x' (₹ per kg)	Quantity Demanded of Commodity 'x' (in kgs)
50	1
40	2
30	3
20	4
10	5

6. Explanation of the Schedule and Demand Curve:

As shown in the table, when the price of commodity 'x' is ₹ 50 per kg, the quantity demanded is 1 kg.
When the price falls from ₹ 50 to ₹ 40, ₹ 30, ₹ 20, and ₹ 10 per kg, the quantity demanded rises to 2 kgs, 3 kgs, 4 kgs, and 5 kgs respectively.
When these price-quantity combinations are plotted on a graph with Quantity Demanded on the X-axis and Price on the Y-axis, we get the demand curve DD, which slopes downwards from left to right, indicating an inverse relationship between price and quantity demanded.

7. Exceptions to the Law of Demand:

There are certain exceptional cases where the Law of Demand does not apply. In these situations, a rise in price leads to a rise in demand and a fall in price leads to a fall in demand, resulting in an upward-sloping exceptional demand curve from left to right:

Giffen's Paradox: Inferior goods or low-quality goods are those goods whose demand does not rise even if their price falls. Sir Robert Giffen observed in England that when the price of bread declined, low-paid workers did not buy more bread; instead, they saved money and used their increased real income to purchase superior food like meat. Such inferior goods are known as Giffen goods.
Prestige Goods (Veblen Effect): Expensive goods like diamonds, gold, luxury cars, and antique paintings are status symbols. Rich people buy more of these goods when their prices are high to display their wealth and social prestige.
Speculation: The law of demand does not hold true when people expect prices to rise still further in the near future. In such cases, even though prices have risen today, consumers buy more now to avoid paying even higher prices later (e.g., shares in the stock market, or oil and sugar before festive seasons).
Price Illusion: Consumers often have an illusion that high-priced goods are of better quality. Therefore, the demand for branded and high-priced products increases even when their prices rise.
Ignorance: Sometimes consumers buy more of a commodity at a higher price out of sheer ignorance because they are unaware of the lower prices of the same commodity prevailing elsewhere in the market.
Habitual Goods: Due to habit of consumption, certain goods like tea, coffee, cigarettes, or tobacco are purchased in required quantities even if their prices increase significantly.

(ii) Define National Income and explain its features. (8 Marks)

1. Meaning of National Income:

National Income is one of the most important subject matters of macroeconomics. The total income of the nation is called National Income. In real terms, national income is the flow of goods and services produced in an economy during a year.

2. Definitions of National Income:

National Income Committee (NIC, 1949): "A national estimate measures the volume of commodities and services turned out during a given period counted without duplication."
Prof. A. C. Pigou: "National dividend is that part of objective income of the community including of course income derived from abroad which can be measured in money."
Prof. Irving Fisher: "National dividend or income consists solely of services as received by ultimate consumers, whether from their material or from their human environments."

3. Features of National Income:

Macro Economic Concept: National income represents the income of the economy as a whole rather than that of an individual unit. Hence, it is a macroeconomic concept.
Value of Only Final Goods and Services: In order to avoid double counting in national income, the value of only final goods and services produced in the economy is considered. The value of intermediate goods or raw materials is not considered separately. For example, while estimating the production of shirts, the value of cotton is not counted separately as it is already included in the price of the shirt.
Net Aggregate Value: National income includes the net value of goods and services produced and does not include depreciation cost (i.e., wear and tear of capital assets used in the process of production).
Net Income from Abroad: National income includes net income from abroad, i.e., the difference between export value and import value $(X - M)$ and net difference between receipts from abroad and payments made abroad $(R - P)$.
Financial Year: National income is always expressed with reference to a specific time period. In India, the financial year runs from 1st April to 31st March.
Flow Concept: National income is a flow concept as it shows the continuous flow of goods and services produced in the economy during the course of a year.
Money Value: National income is always expressed in monetary terms. It represents only those goods and services which are exchanged for money.

(iii) Explain the role of money market in India. (8 Marks)

1. Meaning of Money Market:

The money market is a financial market wherein lending and borrowing of short-term funds take place. It is a market for "near money", i.e., short-term instruments such as trade bills, government securities, promissory notes, treasury bills, etc. Such instruments are highly liquid, less risky, and easily marketable with a maturity period of one year or less than one year.

2. Role of Money Market in India:

The following points outline the vital role of the money market in India:

Short-Term Requirements of Borrowers: The money market provides reasonable access for meeting the short-term financial needs of borrowers (such as traders, industrialists, and farmers) at realistic prices.
Liquidity Management: The money market is a dynamic market. It facilitates better management of liquidity and money in the economy by the monetary authorities. This, in turn, leads to economic stability and development of the country.
Portfolio Management: The money market deals with different types of financial instruments that are designed to suit the risk and return preferences of investors. This enables the investors to hold a balanced portfolio of different financial assets, which helps in minimizing risk and maximizing returns.
Equilibrating Mechanism: Through the rational allocation of resources and mobilization of savings into investment channels, the money market helps to establish an equilibrium between the demand for and supply of short-term funds.
Financial Requirements of the Government: The money market helps the Government to fulfill its short-term financial requirements on the basis of Treasury Bills.
Implementation of Monetary Policy: Monetary policy is implemented by the central bank (Reserve Bank of India). It aims at managing the quantity of money in order to meet the requirements of different sectors of the economy and to increase the pace of economic growth. A well-developed money market ensures the successful implementation of the monetary policy and guides the central bank in developing an appropriate interest rate policy.
Economizes the Use of Cash: The money market deals with various financial instruments that are close substitutes for money and not actual money. Therefore, it economizes the use of cash in the economy.
Growth of Commerce, Industry and Trade: The money market facilitates discounting of bills of exchange to local and international traders who are in urgent need of short-term funds. It also provides working capital for agriculture and small-scale industries, thereby promoting the overall growth of commerce, industry, and trade.

HSC Economics June/July 2026 Board Question Paper Images (Subject Code: J-320)

Below are the official question paper scans of the Maharashtra State Board HSC Class 12 Economics June/July 2026 Supplementary Exam (Paper Code: J-320, Subject Code: 49) held on 1st July 2026:

HSC Mathematics & Statistics (40) March 2026 Board Paper (J-165) Solution (Class 12 Maharashtra Board)

aaminomtex@gmail.com (Amin) — Mon, 26 Oct 2026 07:27:52 +0530

Prepare effectively for your upcoming board exams with our comprehensive, step-by-step HSC Mathematics & Statistics (40) 12th Standard Board Paper Solution - March 2026 (Paper Code: J-165). Designed specifically for Maharashtra State Board Class 12 Arts and Science students, this complete Board Question Paper Solution covers all sections from multiple-choice questions to 4-mark theorems and numericals, complete with marking-scheme steps, key formulas, and examiner tips to help you score 80/80 in Mathematics.

HSC Mathematics & Statistics (40) 12th Standard Board Paper Solution - March 2026 (J-165)

SECTION - A

Q. 1. Select and write the correct answer of the following multiple choice type of questions: [16]

(i) The converse of contrapositive of $\sim p \rightarrow q$ is ____. (2)

(a) $q \rightarrow p$
(b) $\sim q \rightarrow p$
(c) $p \rightarrow \sim q$
(d) $\sim q \rightarrow \sim p$

Answer: (c) $p \rightarrow \sim q$

Solution:

Given statement: $\sim p \rightarrow q$
Contrapositive of $\sim p \rightarrow q$ is $\sim q \rightarrow \sim(\sim p)$, which simplifies to $\sim q \rightarrow p$.
Converse of $\sim q \rightarrow p$ is $p \rightarrow \sim q$.

Memory Trick: The converse of the contrapositive of any conditional statement is always logically equivalent to its inverse. For $P \rightarrow Q$, the inverse is $\sim P \rightarrow \sim Q$. Here, $\sim(\sim p) \rightarrow \sim q$ directly gives $p \rightarrow \sim q$.

(ii) If $A = \begin{bmatrix} 2 & -4 \\ 3 & 1 \end{bmatrix}$, then the adjoint of matrix $A$ is ____. (2)

(a) $\begin{bmatrix} 1 & 3 \\ 4 & -2 \end{bmatrix}$
(b) $\begin{bmatrix} -1 & 3 \\ -4 & 1 \end{bmatrix}$
(c) $\begin{bmatrix} -1 & -3 \\ -4 & 2 \end{bmatrix}$
(d) $\begin{bmatrix} 1 & 4 \\ -3 & 2 \end{bmatrix}$

Answer: (d) $\begin{bmatrix} 1 & 4 \\ -3 & 2 \end{bmatrix}$

Solution:

Formula: For a $2 \times 2$ matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the adjoint is obtained by interchanging the principal diagonal elements and changing the signs of the off-diagonal elements: \[\text{adj}(A) = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}\]
Calculation: For $A = \begin{bmatrix} 2 & -4 \\ 3 & 1 \end{bmatrix}$, we get: \[\text{adj}(A) = \begin{bmatrix} 1 & 4 \\ -3 & 2 \end{bmatrix}\]

(iii) If $\tan^{-1}(2x) + \tan^{-1}(3x) = \frac{\pi}{4}$, then $x =$ ____. (2)

(a) $-1$
(b) $\frac{1}{6}$
(c) $\frac{1}{3}$
(d) $\frac{3}{2}$

Answer: (b) $\frac{1}{6}$

Solution:

Given: $\tan^{-1}(2x) + \tan^{-1}(3x) = \frac{\pi}{4}$
Formula: $\tan^{-1}A + \tan^{-1}B = \tan^{-1}\left(\frac{A + B}{1 - AB}\right)$ for $AB < 1$
Calculation: \[\tan^{-1}\left(\frac{2x + 3x}{1 - (2x)(3x)}\right) = \frac{\pi}{4}\] \[\frac{5x}{1 - 6x^2} = \tan\left(\frac{\pi}{4}\right) = 1\] \[5x = 1 - 6x^2 \implies 6x^2 + 5x - 1 = 0\] \[(6x - 1)(x + 1) = 0 \implies x = \frac{1}{6} \text{ or } x = -1\]
Since $x = -1$ makes the LHS negative (which cannot equal $\frac{\pi}{4} > 0$), $x = -1$ is rejected. Therefore, $x = \frac{1}{6}$.

(iv) The angle between the line $\overline{r} = (\hat{i} + 2\hat{j} + \hat{k}) + \lambda(\hat{i} + \hat{j} + \hat{k})$ and the plane $\overline{r} \cdot (2\hat{i} - \hat{j} + \hat{k}) = 8$ is ____. (2)

(a) $\sin^{-1}\left(\frac{\sqrt{2}}{3}\right)$
(b) $\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)$
(c) $\sin^{-1}\left(\frac{1}{2}\right)$
(d) $\sin^{-1}\left(\frac{1}{\sqrt{2}}\right)$

Answer: (a) $\sin^{-1}\left(\frac{\sqrt{2}}{3}\right)$

Solution:

Given: Vector parallel to the line is $\overline{b} = \hat{i} + \hat{j} + \hat{k}$ and normal vector to the plane is $\overline{n} = 2\hat{i} - \hat{j} + \hat{k}$.
Formula: The acute angle $\theta$ between a line and a plane is given by: \[\sin\theta = \left|\frac{\overline{b} \cdot \overline{n}}{|\overline{b}| |\overline{n}|}\right|\]
Calculation: \[\overline{b} \cdot \overline{n} = (1)(2) + (1)(-1) + (1)(1) = 2 - 1 + 1 = 2\] \[|\overline{b}| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3}\] \[|\overline{n}| = \sqrt{2^2 + (-1)^2 + 1^2} = \sqrt{4 + 1 + 1} = \sqrt{6}\] \[\sin\theta = \frac{2}{\sqrt{3}\sqrt{6}} = \frac{2}{\sqrt{18}} = \frac{2}{3\sqrt{2}} = \frac{\sqrt{2}}{3} \implies \theta = \sin^{-1}\left(\frac{\sqrt{2}}{3}\right)\]

(v) If $y = \sec(\tan^{-1}x)$, then $\frac{dy}{dx}$ at $x = 1$ is ____. (2)

(a) $\frac{1}{2}$
(b) $1$
(c) $\frac{1}{\sqrt{2}}$\)
(d) $\sqrt{2}$

Answer: (c) $\frac{1}{\sqrt{2}}$

Solution:

Given: $y = \sec(\tan^{-1}x)$
Simplification: Let $\tan^{-1}x = \theta \implies \tan\theta = x$. Using the identity $\sec\theta = \sqrt{1 + \tan^2\theta} = \sqrt{1 + x^2}$, we get: \[y = \sqrt{1 + x^2}\]
Differentiation: \[\frac{dy}{dx} = \frac{1}{2\sqrt{1 + x^2}} \cdot (2x) = \frac{x}{\sqrt{1 + x^2}}\]
At $x = 1$: \[\left(\frac{dy}{dx}\right)_{x=1} = \frac{1}{\sqrt{1 + 1^2}} = \frac{1}{\sqrt{2}}\]

(vi) The approximate value of the function $f(x) = x^3 - 3x + 5$ at $x = 1.99$ is ____. (2)

(a) $6.09$
(b) $6.91$
(c) $7.09$
(d) $7.91$

Answer: (b) $6.91$

Solution:

Given: $f(x) = x^3 - 3x + 5$, $a = 2$, and $h = -0.01$ such that $a + h = 1.99$.
Formula: $f(a + h) \approx f(a) + h \cdot f'(a)$
Calculation:
- $f(a) = f(2) = (2)^3 - 3(2) + 5 = 8 - 6 + 5 = 7$
- $f'(x) = 3x^2 - 3 \implies f'(2) = 3(2)^2 - 3 = 12 - 3 = 9$
- $f(1.99) \approx 7 + (-0.01)(9) = 7 - 0.09 = 6.91$

(vii) $\int_{1}^{2} \frac{1}{x^2} \cdot e^{\frac{1}{x}} dx =$ ____. (2)

(a) $\sqrt{e} + 1$
(b) $\sqrt{e} - 1$
(c) $\sqrt{e}(\sqrt{e} - 1)$
(d) $\frac{\sqrt{e} - 1}{e}$

Answer: (c) $\sqrt{e}(\sqrt{e} - 1)$

Solution:

Substitution: Put $\frac{1}{x} = t \implies -\frac{1}{x^2} dx = dt \implies \frac{1}{x^2} dx = -dt$.
Limits: When $x = 1$, $t = 1$; when $x = 2$, $t = \frac{1}{2}$.
Calculation: \[I = \int_{1}^{1/2} e^t (-dt) = \int_{1/2}^{1} e^t dt = \left[e^t\right]_{1/2}^{1} = e^1 - e^{1/2} = e - \sqrt{e} = \sqrt{e}(\sqrt{e} - 1)\]

(viii) If the p.d.f. of a continuous r.v. $X$ is $f(x) = \frac{x + 2}{18}$ for $-2 < x < 4$ and $= 0$ otherwise, then $P(|X| < 1) =$ ____. (2)

(a) $\frac{1}{9}$
(b) $\frac{2}{9}$
(c) $\frac{1}{27}$
(d) $\frac{2}{27}$

Answer: (b) $\frac{2}{9}$

Solution:

Formula: $P(|X| < 1) = P(-1 < X < 1) = \int_{-1}^{1} f(x) dx$
Calculation: \[P(|X| < 1) = \int_{-1}^{1} \frac{x + 2}{18} dx = \frac{1}{18} \left[\frac{x^2}{2} + 2x\right]_{-1}^{1}\] \[= \frac{1}{18} \left[\left(\frac{1}{2} + 2\right) - \left(\frac{1}{2} - 2\right)\right] = \frac{1}{18}(4) = \frac{4}{18} = \frac{2}{9}\]

Q. 2. Answer the following questions: [4]

(i) Write the dual of $(p \vee q) \vee r \equiv p \vee (q \vee r)$. (1)

Answer:

To obtain the dual of a statement pattern, we replace $\vee$ with $\wedge$ and $\wedge$ with $\vee$:

Dual: $(p \wedge q) \wedge r \equiv p \wedge (q \wedge r)$

(ii) Evaluate: $\cos^{-1}\left(\frac{1}{2}\right) + 2\sin^{-1}\left(\frac{1}{2}\right)$. (1)

Answer:

Principal Values: $\cos^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{3}$ and $\sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6}$
Calculation: \[\cos^{-1}\left(\frac{1}{2}\right) + 2\sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{3} + 2\left(\frac{\pi}{6}\right) = \frac{\pi}{3} + \frac{\pi}{3} = \frac{2\pi}{3}\]
Final Answer: $\frac{2\pi}{3}$

(iii) Evaluate: $\int \frac{2x}{1 + x^2} dx$. (1)

Answer:

Formula: Using the standard corollary $\int \frac{f'(x)}{f(x)} dx = \log|f(x)| + c$
Calculation: Since $\frac{d}{dx}(1 + x^2) = 2x$, we get: \[\int \frac{2x}{1 + x^2} dx = \log(1 + x^2) + c\]

(iv) Write the degree of the differential equation $(y''')^2 + 3(y'')^3 + 3xy' + 5y = 0$. (1)

Answer:

The given differential equation is a polynomial equation in its derivatives $y''', y''$, and $y'$.
The highest order derivative present in the equation is $y'''$ (order $= 3$), and its exponent (power) is $2$.
Final Answer: Therefore, the degree of the differential equation is $2$.

SECTION - B

Attempt any EIGHT of the following questions: [16]

Q. 3. Construct the switching circuit of the statement pattern $(\sim p \wedge q) \vee (p \wedge \sim r)$. (2)

Answer:

Let the simple statements represent the switches as follows:

$p$: The switch $S_1$ is closed.
$q$: The switch $S_2$ is closed.
$r$: The switch $S_3$ is closed.
$\sim p$: The switch $S_1'$ is closed.
$\sim r$: The switch $S_3'$ is closed.

Construction of Switching Circuit:

Upper Branch $(\sim p \wedge q)$: Switch $S_1'$ and switch $S_2$ are connected in series.
Lower Branch $(p \wedge \sim r)$: Switch $S_1$ and switch $S_3'$ are connected in series.
Complete Circuit: The upper branch and lower branch are connected in parallel with each other, and the combination is connected in series with a battery and a lamp $L$.

Exam Tip: In the board exam, draw the two parallel branches neatly using a ruler, label the switches clearly as $S_1', S_2$ in the top branch and $S_1, S_3'$ in the bottom branch, and show the current flow arrows from the battery to the lamp $L$.

Q. 4. In $\triangle ABC$, prove that $a(b \cos C - c \cos B) = b^2 - c^2$. (2)

Answer:

Proof:

Consider LHS: \[\text{LHS} = a(b \cos C - c \cos B) = ab \cos C - ac \cos B\]
Formula (Cosine Rule): \[\cos C = \frac{a^2 + b^2 - c^2}{2ab} \quad \text{and} \quad \cos B = \frac{a^2 + c^2 - b^2}{2ac}\]
Substitution & Calculation: \[\text{LHS} = ab\left(\frac{a^2 + b^2 - c^2}{2ab}\right) - ac\left(\frac{a^2 + c^2 - b^2}{2ac}\right)\] \[\text{LHS} = \frac{a^2 + b^2 - c^2}{2} - \frac{a^2 + c^2 - b^2}{2}\] \[\text{LHS} = \frac{a^2 + b^2 - c^2 - a^2 - c^2 + b^2}{2}\] \[\text{LHS} = \frac{2b^2 - 2c^2}{2} = \frac{2(b^2 - c^2)}{2} = b^2 - c^2 = \text{RHS}\]

Therefore, $a(b \cos C - c \cos B) = b^2 - c^2$. Hence proved.

Q. 5. Find the general solution of $4\cos^2\theta = 3$. (2)

Answer:

Given Equation: \[4\cos^2\theta = 3 \implies \cos^2\theta = \frac{3}{4}\]
Formula: The general solution of $\cos^2\theta = \cos^2\alpha$ is given by: \[\theta = n\pi \pm \alpha, \quad n \in \mathbb{Z}\]
Calculation: \[\cos^2\theta = \left(\frac{\sqrt{3}}{2}\right)^2\] Since $\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$, we have: \[\cos^2\theta = \cos^2\left(\frac{\pi}{6}\right)\]
Final Answer: The required general solution is $\theta = n\pi \pm \frac{\pi}{6}$, where $n \in \mathbb{Z}$.

Q. 6. Find $k$, if the sum of the slopes of the lines represented by $x^2 + kxy - 3y^2 = 0$ is twice their product. (2)

Answer:

Given: Homogeneous equation of lines is $x^2 + kxy - 3y^2 = 0$. Comparing with $ax^2 + 2hxy + by^2 = 0$, we get: \[a = 1, \quad 2h = k, \quad b = -3\]
Formula: If $m_1$ and $m_2$ are the slopes of the lines, then:
- Sum of slopes: $m_1 + m_2 = -\frac{2h}{b} = -\frac{k}{-3} = \frac{k}{3}$
- Product of slopes: $m_1 m_2 = \frac{a}{b} = \frac{1}{-3} = -\frac{1}{3}$
Substitution & Calculation: Given that $m_1 + m_2 = 2(m_1 m_2)$: \[\frac{k}{3} = 2\left(-\frac{1}{3}\right) \implies \frac{k}{3} = -\frac{2}{3} \implies k = -2\]
Final Answer: $k = -2$.

Q. 7. Find the value of $p$, for which the vectors $\overline{a} = 3\hat{i} + 2\hat{j} + 9\hat{k}$ and $\overline{b} = \hat{i} + p\hat{j} + 3\hat{k}$ are perpendicular to each other. (2)

Answer:

Given: $\overline{a} = 3\hat{i} + 2\hat{j} + 9\hat{k}$ and $\overline{b} = \hat{i} + p\hat{j} + 3\hat{k}$ are perpendicular ($\overline{a} \perp \overline{b}$).
Formula: Two non-zero vectors $\overline{a}$ and $\overline{b}$ are perpendicular if and only if their scalar (dot) product is zero: \[\overline{a} \cdot \overline{b} = 0\]
Substitution & Calculation: \[(3\hat{i} + 2\hat{j} + 9\hat{k}) \cdot (\hat{i} + p\hat{j} + 3\hat{k}) = 0\] \[(3)(1) + (2)(p) + (9)(3) = 0\] \[3 + 2p + 27 = 0 \implies 2p + 30 = 0 \implies p = -15\]
Final Answer: $p = -15$.

Q. 8. Find the vector equation of the line passing through the points $A(1, 2, 3)$ and $B(2, 3, 4)$. (2)

Answer:

Given: Position vectors of points $A(1, 2, 3)$ and $B(2, 3, 4)$ with respect to the origin are: \[\overline{a} = \hat{i} + 2\hat{j} + 3\hat{k} \quad \text{and} \quad \overline{b} = 2\hat{i} + 3\hat{j} + 4\hat{k}\]
Formula: The vector equation of the line passing through $A(\overline{a})$ and $B(\overline{b})$ is: \[\overline{r} = \overline{a} + \lambda(\overline{b} - \overline{a})\]
Substitution & Calculation: \[\overline{b} - \overline{a} = (2\hat{i} + 3\hat{j} + 4\hat{k}) - (\hat{i} + 2\hat{j} + 3\hat{k}) = \hat{i} + \hat{j} + \hat{k}\] \[\overline{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(\hat{i} + \hat{j} + \hat{k})\]
Final Answer: The vector equation of the line is $\overline{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(\hat{i} + \hat{j} + \hat{k})$.

Q. 9. Find $\frac{dy}{dx}$, if $\sqrt{x} + \sqrt{y} = \sqrt{a}$. (2)

Answer:

Given: $\sqrt{x} + \sqrt{y} = \sqrt{a}$
Formula: $\frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}}$ and $\frac{d}{dx}(\text{constant}) = 0$
Differentiation: Differentiating both sides with respect to $x$: \[\frac{d}{dx}(\sqrt{x}) + \frac{d}{dx}(\sqrt{y}) = \frac{d}{dx}(\sqrt{a})\] \[\frac{1}{2\sqrt{x}} + \frac{1}{2\sqrt{y}} \cdot \frac{dy}{dx} = 0\] \[\frac{1}{2\sqrt{y}} \cdot \frac{dy}{dx} = -\frac{1}{2\sqrt{x}}\] \[\frac{dy}{dx} = -\frac{\sqrt{y}}{\sqrt{x}} = -\sqrt{\frac{y}{x}}\]
Final Answer: $\frac{dy}{dx} = -\sqrt{\frac{y}{x}}$

Q. 10. Find $\frac{d^2y}{dx^2}$, if $y = x^3 + 7x^2 - 2x - 9$. (2)

Answer:

Given: $y = x^3 + 7x^2 - 2x - 9$
First Derivative: Differentiating with respect to $x$: \[\frac{dy}{dx} = \frac{d}{dx}(x^3 + 7x^2 - 2x - 9) = 3x^2 + 14x - 2\]
Second Derivative: Differentiating again with respect to $x$: \[\frac{d^2y}{dx^2} = \frac{d}{dx}(3x^2 + 14x - 2) = 6x + 14\]
Final Answer: $\frac{d^2y}{dx^2} = 6x + 14$

Q. 11. Test whether the function $f(x) = x^3 + 6x^2 + 12x - 7$ is increasing or decreasing for all $x \in \mathbb{R}$. (2)

Answer:

Given Function: $f(x) = x^3 + 6x^2 + 12x - 7$
Condition for Increasing Function: A function $f(x)$ is increasing on $\mathbb{R}$ if $f'(x) \ge 0$ for all $x \in \mathbb{R}$.
Calculation: Differentiating with respect to $x$: \[f'(x) = 3x^2 + 12x + 12\] \[f'(x) = 3(x^2 + 4x + 4) = 3(x + 2)^2\]
Since the square of any real number is always non-negative, $(x + 2)^2 \ge 0$ for all $x \in \mathbb{R}$. Therefore: \[f'(x) = 3(x + 2)^2 \ge 0 \quad \forall x \in \mathbb{R}\]
Final Answer: The function $f(x) = x^3 + 6x^2 + 12x - 7$ is an increasing function for all $x \in \mathbb{R}$.

Q. 12. A stone is dropped into a quiet lake and waves in the form of circles are generated. Radius of the circular wave increases at the rate of $3\text{ cm/sec}$. How fast the area enclosed is increasing when the radius is $8\text{ cm}$? (2)

Answer:

Given: Let $r$ be the radius and $A$ be the area of the circular wave at time $t$.
- Rate of increase of radius: $\frac{dr}{dt} = 3\text{ cm/sec}$
- Instantaneous radius: $r = 8\text{ cm}$
Formula: Area of a circle is $A = \pi r^2$. Differentiating with respect to $t$: \[\frac{dA}{dt} = 2\pi r \frac{dr}{dt}\]
Substitution & Calculation: \[\frac{dA}{dt} = 2\pi (8)(3) = 48\pi\text{ cm}^2\text{/sec}\]
Final Answer: The area enclosed by the circular wave is increasing at the rate of $48\pi\text{ cm}^2\text{/sec}$ (or $48\pi\text{ sq.cm/sec}$).

Q. 13. Evaluate: $\int \sqrt{1 + \sin 2x} \cdot dx$. (2)

Answer:

Given: Let $I = \int \sqrt{1 + \sin 2x} \, dx$
Formula: Using trigonometric identities $\sin^2 x + \cos^2 x = 1$ and $\sin 2x = 2\sin x \cos x$: \[1 + \sin 2x = \sin^2 x + \cos^2 x + 2\sin x \cos x = (\sin x + \cos x)^2\]
Calculation: \[I = \int \sqrt{(\sin x + \cos x)^2} \, dx = \int (\sin x + \cos x) \, dx\] \[I = -\cos x + \sin x + c = \sin x - \cos x + c\]
Final Answer: $\int \sqrt{1 + \sin 2x} \cdot dx = \sin x - \cos x + c$

Q. 14. Given that, $X \sim B(n, p)$, if $n = 10$, $E(X) = 8$ then find $\text{Var}(X)$. (2)

Answer:

Given: For binomial distribution $X \sim B(n, p)$, $n = 10$ and mean $E(X) = 8$.
Formulas:
- Mean: $E(X) = np$
- Variance: $\text{Var}(X) = npq$, where $q = 1 - p$
Substitution & Calculation: \[E(X) = np \implies 8 = 10p \implies p = \frac{8}{10} = 0.8\] \[q = 1 - p = 1 - 0.8 = 0.2\] \[\text{Var}(X) = npq = 10 \times 0.8 \times 0.2 = 1.6\]
Final Answer: $\text{Var}(X) = 1.6$ (or $\frac{8}{5}$).

SECTION - C

Attempt any EIGHT of the following questions: [24]

Q. 15. Examine whether the statement pattern $(p \wedge q) \wedge (\sim p \vee \sim q)$ is a tautology or contradiction or contingency. (3)

Answer:

Truth Table for $(p \wedge q) \wedge (\sim p \vee \sim q)$:

$p$	$q$	$\sim p$	$\sim q$	$p \wedge q$	$\sim p \vee \sim q$	$(p \wedge q) \wedge (\sim p \vee \sim q)$
T	T	F	F	T	F	F
T	F	F	T	F	T	F
F	T	T	F	F	T	F
F	F	T	T	F	T	F

Conclusion: Since all the truth values in the last column are F (False), the given statement pattern $(p \wedge q) \wedge (\sim p \vee \sim q)$ is a contradiction.

Q. 16. In $\triangle ABC$, if $A = 45^{\circ}$, $B = 60^{\circ}$ then find the ratio of its sides. (3)

Answer:

Given: In $\triangle ABC$, $A = 45^{\circ}$ and $B = 60^{\circ}$.
Formula:
- Sum of angles of a triangle: $A + B + C = 180^{\circ}$
- Sine Rule: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \implies a : b : c = \sin A : \sin B : \sin C$
Step 1: Find angle $C$: \[45^{\circ} + 60^{\circ} + C = 180^{\circ} \implies C = 180^{\circ} - 105^{\circ} = 75^{\circ}\]
Step 2: Find the values of $\sin 45^{\circ}, \sin 60^{\circ}$, and $\sin 75^{\circ}$:
- $\sin 45^{\circ} = \frac{1}{\sqrt{2}} = \frac{2}{2\sqrt{2}}$
- $\sin 60^{\circ} = \frac{\sqrt{3}}{2} = \frac{\sqrt{6}}{2\sqrt{2}}$
- $\sin 75^{\circ} = \sin(45^{\circ} + 30^{\circ}) = \sin 45^{\circ}\cos 30^{\circ} + \cos 45^{\circ}\sin 30^{\circ}$ \[\sin 75^{\circ} = \left(\frac{1}{\sqrt{2}}\right)\left(\frac{\sqrt{3}}{2}\right) + \left(\frac{1}{\sqrt{2}}\right)\left(\frac{1}{2}\right) = \frac{\sqrt{3} + 1}{2\sqrt{2}}\]
Step 3: Ratio of sides: \[a : b : c = \sin 45^{\circ} : \sin 60^{\circ} : \sin 75^{\circ}\] \[a : b : c = \frac{1}{\sqrt{2}} : \frac{\sqrt{3}}{2} : \frac{\sqrt{3} + 1}{2\sqrt{2}}\] Multiplying each term of the ratio by $2\sqrt{2}$: \[a : b : c = 2 : \sqrt{6} : (\sqrt{3} + 1)\]
Final Answer: The ratio of the sides of $\triangle ABC$ is $2 : \sqrt{6} : (\sqrt{3} + 1)$.

Q. 17. If two vertices of a triangle are $A(3, 1, 4)$ and $B(-4, 5, -3)$ and the centroid of the triangle is $G(-1, 2, 1)$, then find the coordinates of the third vertex $C$ of the triangle. (3)

Answer:

Given: Let the vertices of $\triangle ABC$ be $A(x_1, y_1, z_1) \equiv (3, 1, 4)$, $B(x_2, y_2, z_2) \equiv (-4, 5, -3)$, and $C(x_3, y_3, z_3) \equiv (x, y, z)$. The centroid is $G(-1, 2, 1)$.
Formula (Centroid Formula): \[G(x, y, z) = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}, \frac{z_1 + z_2 + z_3}{3}\right)\]
Substitution & Calculation: \[(-1, 2, 1) = \left(\frac{3 - 4 + x}{3}, \frac{1 + 5 + y}{3}, \frac{4 - 3 + z}{3}\right)\] Equating corresponding coordinates:
- $\frac{-1 + x}{3} = -1 \implies -1 + x = -3 \implies x = -2$
- $\frac{6 + y}{3} = 2 \implies 6 + y = 6 \implies y = 0$
- $\frac{1 + z}{3} = 1 \implies 1 + z = 3 \implies z = 2$
Final Answer: The coordinates of the third vertex $C$ are $(-2, 0, 2)$.

Q. 18. If $D, E, F$ are the mid-points of the sides $BC, CA, AB$ respectively of $\triangle ABC$, then prove that $\overline{AD} + \overline{BE} + \overline{CF} = \overline{0}$. (3)

Answer:

Proof:

Let $\overline{a}, \overline{b}, \overline{c}$ be the position vectors of the vertices $A, B, C$ of $\triangle ABC$ and $\overline{d}, \overline{e}, \overline{f}$ be the position vectors of the mid-points $D, E, F$ of sides $BC, CA, AB$ respectively.
By Mid-point Formula: \[\overline{d} = \frac{\overline{b} + \overline{c}}{2}, \quad \overline{e} = \frac{\overline{c} + \overline{a}}{2}, \quad \overline{f} = \frac{\overline{a} + \overline{b}}{2}\]
Consider LHS: \[\text{LHS} = \overline{AD} + \overline{BE} + \overline{CF}\] \[\text{LHS} = (\overline{d} - \overline{a}) + (\overline{e} - \overline{b}) + (\overline{f} - \overline{c})\] \[\text{LHS} = (\overline{d} + \overline{e} + \overline{f}) - (\overline{a} + \overline{b} + \overline{c})\] Substituting $\overline{d}, \overline{e}, \overline{f}$: \[\text{LHS} = \left(\frac{\overline{b} + \overline{c}}{2} + \frac{\overline{c} + \overline{a}}{2} + \frac{\overline{a} + \overline{b}}{2}\right) - (\overline{a} + \overline{b} + \overline{c})\] \[\text{LHS} = \frac{2(\overline{a} + \overline{b} + \overline{c})}{2} - (\overline{a} + \overline{b} + \overline{c}) = (\overline{a} + \overline{b} + \overline{c}) - (\overline{a} + \overline{b} + \overline{c}) = \overline{0} = \text{RHS}\]

Therefore, $\overline{AD} + \overline{BE} + \overline{CF} = \overline{0}$. Hence proved.

Q. 19. Show that the lines $\overline{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(2\hat{i} - 2\hat{j} + \hat{k})$ and $\overline{r} = (4\hat{i} - 3\hat{j} + 2\hat{k}) + \mu(\hat{i} - 2\hat{j} + 2\hat{k})$ intersect each other. (3)

Answer:

Given: Comparing the given lines with $\overline{r} = \overline{a}_1 + \lambda\overline{b}_1$ and $\overline{r} = \overline{a}_2 + \mu\overline{b}_2$:
- $\overline{a}_1 = \hat{i} + \hat{j} - \hat{k}, \quad \overline{b}_1 = 2\hat{i} - 2\hat{j} + \hat{k}$
- $\overline{a}_2 = 4\hat{i} - 3\hat{j} + 2\hat{k}, \quad \overline{b}_2 = \hat{i} - 2\hat{j} + 2\hat{k}$
Condition for Intersection of Two Lines: Two non-parallel lines intersect each other if and only if: \[(\overline{a}_2 - \overline{a}_1) \cdot (\overline{b}_1 \times \overline{b}_2) = 0\]
Calculation: \[\overline{a}_2 - \overline{a}_1 = (4 - 1)\hat{i} + (-3 - 1)\hat{j} + (2 - (-1))\hat{k} = 3\hat{i} - 4\hat{j} + 3\hat{k}\] \[(\overline{a}_2 - \overline{a}_1) \cdot (\overline{b}_1 \times \overline{b}_2) = \begin{vmatrix} 3 & -4 & 3 \\ 2 & -2 & 1 \\ 1 & -2 & 2 \end{vmatrix}\] \[= 3(-4 + 2) - (-4)(4 - 1) + 3(-4 + 2)\] \[= 3(-2) + 4(3) + 3(-2) = -6 + 12 - 6 = 0\]
Conclusion: Since $(\overline{a}_2 - \overline{a}_1) \cdot (\overline{b}_1 \times \overline{b}_2) = 0$ and $\overline{b}_1$ is not parallel to $\overline{b}_2$, the given lines intersect each other.

Q. 20. Find the cartesian equation of the plane $\overline{r} = (\hat{i} - \hat{j}) + \lambda(\hat{i} + \hat{j} + \hat{k}) + \mu(\hat{i} - 2\hat{j} + 3\hat{k})$. (3)

Answer:

Given: Comparing with the parametric vector equation $\overline{r} = \overline{a} + \lambda\overline{b} + \mu\overline{c}$:
- $\overline{a} = \hat{i} - \hat{j} + 0\hat{k}$ (point $A(1, -1, 0)$ on the plane)
- $\overline{b} = \hat{i} + \hat{j} + \hat{k}$ and $\overline{c} = \hat{i} - 2\hat{j} + 3\hat{k}$ (vectors parallel to the plane)
Formula: Normal to the plane is $\overline{n} = \overline{b} \times \overline{c}$, and the vector equation of the plane is $\overline{r} \cdot \overline{n} = \overline{a} \cdot \overline{n}$.
Calculation: \[\overline{n} = \overline{b} \times \overline{c} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 1 & 1 \\ 1 & -2 & 3 \end{vmatrix} = \hat{i}(3 + 2) - \hat{j}(3 - 1) + \hat{k}(-2 - 1) = 5\hat{i} - 2\hat{j} - 3\hat{k}\] \[\overline{a} \cdot \overline{n} = (\hat{i} - \hat{j} + 0\hat{k}) \cdot (5\hat{i} - 2\hat{j} - 3\hat{k}) = (1)(5) + (-1)(-2) + (0)(-3) = 5 + 2 + 0 = 7\] Put $\overline{r} = x\hat{i} + y\hat{j} + z\hat{k}$ in $\overline{r} \cdot \overline{n} = \overline{a} \cdot \overline{n}$: \[(x\hat{i} + y\hat{j} + z\hat{k}) \cdot (5\hat{i} - 2\hat{j} - 3\hat{k}) = 7 \implies 5x - 2y - 3z = 7\]
Final Answer: The Cartesian equation of the plane is $5x - 2y - 3z = 7$ (or $5x - 2y - 3z - 7 = 0$).

Q. 21. If $y = f(u)$ is a differentiable function of $u$ and $u = g(x)$ is a differentiable function of $x$ then prove that $y$ is a differentiable function of $x$ and $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$. (3)

Answer:

Proof (Chain Rule):

Let $\delta x$ be a small increment in $x$, and let $\delta u$ and $\delta y$ be the corresponding increments in $u$ and $y$ respectively, such that $\delta x \neq 0$ and $\delta u \neq 0$.
Consider the incrementary ratio: \[\frac{\delta y}{\delta x} = \frac{\delta y}{\delta u} \times \frac{\delta u}{\delta x}\]
Taking limits on both sides as $\delta x \to 0$: \[\lim_{\delta x \to 0} \frac{\delta y}{\delta x} = \lim_{\delta x \to 0} \left(\frac{\delta y}{\delta u} \times \frac{\delta u}{\delta x}\right) = \left(\lim_{\delta x \to 0} \frac{\delta y}{\delta u}\right) \times \left(\lim_{\delta x \to 0} \frac{\delta u}{\delta x}\right)\]
Since $u = g(x)$ is differentiable, it is also continuous; hence as $\delta x \to 0$, $\delta u \to 0$: \[\lim_{\delta x \to 0} \frac{\delta y}{\delta x} = \left(\lim_{\delta u \to 0} \frac{\delta y}{\delta u}\right) \times \left(\lim_{\delta x \to 0} \frac{\delta u}{\delta x}\right) \quad \text{...(Equation 1)}\]
Since $y$ is a differentiable function of $u$ and $u$ is a differentiable function of $x$, both limits on the RHS of Equation 1 exist and are finite: \[\lim_{\delta u \to 0} \frac{\delta y}{\delta u} = \frac{dy}{du} \quad \text{and} \quad \lim_{\delta x \to 0} \frac{\delta u}{\delta x} = \frac{du}{dx}\]
Therefore, the limit on the LHS of Equation 1 also exists and is finite, meaning $y$ is a differentiable function of $x$ and $\lim_{\delta x \to 0} \frac{\delta y}{\delta x} = \frac{dy}{dx}$.
Substituting these into Equation 1, we get: \[\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}\]

Hence proved.

Q. 22. Verify LMVT for the function $f(x) = \log x$, on $[1, e]$. (3)

Answer:

Step 1: Verify Conditions of LMVT:
1. Since $f(x) = \log x$ is defined and continuous for all $x > 0$, $f(x)$ is continuous on the closed interval $[1, e]$.
2. $f'(x) = \frac{1}{x}$ exists finitely for all $x \in (1, e)$, so $f(x)$ is differentiable on the open interval $(1, e)$.
Thus, all conditions of Lagrange's Mean Value Theorem (LMVT) are satisfied.
Step 2: Find $c \in (1, e)$: By LMVT, there exists at least one $c \in (1, e)$ such that: \[f'(c) = \frac{f(b) - f(a)}{b - a}\] Here $a = 1, b = e$, $f(1) = \log 1 = 0$, $f(e) = \log e = 1$, and $f'(c) = \frac{1}{c}$: \[\frac{1}{c} = \frac{1 - 0}{e - 1} \implies c = e - 1\]
Step 3: Check Interval Membership: Since $2 < e < 3 \implies 1 < e - 1 < 2 < e$, we have $c = e - 1 \in (1, e)$.
Final Answer: Lagrange's Mean Value Theorem (LMVT) is verified with $c = e - 1$.

Q. 23. Evaluate: $\int \frac{\sin x}{\sin 3x} dx$. (3)

Answer:

Given: Let $I = \int \frac{\sin x}{\sin 3x} dx$
Formula: $\sin 3x = 3\sin x - 4\sin^3 x$
Calculation: \[I = \int \frac{\sin x}{3\sin x - 4\sin^3 x} dx = \int \frac{1}{3 - 4\sin^2 x} dx\] Dividing numerator and denominator by $\cos^2 x$: \[I = \int \frac{\sec^2 x}{3\sec^2 x - 4\tan^2 x} dx = \int \frac{\sec^2 x}{3(1 + \tan^2 x) - 4\tan^2 x} dx = \int \frac{\sec^2 x}{3 - \tan^2 x} dx\] Put $\tan x = t \implies \sec^2 x \, dx = dt$: \[I = \int \frac{1}{(\sqrt{3})^2 - t^2} dt\] Using $\int \frac{1}{a^2 - x^2} dx = \frac{1}{2a}\log\left|\frac{a + x}{a - x}\right| + c$: \[I = \frac{1}{2\sqrt{3}}\log\left|\frac{\sqrt{3} + t}{\sqrt{3} - t}\right| + c = \frac{1}{2\sqrt{3}}\log\left|\frac{\sqrt{3} + \tan x}{\sqrt{3} - \tan x}\right| + c\]
Final Answer: $\int \frac{\sin x}{\sin 3x} dx = \frac{1}{2\sqrt{3}}\log\left|\frac{\sqrt{3} + \tan x}{\sqrt{3} - \tan x}\right| + c$

Q. 24. Solve the D.E. $3e^x \tan y \, dx + (1 + e^x)\sec^2 y \, dy = 0$. (3)

Answer:

Given Differential Equation: \[3e^x \tan y \, dx + (1 + e^x)\sec^2 y \, dy = 0\]
Separating Variables: Dividing throughout by $(1 + e^x)\tan y$: \[\frac{3e^x}{1 + e^x} dx + \frac{\sec^2 y}{\tan y} dy = 0\]
Integration: Integrating both sides: \[3\int \frac{e^x}{1 + e^x} dx + \int \frac{\sec^2 y}{\tan y} dy = c_1\] Using $\int \frac{f'(x)}{f(x)} dx = \log|f(x)|$: \[3\log(1 + e^x) + \log|\tan y| = \log c \quad (\text{where } c_1 = \log c)\] \[\log\left((1 + e^x)^3 \cdot \tan y\right) = \log c\] \[(1 + e^x)^3 \tan y = c\]
Final Answer: The general solution is $(1 + e^x)^3 \tan y = c$.

Q. 25. Find $E(X)$ and $V(X)$, where $X$ is the number obtained on uppermost face, when a fair die is thrown. (3)

Answer:

Given: Sample space $S = \{1, 2, 3, 4, 5, 6\}$. Since the die is fair, $P(X = x_i) = \frac{1}{6}$ for each $x_i \in \{1, 2, 3, 4, 5, 6\}$.

Probability Distribution Table:

$x_i$	$1$	$2$	$3$	$4$	$5$	$6$	Total
$p_i$	$\frac{1}{6}$	$\frac{1}{6}$	$\frac{1}{6}$	$\frac{1}{6}$	$\frac{1}{6}$	$\frac{1}{6}$	$1$
$x_i p_i$	$\frac{1}{6}$	$\frac{2}{6}$	$\frac{3}{6}$	$\frac{4}{6}$	$\frac{5}{6}$	$\frac{6}{6}$	$\frac{21}{6}$
$x_i^2 p_i$	$\frac{1}{6}$	$\frac{4}{6}$	$\frac{9}{6}$	$\frac{16}{6}$	$\frac{25}{6}$	$\frac{36}{6}$	$\frac{91}{6}$

1. Expected Value $E(X)$: \[E(X) = \sum_{i=1}^{6} x_i p_i = \frac{1 + 2 + 3 + 4 + 5 + 6}{6} = \frac{21}{6} = \frac{7}{2} = 3.5\]
2. Variance $V(X)$: \[E(X^2) = \sum_{i=1}^{6} x_i^2 p_i = \frac{1 + 4 + 9 + 16 + 25 + 36}{6} = \frac{91}{6}\] \[V(X) = E(X^2) - [E(X)]^2 = \frac{91}{6} - \left(\frac{7}{2}\right)^2 = \frac{91}{6} - \frac{49}{4} = \frac{182 - 147}{12} = \frac{35}{12} \approx 2.92\]
Final Answer: $E(X) = \frac{7}{2}$ (or $3.5$) and $V(X) = \frac{35}{12}$ (or $2.92$).

Q. 26. A pair of dice is thrown 4 times. If getting a doublet is considered as success, find the probability of two successes. (3)

Answer:

Given: Number of trials $n = 4$. When a pair of dice is thrown, $n(S) = 36$.
- Doublets are $\{(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)\} \implies 6$ outcomes.
- Probability of getting a doublet in a single throw: $p = \frac{6}{36} = \frac{1}{6}$
- Probability of failure: $q = 1 - p = 1 - \frac{1}{6} = \frac{5}{6}$
Formula: Let $X$ be the number of successes, so $X \sim B\left(4, \frac{1}{6}\right)$. The p.m.f. is: \[P(X = x) = \binom{n}{x} p^x q^{n-x}\]
Substitution & Calculation (for $x = 2$): \[P(X = 2) = \binom{4}{2} \left(\frac{1}{6}\right)^2 \left(\frac{5}{6}\right)^{4-2} = 6 \times \frac{1}{36} \times \frac{25}{36} = \frac{25}{216}\]
Final Answer: The probability of getting two successes is $\frac{25}{216}$.

SECTION - D

Attempt any FIVE of the following questions: [20]

Q. 27. If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, prove that $A \cdot (\text{adj } A) = (\text{adj } A) \cdot A = |A| \cdot I$. (4)

Answer:

Step 1: Find $|A|$ and $|A| \cdot I$: \[|A| = \begin{vmatrix} 1 & 2 \\ 3 & 4 \end{vmatrix} = (1)(4) - (2)(3) = 4 - 6 = -2\] \[|A| \cdot I = -2 \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} -2 & 0 \\ 0 & -2 \end{bmatrix} \quad \text{...(Equation 1)}\]
Step 2: Find $\text{adj } A$: Cofactors of elements of matrix $A$:
- $A_{11} = (-1)^{1+1}(4) = 4$
- $A_{12} = (-1)^{1+2}(3) = -3$
- $A_{21} = (-1)^{2+1}(2) = -2$
- $A_{22} = (-1)^{2+2}(1) = 1$
\[\text{adj } A = \begin{bmatrix} A_{11} & A_{21} \\ A_{12} & A_{22} \end{bmatrix} = \begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix}\]
Step 3: Compute $A \cdot (\text{adj } A)$: \[A \cdot (\text{adj } A) = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix} = \begin{bmatrix} 4 - 6 & -2 + 2 \\ 12 - 12 & -6 + 4 \end{bmatrix} = \begin{bmatrix} -2 & 0 \\ 0 & -2 \end{bmatrix} \quad \text{...(Equation 2)}\]
Step 4: Compute $(\text{adj } A) \cdot A$: \[(\text{adj } A) \cdot A = \begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix} \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 4 - 6 & 8 - 8 \\ -3 + 3 & -6 + 4 \end{bmatrix} = \begin{bmatrix} -2 & 0 \\ 0 & -2 \end{bmatrix} \quad \text{...(Equation 3)}\]

Conclusion: From Equations 1, 2, and 3, $A \cdot (\text{adj } A) = (\text{adj } A) \cdot A = |A| \cdot I$. Hence proved.

Q. 28. Show that every homogeneous equation of degree two in $x$ and $y$, i.e., $ax^2 + 2hxy + by^2 = 0$, represents a pair of lines passing through the origin, if $h^2 - ab \ge 0$. (4)

Answer:

Proof:

Consider the homogeneous equation of degree two in $x$ and $y$:

\[ax^2 + 2hxy + by^2 = 0 \quad \text{...(Equation 1)}\]

where $a, h, b$ are not all zero. We consider two cases:

Case 1: When $b = 0$ (so $a$ and $h$ are not both zero) Substituting $b = 0$ in Equation 1: \[ax^2 + 2hxy = 0 \implies x(ax + 2hy) = 0\] \[\implies x = 0 \quad \text{and} \quad ax + 2hy = 0\] Both of these are linear equations in $x$ and $y$ without constant terms, so they represent two straight lines passing through the origin.
Case 2: When $b \neq 0$ Multiplying Equation 1 by $b$: \[abx^2 + 2hbxy + b^2y^2 = 0 \implies b^2y^2 + 2hbxy = -abx^2\] Adding $h^2x^2$ to both sides to complete the square: \[b^2y^2 + 2hbxy + h^2x^2 = h^2x^2 - abx^2\] \[(by + hx)^2 = (h^2 - ab)x^2\] Since $h^2 - ab \ge 0$, taking the square root on both sides gives real linear factors: \[by + hx = \pm\sqrt{h^2 - ab}\,x\] \[\implies \left(h + \sqrt{h^2 - ab}\right)x + by = 0 \quad \text{and} \quad \left(h - \sqrt{h^2 - ab}\right)x + by = 0\] Both equations are of the first degree in $x$ and $y$ with no constant term, so they represent two straight lines passing through the origin.

Conclusion: In both cases, $ax^2 + 2hxy + by^2 = 0$ represents a pair of lines passing through the origin if $h^2 - ab \ge 0$. Hence proved.

Q. 29. If $A(\overline{a})$ and $B(\overline{b})$ are any two points in space and $R(\overline{r})$ be a point on the line segment $AB$ dividing internally in the ratio $m : n$ then prove that $\overline{r} = \frac{m\overline{b} + n\overline{a}}{m + n}$. (4)

Answer:

Proof (Section Formula for Internal Division):

Let $O$ be the origin, and let $\overline{OA} = \overline{a}$, $\overline{OB} = \overline{b}$, and $\overline{OR} = \overline{r}$ be the position vectors of points $A, B$, and $R$ respectively.
Since point $R$ divides segment $AB$ internally in the ratio $m : n$, we have: \[\frac{l(AR)}{l(RB)} = \frac{m}{n} \implies n \cdot l(AR) = m \cdot l(RB)\]
Since $R$ lies between $A$ and $B$, the directed vectors $\overline{AR}$ and $\overline{RB}$ have the same direction and sense: \[n\overline{AR} = m\overline{RB}\]
Expressing $\overline{AR}$ and $\overline{RB}$ in terms of position vectors $\overline{a}, \overline{b}, \overline{r}$: \[n(\overline{r} - \overline{a}) = m(\overline{b} - \overline{r})\] \[n\overline{r} - n\overline{a} = m\overline{b} - m\overline{r}\] \[m\overline{r} + n\overline{r} = m\overline{b} + n\overline{a}\] \[(m + n)\overline{r} = m\overline{b} + n\overline{a}\] \[\overline{r} = \frac{m\overline{b} + n\overline{a}}{m + n}\]

Hence proved.

Q. 30. Solve the L.P.P. graphically: Minimize: $z = 5x + 2y$, Subject to, $5x + y \ge 10$, $x + y \ge 6$, $x \ge 0$, $y \ge 0$. (4)

Answer:

Step 1: Table of Boundary Lines and Half-Planes:

Inequality	Boundary Equation	Points on Coordinate Axes	Origin Test $(0, 0)$	Feasible Half-Plane
$5x + y \ge 10$	$5x + y = 10$	$A(0, 10)$, $C(2, 0)$	$0 \ge 10$ (False)	Non-origin side
$x + y \ge 6$	$x + y = 6$	$D(0, 6)$, $B(6, 0)$	$0 \ge 6$ (False)	Non-origin side

Since $x \ge 0$ and $y \ge 0$, the feasible region lies in the first quadrant.

Step 2: Point of Intersection $P$ of $5x + y = 10$ and $x + y = 6$:

Subtracting $x + y = 6$ from $5x + y = 10$: \[4x = 4 \implies x = 1\]
Substituting $x = 1$ in $x + y = 6$: \[1 + y = 6 \implies y = 5\]
Thus, the point of intersection is $P(1, 5)$.

Step 3: Corner Points and Evaluation of Objective Function $z = 5x + 2y$:

The unbounded feasible region is shaded above the segments $AP$ and $PB$ in the first quadrant, with corner points $A(0, 10)$, $P(1, 5)$, and $B(6, 0)$:

Corner Point	Value of Objective Function $z = 5x + 2y$
$A(0, 10)$	$z(A) = 5(0) + 2(10) = 20$
$P(1, 5)$	$z(P) = 5(1) + 2(5) = 5 + 10 = \mathbf{15}$ (Minimum)
$B(6, 0)$	$z(B) = 5(6) + 2(0) = 30$

Final Answer: The minimum value of $z$ is $15$, which occurs at $x = 1, y = 5$.

Q. 31. Evaluate: $\int \frac{3x^2 + 4x - 5}{(x^2 - 1)(x + 2)} \cdot dx$. (4)

Answer:

Given: Let $I = \int \frac{3x^2 + 4x - 5}{(x^2 - 1)(x + 2)} dx = \int \frac{3x^2 + 4x - 5}{(x - 1)(x + 1)(x + 2)} dx$
Step 1: Partial Fraction Decomposition: Let: \[\frac{3x^2 + 4x - 5}{(x - 1)(x + 1)(x + 2)} = \frac{A}{x - 1} + \frac{B}{x + 1} + \frac{C}{x + 2}\] Multiplying both sides by $(x - 1)(x + 1)(x + 2)$: \[3x^2 + 4x - 5 = A(x + 1)(x + 2) + B(x - 1)(x + 2) + C(x - 1)(x + 1)\]
- Put $x = 1$: \[3(1)^2 + 4(1) - 5 = A(1 + 1)(1 + 2) \implies 2 = 6A \implies A = \frac{1}{3}\]
- Put $x = -1$: \[3(-1)^2 + 4(-1) - 5 = B(-1 - 1)(-1 + 2) \implies -6 = -2B \implies B = 3\]
- Put $x = -2$: \[3(-2)^2 + 4(-2) - 5 = C(-2 - 1)(-2 + 1) \implies 12 - 8 - 5 = 3C \implies -1 = 3C \implies C = -\frac{1}{3}\]
Step 2: Integration: Substituting $A = \frac{1}{3}, B = 3$, and $C = -\frac{1}{3}$: \[I = \int \left(\frac{1/3}{x - 1} + \frac{3}{x + 1} - \frac{1/3}{x + 2}\right) dx\] \[I = \frac{1}{3}\int \frac{1}{x - 1} dx + 3\int \frac{1}{x + 1} dx - \frac{1}{3}\int \frac{1}{x + 2} dx\] \[I = \frac{1}{3}\log|x - 1| + 3\log|x + 1| - \frac{1}{3}\log|x + 2| + c\]
Final Answer: $\int \frac{3x^2 + 4x - 5}{(x^2 - 1)(x + 2)} dx = \frac{1}{3}\log|x - 1| + 3\log|x + 1| - \frac{1}{3}\log|x + 2| + c$

Q. 32. Prove that: $\int_{a}^{b} f(x) dx = \int_{a}^{b} f(a + b - x) \cdot dx$. Hence, find $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \sin^2 x \cdot dx$. (4)

Answer:

Part 1: Proof of Property:

Consider $\text{RHS} = \int_{a}^{b} f(a + b - x) dx$.
Put $a + b - x = t \implies -dx = dt \implies dx = -dt$.
Change of Limits:
- When $x = a$, $t = a + b - a = b$
- When $x = b$, $t = a + b - b = a$
Substituting in RHS: \[\text{RHS} = \int_{b}^{a} f(t)(-dt) = -\int_{b}^{a} f(t) dt\] Using the property $\int_{a}^{b} f(t) dt = -\int_{b}^{a} f(t) dt$ and $\int_{a}^{b} f(t) dt = \int_{a}^{b} f(x) dx$: \[\text{RHS} = \int_{a}^{b} f(t) dt = \int_{a}^{b} f(x) dx = \text{LHS}\] Hence proved.

Part 2: Evaluation of $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \sin^2 x \cdot dx$:

Let: \[I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \sin^2 x \, dx \quad \text{...(Equation 1)}\]
Using $\int_{a}^{b} f(x) dx = \int_{a}^{b} f(a + b - x) dx$, with $a + b = \frac{\pi}{6} + \frac{\pi}{3} = \frac{\pi}{2}$: \[I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \sin^2\left(\frac{\pi}{2} - x\right) dx = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \cos^2 x \, dx \quad \text{...(Equation 2)}\]
Adding Equation 1 and Equation 2: \[2I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} (\sin^2 x + \cos^2 x) dx = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} 1 \, dx = [x]_{\frac{\pi}{6}}^{\frac{\pi}{3}} = \frac{\pi}{3} - \frac{\pi}{6} = \frac{\pi}{6}\] \[I = \frac{\pi}{12}\]
Final Answer: $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \sin^2 x \cdot dx = \frac{\pi}{12}$

Q. 33. Find the area enclosed between the circle $x^2 + y^2 = 1$ and the line $x + y = 1$ lying in the first quadrant. (4)

Answer:

Given Curves:
- Circle: $x^2 + y^2 = 1 \implies y_1 = \sqrt{1 - x^2}$ (in the first quadrant)
- Line: $x + y = 1 \implies y_2 = 1 - x$
Points of Intersection: The circle $x^2 + y^2 = 1$ and the line $x + y = 1$ intersect at $A(1, 0)$ on the X-axis and $B(0, 1)$ on the Y-axis.
Formula: Required area $A$ enclosed between the circle and the line from $x = 0$ to $x = 1$ is: \[A = \int_{0}^{1} (y_1 - y_2) dx = \int_{0}^{1} \sqrt{1 - x^2} \, dx - \int_{0}^{1} (1 - x) \, dx\]
Substitution & Calculation: Using $\int \sqrt{a^2 - x^2} dx = \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2}\sin^{-1}\left(\frac{x}{a}\right)$: \[A = \left[\frac{x}{2}\sqrt{1 - x^2} + \frac{1}{2}\sin^{-1}(x)\right]_{0}^{1} - \left[x - \frac{x^2}{2}\right]_{0}^{1}\] \[A = \left[\left(0 + \frac{1}{2}\sin^{-1}(1)\right) - (0 + 0)\right] - \left[\left(1 - \frac{1}{2}\right) - 0\right]\] \[A = \left(\frac{1}{2} \cdot \frac{\pi}{2}\right) - \frac{1}{2} = \frac{\pi}{4} - \frac{1}{2} = \left(\frac{\pi - 2}{4}\right)\text{ sq. units}\]
Final Answer: The required enclosed area is $\left(\frac{\pi}{4} - \frac{1}{2}\right)\text{ sq. units}$ (or $\frac{\pi - 2}{4}\text{ sq. units}$).

Q. 34. Solve the differential equation $x^2 \cdot \frac{dy}{dx} = x^2 + xy + y^2$. (4)

Answer:

Given Differential Equation: \[x^2 \frac{dy}{dx} = x^2 + xy + y^2\] Dividing both sides by $x^2$: \[\frac{dy}{dx} = 1 + \frac{y}{x} + \left(\frac{y}{x}\right)^2 \quad \text{...(Equation 1)}\] This is a homogeneous differential equation.
Substitution: Put $y = vx \implies v = \frac{y}{x}$. Differentiating with respect to $x$: \[\frac{dy}{dx} = v + x \frac{dv}{dx} \quad \text{...(Equation 2)}\]
Separating Variables & Integrating: Substituting Equation 2 and $\frac{y}{x} = v$ into Equation 1: \[v + x \frac{dv}{dx} = 1 + v + v^2 \implies x \frac{dv}{dx} = 1 + v^2\] \[\frac{1}{1 + v^2} dv = \frac{1}{x} dx\] Integrating both sides: \[\int \frac{1}{1 + v^2} dv = \int \frac{1}{x} dx\] \[\tan^{-1}(v) = \log|x| + c\] Resubstituting $v = \frac{y}{x}$: \[\tan^{-1}\left(\frac{y}{x}\right) = \log|x| + c\]
Final Answer: The general solution of the differential equation is $\tan^{-1}\left(\frac{y}{x}\right) = \log|x| + c$.

HSC Mathematics & Statistics (40) 12th Standard March 2026 Board Question Paper

Maharashtra HSC Class 12 English June 2026 Board Question Paper with Complete Solutions (J-201)

aaminomtex@gmail.com (Amin) — Sun, 25 Oct 2026 20:24:24 +0530

Welcome to the complete and expertly solved Maharashtra HSC Board Class 12 English Board Question Paper (June 2026, Code J-201). In this comprehensive study guide, we have provided 100% accurate, board-level Question Paper Solutions across all four sections: Prose Comprehension, Poetry & Appreciation, Non-Textual Grammar, Writing Skills (Virtual Message, SOP, Group Discussion, E-mail, Report, Interview, Speech, Compering, Expansion of Idea, Film Review, Blog, and Appeal), and the Literary Genre (Novel) section covering To Sir, With Love, Around the World in Eighty Days, and The Sign of Four. Use these step-by-step model answers and exam tips to master the official MSBSHSE marking scheme and score maximum marks in your upcoming board exams.

SECTION I: PROSE (Reading for Comprehension, Language Study, Summary and Mind Mapping)

Q. 1. (A) Read the extract and complete the activities given below: (12 Marks)

A1. Read the following statements and write TWO correct statements that state the theme of the extract: (2)

The two correct statements that state the theme of the extract are:

(i) Big Data helps educational institutes in decision making and improving efficiency.
(iii) Big Data helps entertainment and media to understand and predict audience preferences.

A2. Netflix and Youtube used big data to increase engagement of the viewers and earn more revenues. Support your answer with proper evidences from the extract. (2)

Answer: In the entertainment and media industry, Big Data focuses on targeting people with the right content at the right time. The extract states that based on a viewer's past views and online behaviour, they are shown personalized recommendations. This specific technique of analysing user data to customize content recommendations is popularly used by Netflix and YouTube to increase viewer engagement and drive higher revenues.

A3. Complete the flow chart on the role of big data in educational industry. One is done for you. Find information from the extract and complete the flow chart: (2)

To prepare customized and dynamic learning programmes according to the need of an individual student. (Given in Question Paper)
To design course material catering to different requirements and comprehension levels of the students.
To gauge the progress of the students from their results and understand their strengths and weaknesses.
To guide the student regarding the best career for him based on his mental make-up and abilities.

A4. Write the precautions one should take while using social media. (2)

Answer: While using social media, one should strictly avoid sharing sensitive personal information such as home addresses, phone numbers, or financial details publicly. Users must verify the authenticity of news before forwarding it to prevent the spread of misinformation. Additionally, it is essential to use strong passwords, enable two-factor authentication, keep privacy settings updated, and avoid interacting with unknown or suspicious links to stay safe from cyberbullying and online fraud.

Exam Tip: In Personal Response questions (A4), always write your answer in 4 to 5 well-structured sentences. Start with a direct topic sentence, provide 2 to 3 logical supporting points, and end with a concluding thought to secure full marks (2/2).

A5. Do as directed: (2)

(i) Every student's comprehension level is different. (Choose the correct 'Negative Sentence' from the options given below and write it.)

Answer: (b) Every student's comprehension level is not the same.

(ii) This will help in guiding the students for the best career. (Choose the correct option replacing the modal auxiliary expressing 'Advice' and write it.)

Answer: (d) This should help in guiding the students for the best career.

A6. Vocabulary: Find out the words/phrases from the extract which mean: (2)

(a) Enrich: enhance
(b) Income: revenues / revenue
(c) Engulfed: inundated
(d) Rapidly: in leaps and bounds

Q. 1. (B) Language Study (Non-Textual Grammar) (4 Marks)

B1. Do as directed: (3)

(i) There are no unemployed, no slackers in a civilised society. (Choose the correct option from the given alternatives showing the correct use of 'Neither... nor' and write it.) (1)

Answer: (a) There are neither unemployed nor slackers in a civilised society.

(ii) Sandip said to Rushika, "Do you like to take part in sports competition?" (Choose the correct 'Indirect Speech' from the given options and write it.) (1)

Answer: (a) Sandip asked Rushika if / whether she liked to take part in sports competition.

(iii) When the officer fired his assistant, he heaved a sigh of relief. (Choose the correct option from the given alternatives showing the correct use of 'No sooner... than' and write it.) (1)

Answer: (c) No sooner did the officer fire his assistant than he heaved a sigh of relief.

B2. Spot the error and rewrite the correct sentence: (1)

Given Sentence: Rutuja is one of the best student in using new technology.
Corrected Sentence: Rutuja is one of the best students in using new technology.

Grammar Rule: The phrase "one of the..." is always followed by a plural countable noun (e.g., one of the best students) and takes a singular verb if it functions as the subject.

Q. 2. (A) Read the extract and complete the activities given below: (12 Marks)

A1. Read the following statements and arrange them in proper sequence: (2)

Answer: The proper chronological sequence of the statements is:

(d) Class assignment was given by Mrs. Brauch.
(b) Dalkoff wrote a short story for the first time.
(a) Dalkoff contributed as co-editor to his high school newspaper.
(c) Dalkoff met Mrs. Brauch after her retirement.

A2. Describe Dalkoff's visit to his High-school's 30th reunion. (2)

Answer: During his 30th high-school reunion, Dalkoff went back and visited his retired English teacher, Mrs. Ruth Brauch. He expressed his heartfelt gratitude and told her what her four words of praise had done for his life. He explained that because she gave him the confidence to be a writer, he was able to pass that same confidence on to the woman who became his wife, inspiring her to become a writer as well.

A3. 'They changed his life.' Here 'They' means _______ and write 'the change' they brought in his life. (2)

Answer: Here, 'They' refers to the four words—"This is good writing"—that Mrs. Brauch wrote in the margin of Dalkoff's assignment paper. These words brought a profound transformation in his life: they gave the terribly insecure and shy boy self-confidence, inspired him to write his first short story, helped him become the co-editor of his high-school newspaper, broadened his horizons, and set him off on a successful, fulfilling life.

A4. 'The reward or the words of praise,' what you will remember the most. Justify your view point. (2)

Answer: I will always remember words of praise the most. While a material reward offers only temporary excitement and fades over time, genuine words of appreciation leave an everlasting emotional impact. Sincere encouragement from a mentor builds self-belief, eliminates self-doubt, and—just like Mrs. Brauch's feedback to Dalkoff—can spark a lifelong passion and positively transform a person's destiny.

A5. Do as directed: (2)

(i) The students had been reading 'To Kill a Mockingbird.' (Choose the correct option from the given alternatives showing the correct use of 'Future Perfect Tense' and write it.)

Answer: (d) The students will have read 'To Kill a Mockingbird.'

(ii) He was named co-editor of his high-school newspaper. (Choose the correct option from the given alternatives showing the correct use of 'Active Voice' and write it.)

Answer: (b) They named him co-editor of his high-school newspaper.

A6. Match the column: Match the words / phrases in column 'A' with their meaning in column 'B': (2)

Column 'A' (Words / Phrases)	Column 'B' (Correct Meaning)
(1) recall	(d) remember
(2) shy	(c) timid
(3) pass on	(a) give to another person
(4) convince	(b) assure

Q. 2. (B) Summarising: (3 Marks)

Write a 'Summary' of the above extract by using the following points: (Class assignment — teacher's encouraging feedback in the margin — fulfilment of the dream — Dalkoff as co-editor — School reunion experience.)

The Transformative Power of Encouragement

As a shy and insecure boy, Dalkoff lacked self-confidence until his high-school English teacher, Mrs. Brauch, assigned the class to write a follow-up chapter to a novel. On his submitted assignment, she wrote four encouraging words in the margin: "This is good writing." This simple feedback transformed his life, instilling the self-belief needed to fulfill his dream of writing short stories. As his confidence grew, he became the co-editor of his high-school newspaper and embarked on a successful career. Thirty years later, during his school reunion, a grateful Dalkoff visited the retired Mrs. Brauch to share how her encouragement not only made him a writer but also inspired his wife to become one.

Q. 2. (C) Mind Mapping: (3 Marks)

Prepare a mind map on the topic 'Role of Newspapers in Society,' expressing and presenting your own ideas, flow, design and structure.

Central Idea: ROLE OF NEWSPAPERS IN SOCIETY

Branch 1: Information & News Dissemination
- Local, national, and international daily news updates
- Coverage of politics, economy, business, and sports
- Weather forecasts, civic announcements, and public notices
Branch 2: Education & Intellectual Growth
- Editorial columns and expert opinion pieces
- Enhancement of vocabulary, language skills, and reading habits
- Informative supplements on science, health, career, and technology
Branch 3: Social & Democratic Pillar (Fourth Estate)
- Shaping public opinion and raising awareness on social issues
- Investigative journalism exposing corruption and ensuring accountability
- Platform for citizens' grievances through 'Letters to the Editor'
Branch 4: Entertainment & Utility Services
- Crosswords, Sudoku puzzles, cartoons, and comic strips
- Classified advertisements for jobs, real estate, and matrimonials
- Reviews of books, cinema, art, and cultural events

Exam Tip: In the board exam, draw your Mind Map using neat boxes or circles connected by arrows radiating from the central title box using only a black or blue pen.

SECTION II: POETRY (Poetry and Appreciation)

Q. 3. (A) Read the extract and complete the activities given below: (10 Marks)

A1. Identify the lines from the extract that indicate the father's 'dull, monotonous routine' and 'his least value for other': (2)

Aspect	Lines from the Extract
Dull, Monotonous routine	(1) "My father travels on the late evening train / Standing among silent commuters in the yellow light" (OR "Suburbs slide past his unseeing eyes")
Least value for other	(1) "Like a word dropped from a long sentence."

A2. The passengers are described as 'silent.' Give reasons. (2)

Answer: The passengers on the late evening train are described as 'silent' because they are physically and mentally exhausted after a long, hard day of work. Furthermore, the monotonous, fast-paced routine of urban life leaves commuters absorbed in their own worries and fatigue, creating a deep sense of emotional detachment and alienation even while standing in a crowded compartment.

A3. Write the difficulties you face while commuting to and from the college by public transport. (2)

Answer: While commuting to and from college by public transport, I frequently face challenges such as severe overcrowding during peak hours, irregular bus or train schedules, and heavy traffic congestion. Standing for long durations with a heavy bag is physically tiring, and the constant noise, heat, and air pollution make the daily commute even more stressful, particularly during the monsoon season.

A4. Name and explain the figure of speech in the following line: 'Suburbs slide past his unseeing eyes.' (2)

Alliteration: The consonant sound 's' is pleasingly repeated at the beginning of the words 'Suburbs' and 'slide' for poetic and musical effect.
Transferred Epithet: The adjective 'unseeing' is transferred from the father to his 'eyes'—his eyes can physically see, but it is the father's mind that is indifferent and preoccupied with his own thoughts.
Paradox: The phrase 'unseeing eyes' combines two contradictory ideas (eyes that do not see) to highlight how familiar and monotonous the scenery has become to the father.

A5. Compose four poetic lines on the title 'My Father.' (2)

My Father
My father is our family's guiding light,
Who toils tirelessly from morning to night.
He silently sacrifices his own little joys,
To bless our lives with courage and poise.

Q. 3. (B) Appreciation: (4 Marks)

Read the extract and write the appreciation of the poem 'Song of the Open Road'.

Appreciation of the Poem 'Song of the Open Road'

About the Poem, Poet, and Title: 'Song of the Open Road' is an inspiring motivational poem written by the celebrated nineteenth-century American poet Walt Whitman, famously known as the 'Father of Free Verse'. Taken from his classic collection Leaves of Grass, the title metaphorically represents the journey of life marked by freedom, mobility, and endless possibilities.
Theme: The central theme of the poem is freedom, self-reliance, and optimism. The speaker sets out on the 'open road' feeling "afoot and light-hearted," choosing to leave behind domestic routines and societal pressures in order to experience life on his own terms.
Poetic Style and Devices: The poem is written in free verse without a fixed rhyme scheme or meter, which aptly reflects the poet's desire for unrestrained freedom. It is narrated as a dramatic monologue in the first person ('I'). Whitman enriches the poem with figures of speech such as Metaphor (the 'open road' symbolizing the journey of life), Repetition ("Henceforth I ask not... Henceforth I whimper no more"), Climax, Inversion, and Paradox ("old delicious burdens").
Special Features and Message: A striking structural feature of the poem is the final stanza enclosed in parentheses (brackets), where the poet pauses to reflect that although he seeks independence, he lovingly carries his "old delicious burdens"—sweet memories and relationships with people. The poem delivers a powerful message that we are the makers of our own destiny ("I myself am good-fortune").
My Opinion: In my opinion, this is a timeless and uplifting poem that encourages readers to stop complaining, trust their inner strength, and walk the path of life with confidence and joy.

Memory Trick: While writing a poetic appreciation in Paragraph format for the HSC exam, remember the acronym ATPM (About, Theme, Poetic Devices/Style, Message & Opinion) to cover every marking criterion seamlessly.

SECTION III: WRITING SKILLS

Q. 4. Complete the activities as per the instructions given below: (16 Marks)

(A) Attempt 'Any One' of the following activities: (4 Marks)

(I) Virtual Message:

MESSAGE

Date: 16 June 2026
Time: 9:30 a.m.

Dear Dad,

After you and Mom left for the gym, I suddenly started feeling very uneasy, sick, and weak. Since I have to attend a very important interview for a new job today, I am leaving home early. As I am feeling unwell, could you please pick me up at around 2:00 p.m. from the interview site?

Jeevan

(II) Statement of Purpose (SOP):

Statement of Purpose

My fascination with literature, critical thinking, and human communication began during my early school years, inspiring me to actively participate in debate clubs, elocution contests, and creative writing workshops. Having completed my Higher Secondary Certificate (HSC) in the Arts stream with distinction, I now wish to pursue a Bachelor of Arts (Honours) in English Literature at your esteemed university.

My long-term career goal is to become an academic researcher, university professor, and author exploring how contemporary literature shapes social consciousness. Throughout junior college, I consistently secured top grades in English, served as the Student Editor of our college magazine, and won the Inter-School Essay Championship.

Your university's interdisciplinary curriculum, distinguished faculty, and vibrant literary research ecosystem align perfectly with my academic aspirations. I am confident that studying at your prestigious institution will sharpen my analytical abilities and empower me to contribute meaningfully to the academic community.

(III) Group Discussion:

Topic: Is Money the Key to Happiness?

Evaluator (Teacher): Good morning, everyone. Welcome to today's group discussion. Our topic is 'Is Money the Key to Happiness?' You may now share your views.
Participant 1 (Rahul): Good morning, all. I firmly believe that money plays a vital role in achieving happiness. In today's world, financial stability is essential to fulfill basic human needs such as nutritious food, quality healthcare, shelter, and good education. Money provides security and reduces daily anxiety, which forms the foundation of a happy life.
Participant 2 (Sneha): While I agree with Rahul that money is necessary for basic survival, I disagree that it is the ultimate 'key' to happiness. We often see wealthy individuals suffering from stress, loneliness, and broken relationships. True happiness stems from inner peace, good health, contentment, and meaningful bonds with family and friends—none of which can be bought with money.
Participant 3 (Amit): I would like to bridge both points made by Rahul and Sneha. Money can certainly buy material comfort, convenience, and opportunities to pursue hobbies or help the needy, which brings immense joy. However, beyond a certain threshold, extra wealth does not guarantee emotional fulfilment. Therefore, money is an important facilitator of comfort, but gratitude and positive relationships are the true keys to happiness.
Evaluator (Teacher): Thank you, participants. That was an insightful and balanced discussion concluding that while money is indispensable for security and comfort, true happiness ultimately depends on emotional well-being and human relationships.

(B) Attempt 'Any One' from the given activities: (4 Marks)

(I) E-mail Writing:

To: manager@lonavalatouristhub.com
Subject: Inquiry regarding family room bookings, tariff, and facilities

Respected Sir/Madam,

I am planning a three-day family vacation to Lonavala with four family members from 25th June to 27th June 2026, and we would like to stay at your esteemed tourist centre.

Could you kindly share the details regarding the types of rooms available (preferably an AC Family Suite or two interconnected deluxe rooms) along with their per-night tariff and check-in/check-out timings? Additionally, please let me know about the in-house food arrangements, whether complimentary breakfast is included in the package, and if your centre organizes local sightseeing tours and pick-up facilities from Lonavala Railway Station.

Kindly email the brochure and tariff details at your earliest convenience so that we can finalize our reservation.

Thank you.

Warm regards,
ABC
Contact: 98XXXXXXXX

(II) Report Writing:

Grand Success at Inter-College Youth Festival 'Tarang 2026'
— By Cultural Secretary, XYZ Junior College

Mumbai, 16th June 2026: XYZ Junior College successfully organized a vibrant two-day Inter-College Youth Festival, 'Tarang 2026', on the 14th and 15th of June in the college main auditorium. The grand cultural extravaganza witnessed enthusiastic participation from over 25 prominent colleges across Mumbai.

The festival was inaugurated by the renowned playback singer Mr. Shaan, who graced the occasion as the Chief Guest. After lighting the ceremonial lamp, he inspired the youth with a melodious performance and an encouraging address on nurturing artistic passion alongside academics.

The event featured fierce yet healthy competitions across Classical and Western Singing, Elocution and Debate, Folk and Contemporary Dance, and Fine Arts exhibitions. Our host college clinched the First Prize in the Group Dance competition, while St. Xavier's College won the Best Debate Trophy. The coveted Overall Championship Trophy was bagged by Jai Hind College. The festival concluded with a prize distribution ceremony and the National Anthem, leaving everyone with cherished memories.

(III) Interview Questions:

Good morning, Doctor! It is an absolute privilege to welcome you to our college studio today. Our students are eager to learn from your inspiring journey in the medical field.

Field / Point	Details / Interview Questions
Name of the interviewee	Dr. Rajesh Sharma (MD, DM Neurology)
Area of success / Reputation	Renowned Neurologist and Social Healthcare Reformer
Date / Venue / Time	16th June 2026 / College Auditorium / 10:30 a.m.
Duration of Interview	30 Minutes
(1) Choice of profession	What inspired you during your formative years to choose the noble profession of medicine over other career paths?
(2) Area of specialization	Medicine has many branches; what motivated you to specialize specifically in a complex field like Neurology?
(3) Most challenging case	Could you share your experience of the most critical and challenging medical case you have successfully handled so far?
(4) Success / Achievement	Looking back at your illustrious career, which milestone do you consider your greatest professional achievement?
(5) Stress management	Handling life-and-death situations daily is emotionally exhausting; how do you manage occupational stress and stay calm under pressure?
(6) Work balance	With emergency calls and long hospital hours, how do you strike a healthy balance between your family life and professional commitments?
(7) Goals / Dreams	What are your future goals or dreams regarding making advanced healthcare accessible to underprivileged rural patients?
(8) Advice	Finally, what message or advice would you like to give to young NEET aspirants and students wishing to serve in the medical field?

Thank you so much, Doctor, for sharing your valuable time and wisdom with us today!

(C) Attempt 'Any One' of the following activities: (4 Marks)

(I) Speech Writing:

Human Future with AI (Artificial Intelligence)

Respected teacher and my dear friends, a very good morning to one and all!

Today, I stand before you to share my views on a revolutionary force that is redefining human civilization: 'Human Future with Artificial Intelligence.' AI is no longer a distant concept confined to science-fiction films; it is actively shaping our present and building our tomorrow.

From early cancer detection and climate modelling to personalized education and autonomous transport, AI has enhanced human efficiency beyond imagination. It processes massive data in seconds, freeing human minds for higher creativity and innovation. However, our future with AI is a double-edged sword. Rapid automation raises valid concerns regarding job displacement, deepfakes, data privacy, and ethical accountability. We must ask ourselves: Are we guiding technology, or allowing technology to govern us?

My friends, the key to a prosperous future lies in human-AI collaboration. AI possesses artificial computation, but humans possess emotional intelligence, empathy, and moral conscience. If we regulate AI ethically and upskill ourselves continuously, AI will serve as humanity's greatest ally rather than a replacement.

Thank you!

(II) Compering:

Compering Script: Independence Day Celebration

1. Greeting with Patriotic Spirit:
"Freedom in the mind, faith in the words, pride in our souls!"
A very warm, energetic, and patriotic morning to our respected Principal, honourable Chief Guest, dedicated teachers, and my dear fellow students. Today, as the tricolour breeze fills the air, we gather here with immense pride to celebrate the Independence Day of our great nation, India!
2. Welcoming Dignitaries:
It is my honour to extend a floral and heartfelt welcome to our Chief Guest for today, Colonel Vikram Rathod (Retd.), a decorated war veteran who has dedicated three decades of his life to guarding our borders. I request our Principal to felicitate Colonel Rathod with a sapling and a shawl. Please put your hands together!
3. Flag Hoisting and National Anthem:
Now comes the most solemn and proud moment of the morning. May I request our honourable Chief Guest and respected Principal to step forward for the unfurling of the National Flag. I request everyone to stand in attention as we salute the Tiranga and sing our National Anthem in unison.
[After the National Anthem] Bharat Mata ki Jai! Vande Mataram!
4. Cultural Fest Programme:
Thank you, everyone, please be seated. To celebrate the rich heritage and sacrifices of our freedom fighters, our talented students have curated a vibrant cultural programme. First, let us welcome the college choir group with a soul-stirring patriotic medley, followed by an energetic dance drama titled 'Udaan: The Spirit of Young India'. Let us encourage them with a thunderous round of applause!
5. Patriotic Message and Vote of Thanks:
What a mesmerizing performance! As we conclude today's ceremony, let us remember that freedom came at a heavy price. Let us pledge to uphold unity, integrity, and civic responsibility to build a developed and harmonious India. I extend my sincere gratitude to our Chief Guest, Principal, staff, and students for making this morning unforgettable. Jai Hind! Jai Bharat!

(III) Expansion of Idea:

Forgiveness is Often Better Than Punishment

The timeless proverb 'Forgiveness is often better than punishment' highlights the transformative moral power of mercy over retribution. As Alexander Pope famously wrote, "To err is human; to forgive, divine." While punishment is the conventional way to penalize a wrongdoer, it frequently breeds resentment, rebellion, and an endless cycle of hatred. Punishment may enforce temporary fear, but it rarely reforms a person's conscience.

In contrast, forgiveness is an act of supreme humanity and inner strength. As Mahatma Gandhi rightly observed, "The weak can never forgive. Forgiveness is the attribute of the strong." When we choose to forgive someone who has erred, we awaken their sense of guilt and self-reflection, offering them a genuine opportunity to reform and become a better human being. Furthermore, forgiveness liberates the victim from the toxic burden of anger and bitterness, restoring mental peace.

History bears witness to leaders like Nelson Mandela, who chose forgiveness and reconciliation over vengeance to unite a divided South Africa. Thus, while severe crimes require legal justice, in human relationships and society, compassion and forgiveness heal wounds far more effectively than punishment ever can.

(D) Attempt 'Any One' of the following activities: (4 Marks)

(I) Film Review:

Film Review: '3 Idiots' — A Timeless Masterpiece on True Education

Title, Subject, and Story: '3 Idiots' is an iconic Indian coming-of-age comedy-drama that examines the immense academic pressure and rote-learning culture in the Indian education system. The story revolves around three engineering students—Rancho, Farhan, and Raju—at the prestigious Imperial College of Engineering, tracing their heartwarming bond and their quest to follow their true passions over the rat race for grades.
Cinematography, Music, and Presentation: Muraleedharan C. K.'s cinematography breathtakingly captures both the bustling campus life and the serene landscapes of Ladakh. Shantanu Moitra's music—especially tracks like 'All Izz Well' and 'Give Me Some Sunshine'—perfectly echoes the youth's emotions. The screenplay seamlessly blends rib-tickling humour with deeply moving social commentary.
Producer, Director, and Characters: Produced by Vidhu Vinod Chopra and masterfully directed by Rajkumar Hirani, the film features stellar performances. Aamir Khan shines as the unconventional genius Rancho, R. Madhavan and Sharman Joshi deliver relatable performances, and Boman Irani is unforgettable as the strict college director 'Virus'.
Personal Views, Evaluation, and Recommendation: '3 Idiots' is more than just cinema; it is an eye-opener that teaches us: "Pursue excellence, and success will follow." I rate this film 5/5 stars and strongly recommend it to every student, teacher, and parent.

(II) Blog Writing:

STUDENT HOBBY CLUB BLOG | www.studenthobbyclub.org/blog
Home | Book Club | Archive | Contact Us

Read for Survival: Why Books Are Our Ultimate Lifeline

Posted by: Aryan Deshmukh | Date: 16 June 2026

Have you ever felt overwhelmed by the constant noise and endless scrolling of the digital world? In an era of shrinking attention spans, reading is no longer just a leisure pastime—it is a vital tool for mental and intellectual survival!

Personally, I love immersing myself in biographies, mystery thrillers, and classic science fiction, as they transport me across time and space. Beyond sheer entertainment, reading offers extraordinary cognitive benefits. It sharpens concentration, enriches vocabulary, stimulates critical thinking, and significantly lowers stress levels. Most importantly, books nurture empathy by allowing us to walk in the shoes of diverse characters.

On a personal note, a good book has always been my safest sanctuary during tough days. Whether you prefer the aroma of a paperback or the convenience of an e-reader, dedicating just twenty minutes a day to reading can transform your perspective on life. So, pick up a book today and read to thrive!

Liked this post? Share your favourite book title in the comments below! | Share | Subscribe

(III) Appeal Writing:

SAY NO TO PLASTIC: SAVE OUR MOTHER EARTH!

"Plastic is Drastic! Refuse Single-Use Plastic Today for a Greener Tomorrow!"

[Logo / Visual Illustration: A green Planet Earth breaking free from a suffocating plastic bag]

Did you know that a single plastic carry bag takes over 500 years to decompose? Every single day, tons of non-biodegradable plastic choke our oceans, kill innocent cattle and marine life, clog urban drainage systems, and release toxic microplastics into our food chain!

How Can YOU Be a Planet Hero?

Carry Your Own Bag: Always keep a reusable cloth or jute bag handy while shopping.

Ditch Single-Use Plastic: Say a firm NO to plastic straws, disposable cutlery, thermocol plates, and bottled water.

Switch to Eco-Friendly Alternatives: Use steel water bottles, bamboo toothbrushes, and biodegradable packaging.

Spread Awareness: Encourage your family, neighbours, and shopkeepers to adopt a zero-plastic lifestyle.

Take the 'Plastic-Free Campus & City Pledge' Today!
Venue: College Quadrangle | Date: 20th June 2026 | Time: 10:00 a.m.
Issued in Public Interest by: The NSS Unit & Eco-Club, XYZ Junior College

SECTION IV: LITERARY GENRE (NOVEL)

Q. 5. (A) Complete the activities given below as per the instructions: (4 Marks)

(I) Match the columns given below in column A 'Types of novel' to column B 'Language origin': (2)

Column 'A' (Types of Novel)	Column 'B' (Correct Language Origin)
(1) Picaresque	(b) Spanish (from 'picaro')
(2) Epistolary	(a) Latin (from 'epistola')
(3) Bildungsroman	(d) German
(4) Novella	(c) Italian (from 'novella')

(II) State whether the following statements are 'True or False' and write them: (2)

(a) Psychological novel means the flow of thought. — False
(Textbook Note: The phrase 'Stream of Consciousness' coined by William James means the flow of thoughts, whereas a psychological novel treats the internal life and motives of the protagonist.)
(b) The pulp fiction era provided a building ground for a detective fiction. — True
(c) Realistic novel narrates the adventures of the protagonist. — False
(Textbook Note: A Picaresque novel narrates the adventures of an eccentric protagonist, whereas a Realistic novel gives the effect of realism and complex everyday character motives.)
(d) Hercule Poirot is the evergreen detective character created by Agatha Christie. — True

Q. 5. (B) Answer in about 50 words to the questions given below ('To Sir, With Love'): (4 Marks)

(I) Write a short note on Denham's viewpoint on P.T. in 'To Sir, With Love.' (2)

Answer: During the Half-Yearly Report of the Students' Council, Denham bluntly criticizes the mandatory P.T. (Physical Training) classes as ill-conceived, pointless, and monotonous. He argues that due to space constraints and lack of proper equipment at Greenslade School, twenty minutes of routine physical exercises twice a week offer no real physical benefit. He asserts that students would much rather play invigorating outdoor games and suggests that P.T. should be compulsory only for weak or sedentary students who genuinely need it.

(II) When the turn of Mr. Braithwaite's class came, he sat up anxiously. Explain with reference to the novel, 'To Sir, With Love.' (2)

Answer: Mr. Braithwaite sat up anxiously when the senior class was called to present their report during the Students' Council programme because it was the ultimate test of his teaching efforts. Previously notorious for being unruly and disrespectful, his students were now addressing the entire school and faculty independently. He felt nervous yet deeply hopeful to see whether they would demonstrate the maturity, courtesy, and self-confidence he had patiently instilled in them.

Q. 5. (C) Answer in about 50 words to the questions given below ('Around the World in Eighty Days'): (4 Marks)

(I) Passepartout blamed himself for his master's misfortune in 'Around the World in Eighty Days.' Give reasons. (2)

Answer: Passepartout was overwhelmed with guilt and blamed himself for Phileas Fogg's apparent ruin because he had known Inspector Fix's true identity and his mission to arrest Fogg much earlier in the journey, yet kept it a secret from his master. Passepartout felt that if he had warned Fogg in time, Fogg could have proven his innocence or avoided Fix's trap at Liverpool, preventing the wrongful arrest that delayed their arrival in London.

(II) Explain the significance of the statement 'You have made a mistake of one day!' in giving a justifiable ending in 'Around the World in Eighty Days.' (2)

Answer: Passepartout's breathless exclamation, "You have made a mistake of one day!", serves as the thrilling climax and turning point of the novel. Believing he arrived in London five minutes late on Saturday night, Fogg accepts defeat. However, because Fogg had travelled continuously eastward towards the sun, he had unconsciously gained four minutes for every degree of longitude crossed—totalling a full twenty-four hours. Discovering it is actually Saturday and not Sunday enables Fogg to reach the Reform Club just in time, winning the wager and marrying Aouda.

Q. 5. (D) Answer in about 50 words to the questions given below ('The Sign of Four'): (4 Marks)

(I) Sherlock Holmes established a connection between Captain Morstan's disappearance and receipt of pearls in 'The Sign of Four.' Elaborate. (2)

Answer: Through sharp deductive reasoning, Sherlock Holmes links Captain Morstan's mysterious disappearance on 3rd December 1878 with the six rare pearls Mary Morstan received. In 1882—four years after her father vanished—an anonymous newspaper inquiry asked for Mary's address, after which she received a precious pearl every year on the exact same date. Holmes deduces that the anonymous sender was aware of a past injustice done to Captain Morstan and was trying to make partial restitution to Mary from a newly acquired treasure.

(II) Describe a character sketch of Mary Morstan as a model client in 'The Sign of Four.' (2)

Answer: In 'The Sign of Four', Mary Morstan is portrayed as an exemplary and model client. Despite enduring the trauma of her father's disappearance and receiving a mysterious summons, she remains remarkably poised, calm, and articulate. She arrives at Baker Street thoroughly prepared, presenting Sherlock Holmes with a clear chronological account along with crucial documentary evidence—newspaper clippings, the anonymous letter, the six pearls, and Captain Morstan's strange map—which earns Holmes's admiration and enables him to investigate the case immediately.

Official Maharashtra HSC June 2026 English Board Question Paper (J-201)

HSC 12th Chemistry June 2026 Board Paper with Complete Solution (J-261) | Maharashtra Board

aaminomtex@gmail.com (Amin) — Sun, 25 Oct 2026 14:21:52 +0530

Here is the complete HSC 12th Chemistry June 2026 Board Question Paper (Subject Code: J-261) along with step-by-step model solutions, numerical calculations, and chemical reactions prepared strictly as per the Maharashtra State Board marking scheme.

SECTION - A

Q. 1. Select and write the correct answers for the following multiple choice type of questions: [10 Marks]

Which of the following is NOT an elastomer?
Answer: (d) polyethene
Anisole reacts with $\mathrm{HI}$ at $398\text{ K}$ to form
Answer: (b) benzenol
Unit of Henry's constant is
Answer: (a) $\mathrm{mol\ L^{-1}\ bar^{-1}}$
The shape of bromine penta-fluoride is
Answer: (d) Square pyramidal
The relation between edge length and radius for body centered cubic structure is -
Answer: (b) $r = \frac{\sqrt{3}a}{4}$
Atomic number of post actinoid element range from -
Answer: (d) $104\text{–}118$
The $\mathrm{pH}$ of buffer solution containing $0.05\text{ mol }\mathrm{NaF}$ per litre and $0.01\text{ mol }\mathrm{HF}$ per litre ($\mathrm{p}K_a = 3.1427$) is.
Answer: (a) $3.8417$
[Hint: $\mathrm{pH} = \mathrm{p}K_a + \log_{10}\frac{[\text{Salt}]}{[\text{Acid}]} = 3.1427 + \log_{10}\left(\frac{0.05}{0.01}\right) = 3.1427 + 0.6990 = 3.8417$]
Preparation of ethanamine from acetonitrile represents -
Answer: (b) Mendius reduction
The source of dichlorocarbene is
Answer: (c) chloroform
The reagent used to convert acetyl chloride to acetaldehyde is -
Answer: (d) $\mathrm{H_2 / Pd\text{-}BaSO_4}$

Q. 2. Answer the following questions: [8 Marks]

Write the relation between conductivity and molar conductivity.
Answer: The relation between conductivity ($k$) and molar conductivity ($\Lambda$) is: \[\Lambda = \frac{1000\,k}{c}\] where $\Lambda$ is molar conductivity (in $\mathrm{\Omega^{-1}\ cm^2\ mol^{-1}}$), $k$ is conductivity (in $\mathrm{\Omega^{-1}\ cm^{-1}}$), and $c$ is molar concentration (in $\mathrm{mol\ L^{-1}}$ or $\mathrm{mol\ dm^{-3}}$).
(In SI units, $\Lambda = \dfrac{k}{c}$, where $k$ is in $\mathrm{S\ m^{-1}}$ and $c$ is in $\mathrm{mol\ m^{-3}}$).
Write the oxidation state of Nickel in $[\mathrm{Ni(Cl)_4}]^{2\ominus}$ complex ion.
Answer: Let the oxidation state of $\mathrm{Ni}$ be $x$.
$x + 4(-1) = -2 \implies x = +2$.
Therefore, the oxidation state of Nickel is $+2$.
Write the name of nanostructured material used in tyres of car to increase their life.
Answer: Carbon black
Write the glycosidic linkage present in maltose.
Answer: $\alpha\text{-}1,4\text{-glycosidic linkage}$
Draw the structure of perchloric acid.
Answer: Perchloric acid ($\mathrm{HClO_4}$) has a tetrahedral structure with a central chlorine atom bonded to three oxygen atoms via double bonds and one hydroxyl ($\mathrm{-OH}$) group via a single bond: \[\begin{array}{c} \mathrm{O} \\ \parallel \\ \mathrm{O = Cl - OH} \\ \parallel \\ \mathrm{O} \end{array}\]
Write the trade name of most common thermoplastic polyester.
Answer: Terylene or Dacron (Chemical name: Polyethylene terephthalate, PET).
Calculate change over temperature for a certain reaction, whose $\Delta H^\circ$ is $-60\text{ kJ}$ and $\Delta S^\circ$ is $-150\text{ J K}^{-1}$.
Answer:
Given: $\Delta H^\circ = -60\text{ kJ} = -60,000\text{ J}$, $\Delta S^\circ = -150\text{ J K}^{-1}$. \[T = \frac{\Delta H^\circ}{\Delta S^\circ} = \frac{-60,000\text{ J}}{-150\text{ J K}^{-1}} = 400\text{ K}\]
Write the unit of rate constant of zero order reaction.
Answer: $\mathrm{mol\ L^{-1}\ s^{-1}}$ or $\mathrm{mol\ dm^{-3}\ s^{-1}}$ (or $\mathrm{mol\ dm^{-3}\ time^{-1}}$).

SECTION - B

Attempt any EIGHT of the following questions: [16 Marks]

Q. 3. Write reactions involved during preparation of Nylon 6, 6 fibre.

Answer:
Nylon 6,6 is prepared by the condensation polymerization of equimolar aqueous solutions of adipic acid and hexamethylenediamine:

Step 1: Formation of Nylon salt

\[n\,\mathrm{HOOC-(CH_2)_4-COOH} \text{ (Adipic acid)} + n\,\mathrm{H_2N-(CH_2)_6-NH_2} \text{ (Hexamethylenediamine)} \longrightarrow n\left[\mathrm{^\ominus OOC-(CH_2)_4-COO^\ominus\ H_3\overset{\oplus}{N}-(CH_2)_6-\overset{\oplus}{N}H_3}\right] \text{ (Nylon salt)}\]

Step 2: Condensation polymerization to Nylon 6,6

\[n\left[\mathrm{^\ominus OOC-(CH_2)_4-COO^\ominus\ H_3\overset{\oplus}{N}-(CH_2)_6-\overset{\oplus}{N}H_3}\right] \xrightarrow[\text{High pressure}]{553\text{ K}} \left[\mathrm{-OC-(CH_2)_4-CO-NH-(CH_2)_6-NH-}\right]_n \text{ (Nylon 6,6)} + 2n\,\mathrm{H_2O}\]

Q. 4. Derive Ostwald's dilution law equation for a weak acid.

Answer:
Consider $1\text{ mol}$ of a weak monobasic acid $\mathrm{HA}$ dissolved in $V\text{ dm}^3$ of solution (initial concentration $c = \frac{1}{V}\text{ mol dm}^{-3}$). The acid dissociates partially in aqueous solution as:

\[\mathrm{HA}_{(aq)} \rightleftharpoons \mathrm{H}^+_{(aq)} + \mathrm{A}^-_{(aq)}\]

Let $\alpha$ be the degree of dissociation of the weak acid at equilibrium:

Equilibrium concentration of undissociated acid, $[\mathrm{HA}] = c(1 - \alpha)\text{ mol dm}^{-3}$
Equilibrium concentration of $\mathrm{H}^+$ ions, $[\mathrm{H}^+] = c\alpha\text{ mol dm}^{-3}$
Equilibrium concentration of $\mathrm{A}^-$ ions, $[\mathrm{A}^-] = c\alpha\text{ mol dm}^{-3}$

Applying the law of mass action, the acid dissociation constant ($K_a$) is given by:

\[K_a = \frac{[\mathrm{H}^+][\mathrm{A}^-]}{[\mathrm{HA}]} = \frac{(c\alpha)(c\alpha)}{c(1 - \alpha)} = \frac{c\alpha^2}{1 - \alpha}\]

For a weak acid, $\alpha$ is very small compared to $1$ ($\alpha \ll 1$), so $1 - \alpha \approx 1$. Therefore:

\[K_a = c\alpha^2 \quad \text{or} \quad K_a = \frac{\alpha^2}{V}\] \[\alpha = \sqrt{\frac{K_a}{c}} = \sqrt{K_a \cdot V}\]

This equation represents Ostwald's dilution law for a weak acid.

Q. 5. Draw the structures of two forms of alanine having $\mathrm{pH} \le 2$ and $\mathrm{pH} \ge 10$

Answer:
Alanine ($\mathrm{CH_3-CH(NH_2)-COOH}$) exists in different ionic forms depending on the $\mathrm{pH}$ of the aqueous medium:

1. At $\mathrm{pH} \le 2$ (Cationic form in strongly acidic medium):

\[\begin{array}{c} \mathrm{CH_3} \\ | \\ \mathrm{H_3\overset{\oplus}{N} - CH - COOH} \end{array}\]

2. At $\mathrm{pH} \ge 10$ (Anionic form in strongly basic medium):

\[\begin{array}{c} \mathrm{CH_3} \\ | \\ \mathrm{H_2N - CH - COO^\ominus} \end{array}\]

Q. 6. Define the following terms: (a) Intensive property (b) Standard enthalpy of combustion

Answer:

(a) Intensive property: Any property of a system whose magnitude is independent of the amount of matter present in the system is called an intensive property. Examples: Temperature, pressure, density, refractive index, viscosity.
(b) Standard enthalpy of combustion ($\Delta_c H^\circ$): It is defined as the enthalpy change accompanying a reaction in which one mole of a substance in its standard state is completely oxidized (burnt) in excess of oxygen at standard conditions ($T = 298\text{ K}$ and $P = 1\text{ bar}$).

Q. 7. What is the action of following reagents on ethyl amine? (Indicate by reaction) (a) Nitrous acid (b) Chloroform and $\mathrm{KOH}$

Answer:

(a) Action of Nitrous acid ($\mathrm{HNO_2}$):
Ethyl amine reacts with nitrous acid (prepared in situ from $\mathrm{NaNO_2}$ and $\mathrm{HCl}$ at $273\text{–}278\text{ K}$) to form an unstable aliphatic diazonium salt which decomposes immediately to yield ethyl alcohol (ethanol) and liberate nitrogen gas:

\[\mathrm{C_2H_5-NH_2} + \mathrm{HNO_2} \xrightarrow[273\text{–}278\text{ K}]{\mathrm{NaNO_2 + HCl}} \left[\mathrm{C_2H_5-\overset{\oplus}{N}_2Cl^\ominus}\right] \xrightarrow{\mathrm{H_2O}} \mathrm{C_2H_5-OH} \text{ (Ethanol)} + \mathrm{N_2}\uparrow + \mathrm{HCl}\]

(Overall reaction: $\mathrm{C_2H_5NH_2} + \mathrm{HNO_2} \xrightarrow{\mathrm{NaNO_2 + HCl}} \mathrm{C_2H_5OH} + \mathrm{N_2}\uparrow + \mathrm{H_2O}$)

(b) Action of Chloroform and $\mathrm{KOH}$ (Carbylamine reaction):
When ethyl amine is heated with chloroform and alcoholic potassium hydroxide, it forms ethyl isocyanide (ethyl carbylamine) having a foul, offensive smell:

\[\mathrm{C_2H_5-NH_2} + \mathrm{CHCl_3} + 3\mathrm{KOH}_{\text{(alc.)}} \xrightarrow{\Delta} \mathrm{C_2H_5-NC} \text{ (Ethyl isocyanide)} + 3\mathrm{KCl} + 3\mathrm{H_2O}\]

Q. 8. (a) State Kohlrausch law of independent migration of ions. (b) Write Nernst equation to calculate electrode potential.

Answer:

(a) Kohlrausch law of independent migration of ions:
It states that at infinite dilution, each ion migrates independently of its co-ion and makes its own contribution to the total molar conductivity of an electrolyte, irrespective of the nature of the other ion with which it is associated.

\[\Lambda_0 = n_+ \lambda^0_+ + n_- \lambda^0_-\]

(b) Nernst equation for electrode potential:
For a general reduction electrode reaction $\mathrm{M}^{n+}_{(aq)} + n\mathrm{e}^- \longrightarrow \mathrm{M}_{(s)}$, the Nernst equation to calculate the electrode potential is:

\[E_{\text{elec}} = E^\circ_{\text{elec}} - \frac{2.303\,RT}{nF}\log_{10}\left(\frac{1}{[\mathrm{M}^{n+}]}\right) \quad \text{or} \quad E_{\text{elec}} = E^\circ_{\text{elec}} + \frac{2.303\,RT}{nF}\log_{10}[\mathrm{M}^{n+}]\]

(At $298\text{ K}$, $E_{\text{elec}} = E^\circ_{\text{elec}} + \dfrac{0.0592}{n}\log_{10}[\mathrm{M}^{n+}]$), where $E^\circ_{\text{elec}}$ is the standard electrode potential, $R$ is the gas constant, $T$ is absolute temperature, $n$ is the moles of electrons transferred, and $F$ is Faraday's constant.

Q. 9. (a) Write general electronic configuration of $3d$ series elements. (b) Write any two points which explain about the colour of a transition metal ion.

Answer:

(a) General electronic configuration of $3d$ series elements:

\[[\mathrm{Ar}]\,3d^{1\text{–}10}\,4s^{1\text{–}2}\]

(b) Explanation for the colour of transition metal ions:

Transition metal ions generally possess one or more unpaired electrons in their incompletely filled $(n-1)d$ orbitals ($d^1$ to $d^9$ configurations).
In the presence of ligands or solvent molecules, the degenerate $d$-orbitals split into two energy levels. When visible light falls on the ion, an electron is promoted from a lower energy $d$-orbital to a higher energy $d$-orbital ($d\text{-}d$ transition) by absorbing a specific wavelength of visible light, and the transmitted light exhibits the complementary colour.

Q. 10. Explain aldol condensation reaction with respect to ethanal.

Answer:
When two molecules of ethanal (acetaldehyde), which contains $\alpha$-hydrogen atoms, are warmed with a dilute base such as dilute $\mathrm{NaOH}$ (or $\mathrm{KOH}$), they undergo nucleophilic addition to form a $\beta$-hydroxy aldehyde called aldol ($3\text{-hydroxybutanal}$).

Step 1: Formation of Aldol (Addition step)

\[2\,\mathrm{CH_3-CHO} \text{ (Ethanal)} \underset{\text{warm}}{\stackrel{\text{dil. }\mathrm{NaOH}}{\rightleftharpoons}} \mathrm{CH_3-CH(OH)-CH_2-CHO} \text{ (3-Hydroxybutanal / Aldol)}\]

Step 2: Dehydration of Aldol (Condensation step)
On heating, $3\text{-hydroxybutanal}$ readily eliminates a water molecule to form an $\alpha,\beta$-unsaturated aldehyde called crotonaldehyde ($\text{but-2-enal}$).

\[\mathrm{CH_3-CH(OH)-CH_2-CHO} \xrightarrow{\Delta} \mathrm{CH_3-CH=CH-CHO} \text{ (But-2-enal / Crotonaldehyde)} + \mathrm{H_2O}\]

Q. 11. Write a note on Kolbe reaction.

Answer:
When phenol is treated with aqueous sodium hydroxide, it forms sodium phenoxide, in which the phenoxide ion is more reactive than phenol towards electrophilic aromatic substitution. When sodium phenoxide is heated with carbon dioxide ($\mathrm{CO_2}$, a weak electrophile) at $398\text{ K}$ under a pressure of $6\text{ atm}$ (or $4\text{–}7\text{ atm}$), it undergoes electrophilic substitution predominantly at the ortho-position to form sodium salicylate. Subsequent acidification with dilute hydrochloric acid yields salicylic acid ($2\text{-hydroxybenzoic acid}$).

\[\mathrm{C_6H_5OH} \text{ (Phenol)} + \mathrm{NaOH} \longrightarrow \mathrm{C_6H_5ONa} \text{ (Sodium phenoxide)} + \mathrm{H_2O}\] \[\mathrm{C_6H_5ONa} + \mathrm{CO_2} \xrightarrow{398\text{ K},\ 6\text{ atm}} o\text{-}\mathrm{HO-C_6H_4-COONa} \text{ (Sodium salicylate)} \xrightarrow{\mathrm{H_3O^+}} o\text{-}\mathrm{HO-C_6H_4-COOH} \text{ (Salicylic acid)}\]

Q. 12. (a) Write the relation between standard cell potential and equilibrium constant. (b) Write the representation of sodium hexacyanoferrate (III) complex.

Answer:

(a) Relation between standard cell potential ($E^\circ_{\text{cell}}$) and equilibrium constant ($K$):

\[E^\circ_{\text{cell}} = \frac{2.303\,RT}{nF}\log_{10} K\]

At $298\text{ K}$:

\[E^\circ_{\text{cell}} = \frac{0.0592}{n}\log_{10} K\]

(b) Formula of sodium hexacyanoferrate (III):

\[\mathrm{Na_3[Fe(CN)_6]}\]

Q. 13. In first order reaction $60\%$ of reactant decomposes in $40\text{ min}$. Calculate the rate constant of the reaction.

Answer:

Given:
- Initial concentration of reactant, $[\mathrm{A}]_0 = 100$
- Amount decomposed in $t = 40\text{ min}$ is $60\%$
- Concentration of reactant remaining at time $t$, $[\mathrm{A}]_t = 100 - 60 = 40$
- Time, $t = 40\text{ min}$
Formula: \[k = \frac{2.303}{t}\log_{10}\left(\frac{[\mathrm{A}]_0}{[\mathrm{A}]_t}\right)\]
Calculation: \[k = \frac{2.303}{40\text{ min}}\log_{10}\left(\frac{100}{40}\right) = \frac{2.303}{40}\log_{10}(2.5)\] \[k = \frac{2.303 \times 0.3979}{40} = 0.02291\text{ min}^{-1} \text{ (or } 2.291 \times 10^{-2}\text{ min}^{-1}\text{)}\]

Final Answer: The rate constant of the reaction is $0.02291\text{ min}^{-1}$ (or $3.818 \times 10^{-4}\text{ s}^{-1}$).

Q. 14. Explain optical activity of 2-chlorobutane.

Answer:

In $2\text{-chlorobutane}$ ($\mathrm{CH_3-\overset{*}{C}H(Cl)-CH_2-CH_3}$), the second carbon atom ($\mathrm{C_2}$) is bonded to four different substituent groups: $\mathrm{-H}$, $\mathrm{-Cl}$, $\mathrm{-CH_3}$ (methyl), and $\mathrm{-C_2H_5}$ (ethyl). Hence, $\mathrm{C_2}$ is an asymmetric (chiral) carbon atom.
Due to the presence of this chiral carbon, the $2\text{-chlorobutane}$ molecule is unsymmetrical and lacks a plane of symmetry.
As a result, $2\text{-chlorobutane}$ exists as a pair of non-superimposable mirror-image stereoisomers called enantiomers: \[\begin{array}{ccc} \mathrm{C_2H_5} & \vdots & \mathrm{C_2H_5} \\ | & \vdots & | \\ \mathrm{H - \overset{*}{C} - Cl} & \vdots & \mathrm{Cl - \overset{*}{C} - H} \\ | & \vdots & | \\ \mathrm{CH_3} & \vdots & \mathrm{CH_3} \\ \text{(I)} & \text{Mirror} & \text{(II)} \end{array}\]
These two enantiomers are optically active: one isomer rotates the plane of plane-polarized light clockwise (dextrorotatory, $d\text{-}$ or $(+)\text{-isomer}$), while the other rotates it anticlockwise by an equal magnitude (laevorotatory, $l\text{-}$ or $(-)\text{-isomer}$).

SECTION - C

Attempt any EIGHT of the following questions: [24 Marks]

Q. 15. (a) Derive a relationship between molar mass, density of the substance and unit cell edge length. (b) Define - paramagnetic solids.

Answer:

(a) Derivation of relationship between molar mass ($M$), density ($\rho$), and edge length ($a$):

If the edge length of a cubic unit cell is $a$, then the volume of the unit cell is $V = a^3$.
Suppose the mass of one particle is $m$ and there are $n$ (or $z$) particles per unit cell.
\[\text{Mass of unit cell} = m \times n\]
The density of the unit cell ($\rho$), which is the same as the density of the crystalline solid, is given by: \[\rho = \frac{\text{Mass of unit cell}}{\text{Volume of unit cell}} = \frac{m \times n}{a^3} \quad \text{--- (Equation 1)}\]
The molar mass ($M$) of the substance is given by: \[M = \text{mass of one particle } (m) \times \text{Avogadro's number } (N_A) \implies m = \frac{M}{N_A}\]
Substituting $m = \dfrac{M}{N_A}$ into Equation 1 gives: \[\rho = \frac{n \times M}{a^3 \times N_A}\]

(b) Paramagnetic solids:
The substances which contain one or more unpaired electrons in their atoms, ions, or molecules and are weakly attracted by an external magnetic field are called paramagnetic solids. Examples: $\mathrm{O_2}$, $\mathrm{Cu^{2+}}$, $\mathrm{Fe^{3+}}$, $\mathrm{Cr^{3+}}$.

Q. 16. (a) Mention the names of four principles of green chemistry. (b) Write advantages of nanoparticles and nanotechnology in the field of energy sector.

Answer:

(a) Names of four principles of Green Chemistry (Any four):

Prevention of waste or by-products
Atom economy
Less hazardous chemical syntheses
Designing safer chemicals
Use of safer solvents and auxiliaries
Use of catalysis

(b) Advantages of nanoparticles and nanotechnology in the energy sector:

Nanotechnology makes solar power more economical and significantly increases the efficiency of solar cells.
Energy storage devices (such as rechargeable batteries and supercapacitors) built with nanomaterials become lighter, faster-charging, and have higher power density.
Nanocatalysts improve the efficiency of fuel cells and biofuel production.

Q. 17. (a) State second law of thermodynamics. (b) Define - Nucleotides. (c) Write chemical reaction for conversion of benzene diazonium chloride into benzene.

Answer:

(a) Second law of thermodynamics:
It states that the total entropy of a system and its surroundings (the universe) increases in any spontaneous process:

\[\Delta S_{\text{total}} = \Delta S_{\text{sys}} + \Delta S_{\text{surr}} > 0\]

(b) Nucleotides:
Nucleotides are the repeating monomer units of nucleic acids ($\mathrm{DNA}$ and $\mathrm{RNA}$). Chemically, a nucleotide is a phosphate ester of a nucleoside and consists of three components: a pentose sugar, a nitrogenous base, and a phosphate group.

(c) Conversion of benzene diazonium chloride into benzene:
Benzene diazonium chloride on treatment with hypophosphorous acid ($\mathrm{H_3PO_2}$, phosphinic acid) in the presence of water (and $\mathrm{Cu^+}$ catalyst) is reduced to benzene:

\[\mathrm{C_6H_5-\overset{\oplus}{N}_2Cl^\ominus} + \mathrm{H_3PO_2} + \mathrm{H_2O} \xrightarrow{\mathrm{Cu^+}} \mathrm{C_6H_6} \text{ (Benzene)} + \mathrm{N_2}\uparrow + \mathrm{H_3PO_3} + \mathrm{HCl}\]

(Alternatively, using ethanol: $\mathrm{C_6H_5N_2^+Cl^-} + \mathrm{CH_3CH_2OH} \longrightarrow \mathrm{C_6H_6} + \mathrm{N_2}\uparrow + \mathrm{CH_3CHO} + \mathrm{HCl}$)

Q. 18. The solubility product of $\mathrm{AgBr}$ is $5.2 \times 10^{-13}$. Calculate its solubility in $\mathrm{mol\ dm^{-3}}$ and $\mathrm{g\ dm^{-3}}$. (Molar mass of $\mathrm{AgBr} = 187.8\text{ g mol}^{-1}$)

Answer:

Given: $K_{sp} = 5.2 \times 10^{-13}$, Molar mass of $\mathrm{AgBr} = 187.8\text{ g mol}^{-1}$.
Solubility equilibrium: \[\mathrm{AgBr}_{(s)} \rightleftharpoons \mathrm{Ag}^+_{(aq)} + \mathrm{Br}^\ominus_{(aq)}\] Here, $x = 1$ and $y = 1$. Therefore: \[K_{sp} = [\mathrm{Ag}^+][\mathrm{Br}^\ominus] = S \times S = S^2\]
(1) Molar solubility ($S$) in $\mathrm{mol\ dm^{-3}}$: \[S = \sqrt{K_{sp}} = \sqrt{5.2 \times 10^{-13}} = \sqrt{52 \times 10^{-14}} = 7.211 \times 10^{-7}\text{ mol dm}^{-3}\]
(2) Solubility in $\mathrm{g\ dm^{-3}}$: \[\text{Solubility (in }\mathrm{g\ dm^{-3}}\text{)} = S\text{ (in }\mathrm{mol\ dm^{-3}}\text{)} \times \text{Molar mass}\] \[\text{Solubility} = 7.211 \times 10^{-7}\text{ mol dm}^{-3} \times 187.8\text{ g mol}^{-1} = 1.354 \times 10^{-4}\text{ g dm}^{-3}\]

Final Answer: The solubility of $\mathrm{AgBr}$ is $7.211 \times 10^{-7}\text{ mol dm}^{-3}$ and $1.354 \times 10^{-4}\text{ g dm}^{-3}$.

Q. 19. Convert the following: (a) Ethanol to ethoxyethane. (b) Phenol to p-benzoquinone (c) Salicylic acid to aspirin

Answer:

(a) Ethanol to ethoxyethane (Continuous etherification):

\[2\,\mathrm{C_2H_5-OH} \text{ (Ethanol)} \xrightarrow[413\text{ K}]{\text{Conc. }\mathrm{H_2SO_4}} \mathrm{C_2H_5-O-C_2H_5} \text{ (Ethoxyethane)} + \mathrm{H_2O}\]

(b) Phenol to p-benzoquinone (Oxidation of phenol):

\[\mathrm{C_6H_5OH} \text{ (Phenol)} + 2[\mathrm{O}] \xrightarrow{\mathrm{Na_2Cr_2O_7 + \text{Conc. }H_2SO_4}} \mathrm{O=C_6H_4=O} \text{ (p-Benzoquinone)} + \mathrm{H_2O}\]

(c) Salicylic acid to aspirin (Acetylation):

\[o\text{-}\mathrm{HO-C_6H_4-COOH} \text{ (Salicylic acid)} + (\mathrm{CH_3CO})_2\mathrm{O} \text{ (Acetic anhydride)} \xrightarrow[\Delta]{\text{Conc. }\mathrm{H_2SO_4}} o\text{-}\mathrm{CH_3COO-C_6H_4-COOH} \text{ (Aspirin / Acetylsalicylic acid)} + \mathrm{CH_3COOH}\]

Q. 20. (a) State Raoult's law. (b) Derive the relationship to determine molar mass of non volatile solute by depression of freezing point.

Answer:

(a) Raoult's law:

General statement: It states that the partial vapour pressure of any volatile component of a solution is equal to the vapour pressure of the pure component multiplied by its mole fraction in the solution ($P_1 = P_1^\circ x_1$).
For a solution containing a non-volatile solute: At a given temperature, the relative lowering of vapour pressure of a solution containing a non-volatile solute is equal to the mole fraction of the solute in the solution.

(b) Derivation of molar mass of non-volatile solute from depression in freezing point:

For a dilute solution, the depression in freezing point ($\Delta T_f$) is directly proportional to the molality ($m$) of the solution: \[\Delta T_f \propto m \implies \Delta T_f = K_f\,m \quad \text{--- (Equation 1)}\] where $K_f$ is the molal depression constant (cryoscopic constant).
Suppose $W_2\text{ g}$ of a non-volatile solute of molar mass $M_2\text{ g mol}^{-1}$ is dissolved in $W_1\text{ g}$ (i.e., $\dfrac{W_1}{1000}\text{ kg}$) of solvent.
\[\text{Number of moles of solute} = \frac{W_2}{M_2}\] \[\text{Molality }(m) = \frac{\text{Moles of solute}}{\text{Mass of solvent in kg}} = \frac{W_2 / M_2}{W_1 / 1000} = \frac{1000\,W_2}{M_2\,W_1} \quad \text{--- (Equation 2)}\]
Substituting Equation 2 into Equation 1 gives: \[\Delta T_f = \frac{1000\,K_f\,W_2}{M_2\,W_1}\] \[M_2 = \frac{1000\,K_f\,W_2}{\Delta T_f\,W_1}\]

Q. 21. (a) Explain the formation of $[\mathrm{CoF_6}]^{3\ominus}$ complex ion with respect to: (i) Oxidation State of Cobalt (ii) Type of hybridisation (iii) Magnetic property (iv) Geometrical shape (b) Write the name of ligand used to estimate hardness of water.

Answer:

(a) Formation of $[\mathrm{CoF_6}]^{3\ominus}$ complex ion:

(i) Oxidation State of Cobalt:
Let oxidation state of $\mathrm{Co}$ be $x$. Then $x + 6(-1) = -3 \implies x = +3$. Thus, Cobalt is in the $+3$ oxidation state ($\mathrm{Co^{3+}}$).
(ii) Type of hybridisation:
Electronic configuration of $\mathrm{Co}$ ($Z = 27$): $[\mathrm{Ar}]\,3d^7\,4s^2$.
Electronic configuration of $\mathrm{Co^{3+}}$ ion: $[\mathrm{Ar}]\,3d^6\,4s^0\,4p^0\,4d^0$: \[3d^6: [\uparrow\downarrow][\uparrow][\uparrow][\uparrow][\uparrow] \quad 4s^0: [\ \ ] \quad 4p^0: [\ \ ][\ \ ][\ \ ] \quad 4d^0: [\ \ ][\ \ ][\ \ ][\ \ ][\ \ ]\] Since $\mathrm{F^\ominus}$ is a weak field ligand, it does not cause pairing of $3d$ electrons. One vacant $4s$, three vacant $4p$, and two vacant $4d$ orbitals undergo $sp^3d^2$ hybridisation (outer orbital / high-spin complex) to form six equivalent $sp^3d^2$ hybrid orbitals, which accept six lone pairs from six $\mathrm{F^\ominus}$ ligands.
(iii) Magnetic property:
Due to the presence of $4$ unpaired electrons in the $3d$ orbitals, the complex is paramagnetic ($\mu = \sqrt{4(4+2)} = \sqrt{24} = 4.90\text{ BM}$).
(iv) Geometrical shape:
Octahedral geometry.

(b) Name of ligand used to estimate hardness of water:
$\mathrm{EDTA}$ (Ethylenediaminetetraacetate ion).

Q. 22. (a) Write cell reactions of hydrogen-oxygen fuel cell at anode and cathode. (b) Write an application of electrochemical series.

Answer:

(a) Cell reactions of $\mathrm{H_2\text{-}O_2}$ fuel cell:

Oxidation at Anode ($-$): \[2\mathrm{H}_{2(g)} + 4\mathrm{OH}^\ominus_{(aq)} \longrightarrow 4\mathrm{H_2O}_{(l)} + 4\mathrm{e}^\ominus\]
Reduction at Cathode ($+$): \[\mathrm{O}_{2(g)} + 2\mathrm{H_2O}_{(l)} + 4\mathrm{e}^\ominus \longrightarrow 4\mathrm{OH}^\ominus_{(aq)}\]
Net Cell Reaction: \[2\mathrm{H}_{2(g)} + \mathrm{O}_{2(g)} \longrightarrow 2\mathrm{H_2O}_{(l)}\]

(b) Application of electrochemical series (Any one):

Predicting spontaneity of redox reactions: A redox reaction is spontaneous if the standard cell potential $E^\circ_{\text{cell}} = E^\circ_{\text{cathode}} - E^\circ_{\text{anode}}$ calculated from the series is positive.
Comparing relative strength of oxidizing and reducing agents: Species with higher positive standard reduction potentials are stronger oxidizing agents, while those with more negative standard reduction potentials are stronger reducing agents.

Q. 23. (a) Explain anomalous behaviour of oxygen with respect to: (i) Atomicity (ii) Magnetic property (iii) Oxidation state (iv) Nature of hydride (b) Write any two uses of helium.

Answer:

(a) Anomalous behaviour of oxygen:

(i) Atomicity: Oxygen is a diatomic gas ($\mathrm{O_2}$) due to its small size and ability to form strong $p\pi\text{-}p\pi$ double bonds, whereas other elements of Group 16 are polyatomic solids (e.g., $\mathrm{S_8}$, $\mathrm{Se_8}$).
(ii) Magnetic property: Molecular oxygen ($\mathrm{O_2}$) is paramagnetic due to two unpaired electrons in its $\pi^*$ antibonding molecular orbitals, whereas other elements of Group 16 are diamagnetic.
(iii) Oxidation state: Oxygen generally exhibits a $-2$ oxidation state (except $-1$ in peroxides and $+2$ in $\mathrm{OF_2}$) and cannot expand its covalency beyond $4$ due to the absence of vacant $d$-orbitals in its valence shell. Other Group 16 elements exhibit $-2, +2, +4,$ and $+6$ oxidation states.
(iv) Nature of hydride: The hydride of oxygen ($\mathrm{H_2O}$) is a liquid at room temperature due to extensive intermolecular hydrogen bonding, whereas the hydrides of other Group 16 elements ($\mathrm{H_2S, H_2Se, H_2Te}$) are pungent, foul-smelling gases.

(b) Uses of Helium (Any two):

A mixture of helium ($85\%$) and oxygen ($15\%$) is used by deep-sea divers for respiration in place of nitrogen because helium has very low solubility in blood.
Being non-inflammable and light, it is used for filling meteorological observation balloons.
Liquid helium is used as a cryogenic fluid to produce low temperatures required for superconducting magnets in $\mathrm{MRI}$ systems.

Q. 24. What is the energy of activation of a reaction whose rate constant doubles when the temperature changes from $303\text{ K}$ to $313\text{ K}$?

Answer:

Given: $T_1 = 303\text{ K}$, $T_2 = 313\text{ K}$, $k_2 = 2k_1 \implies \dfrac{k_2}{k_1} = 2$, $R = 8.314\text{ J K}^{-1}\text{ mol}^{-1}$.
Formula: \[\log_{10}\left(\frac{k_2}{k_1}\right) = \frac{E_a}{2.303\,R}\left(\frac{T_2 - T_1}{T_1\,T_2}\right)\]
Calculation: \[\log_{10}(2) = \frac{E_a}{2.303 \times 8.314}\left(\frac{313 - 303}{303 \times 313}\right)\] \[0.3010 = \frac{E_a}{19.147}\left(\frac{10}{94839}\right)\] \[E_a = \frac{0.3010 \times 19.147 \times 94839}{10} = 54657.4\text{ J mol}^{-1} = 54.66\text{ kJ mol}^{-1}\]

Final Answer: The energy of activation of the reaction is $54.66\text{ kJ mol}^{-1}$ (or $54657.4\text{ J mol}^{-1}$).

Q. 25. (a) Write chemical reactions taking place during extraction of iron in zone of reduction of the blast furnace. (b) Write observed electronic configuration of gadolinium. ($Z=64$)

Answer:

(a) Chemical reactions in the zone of reduction of the blast furnace ($\approx 900\text{ K}$):
In the zone of reduction, ferric oxide ($\mathrm{Fe_2O_3}$) is reduced to spongy iron by carbon monoxide ($\mathrm{CO}$):

\[\mathrm{Fe_2O_3} + 3\mathrm{CO} \xrightarrow{900\text{ K}} 2\mathrm{Fe} \text{ (Spongy iron)} + 3\mathrm{CO_2}\uparrow\]

Stepwise reduction reactions:

\[3\mathrm{Fe_2O_3} + \mathrm{CO} \longrightarrow 2\mathrm{Fe_3O_4} + \mathrm{CO_2}\] \[\mathrm{Fe_3O_4} + \mathrm{CO} \longrightarrow 3\mathrm{FeO} + \mathrm{CO_2}\] \[\mathrm{FeO} + \mathrm{CO} \longrightarrow \mathrm{Fe} + \mathrm{CO_2}\]

(At higher temperature around $1500\text{ K}$, reduction also occurs directly by carbon: $\mathrm{Fe_2O_3} + 3\mathrm{C} \xrightarrow{1500\text{ K}} 2\mathrm{Fe} + 3\mathrm{CO}$)

(b) Observed electronic configuration of Gadolinium ($\mathrm{Gd}, Z = 64$):

\[[\mathrm{Xe}]\,4f^7\,5d^1\,6s^2\]

Q. 26. Convert the following: (a) Acetic acid to ethyl alcohol (b) Acetic acid to acetic anhydride (c) Acetyl chloride to acetamide

Answer:

(a) Acetic acid to ethyl alcohol (Reduction):

\[\mathrm{CH_3-COOH} \text{ (Acetic acid)} \xrightarrow[\text{(ii) }\mathrm{H_3O^+}]{\text{(i) }\mathrm{LiAlH_4 / \text{dry ether}}} \mathrm{CH_3-CH_2-OH} \text{ (Ethyl alcohol)} + \mathrm{H_2O}\]

(b) Acetic acid to acetic anhydride (Dehydration):

\[2\,\mathrm{CH_3-COOH} \text{ (Acetic acid)} \xrightarrow{\mathrm{P_2O_5},\ \Delta} (\mathrm{CH_3CO})_2\mathrm{O} \text{ (Acetic anhydride)} + \mathrm{H_2O}\]

(c) Acetyl chloride to acetamide (Ammonolysis):

\[\mathrm{CH_3-CO-Cl} \text{ (Acetyl chloride)} + 2\mathrm{NH_3} \longrightarrow \mathrm{CH_3-CO-NH_2} \text{ (Acetamide)} + \mathrm{NH_4Cl}\]

SECTION - D

Attempt any THREE of the following questions: [12 Marks]

Q. 27. (a) Define: (i) Co-ordination numbers (ii) Ambidentate ligand (b) Write a note on Clemmensen reduction of propanone.

Answer:

(a) Definitions:

(i) Co-ordination number: The coordination number of a central metal atom or ion in a coordination complex is the total number of ligand donor atoms directly attached to it (or the number of electron pairs donated by the ligands to form coordinate covalent bonds with the central metal ion).
(ii) Ambidentate ligand: A monodentate ligand which possesses two or more different donor atoms and can link to the central metal atom or ion through either of the donor atoms at a time is called an ambidentate ligand. Examples: $\mathrm{NO_2^\ominus}$ (binds via $\mathrm{N}$ or $\mathrm{O}$), $\mathrm{SCN^\ominus}$ (binds via $\mathrm{S}$ or $\mathrm{N}$).

(b) Clemmensen reduction of propanone:
When propanone (acetone) is heated with zinc-amalgam ($\mathrm{Zn\text{-}Hg}$) and concentrated hydrochloric acid ($\mathrm{Conc.\ HCl}$), its carbonyl group ($\mathrm{>C=O}$) is reduced to a methylene group ($\mathrm{-CH_2-}$) to form propane.

\[\mathrm{CH_3-\overset{\underset{\parallel}{O}}{C}-CH_3} \text{ (Propanone)} + 4[\mathrm{H}] \xrightarrow[\Delta]{\mathrm{Zn\text{-}Hg\ /\ \text{Conc. }HCl}} \mathrm{CH_3-CH_2-CH_3} \text{ (Propane)} + \mathrm{H_2O}\]

Q. 28. (a) Derive relationship between $\Delta H$ and $\Delta U$ for a gaseous chemical reaction. (b) Give reason Ozone is a good bleaching agent. (c) Write any two uses of $\mathrm{SO_2}$.

Answer:

(a) Derivation of relationship between $\Delta H$ and $\Delta U$:

Enthalpy ($H$) is defined as $H = U + PV$.
At constant pressure $P$, let $H_1, U_1, V_1$ represent the initial state (reactants) and $H_2, U_2, V_2$ represent the final state (products): \[H_1 = U_1 + PV_1 \quad \text{and} \quad H_2 = U_2 + PV_2\] \[\Delta H = H_2 - H_1 = (U_2 - U_1) + P(V_2 - V_1) = \Delta U + P\Delta V \quad \text{--- (Equation 1)}\]
Let $n_1$ be the number of moles of gaseous reactants occupying volume $V_1$, and $n_2$ be the number of moles of gaseous products occupying volume $V_2$ at constant temperature $T$ and pressure $P$. Assuming ideal gas behaviour ($PV = nRT$): \[PV_1 = n_1 RT \quad \text{and} \quad PV_2 = n_2 RT\] \[P\Delta V = P(V_2 - V_1) = n_2 RT - n_1 RT = (n_2 - n_1)RT = \Delta n_g RT \quad \text{--- (Equation 2)}\]
Substituting Equation 2 into Equation 1 gives: \[\Delta H = \Delta U + \Delta n_g RT\] where $\Delta n_g = n_2 - n_1$ is the difference between the number of moles of gaseous products and gaseous reactants.

(b) Reason Ozone is a good bleaching agent:
Ozone ($\mathrm{O_3}$) is thermodynamically unstable and readily decomposes to liberate nascent oxygen ($[\mathrm{O}]$):

\[\mathrm{O_3} \longrightarrow \mathrm{O_2} + [\mathrm{O}] \text{ (Nascent oxygen)}\]

This nascent oxygen is a powerful oxidizing agent that oxidizes coloured organic matter into colourless substances:

\[\text{Coloured matter} + [\mathrm{O}] \longrightarrow \text{Colourless matter}\]

Because the bleaching action of ozone is due to oxidation, it acts as a permanent bleaching agent (dry bleach).

(c) Uses of $\mathrm{SO_2}$ (Any two):

Used in the manufacture of sulphuric acid ($\mathrm{H_2SO_4}$) by the Contact process.
Used as a mild bleaching agent for delicate materials such as wool, silk, straw, and starch.
Used in refining petroleum and sugar, and as an anti-chlor, disinfectant, and preservative.

Q. 29. (a) Define - Polymorphism. (b) State and explain Hess's law of constant summation. Write its one application.

Answer:

(a) Polymorphism:
The phenomenon in which a single chemical substance (compound) crystallizes in two or more distinct crystalline structures under different conditions is called polymorphism, and the different crystalline forms are called polymorphic forms. Example: Calcium carbonate ($\mathrm{CaCO_3}$) exists as calcite and aragonite; silica ($\mathrm{SiO_2}$) exists as $\alpha\text{-quartz}$, $\beta\text{-quartz}$, and cristobalite.

(b) Hess's Law of Constant Heat Summation:

Statement: The law states that the overall enthalpy change accompanying a chemical reaction is the same regardless of whether the reaction takes place in one step or in a series of multiple steps.
Explanation: Hess's law is a direct consequence of the fact that enthalpy is a state function (it depends only on the initial and final states, not on the path).
Consider the conversion of reactant $\mathrm{A}$ into product $\mathrm{D}$ in a single step: \[\mathrm{A} \longrightarrow \mathrm{D}, \quad \Delta_r H^\circ\] Suppose the same reaction occurs in three consecutive steps:
- Step 1: $\mathrm{A} \longrightarrow \mathrm{B}, \quad \Delta_r H^\circ_1$
- Step 2: $\mathrm{B} \longrightarrow \mathrm{C}, \quad \Delta_r H^\circ_2$
- Step 3: $\mathrm{C} \longrightarrow \mathrm{D}, \quad \Delta_r H^\circ_3$
According to Hess's law, the total enthalpy change is the algebraic sum of the enthalpy changes of the individual steps: \[\Delta_r H^\circ = \Delta_r H^\circ_1 + \Delta_r H^\circ_2 + \Delta_r H^\circ_3\]
Application: Hess's law is used to calculate the enthalpy of formation, combustion, or ionization of reactions whose enthalpy changes cannot be measured directly by experiment (by treating thermochemical equations algebraically).

Q. 30. (a) $0.01\text{ m}$ aqueous formic acid solution freezes at $-0.021^\circ\text{C}$. Calculate its degree of dissociation. ($K_f = 1.86\text{ K kg mol}^{-1}$) (b) Write properties of interstitial compounds.

Answer:

(a) Calculation of degree of dissociation ($\alpha$):

Given: Molality $m = 0.01\text{ m}$, Freezing point of solution $T_f = -0.021^\circ\text{C}$, $K_f = 1.86\text{ K kg mol}^{-1}$.
Observed depression in freezing point: \[\Delta T_f = T_f^\circ - T_f = 0^\circ\text{C} - (-0.021^\circ\text{C}) = 0.021\text{ K}\]
Theoretical depression in freezing point ($\Delta T_{f(\text{theo})}$) and van't Hoff factor ($i$): \[\Delta T_f = i \times K_f \times m \implies 0.021 = i \times 1.86 \times 0.01 = i \times 0.0186\] \[i = \frac{0.021}{0.0186} = 1.129\]
Degree of dissociation ($\alpha$):
Formic acid dissociates into $n = 2$ ions per formula unit: $\mathrm{HCOOH}_{(aq)} \rightleftharpoons \mathrm{H}^+_{(aq)} + \mathrm{HCOO}^\ominus_{(aq)}$. \[\alpha = \frac{i - 1}{n - 1} = \frac{1.129 - 1}{2 - 1} = 0.129 \text{ (or } 12.9\%\text{)}\]

Final Answer: The degree of dissociation of formic acid is $0.129$ (or $12.9\%$).

(b) Properties of interstitial compounds (Any four):

They are hard and show good electrical and thermal conductivity (they retain metallic conductivity).
They have very high melting points, higher than those of the pure transition metals.
They are chemically inert.
Their densities are lower than those of the parent metals, and metallic hydrides can be used as powerful reducing agents.

Q. 31. (a) What is the action of following reagents on Bromoethane? (i) $\mathrm{KNO_2}$ (ii) '$\mathrm{Mg}$' metal (iii) Silver acetate (b) Draw structure - DDT insecticide.

Answer:

(a) Action of reagents on Bromoethane ($\mathrm{CH_3-CH_2-Br}$):

(i) Action of Potassium nitrite ($\mathrm{KNO_2}$):
Bromoethane on reaction with aqueous/alcoholic potassium nitrite forms ethyl nitrite as the major product: \[\mathrm{CH_3-CH_2-Br} + \mathrm{KNO_2} \longrightarrow \mathrm{CH_3-CH_2-O-N=O} \text{ (Ethyl nitrite)} + \mathrm{KBr}\]
(ii) Action of Magnesium ($\mathrm{Mg}$) metal:
Bromoethane reacts with magnesium metal in the presence of dry ether as solvent to form ethylmagnesium bromide (Grignard reagent): \[\mathrm{CH_3-CH_2-Br} + \mathrm{Mg} \xrightarrow{\text{Dry ether}} \mathrm{CH_3-CH_2-MgBr} \text{ (Ethylmagnesium bromide)}\]
(iii) Action of Silver acetate ($\mathrm{CH_3COOAg}$):
Bromoethane on heating with alcoholic silver acetate undergoes nucleophilic substitution to form ethyl acetate (ethyl ethanoate): \[\mathrm{CH_3-CH_2-Br} + \mathrm{CH_3COOAg} \xrightarrow{\text{Alcohol},\ \Delta} \mathrm{CH_3-COOCH_2CH_3} \text{ (Ethyl acetate)} + \mathrm{AgBr}\downarrow\]

(b) Structure of DDT (Dichlorodiphenyltrichloroethane / $1,1,1\text{-trichloro-}2,2\text{-bis(4-chlorophenyl)ethane}$):

\[\begin{array}{c} \mathrm{Cl - C_6H_4} \\ \backslash \\ \mathrm{CH - CCl_3} \\ / \\ \mathrm{Cl - C_6H_4} \end{array}\]

(Two para-chlorophenyl rings and one $\mathrm{-CCl_3}$ group attached to the $\mathrm{>CH-}$ carbon atom.)

HSC 12th Chemistry March 2026 Board Paper Solution with Question Paper | Maharashtra Board

aaminomtex@gmail.com (Amin) — Sun, 25 Oct 2026 13:35:05 +0530

Here is the complete HSC 12th Chemistry March 2026 Board Question Paper (Subject Code: J-155) along with step-by-step model solutions, numerical calculations, and chemical reactions prepared strictly as per the Maharashtra State Board marking scheme.

HSC Chemistry 12th Standard March 2026 Board Paper Solution

SECTION - A

Q. 1. Select and write the correct answer for the following multiple choice type of questions: [10 Marks]

(Examiner's Tip: Always write the sub-question number, the correct option letter, and the complete option text to secure full marks.)

The electrolyte used in $\ce{H2-O2}$ fuel cell is

Answer: (b) aqueous $\ce{KOH}$
The oxidation state of bromine in $\ce{HOBrO2}$ oxoacid is

Answer: (b) $+5$
Acetonitrile may be prepared by heating the following reactants:

Answer: (c) Methyl chloride with alcoholic $\ce{KCN}$
The carbonated water is an example of

Answer: (c) gas in liquid solution
The unit of rate constant is per second, the order of reaction is-

Answer: (b) First
A system releases $10\text{ kJ}$ of heat and performs $15\text{ kJ}$ of work on the surrounding. Hence the change in internal energy is:

Answer: (d) $-25\text{ kJ}$
The co-ordination complex ions $\ce{[Co(NH3)5(NO2)]^{2+}}$ and $\ce{[Co(NH3)5(ONO)]^{2+}}$ are __________ of each other.

Answer: (c) linkage isomers
The correct order for basic strength of amines and ammonia is

Answer: (d) $\ce{NH3} < \ce{R-NH2} < \ce{R2NH} > \ce{R3N}$
Amongst the following $3d$-series elements, having highest value of first ionization enthalpy is

Answer: (a) $\ce{Zn}$
The $\text{pOH}$ of $0.01\text{ M}$ $\ce{HCl}$ solution is

Answer: (d) $12$

Q. 2. Answer the following questions: [8 Marks]

Write the name of product obtained, when selenium is treated with magnesium metal.

Answer: Magnesium selenide ($\ce{MgSe}$)
\[\ce{Mg + Se ->[\Delta] MgSe}\]
Write the SI unit of cell constant.

Answer: $\text{m}^{-1}$
Write the name of radioactive element in lanthanide series.

Answer: Promethium ($\ce{Pm}$)
Complete the following reaction: $\ce{Phenol + 3 H2 ->[\text{Nickel, } 433\text{ K}] ?}$

Answer:
\[\ce{C6H5OH (Phenol) + 3 H2 ->[\text{Ni, } 433\text{ K}] C6H11OH (Cyclohexanol)}\]
Write the IUPAC name of: (Benzene ring with $\ce{-CHO}$ and $\ce{-OH}$ at adjacent ortho positions)

Answer: 2-Hydroxybenzaldehyde (or 2-Hydroxybenzenecarbaldehyde)
Copper has fcc structure with edge length $495\text{ pm}$. What is the radius of copper atom in $\text{pm}$?

Answer: For an fcc unit cell, atomic radius is given by:
\[r = \frac{a}{2\sqrt{2}} = \frac{495\text{ pm}}{2 \times 1.414} = \frac{495}{2.828} = 175.03\text{ pm} \approx 175\text{ pm}\]
Write the chemical name of Teflon.

Answer: Polytetrafluoroethylene (or Polytetrafluoroethene, $\text{PTFE}$)
Write the name of nanoparticle which acts as highly effective bacterial disinfectant.

Answer: Silver nanoparticles ($\text{AgNPs}$)

SECTION - B

Attempt any EIGHT of the following questions: [16 Marks]

Q. 3. Define:

Isomorphism:

The phenomenon in which two or more crystalline substances exhibit the same crystal structure is called isomorphism, and the substances are said to be isomorphous. Isomorphous substances have the same atomic ratio in their chemical compositions.
Example: $\ce{NaF}$ and $\ce{MgO}$ (atomic ratio $1:1$); $\ce{NaNO3}$ and $\ce{CaCO3}$ (atomic ratio $1:1:3$).
Unit Cell:

The smallest repeating structural unit of a crystalline solid which, when repeated in three dimensions, generates the entire crystal lattice is called a unit cell.

Q. 4. Explain the trends in the following properties of group 17 elements:

Atomic radii:

Group 17 elements (halogens) have the smallest atomic radii in their respective periods due to the maximum effective nuclear charge. Down the group from fluorine ($\ce{F}$) to iodine ($\ce{I}$), the atomic and ionic radii increase gradually because a new principal quantum shell is added at each successive element, which outweighs the effect of increased nuclear charge.
Electronegativity:

Group 17 elements have very high electronegativities, with fluorine being the most electronegative element in the periodic table. Down the group from fluorine ($\ce{F}$) to iodine ($\ce{I}$), electronegativity decreases gradually due to an increase in atomic size and shielding effect, which reduces the tendency of the nucleus to attract a shared pair of electrons.

Q. 5. Explain Homoleptic and Heteroleptic complexes with example.

Homoleptic complexes:

The coordination complexes in which the central metal atom or ion is bound to only one type of donor group (ligand) are called homoleptic complexes.
Example: $\ce{[Co(NH3)6]^{3+}}$ (where $\ce{Co^{3+}}$ is attached only to $\ce{NH3}$ ligands) or $\ce{[Fe(CN)6]^{4-}}$.
Heteroleptic complexes:

The coordination complexes in which the central metal atom or ion is bound to more than one type of donor groups (ligands) are called heteroleptic complexes.
Example: $\ce{[Co(NH3)4Cl2]^+}$ (where $\ce{Co^{3+}}$ is attached to two types of ligands: $\ce{NH3}$ and $\ce{Cl^-}$).

Q. 6.

Write the reaction of bromine water on glucose.

Answer: When glucose is warmed with bromine water (a mild oxidizing agent), the aldehyde group ($\ce{-CHO}$) is oxidized to a carboxylic acid group ($\ce{-COOH}$) to form gluconic acid. This reaction confirms that the carbonyl group in glucose is an aldehydic group.
\[\ce{CHO-(CHOH)4-CH2OH (D-Glucose) + [O] ->[\text{Bromine water}] COOH-(CHOH)4-CH2OH (D-Gluconic acid)}\]
Define Enantiomers.

Answer: Stereoisomers (optical isomers) which are non-superimposable mirror images of each other are called enantiomers (or enantiomorphs or optical antipodes). They have identical physical properties (such as melting point and boiling point) except for the direction of rotation of plane-polarized light.

Q. 7.

Write example of one dimensional nanostructure of size less than $100\text{ nm}$.

Answer: Nanowires, Carbon Nanotubes ($\text{CNTs}$), and Nanorods (any one example carries full credit).
Write the hydrolysis reaction of ethyl methyl ether.

Answer: When ethyl methyl ether (methoxyethane) is heated with dilute sulfuric acid under high pressure, it undergoes acid-catalyzed hydrolysis to form methanol and ethanol:
\[\ce{CH3-O-C2H5 (Ethyl methyl ether) + H2O ->[\ce{dil. H2SO4}][\Delta,\text{ Pressure}] CH3OH (Methanol) + C2H5OH (Ethanol)}\]

Q. 8. Calculate the osmotic pressure of solution containing $0.822\text{ gm}$ of sucrose in $300\text{ mL}$ of water at $298\text{ K}$.

Answer:

Given:

Mass of solute (sucrose, $W_2$) $= 0.822\text{ g}$
Volume of solution ($V$) $= 300\text{ mL} = 0.3\text{ dm}^3$ (or $\text{L}$)
Temperature ($T$) $= 298\text{ K}$
Molar mass of sucrose ($\ce{C12H22O11}$, $M_2$) $= 12(12) + 22(1) + 11(16) = 342\text{ g mol}^{-1}$
Universal gas constant ($R$) $= 0.08205\text{ dm}^3\text{ atm K}^{-1}\text{ mol}^{-1}$

Formula:

\[\pi = \frac{W_2 \times R \times T}{M_2 \times V}\]

Calculation:

\[\pi = \frac{0.822 \times 0.08205 \times 298}{342 \times 0.3}\] \[\pi = \frac{20.0986}{102.6} = 0.1959\text{ atm}\]

Result: The osmotic pressure of the sucrose solution is $0.1959\text{ atm}$ (or $0.196\text{ atm}$ using $R = 0.0821\text{ dm}^3\text{ atm K}^{-1}\text{ mol}^{-1}$).

Q. 9. Write the mathematical equation of first law of thermodynamics for following processes:

Answer: According to the first law of thermodynamics, $\Delta U = q + W = q - P_{\text{ext}}\Delta V$.

Isochoric process:

In an isochoric process, volume remains constant ($\Delta V = 0$), so mechanical work done is zero ($W = -P_{\text{ext}}\Delta V = 0$).
Therefore, the mathematical equation becomes:
\[\Delta U = q_V\]
(where $q_V$ is the heat supplied at constant volume).
Adiabatic process:

In an adiabatic process, no heat enters or leaves the system ($q = 0$).
Therefore, the mathematical equation becomes:
\[\Delta U = W \quad (\text{or } -\Delta U = -W)\]

Q. 10. Write two similarities and two differences between lanthanoids and actinoids.

Answer:

Similarities between Lanthanoids and Actinoids:

Both series involve the progressive filling of $(n-2)f$ orbitals and exhibit $+3$ as their most common and stable oxidation state.
Both exhibit a regular decrease in atomic and ionic radii with increasing atomic number across the series (known as lanthanoid contraction and actinoid contraction) due to poor shielding by $f$-electrons.

Differences between Lanthanoids and Actinoids:

Lanthanoids	Actinoids
1. Except promethium ($\ce{Pm}$), all lanthanoids are non-radioactive.	1. All actinoids are radioactive elements.
2. Besides $+3$, they show a maximum oxidation state of $+4$ (and occasionally $+2$).	2. Besides $+3$, they exhibit higher oxidation states such as $+4, +5, +6,$ and $+7$.
3. They have a relatively lesser tendency to form coordination complexes.	3. They have a much greater tendency to form coordination complexes.

Q. 11. What is the action of following reagents on chlorobenzene?

$\ce{conc. HNO3}$:

Answer: When chlorobenzene is heated with a nitrating mixture (concentrated $\ce{HNO3}$ and concentrated $\ce{H2SO4}$), it undergoes electrophilic nitration to yield a mixture of 1-chloro-2-nitrobenzene (o-chloronitrobenzene, minor product) and 1-chloro-4-nitrobenzene (p-chloronitrobenzene, major product).
\[\ce{2 C6H5Cl + 2 HNO3 (conc.) ->[\ce{conc. H2SO4}][\Delta] \underset{\text{1-Chloro-4-nitrobenzene (Major)}}{p-Cl-C6H4-NO2} + \underset{\text{1-Chloro-2-nitrobenzene (Minor)}}{o-Cl-C6H4-NO2} + 2 H2O}\]
$\ce{fuming H2SO4}$:

Answer: When chlorobenzene is heated with fuming sulfuric acid ($\ce{H2SO4 + SO3}$), it undergoes sulfonation to form 4-chlorobenzenesulfonic acid (major product) and 2-chlorobenzenesulfonic acid (minor product).
\[\ce{2 C6H5Cl + 2 H2SO4 (fuming) ->[\Delta] \underset{\text{4-Chlorobenzenesulfonic acid (Major)}}{p-Cl-C6H4-SO3H} + \underset{\text{2-Chlorobenzenesulfonic acid (Minor)}}{o-Cl-C6H4-SO3H} + 2 H2O}\]

Q. 12. Explain Cannizzaro reaction with suitable example.

Answer:

Explanation: Aldehydes that do not possess any $\alpha$-hydrogen atom undergo self-oxidation and reduction (disproportionation reaction) when heated with concentrated alkali ($50\%$ aqueous $\ce{NaOH}$ or $\ce{KOH}$). In this reaction, one molecule of the aldehyde is reduced to the corresponding alcohol, and the other molecule is simultaneously oxidized to the carboxylic acid salt. This reaction is called the Cannizzaro reaction.

Example: Formaldehyde (methanal), which lacks an $\alpha$-hydrogen, reacts with $50\%$ $\ce{NaOH}$ to form methanol and sodium formate:

\[\ce{2 HCHO (Formaldehyde) + NaOH (50\%, conc.) ->[\Delta] CH3OH (Methanol) + HCOONa (Sodium formate)}\]

Q. 13. A chemical reaction occurs in two steps:

(i) $\ce{NO2Cl(g) -> NO2(g) + Cl(g)}$ (Slow step)
(ii) $\ce{NO2Cl(g) + Cl(g) -> NO2(g) + Cl2(g)}$ (Fast step)

Write down the rate law.

Answer: Since the rate of an overall reaction depends on the slowest elementary step (rate-determining step), the rate law is:
\[\text{Rate} = k[\ce{NO2Cl}]\]
Identify the reaction intermediate.

Answer: The reaction intermediate is $\ce{Cl(g)}$ (chlorine atom), as it is produced in the first step and consumed in the second step.

Q. 14. The standard potential of electrode $\ce{Cu^{++}}(0.02\text{ M})|\ce{Cu(s)}$ is $0.337\text{ volt}$. Calculate its potential in volt.

Answer:

Given:

Standard electrode potential ($E^\circ_{\ce{Cu^{2+}/Cu}}$) $= 0.337\text{ V}$
Concentration of $\ce{Cu^{2+}}$ ($[\ce{Cu^{2+}}]$) $= 0.02\text{ M}$
Electrode reduction reaction: $\ce{Cu^{2+}(aq) + 2e^- -> Cu(s)}$, hence $n = 2$

Formula (Nernst Equation at $298\text{ K}$):

\[E_{\text{cell}} = E^\circ - \frac{0.0592}{n} \log_{10} \left(\frac{1}{[\ce{Cu^{2+}}]}\right)\]

Calculation:

\[E = 0.337 - \frac{0.0592}{2} \log_{10} \left(\frac{1}{0.02}\right)\] \[E = 0.337 - 0.0296 \times \log_{10}(50)\] \[E = 0.337 - (0.0296 \times 1.6990)\] \[E = 0.337 - 0.0503 = 0.2867\text{ V}\]

Result: The electrode potential is $0.2867\text{ V}$.

SECTION - C

Attempt any EIGHT of the following questions: [24 Marks]

Q. 15.

Derive relationship between relative lowering of vapour pressure and molar mass of non volatile solute.

Answer: According to Raoult's law, for a solution of a non-volatile solute in a volatile solvent, the relative lowering of vapour pressure is equal to the mole fraction of the solute ($x_2$):
\[\frac{P_1^\circ - P_1}{P_1^\circ} = x_2\]
where $P_1^\circ$ is the vapour pressure of pure solvent and $P_1$ is the vapour pressure of the solution.

If $n_1$ and $n_2$ are the number of moles of solvent and solute respectively in the solution, then:
\[x_2 = \frac{n_2}{n_1 + n_2}\]
For a very dilute solution, $n_2 \ll n_1$, so $n_1 + n_2 \approx n_1$. Hence:
\[\frac{P_1^\circ - P_1}{P_1^\circ} = \frac{n_2}{n_1}\]
If $W_1$ and $W_2$ are the masses and $M_1$ and $M_2$ are the molar masses of the solvent and solute respectively, then $n_1 = \frac{W_1}{M_1}$ and $n_2 = \frac{W_2}{M_2}$. Substituting these values:
\[\frac{P_1^\circ - P_1}{P_1^\circ} = \frac{W_2 / M_2}{W_1 / M_1} = \frac{W_2 \times M_1}{W_1 \times M_2}\]
Rearranging for the molar mass of the non-volatile solute ($M_2$):
\[M_2 = \frac{W_2 \times M_1 \times P_1^\circ}{W_1 \times (P_1^\circ - P_1)}\]
Write statement of second law of thermodynamics.

Answer: The second law of thermodynamics states that the total entropy of a system and its surroundings (an isolated system) always increases in a spontaneous process:
\[\Delta S_{\text{total}} = \Delta S_{\text{sys}} + \Delta S_{\text{surr}} > 0\]

Q. 16.

Convert, acetamide into ethyl amine.

Answer: Acetamide on reduction with lithium aluminium hydride ($\ce{LiAlH4}$) in dry ether followed by acid hydrolysis (or with $\ce{Na}$ / absolute ethanol) gives ethylamine (ethanamine):
\[\ce{CH3-CO-NH2 (Acetamide) + 4[H] ->[\text{(i) }\ce{LiAlH4}\text{ / dry ether}][\text{(ii) }\ce{H3O^+}] CH3-CH2-NH2 (Ethylamine) + H2O}\]
Explain amphoteric nature of water.

Answer: According to the Bronsted-Lowry theory, water has the ability to act both as a proton donor (acid) and as a proton acceptor (base). This dual behaviour is known as the amphoteric nature of water.
- Water as a Bronsted acid (proton donor) towards $\ce{NH3}$:
  \[\ce{H2O(l) + NH3(aq) <=> OH^-(aq) + NH4^+(aq)}\]
- Water as a Bronsted base (proton acceptor) towards $\ce{HCl}$:
  \[\ce{H2O(l) + HCl(aq) <=> H3O^+(aq) + Cl^-(aq)}\]

Q. 17. Convert the following:

Ethanol into sodium ethoxide.

Answer: Ethanol reacts with active sodium metal to form sodium ethoxide and liberates hydrogen gas:
\[\ce{2 C2H5OH (Ethanol) + 2 Na -> 2 C2H5ONa (Sodium ethoxide) + H2 ^}\]
Phenol into o-phenol sulfonic acid.

Answer: Phenol reacts with concentrated sulfuric acid at room temperature ($293\text{ K}$) to yield o-phenolsulfonic acid (2-hydroxybenzenesulfonic acid) as the major product:
\[\ce{C6H5OH (Phenol) + H2SO4 (conc.) ->[293\text{ K}] o-HO-C6H4-SO3H (\textit{o}-Phenolsulfonic acid) + H2O}\]
Bromomethane into methoxyethane.

Answer: When bromomethane is heated with alcoholic sodium ethoxide (Williamson synthesis), methoxyethane (ethyl methyl ether) is formed:
\[\ce{CH3Br (Bromomethane) + C2H5ONa (Sodium ethoxide) ->[\Delta] CH3-O-C2H5 (Methoxyethane) + NaBr}\]

Q. 18.

What is peptide linkage?

Answer: The amide bond ($\ce{-CO-NH-}$) formed by the condensation of the carboxyl group ($\ce{-COOH}$) of one $\alpha$-amino acid molecule with the amino group ($\ce{-NH2}$) of another $\alpha$-amino acid molecule by the elimination of a water molecule is called a peptide linkage (or peptide bond).
\[\ce{H2N-CHR-COOH + H-NH-CHR'-COOH ->[-\ce{H2O}] H2N-CHR-\underset{\text{Peptide linkage}}{\mathbf{CO-NH}}-CHR'-COOH}\]
Write general characteristics of interhalogen compounds.

Answer:
- An interhalogen compound has the general formula $\ce{XX'_n}$ (where $n = 1, 3, 5, 7$), where $\ce{X}$ is the halogen of larger size (more electropositive) showing a positive oxidation state, and $\ce{X'}$ is the halogen of smaller size (more electronegative) with an oxidation state of $-1$.
- All interhalogen compounds have an even number of atoms ($2, 4, 6, 8$) and are covalent and diamagnetic in nature.
- They are generally more reactive than regular diatomic halogens (except $\ce{F2}$) because the $\ce{X-X'}$ bond is polar and weaker than the homonuclear $\ce{X-X}$ bond.
- Their physical properties (such as colour, melting point, and boiling point) are intermediate between those of their constituent halogens.

Q. 19.

Write observed electronic configuration of chromium ($Z=24$).

Answer:
\[\ce{Cr} (Z = 24): 1s^2\, 2s^2\, 2p^6\, 3s^2\, 3p^6\, 3d^5\, 4s^1 \quad \text{or} \quad [\ce{Ar}]\, 3d^5\, 4s^1\]
Calculate magnetic moment of $\ce{Ti^{3+}}$ by using spin only formula. ($Z\text{ of Ti}=22$)

Answer:
- Electronic configuration of $\ce{Ti} (Z = 22) = [\ce{Ar}]\, 3d^2\, 4s^2$
- Electronic configuration of $\ce{Ti^{3+}} = [\ce{Ar}]\, 3d^1$
- Number of unpaired electrons ($n$) $= 1$
- Spin-only formula for effective magnetic moment:
\[\mu = \sqrt{n(n+2)}\text{ BM} = \sqrt{1(1+2)} = \sqrt{3} = 1.732\text{ BM}\]
Define Green chemistry.

Answer: Green chemistry is the use of chemistry for pollution prevention by environmentally conscious design of chemical products and processes that reduce or eliminate the use or generation of hazardous substances.

Q. 20.

Write examples of coordination metal complexes in biology.

Answer:
1. Chlorophyll: A green pigment in plants which is a coordination complex of Magnesium ($\ce{Mg^{2+}}$) with a porphyrin (chlorin) ring.
2. Haemoglobin: An oxygen-carrier in blood which is a coordination complex of Iron ($\ce{Fe^{2+}}$).
3. Vitamin $\ce{B12}$ (Cyanocobalamin): A coordination complex of Cobalt ($\ce{Co}$).
Calculate the work done in oxidation of $2\text{ moles}$ of $\ce{SO2}$ at $298\text{ K}$, if $\ce{SO2(g) + 1/2 O2(g) -> SO3(g)}$. [Given: $R = 8.314\text{ J/K/mol}$]

Answer:

The balanced equation for the oxidation of $2\text{ moles}$ of $\ce{SO2}$ is:
\[\ce{2 SO2(g) + O2(g) -> 2 SO3(g)}\]
- Change in number of moles of gaseous species: $\Delta n_g = n_p - n_r = 2 - (2 + 1) = -1\text{ mol}$
- Temperature ($T$) $= 298\text{ K}$
- Gas constant ($R$) $= 8.314\text{ J K}^{-1}\text{ mol}^{-1}$
Formula:
\[W = -\Delta n_g R T\]
Calculation:
\[W = -(-1\text{ mol}) \times 8.314\text{ J K}^{-1}\text{ mol}^{-1} \times 298\text{ K} = +2477.57\text{ J} = +2.478\text{ kJ}\]
Result: The work done is $+2477.57\text{ J}$ (or $+2.478\text{ kJ}$). The positive sign indicates that work is done on the system by the surroundings.

Q. 21.

Write the formula to measure $\%$ atom economy according to green chemistry.

Answer:
\[\%\text{ Atom Economy} = \frac{\text{Formula weight of the desired product}}{\text{Sum of formula weights of all the reactants used in the reaction}} \times 100\]
Convert:

(a) Benzene into benzaldehyde
Answer: When benzene is treated with carbon monoxide and hydrogen chloride gas in the presence of anhydrous aluminium chloride and cuprous chloride under high pressure (Gattermann-Koch formylation), benzaldehyde is formed:
\[\ce{C6H6 (Benzene) + CO + HCl ->[\ce{Anhyd. AlCl3 / CuCl}][\text{High pressure}] C6H5CHO (Benzaldehyde)}\]
(b) Cyclohexene into adipic acid
Answer: When cyclohexene is heated with acidified potassium permanganate ($\ce{KMnO4}$ in dilute $\ce{H2SO4}$), vigorous oxidative cleavage of the double bond takes place to yield adipic acid (hexane-1,6-dioic acid):
\[\ce{C6H10 (Cyclohexene) + 4[O] ->[\ce{KMnO4 / dil. H2SO4}][\Delta] HOOC-(CH2)4-COOH (Adipic acid)}\]

Q. 22.

Write a reaction for preparation of Nylon-6.

Answer: When $\varepsilon$-caprolactam is heated with water at high temperature ($533\text{ K} - 543\text{ K}$), it undergoes ring-opening polymerization to give the polyamide fibre Nylon-6:
\[n\,\ce{C6H11NO (\varepsilon\text{-Caprolactam})} \xrightarrow{\ce{H2O},\ 533-543\text{ K}} \ce{[-NH-(CH2)5-CO-]_n (Nylon-6)}\]
The salt of $\ce{Sc^{3+}}$ ion is colourless and the salt of $\ce{Mn^{3+}}$ ion is coloured. Explain. [$Z\text{ of Sc}=21$ and $Z\text{ of Mn}=25$]

Answer:
- The colour of transition metal ions is due to the presence of unpaired electrons in $(n-1)d$ orbitals and the resulting $d\text{-}d$ electronic transitions in the visible region.
- For $\ce{Sc^{3+}}$: Atomic number of $\ce{Sc} = 21$, electronic configuration is $[\ce{Ar}]\, 3d^1\, 4s^2$. In $\ce{Sc^{3+}}$ ion, the configuration is $[\ce{Ar}]\, 3d^0$. Since it has no unpaired electrons in the $3d$ subshell ($n = 0$), $d\text{-}d$ transition cannot take place. Hence, $\ce{Sc^{3+}}$ salts are colourless.
- For $\ce{Mn^{3+}}$: Atomic number of $\ce{Mn} = 25$, electronic configuration is $[\ce{Ar}]\, 3d^5\, 4s^2$. In $\ce{Mn^{3+}}$ ion, the configuration is $[\ce{Ar}]\, 3d^4$, which contains $4$ unpaired electrons. When visible light falls on the salt, these unpaired $3d$ electrons absorb radiation and undergo $d\text{-}d$ transition, transmitting the complementary colour. Hence, $\ce{Mn^{3+}}$ salts are coloured.

Q. 23.

Write the postulates of Werner's theory of co-ordination complexes.

Answer: The postulates of Werner's coordination theory are:
1. Unlike simple salts, the metal atom in a coordination complex possesses two types of valencies: primary (ionizable) valency and secondary (non-ionizable) valency.
2. The ionizable sphere (outer sphere) consists of entities that satisfy the primary valency (oxidation state), which are always satisfied by anions.
3. The coordination sphere (inner sphere) consists of entities that satisfy the secondary valency (coordination number). The secondary valencies of a metal ion are fixed and are satisfied by either negative ions or neutral molecules (ligands).
4. The secondary valencies have fixed spatial orientations around the central metal ion, which gives a definite geometry (such as octahedral, tetrahedral, or square planar) to the coordination complex.
What is the action of ethane-1, 2-diol on acetone?

Answer: Acetone reacts reversibly with ethane-1,2-diol (ethylene glycol) in the presence of dry $\ce{HCl}$ gas (or $p$-toluenesulfonic acid) to form a cyclic ketal called acetone ethylene ketal (2,2-dimethyl-1,3-dioxolane):
\[\ce{(CH3)2C=O (Acetone) + HO-CH2-CH2-OH (Ethane-1,2-diol) <=>[\ce{dry HCl gas}] (CH3)2C(-O-CH2-CH2-O-) (Acetone ethylene ketal) + H2O}\]

Q. 24.

Write Tollen's reagent test for acetaldehyde.

Answer: When acetaldehyde is warmed with freshly prepared Tollens' reagent (ammoniacal silver nitrate solution), acetaldehyde is oxidized to acetate ion and $\ce{Ag^+}$ ions are reduced to metallic silver ($\ce{Ag}$), which deposits as a bright silver mirror on the inner wall of the test tube:
\[\ce{CH3-CHO + 2[Ag(NH3)2]^+ + 3OH^- ->[\Delta] CH3-COO^- + 2Ag v (Silver mirror) + 4NH3 ^ + 2H2O}\]
Write the structure of zwitter ion of sulfanilic acid. Write the reaction between benzene diazonium chloride and phenol in alkaline medium.

Answer:

(a) Zwitter ion of sulfanilic acid (p-aminobenzenesulfonic acid):
\[\ce{^+H3N-C6H4-SO3^-} \quad (\text{with }\ce{-NH3^+}\text{ and }\ce{-SO3^-}\text{ at }1,4\text{ or }\textit{para}\text{ positions})\]
(b) Coupling reaction of benzenediazonium chloride with phenol:
Benzenediazonium chloride reacts with phenol in cold, mildly alkaline medium ($\text{pH } 9\text{--}10$ at $273\text{--}278\text{ K}$) to form an orange azo dye, p-hydroxyazobenzene:
\[\ce{C6H5-N2^+Cl^- + H-C6H4-OH (Phenol) ->[\ce{OH^-}\text{ (pH 9-10)}][273-278\text{ K}] C6H5-N=N-C6H4-OH (\textit{p}-Hydroxyazobenzene, Orange dye) + Cl^- + H2O}\]

Q. 25.

Define: Common ion effect.

Answer: The common ion effect states that the ionization (dissociation) of a weak electrolyte is suppressed in the presence of a strong electrolyte containing an ion common to the weak electrolyte.
Example: The degree of dissociation of weak acetic acid ($\ce{CH3COOH}$) is suppressed by adding strong electrolyte sodium acetate ($\ce{CH3COONa}$), which furnishes the common $\ce{CH3COO^-}$ ion.
Write preparation of glucose from starch.

Answer: On commercial scale, glucose is prepared by boiling starch with dilute sulfuric acid at $393\text{ K}$ under $2\text{ to }3\text{ atm}$ pressure:
\[\ce{(C6H10O5)_n (Starch) + n H2O ->[\ce{dil. H2SO4}][393\text{ K, } 2\text{--}3\text{ atm}] n C6H12O6 (D-Glucose)}\]

Q. 26.

Draw the structure of 2, 4 dinitrophenylhydrazone of acetaldehyde.

Answer: The chemical structure of acetaldehyde 2,4-dinitrophenylhydrazone is:
\[\ce{CH3-CH=N-NH}-\text{C}_6\text{H}_3(\text{2-NO}_2)(\text{4-NO}_2)\]
(Structural description: A benzene ring attached to $\ce{-NH-N=CH-CH3}$ at position $1$, with nitro groups $\ce{-NO2}$ at positions $2$ and $4$ of the ring.)
The rate of $\ce{A + B -> P}$ is $3.6 \times 10^{-2}\text{ mol/dm}^3\text{/s}$. When $[\ce{A}] = 0.2\text{ M}$ and $[\ce{B}] = 0.1\text{ M}$. Calculate the rate constant if reaction is first order in $\ce{B}$ and second order in $\ce{A}$.

Answer:

Given:
- $\text{Rate} = 3.6 \times 10^{-2}\text{ mol dm}^{-3}\text{ s}^{-1}$
- $[\ce{A}] = 0.2\text{ M}$, $[\ce{B}] = 0.1\text{ M}$
- Order with respect to $\ce{A} = 2$, Order with respect to $\ce{B} = 1$ (Overall order $= 3$)
Rate Law Formula:
\[\text{Rate} = k[\ce{A}]^2[\ce{B}]^1 \implies k = \frac{\text{Rate}}{[\ce{A}]^2[\ce{B}]}\]
Calculation:
\[k = \frac{3.6 \times 10^{-2}}{(0.2)^2 \times (0.1)} = \frac{3.6 \times 10^{-2}}{0.04 \times 0.1} = \frac{3.6 \times 10^{-2}}{4.0 \times 10^{-3}} = 9.0\text{ dm}^6\text{ mol}^{-2}\text{ s}^{-1}\]
Result: The rate constant of the reaction is $9.0\text{ dm}^6\text{ mol}^{-2}\text{ s}^{-1}$ (or $9.0\text{ mol}^{-2}\text{ dm}^6\text{ s}^{-1}$).

SECTION - D

Attempt any THREE of the following questions: [12 Marks]

Q. 27.

Define:

(a) Isotonic solution:
Two or more solutions having the same osmotic pressure at a given temperature are called isotonic solutions. Such solutions have the same molar concentration of solute particles, and no net osmosis occurs when they are separated by a semipermeable membrane.

(b) Molecularity of reaction:
The molecularity of an elementary reaction is defined as the total number of reactant particles (atoms, ions, or molecules) taking part in that elementary chemical reaction which must collide simultaneously to bring about the reaction.
State and explain Hess's law of constant heat summation.

Statement: Hess's law of constant heat summation states that the overall enthalpy change ($\Delta H^\circ$) for a chemical reaction is the same whether the reaction takes place in a single step or in a series of steps, provided the initial and final states of the reactants and products remain the same.

Explanation with Example: Consider the formation of $\ce{CO2(g)}$ from $\ce{C(graphite)}$ and $\ce{O2(g)}$:
- Single-step reaction: \[\ce{C(graphite) + O2(g) -> CO2(g)}, \quad \Delta H^\circ = -393.5\text{ kJ}\]
- Two-step reaction:
  Step 1: $\ce{C(graphite) + 1/2 O2(g) -> CO(g)}, \quad \Delta H_1^\circ = -110.5\text{ kJ}$
  
  Step 2: $\ce{CO(g) + 1/2 O2(g) -> CO2(g)}, \quad \Delta H_2^\circ = -283.0\text{ kJ}$
According to Hess's law, the overall enthalpy change is the algebraic sum of the enthalpy changes of individual steps:
\[\Delta H^\circ = \Delta H_1^\circ + \Delta H_2^\circ = (-110.5\text{ kJ}) + (-283.0\text{ kJ}) = -393.5\text{ kJ}\]

Q. 28.

Define:

(a) Schottky defect:
The vacancy defect in an ionic solid in which an equal number of cations and anions are missing from their regular lattice positions, thereby maintaining electrical neutrality and lowering the density of the crystal, is called Schottky defect (e.g., $\ce{NaCl}$, $\ce{KCl}$).

(b) Ferromagnetism:
The property of substances containing a large number of unpaired electrons that are strongly attracted by an external magnetic field and can be permanently magnetized due to the spontaneous alignment of magnetic moments (domains) in the same direction is called ferromagnetism (e.g., $\ce{Fe}$, $\ce{Co}$, $\ce{Ni}$, $\ce{CrO2}$).
Write the name of catalyst used in preparation of HDP.

Answer: Ziegler-Natta catalyst, which is a coordination catalyst mixture of triethylaluminium $\ce{Al(C2H5)3}$ and titanium tetrachloride $\ce{TiCl4}$.
Draw structure of isoprene unit of natural rubber.

Answer:
- Monomer Isoprene (2-methylbuta-1,3-diene): \[\ce{CH2=C(CH3)-CH=CH2}\]
- Repeating Isoprene Unit in Natural Rubber (cis-1,4-polyisoprene): \[\ce{[-CH2-C(CH3)=CH-CH2-]_n}\]

Q. 29.

Calculate effective atomic number of $\ce{Fe^{2+}}$ in $\ce{[Fe(CN)6]^{4-}}$ [Given: ($Z=26$)]

Answer:
- Atomic number of $\ce{Fe}$ ($Z$) $= 26$
- Oxidation state of $\ce{Fe}$ in $\ce{[Fe(CN)6]^{4-}}$ ($X$) $= +2$ (electrons lost $= 2$)
- Number of electrons donated by $6$ $\ce{CN^-}$ ligands ($Y$) $= 6 \times 2 = 12$
Formula:
\[\text{EAN} = Z - X + Y = 26 - 2 + 12 = 36\]
Result: The effective atomic number (EAN) of $\ce{Fe^{2+}}$ in $\ce{[Fe(CN)6]^{4-}}$ is $36$ (which equals the atomic number of Krypton, $\ce{Kr}$).
Draw neat and labelled diagram of lead accumulator.

Answer: (Examiner's Note: Students must draw a neat diagram of the Lead Storage Battery with the following mandatory labels to score full marks:)
- Anode (Negative plates): Lead ($\ce{Pb}$) grids packed with spongy lead ($\ce{Pb}$).
- Cathode (Positive plates): Lead ($\ce{Pb}$) grids packed with lead dioxide ($\ce{PbO2}$).
- Electrolyte: $38\%$ by mass aqueous sulfuric acid ($\ce{H2SO4}$) having density $1.2\text{ g/mL}$.
- Container: Insulating acid-proof container with alternating anode and cathode plates connected in parallel.
Write two advantages of hydrogen - oxygen fuel cell.

Answer:
1. High Efficiency: $\ce{H2-O2}$ fuel cells convert chemical energy directly into electrical energy with a high efficiency of about $70\%$, compared to about $40\%$ for conventional thermal power plants.
2. Non-polluting & Continuous Operation: They are completely eco-friendly because the only exhaust product is pure water ($\ce{H2O}$), and they operate continuously without discharging as long as $\ce{H2}$ and $\ce{O2}$ gases are supplied.

Q. 30.

Convert, ethanoic acid to ethanol.

Answer: Ethanoic acid is reduced to ethanol using a strong reducing agent like lithium aluminium hydride ($\ce{LiAlH4}$) in dry ether followed by acid hydrolysis:
\[\ce{CH3-COOH (Ethanoic acid) ->[\text{(i) }\ce{LiAlH4}\text{ / dry ether}][\text{(ii) }\ce{H3O^+}] CH3-CH2-OH (Ethanol) + H2O}\]
Explain the mechanism of alkaline hydrolysis of bromomethane.

Answer:

1. Chemical Reaction & Kinetics: Alkaline hydrolysis of bromomethane is a bimolecular nucleophilic substitution ($\text{S}_{\text{N}}2$) reaction:
\[\ce{CH3Br + OH^-(aq) ->[\Delta] CH3OH + Br^-(aq)}\]
It follows second-order kinetics, meaning the rate depends on the concentration of both the substrate ($\ce{CH3Br}$) and the nucleophile ($\ce{OH^-}$):
\[\text{Rate} = k[\ce{CH3Br}][\ce{OH^-}]\]
2. Mechanism Representation:
\[\ce{HO^- + C(H)3-Br ->[\text{Slow}] \left[ \overset{\delta -}{HO} \cdots C(H)3 \cdots \overset{\delta -}{Br} \right]^{\ddagger} (\text{Transition state}) ->[\text{Fast}] HO-C(H)3 + Br^-}\]
3. Salient Features of the $\text{S}_{\text{N}}2$ Mechanism:
- Single-step concerted mechanism: Bond breaking ($\ce{C-Br}$) and bond forming ($\ce{C-OH}$) occur simultaneously in a single step without any reaction intermediate.
- Backside attack: The incoming nucleophile ($\ce{OH^-}$) approaches the electrophilic carbon atom from the side directly opposite ($180^\circ$) to the leaving group ($\ce{Br^-}$) to minimize steric hindrance and electrostatic repulsion.
- Pentacoordinate Transition State: In the transition state, the carbon atom is simultaneously bonded to three hydrogen atoms via $\sigma$ bonds and partially bonded to both $\ce{OH}$ and $\ce{Br}$ (carrying partial negative charges $\delta-$). The three $\ce{C-H}$ bonds lie in a single plane.
- Walden Inversion: As the $\ce{Br^-}$ ion departs, the configuration of the carbon atom flips inside out (like an umbrella in a strong gale), resulting in $100\%$ inversion of configuration.

Q. 31.

Acetic acid is $5\%$ ionised in its decimolar solution. Calculate the dissociation constant of acetic acid.

Answer:

Given:
- Concentration of decimolar solution ($c$) $= 0.1\text{ M}$
- Percent dissociation $= 5\% \implies$ Degree of dissociation ($\alpha$) $= \frac{5}{100} = 0.05$
Formula (Ostwald's Dilution Law for a Weak Monobasic Acid):
\[K_a = \frac{c\alpha^2}{1 - \alpha}\]
Calculation:
- Method 1 (Using small $\alpha$ approximation $1 - \alpha \approx 1$): \[K_a = c\alpha^2 = 0.1 \times (0.05)^2 = 0.1 \times 2.5 \times 10^{-3} = 2.5 \times 10^{-4}\]
- Method 2 (Exact substitution without approximation): \[K_a = \frac{0.1 \times (0.05)^2}{1 - 0.05} = \frac{2.5 \times 10^{-4}}{0.95} = 2.63 \times 10^{-4}\]
Result: The dissociation constant ($K_a$) of acetic acid is $2.5 \times 10^{-4}$ (or $2.63 \times 10^{-4}$).
Write chemical formula of epsom salt.

Answer: $\ce{MgSO4.7H2O}$ (Magnesium sulfate heptahydrate)
Give two uses of $\ce{H2SO4}$.

Answer:
1. It is used in the manufacture of chemical fertilizers such as ammonium sulfate and superphosphate of lime.
2. It is used as an electrolyte in lead storage batteries, in petroleum refining, and in the manufacture of pigments, paints, dyes, and detergents.

HSC 12th Physics June 2026 Board Question Paper with Solutions (J-229)

aaminomtex@gmail.com (Amin) — Sun, 25 Oct 2026 09:47:00 +0530

Maharashtra State Board HSC 12th Standard Physics (Subject Code: 54) June / July 2026 Supplementary Exam Question Paper (J-229) with Complete Solutions and Model Answer Key.

SECTION - A

Q. 1. Select and write the correct answers for the following multiple choice type of questions:

The MI of ring of radius 'R' and mass 'M' about its central axis is

Answer: (c) $ MR^2 $
For a monoatomic gas enclosed in a container at a constant temperature has degrees of freedom

Answer: (b) $ 3 $
The equation of the first law of thermodynamics for isochoric process is

Answer: (a) $ Q = \Delta U $
Cyclotron can not accelerate

Answer: (c) neutrons
In a purely inductive and capacitive circuit, power factor is,

Answer: (b) zero
The Boolean equation for NOR gate is

Answer: (d) $ Y = \overline{A + B} $
When n number of capacitors having each of capacity 'C' are connected in series, their equivalent capacity is

Answer: (c) $ \frac{C}{n} $
A polarising angle for an optical material is $ 35^\circ $. Then refractive index of the material is

Answer: (b) $ 0.7002 $
For a streamline flow, the relative velocity between the two liquid layers is $ 10\text{ cm/s} $. The distance between them is $ 5\text{ cm} $. The velocity gradient is

Answer: (d) $ 2\text{ s}^{-1} $
The wavelength of a stationary wave on a string is $ 0.75\text{ m} $ with frequency of $ 64\text{ Hz} $. Then, the velocity of wave is

Answer: (a) $ 48\text{ m/s} $

Q. 2. Answer the following questions:

Absorption of ink by a blotting paper is an example of which phenomenon of fluids?

Answer: Capillarity (or Capillary action).
State Kirchhoff's first law for electrical network.

Answer: Kirchhoff's first law (Current Law) states that the algebraic sum of the currents meeting at any junction (node) in an electrical network is zero, i.e., $ \sum I = 0 $.
Define 'photoelectric work function.'

Answer: The minimum amount of energy required to be provided to an electron to just pull it out of the metal surface (with zero kinetic energy) is called the photoelectric work function of that metal.
Which series of spectral lines belongs to the ultraviolet region of the hydrogen spectrum?

Answer: Lyman series.
Which semiconductor device converts solar energy into electrical energy?

Answer: Solar cell (or Photovoltaic cell).
Define the term wavefront.

Answer: The locus of all points in a medium to which light waves reach simultaneously, so that all the points are in the same phase, is called a wavefront.
A motorcyclist performs horizontal circular motion inside a well of inner radius $ 3\text{ m} $. Calculate the minimum speed to perform the stunt if coefficient of static friction is $ 0.3 $. $ [g = 10\text{ m/s}^2] $.

Answer:
Given: $ r = 3\text{ m} $, $ \mu_s = 0.3 $, $ g = 10\text{ m/s}^2 $
Formula: $ v_{\min} = \sqrt{\frac{rg}{\mu_s}} $
\[ v_{\min} = \sqrt{\frac{3 \times 10}{0.3}} = \sqrt{\frac{30}{0.3}} = \sqrt{100} = 10\text{ m/s} \] $ v_{\min} = 10\text{ m/s} $
Calculate the maximum velocity of a particle having amplitude of oscillation $ 4\text{ cm} $ and period $ 4\text{ seconds} $.

Answer:
Given: $ A = 4\text{ cm} = 4 \times 10^{-2}\text{ m} $, $ T = 4\text{ s} $, $ \pi = 3.142 $
Formula: $ v_{\max} = A\omega = A\left(\frac{2\pi}{T}\right) $
\[ v_{\max} = 4 \times 10^{-2} \times \left(\frac{2 \times 3.142}{4}\right) = 2 \times 3.142 \times 10^{-2} = 6.284 \times 10^{-2}\text{ m/s} \] $ v_{\max} = 6.284 \times 10^{-2}\text{ m/s} $ (or $ 6.284\text{ cm/s} $)

SECTION - B

Attempt any EIGHT questions from the following:

Q. 3. Obtain an expression for the maximum speed of a vehicle along horizontal circular curved road.

Answer:

(Note to student: Draw a neat diagram of a vehicle on a horizontal curved road of radius $ r $ showing weight $ mg $ acting vertically downwards, normal reaction $ N $ acting vertically upwards, and force of static friction $ f_s $ directed horizontally towards the centre $ C $ of the circular track.)

Consider a vehicle of mass $ m $ moving with speed $ v $ along a horizontal circular track of radius $ r $.

In the reference frame of an inertial observer, the forces acting on the vehicle are:

Weight ($ mg $): acting vertically downwards.
Normal reaction ($ N $): acting vertically upwards, perpendicular to the road surface.
Force of static friction ($ f_s $): acting horizontally between the road and the tyres, directed towards the centre of the circular track.

Since there is no vertical motion of the vehicle, the normal reaction balances the weight of the vehicle:

\[ N = mg \quad \text{... (Equation 1)} \]

The force of static friction provides the necessary centripetal force for circular motion:

\[ f_s = \frac{mv^2}{r} \quad \text{... (Equation 2)} \]

Dividing Equation 2 by Equation 1, we get:

\[ \frac{f_s}{N} = \frac{v^2}{rg} \quad \text{... (Equation 3)} \]

For a given track, radius $ r $ is constant. As speed $ v $ increases, the force of static friction $ f_s $ also increases up to its limiting value:

\[ (f_s)_{\max} = \mu_s N \quad \text{(where } \mu_s \text{ is the coefficient of static friction between the tyres and the road)} \]

Substituting $ (f_s)_{\max} = \mu_s N $ for maximum possible speed ($ v_{\max} $) in Equation 3:

\[ \frac{\mu_s N}{N} = \frac{v_{\max}^2}{rg} \implies \mu_s = \frac{v_{\max}^2}{rg} \]

\[ v_{\max}^2 = \mu_s rg \]

\[ \mathbf{v_{\max} = \sqrt{\mu_s rg}} \]

This is the required expression for the maximum safe speed of a vehicle along a horizontal circular curved road.

Q. 4. For a thermodynamic system: Define: (a) Mechanical equilibrium (b) Chemical equilibrium

Answer:

Mechanical equilibrium: When there are no unbalanced forces within the system and between the system and its surroundings, the system is said to be in mechanical equilibrium. In this state, the pressure throughout the system and between the system and its surroundings is the same and does not change with time.
Chemical equilibrium: A system is said to be in chemical equilibrium when there are no chemical reactions going on within the system, or there is no transfer of matter from one part of the system to the other due to diffusion. In this state, the chemical composition of the system is the same throughout and does not change with time.

Q. 5. What are polar molecules and non polar molecules?

Answer:

Polar molecules: A molecule in which the centre of mass of positive charges (protons) does not coincide with the centre of mass of negative charges (electrons), because of the asymmetric shape of the molecules, is called a polar molecule. They possess a permanent electric dipole moment. Examples: $ \text{HCl}, \text{H}_2\text{O}, \text{NH}_3, \text{N}_2\text{O} $.
Non-polar molecules: A molecule in which the centre of mass of positive charges coincides with the centre of mass of negative charges is called a non-polar molecule. These molecules have symmetrical shapes and have zero dipole moment in the normal state. Examples: $ \text{H}_2, \text{N}_2, \text{O}_2, \text{CO}_2, \text{CH}_4 $.

Q. 6. Explain how will you convert moving coil galvanometer into voltmeter?

Answer:

(Note to student: Draw a neat circuit diagram showing a Moving Coil Galvanometer of resistance $ G $ connected in series with a high resistance $ X $ across potential difference $ V $, carrying full-scale deflection current $ I_g $.)

To convert a moving coil galvanometer (MCG) into a voltmeter, its effective resistance must be increased and its voltage measuring capacity must be increased to the desired value. This is achieved by connecting a very high resistance in series with the galvanometer.

Let $ G $ be the resistance of the galvanometer coil.
Let $ I_g $ be the current required for full-scale deflection of the galvanometer.
Let $ V $ be the maximum potential difference (voltage) to be measured.
Let $ X $ (or $ R $) be the high resistance connected in series with the galvanometer.

When resistance $ X $ is connected in series with the galvanometer, the total resistance of the circuit becomes $ (G + X) $. By Ohm's law, if $ V $ is the potential difference across the series combination when current $ I_g $ flows through it:

\[ V = I_g (G + X) \]

\[ \frac{V}{I_g} = G + X \]

\[ \mathbf{X = \frac{V}{I_g} - G} \]

This equation gives the value of the high series resistance required to convert a moving coil galvanometer into a voltmeter of range $ 0 $ to $ V $.

Q. 7. Distinguish between free and forced vibrations.

Answer:

Free Vibrations	Forced Vibrations
1. Free vibrations are produced when a body is disturbed from its equilibrium position and released to oscillate on its own.	1. Forced vibrations are produced by the continuous action of an external periodic force on a body.
2. The body vibrates with its own natural frequency, which depends on the intrinsic elastic and inertial properties of the body.	2. The body vibrates with a frequency equal to the frequency of the applied external periodic force.
3. No external periodic force is required to maintain the vibrations once initiated.	3. A continuous external periodic force is required to maintain the vibrations.
4. The amplitude of vibrations continuously decreases with time due to damping forces.	4. The amplitude of vibrations remains constant as long as the external periodic force acts on the body.

Q. 8. State any two postulates of Bohr's theory of hydrogen atom.

Answer:

Any two of the following postulates:

First Postulate (Circular Orbit): In a hydrogen atom, the electron revolves around the nucleus in a fixed circular orbit with constant speed. The necessary centripetal force is provided by the electrostatic force of attraction between the positively charged nucleus and the negatively charged electron: \[ \frac{m_e v^2}{r} = \frac{e^2}{4\pi\varepsilon_0 r^2} \]
Second Postulate (Quantization of Angular Momentum): The radius of the orbit of an electron can only take certain fixed values such that the angular momentum of the electron in these stationary orbits is an integral multiple of $ \frac{h}{2\pi} $, where $ h $ is Planck's constant: \[ L = m_e v r = \frac{nh}{2\pi} \quad (\text{where } n = 1, 2, 3, \dots) \]
Third Postulate (Energy Radiation): An electron does not radiate energy while revolving in a stationary orbit. Energy is radiated in the form of a photon only when the electron jumps from a higher energy orbit ($ E_n $) to a lower energy orbit ($ E_p $): \[ h\nu = E_n - E_p \]

Q. 9. Explain the cyclic process with the help of a neat p-V diagram.

Answer:

(Note to student: Draw a $ p\text{-}V $ diagram with Volume $ V $ on the X-axis and Pressure $ p $ on the Y-axis showing a closed loop formed by two paths between state $ A(p_1, V_1) $ and state $ B(p_2, V_2) $ with directional arrows indicating the cycle.)

Cyclic Process: A thermodynamic process that returns a system to its initial state is called a cyclic process.

In a cyclic process, the initial and the final states are the same. Since internal energy is a state variable, the total change in internal energy of the system for a cyclic process is zero ($ \Delta U = 0 $).
According to the first law of thermodynamics: $ Q = \Delta U + W $.
Since $ \Delta U = 0 $, we get $ \mathbf{Q = W} $. Thus, the total heat supplied to the system in a cyclic process is completely converted into work done by the system (or vice versa).
$ p\text{-}V $ Diagram Representation: On a pressure-volume ($ p\text{-}V $) diagram, a cyclic process is represented by a closed curve (loop), and the area enclosed by the closed curve gives the net work done during the cycle.

Q. 10. A star is emitting light at the wavelength of $ 4000\text{ \AA} $. Determine the limit of angular resolution of a telescope having an objective of diameter of $ 400\text{ cm} $.

Answer:

Given:
Wavelength of light ($ \lambda $) $ = 4000\text{ \AA} = 4000 \times 10^{-10}\text{ m} = 4 \times 10^{-7}\text{ m} $
Diameter of objective ($ D $) $ = 400\text{ cm} = 4\text{ m} $

Formula:
Limit of angular resolution: \[ d\theta = \frac{1.22 \lambda}{D} \]

Calculation: \[ d\theta = \frac{1.22 \times 4 \times 10^{-7}}{4} = 1.22 \times 10^{-7}\text{ rad} \]

The limit of angular resolution of the telescope is $ 1.22 \times 10^{-7}\text{ rad} $.

Q. 11. Calculate the value of $ \lambda_{\max} $ for solar radiation assuming the temperature on the surface of the sun is $ 5800\text{ K} $. [Wien's constant $ = 2.897 \times 10^{-3}\text{ m}\cdot\text{K} $]

Answer:

Given:
Temperature ($ T $) $ = 5800\text{ K} $
Wien's constant ($ b $) $ = 2.897 \times 10^{-3}\text{ m}\cdot\text{K} $

Formula:
According to Wien's displacement law: \[ \lambda_{\max} = \frac{b}{T} \]

Calculation: \[ \lambda_{\max} = \frac{2.897 \times 10^{-3}}{5800} = 0.00049948 \times 10^{-3}\text{ m} = 4.995 \times 10^{-7}\text{ m} \text{ (or } 4995\text{ \AA)} \]

The value of $ \lambda_{\max} $ for solar radiation is $ 4.995 \times 10^{-7}\text{ m} $ (or $ 4995\text{ \AA} $).

Q. 12. Calculate the radius of water drop, so that the excess pressure inside it is $ 24\text{ N/m}^2 $. [Surface tension of water $ = 7.2 \times 10^{-2}\text{ N/m} $]

Answer:

Given:
Excess pressure ($ \Delta P $) $ = 24\text{ N/m}^2 $
Surface tension ($ T $) $ = 7.2 \times 10^{-2}\text{ N/m} $

Formula:
Excess pressure inside a liquid drop: \[ \Delta P = \frac{2T}{r} \]

Calculation: \[ r = \frac{2T}{\Delta P} = \frac{2 \times 7.2 \times 10^{-2}}{24} = \frac{14.4 \times 10^{-2}}{24} = 0.6 \times 10^{-2}\text{ m} = 6 \times 10^{-3}\text{ m} \]

The radius of the water drop is $ 6 \times 10^{-3}\text{ m} $ (or $ 6\text{ mm} $).

Q. 13. The susceptibility of magnetic material is $ \chi $ at $ 27^\circ\text{C} $. At what temperature its susceptibility will be $ \chi/3 $?

Answer:

Given:
Initial susceptibility ($ \chi_1 $) $ = \chi $
Initial temperature ($ T_1 $) $ = 27^\circ\text{C} = 27 + 273 = 300\text{ K} $
Final susceptibility ($ \chi_2 $) $ = \frac{\chi}{3} $

Formula:
According to Curie's law: \[ \chi \propto \frac{1}{T} \implies \chi_1 T_1 = \chi_2 T_2 \]

Calculation: \[ T_2 = \frac{\chi_1 \times T_1}{\chi_2} = \frac{\chi \times 300}{\chi / 3} = 300 \times 3 = 900\text{ K} \] In Celsius: $ T_2 = 900 - 273 = 627^\circ\text{C} $

Its susceptibility will be $ \chi/3 $ at $ 900\text{ K} $ (or $ 627^\circ\text{C} $).

Q. 14. An alternating voltage source given by $ e = 140 \sin(314.2 t) $ is connected across a pure resistor of $ 50\text{ }\Omega $. Determine: (i) Frequency of the source. (ii) RMS value of emf.

Answer:

Given:
Equation of alternating voltage: $ e = 140 \sin(314.2 t) $
Comparing with the standard equation $ e = e_0 \sin(\omega t) $:
Peak emf ($ e_0 $) $ = 140\text{ V} $
Angular frequency ($ \omega $) $ = 314.2\text{ rad/s} $

(i) Frequency of the source ($ f $):
Formula: $ \omega = 2\pi f $ \[ f = \frac{\omega}{2\pi} = \frac{314.2}{2 \times 3.142} = \frac{314.2}{6.284} = 50\text{ Hz} \] Frequency of the source is $ 50\text{ Hz} $.

(ii) RMS value of emf ($ e_{\text{rms}} $):
Formula: \[ e_{\text{rms}} = \frac{e_0}{\sqrt{2}} = 0.707 \times e_0 = 0.707 \times 140 = 98.98\text{ V} \] RMS value of emf is $ 98.98\text{ V} $.

SECTION - C

Attempt any EIGHT questions of the following:

Q. 15. State and prove perpendicular axes theorem.

Answer:

Statement: The moment of inertia ($ I_z $) of a laminar object about an axis ($ Z $) perpendicular to its plane is the sum of its moments of inertia about two mutually perpendicular axes ($ X $ and $ Y $) in its plane, all the three axes being concurrent.

Mathematically: $ \mathbf{I_z = I_x + I_y} $

(Note to student: Draw a neat diagram of a plane lamina in the X-Y plane showing mutually perpendicular axes $ OX, OY $ in the plane and $ OZ $ perpendicular to the plane, with a mass element $ dm $ at point $ P(x, y) $ at perpendicular distances $ y $ from $ OX $, $ x $ from $ OY $, and $ r $ from $ OZ $.)

Proof:

Consider a rigid laminar object lying in the X-Y plane. Let $ OX $ and $ OY $ be two mutually perpendicular axes in the plane of the lamina, and $ OZ $ be the axis perpendicular to its plane intersecting at origin $ O $.
Let $ I_x, I_y, $ and $ I_z $ be the moments of inertia of the lamina about the $ X, Y, $ and $ Z $ axes respectively.
Consider an infinitesimal mass element $ dm $ situated at point $ P(x, y) $.
The perpendicular distance of $ P $ from the X-axis is $ y $, from the Y-axis is $ x $, and from the Z-axis is $ OP = r $.
By definitions of moment of inertia: \[ I_x = \int y^2 \, dm, \quad I_y = \int x^2 \, dm, \quad I_z = \int r^2 \, dm \]
From Pythagoras theorem in the right-angled triangle in the X-Y plane: \[ r^2 = x^2 + y^2 \]
Substituting $ r^2 = x^2 + y^2 $ in the expression for $ I_z $: \[ I_z = \int (x^2 + y^2) \, dm = \int y^2 \, dm + \int x^2 \, dm \]
Therefore, $ \mathbf{I_z = I_x + I_y} $.

Hence, the theorem of perpendicular axes is proved.

Q. 16. On the basis of molecular theory, explain the phenomenon of surface tension.

Answer:

(Note to student: Draw a neat diagram of a liquid in a container showing the free surface $ XY $, surface film $ XY\text{-}X'Y' $ of thickness equal to the molecular range, and three molecules $ A, B, $ and $ C $ along with their spheres of influence.)

The phenomenon of surface tension arises due to intermolecular cohesive forces and can be explained by considering three molecules $ A, B, $ and $ C $ in a liquid:

Molecule A (deep inside the liquid): The sphere of influence of molecule $ A $ lies completely inside the liquid. As a result, it is acted upon by equal cohesive forces in all directions. Therefore, the net cohesive force acting on molecule $ A $ is zero.
Molecule B (just below the free surface, within the surface film): A larger part of its sphere of influence lies inside the liquid and a smaller part lies in air. Since the number of air molecules is negligible compared to liquid molecules, the downward cohesive force due to liquid molecules is much stronger than the upward adhesive force. Hence, molecule $ B $ experiences a net downward force.
Molecule C (on the free surface of the liquid): Exactly half of its sphere of influence lies in the liquid and the upper half lies in air. Therefore, molecule $ C $ experiences the maximum net downward force directed into the liquid.

Conclusion: Thus, all molecules lying within the surface film experience a net inward (downward) pull. Whenever a molecule is brought from the interior of the liquid to the surface film, work has to be done against this inward force. This work is stored in the surface molecules in the form of potential energy. To achieve stable equilibrium with minimum potential energy, the free surface of the liquid tends to contract to have the minimum possible surface area and behaves like a stretched elastic membrane under tension. This phenomenon is called surface tension.

Q. 17. Show that all harmonics are present in the case of an air column vibrating in a pipe open at both the ends.

Answer:

(Note to student: Draw neat diagrams showing the first three modes of vibration of an air column in a pipe open at both ends: (a) Fundamental mode with 1 node and 2 antinodes, (b) First overtone with 2 nodes and 3 antinodes, and (c) Second overtone with 3 nodes and 4 antinodes.)

When a longitudinal stationary wave is set up in an air column inside a pipe open at both ends, antinodes are formed at both open ends and nodes are formed in between. Let $ L $ be the corrected length of the air column and $ v $ be the speed of sound in air.

Fundamental mode (First harmonic):
In the simplest mode of vibration, two antinodes are formed at the open ends and one node is formed at the centre.
Length of air column: $ L = \frac{\lambda_1}{2} \implies \lambda_1 = 2L $
Fundamental frequency ($ n $) is: \[ n = \frac{v}{\lambda_1} = \mathbf{\frac{v}{2L}} \quad \text{... (Equation 1)} \]
First overtone (Second harmonic):
In the next mode of vibration, three antinodes and two nodes are formed.
Length of air column: $ L = \lambda_2 \implies \lambda_2 = L $
Frequency of first overtone ($ n_1 $) is: \[ n_1 = \frac{v}{\lambda_2} = \frac{v}{L} = 2\left(\frac{v}{2L}\right) = \mathbf{2n} \quad \text{... (Equation 2)} \]
Second overtone (Third harmonic):
In this mode of vibration, four antinodes and three nodes are formed.
Length of air column: $ L = \frac{3\lambda_3}{2} \implies \lambda_3 = \frac{2L}{3} $
Frequency of second overtone ($ n_2 $) is: \[ n_2 = \frac{v}{\lambda_3} = \frac{3v}{2L} = 3\left(\frac{v}{2L}\right) = \mathbf{3n} \quad \text{... (Equation 3)} \]

Conclusion: From Equations 1, 2, and 3, the frequencies of the fundamental tone and overtones are in the ratio $ n : n_1 : n_2 : \dots = 1 : 2 : 3 : \dots $ Thus, both even and odd harmonics (i.e., all harmonics) are present as overtones in the vibrations of an air column in a pipe open at both ends.

Q. 18. What is interference of light? State any four conditions for obtaining well defined and steady interference pattern.

Answer:

Interference of light: The modification in the intensity of light (redistribution of light energy) produced by the superposition of two or more light waves from coherent sources is called interference of light.

Conditions for obtaining a well-defined and steady interference pattern (Any four):

The two sources of light must be coherent (i.e., they must maintain a constant phase difference).
The two sources of light must be monochromatic (having the same wavelength and frequency).
The two interfering waves must have the same amplitude (or equal intensity) for good contrast between bright and dark fringes.
The two light sources must be narrow.
The separation between the two light sources should be small, and the distance between the sources and the screen should be large.
The two interfering light waves must be in the same state of polarization.

Q. 19. With the help of a neat circuit diagram, explain the working of a half wave rectifier. Draw input and output waveforms.

Answer:

(Note to student: Draw (1) the circuit diagram of a half-wave rectifier showing AC mains connected to primary coil $ P_1P_2 $ of a transformer, secondary coil $ S_1S_2 $ connected in series with junction diode $ D $ and load resistor $ R_L $, and (2) the vertically aligned Input AC voltage waveform and Output pulsating DC voltage waveform plotted against time.)

Working of a Half Wave Rectifier:

The AC input voltage to be rectified is applied across the primary coil ($ P_1P_2 $) of a transformer. The secondary coil ($ S_1S_2 $) is connected in series with a p-n junction diode ($ D $) and a load resistance ($ R_L $).
During the positive half cycle of input AC: Terminal $ S_1 $ of the secondary coil is positive with respect to terminal $ S_2 $. Hence, the diode $ D $ is forward-biased and conducts electric current. Current flows through the load resistance $ R_L $ from $ A $ to $ B $, and an output voltage is obtained across $ R_L $.
During the negative half cycle of input AC: Terminal $ S_1 $ becomes negative with respect to terminal $ S_2 $. Hence, the diode $ D $ is reverse-biased and does not conduct any current. No current flows through $ R_L $, and the output voltage across $ R_L $ is zero.
Since the diode conducts only during the positive half cycles of the AC input, the current through the load resistance flows only in one direction. Thus, the circuit acts as a half-wave rectifier producing a pulsating DC output.

Q. 20. Obtain an expression for orbital magnetic moment of an electron revolving around the nucleus of an atom.

Answer:

(Note to student: Draw a circular orbit of radius $ r $ showing an electron revolving anticlockwise with linear speed $ v $, conventional current $ I $ clockwise, orbital angular momentum vector $ \vec{L} $ upwards, and orbital magnetic moment vector $ \vec{M}_o $ downwards.)

Consider an electron of mass $ m_e $ and charge $ e $ revolving with constant tangential speed $ v $ in a circular orbit of radius $ r $ around the positively charged nucleus.

The revolving electron constitutes a tiny circular current loop. The magnitude of current $ I $ associated with the revolving electron is:

\[ I = \frac{e}{T} \quad \text{... (Equation 1)} \]

Where $ T $ is the time period of revolution of the electron. Since $ T = \frac{2\pi r}{v} $, substituting this in Equation 1 gives:

\[ I = \frac{e}{2\pi r / v} = \frac{ev}{2\pi r} \quad \text{... (Equation 2)} \]

The area enclosed by the circular orbit is:

\[ A = \pi r^2 \quad \text{... (Equation 3)} \]

The orbital magnetic dipole moment ($ M_o $) associated with the current loop has a magnitude given by:

\[ M_o = I \times A \]

Substituting Equations 2 and 3:

\[ M_o = \left(\frac{ev}{2\pi r}\right) \times (\pi r^2) \]

\[ \mathbf{M_o = \frac{evr}{2}} \quad \text{... (Equation 4)} \]

Multiplying and dividing the right-hand side of Equation 4 by the mass of the electron ($ m_e $):

\[ M_o = \frac{e}{2m_e} (m_e v r) \]

Since the orbital angular momentum of the electron is $ L = m_e v r $:

\[ \mathbf{M_o = \frac{eL}{2m_e}} \]

In vector form, $ \mathbf{\vec{M}_o = -\frac{e}{2m_e} \vec{L}} $, showing that the orbital magnetic moment is directed opposite to the orbital angular momentum.

Q. 21. Define the terms: (a) Inductive reactance (b) Capacitive reactance (c) Impedance

Answer:

Inductive reactance ($ X_L $): The effective opposition offered by an inductor to the flow of alternating current in an AC circuit is called inductive reactance. It is given by $ X_L = \omega L = 2\pi f L $, and its SI unit is ohm ($ \Omega $).
Capacitive reactance ($ X_C $): The effective opposition offered by a capacitor to the flow of alternating current in an AC circuit is called capacitive reactance. It is given by $ X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C} $, and its SI unit is ohm ($ \Omega $).
Impedance ($ Z $): The total effective opposition offered by an AC circuit containing a resistor, an inductor, and a capacitor (or their combination) to the flow of alternating current is called impedance. It is the ratio of the RMS voltage to the RMS current, i.e., $ Z = \frac{e_{\text{rms}}}{i_{\text{rms}}} = \sqrt{R^2 + (X_L - X_C)^2} $, and its SI unit is ohm ($ \Omega $).

Q. 22. A parallel plate capacitor filled with air has an area $ 6\text{ cm}^2 $ and the plate separation is $ 2\text{ mm} $. Calculate its capacity.

Answer:

Given:
Area of each plate ($ A $) $ = 6\text{ cm}^2 = 6 \times 10^{-4}\text{ m}^2 $
Plate separation ($ d $) $ = 2\text{ mm} = 2 \times 10^{-3}\text{ m} $
Permittivity of free space ($ \varepsilon_0 $) $ = 8.85 \times 10^{-12}\text{ C}^2/\text{Nm}^2 $

Formula:
Capacity of an air-filled parallel plate capacitor: \[ C = \frac{\varepsilon_0 A}{d} \]

Calculation: \[ C = \frac{8.85 \times 10^{-12} \times 6 \times 10^{-4}}{2 \times 10^{-3}} = 8.85 \times 10^{-12} \times 3 \times 10^{-1} = 26.55 \times 10^{-13}\text{ F} = 2.655 \times 10^{-12}\text{ F} \]

The capacity of the parallel plate capacitor is $ 2.655 \times 10^{-12}\text{ F} $ (or $ 2.655\text{ pF} $).

Q. 23. Two resistances $ X $ and $ Y $ in the two gaps of a meterbridge give a null point dividing the wire in the ratio $ 4:6 $. If both resistances are increased by $ 30\text{ }\Omega $, the null point divides the wire in the ratio $ 5:6 $. Calculate the value of $ X $ and $ Y $.

Answer:

Given:
First balancing ratio: $ \frac{l_x}{l_y} = \frac{4}{6} = \frac{2}{3} $
Second balancing ratio (when $ X' = X + 30 $ and $ Y' = Y + 30 $): $ \frac{l'_x}{l'_y} = \frac{5}{6} $

Formula:
For a balanced meter bridge: \[ \frac{X}{Y} = \frac{l_x}{l_y} \]

Calculation:
Case 1: \[ \frac{X}{Y} = \frac{2}{3} \implies Y = 1.5 X \quad \text{... (Equation 1)} \]

Case 2: \[ \frac{X + 30}{Y + 30} = \frac{5}{6} \] \[ 6(X + 30) = 5(Y + 30) \] \[ 6X + 180 = 5Y + 150 \] \[ 5Y - 6X = 30 \quad \text{... (Equation 2)} \]

Substituting $ Y = 1.5 X $ from Equation 1 into Equation 2: \[ 5(1.5 X) - 6X = 30 \implies 7.5 X - 6X = 30 \implies 1.5 X = 30 \implies X = \frac{30}{1.5} = 20\text{ }\Omega \]

Substituting $ X = 20\text{ }\Omega $ in Equation 1: \[ Y = 1.5 \times 20 = 30\text{ }\Omega \]

The values of the two resistances are $ X = 20\text{ }\Omega $ and $ Y = 30\text{ }\Omega $.

Q. 24. Calculate the shortest wavelength of Paschen series and the longest wavelength of Balmer series for hydrogen atom.

Answer:

Given:
Rydberg's constant ($ R $) $ = 1.097 \times 10^7\text{ m}^{-1} $

Formula: \[ \frac{1}{\lambda} = R \left(\frac{1}{p^2} - \frac{1}{n^2}\right) \]

1. Shortest wavelength of Paschen series ($ \lambda_s $):
For the Paschen series, $ p = 3 $, and for the shortest wavelength (series limit), $ n = \infty $. \[ \frac{1}{\lambda_s} = R \left(\frac{1}{3^2} - \frac{1}{\infty^2}\right) = \frac{R}{9} \] \[ \lambda_s = \frac{9}{R} = \frac{9}{1.097 \times 10^7} = 8.204 \times 10^{-7}\text{ m} \text{ (or } 8204\text{ \AA)} \]

2. Longest wavelength of Balmer series ($ \lambda_l $):
For the Balmer series, $ p = 2 $, and for the longest wavelength, $ n = 3 $. \[ \frac{1}{\lambda_l} = R \left(\frac{1}{2^2} - \frac{1}{3^2}\right) = R \left(\frac{1}{4} - \frac{1}{9}\right) = \frac{5R}{36} \] \[ \lambda_l = \frac{36}{5R} = \frac{36}{5 \times 1.097 \times 10^7} = \frac{36}{5.485 \times 10^7} = 6.563 \times 10^{-7}\text{ m} \text{ (or } 6563\text{ \AA)} \]

The shortest wavelength of the Paschen series is $ 8.204 \times 10^{-7}\text{ m} $ ($ 8204\text{ \AA} $) and the longest wavelength of the Balmer series is $ 6.563 \times 10^{-7}\text{ m} $ ($ 6563\text{ \AA} $).

Q. 25. Calculate the current flowing through two long parallel wires carrying equal currents and separated by a distance of $ 1.35\text{ cm} $ experiencing a force per unit length of $ 4.76 \times 10^{-2}\text{ N/m} $.

Answer:

Given:
Distance between the wires ($ d $) $ = 1.35\text{ cm} = 1.35 \times 10^{-2}\text{ m} $
Force per unit length ($ \frac{F}{L} $) $ = 4.76 \times 10^{-2}\text{ N/m} $
Permeability of free space ($ \mu_0 $) $ = 4\pi \times 10^{-7}\text{ T}\cdot\text{m/A} $
Currents in the wires: $ I_1 = I_2 = I $

Formula: \[ \frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d} = \frac{\mu_0 I^2}{2\pi d} \]

Calculation: \[ 4.76 \times 10^{-2} = \frac{4\pi \times 10^{-7} \times I^2}{2\pi \times 1.35 \times 10^{-2}} = \frac{2 \times 10^{-7} \times I^2}{1.35 \times 10^{-2}} \] \[ I^2 = \frac{4.76 \times 10^{-2} \times 1.35 \times 10^{-2}}{2 \times 10^{-7}} = 3.213 \times 10^3 = 3213 \] \[ I = \sqrt{3213} = 56.68\text{ A} \]

The current flowing through each wire is $ 56.68\text{ A} $.

Q. 26. An emf of $ 90\text{ mV} $ is induced in the winding of a coil, when current in a nearby coil is increasing at the rate of $ 4.5\text{ A/s} $. What is the mutual inductance of the two coils?

Answer:

Given:
Induced emf ($ e $) $ = 90\text{ mV} = 90 \times 10^{-3}\text{ V} $
Rate of change of current ($ \frac{dI}{dt} $) $ = 4.5\text{ A/s} $

Formula: \[ e = M \frac{dI}{dt} \]

Calculation: \[ M = \frac{e}{dI / dt} = \frac{90 \times 10^{-3}}{4.5} = 20 \times 10^{-3}\text{ H} = 20\text{ mH} \]

The mutual inductance of the two coils is $ 20 \times 10^{-3}\text{ H} $ (or $ 20\text{ mH} $).

SECTION - D

Attempt any THREE questions of the following:

Q. 27. Derive an expression for pressure exerted by a gas on the basis of kinetic theory of gases.

Answer:

(Note to student: Draw a neat diagram of a cubical box of side $ L $ along the $ X, Y, $ and $ Z $ axes with a shaded wall perpendicular to the X-axis and a gas molecule moving with velocity $ \vec{v} $.)

Consider an ideal gas enclosed in a cubical container of side $ L $ and volume $ V = L^3 $. Let $ N $ be the total number of gas molecules, each of mass $ m $.

Let a molecule move with velocity $ v $ having components $ v_x, v_y, $ and $ v_z $ along the $ X, Y, $ and $ Z $ axes respectively. Therefore, $ v^2 = v_x^2 + v_y^2 + v_z^2 $.

Change in Momentum: When this molecule collides elastically with the shaded wall perpendicular to the X-axis, its X-component of velocity reverses to $ -v_x $ while $ v_y $ and $ v_z $ remain unchanged.
Initial momentum along X-axis $ = m v_x $
Final momentum along X-axis $ = -m v_x $
Change in momentum of the molecule ($ \Delta p_x $) $ = -m v_x - m v_x = -2m v_x $
Therefore, momentum imparted to the wall during one collision $ = +2m v_x $.
Time between successive collisions: The molecule travels a distance of $ 2L $ (to the opposite wall and back) before colliding with the same wall again. The time interval between two successive collisions with the same wall is: \[ \Delta t = \frac{2L}{v_x} \]
Average force exerted by one molecule: By Newton's second law, the average force exerted by this single molecule on the wall is: \[ f_{x1} = \frac{\Delta p_x}{\Delta t} = \frac{2m v_{x1}}{2L / v_{x1}} = \frac{m v_{x1}^2}{L} \]
Total force along X-axis: The total force exerted by all $ N $ molecules on the wall is: \[ F_x = \frac{m}{L} (v_{x1}^2 + v_{x2}^2 + \dots + v_{xN}^2) \]
Pressure along X-axis: Pressure is force per unit area ($ \text{Area} = L^2 $): \[ P = \frac{F_x}{L^2} = \frac{m}{L^3} (v_{x1}^2 + v_{x2}^2 + \dots + v_{xN}^2) \] Since $ V = L^3 $ and mean square velocity component $ \langle v_x^2 \rangle = \frac{v_{x1}^2 + v_{x2}^2 + \dots + v_{xN}^2}{N} $: \[ P = \frac{Nm}{V} \langle v_x^2 \rangle \]
Symmetry of motion: Since the gas molecules move randomly in all directions, by symmetry: \[ \langle v_x^2 \rangle = \langle v_y^2 \rangle = \langle v_z^2 \rangle \] Since $ \langle v^2 \rangle = \langle v_x^2 \rangle + \langle v_y^2 \rangle + \langle v_z^2 \rangle = 3\langle v_x^2 \rangle $, we get: \[ \langle v_x^2 \rangle = \frac{1}{3} \langle v^2 \rangle = \frac{1}{3} v_{\text{rms}}^2 \]
Final Expression: Substituting $ \langle v_x^2 \rangle $ into the pressure equation: \[ \mathbf{P = \frac{1}{3} \frac{Nm}{V} v_{\text{rms}}^2} \] Since total mass of the gas $ M = Nm $ and density $ \rho = \frac{M}{V} $: \[ \mathbf{P = \frac{1}{3} \rho v_{\text{rms}}^2} \]

Q. 28. State differential equation of linear S.H.M. hence obtain an expression for acceleration, velocity and displacement of a particle performing S.H.M.

Answer:

Differential Equation of Linear S.H.M.: \[ \mathbf{\frac{d^2x}{dt^2} + \omega^2 x = 0} \] (where $ x $ is the displacement from the mean position at time $ t $, and $ \omega = \sqrt{\frac{k}{m}} $ is the angular frequency).

1. Expression for Acceleration ($ a $):
From the differential equation of linear S.H.M.: \[ \frac{d^2x}{dt^2} = -\omega^2 x \] Since $ \frac{d^2x}{dt^2} $ is the acceleration ($ a $) of the particle: \[ \mathbf{a = -\omega^2 x} \] (The negative sign shows that acceleration is directed opposite to displacement).

2. Expression for Velocity ($ v $):
Acceleration can be written as: \[ a = \frac{dv}{dt} = \frac{dv}{dx} \cdot \frac{dx}{dt} = v \frac{dv}{dx} \] Substituting $ a = -\omega^2 x $: \[ v \frac{dv}{dx} = -\omega^2 x \implies v \, dv = -\omega^2 x \, dx \] Integrating both sides: \[ \int v \, dv = -\omega^2 \int x \, dx \] \[ \frac{v^2}{2} = -\frac{\omega^2 x^2}{2} + C \quad \text{... (Equation 1)} \] Where $ C $ is the constant of integration. At the extreme position, $ x = \pm A $ and $ v = 0 $: \[ 0 = -\frac{\omega^2 A^2}{2} + C \implies C = \frac{\omega^2 A^2}{2} \] Substituting the value of $ C $ in Equation 1: \[ \frac{v^2}{2} = -\frac{\omega^2 x^2}{2} + \frac{\omega^2 A^2}{2} \implies v^2 = \omega^2(A^2 - x^2) \] \[ \mathbf{v = \pm \omega \sqrt{A^2 - x^2}} \]

3. Expression for Displacement ($ x $):
Since $ v = \frac{dx}{dt} $, we can write: \[ \frac{dx}{dt} = \omega \sqrt{A^2 - x^2} \implies \frac{dx}{\sqrt{A^2 - x^2}} = \omega \, dt \] Integrating both sides: \[ \int \frac{dx}{\sqrt{A^2 - x^2}} = \int \omega \, dt \] \[ \sin^{-1}\left(\frac{x}{A}\right) = \omega t + \phi \] (where $ \phi $ is the constant of integration representing the initial phase or epoch) \[ \frac{x}{A} = \sin(\omega t + \phi) \] \[ \mathbf{x = A \sin(\omega t + \phi)} \]

Q. 29. With the help of neat diagram explain construction of a transformer. Calculate energy required to build up current of $ 1\text{ A} $ in an inductor of inductance $ 20\text{ mH} $.

Answer:

Construction of a Transformer:

(Note to student: Draw a neat labeled diagram of a transformer showing the laminated soft iron core, primary coil $ P $ with input AC source, and secondary coil $ S $ with load impedance.)

A transformer consists of two sets of insulated copper coils, known as the primary coil ($ P $) and the secondary coil ($ S $), wound on a soft iron core.
Soft Iron Core: The core is constructed from thin rectangular laminated sheets (stampings) of soft iron insulated from each other by a layer of varnish. Soft iron offers high magnetic permeability and reduces hysteresis loss, while the laminations reduce eddy current losses.
Primary Coil ($ P $): The coil to which the alternating input voltage source is connected is called the primary coil. Let $ N_p $ be the number of turns in the primary coil.
Secondary Coil ($ S $): The coil across which the transformed alternating output voltage is obtained is called the secondary coil. Let $ N_s $ be the number of turns in the secondary coil.
The primary and secondary coils are electrically insulated from each other and from the iron core, but they are magnetically coupled through the core.

Numerical — Calculation of Energy in Inductor:
Given:
Inductance ($ L $) $ = 20\text{ mH} = 20 \times 10^{-3}\text{ H} $
Current ($ I $) $ = 1\text{ A} $

Formula:
Energy stored in the magnetic field of an inductor: \[ U = \frac{1}{2} L I^2 \]

Calculation: \[ U = \frac{1}{2} \times 20 \times 10^{-3} \times (1)^2 = 10 \times 10^{-3}\text{ J} = 10\text{ mJ} \]

The energy required to build up the current is $ 10 \times 10^{-3}\text{ J} $ (or $ 10\text{ mJ} $).

Q. 30. Draw a neat schematic diagram of a photo cell. The photoelectric work function for a metal is $ 4.2\text{ eV} $. Find threshold wavelength.

Answer:

Schematic Diagram of a Photo Cell:
(Note to student: Draw a neat labeled schematic diagram of a photo cell showing and labeling: (1) Evacuated glass/quartz bulb, (2) Semi-cylindrical photosensitive cathode/emitter $ C $, (3) Wire anode/collector $ A $, (4) Incident light/UV radiation, and (5) External circuit containing a microammeter ($ \mu\text{A} $) and DC source/battery.)

Numerical — Calculation of Threshold Wavelength ($ \lambda_0 $):
Given:
Photoelectric work function ($ \phi_0 $) $ = 4.2\text{ eV} = 4.2 \times 1.6 \times 10^{-19}\text{ J} = 6.72 \times 10^{-19}\text{ J} $
Planck's constant ($ h $) $ = 6.63 \times 10^{-34}\text{ J}\cdot\text{s} $
Speed of light ($ c $) $ = 3 \times 10^8\text{ m/s} $

Formula: \[ \phi_0 = \frac{hc}{\lambda_0} \implies \lambda_0 = \frac{hc}{\phi_0} \]

Calculation: \[ \lambda_0 = \frac{6.63 \times 10^{-34} \times 3 \times 10^8}{6.72 \times 10^{-19}} = \frac{19.89 \times 10^{-26}}{6.72 \times 10^{-19}} = 2.9598 \times 10^{-7}\text{ m} \approx 2.96 \times 10^{-7}\text{ m} \text{ (or } 2960\text{ \AA)} \]

The threshold wavelength for the metal is $ 2.96 \times 10^{-7}\text{ m} $ (or $ 2960\text{ \AA} $).

Q. 31. Obtain an expression for magnetic induction of a toroid of $ N $ no. of turns about an axis passing through its centre and perpendicular to plane. A Carnot refrigerator operates between $ 150\text{ K} $ and $ 200\text{ K} $, calculate its coefficient of performance.

Answer:

Expression for Magnetic Induction of a Toroid:

(Note to student: Draw a neat diagram of a toroid carrying current $ I $ showing the concentric circular Amperian loop of radius $ r $ along its central axis and the tangential magnetic field $ \vec{B} $.)

A toroid is a hollow circular ring of finite thickness on which a large number ($ N $) of insulated wire turns are closely wound.
Let $ I $ be the current flowing through the toroidal winding. To find the magnetic induction ($ B $) inside the toroid at a distance $ r $ from its centre $ O $, consider a concentric circular Amperian loop of radius $ r $ passing through the centre of the turns.
By symmetry, the magnitude of the magnetic field $ B $ is the same at all points on the Amperian loop and is directed tangentially along the loop at every point (i.e., angle $ \theta $ between $ \vec{B} $ and $ d\vec{l} $ is $ 0^\circ $).
Applying Ampere's Circuital Law to the Amperian loop: \[ \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enclosed}} \]
Here, $ \oint \vec{B} \cdot d\vec{l} = \oint B \, dl \cos 0^\circ = B \oint dl = B (2\pi r) $.
Since there are $ N $ turns and each turn carries current $ I $, the total current enclosed by the Amperian loop is: \[ I_{\text{enclosed}} = NI \]
Substituting these into Ampere's Circuital Law: \[ B (2\pi r) = \mu_0 NI \] \[ \mathbf{B = \frac{\mu_0 NI}{2\pi r}} \]
If $ n $ is the number of turns per unit length of the toroid ($ n = \frac{N}{2\pi r} $), then: \[ \mathbf{B = \mu_0 n I} \]

Numerical — Coefficient of Performance of Carnot Refrigerator:
Given:
Temperature of cold reservoir ($ T_C $) $ = 150\text{ K} $
Temperature of hot reservoir ($ T_H $) $ = 200\text{ K} $

Formula:
Coefficient of performance: \[ K = \frac{T_C}{T_H - T_C} \]

Calculation: \[ K = \frac{150}{200 - 150} = \frac{150}{50} = 3 \]

The coefficient of performance of the Carnot refrigerator is $ 3 $.

HSC 12th History June 2026 Board Paper Solution (J-270) | Maharashtra Board

aaminomtex@gmail.com (Amin) — Sat, 24 Oct 2026 23:47:38 +0530

Welcome to Omtex Classes! Are you preparing for your upcoming Maharashtra State Board 12th Arts exams? Practicing official board question papers is the best strategy to score high marks in History. Below is the complete, point-by-point model solution for the HSC 12th Standard History June 2026 Board Paper (Subject Code: J-270). We have solved all questions—including every option in the Concept Maps, Short Notes, Give Reasons, State Your Opinion, and Answer in Detail sections—strictly following the Maharashtra State Board (Balbharati) textbook and official evaluation scheme.

Q. 1. (A) Choose the correct alternatives and rewrite the statements : (6 Marks)

The first explorer to circumnavigate the African continent was (c) Bartholomew Dias.
The trading licence to do trade with Eastern countries was given to the company by (b) Queen Elizabeth I of England.
The political relations between Chhatrapati Shivaji Maharaj and the British began due to the (b) Afzal Khan episode.
Hyderabad's struggle of freedom was led skillfully by (b) Swami Ramanand Tirth.
H.S. Williams was the first person to think of (d) African unity.
A legal agreement was signed in Geneva in 1947 which is known as (c) GATT.

Q. 1. (B) Find the incorrect pairs from every set in group 'B' and write the correct ones : (4 Marks)

(i) Set 1:

Incorrect Pair: (d) Declaration of Independence of America - George Washington
Correct Pair: (d) Declaration of Independence of America - Thomas Jefferson

(ii) Set 2:

Incorrect Pair: (d) Chauthai - One tenth income of the tax payer
Correct Pair: (d) Chauthai - One fourth of the total income of the tax payer

(iii) Set 3:

Incorrect Pair: (a) Bandung - Berlin Conference
Correct Pair: (a) Bandung - Bandung Conference

(iv) Set 4:

Incorrect Pair: (b) Bhau Katdare - Raigad Nisarg Mitra
Correct Pair: (b) Bhau Katdare - Sahyadri Nisarg Mitra

Q. 2. (A) Write the names of historical places / persons / events / countries : (4 Marks)

The treatise written by Varahmihir - Brihatsamhita
Charles II, King of England was engaged to - Princess Braganza (Princess Catherine of Braganza of Portugal)
The name of the group of nations including England, France, Russia, Italy and America in the First World War - Allied Powers (Allies)
The idea of Commonwealth Games was introduced by - Reverend Astley Cooper

Q. 2. (B) Choose the correct reasons and complete the sentences : (4 Marks)

The British wanted to gain control over Myanmar, because (b) it was very important for them to control the natural resources and market in Myanmar.
King Hari Singh asked India for help, because (a) to protect Kashmir from armed intruders from the Pakistan border.
United Nations Organization was established after the Second World War, (c) to maintain peace in all over the world.
India became the member of World Trade Organization, because (c) India adopted the policy of liberalization.

Q. 3. (A) Observe the given map and answer the questions based on it : (5 Marks)

What information does this map give?
Ans: The given map provides information about the Freedom Struggle of 1857, showing the territories under British control and the important centres of the Freedom Struggle of 1857 across India.
Name the important centres of freedom struggle located in present Maharashtra.
Ans: Nagpur, Aurangabad, Satara, and Kolhapur are the important centres of the freedom struggle of 1857 located in present-day Maharashtra.
Name the important centres of freedom struggle located in present Pakistan.
Ans: Peshawar, Lahore, Shikarpur, and Karachi are the important centres of the freedom struggle of 1857 located in present-day Pakistan.
Name the neighbouring country situated in southern part of India.
Ans: Sri Lanka is the neighbouring country situated in the southern part of India.
Name the island situated in Arabian Sea.
Ans: Lakshadweep Islands are situated in the Arabian Sea.

Q. 3. (B) Complete the concept maps (Any FOUR - All 6 Solved) : (8 Marks)

(i) European powers established colonies in India:

European Powers Established Colonies in India
1. Portuguese	2. Dutch	3. British	4. French

(ii) Social reformers and their organisations:

Social Reformer	Organisation
1. Maharshi Vitthal Ramji Shinde	Depressed Classes Mission
Dr. Atmaram Pandurang	2. Prarthana Samaj
Sir Sayyad Ahmad Khan	3. Mohammedan Anglo-Oriental College (Scientific Society)
4. Dr. Babasaheb Ambedkar	Bahishkrut Hitkarni Sabha

(iii) Persons who participated in the liberation movement of Dadra and Nagar-Haveli:

Persons who Participated in the Liberation Movement of Dadra and Nagar-Haveli
1. Vishwanath Lavande	2. Rajabhau Wakankar	3. Sudhir Phadke	4. Nana Kajrekar

(iv) Founder countries of Non-Aligned Movement (NAM):

Founder Countries of Non-Aligned Movement (NAM)
Yugoslavia (Given)	1. India	2. Egypt	3. Indonesia	4. Ghana

(v) Cities joined through Golden Quadrilateral (Suvarna Chatushkon Yojana):

Cities Joined through Golden Quadrilateral
1. Delhi	2. Mumbai	3. Chennai	4. Kolkata

(vi) Sponsorers working for eradication of Polio:

Sponsorers Working for Eradication of Polio
1. Government of India	2. World Health Organization (WHO)	3. UNICEF	4. Rotary International

Q. 4. (A) Write short notes (Any THREE - All 5 Solved) : (6 Marks)

(i) Cartaz

Meaning and Origin: Once the Portuguese established firm naval control over the Indian Ocean in the 16th century, they made it compulsory for Indian rulers and merchants to purchase a naval license from them before sailing on the Indian Ocean. This license document was called a 'Cartaz'.
Details Included: A Cartaz generally carried details such as the name of the vessel (ship), the name of the captain of the ship, the port of departure and arrival, and the list of weaponry carried on the ship for self-defence.
Consequences: If any Indian ship ventured into the sea without a Portuguese Cartaz, the Portuguese navy either seized the ship by force or sank it. The Portuguese navy was so powerful that even the mighty Mughals and the Sultanates in the south had to buy a Cartaz from them.

(ii) Vykom Satyagraha

Background: In Travancore (Kerala), people belonging to the scheduled castes (untouchables) were prohibited not only from entering the Shiva temple at Vykom (Vaikom) but also from walking on the public streets surrounding the temple.
Participation of Ramaswamy Naikar (Periyar) and Leaders: In 1924, an anti-untouchability movement known as the Vykom Satyagraha was launched by leaders like T. K. Madhavan, K. P. Kesava Menon, and K. Kelappan. Prominent social reformer Ramaswamy Naikar (Periyar) actively participated in the Vykom Satyagraha in Travancore and fought tirelessly against untouchability, earning the title 'Vaikom Veerar' (Hero of Vaikom).
Outcome: Mahatma Gandhi also visited Vykom in 1925 to support the satyagrahis and negotiated with the authorities. Eventually, the public roads around the temple were opened to all castes, marking a historic victory against untouchability.

(iii) Anusheelan Samiti

Foundation and Leaders: Anusheelan Samiti was an active secret revolutionary organization in Bengal that fought against British colonial rule. Aurobindo Ghosh and his brother Barindrakumar Ghosh were at the head of this organization, while Pramathanath Mitra served as its president.
Bomb-Making Centre at Maniktala: The organization had more than 500 branches and operated a secret bomb-manufacturing centre at Maniktala near Kolkata.
Assassination Attempt on Kingsford (1908): In 1908, two young revolutionaries of the Anusheelan Samiti, Khudiram Bose and Prafulla Chaki, planned to assassinate an oppressive British magistrate named Kingsford at Muzaffarpur. However, the horse carriage on which they threw the bomb did not carry Kingsford, but two British women instead, who both died. Prafulla Chaki shot himself to avoid capture by the British police, while Khudiram Bose was arrested and hanged.

(iv) Cold War

Concept: Following the end of the Second World War in 1945, an era of intense ideological conflict, political distrust, and military rivalry began between two superpowers—the capitalist United States of America (USA) and the communist Soviet Union (USSR). Because there was no direct armed war fought between the two superpowers, American journalist Walter Lippmann termed this period the 'Cold War'.
Division of the World and Arms Race: Europe and the world were divided into two rival military camps—NATO (1949) led by the USA, and the Warsaw Pact (1955) led by Soviet Russia. The Cold War witnessed a dangerous nuclear arms race, space race, espionage, and proxy wars.
End of the Cold War (1991): In the late 1980s, Soviet President Mikhail Gorbachev introduced liberal policies of Perestroika (restructuring) and Glasnost (openness). With the disintegration of the Soviet Union in 1991, the Cold War came to an end.

(v) AYUSH

Establishment: In November 2014, the Government of India established the independent Ministry of AYUSH for the development and propagation of traditional Indian and alternative systems of medicine.
Full Form: The acronym AYUSH stands for Ayurveda, Yoga and Naturopathy, Unani, Siddha, and Homeopathy.
Objectives: The department works to upgrade educational standards in Indian systems of medicine, promote advanced research in medicinal plants, standardize drug quality control, and provide affordable, holistic healthcare to citizens.

Q. 4. (B) Explain the following statements with reasons (Any THREE - All 5 Solved) : (9 Marks)

(i) The incident known as 'Boston Tea Party' happened during the American War of Independence.

The British Parliament imposed several unjust restrictions and heavy taxes—such as the Stamp Act (1765) and the Townshend Acts—on the 13 American colonies without giving the colonists any representation in the British Parliament.
The American colonies strongly resisted this economic exploitation, raised the slogan "No Taxation without Representation", and boycotted British goods.
In 1773, the British Parliament gave the East India Company the monopoly to sell tea to the American colonies. To protest against this monopoly and tax on tea, local residents in Boston disguised as Native Americans boarded British merchant ships at Boston port in 1773 and dumped boxes of tea into the sea. This historic protest event is known as the 'Boston Tea Party'.

(ii) The 'Carnatic Wars' were fought between British and French.

England and France were arch-rivals in Europe and competed fiercely with each other to establish a monopoly over Indian trade and political supremacy in India.
Whenever war broke out between England and France in Europe, the British East India Company and the French East India Company clashed with each other in India as well.
Taking advantage of internal family disputes for the throne in the Carnatic region and Hyderabad, both the British and the French intervened on opposite sides to increase their political dominance. This led to three battles between them known as the 'Carnatic Wars' (1746–1763), in which the British ultimately defeated the French in the Third Carnatic War (Battle of Wandiwash, 1760).

(iii) The Second World War occurred.

Humiliating Treaty of Versailles: After the First World War, the victorious Allied nations imposed the harsh Treaty of Versailles (1919) on defeated Germany, seizing its colonies, reducing its armed forces, and imposing huge war reparations, which fueled intense resentment in Germany.
Rise of Militarism and Dictatorships: The rise of Nazism under Adolf Hitler in Germany and Fascism under Benito Mussolini in Italy promoted aggressive nationalism and expansionism. Meanwhile, Japan invaded Manchuria and China, and Italy attacked Ethiopia.
Failure of the League of Nations and Attack on Poland: The League of Nations proved too weak to stop the aggressive expansion of Germany, Italy, and Japan (the Axis Powers). Finally, when Germany invaded Poland on 1st September 1939, England and France declared war on Germany, sparking the Second World War.

(iv) The British granted independence to Myanmar in 1948.

The British had annexed Myanmar (Burma) after fighting three Anglo-Burmese Wars in the 19th century and brought its natural wealth and forest resources under British control.
During the Second World War, Japan captured Myanmar. Burmese nationalist leader Aung San initially raised the 'Burma Independence Army', but later realizing Japan's oppressive rule, he turned against the Japanese and helped the British recapture Myanmar in 1945.
After the Second World War, the British realized that they could no longer maintain their colonial hold in the face of strong nationalist resistance led by Aung San and the people of Myanmar. Therefore, the British granted independence to Myanmar on 4th January 1948.

(v) The Speed Post service has become more reliable and speedy.

The Indian Postal Department started the 'Speed Post' service in 1986, which transformed the nature of postal delivery across the country.
Under this service, more than three crore letters and parcels are delivered every month. When a customer posts a consignment, an SMS notification is sent to their mobile phone, and the customer can easily track the real-time location of their letter or parcel on the internet until final delivery.
Additionally, modern facilities such as online postage payment, custom packaging, free pick-up for bulk corporate customers, "Book Now, Pay Later", and passport delivery have made Speed Post extremely reliable and speedy.

Q. 5. State your opinion on the following statements (Any THREE - All 5 Solved) : (9 Marks)

(i) The European natural scientists of the 17th century laid the foundation of modern science.

Ans: I agree with this statement because the natural scientists of the 17th century revolutionized human thought:

Emphasis on Empirical Observation: Scientists of the 17th century broke away from medieval religious dogmas and emphasized proving scientific principles through actual empirical experiments (observation and experimentation) that remain valid regardless of time and space.
Formulation of Scientific Laws and Terminology: They converted newly discovered scientific rules into universal mathematical and scientific formulae and created a new scientific terminology.
Inventions and Scientific Institutions: Pioneers like Galileo, Kepler, and Newton advanced astronomy, physics, and mathematics, while institutions like the Royal Society (London) and Academy of Sciences (Paris) published research journals and encouraged scientific inquiry, thereby laying the solid foundation of modern science.

(ii) Portuguese established colonies in India.

Ans: Yes, the Portuguese were the first European colonial power to establish colonies in India:

Following Vasco da Gama's arrival at Calicut in 1498, the Portuguese took advantage of the lack of unity among Indian kings and the absence of navies among local rulers to establish their strongholds along the Indian coast.
By 1608, the Portuguese had established colonies on the western coast of India at Diu, Daman, Chaul, Goa (including Sashti and Bardesh), Honnavar, Gangoli, Basrur, Mangalore, Kannur, Kodungallur, Kochi, and Kollam, and on the eastern coast at Nagapattinam, Mylapore (San Thome), and Hugli in Bengal.
Goa served as the capital of the Portuguese empire in the East, headed by a Viceroy and Commander-in-Chief (Capitao-i-Geral), and they maintained control over the Indian Ocean through their strong navy and the Cartaz system.

(iii) The policies adopted by Maharaja Sayajirao Gaikwad are considered progressive.

Ans: The policies implemented by Maharaja Sayajirao Gaikwad of Baroda Princely State were truly progressive and ahead of his time:

Free and Compulsory Education and Support to Untouchables: He made primary education free and compulsory in Baroda State, opened schools and hostels for untouchable and tribal students, and provided financial scholarships to deserving students like Dr. Babasaheb Ambedkar for higher studies.
Progressive Social Laws: He legally banned child marriage and polygamy, legalized widow remarriage, enacted legislation granting women the right to divorce and a share in their father's and husband's property, and encouraged inter-caste dining.
Village Administration and Libraries: He passed a resolution that every village in his state must have a Gram Panchayat and a library, and started physical training, technical schools, and fine arts institutions.

(iv) The process of decolonisation began in the African continent.

Ans: The process of decolonisation in the African continent in the 20th century liberated millions of people from European colonial exploitation:

During the 19th century, European imperial powers had partitioned the entire African continent among themselves at the Berlin Conference (1884–85) to exploit its minerals, natural resources, and people.
The spread of Western education in Africa produced nationalist leaders who became aware of democratic values and self-rule. Moreover, the five Pan-African Congresses (from 1919 in Paris to the 1945 African Unity Conference at Manchester) united African leaders against colonialism.
After the Second World War weakened European imperial powers like Britain and France, independence movements intensified across Africa, leading to the gradual decolonisation and independence of African nations one after another between the 1950s and 1990s.

(v) Tourism industry is a continuously developing field.

Ans: I strongly agree that the tourism industry in India is a continuously developing and dynamic sector:

Rich Cultural and Natural Heritage: India is blessed with ancient temples, forts, caves, UNESCO World Heritage sites, pilgrimage centres, hill stations, coastlines, and biodiversity hotspots that attract millions of domestic and international tourists.
Institutional Support and Facilities: In 1966, the Government of India established the Indian Tourism Development Corporation (ITDC) to provide quality hotels, transport, duty-free shops, and entertainment facilities for tourists. Campaigns like "Incredible India" and "Atithi Devo Bhava" have boosted global tourism to India.
Economic Growth and New Avenues: Modern developments such as heritage walks, eco-tourism, agro-tourism, medical tourism, and adventure sports have made tourism a major source of foreign exchange and employment for guides, translators, hotel staff, transport operators, and local artisans.

Q. 6. Answer the following questions in detail (Any TWO - All 3 Solved) : (10 Marks)

(i) Explain in detail about the causes of the rise of colonialism.

Ans: Colonialism refers to the occupation of the land of a less developed distant nation by a powerful developed nation to establish political and economic control over it. The major causes of the rise of colonialism are as follows:

1. Industrial Revolution: The Industrial Revolution began in the 18th century. With the use of machines, production increased manifold. Domestic markets in European countries were not sufficient to consume the massive surplus of manufactured goods, compelling Europeans to find new overseas markets.
2. Requirement of Raw Materials: To keep their factories running continuously and to compete with rival nations, European industrial nations needed a steady, cheap supply of raw materials (such as cotton, iron, coal, and rubber). Establishing direct control over colonies guaranteed a monopoly over raw materials.
3. Investing Surplus Capital: The capitalist class in Europe accumulated immense profits from trade and industrialization. Since markets in Asian and African countries were less developed, European capitalists looked for secure colonies where they could invest their surplus capital profitably.
4. Sources of Minerals: Countries in Asia and Africa had rich natural deposits of gold, diamonds, silver, coal, and iron ore, which attracted European powers to colonize these regions.
5. Geographic Importance: European nations realized that certain strategic ports and islands in Asia and Africa—such as Malta, Gibraltar, Aden, Singapore, and Andaman-Nicobar—were vital for controlling international sea trade routes. Hence, they conquered these strategic locations.
6. Availability of Labour: European merchants needed large numbers of labourers at very low wages. As the slave trade declined, establishing colonies in densely populated Asia and Africa ensured an abundant supply of cheap local labour.
7. Feeling of Racial Superiority: European colonizers claimed a racial superiority and assumed the self-proclaimed responsibility of "civilizing" the people of Asia and Africa, while also using colonization to propagate Christianity.

(ii) Write in detail about the 'Goa liberation movement.'

Ans: Goa had been under Portuguese colonial rule for nearly 450 years since 1510. Even after India gained independence from the British in 1947, Portugal refused to leave Goa. The Goa Liberation Movement unfolded through the following major phases:

1. Early Awakening and Dr. T. B. Kunha (1928): In 1928, the Goa Congress Committee was founded in Mumbai under the leadership of Dr. T. B. Kunha (Father of Goan Nationalism) to fight against Portuguese imperialism and create political awareness among Goans.
2. Civil Disobedience Movement of 1946: In 1946, Dr. Ram Manohar Lohia launched a Civil Disobedience Movement in Goa by defying the Portuguese ban on public meetings at Margao. He was arrested and deported by the Portuguese. Dr. T. B. Kunha, Purushottam Kakodkar, and Dr. Ram Hegde were also arrested; Dr. Kunha was deported to Portugal for 8 years. After returning to Mumbai, he started the newspapers Azad Goa and Swatantra Goa.
3. Goa Vimochan Sahayak Samiti (1954): In 1954, the Goa Vimochan Sahayak Samiti was formed in Pune under the leadership of Keshavrao Jedhe, N. G. Gore, and Jayantrao Tilak to support the liberation struggle. Acharya P. K. Atre used his daily newspaper 'Maratha' to give momentum to the movement.
4. Peaceful Satyagrahas (1954–1955): Volunteers under the leadership of N. G. Gore and Senapati Bapat entered Goa and hoisted the Indian national flag at the Panaji fort. Pandit Mahadevshastri Joshi, Shirubhau Limaye, Peter Alvares, and Sudha Joshi also led satyagrahas in Goa. The Portuguese police brutally shot dead several unarmed satyagrahis, including Karnail Singh Benipal and Hirve Guruji.
5. Armed Struggle by Azad Gomantak Dal: Revolutionary youth formed the Azad Gomantak Dal under Vishwanath Lavande, Rajabhau Wakankar, Sudhir Phadke, Nana Kajrekar, Prabhakar Sinari, and Mohan Ranade. After liberating Dadra and Nagar-Haveli in 1954, they launched armed attacks on Portuguese police stations in Goa. Mohan Ranade was wounded, captured by the Portuguese, and kept in prison in Portugal until 1969.
6. Operation Vijay (December 1961): When peaceful negotiations failed and public outrage grew over Portuguese atrocities, the Government of India launched military action code-named 'Operation Vijay' in December 1961. The Indian Army entered Goa, and aided by local people who showed them the routes and prevented the Portuguese from blowing up bridges, the Portuguese forces surrendered within 48 hours. On 19th December 1961, Goa was liberated and became an integral part of India.

(iii) Write the information about South Asian Association for Regional Co-Operation (SAARC).

Ans: The South Asian Association for Regional Co-operation (SAARC) is an important regional organization of South Asian nations:

1. Background and Establishment: In 1980, the then President of Bangladesh, Ziaur Rahman, initiated the idea of forming a regional organization of South Asian countries. In 1981, the foreign secretaries of seven nations met in Colombo (Sri Lanka), followed by a meeting of foreign ministers in New Delhi in 1983. Finally, on 8th December 1985, SAARC was formally established at the first summit held in Dhaka (Bangladesh).
2. Member Nations, Observers, and Headquarters:
- The seven founding members were India, Bangladesh, Pakistan, Nepal, Bhutan, Sri Lanka, and Maldives.
- In 2007, Afghanistan joined as the 8th member nation.
- Nine countries/organizations—Australia, China, the European Union, Iran, Japan, Mauritius, Myanmar, South Korea, and the USA—have been granted Observer status.
- The permanent Secretariat (Headquarters) of SAARC is situated in Kathmandu, Nepal.
3. Main Objectives of SAARC:
- To promote the welfare of the people of South Asia and raise their standard of living.
- To accelerate economic growth, social progress, and cultural development in the region.
- To foster mutual trust, understanding, and collective self-reliance among South Asian nations.
- To cooperate with international and regional organizations with similar aims.
4. Key Programmes and Achievements:
- Under SAARC, an Agricultural Information Centre was set up in Bangladesh, a Meteorological Research Centre in Dhaka, a Tuberculosis Eradication Centre in Kathmandu, and a Documentation Centre in New Delhi.
- To boost regional trade, SAARC nations signed the SAPTA (SAARC Preferential Trading Arrangement) in 1993 and later signed the SAFTA (South Asian Free Trade Area) agreement in 2004.
- To tackle social challenges, SAARC declared 1990 as the 'SAARC Year of the Girl Child' and 1991–2000 as the 'SAARC Decade of the Girl Child'.

Q. 7. Answer the following questions in detail with the help of given points (Any THREE - All 5 Solved) : (15 Marks)

(i) Write the information about geographical discoveries and explorers :

(a) Ibn Batuta

Ibn Batuta was a renowned 14th-century scholar and famous medieval traveller from Morocco who travelled across Africa, West Asia, India, Maldives, Sumatra, and China for 30 years.
He gained fame for following his personal rule of never travelling twice on the same route.
His accounts and travelogue were extremely helpful in understanding medieval trade routes, geography, administration, and social life across Asia and Africa.

(b) Captain James Cook

Captain James Cook of the British Royal Navy was a famous 18th-century explorer who set foot on the lands of New Zealand and Australia.
He inspected and accurately mapped the islands of the Pacific Ocean and prepared detailed nautical maps.
Within a period of three years, Captain Cook sailed nearly 60,000 nautical miles, paving the way for British colonial expansion in Australia and New Zealand.

(c) Bartholomew Dias

Bartholomew Dias was a Portuguese explorer who set out in 1487 under the patronage of King John II of Portugal to find a sea route to India.
In 1488, he became the first explorer to circumnavigate the African continent by reaching its southern tip.
He initially named the southern tip of Africa the 'Cape of Storms' due to severe weather, but the King of Portugal later changed its name to the 'Cape of Good Hope' because it offered hope of reaching India by sea.

(ii) Explain the relation between Portuguese and Marathas :

(a) Chhatrapati Shivaji Maharaj

Chhatrapati Shivaji Maharaj first came into contact with the Portuguese when he established control over Kalyan and Bhiwandi. Alarmed by his creation of a strong Maratha Navy, the Portuguese initially helped Siddi of Janjira, though they later signed treaties of friendship with the Marathas in 1663 and 1670.
Portuguese Governor General Gonsalo Martez even remarked regarding Chhatrapati Shivaji Maharaj's military genius during the siege of Chakan: "He is much more to be feared than the Mughal Emperor."
To protect the local salt trade in Swarajya from cheap Portuguese salt, Shivaji Maharaj levied heavy octroi on Portuguese salt and defeated the Portuguese during the siege of Phonda fort (Goa) in 1675.

(b) Chhatrapati Sambhaji Maharaj

Because the Portuguese joined hands with Mughal Emperor Aurangzeb against the Marathas, Chhatrapati Sambhaji Maharaj decided to teach the Portuguese a lesson and attacked Revdanda in 1683.
While the Portuguese laid siege to the Maratha fort of Phonda, Maratha commander Yesaji Kank defended it heroically, and Chhatrapati Sambhaji Maharaj launched a fierce counter-attack that forced the Portuguese Governor to retreat.
Pursuing the retreating Portuguese army, Sambhaji Maharaj invaded Goa, captured the fort of St. Estevao (Juve), and brought the Portuguese to the verge of defeat. However, on receiving news that the Mughals had attacked South Konkan, he had to return from the doorstep of Goa to defend Raigad.

(c) Chhatrapati Shahu Maharaj

During the reign of Chhatrapati Shahu Maharaj, the conflict between the Marathas and the Portuguese was renewed when Husain Ali Sayyad (representative of the Mughals in the Deccan) granted the Marathas the right to collect Chauthai from the Kalyan-Bhiwandi region, which the Portuguese refused to honour.
Chimaji Appa, the younger brother of Peshwa Bajirao I, was assigned the responsibility of leading the campaign against the Portuguese in North Konkan (1737–1739).
Chimaji Appa conquered the fort of Thane and Sashti (Salsette) island in March 1737, while Shankaraji Phadke entered Vasai island. After a heroic two-year siege of the heavily fortified Vasai Fort, the Marathas defeated the Portuguese completely in 1739.

(iii) Write about Indian National Congress :

(a) Background

After the Freedom Struggle of 1857, British economic exploitation, heavy taxation, famines, and racial discrimination created widespread discontent across India.
At the same time, the spread of Western education, railway and telegraph networks, and nationalist newspapers inspired educated Indians to form political organizations such as the Bombay Association, East India Association, Madras Native Association, and Indian Association, which paved the way for an all-India political organization.

(b) Foundation

On 28th December 1885, the first session of the Indian National Congress (INC) was held at Gokuldas Tejpal Sanskrit Pathshala in Mumbai.
It was attended by 72 delegates from different provinces of India. Vyomeshchandra Bannerjee (W. C. Bonnerjee) was the first President of this session, and retired British officer Allan Octavian Hume (A. O. Hume) played a significant role in its establishment.

(c) Moderate and Extremist Thought Process

Moderate Thought Process: Early leaders of the Congress like Gopal Krishna Gokhale, Pherozeshah Mehta, and Surendranath Banerjee believed that if they placed the grievances and demands of Indians (such as representation in legislature, reduction in military expenditure, and employment of Indians in civil services) before the British government through constitutional means like petitions and resolutions, the British would grant them justice.
Extremist Thought Process: Leaders like Lokmanya Bal Gangadhar Tilak, Lala Lajpat Rai, and Bipin Chandra Pal (Lal-Bal-Pal) felt that mere petitions and resolutions were ineffective and insisted that Indians must actively agitate for Swarajya. Following the Partition of Bengal (1905), they championed the four-point programme of Swarajya, Swadeshi, Boycott, and National Education. Ideological differences between the Moderates and Extremists reached a peak at the Surat Session in 1907, leading to a split before they reunited at the Lucknow Session in 1916.

(iv) Write information about the impact of World Wars on India :

(a) First World War (1914–1918)

Contribution of Soldiers and Money: Britain dragged India into the First World War without consulting Indian leaders. A total of 10,96,013 Indian soldiers were sent by the British to fight on various fronts across Europe, West Asia, and Africa, and 17 Indian soldiers (including Khudadad Khan) were honoured with the prestigious Victoria Cross for their bravery. India provided ₹19 billion worth of war assistance in the form of food grains, clothes, ammunition, and funding.
Industrial and Economic Impact: India's Tata Iron and Steel Company (TISCO) supplied steel rails for constructing railway tracks in the war zones, and Indian trade in jute, textiles, hides, and minerals grew rapidly. However, ordinary citizens suffered from severe price rises (inflation), shortage of daily essentials, and heavy war taxes.
Political Impact: Indian soldiers returning from Europe gained confidence after seeing that European armies were not invincible. In 1917, Edwin Montagu declared that Indians would gradually be given self-governance, leading to the Montagu-Chelmsford Reforms (1919), though its inadequacies fueled the Non-Cooperation Movement.

(b) Second World War (1939–1945)

Forced Involvement and Congress Resignation: In September 1939, Viceroy Lord Linlithgow announced that India had joined the Second World War on the side of England without consulting Indian representatives. Protesting this unilateral decision, the Indian National Congress ministries in the provinces resigned.
Military Contribution and Wartime Production: Over 25 lakh Indian soldiers fought against the Axis forces across North Africa, Europe, and Asia. 31 Indian soldiers and officers—including Yeshwant Ghadge and Namdeo Jadhav from Maharashtra—were awarded the Victoria Cross for extraordinary bravery. To meet wartime demands, the British established naval and air force training centres and munitions factories in India, though hoarding and war expenditure caused acute food scarcity and inflation.
Final Push for Independence: During the war, Mahatma Gandhi launched the Quit India Movement (1942) inside India, while Netaji Subhash Chandra Bose led the Azad Hind Sena (INA) against the British on India's eastern border. Although Britain won the Second World War, its economy and military power were exhausted. Following the Royal Indian Navy (RIN) uprising in 1946, Britain realized it could no longer rule India and granted independence on 15th August 1947.

(v) Write about India's defence affairs policy :

(a) Kargil War (1999)

In 1999, anundeclared war broke out between India and Pakistan when Pakistani troops infiltrators crossed the Line of Control (LoC) and occupied strategic mountain peaks in the Kargil-Dras sector of Ladakh.
The Indian Army launched 'Operation Vijay' (supported by the Indian Air Force) and fought heroically in extreme high-altitude terrain to evict the Pakistani intruders and recapture all Indian peaks by July 1999.
The Kargil War exposed the need for modernization in India's defence preparedness, intelligence gathering, military training, and rapid equipment procurement, leading to major defence reforms.

(b) Make in India

Historically, India depended heavily on importing weapons, aircraft, and defence hardware from foreign countries. To achieve self-reliance in the defence sector, the Government of India emphasized indigenous defence manufacturing under the 'Make in India' policy.
Under this policy, Indian public sector undertakings (like DRDO and HAL) and private Indian industries are encouraged to design and manufacture modern weapons, missiles, fighter aircraft, naval ships, and artillery within India.
This reduces India's dependence on foreign nations during wartime, saves valuable foreign exchange, and enables India to export defence equipment to other countries.

(c) Job Opportunities

The promotion of indigenous defence production under 'Make in India' and the establishment of defence industrial corridors have generated thousands of new job opportunities for Indian youth in engineering, technology, manufacturing, and scientific research.
Furthermore, the Indian Army, Navy, Air Force, Coast Guard, and paramilitary forces (such as BSF, CRPF, ITBP, and CISF) provide honourable and stable career opportunities for young men and women across the country to serve the nation.

HSC 12th History June 2026 Question Paper (J-270)

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HSC 12th Book Keeping & Accountancy Board Paper Solution (June 2026) | Maharashtra Board

aaminomtex@gmail.com (Amin) — Sat, 24 Oct 2026 23:15:04 +0530

HSC Book Keeping and Accountancy - June 2026 Board Paper Solution

Welcome to Omtex Classes! Are you preparing for your upcoming Maharashtra Board 12th Commerce exams? Practicing previous year question papers is one of the most effective ways to score high marks in Book Keeping and Accountancy. Below, you will find the complete, step-by-step solution for the HSC 12th Standard Book Keeping and Accountancy June 2026 Board Paper (Subject Code: J-309). We have solved all the practical problems, including Partnership Final Accounts, Not-for-Profit Concerns, Admission/Retirement of a Partner, Death of a Partner, Dissolution, Issue of Shares, and Bills of Exchange, following the exact Maharashtra State Board formatting and working notes required to secure full marks.

Q. 1. All sub-questions are compulsory :

(A) Select the most appropriate alternative from the following and rewrite the sentences:

Goodwill is an Intangible Asset.
Excess of income over expenditure is termed as Surplus.
Decrease in the value of Assets should be Debited to Revaluation Account.
Notary public is Govt. Officer.
Fixed Deposit Account comes under Investment group.

(B) Write a word / term / phrase which can substitute each of the following statements:

Expenses which are paid before they are due: Prepaid Expenses (Unexpired Expenses)
The receipts which are not recurring in nature: Capital Receipts
The ratio in which general reserve is distributed to the old partners in case of admission of a partner: Old Profit Sharing Ratio
Winding up of partnership business: Dissolution of Partnership Firm
Shares issued at its face value: Issued at Par

(C) Complete the sentences:

Liabilities of partners in partnership firm is Unlimited.
Pune University prepares Income and Expenditure Account instead of a Profit and Loss Account.
A person who draws the bill of exchange is known as Drawer.
Revaluation Account is also known as Profit and Loss Adjustment Account.
Current Ratio = Current Assets / Current Liabilities

(D) Do you AGREE / DISAGREE with the following Statements:

Free distribution of goods is debited to Trading Account. - I Disagree
Non-trading concerns do not have profit motive. - I Agree
When the Goodwill is written off goodwill account is debited. - I Disagree
On Dissolution of partnership firm Cash / Bank Account is closed automatically. - I Agree
A deceased partner is not entitled to Goodwill of the Firm. - I Disagree

Q. 2. Admission of a Partner (Kaustubh, Ketan, and Pranav)

In the books of the Firm

(a) Dr. Revaluation Account Cr.

Particulars	Amount (₹)	Particulars	Amount (₹)	Amount (₹)
To R.D.D. A/c (Provision for Bad Debts)	5,200	By Building A/c		40,000
To Stock A/c	16,000	By Partners' Capital A/cs (Loss on Revaluation):
To Machinery A/c	28,000	Kaustubh (9,200 x 1/2)	4,600
		Ketan (9,200 x 1/2)	4,600	9,200
Total	49,200	Total		49,200

(b) Dr. Partners' Capital Accounts Cr.

Particulars	Kaustubh (₹)	Ketan (₹)	Pranav (₹)	Particulars	Kaustubh (₹)	Ketan (₹)	Pranav (₹)
To Revaluation A/c (Loss)	4,600	4,600	-	By Balance b/d	2,00,000	2,80,000	-
To Balance c/d	2,35,400	3,15,400	2,00,000	By Cash A/c (Capital)	-	-	2,00,000
				By Goodwill A/c	40,000	40,000	-
Total	2,40,000	3,20,000	2,00,000	Total	2,40,000	3,20,000	2,00,000

(c) Balance Sheet as on 1st April 2023

Liabilities	Amount (₹)	Amount (₹)	Assets	Amount (₹)	Amount (₹)
Capital Accounts:			Building	1,20,000
Kaustubh	2,35,400		Add: Appreciation	40,000	1,60,000
Ketan	3,15,400		Machinery	1,40,000
Pranav	2,00,000	7,50,800	Less: 20% Depreciation	28,000	1,12,000
Creditors		68,000	Investment		48,000
Bills Payable		52,000	Debtors	52,000
			Less: 10% R.D.D.	5,200	46,800
			Stock	80,000
			Less: 20% Reduction	16,000	64,000
			Cash		4,40,000
Total		8,70,800	Total		8,70,800

Working Notes:

1. Cash Account Balance: Opening Balance ₹1,60,000 + Pranav's Capital ₹2,00,000 + Goodwill brought in Cash ₹80,000 = ₹4,40,000.

2. Distribution of Goodwill: Goodwill of ₹80,000 brought in by Pranav is shared by Kaustubh and Ketan in their Sacrifice Ratio (Equal, i.e., 1:1) = ₹40,000 each.

Q. 2. OR - Retirement of a Partner (Sagar, Suraj, and Sahil)

In the books of the Firm

(a) Dr. Profit and Loss Adjustment Account Cr.

Particulars	Amount (₹)	Amount (₹)	Particulars	Amount (₹)
To Machinery A/c		1,200	By Building A/c	14,000
To Computer A/c		3,800	By R.D.D. A/c	1,800
To Partners' Capital A/cs (Profit):
Sagar (10,800 x 1/3)	3,600
Suraj (10,800 x 1/3)	3,600
Sahil (10,800 x 1/3)	3,600	10,800
Total		15,800	Total	15,800

(b) Dr. Partners' Capital Accounts Cr.

Particulars	Sagar (₹)	Suraj (₹)	Sahil (₹)	Particulars	Sagar (₹)	Suraj (₹)	Sahil (₹)
To Sahil's Loan A/c	-	-	80,600	By Balance b/d	87,200	70,000	64,000
To Balance c/d	1,03,800	86,600	-	By Reserve Fund A/c	9,000	9,000	9,000
				By Goodwill A/c (Revalued Increase)	4,000	4,000	4,000
				By Profit and Loss Adjustment A/c (Profit)	3,600	3,600	3,600
Total	1,03,800	86,600	80,600	Total	1,03,800	86,600	80,600

(c) Balance Sheet as on 1st April 2023

Liabilities	Amount (₹)	Amount (₹)	Assets	Amount (₹)	Amount (₹)
Capital Accounts:			Building	1,40,000
Sagar	1,03,800		Add: 10% Appreciation	14,000	1,54,000
Suraj	86,600	1,90,400	Machinery	20,000
Sahil's Loan A/c		80,600	Less: Reduction in Value	1,200	18,800
Creditors		61,800	Computer	35,000
			Less: Depreciation	3,800	31,200
			Goodwill	24,000
			Add: Increase in Value	12,000	36,000
			Debtors (All Good)		36,000
			Bank		56,800
Total		3,32,800	Total		3,32,800

Q. 3. Dissolution of Partnership Firm (All Partners Insolvent)

In the books of M/s. Bhumi, Bhupal and Bipin

(a) Dr. Realisation Account Cr.

Particulars	Amount (₹)	Amount (₹)	Particulars	Amount (₹)	Amount (₹)
To Sundry Assets A/c:			By Bank A/c (Assets Realised):
Computer	11,000		Stock	35,000
Furniture	45,000		Furniture	37,000
Stock	53,000	1,09,000	Computer	6,000	78,000
To Bank A/c (Realisation Expenses)		11,000	By Partners' Capital A/cs (Loss on Realisation):
			Bhumi (42,000 x 1/3)	14,000
			Bhupal (42,000 x 1/3)	14,000
			Bipin (42,000 x 1/3)	14,000	42,000
Total		1,20,000	Total		1,20,000

(b) Dr. Bank Account Cr.

Particulars	Amount (₹)	Particulars	Amount (₹)
To Balance b/d	7,000	By Realisation A/c (Expenses)	11,000
To Realisation A/c (Assets Realised)	78,000	By Bills Payable A/c (Available Cash Paid)	82,000
To Bhumi's Capital A/c (Cash brought in)	8,000
Total	93,000	Total	93,000

(c) Dr. Bills Payable Account Cr.

Particulars	Amount (₹)	Particulars	Amount (₹)
To Bank A/c (Final Payment)	82,000	By Balance b/d	85,000
To Deficiency A/c (Unpaid Balance)	3,000
Total	85,000	Total	85,000

(d) Dr. Bhumi's Loan Account Cr.

Particulars	Amount (₹)	Particulars	Amount (₹)
To Deficiency A/c (Unpaid Balance)	25,000	By Balance b/d	25,000
Total	25,000	Total	25,000

(e) Dr. Partners' Capital Accounts Cr.

Particulars	Bhumi (₹)	Bhupal (₹)	Bipin (₹)	Particulars	Bhumi (₹)	Bhupal (₹)	Bipin (₹)
To Balance b/d	-	-	20,000	By Balance b/d	15,000	11,000	-
To Realisation A/c (Loss)	14,000	14,000	14,000	By Bank A/c	8,000	-	-
To Deficiency A/c	9,000	-	-	By Deficiency A/c	-	3,000	34,000
Total	23,000	14,000	34,000	Total	23,000	14,000	34,000

(f) Dr. Deficiency Account Cr.

Particulars	Amount (₹)	Particulars	Amount (₹)
To Bhumi's Capital A/c	9,000	By Bhupal's Capital A/c	3,000
To Bills Payable A/c (Unpaid)	3,000	By Bipin's Capital A/c	34,000
To Bhumi's Loan A/c (Unpaid)	25,000
Total	37,000	Total	37,000

Q. 3. OR - Bills of Exchange

Journal Entries in the books of Surekha

Sr. No.	Particulars	Debit (₹)	Credit (₹)
1	Suman's A/c ...Dr. To Sales A/c (Being goods sold on credit to Suman)	20,000	20,000
2	Bills Receivable A/c ...Dr. To Suman's A/c (Being bill drawn and acceptance received for 3 months)	20,000	20,000
3	Bank A/c ...Dr. Discount A/c (20,000 x 10% x 3/12) ...Dr. To Bills Receivable A/c (Being the bill discounted with the bank @ 10% p.a.)	19,500 500	20,000
4	Suman's A/c ...Dr. To Bank A/c (Being discounted bill dishonoured on the due date)	20,000	20,000
5	Suman's A/c ...Dr. To Interest A/c (Being interest charged for extension of credit period)	1,000	1,000
6	Cash A/c ...Dr. To Suman's A/c (Being part payment of ₹8,000 received along with interest ₹1,000 in cash)	9,000	9,000
7	Bills Receivable A/c ...Dr. To Suman's A/c (Being new bill drawn and accepted for the balance amount for 2 months)	12,000	12,000
8	Cash A/c ...Dr. Rebate A/c ...Dr. To Bills Receivable A/c (Being new bill retired one month before due date and rebate allowed)	11,800 200	12,000
	Total	1,14,000	1,14,000

Q. 4. Death of a Partner (Pravin, Prashant, and Sushil)

In the books of the Firm

(a) Dr. Revaluation Account Cr.

Particulars	Amount (₹)	Amount (₹)	Particulars	Amount (₹)
To Creditors A/c		20,000	By Loose Tools A/c	60,000
To R.D.D. A/c		20,000	By Livestock A/c	40,000
To Partners' Capital A/cs (Profit on Revaluation):
Pravin (60,000 x 5/10)	30,000
Prashant (60,000 x 3/10)	18,000
Sushil (60,000 x 2/10)	12,000	60,000
Total		1,00,000	Total	1,00,000

(b) Dr. Sushil's Capital Account Cr.

Particulars	Amount (₹)	Particulars	Amount (₹)
To Sushil's Executor's Loan A/c	1,46,000	By Balance b/d	1,00,000
		By Reserve Fund A/c (50,000 x 2/10)	10,000
		By Revaluation A/c (Profit)	12,000
		By Goodwill A/c	24,000
Total	1,46,000	Total	1,46,000

(c) Working Note: Calculation of Sushil's Share in Goodwill

Step 1: Calculate Total Profit for 3 Years

Total Profit = Profit of 2019-20 + Profit of 2020-21 - Loss of 2021-22

Total Profit = ₹1,80,000 + ₹1,20,000 - ₹1,20,000 = ₹1,80,000

Step 2: Calculate Average Profit

Average Profit = Total Profit / Number of Years = ₹1,80,000 / 3 = ₹60,000

Step 3: Calculate Firm's Goodwill

Firm's Goodwill = Average Profit x Number of Years' Purchase = ₹60,000 x 2 = ₹1,20,000

Step 4: Calculate Sushil's Share in Goodwill

Sushil's Share = Firm's Goodwill x Sushil's Profit Sharing Ratio = ₹1,20,000 x (2/10) = ₹24,000

Q. 4. OR - Importance of Computerized Accounting System

Meaning: A Computerized Accounting System (CAS) is an accounting information system that processes financial transactions and events using computer hardware and accounting software to generate accurate financial statements and reports as per user requirements.

The importance of a Computerized Accounting System in modern business organizations is explained below:

Automation of Operations: Once a financial transaction is entered through an accounting voucher, all calculations, ledger postings, trial balance, and final accounts are performed automatically by the accounting software without repetitive manual work.
High Speed and Efficiency: Computerized accounting software processes thousands of business transactions and generates complex financial reports (such as Balance Sheet, Profit and Loss Account, and GST reports) within seconds, which saves significant time.
Accuracy and Reliability: Because arithmetic calculations and ledger transfers are automated, human errors like transposition errors, casting mistakes, or omission of postings are minimized, making financial statements highly reliable.
Up-to-Date Information: Accounting records are updated immediately upon voucher entry. Management can access real-time financial positions, stock levels, and debtor/creditor balances whenever required for quick decision-making.
Storage and Quick Retrieval: Huge volumes of financial data spanning multiple accounting years can be stored safely on hard disks or cloud servers and retrieved instantly using simple search queries.
Security and Confidentiality: Financial data in a computerized system is protected through user login IDs, passwords, data encryption, and role-based access controls, preventing unauthorized manipulation or viewing of accounts.
Reduction in Cost: Replacing physical ledgers, registers, and manual clerical work reduces stationery expenses, storage space costs, and administrative overheads over the long run.
Compactness and Easy Transferability: Financial records do not require large physical record rooms and can be easily exported, printed, or shared electronically with auditors, banks, tax authorities, and stakeholders.

Q. 5. Issue of Shares (Finolex Co. Ltd.)

Journal Entries in the books of Finolex Co. Ltd.

Sr. No.	Particulars	Debit (₹)	Credit (₹)
1	Bank A/c ...Dr. To Equity Share Application A/c (Being application money received on 60,000 equity shares @ ₹3 per share)	1,80,000	1,80,000
2	Equity Share Application A/c ...Dr. To Equity Share Capital A/c To Equity Share Allotment A/c (Being application money on 50,000 shares transferred to Share Capital A/c and excess application money on 10,000 shares adjusted towards allotment)	1,80,000	1,50,000 30,000
3	Equity Share Allotment A/c ...Dr. To Equity Share Capital A/c (Being allotment money due on 50,000 equity shares @ ₹5 per share)	2,50,000	2,50,000
4	Bank A/c ...Dr. To Equity Share Allotment A/c (Being balance allotment money received after adjusting excess application money)	2,20,000	2,20,000
5	Equity Share First and Final Call A/c ...Dr. To Equity Share Capital A/c (Being first and final call money due on 50,000 equity shares @ ₹2 per share)	1,00,000	1,00,000
6	Bank A/c ...Dr. Calls in Arrears A/c ...Dr. To Equity Share First and Final Call A/c (Being first and final call money received on 45,000 shares @ ₹2 per share and arrears recorded on 5,000 shares held by Mr. Rahul)	90,000 10,000	1,00,000
	Total	10,30,000	10,30,000

Q. 5. OR - Analysis of Financial Statements

1. Current Ratio

Formula: Current Ratio = Current Assets / Current Liabilities

Calculation of Current Assets:

Debtors = ₹2,50,000
Stock = ₹1,50,000
Cash in hand = ₹2,00,000
Total Current Assets = ₹2,50,000 + ₹1,50,000 + ₹2,00,000 = ₹6,00,000

Calculation of Current Liabilities:

Creditors = ₹1,20,000
Bills Payable = ₹50,000
Bank Overdraft = ₹30,000
Total Current Liabilities = ₹1,20,000 + ₹50,000 + ₹30,000 = ₹2,00,000

Current Ratio = 6,00,000 / 2,00,000 = 3 : 1

2. Gross Profit Ratio

Formula: Gross Profit Ratio = (Gross Profit / Net Sales) x 100

Given:

Gross Profit = ₹8,00,000
Net Sales (Sales) = ₹16,00,000

Gross Profit Ratio = (8,00,000 / 16,00,000) x 100 = 50%

3. Net Profit Ratio

Formula: Net Profit Ratio = (Net Profit / Net Sales) x 100

Given:

Net Profit = ₹4,00,000
Net Sales (Sales) = ₹16,00,000

Net Profit Ratio = (4,00,000 / 16,00,000) x 100 = 25%

Q. 6. Not-for-Profit Concerns (Atharv Art Circle, Sindhudurg)

In the books of Atharv Art Circle, Sindhudurg

Dr. Income and Expenditure Account for the year ended 31st March 2023 Cr.

Expenditure	Amount (₹)	Income	Amount (₹)	Amount (₹)
To Postage	12,000	By Subscriptions	1,20,000
To Office Rent	7,800	Add: Outstanding Subscriptions	15,000	1,35,000
To Newspaper	900	By Donations	54,000
To Electricity Charges	1,680	Less: 50% Capitalised	27,000	27,000
To Stationery	12,000	By Life Membership Fees	66,000
To Salary	10,000	Less: 100% Capitalised	66,000	Nil
To Depreciation on Furniture (10% on ₹36,000)	3,600
To Surplus (Excess of Income over Expenditure)	1,14,020
Total	1,62,000	Total		1,62,000

Balance Sheet as on 31st March 2023

Liabilities	Amount (₹)	Amount (₹)	Assets	Amount (₹)	Amount (₹)
Capital Fund:			Furniture	36,000
Opening Balance (Newly Started)	Nil		Less: 10% Depreciation	3,600	32,400
Add: Life Membership Fees (Capitalised)	66,000		Investment		9,000
Add: Donations (50% Capitalised)	27,000		Laptop		18,000
Add: Surplus	1,14,020	2,07,020	Machinery		20,000
			Outstanding Subscriptions		15,000
			Cash in hand		22,620
			Cash at bank		90,000
Total		2,07,020	Total		2,07,020

Q. 7. Partnership Final Accounts (Jayesh and Rupesh)

In the books of Jayesh and Rupesh

Dr. Trading and Profit and Loss Account for the year ended 31st March 2023 Cr.

Particulars	Amount (₹)	Amount (₹)	Particulars	Amount (₹)
To Opening Stock		35,000	By Sales	2,20,000
To Purchases	1,63,000		By Goods distributed as free samples	2,000
Less: Return Outward	3,000	1,60,000	By Closing Stock	38,000
To Productive Wages		5,000
To Gross Profit c/d		60,000
Total		2,60,000	Total	2,60,000
To General Expenses		8,000	By Gross Profit b/d	60,000
To Rent (10 Months)	20,000
Add: Outstanding Rent (2 Months)	4,000	24,000
To Depreciation on Furniture		3,000
To Advertisement (Free Samples)		2,000
To Interest on Capital:
Jayesh (5% on ₹1,20,000)	6,000
Rupesh (5% on ₹80,000)	4,000	10,000
To Net Profit transferred to Capital A/cs (3:2):
Jayesh (13,000 x 3/5)	7,800
Rupesh (13,000 x 2/5)	5,200	13,000
Total		60,000	Total	60,000

Balance Sheet as on 31st March 2023

Liabilities	Amount (₹)	Amount (₹)	Assets	Amount (₹)	Amount (₹)
Capital Account - Jayesh:			Premises		1,30,000
Opening Balance	1,20,000		Furniture	40,000
Add: Interest on Capital	6,000		Less: 10% Depreciation	3,000	37,000
Add: Net Profit	7,800	1,33,800	Debtors		45,000
Capital Account - Rupesh:			Closing Stock		38,000
Opening Balance	80,000		Cash at Bank		24,000
Add: Interest on Capital	4,000
Add: Net Profit	5,200	89,200
Creditors		47,000
Outstanding Rent (for 2 months)		4,000
Total		2,74,000	Total		2,74,000

Working Notes:

1. Depreciation on Furniture:

Opening Furniture (₹40,000 - ₹20,000 Addition) = ₹20,000. Depreciation for full year (12 months) @ 10% p.a. = ₹2,000.
Additional Furniture purchased on 1st October 2022 = ₹20,000. Depreciation for 6 months (1st Oct 2022 to 31st March 2023) @ 10% p.a. = ₹1,000.
Total Depreciation on Furniture = ₹2,000 + ₹1,000 = ₹3,000.

2. Hidden Adjustment - Outstanding Rent:

Rent paid for 10 months = ₹20,000. Monthly Rent = ₹20,000 / 10 = ₹2,000 per month.
Outstanding Rent for remaining 2 months = ₹2,000 x 2 = ₹4,000.

HSC Book Keeping and Accountancy June 2026 Question Paper (J-309)

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HSC 12th Maths June 2026 Board Paper Solution (J-276) | Class 12 Mathematics & Statistics Answer Key

aaminomtex@gmail.com (Amin) — Sat, 24 Oct 2026 10:07:26 +0530

Are you looking for the complete, step-by-step solutions for the HSC 12th Standard Mathematics and Statistics June 2026 Board Question Paper (Subject Code: J-276) held on 24/06/2026? Below is the full, solved answer key covering Section A, Section B, Section C, and Section D prepared strictly according to the Maharashtra State Board marking scheme to help Class 12 students score full marks.

HSC 12th Maths (Arts & Science) June 2026 Board Paper Solution

SECTION - A

Q. 1. Select and write the correct answer of the following multiple choice type of questions:

If A = {1, 2, 3, 4, 5} then which of the following is not true?

Solution: (c) $x \in A, x+6 \ge 9$

Explanation: If we take $x = 1$ from set A, $1 + 6 = 7$, which is not greater than or equal to 9. Thus, the statement is false.
The value of $\sin[\sin^{-1}(-\frac{1}{\sqrt{2}}) - 3 \sin^{-1}(\frac{\sqrt{3}}{2})] =$

Solution: (b) $\frac{1}{\sqrt{2}}$

Explanation:
$\sin^{-1}(-\frac{1}{\sqrt{2}}) = -\frac{\pi}{4}$
$\sin^{-1}(\frac{\sqrt{3}}{2}) = \frac{\pi}{3}$
Expression becomes: $\sin(-\frac{\pi}{4} - 3(\frac{\pi}{3})) = \sin(-\frac{\pi}{4} - \pi) = \sin(-\frac{5\pi}{4}) = \frac{1}{\sqrt{2}}$.
If $\hat{p}, \hat{q}, \hat{r}$ are mutually perpendicular unit vectors and form a right handed triplet then the value of $\hat{p}\cdot(\hat{q}\times\hat{r}) + \hat{q}\cdot(\hat{p}\times\hat{r}) + \hat{r}\cdot(\hat{p}\times\hat{q}) = $

Solution: (c) 1

Explanation: For a right-handed triplet of mutually perpendicular unit vectors:
$\hat{q} \times \hat{r} = \hat{p}$ (so $\hat{p}\cdot\hat{p} = 1$)
$\hat{p} \times \hat{r} = -\hat{q}$ (so $\hat{q}\cdot(-\hat{q}) = -1$)
$\hat{p} \times \hat{q} = \hat{r}$ (so $\hat{r}\cdot\hat{r} = 1$)
Sum = $1 - 1 + 1 = 1$.
The Cartesian equation of the line passing through the points A(3, 4, -1) and B(2, -1, 3) is:

Solution: (a) $\frac{x-3}{1} = \frac{y-4}{5} = \frac{z+1}{-4}$

Explanation: Direction ratios (DRs) of AB = (2-3, -1-4, 3-(-1)) = (-1, -5, 4). Equivalent DRs are (1, 5, -4). Using point A(3,4,-1), the equation is $\frac{x-3}{1} = \frac{y-4}{5} = \frac{z+1}{-4}$.
If $y = \log_x a$ then $\frac{dy}{dx} =$

Solution: (a) $\frac{-\log a}{x(\log x)^2}$

Explanation: Using the change of base rule, $y = \frac{\log a}{\log x} = \log a \cdot (\log x)^{-1}$. Differentiating with respect to $x$: $\frac{dy}{dx} = \log a \cdot (-1)(\log x)^{-2} \cdot \frac{d}{dx}(\log x) = \log a \cdot \frac{-1}{(\log x)^2} \cdot \frac{1}{x} = \frac{-\log a}{x(\log x)^2}$.
If $f(x) = \frac{\sin^{-1}x}{\sqrt{1-x^2}}$ and $g(x) = e^{\sin^{-1}x}$ then $\int f(x)\cdot g(x)dx = $

Solution: (b) $e^{\sin^{-1}x}(\sin^{-1}x - 1) + c$

Explanation: We need to evaluate $\int \frac{\sin^{-1}x}{\sqrt{1-x^2}} e^{\sin^{-1}x} dx$. Let $t = \sin^{-1}x$, then $dt = \frac{1}{\sqrt{1-x^2}} dx$. The integral becomes $\int t e^t dt$. Using integration by parts: $t \int e^t dt - \int (1 \cdot e^t) dt = t e^t - e^t + c = e^t(t - 1) + c$. Substituting back $t = \sin^{-1}x$, we get $e^{\sin^{-1}x}(\sin^{-1}x - 1) + c$.
The area bounded by the line $y=2x$, X-axis and the lines $x=-1$, $x=4$ is _____ sq. units.

Solution: (b) 17

Explanation: The area is given by $\int_{-1}^{4} |y| dx = \int_{-1}^{0} |2x| dx + \int_{0}^{4} |2x| dx$. Since $2x$ is negative for $x < 0$, we have $\int_{-1}^{0} (-2x) dx + \int_{0}^{4} (2x) dx$. Evaluating the integrals: $[-x^2]_{-1}^{0} + [x^2]_{0}^{4} = (0 - (-1)) + (16 - 0) = 1 + 16 = 17$ sq. units.
In a Binomial Distribution, $E(X)=6$, $Var(X)=4.2$ then the value of n is _____

Solution: 20

Explanation: For a Binomial Distribution, Mean $E(X) = np = 6$ and Variance $Var(X) = npq = 4.2$. Dividing variance by mean: $\frac{npq}{np} = \frac{4.2}{6} \Rightarrow q = 0.7$. Since $p = 1 - q$, we get $p = 1 - 0.7 = 0.3$. Substituting $p$ back into the mean equation: $n(0.3) = 6 \Rightarrow n = \frac{6}{0.3} = 20$.

Q. 2. Answer the following questions:

Write the dual of $p \wedge \sim q$

Solution: $p \vee \sim q$

Explanation: To write the dual of a logical statement, we replace the conjunction symbol ($\wedge$) with the disjunction symbol ($\vee$) and vice versa. Negations ($\sim$) remain unchanged.
What is the distance of point (3, 5, 6) from XZ plane?

Solution: 5 units

Explanation: The perpendicular distance of any point $(x, y, z)$ from the XZ plane is given by the absolute value of its y-coordinate, $|y|$. Here, the y-coordinate is 5, so the distance is $|5| = 5$ units.
The displacement of a particle at time t is given by $S = 3t^2 + 4t + 1$. Find its velocity at time t.

Solution: $v = 6t + 4$

Explanation: Velocity ($v$) is the rate of change of displacement ($S$) with respect to time ($t$).
$v = \frac{dS}{dt} = \frac{d}{dt}(3t^2 + 4t + 1) = 6t + 4$.
Write the order of differential equation $(\frac{d^2y}{dx^2})^{\frac{5}{2}} = \sqrt[3]{\frac{dy}{dx}}$

Solution: Order = 2

Explanation: The order of a differential equation is the order of the highest derivative present in the equation. Here, the highest derivative is $\frac{d^2y}{dx^2}$, which is a second-order derivative. Therefore, the order is 2.

SECTION - B

Attempt any EIGHT of the following questions:

Q. 3. Construct the switching circuit for the statement $(\sim p \wedge q) \vee (p \wedge \sim r)$.

Solution:

Let,
$p$: Switch $S_1$ is closed.
$q$: Switch $S_2$ is closed.
$r$: Switch $S_3$ is closed.
$\sim p$: Switch $S_1'$ is closed.
$\sim r$: Switch $S_3'$ is closed.

The given logical statement pattern is $(\sim p \wedge q) \vee (p \wedge \sim r)$.

The term $(\sim p \wedge q)$ indicates that switches $S_1'$ and $S_2$ are connected in series.
The term $(p \wedge \sim r)$ indicates that switches $S_1$ and $S_3'$ are connected in series.
The disjunction ($\vee$) between the two brackets indicates that these two series branches are connected in parallel to each other.

Q. 4. Find the matrix of cofactors of matrix $\begin{bmatrix} 5 & 2 \\ -1 & 6 \end{bmatrix}$

Solution:

Let the given matrix be $A = \begin{bmatrix} 5 & 2 \\ -1 & 6 \end{bmatrix}$

Here, the elements of the matrix are:
$a_{11} = 5, a_{12} = 2$
$a_{21} = -1, a_{22} = 6$

The cofactors $C_{ij}$ are calculated using the formula $C_{ij} = (-1)^{i+j} M_{ij}$, where $M_{ij}$ is the minor of the element $a_{ij}$.

$C_{11} = (-1)^{1+1} (6) = 1 \times 6 = 6$
$C_{12} = (-1)^{1+2} (-1) = -1 \times -1 = 1$
$C_{21} = (-1)^{2+1} (2) = -1 \times 2 = -2$
$C_{22} = (-1)^{2+2} (5) = 1 \times 5 = 5$

Therefore, the matrix of cofactors is:

$\begin{bmatrix} C_{11} & C_{12} \\ C_{21} & C_{22} \end{bmatrix} = \begin{bmatrix} 6 & 1 \\ -2 & 5 \end{bmatrix}$

Q. 5. Find the general solution of equation $\cos 5\theta = \frac{1}{2}$.

Solution:

Given equation is $\cos 5\theta = \frac{1}{2}$

We know that $\cos(\frac{\pi}{3}) = \frac{1}{2}$

Therefore, $\cos 5\theta = \cos(\frac{\pi}{3})$

The general solution for $\cos x = \cos \alpha$ is $x = 2n\pi \pm \alpha$, where $n \in Z$.

Here, $x = 5\theta$ and $\alpha = \frac{\pi}{3}$.

$5\theta = 2n\pi \pm \frac{\pi}{3}$, where $n \in Z$

Dividing both sides by 5, we get:

$\theta = \frac{2n\pi}{5} \pm \frac{\pi}{15}$, where $n \in Z$

This is the required general solution.

Q. 6. Find the value of k, if $2x+y=0$ is one of the lines represented by $3x^2+kxy+2y^2=0$.

Solution:

The given combined equation of the lines is:
$3x^2 + kxy + 2y^2 = 0$

Comparing this with the standard homogeneous equation $ax^2 + 2hxy + by^2 = 0$, we get:
$a = 3$, $2h = k$, and $b = 2$

The corresponding auxiliary equation is $bm^2 + 2hm + a = 0$.
Substituting the values of $a$, $2h$, and $b$, we get:
$2m^2 + km + 3 = 0$ ...(i)

We are given that $2x + y = 0$ is one of the lines represented by the combined equation.
Writing this in the slope-intercept form ($y = mx + c$), we get:
$y = -2x$

Therefore, the slope of this line is $m = -2$.

Since this line is a part of the given combined equation, its slope $m = -2$ must satisfy the auxiliary equation (i).

Substitute $m = -2$ in equation (i):
$2(-2)^2 + k(-2) + 3 = 0$
$2(4) - 2k + 3 = 0$
$8 - 2k + 3 = 0$
$11 - 2k = 0$
$2k = 11$
$k = \frac{11}{2}$

Therefore, the value of $k$ is $\frac{11}{2}$.

Q. 7. If $\overline{a}$, $\overline{b}$, $\overline{c}$ are the position vectors of points A, B, C respectively and $10\overline{a} = 7\overline{b} + 3\overline{c}$ then find the ratio in which the point C divides the line segment AB.

Solution:

Given equation is:
$10\overline{a} = 7\overline{b} + 3\overline{c}$

We need to find the ratio in which point C divides the line segment AB. Let's express $\overline{c}$ in terms of $\overline{a}$ and $\overline{b}$.

$3\overline{c} = 10\overline{a} - 7\overline{b}$

Dividing both sides by 3, we get:

$\overline{c} = \frac{10\overline{a} - 7\overline{b}}{3}$

This can be written in the form of the section formula for external division:

$\overline{c} = \frac{-7\overline{b} + 10\overline{a}}{-7 + 10}$

Comparing this with the section formula $\overline{c} = \frac{m\overline{b} + n\overline{a}}{m + n}$, we get:

$m = -7$ and $n = 10$

The ratio is $m : n = -7 : 10$.

Since the ratio is negative, the division is external. We can also compare it directly with the external division formula $\overline{c} = \frac{m\overline{b} - n\overline{a}}{m - n}$:

$\overline{c} = \frac{7\overline{b} - 10\overline{a}}{7 - 10} = \frac{10\overline{a} - 7\overline{b}}{3}$

Here, $m = 7$ and $n = 10$.

Therefore, the point C divides the line segment AB externally in the ratio 7:10.

Q. 8. Find the angle between the lines $\overline{r}=(\hat{i}+2\hat{j}+3\hat{k})+\lambda(2\hat{i}-\hat{j}+\hat{k})$ and $\overline{r}=(\hat{i}+2\hat{j}+3\hat{k})+\mu(\hat{i}+2\hat{j}+\hat{k})$

Solution:

Let the given lines be:
$L_1: \overline{r} = \overline{a}_1 + \lambda\overline{b}_1$
$L_2: \overline{r} = \overline{a}_2 + \mu\overline{b}_2$

Comparing the given equations with standard forms, the direction vectors of the lines are:
$\overline{b}_1 = 2\hat{i} - \hat{j} + \hat{k}$
$\overline{b}_2 = \hat{i} + 2\hat{j} + \hat{k}$

Let $\theta$ be the acute angle between the two lines. The formula for the angle is:
$\cos \theta = \frac{|\overline{b}_1 \cdot \overline{b}_2|}{|\overline{b}_1| |\overline{b}_2|}$

Now, let's find the dot product $\overline{b}_1 \cdot \overline{b}_2$:
$\overline{b}_1 \cdot \overline{b}_2 = (2\hat{i} - \hat{j} + \hat{k}) \cdot (\hat{i} + 2\hat{j} + \hat{k})$
$\overline{b}_1 \cdot \overline{b}_2 = (2)(1) + (-1)(2) + (1)(1)$
$\overline{b}_1 \cdot \overline{b}_2 = 2 - 2 + 1 = 1$

Next, find the magnitudes of the direction vectors:
$|\overline{b}_1| = \sqrt{(2)^2 + (-1)^2 + (1)^2} = \sqrt{4 + 1 + 1} = \sqrt{6}$
$|\overline{b}_2| = \sqrt{(1)^2 + (2)^2 + (1)^2} = \sqrt{1 + 4 + 1} = \sqrt{6}$

Substitute these values into the formula:
$\cos \theta = \frac{|1|}{\sqrt{6} \times \sqrt{6}}$
$\cos \theta = \frac{1}{6}$

Therefore, the angle between the lines is:
$\theta = \cos^{-1}(\frac{1}{6})$

Q. 9. Test whether the function $f(x) = x - \frac{1}{x}$, $x \in R, x \ne 0$ is increasing or decreasing.

Solution:

The given function is $f(x) = x - \frac{1}{x}$

Differentiating with respect to $x$, we get:

$f'(x) = \frac{d}{dx}(x) - \frac{d}{dx}(\frac{1}{x})$

$f'(x) = 1 - (-\frac{1}{x^2})$

$f'(x) = 1 + \frac{1}{x^2}$

For any real number $x \ne 0$, the square of $x$ is always positive, i.e., $x^2 > 0$.

Therefore, $\frac{1}{x^2} > 0$

Adding 1 to both sides:

$1 + \frac{1}{x^2} > 1 > 0$

This means $f'(x) > 0$ for all $x \in R, x \ne 0$.

Hence, the given function $f(x)$ is strictly increasing for all $x \in R, x \ne 0$.

Q. 10. Evaluate: $\int_{-1}^{1}|x|dx$

Solution:

Let $I = \int_{-1}^{1}|x|dx$

We know the definition of the absolute value function:

$|x| = -x$ for $x < 0$

$|x| = x$ for $x \ge 0$

We split the definite integral at $x = 0$:

$I = \int_{-1}^{0} (-x) dx + \int_{0}^{1} (x) dx$

Integrating both terms:

$I = \left[ -\frac{x^2}{2} \right]_{-1}^{0} + \left[ \frac{x^2}{2} \right]_{0}^{1}$

Applying the limits:

$I = \left( 0 - \left(-\frac{(-1)^2}{2}\right) \right) + \left( \frac{1^2}{2} - 0 \right)$

$I = \left( 0 - \left(-\frac{1}{2}\right) \right) + \left( \frac{1}{2} \right)$

$I = \frac{1}{2} + \frac{1}{2}$

$I = 1$

(Alternatively, since $f(x) = |x|$ is an even function, $\int_{-1}^{1}|x|dx = 2\int_{0}^{1}xdx = 2 \left[ \frac{x^2}{2} \right]_{0}^{1} = 2(\frac{1}{2}) = 1$)

Question 11

Evaluate: ∫ x log x dx

Solution:

Let I = ∫ x log x dx

We solve this using the method of integration by parts. The formula is:

∫ u · v dx = u ∫ v dx - ∫ [ (du/dx) · ∫ v dx ] dx

According to the LIATE rule (Logarithmic, Inverse, Algebraic, Trigonometric, Exponential), we choose our functions as follows:

u = log x (Logarithmic function)
v = x (Algebraic function)

Applying the integration by parts formula:

I = (log x) ∫ x dx - ∫ [ (d/dx (log x)) · ∫ x dx ] dx

Now, compute the individual derivatives and integrals:

∫ x dx = x² / 2
d/dx (log x) = 1 / x

Substitute these back into the equation:

I = (log x) · (x² / 2) - ∫ [ (1 / x) · (x² / 2) ] dx

I = (x² / 2) log x - (1 / 2) ∫ x dx

I = (x² / 2) log x - (1 / 2) · (x² / 2) + C

I = (x² / 2) log x - (x² / 4) + C

Where 'C' is the constant of integration.

Question 12

Find the area bounded by curve x² + y = 0, X-axis and lines x = 1 and x = 4.

Solution:

The equation of the given curve is:

x² + y = 0 ⇒ y = -x²

Required area (A) bounded by the curve, the X-axis, and the lines x = 1 and x = 4 is given by:

A = ∫₁⁴ |y| dx

A = ∫₁⁴ |-x²| dx

A = ∫₁⁴ x² dx

Integrating with respect to x:

A = [ x³ / 3 ]₁⁴

A = (4³ / 3) - (1³ / 3)

A = (64 / 3) - (1 / 3)

A = 63 / 3

A = 21 sq. units

Question 13

The probability distribution of r.v. X is as follows:

X = x	0	1	2	3	4
P(X = x)	0.1	k	2k	2k	k

Find : (i) k (ii) F(2)

Solution:

Since P(X = x) is the probability distribution of a random variable X, the sum of all probabilities is equal to 1:

∑ P(X = x) = 1

(i) To find k:

P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) = 1

0.1 + k + 2k + 2k + k = 1

0.1 + 6k = 1

6k = 1 - 0.1

6k = 0.9

k = 0.9 / 6

k = 0.15 (or 3/20)

(ii) To find F(2):

By definition of the cumulative distribution function (c.d.f.):

F(2) = P(X ≤ 2)

F(2) = P(X = 0) + P(X = 1) + P(X = 2)

F(2) = 0.1 + k + 2k

F(2) = 0.1 + 3k

Substituting k = 0.15:

F(2) = 0.1 + 3(0.15)

F(2) = 0.1 + 0.45

F(2) = 0.55 (or 11/20)

Question 14

Evaluate: ∫ [ e^x(1 + x) / cos²(xe^x) ] dx

Solution:

Let I = ∫ [ e^x(1 + x) / cos²(xe^x) ] dx

Use the method of substitution by putting:

xe^x = t

Differentiating both sides with respect to x using the product rule:

[ x · (d/dx (e^x)) + e^x · (d/dx (x)) ] dx = dt

(x e^x + e^x · 1) dx = dt

e^x(1 + x) dx = dt

Substitute these values into the integral I:

I = ∫ (1 / cos² t) dt

I = ∫ sec² t dt

I = tan t + C

Resubstituting t = xe^x:

I = tan(xe^x) + C

Where 'C' is the constant of integration.

Section C

Question 15

Problem: Determine using a truth table whether the logical expression [(p ∨ q) ∧ ~p] ∧ ~q is a tautology, a contradiction, or a contingency.

Solution:

p	q	~p	~q	p ∨ q	(p ∨ q) ∧ ~p	[(p ∨ q) ∧ ~p] ∧ ~q
T	T	F	F	T	F	F
T	F	F	T	T	F	F
F	T	T	F	T	T	F
F	F	T	T	F	F	F

All the truth values in the last column are False (F).

Conclusion: The given statement pattern is a contradiction.

Question 16

Problem: Show that sin^-1(3/5) + cos^-1(12/13) = sin^-1(56/65).

Solution:

Let sin^-1(3/5) = x and cos^-1(12/13) = y, where 0 < x, y < π/2.

Therefore, sin x = 3/5 and cos y = 12/13.

Using the trigonometric identity sin²θ + cos²θ = 1:

cos x = √(1 - sin²x) = √(1 - (3/5)²) = √(1 - 9/25) = √(16/25) = 4/5
sin y = √(1 - cos²y) = √(1 - (12/13)²) = √(1 - 144/169) = √(25/169) = 5/13

We know the compound angle formula:

sin(x + y) = sin x cos y + cos x sin y

sin(x + y) = (3/5)(12/13) + (4/5)(5/13)

sin(x + y) = (36/65) + (20/65)

sin(x + y) = 56/65

x + y = sin^-1(56/65)

Resubstituting the values of x and y:

sin^-1(3/5) + cos^-1(12/13) = sin^-1(56/65)

Hence proved.

Question 17

Problem: Given a triangle ABC with sides a = 10, b = 8, and c = 6, compute the values of (i) sin(C/2) and (ii) cos(B/2).

Solution:

Given: a = 10, b = 8, c = 6

First, find the semi-perimeter (s) of ΔABC:

2s = a + b + c = 10 + 8 + 6 = 24

s = 12

Now compute:

s - a = 12 - 10 = 2
s - b = 12 - 8 = 4
s - c = 12 - 6 = 6

(i) To find sin(C/2):

By the half-angle formula:

sin(C/2) = √[ (s - a)(s - b) / (ab) ]

sin(C/2) = √[ (2 × 4) / (10 × 8) ]

sin(C/2) = √(8 / 80) = √(1 / 10)

sin(C/2) = 1 / √10

(ii) To find cos(B/2):

By the half-angle formula:

cos(B/2) = √[ s(s - b) / (ac) ]

cos(B/2) = √[ (12 × 4) / (10 × 6) ]

cos(B/2) = √(48 / 60) = √(4 / 5)

cos(B/2) = 2 / √5

Question 18

Problem: Determine with justification whether the four points A(1, 2, 1), B(2, -3, 4), C(3, 4, -5), and D(2, 3, -2) lie on the same plane.

Solution:

Let the position vectors of the given points be:

a = î + 2ĵ + k̂
b = 2î - 3ĵ + 4k̂
c = 3î + 4ĵ - 5k̂
d = 2î + 3ĵ - 2k̂

Find the vectors AB, AC, and AD:

AB = b - a = (2 - 1)î + (-3 - 2)ĵ + (4 - 1)k̂ = î - 5ĵ + 3k̂
AC = c - a = (3 - 1)î + (4 - 2)ĵ + (-5 - 1)k̂ = 2î + 2ĵ - 6k̂
AD = d - a = (2 - 1)î + (3 - 2)ĵ + (-2 - 1)k̂ = î + ĵ - 3k̂

Now, evaluate the scalar triple product [AB AC AD]:

\| 1	-5	3 \|
\| 2	2	-6 \|
\| 1	1	-3 \|

[AB AC AD] = 1( (2)(-3) - (-6)(1) ) - (-5)( (2)(-3) - (-6)(1) ) + 3( (2)(1) - (2)(1) )

[AB AC AD] = 1(-6 + 6) + 5(-6 + 6) + 3(2 - 2)

[AB AC AD] = 1(0) + 5(0) + 3(0) = 0

Since the scalar triple product [AB AC AD] = 0, the vectors AB, AC, and AD are coplanar.

Conclusion: Yes, the four points A, B, C, and D are coplanar.

Question 19

Problem: Calculate the shortest distance between the parallel lines x/2 = y/(-1) = z/2 and (x + 1)/2 = (y - 1)/(-1) = (z + 1)/2.

Solution:

The equations of the given parallel lines are:

Line 1: x/2 = y/(-1) = z/2
Line 2: (x + 1)/2 = (y - 1)/(-1) = (z + 1)/2

From the equations of the lines:

Position vector of a point on the first line: a₁ = 0î + 0ĵ + 0k̂
Position vector of a point on the second line: a₂ = -î + ĵ - k̂
Direction vector parallel to both lines: b = 2î - ĵ + 2k̂

Now, compute a₂ - a₁:

a₂ - a₁ = -î + ĵ - k̂

Next, compute the cross product (a₂ - a₁) × b:

\| î	ĵ	k̂ \|
\| -1	1	-1 \|
\| 2	-1	2 \|

(a₂ - a₁) × b = î(2 - 1) - ĵ(-2 + 2) + k̂(1 - 2)

(a₂ - a₁) × b = î - 0ĵ - k̂ = î - k̂

Find the magnitudes:

|(a₂ - a₁) × b| = √(1² + 0² + (-1)²) = √2
|b| = √(2² + (-1)² + 2²) = √(4 + 1 + 4) = √9 = 3

The distance (d) between the two parallel lines is given by:

d = |(a₂ - a₁) × b| / |b|

d = (√2 / 3) units

Question 20

Problem: Determine the vector equation of the plane passing through the points A(2, 1, 1), B(0, 2, 3), and C(4, 5, 6).

Solution:

Let the position vectors of points A, B, and C be a, b, and c respectively:

a = 2î + ĵ + k̂
b = 0î + 2ĵ + 3k̂
c = 4î + 5ĵ + 6k̂

Then:

AB = b - a = (0 - 2)î + (2 - 1)ĵ + (3 - 1)k̂ = -2î + ĵ + 2k̂
AC = c - a = (4 - 2)î + (5 - 1)ĵ + (6 - 1)k̂ = 2î + 4ĵ + 5k̂

A normal vector to the plane is given by n = AB × AC:

\| î	ĵ	k̂ \|
\| -2	1	2 \|
\| 2	4	5 \|

AB × AC = î(5 - 8) - ĵ(-10 - 4) + k̂(-8 - 2)

AB × AC = -3î + 14ĵ - 10k̂

Now calculate a · (AB × AC):

a · (AB × AC) = (2î + ĵ + k̂) · (-3î + 14ĵ - 10k̂)

a · (AB × AC) = 2(-3) + 1(14) + 1(-10) = -6 + 14 - 10 = -2

The vector equation of the plane passing through three points is:

r · (AB × AC) = a · (AB × AC)

r · (-3î + 14ĵ - 10k̂) = -2

Or equivalently:

r · (3î - 14ĵ + 10k̂) = 2

Question 21

Problem: Given y = log[ (x + √(x² + a²)) / (√(x² + a²) - x) ], find dy/dx.

Solution:

Given:

y = log[ (x + √(x² + a²)) / (√(x² + a²) - x) ]

Using the logarithmic property log(m / n) = log m - log n:

y = log(x + √(x² + a²)) - log(√(x² + a²) - x)

Differentiating both sides with respect to x:

dy/dx = [ 1 / (x + √(x² + a²)) ] · d/dx (x + √(x² + a²)) - [ 1 / (√(x² + a²) - x) ] · d/dx (√(x² + a²) - x)

dy/dx = [ 1 / (x + √(x² + a²)) ] · [ 1 + (2x / (2√(x² + a²))) ] - [ 1 / (√(x² + a²) - x) ] · [ (2x / (2√(x² + a²))) - 1 ]

dy/dx = [ 1 / (x + √(x² + a²)) ] · [ (√(x² + a²) + x) / √(x² + a²) ] - [ 1 / (√(x² + a²) - x) ] · [ (x - √(x² + a²)) / √(x² + a²) ]

dy/dx = [ 1 / √(x² + a²) ] - [ -(√(x² + a²) - x) / ( (√(x² + a²) - x) √(x² + a²) ) ]

dy/dx = [ 1 / √(x² + a²) ] + [ 1 / √(x² + a²) ]

dy/dx = 2 / √(x² + a²)

Question 22

Problem: A wire of total length 36 cm is bent into the shape of a rectangle. Find the dimensions of the rectangle so that its enclosed area is maximized.

Solution:

Let the length of the rectangle be x cm and the breadth of the rectangle be y cm.

Given that the perimeter of the rectangle is equal to the length of the wire (36 cm):

2(x + y) = 36

x + y = 18

y = 18 - x — (Equation 1)

Let A be the area of the rectangle:

A = x · y

A = x(18 - x)

A = 18x - x²

Differentiating A with respect to x:

dA/dx = 18 - 2x

Differentiating again with respect to x:

d²A/dx² = -2

For maximum or minimum area, put dA/dx = 0:

18 - 2x = 0

2x = 18 ⇒ x = 9

At x = 9, d²A/dx² = -2 < 0.

By the second derivative test, the area A is maximum when x = 9 cm.

Substituting x = 9 in Equation 1:

y = 18 - 9 ⇒ y = 9

Conclusion: For the area to be maximum, the dimensions of the rectangle are length = 9 cm and breadth = 9 cm.

Question 23

Problem: Verify Rolle's theorem for the polynomial function f(x) = x² - 5x + 9 on the interval [1, 4].

Solution:

Given function: f(x) = x² - 5x + 9 on x ∈ [1, 4].

Since f(x) is a polynomial function:

f(x) is continuous on the closed interval [1, 4].
f(x) is differentiable on the open interval (1, 4).

Now check the values of f(x) at the endpoints of the interval:

f(1) = (1)² - 5(1) + 9 = 1 - 5 + 9 = 5
f(4) = (4)² - 5(4) + 9 = 16 - 20 + 9 = 5

Here, f(1) = f(4) = 5.

Thus, all three conditions of Rolle's theorem are satisfied. Therefore, there exists at least one real number c ∈ (1, 4) such that f'(c) = 0.

Differentiating f(x) with respect to x:

f'(x) = 2x - 5

Put f'(c) = 0:

2c - 5 = 0

2c = 5 ⇒ c = 5/2 = 2.5

Since c = 2.5 ∈ (1, 4), Rolle's theorem is verified.

Question 24

Problem: Find the general solution of the differential equation x(dy/dx) - √(x² + y²) = y.

Solution:

Given differential equation:

x(dy/dx) - √(x² + y²) = y

x(dy/dx) = y + √(x² + y²)

dy/dx = [ y + √(x² + y²) ] / x — (Equation 1)

This is a homogeneous differential equation. Put y = vx.

Differentiating with respect to x:

dy/dx = v + x(dv/dx)

Substitute y = vx and dy/dx into Equation 1:

v + x(dv/dx) = [ vx + √(x² + v²x²) ] / x

v + x(dv/dx) = [ vx + x√(1 + v²) ] / x

v + x(dv/dx) = v + √(1 + v²)

x(dv/dx) = √(1 + v²)

Separating the variables:

[ 1 / √(1 + v²) ] dv = (1 / x) dx

Integrating both sides:

∫ [ 1 / √(1 + v²) ] dv = ∫ (1 / x) dx

log|v + √(1 + v²)| = log|x| + log|c|

log|v + √(1 + v²)| = log|cx|

v + √(1 + v²) = cx

Resubstituting v = y / x:

(y / x) + √(1 + y²/x²) = cx

[ y + √(x² + y²) ] / x = cx

y + √(x² + y²) = cx²

This is the required general solution.

Question 25

Problem: For a continuous random variable X with probability density function f(x) = x/8 for 0 < x < 4 (and 0 otherwise), find (i) the cumulative distribution function F(x), and (ii) the values of F(x) at x = 0.5 and x = 5.

Solution:

Given p.d.f. of continuous random variable X:

f(x) = x / 8, for 0 < x < 4
f(x) = 0, otherwise

(i) Expression for c.d.f. F(x):

By definition, F(x) = ∫_-∞^x f(t) dt.

For x ≤ 0: F(x) = 0
For 0 < x < 4:
F(x) = ∫₀^x (t / 8) dt = (1 / 8) [ t² / 2 ]₀^x = x² / 16
For x ≥ 4: F(x) = 1

(ii) Values of F(x) at x = 0.5 and x = 5:

At x = 0.5: Since 0 < 0.5 < 4, we use F(x) = x² / 16:
F(0.5) = (0.5)² / 16 = 0.25 / 16 = (1/4) / 16 = 1 / 64 (or 0.015625)
At x = 5: Since x = 5 ≥ 4, the cumulative probability reaches its maximum value:
F(5) = 1

Question 26

Problem: Given the probability mass function of a discrete random variable X as P(X = x) = ⁴C_x (5/9)^x (4/9)^4-x for x = 0, 1, 2, 3, 4 (and 0 otherwise), calculate the expected value E(X) and variance Var(X).

Solution:

Given p.m.f.:

P(X = x) = ⁴C_x (5/9)^x (4/9)^4-x, for x = 0, 1, 2, 3, 4

Comparing this with the standard Binomial distribution formula P(X = x) = ⁿC_x p^x q^n-x, we identify that X follows a Binomial distribution B(n, p) with:

Number of trials, n = 4
Probability of success, p = 5/9
Probability of failure, q = 4/9

1. To find E(X):

For a binomial distribution, the mean or expected value is given by:

E(X) = n · p

E(X) = 4 × (5/9)

E(X) = 20 / 9

2. To find Var(X):

For a binomial distribution, the variance is given by:

Var(X) = n · p · q

Var(X) = 4 × (5/9) × (4/9)

Var(X) = 80 / 81

Section D

Question 27

Problem: Find the inverse of the matrix:

[ -4	-3	-3 ]
[ 1	0	1 ]
[ 4	4	3 ]

Solution:

Let the given matrix be A:

[ -4	-3	-3 ]
[ 1	0	1 ]
[ 4	4	3 ]

Step 1: Find the determinant |A|

|A| = -4(0 × 3 - 1 × 4) - (-3)(1 × 3 - 1 × 4) + (-3)(1 × 4 - 0 × 4)

|A| = -4(0 - 4) + 3(3 - 4) - 3(4 - 0)

|A| = -4(-4) + 3(-1) - 3(4)

|A| = 16 - 3 - 12 = 1 ≠ 0

Since |A| ≠ 0, the matrix A is non-singular and A^-1 exists.

Step 2: Find the cofactors of the elements of matrix A

A₁₁ = (-1)¹⁺¹ (0 - 4) = -4
A₁₂ = (-1)¹⁺² (3 - 4) = -(-1) = 1
A₁₃ = (-1)¹⁺³ (4 - 0) = 4
A₂₁ = (-1)²⁺¹ (-9 - (-12)) = -(3) = -3
A₂₂ = (-1)²⁺² (-12 - (-12)) = 0
A₂₃ = (-1)²⁺³ (-16 - (-12)) = -(-4) = 4
A₃₁ = (-1)³⁺¹ (-3 - 0) = -3
A₃₂ = (-1)³⁺² (-4 - (-3)) = -(-1) = 1
A₃₃ = (-1)³⁺³ (0 - (-3)) = 3

The matrix of cofactors [A_ij] is:

[ -4	1	4 ]
[ -3	0	4 ]
[ -3	1	3 ]

Step 3: Find the adjoint of matrix A

adj(A) is the transpose of the cofactor matrix:

[ -4	-3	-3 ]
[ 1	0	1 ]
[ 4	4	3 ]

Step 4: Find the inverse A^-1

Using the formula A^-1 = (1 / |A|) · adj(A):

A^-1 = (1 / 1) · adj(A)

Therefore, A^-1 is:

[ -4	-3	-3 ]
[ 1	0	1 ]
[ 4	4	3 ]

Question 28

Problem: Prove that the homogeneous equation of degree two in x and y, ax² + 2hxy + by² = 0, represents a pair of lines passing through the origin if h² - ab ≥ 0.

Solution:

Consider the homogeneous equation of degree two in x and y:

ax² + 2hxy + by² = 0 — (Equation 1)

In this second-degree equation, the coefficients a, h, and b cannot all be zero simultaneously. We consider two cases based on the value of b:

Case 1: When b = 0

If b = 0, then a and h cannot both be zero. Substituting b = 0 into Equation 1 gives:

ax² + 2hxy = 0

x(ax + 2hy) = 0

This gives two linear equations:

x = 0 and ax + 2hy = 0

Since both are first-degree equations in x and y with no constant term, they represent two straight lines passing through the origin.

Case 2: When b ≠ 0

Multiply both sides of Equation 1 by b:

abx² + 2hbxy + b²y² = 0

b²y² + 2hbxy = -abx²

To complete the square on the left-hand side, add h²x² to both sides:

b²y² + 2hbxy + h²x² = h²x² - abx²

(by + hx)² = (h² - ab)x²

Given that h² - ab ≥ 0, we can write (h² - ab) = [√(h² - ab)]²:

(hx + by)² - [x√(h² - ab)]² = 0

Factoring using a² - b² = (a + b)(a - b):

[ (hx + by) + x√(h² - ab) ] [ (hx + by) - x√(h² - ab) ] = 0

[ (h + √(h² - ab))x + by ] [ (h - √(h² - ab))x + by ] = 0

This equation is the combined equation of the two lines:

(h + √(h² - ab))x + by = 0
(h - √(h² - ab))x + by = 0

Since both equations are linear in x and y and lack constant terms, they represent a pair of straight lines passing through the origin.

Hence proved.

Question 29

Problem: Prove that, if a, b, c are three non-coplanar vectors, then any vector r in space can be uniquely expressed as a linear combination of a, b, c.

Solution:

Part 1: Existence of the Linear Combination

Let a, b, c be three non-coplanar vectors in space, and let r be any arbitrary vector in space.

Take a fixed point O in space as the origin. Let the vectors a, b, c, and r be positioned along the directed line segments from O such that OP = r, and the lines along a, b, and c pass through O.

Through point P, draw planes parallel to the planes determined by the pairs of vectors (b, c), (c, a), and (a, b). Let these planes intersect the lines containing a, b, and c at points A, B, and C respectively, forming a parallelepiped with OP as its diagonal and OA, OB, OC as its coterminous edges.

Since OA is collinear with a, there exists a scalar x ∈ R such that OA = xa.
Since OB is collinear with b, there exists a scalar y ∈ R such that OB = yb.
Since OC is collinear with c, there exists a scalar z ∈ R such that OC = zc.

By the parallelepiped law of vector addition:

OP = OA + OB + OC

r = xa + yb + zc

Thus, the vector r can be expressed as a linear combination of a, b, and c.

Part 2: Uniqueness of the Representation

Suppose the vector r can also be expressed as another linear combination of a, b, and c using scalars x', y', z':

r = x'a + y'b + z'c

Equating the two expressions for r:

xa + yb + zc = x'a + y'b + z'c

(x - x')a + (y - y')b + (z - z')c = 0

Since a, b, and c are three non-coplanar vectors, they are linearly independent. If any scalar coefficient were non-zero (for example, if x - x' ≠ 0), we could express a as a linear combination of b and c, which would contradict the fact that a, b, and c are non-coplanar.

Therefore, all coefficients must equal zero:

x - x' = 0 ⇒ x = x'
y - y' = 0 ⇒ y = y'
z - z' = 0 ⇒ z = z'

Hence, the linear combination is unique.

Hence proved.

Question 30

Problem: Solve the linear programming problem (L.P.P.) by graphical method:

Minimize: z = 8x + 10y

Subject to:

2x + y ≥ 7
2x + 3y ≥ 15
x ≥ 0
y ≥ 2

Solution:

Step 1: Convert the given inequations into equations to plot the boundary lines

Inequation	Equation of Line	Points on the Line	Origin Test (0, 0)	Feasible Half-Plane
2x + y ≥ 7	2x + y = 7 (L₁)	(0, 7), (3.5, 0)	0 ≥ 7 (False)	Non-origin side of L₁
2x + 3y ≥ 15	2x + 3y = 15 (L₂)	(0, 5), (7.5, 0)	0 ≥ 15 (False)	Non-origin side of L₂
y ≥ 2	y = 2 (L₃)	(0, 2), line parallel to X-axis	0 ≥ 2 (False)	Non-origin side (above y = 2)
x ≥ 0	x = 0 (Y-axis)	Along Y-axis	—	Right side of Y-axis

Step 2: Determine the corner points of the feasible region

Plotting the lines and shading the common region in the first quadrant gives an unbounded feasible region with vertices A, B, and C:

Point A: Intersection of x = 0 and 2x + y = 7:
Substituting x = 0 gives y = 7. Thus, A = (0, 7).
Point B: Intersection of 2x + y = 7 and 2x + 3y = 15:
Subtracting (2x + y = 7) from (2x + 3y = 15) gives 2y = 8 ⇒ y = 4.

Substituting y = 4 into 2x + y = 7 gives 2x + 4 = 7 ⇒ 2x = 3 ⇒ x = 1.5 (or 3/2).

Thus, B = (1.5, 4).
Point C: Intersection of 2x + 3y = 15 and y = 2:
Substituting y = 2 into 2x + 3y = 15 gives 2x + 6 = 15 ⇒ 2x = 9 ⇒ x = 4.5 (or 9/2).

Thus, C = (4.5, 2).

Step 3: Evaluate the objective function z = 8x + 10y at the corner points

Corner Point	Coordinates (x, y)	Value of z = 8x + 10y
A	(0, 7)	z = 8(0) + 10(7) = 0 + 70 = 70
B	(1.5, 4)	z = 8(1.5) + 10(4) = 12 + 40 = 52 (Minimum)
C	(4.5, 2)	z = 8(4.5) + 10(2) = 36 + 20 = 56

Conclusion: The minimum value of z is 52, which occurs at the point x = 1.5 (or 3/2) and y = 4.

Question 31

Problem: If x = f(t) and y = g(t) are differentiable functions of t, so that y is a differentiable function of x and dx/dt ≠ 0, then prove that dy/dx = (dy/dt) / (dx/dt). Hence find dy/dx, if x = sin t and y = cos t.

Solution:

Part 1: Proof of Parametric Differentiation Theorem

Let x = f(t) and y = g(t) be differentiable functions of a parameter t.

Let δt be a small increment in t, and let δx and δy be the corresponding small increments in x and y respectively, such that δx ≠ 0. As δt → 0, we have δx → 0 and δy → 0.

Since x and y are differentiable functions of t:

lim_{δt → 0} (δx / δt) = dx/dt and lim_{δt → 0} (δy / δt) = dy/dt

Consider the increment ratio:

δy / δx = (δy / δt) / (δx / δt), where (δx / δt) ≠ 0

Taking the limit as δt → 0 on both sides:

lim_{δt → 0} (δy / δx) = [ lim_{δt → 0} (δy / δt) ] / [ lim_{δt → 0} (δx / δt) ]

Since δx → 0 as δt → 0, and y is a differentiable function of x, the left-hand side limit exists and equals dy/dx:

lim_{δx → 0} (δy / δx) = (dy/dt) / (dx/dt)

dy/dx = (dy/dt) / (dx/dt), where dx/dt ≠ 0

Hence proved.

Part 2: To find dy/dx when x = sin t and y = cos t

Given: x = sin t and y = cos t

Differentiating both with respect to t:

dx/dt = d/dt (sin t) = cos t
dy/dt = d/dt (cos t) = -sin t

Using the parametric differentiation formula:

dy/dx = (dy/dt) / (dx/dt)

dy/dx = -sin t / cos t

dy/dx = -tan t

Question 32

Problem: Evaluate: ∫ [ 2 / (3 + 2sin²x + 5cos²x) ] dx

Solution:

Let I = ∫ [ 2 / (3 + 2sin²x + 5cos²x) ] dx

Divide both the numerator and the denominator by cos²x:

I = ∫ [ (2 / cos²x) / ( (3 / cos²x) + 2(sin²x / cos²x) + 5(cos²x / cos²x) ) ] dx

I = ∫ [ 2sec²x / (3sec²x + 2tan²x + 5) ] dx

Use the trigonometric identity sec²x = 1 + tan²x in the denominator:

I = ∫ [ 2sec²x / (3(1 + tan²x) + 2tan²x + 5) ] dx

I = ∫ [ 2sec²x / (3 + 3tan²x + 2tan²x + 5) ] dx

I = ∫ [ 2sec²x / (5tan²x + 8) ] dx

Substitute tan x = t.

Differentiating with respect to x gives: sec²x dx = dt.

Substitute into the integral I:

I = ∫ [ 2 / (5t² + 8) ] dt

I = (2 / 5) ∫ [ 1 / (t² + 8/5) ] dt

I = (2 / 5) ∫ [ 1 / (t² + (√(8/5))²) ] dt

Using the standard integration formula ∫ [ 1 / (x² + a²) ] dx = (1 / a) tan^-1(x / a) + C:

I = (2 / 5) · [ 1 / √(8/5) ] tan^-1[ t / √(8/5) ] + C

I = (2 / 5) · ( √5 / 2√2 ) tan^-1( √5 t / 2√2 ) + C

I = ( 1 / √10 ) tan^-1( √5 t / 2√2 ) + C

Resubstituting t = tan x:

I = ( 1 / √10 ) tan^-1[ ( √5 tan x ) / ( 2√2 ) ] + C

Where 'C' is the constant of integration.

Question 33

Problem: Prove that:

∫_-a^a f(x) dx = 2 ∫₀^a f(x) dx, if f(x) is an even function

∫_-a^a f(x) dx = 0, if f(x) is an odd function

Solution:

Since -a < 0 < a, by the additive property of definite integrals, we can split the integral at 0:

∫_-a^a f(x) dx = ∫_-a⁰ f(x) dx + ∫₀^a f(x) dx — (Equation 1)

Consider the first integral on the right-hand side:

I₁ = ∫_-a⁰ f(x) dx

Put x = -t. Differentiating with respect to t gives dx = -dt.

Changing the limits of integration:

When x = -a, t = a
When x = 0, t = 0

Substituting into I₁:

I₁ = ∫_a⁰ f(-t) (-dt) = -∫_a⁰ f(-t) dt

Using the property -∫_a^b g(t) dt = ∫_b^a g(t) dt:

I₁ = ∫₀^a f(-t) dt

Since a definite integral is independent of the variable of integration (∫₀^a f(-t) dt = ∫₀^a f(-x) dx):

I₁ = ∫₀^a f(-x) dx

Substituting I₁ back into Equation 1:

∫_-a^a f(x) dx = ∫₀^a f(-x) dx + ∫₀^a f(x) dx

∫_-a^a f(x) dx = ∫₀^a [ f(-x) + f(x) ] dx — (Equation 2)

Case 1: If f(x) is an even function

By definition of an even function, f(-x) = f(x). Substituting this into Equation 2:

∫_-a^a f(x) dx = ∫₀^a [ f(x) + f(x) ] dx

∫_-a^a f(x) dx = 2 ∫₀^a f(x) dx

Case 2: If f(x) is an odd function

By definition of an odd function, f(-x) = -f(x). Substituting this into Equation 2:

∫_-a^a f(x) dx = ∫₀^a [ -f(x) + f(x) ] dx

∫_-a^a f(x) dx = 0

Hence proved.

Question 34

Problem: A body cools according to Newton's law from 100°C to 60°C in 20 minutes. The temperature of the surrounding being 20°C. How long will it take to cool down to 40°C?

Solution:

Let θ be the temperature of the body at time t minutes. The temperature of the surrounding medium is θ₀ = 20°C.

According to Newton's Law of Cooling, the rate of change of temperature is directly proportional to the difference between the temperature of the body and the surrounding medium:

dθ/dt ∝ -(θ - θ₀)

dθ/dt = -k(θ - 20), where k > 0 is the constant of proportionality.

Separating the variables:

[ 1 / (θ - 20) ] dθ = -k dt

Integrating both sides:

∫ [ 1 / (θ - 20) ] dθ = -k ∫ dt

log(θ - 20) = -kt + c — (Equation 1)

Step 1: Find the constant c using the initial condition

Initially, at t = 0, θ = 100°C. Substituting into Equation 1:

log(100 - 20) = -k(0) + c

c = log 80

Substituting c = log 80 back into Equation 1:

log(θ - 20) = -kt + log 80

log(θ - 20) - log 80 = -kt

log[ (θ - 20) / 80 ] = -kt — (Equation 2)

Step 2: Find the value of -k using the second condition

Given that at t = 20 minutes, θ = 60°C. Substituting into Equation 2:

log[ (60 - 20) / 80 ] = -20k

log(40 / 80) = -20k

log(1 / 2) = -20k

-k = (1 / 20) log(1 / 2)

Substituting -k into Equation 2 gives:

log[ (θ - 20) / 80 ] = (t / 20) log(1 / 2) — (Equation 3)

Step 3: Find time t when θ = 40°C

Substitute θ = 40 into Equation 3:

log[ (40 - 20) / 80 ] = (t / 20) log(1 / 2)

log(20 / 80) = (t / 20) log(1 / 2)

log(1 / 4) = (t / 20) log(1 / 2)

log[ (1 / 2)² ] = (t / 20) log(1 / 2)

2 log(1 / 2) = (t / 20) log(1 / 2)

Dividing both sides by log(1 / 2):

2 = t / 20

t = 40 minutes

Conclusion: It will take 40 minutes in total for the body to cool down to 40°C.

HSC 12th Maths June 2026 Board Question Paper (J-276) – Original Images

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HSC 12th Secretarial Practice (SP) Board Paper Solution June 2026 | Maharashtra Board

aaminomtex@gmail.com (Amin) — Sat, 24 Oct 2026 08:42:41 +0530

HSC SP 12th Standard June 2026 Board Paper Solution

Q. 1. Objective Questions

(A) Select the correct answer from the options given below and rewrite the sentences:

Company has to pay taxes to government.[cite: 1]
A depository receipt traded in U.S.A. is called American Depository Receipt.[cite: 1]
Transmission of shares takes place due to operation of law.[cite: 1]
Money from Debenture Redemption Reserve account is used for redemption of debentures.[cite: 1]
Deposits are short term loans of the company.[cite: 2]

(B) Write a word or a term or a phrase which can substitute each of the following statements:

The agent of the depository. - Depository Participant (DP)[cite: 2]
The return paid to the creditors by the company. - Interest[cite: 2]
Instrument for payment of dividend. - Dividend Warrant[cite: 2]
A specific place where trading of securities is arranged in an organized method. - Stock Exchange[cite: 2]
A market which provides short term funds. - Money Market[cite: 2]

(C) Match the pairs:

(a) Working capital: (4) Sum of current assets[cite: 2]
(b) Board of Directors: (5) Power to issue debentures[cite: 2]
(c) Tenure of deposits: (1) Maximum 36 months[cite: 2]
(d) Financial Market: (2) Trading of financial securities[cite: 2]
(e) Bad delivery: (3) Problem faced in physical mode[cite: 2]

(D) Find the odd one:

Land and Building, Plant and Machinery, Cash - Cash[cite: 3]
Face value, Redemption value, Market value - Market value[cite: 3]
SEBI, Debentureholder, Shareholder - SEBI[cite: 3]
Dividend account, Dividend mandate, Unpaid/Unclaimed dividend account - Dividend mandate[cite: 3]
New issue market, Secondary market, Call money market - Call money market[cite: 3]

Q. 2. Explain the following terms / concepts (Any FOUR):

(i) Investment Decision

Meaning: Investment decision refers to the careful selection of assets in which a company will invest its funds. It ensures that the funds are utilized effectively to earn maximum possible returns for the investors.

Scope: It primarily involves two major areas: investing in fixed or long-term assets (known as Capital Budgeting) and investing in current or short-term assets (known as Working Capital Management).

(ii) Trade Credit

Meaning: Trade credit is a short-term financing facility where a manufacturer or wholesaler sells goods to a retailer or another business on credit. It allows the buyer to purchase goods now and pay for them at a later scheduled date.

Significance: It is one of the most common and easy sources of working capital. It does not involve the actual borrowing of cash but rather delays the cash payment, which helps businesses manage their short-term cash flows efficiently.

(iii) Initial Public Offer (IPO)

Meaning: Initial Public Offer (IPO) is the process of offering shares of an unlisted company to the general public for the very first time.

Purpose: By issuing an IPO, a privately held company transforms into a public company to raise fresh capital from the public. This method is used by companies for expansion, growth, or to pay off debts.

(iv) Debenture Certificate

Meaning: A debenture certificate is a formal document issued by a company under its common seal, acting as an official acknowledgment of debt owed to the debenture holder.

Content and Time Limit: It contains details such as the name of the holder, number of debentures allotted, interest rate, and the maturity date. A company must issue debenture certificates within six months from the date of allotment of debentures.

(v) Rematerialization

Meaning: Rematerialization (Remat) is the reverse process of dematerialization. It is the process by which a client can get their electronic (demat) holdings of securities converted back into physical certificates.

Procedure: To initiate this process, the investor has to fill out and submit a Remat Request Form (RRF) to their Depository Participant (DP). Once processed, the electronic balance is reduced, and physical certificates are dispatched to the investor.

(vi) Stock Exchange

Meaning: A stock exchange is an organized and regulated market where financial securities such as shares, debentures, bonds, and mutual funds are bought and sold.

Function: It provides a safe and transparent platform for investors to trade existing securities. In India, stock exchanges like the BSE and NSE function under the strict rules and guidelines formulated by the Securities and Exchange Board of India (SEBI).

Q. 3. Study the following cases / situations and express your opinion (Any TWO):

(i) ALPHA Company Ltd. has made a demand on its shareholders to pay the balance unpaid amount of ₹20/- per share...

Are the shareholders liable to pay ₹20/- for the shares held by them?
Yes, the shareholders are legally liable to pay the unpaid balance amount of ₹20/- per share when the company makes a valid call.
Name the letter sent by the company to its shareholders asking them to pay ₹20/-.
The company sends a Call Letter (or Letter of Call on Shares) to the shareholders.
What happens if a shareholder fails to pay the money within the specified time?
If a shareholder fails to pay the call money within the specified time along with any interest due, the company’s Board of Directors has the right to forfeit (cancel) their shares.

(ii) Mr. Rajesh holds 100 shares of STAR Company Ltd. in demat mode:

He wants to transfer one share each to his wife, daughter and son. Can he do so?
Yes, he can do so. Shares of a public limited company are freely transferable, and he can transfer his holdings in any desired quantity to his family members.
Does he need to submit DRF or DIS if he wants to transfer his shares?
He needs to submit a DIS (Delivery Instruction Slip) to his Depository Participant (DP) to execute the transfer of shares. DRF (Dematerialization Request Form) is used for converting physical shares into electronic form, not for transferring them.
Can he nominate his wife in his demat account?
Yes, he can easily nominate his wife in his demat account. The depository system provides a nomination facility for individual investors.

(iii) DONALD Company Ltd. declares a dividend of ₹10/- per share for F.Y. 2024-25.

Is company under default, if dividend was not paid within 30 days of its declaration?
Yes, the company is in default. As per the provisions of the Companies Act, 2013, a declared dividend must be paid to the registered shareholders within 30 days from the date of its declaration.
Is company right in transferring the unpaid dividend to its Debenture Reserve Account?
No, the company is incorrect. The total amount of dividend which remains unpaid or unclaimed within 30 days must be transferred to a special bank account called the Unpaid Dividend Account, not the Debenture Reserve Account.
Does the company have to transfer the amount of unpaid dividend to IEPF after 30 days?
No. After the expiry of the 30-day payment period, the unpaid amount is first transferred to the Unpaid Dividend Account within the next 7 days. The amount is transferred to the IEPF (Investor Education and Protection Fund) only if it remains unpaid or unclaimed for seven consecutive years in the Unpaid Dividend Account.

Q. 4. Distinguish between the following (Any THREE):

(i) Owned Capital and Borrowed Capital

Points of Difference	Owned Capital	Borrowed Capital
1. Meaning	It is that capital which is contributed by the shareholders or owners of the company.	It is that capital which is borrowed by the company from creditors, banks, or the public.
2. Sources	It is collected by issuing equity shares, preference shares, and retaining earnings.	It is collected by issuing debentures, accepting public deposits, and taking bank loans.
3. Status of Provider	The providers of owned capital (shareholders) are the owners of the company.	The providers of borrowed capital (e.g., debenture holders) are the creditors of the company.
4. Return on Investment	The return paid on owned capital is called 'Dividend', which fluctuates with profits.	The return paid on borrowed capital is called 'Interest', which is paid at a fixed rate.
5. Voting Rights	Equity shareholders (providers of owned capital) enjoy normal voting rights in company meetings.	Creditors do not have voting rights in the general meetings of the company.
6. Security	It is generally unsecured capital. Assets are not charged against it.	It is generally secured capital. Fixed assets are usually kept as a security.

(ii) Rights Shares and Bonus Shares

Points of Difference	Rights Shares	Bonus Shares
1. Meaning	Shares offered by a company to its existing equity shareholders in proportion to their existing holdings.	Shares issued free of cost by a company to its existing equity shareholders out of accumulated profits.
2. Payment	Shareholders have to pay for the rights shares at a price usually lower than the market value.	Shareholders do not have to pay anything as these are issued free of cost.
3. Purpose	The main purpose is to raise fresh working capital or funds for expansion.	The main purpose is the capitalization of accumulated profits or free reserves.
4. Right to Renounce	Shareholders can renounce (transfer) their rights to purchase shares in favor of another person.	Shareholders cannot renounce bonus shares in favor of another person.
5. Minimum Subscription	The company must collect the minimum subscription amount as stated in the prospectus.	There is no question of minimum subscription as no money is collected.

(iii) Interim Dividend and Final Dividend

Points of Difference	Interim Dividend	Final Dividend
1. Meaning	Dividend declared and paid between two Annual General Meetings (AGMs) of the company.	Dividend declared and paid after the close of the financial year at the Annual General Meeting (AGM).
2. Who Declares	It is declared directly by the Board of Directors in a Board Meeting.	It is declared by the shareholders in the AGM upon the recommendation of the Board of Directors.
3. Time of Declaration	It is declared before the finalization of the annual accounts of the company.	It is declared only after the final accounts are prepared, audited, and approved.
4. Source	It is generally paid out of the profits of the current financial year up to the date of declaration.	It is paid out of the current year's profits, free reserves, or capital profits.
5. Rate of Dividend	The rate of interim dividend is generally lower than the final dividend.	The rate of final dividend is generally higher than the interim dividend.

(iv) Jobber and Broker

Points of Difference	Jobber	Broker
1. Meaning	A jobber is a dealer in securities who acts as a market maker and trades on his own behalf.	A broker is a member of the stock exchange who acts as an agent and trades on behalf of clients.
2. Nature of Dealing	A jobber is prohibited from dealing directly with the general public or investors.	A broker deals directly with the general public (investors) and executes their orders.
3. Source of Income	The income of a jobber is the profit earned from the price difference between buying and selling securities.	The income of a broker is the commission or brokerage charged to the clients for services rendered.
4. Role	A jobber acts as a principal who buys and sells securities in his own name.	A broker acts as an intermediary or link between the investor and the jobber.
5. Risk Involved	A jobber assumes high financial risk as he buys securities and holds them in his own inventory.	A broker bears almost no financial risk as he only executes trades on behalf of clients.

Q. 5. Answer in brief (Any TWO) :

(i) State any four provisions of Companies Act, 2013 for issue of debentures.

The Companies Act, 2013 contains several provisions regulating the issue of debentures. Any four provisions are as follows:

No Voting Rights: A company cannot issue any debentures carrying voting rights. Debenture holders are creditors, not owners, and therefore cannot vote in the general meetings of the company.
Types of Debentures: A company can issue secured or unsecured debentures, as well as fully or partly convertible debentures. However, the issue of convertible debentures requires a special resolution passed in a general meeting.
Payment of Interest and Redemption: A company is legally bound to pay interest and redeem the debentures according to the terms and conditions of their issue. It is a compulsory financial obligation.
Creation of Debenture Redemption Reserve (DRR): The company must create a Debenture Redemption Reserve out of its profits available for dividend payment. The money in this account can only be used for the redemption of debentures.

(ii) Explain the features of Interest.

Interest is the return paid by a company to its creditors for using their funds. Its main features are:

Fixed Rate: Interest is paid at a predetermined and fixed rate agreed upon at the time of borrowing, regardless of the company's profit levels.
Compulsory Payment: It is a compulsory obligation for the company. Interest must be paid to creditors (like debenture holders or banks) on time, even if the company suffers a loss.
Charge against Profit: Interest is considered an expense and is a charge against the profits of the company. It is deducted from the company's income before calculating the tax liability.
Prioritized Status: In the event of the winding up of a company, the payment of interest and the repayment of borrowed capital are given priority over the payment of dividends to shareholders.

(iii) State any four functions of financial market.

A financial market plays a crucial role in the economic development of a country. Any four of its functions are:

Mobilization of Savings: The financial market facilitates the transfer of savings from individuals and households (investors) to the corporate sector (borrowers) for productive investments.
Price Discovery: Financial markets help in establishing the price of financial assets. The price is determined by the forces of demand and supply of securities in the market.
Providing Liquidity: It provides liquidity to financial instruments by offering a ready market. Investors can easily and quickly convert their securities into cash whenever required by selling them in the market.
Reduces Cost of Transactions: Financial markets provide valuable information regarding securities and trading, which saves the time, effort, and money of both buyers and sellers, thereby reducing transaction costs.

Q. 6. Justify the following statements (Any TWO):

(i) There are various factors affecting the requirement of fixed capital.

Justification:

Nature of Business: Manufacturing and public utility companies require a huge amount of fixed capital to acquire plant, machinery, and infrastructure, whereas trading companies need relatively less fixed capital.
Size of Business: A firm operating on a large scale requires massive fixed capital for expansive operational facilities compared to a small-scale business.
Scope of Business: If a business undertakes both production and distribution of its goods, its fixed capital requirement will naturally be higher.
Lease or Rent arrangements: If an entrepreneur decides to acquire assets on a lease or rental basis, the need for outright purchase diminishes, thereby reducing the fixed capital requirement.
Thus, the requirement of fixed capital is not uniform for all organizations and is influenced by these various dynamic factors.

(ii) Debenture trustees are appointed by a company issuing debentures.

Justification:

Legal Requirement: As per the Companies Act, 2013, a company issuing a prospectus or inviting more than 500 persons to subscribe to its debentures must appoint one or more Debenture Trustees.
Protection of Interest: Debenture trustees are appointed to protect the interests and rights of the debenture holders. They ensure the company does not take actions detrimental to the investors.
Trust Deed: The company enters into a contract with them called a 'Debenture Trust Deed', which creates a legal charge on the company's assets in favor of the trustees.
Grievance Redressal: In case the company defaults in paying interest or repaying the principal, the trustees have the legal right to sell the charged assets to recover the money. Therefore, their appointment is essential for safeguarding investors.

(iii) All companies cannot accept deposits from public.

Justification:

Private Companies: According to the Companies Act, 2013, a private company is strictly prohibited from inviting or accepting deposits from the general public. It can only accept deposits from its members, directors, or their relatives.
Eligible Public Companies: Only 'Eligible Public Companies' are permitted to accept deposits from the public. To be eligible, a public company must have a net worth of not less than ₹100 crores or a turnover of not less than ₹500 crores.
Compliance and Rating: Even eligible companies must obtain the prior consent of their shareholders via a special resolution and secure a credit rating from a recognized agency.
RBI Regulations: Banking companies, Non-Banking Financial Companies (NBFCs), and housing finance companies are governed by separate RBI regulations for accepting deposits. Thus, general public deposit acceptance is restricted and highly regulated.

(iv) Electronic holding of securities is safer than physical holding.

Justification:

Elimination of Physical Risks: Electronic holding (dematerialization) completely eliminates the risks associated with physical certificates, such as loss in transit, theft, mutilation, or damage by fire and water.
No Bad Deliveries: In the physical system, trading can be halted due to 'bad deliveries' caused by torn certificates or signature mismatches. Electronic holdings guarantee clean and precise transfers.
Instant Transfer: Transferring securities electronically is instantaneous and seamless, saving the time that was previously wasted in sending physical documents to company registrars.
Automatic Corporate Benefits: In demat holding, corporate benefits like dividends, bonus shares, and rights issues are automatically and directly credited to the investor's bank or demat account. Hence, it is significantly safer and far more efficient.

Q. 7. Attempt the following (Any TWO) :

(i) Write a letter to the member for the issue of Share certificate.

TATA MOTORS LTD.
Registered Office: Bombay House, 24, Homi Mody Street, Fort, Mumbai - 400 001.
CIN: L28920MH1945PLC004520
Phone: 022 - 2204 4356
Website: www.tatamotors.com
Email: investors@tatamotors.com

Ref. No.: TM/SC/102/2025-26
Date: 25th July, 2026

To,
Mr. Rohan Sharma,
15, Vasant Vihar,
Pune - 411 004.

Subject: Issue of Share Certificate.

Dear Sir,

This is to inform you that, as per your application No. 4567 and the subsequent Board Resolution passed in the Board Meeting held on 15th July 2026, you have been allotted 100 (One Hundred) Equity Shares of ₹10/- each fully paid up.

Your Share Certificate is enclosed herewith. The details of your shareholding are as follows:

Folio No.	Share Certificate No.	Distinctive Numbers (From - To)	Total No. of Shares
R/304	9087	45001 - 45100	100

Please keep this Share Certificate safely, as it is the primary evidence of your title to the shares.

Thanking you,

Yours faithfully,
For Tata Motors Ltd.
(Sign)
Company Secretary

Encl: Share Certificate No. 9087

(ii) Draft a letter of allotment to debentureholder.

RELIANCE INDUSTRIES LTD.
Registered Office: Maker Chambers IV, Nariman Point, Mumbai - 400 021.
CIN: L25111MH1973PLC019278
Phone: 022 - 2278 5000
Website: www.ril.com
Email: investor.relations@ril.com

Ref. No.: RIL/DA/441/2025-26
Date: 10th August, 2026

To,
Mrs. Sunita Verma,
42, Park Street,
Kolkata - 700 016.

Subject: Allotment of Debentures.

Dear Madam,

In response to your application No. D-8890 dated 20th July 2026, I am directed by the Board of Directors to inform you that, at the Board Meeting held on 5th August 2026, you have been allotted 50 (Fifty) 9% Non-Convertible Debentures of ₹100/- each, for a tenure of 5 years.

The details of the allotment are given below:

Folio No.	No. of Debentures Applied	No. of Debentures Allotted	Distinctive Numbers (From - To)	Amount Received (₹)
D/112	50	50	10201 - 10250	5,000/-

The Debenture Certificate will be dispatched to your registered address within the stipulated time limit of six months from the date of allotment.

Thanking you,

Yours faithfully,
For Reliance Industries Ltd.
(Sign)
Company Secretary

(iii) Draft a letter to depositor informing him about payment of interest through Interest Warrant.

INFOSYS LIMITED
Registered Office: Electronics City, Hosur Road, Bengaluru - 560 100.
CIN: L85110KA1981PLC013115
Phone: 080 - 2852 0261
Website: www.infosys.com
Email: investorservices@infosys.com

Ref. No.: INF/FD/INT/99/2025-26
Date: 5th September, 2026

To,
Mr. Amit Patel,
10, Navrangpura,
Ahmedabad - 380 009.

Subject: Payment of Interest on Fixed Deposit.

Dear Sir,

I am instructed by the Board of Directors to convey to you that the interest on your Fixed Deposit with our company is due for payment. The interest has been calculated @ 8.5% p.a. for the year ending 31st August 2026.

The details of the interest payable to you are as follows:

FDR No.	Deposit Amount (₹)	Gross Interest (₹)	TDS (₹) @ 10%	Net Interest Amount (₹)	Interest Warrant No.
FD-404	1,00,000/-	8,500/-	850/-	7,650/-	IW-998877

Please find enclosed herewith the Interest Warrant No. IW-998877 dated 1st September 2026, drawn on State Bank of India, for ₹7,650/-.

Thanking you,

Yours faithfully,
For Infosys Limited
(Sign)
Company Secretary

Encl: Interest Warrant No. IW-998877

Q. 8. Answer the following question (Any ONE) :

(i) What is Debenture? Explain the different types of debentures.

Meaning: The word 'debenture' is derived from the Latin word 'debere' which means 'to owe'. A debenture is a document issued by a company under its common seal, acknowledging a debt to the holders. It is a long-term source of borrowed capital.

Definition: According to Section 2(30) of the Companies Act, 2013, "Debenture includes debenture stock, bonds, and any other instrument of a company evidencing a debt, whether constituting a charge on the assets of the company or not."

Types of Debentures:

On the basis of Security:
- Secured Debentures: These debentures are secured by a charge on the assets of the company. If the company fails to pay the interest or principal, the debenture holders can recover their money by selling the mortgaged assets.
- Unsecured Debentures: These debentures are not backed by any security or charge on the company's assets. (Note: Under the Companies Act, 2013, companies are generally prohibited from issuing unsecured debentures).
On the basis of Transferability:
- Registered Debentures: The details of these debenture holders (name, address, etc.) are recorded in the 'Register of Debentureholders'. They can only be transferred through a formal transfer deed.
- Bearer Debentures: The names of the holders are not recorded in the company's register. They are easily transferable by mere delivery, similar to currency notes. Interest is paid to the person who produces the attached coupons.
On the basis of Repayment:
- Redeemable Debentures: These are issued for a specific period and are payable at the end of that period (maturity date) as mentioned on the debenture certificate.
- Irredeemable Debentures: These are not repayable during the lifetime of the company. They are repaid only at the time of the liquidation (winding up) of the company.
On the basis of Convertibility:
- Convertible Debentures: These debentures give the holder the option to convert them into equity shares after a specific period. This issue requires a special resolution in the general meeting.
- Non-Convertible Debentures (NCDs): These debentures cannot be converted into equity shares upon maturity. They are redeemed in cash.

(ii) Explain the statutory provisions for allotment of shares.

Meaning: Allotment of shares means distributing shares among the applicants based on their applications submitted in response to the prospectus. A public company must follow the statutory provisions of the Companies Act, 2013, and SEBI guidelines.

Statutory Provisions for Allotment of Shares:

Registration of Prospectus: Before issuing the prospectus to the general public, a copy of it must be filed with and registered by the Registrar of Companies (ROC).
Application Money: The Companies Act states that the application money must be at least 5% of the nominal (face) value of the share. However, SEBI specifies that for public issues, the application money should not be less than 25% of the issue price.
Minimum Subscription: It is the minimum amount that must be collected by the company from the public to proceed with the allotment. SEBI guidelines state that the minimum subscription must be 90% of the entire issue size. If 90% is not received within the stipulated time, the entire application money must be refunded within 15 days.
Closing of Subscription List: According to SEBI, the subscription list must be kept open for a minimum of 3 working days and a maximum of 10 working days. Applicants can apply for shares only during this period.
Basis of Allotment: The allotment must be strictly made based on the predetermined rules and categories. Once the allotment is finalized by the Allotment Committee, it requires the approval of the Board of Directors.
Over Subscription: In case a company receives applications for more shares than it has offered, it is called over-subscription. SEBI does not allow the company to keep any excess money; it must be refunded, or in certain cases, adjusted towards allotment/call money if permitted.
Permission to Deal on Stock Exchange: Every company making a public offer must apply to one or more recognized stock exchanges to list its shares. The prospectus must state the name of the stock exchange where the application has been made.
Appointment of Intermediaries: To process the share allotment smoothly, the company must appoint various intermediaries like Merchant Bankers, Lead Managers, Syndicate Members, Bankers to the Issue, and a Registrar to the Issue.

HSC 12th Secretarial Practice (SP) Board Paper Solution March 2026 | Maharashtra Board

aaminomtex@gmail.com (Amin) — Sat, 24 Oct 2026 08:34:33 +0530

Q. 1. Objective Questions

(A) Select the proper option from the options given below and rewrite the sentences:

Letter of regret is sent to applicants who have not been given shares by the company.
Finance is the management of monetary affairs of the company.
Shares issued free of cost to the shareholders are known as bonus shares.
A company cannot accept deposits for more than 36 months.
Equity shareholders participate in the management of the company.

(B) Write a word or term or phrase which can substitute each of the following sentences:

The return on investment paid to the creditors by the company.
Answer: Interest
The internal source of financing.
Answer: Retained Earnings (or Ploughing back of profit)
A bill which is issued by Reserve Bank of India on behalf of the Government of India.
Answer: Treasury Bill
Transfer of shares due to operation of law.
Answer: Transmission of shares
Documentary evidence of holding the debentures.
Answer: Debenture Certificate

(C) Correct the underlined word/s and rewrite the following sentences:

A Bear expects the prices of shares to rise in future.
Answer: A Bull expects the prices of shares to rise in future.
Bond is a source of Short term finance.
Answer: Bond is a source of Long term finance.
Dividend is payable every year irrespective of profits made by company.
Answer: Interest is payable every year irrespective of profits made by company.
Securities are fungible in Physical mode.
Answer: Securities are fungible in Demat (or Electronic) mode.
The details of debentureholders are entered in Register of Members.
Answer: The details of debentureholders are entered in Register of Debentureholders.

(D) Select the correct option from the bracket:

(Short term loan, Private company, NSDL, Index of NSE, Funds for long term.)

(a) Private company - (1) Deposit from Members
(b) First Depository in India - (2) NSDL
(c) Funds for long term - (3) Capital Market
(d) Public Deposit - (4) Short term loan
(e) Index of NSE - (5) Nifty

Q. 2. Explain the following terms / concepts (Any FOUR) :

(1) Debenture Certificate

Meaning: A debenture certificate is a formal document issued by a company to its debenture holders as documentary evidence of the debt.
Time of Issue: The company is legally required to issue the debenture certificate within six months from the date of allotment of the debentures.
Contents: It contains important details such as the name and address of the debenture holder, the number of debentures held, the face value, the rate of interest, and the maturity or redemption date.

(2) Fixed Capital

Meaning: Fixed capital refers to the portion of total capital that is invested in acquiring fixed assets. These assets are used for a longer period in the business and are not meant for immediate resale.
Nature: It stays in the business almost permanently and forms the foundation of the company's infrastructure.
Sources and Examples: It is usually raised through long-term sources of finance such as equity shares, preference shares, and long-term loans. Examples of fixed assets include land, building, plant, and machinery.

(3) Register of Deposit

Meaning: The Register of Deposit is a statutory register that every company accepting deposits from the public or its members is required to maintain at its registered office.
Time of Entry: The company must enter the details of the deposits in this register within seven days from the date of issuance of the deposit receipt.
Contents: It includes the name and address of the depositor, the amount of the deposit, the date of deposit, the duration, the rate of interest, and the maturity date.

(4) Dematerialization

Meaning: Dematerialization (often called Demat) is the process by which a client gets their physical security certificates converted into an electronic or digital format.
Procedure: To dematerialize securities, an investor must open a Demat account with a Depository Participant (DP) and submit a Dematerialization Request Form (DRF) along with the physical certificates.
Advantages: It eliminates extensive paperwork and completely removes the risks associated with holding physical certificates, such as loss, theft, mutilation, or forgery.

(5) Stock Exchange

Meaning: A Stock Exchange is a highly organized market where existing financial securities like shares, debentures, and bonds of corporate bodies, as well as government securities, are bought and sold.
Role: It provides high liquidity to investors, allowing them to easily convert their securities into cash and vice versa. It also acts as an economic barometer of a country's financial health.
Regulation: In India, the functioning of all recognized stock exchanges is closely monitored and regulated by the Securities and Exchange Board of India (SEBI).

(6) Overdraft

Meaning: An overdraft is a short-term credit facility provided by commercial banks exclusively to their current account holders (usually businessmen and corporate entities).
Feature: It allows the account holder to withdraw funds from their current account up to a pre-approved limit, even if the actual bank balance is zero or insufficient to cover the withdrawal.
Interest: Interest is charged by the bank only on the actual amount overdrawn by the customer and only for the exact number of days the funds are utilized.

Q. 3. Study the following cases / situations and express your opinion (Any TWO):

(1) Mrs. Khushi wishes to open a Demat Account in her name.

Can she open the account going to Mumbai office of NSDL?
Answer: No. An investor cannot directly open a Demat account with a Depository like NSDL. She must open the account through a Depository Participant (DP) registered with the Depository.
Is she required to pay for opening of account and its maintenance?
Answer: Yes. Mrs. Khushi is required to pay account opening charges, Annual Maintenance Charges (AMC), and transaction fees to the Depository Participant (DP) for the services provided.
Does she have to send the shares to respective company for demating?
Answer: No. She does not need to send the shares directly to the company. She must submit a Dematerialization Request Form (DRF) along with her physical share certificates to her Depository Participant (DP), who will then process the request with the company or its Registrar and Transfer Agent (RTA).

(2) Harsh Company Limited decides to pay interim dividend.

Is the board justified to decide interim dividend of ₹10 per share even though profits till date are insufficient?
Answer: No. The Board of Directors is not justified in declaring an interim dividend if the profits generated during the current financial year till date are insufficient.
Can the board declare dividend out of free reserves?
Answer: No. An interim dividend cannot be declared out of free reserves. It can only be paid out of the surplus in the profit and loss account or out of the profits of the financial year in which such dividend is sought to be declared.
In how many days company must pay dividend to shareholders from the date of declaration?
Answer: The company must pay the declared interim dividend to its shareholders within 30 days from the date of its declaration.

(3) Angel Company Ltd. plans to raise funds through public deposits. Its net worth is ₹250 crores.

Can they accept deposits from the public?
Answer: Yes. Angel Company Ltd. can accept deposits from the public. It qualifies as an "Eligible Public Company" since its net worth of ₹250 crores exceeds the statutory requirement of ₹100 crores.
Can they accept deposits which mature after 5 years?
Answer: No. A company cannot accept deposits with a maturity period of 5 years. As per the provisions of the Companies Act, public deposits can only be accepted for a minimum period of 6 months and a maximum period of 36 months (3 years).
Within what period should the company issue deposit receipt to its depositors?
Answer: The company must issue the deposit receipt to its depositors within 21 days from the date of receipt of the deposit amount or the realization of the cheque.

Q. 4. Distinguish between the following (Any THREE) :

(1) Equity shares and Preference shares

Points of Difference	Equity Shares	Preference Shares
1. Meaning	Shares that do not enjoy any preferential right regarding the payment of dividend and repayment of capital are called equity shares.	Shares that carry preferential rights regarding the payment of dividend and repayment of capital are called preference shares.
2. Rate of Dividend	The rate of dividend is fluctuating. It depends upon the amount of profit earned by the company.	The rate of dividend is fixed and predetermined at the time of issue.
3. Voting Rights	Equity shareholders enjoy normal voting rights and can participate in the management of the company.	Preference shareholders do not enjoy normal voting rights. They can vote only on matters directly affecting their rights.
4. Return of Capital	At the time of winding up of the company, equity capital is paid back last, after all other claims are settled.	At the time of winding up, preference capital is paid back before equity capital.
5. Risk	Equity shares are subject to higher risk as there is no guarantee of dividend or capital refund.	Preference shares are subject to lower risk as they guarantee a fixed dividend and priority in capital repayment.

(2) Dividend and Interest

Points of Difference	Dividend	Interest
1. Meaning	It is the return on investment paid to the shareholders of the company out of divisible profits.	It is the return paid to the creditors of the company for the borrowed capital (e.g., loans, debentures, deposits).
2. Given to	It is paid to the owners (shareholders) of the company.	It is paid to the creditors (debentureholders, depositors, lenders) of the company.
3. Obligation	Payment of dividend is not compulsory. It is paid only when the company earns sufficient profits.	Payment of interest is a fixed and legal obligation. It must be paid irrespective of the company's profits or losses.
4. Rate	The rate of dividend on equity shares fluctuates according to the profits.	The rate of interest is fixed and predetermined at the time of borrowing.
5. Charge on Profit	Dividend is an appropriation of profit.	Interest is a charge against profit.

(3) Jobber and Broker

Points of Difference	Jobber	Broker
1. Meaning	A jobber is an independent dealer on the stock exchange who buys and sells securities in his own name.	A broker is an agent who buys and sells securities on behalf of his clients (investors).
2. Dealings	A jobber cannot deal directly with the general public or investors. He deals only with brokers or other jobbers.	A broker deals directly with the general public and acts as a link between the investors and the jobbers.
3. Reward / Income	The income of a jobber is the profit he earns, which is the price difference between buying and selling prices (called "turn" or "spread").	The income of a broker is the commission or brokerage charged to the clients for the services rendered.
4. Role	A jobber acts as a principal trader.	A broker acts strictly as an agent for his clients.

(4) Initial Public Offer (IPO) and Further Public Offer (FPO)

Points of Difference	Initial Public Offer (IPO)	Further Public Offer (FPO)
1. Meaning	IPO refers to the process where an unlisted company offers its shares to the general public for the very first time.	FPO refers to the process where an already listed company issues additional shares to the general public.
2. Type of Company	It is issued by an unlisted company.	It is issued by a listed company.
3. Time of Issue	It is the first sale of shares to the public. It happens before the company gets listed on the stock exchange.	It happens after the IPO, once the company is already established and listed on the stock exchange.
4. Risk Factor	It carries a higher risk for investors as the company does not have a past trading record on the stock exchange.	It carries comparatively lower risk because the company’s past market performance and financial records are available for analysis.
5. Objective	The main objective is usually to raise capital to start or launch a new business.	The main objective is to raise additional capital for business expansion, diversification, or clearing debts.

Q. 5. Answer in brief (Any TWO):

(1) State any four terms and conditions regarding acceptance of deposit.

The following are four important terms and conditions regarding the acceptance of deposits by a company:

Period of Deposit: A company can accept or renew deposits for a minimum period of 6 months and a maximum period of 36 months. A company cannot accept deposits repayable on demand.
Amount of Deposit: An eligible public company cannot accept fresh deposits if the amount of such deposits, together with previous outstanding deposits, exceeds 35% of the aggregate of the paid-up share capital and free reserves of the company.
Secured or Unsecured Deposits: A company can accept both secured and unsecured deposits. However, it must clearly state whether the deposit is secured or unsecured in every circular, form, advertisement, or any other document related to the invitation or acceptance of deposits.
Application Form: A company must accept deposits only through an application in the prescribed form supplied by the company. The form must contain a declaration by the depositor stating that the deposit is not being made out of any borrowed money.

(2) State the features of Interim Dividend.

The main features of Interim Dividend are as follows:

Time of Declaration: Interim dividend is declared by the Board of Directors between two Annual General Meetings (AGMs) of the company.
Authority: The Board of Directors has the exclusive power to declare an interim dividend, provided it is authorized by the Articles of Association. It does not require the approval of shareholders.
Source of Payment: It is paid out of the surplus in the profit and loss account or out of the profits generated in the current financial year till the quarter preceding the declaration. It cannot be paid out of free reserves.
Irrevocability: Once an interim dividend is declared by the Board, it becomes a debt of the company and cannot be canceled or revoked.
Separate Bank Account: The company must deposit the interim dividend amount into a separate bank account within 5 days of its declaration.
Time of Payment: The interim dividend must be paid to the shareholders within 30 days from the date of its declaration.

(3) State any four factors affecting working capital requirement.

The working capital requirement of a business is influenced by the following factors:

Nature of Business: Manufacturing companies require a larger amount of working capital to maintain raw materials, work-in-progress, and finished goods inventory. In contrast, public utility services or trading firms need less working capital as they deal with immediate cash sales.
Size of Business: The size of the business dictates the scale of its operations. A large-scale firm with massive production and sales volumes will naturally require a higher amount of working capital compared to a small-scale firm.
Production Cycle: The production cycle is the time required to convert raw materials into finished goods. If the production cycle is long, funds will be tied up for a longer period, resulting in a higher working capital requirement. A shorter cycle requires less working capital.
Credit Policy: The credit terms granted by the company to its customers directly impact working capital. A liberal credit policy (allowing a longer time for payment) increases debtors and the need for working capital. A strict credit policy reduces the working capital requirement.

Q. 6. Justify the following statements (Any TWO):

(1) Debenture trustees are appointed by a company issuing debentures.

Legal Requirement: According to the Companies Act, if a company issues a prospectus or invites more than 500 persons to buy its debentures, it is mandatory to appoint one or more Debenture Trustees.
Protection of Interest: Debenture trustees are typically financial institutions appointed to protect the interests and rights of the debentureholders.
Trust Deed: The company enters into a legal contract with the trustees known as the 'Debenture Trust Deed' before the prospectus or letter of offer is issued.
Creation of Charge: Through this deed, the company creates a charge on its assets in favor of the debenture trustees, providing security to the debentureholders.
Redressal of Grievances: The trustees monitor the company's financial health, ensure that it fulfills its obligations regarding the payment of interest and principal, and redress any grievances of the debentureholders.
Conclusion: Thus, to safeguard the investment and rights of the creditors, debenture trustees are appointed by a company issuing debentures.

(2) Depository system is very similar to the banking system.

Safe Custody: Just as a bank keeps the funds of its customers in safe custody, a depository keeps the financial securities (shares, bonds, etc.) of investors in safe custody.
Electronic Form: In a bank, money is held as an account balance without physical handling. Similarly, in a depository system, securities are held in a dematerialized (electronic) form, eliminating physical certificates.
Transferability: A bank allows the easy transfer of funds from one account to another. The depository system allows for the easy and immediate transfer of securities between accounts without extensive paperwork.
Intermediaries: A customer interacts with a bank branch, whereas an investor interacts with a Depository Participant (DP), which acts as an agent of the Depository.
Account Statements: Both banks and depositories provide regular account statements to their clients, reflecting their current balance and recent transactions.
Conclusion: Thus, the depository system functions exactly like the banking system, but it deals with financial securities instead of money.

(3) There are many participants in money market.

Nature of Market: The money market is a financial market for borrowing and lending short-term funds, usually with a maturity period of up to one year.
Wholesale Market: It is predominantly a wholesale market where the volume of transactions is extremely large.
Central Bank Role: The Reserve Bank of India (RBI) is the most important participant, acting as the regulator and controller of the money market.
Institutional Participants: Due to the large ticket size of transactions, the market is dominated by large institutional participants such as Commercial Banks, Cooperative Banks, and Primary Dealers.
Other Entities: Financial Institutions, Mutual Funds, Non-Banking Financial Companies (NBFCs), Insurance Companies, and large Corporate Houses also actively participate to manage their short-term liquidity.
Conclusion: Therefore, because it involves the management of massive short-term funds across the economy, there are many varied institutional participants in the money market.

(4) Company has to fulfill certain provisions while making Right Issue.

Meaning: A Rights Issue is an offering of new equity shares by a company to its existing equity shareholders in proportion to their current shareholdings.
Letter of Offer: The company must send a formal 'Letter of Offer' to all existing shareholders, detailing the number of shares offered, the issue price, and the validity period of the offer.
Time Limit: The offer must be kept open for a minimum of 15 days and a maximum of 30 days, giving shareholders adequate time to decide.
Right of Renunciation: Unless restricted by the Articles of Association, shareholders possess the right to renounce (transfer) their offered shares in favor of any other person.
Unsubscribed Shares: If a shareholder does not respond within the stipulated time, it is deemed rejected. The Board of Directors then has the right to dispose of these unsubscribed shares in a manner most beneficial to the company.
Conclusion: Thus, to ensure fairness and protect the pre-emptive rights of existing shareholders, a company has to strictly fulfill these provisions while making a Right Issue.

Q. 7. Attempt the following (Any TWO) :

(1) Draft a reply letter resolving the query of the member on low rate of dividend.

RELIANCE INDUSTRIES LTD.
Registered Office: Reliance Centre, Nariman Point, Mumbai - 400 021.
CIN: L28920MH1985PLC056636

Phone: 022-22876543
Website: www.relianceindustries.com
Email: investors@relianceindustries.com
Date: 24th October 2026

To,
Mr. Ramesh Sharma
102, Shanti Niwas,
M. G. Road, Pune - 411 001.

Sub: Reply to your query regarding the low rate of dividend.

Dear Sir,

This is in response to your letter dated 15th October 2026, expressing your concern over the low rate of dividend declared by the company for the financial year 2025-26.

We appreciate your interest in the company's financial performance. The Board of Directors has recommended a dividend of 10% for the current year, which is lower than the 15% declared last year. We wish to bring to your notice the following reasons for this decision:

The company has undertaken a massive expansion project to set up a new manufacturing plant in Gujarat. A significant portion of the profits has been retained (ploughed back) to finance this project internally.
During the year, the company faced unavoidable operational delays and increased raw material costs due to global supply chain disruptions, which marginally affected our net profit.

We assure you that the retained earnings invested in the expansion project will lead to higher production capacity, increased sales, and ultimately greater profits in the coming years. This strategic decision will maximize shareholder wealth in the long run.

We hope this clarifies your query. We value your continued support and trust in the company.

Thanking you,

Yours faithfully,
For Reliance Industries Ltd.
Sd/-
(Company Secretary)

(2) Write a letter to debentureholder informing him/her about conversion of debentures into equity shares.

TATA MOTORS LTD.
Registered Office: Bombay House, 24 Homi Mody Street, Mumbai - 400 001.
CIN: L28920MH1945PLC004520

Phone: 022-66658282
Website: www.tatamotors.com
Email: secretarial@tatamotors.com
Date: 24th October 2026

To,
Ms. Sunita Patil
45, Gokul Dham Society,
Thane (W) - 400 601.

Sub: Conversion of Debentures into Equity Shares.

Dear Madam,

We are pleased to inform you that, in accordance with the terms of the issue of 10% Convertible Debentures and the resolution passed in the Extraordinary General Meeting held on 5th October 2026, your debentures are due for conversion.

As per the scheme of conversion, your 100 (One Hundred) 10% Convertible Debentures of ₹100/- each are being converted into 100 (One Hundred) Equity Shares of ₹10/- each at a premium of ₹90/- per share.

The details of the conversion are as follows:

Folio No. / DP ID & Client ID: IN300123 / 45678912
Number of Debentures Held: 100
Number of Equity Shares Allotted: 100
Distinctive Numbers: 45001 to 45100

Since your securities are held in Demat form, the corresponding equity shares have been credited to your Demat account mentioned above on 22nd October 2026.

Your name has been entered in the Register of Members. The company assures you of its best services at all times.

Thanking you,

Yours faithfully,
For Tata Motors Ltd.
Sd/-
(Company Secretary)

(3) Draft a letter of thanks to the depositor of a company.

INFOSYS LIMITED
Registered Office: Electronics City, Hosur Road, Bengaluru - 560 100.
CIN: L85110KA1981PLC013115

Phone: 080-28520261
Website: www.infosys.com
Email: deposits@infosys.com
Date: 24th October 2026

To,
Mr. Anil Deshmukh
78, Saraswati Baug,
Andheri (E), Mumbai - 400 069.

Sub: Letter of thanks for your Fixed Deposit.

Dear Sir,

We are in receipt of your application form dated 20th October 2026, along with the cheque of ₹1,00,000/- (Rupees One Lakh Only) drawn on State Bank of India, towards an unsecured fixed deposit with our company.

We express our sincere thanks and gratitude for the trust and confidence you have placed in our company by keeping your funds with us.

The details of your deposit are as under:

Deposit Receipt No.: FD/2026/1452
Amount of Deposit: ₹1,00,000/-
Period of Deposit: 24 Months
Rate of Interest: 8.5% p.a.
Date of Maturity: 19th October 2028

The Fixed Deposit Receipt (FDR) is enclosed herewith. We assure you that the interest warrants will be sent to you periodically as per the mandate given in your application form.

We assure you of our best services always.

Thanking you,

Yours faithfully,
For Infosys Limited
Sd/-
(Company Secretary)

Encl: Fixed Deposit Receipt No. FD/2026/1452

Q. 8. Answer the following questions (Any ONE) :

(1) What is debenture? Explain the different types of debentures.

Meaning of Debenture:
The word 'debenture' is derived from the Latin word 'debere', which means to owe. A debenture is an instrument issued by a company under its common seal acknowledging a debt to the holder. It is a document that shows that the company has borrowed a specific amount of money from the public and promises to repay it along with fixed interest after a certain period.

Types of Debentures:

On the basis of Security:
- Secured Debentures: These debentures are secured by a charge on the assets of the company. If the company fails to repay the principal or interest, the debentureholders can recover their dues by selling the mortgaged assets.
- Unsecured Debentures: These debentures are not secured by any charge on the company's assets. They are merely naked promises to pay. The issue of unsecured debentures is now prohibited by the Companies Act, 2013.
On the basis of Transferability:
- Registered Debentures: These debentures are recorded in the 'Register of Debentureholders' maintained by the company. The name, address, and holding details are mentioned. They can only be transferred through a proper transfer deed.
- Bearer Debentures: The names of the holders are not recorded in the company's register. They are easily transferable by mere delivery. Interest coupons are attached to the debenture certificate, and payment is made to the bearer.
On the basis of Repayment:
- Redeemable Debentures: These debentures are repayable by the company at the end of a specific period mentioned on the debenture certificate. The principal amount is returned to the holders.
- Irredeemable Debentures: These debentures are not repayable during the lifetime of the company. They are repaid only at the time of the liquidation (winding up) of the company or upon the breach of any condition.
On the basis of Convertibility:
- Convertible Debentures: These debentures give the holder the right to convert their debentures into equity shares of the company after a specific period, at a pre-determined rate.
- Non-Convertible Debentures (NCDs): These debentures cannot be converted into equity shares. They remain as a debt until they are redeemed upon maturity.

(2) Explain briefly the procedure for allotment of shares.

Meaning: Allotment of shares means distributing shares among the applicants who have submitted a written application along with the application money. The brief procedure for the allotment of shares is as follows:

Appointment of Allotment Committee:
When the subscription list is closed, the company secretary informs the Board of Directors. The Board appoints an Allotment Committee to prepare a report on the allotment and formulate an allotment formula, especially in cases of oversubscription.
Hold Board Meeting to Decide Allotment Formula:
The Board of Directors holds a meeting to review the Allotment Committee's report. They finalize the formula for allotment. In case of oversubscription, the allotment formula requires the approval of the Securities and Exchange Board of India (SEBI).
Approval of SEBI and Stock Exchange:
The finalized allotment formula is submitted to the relevant Stock Exchange authorities and SEBI for formal approval to ensure fair distribution among applicants.
Dispatch of Letters of Allotment or Regret:
After finalizing the list, the secretary dispatches the 'Letter of Allotment' to the successful applicants, detailing the number of shares allotted and the allotment money due. For applicants who are not allotted any shares, a 'Letter of Regret' is sent along with a refund order or electronic refund of their application money.
Collection of Allotment Money:
The applicants who receive the Letter of Allotment must pay the due allotment money into the company's designated bank account within the specified time limit.
Arrangement Relating to Letters of Renunciation:
If the Articles of Association permit, an applicant can renounce (transfer) their allotted shares to another person. The original allottee submits a formal 'Letter of Renunciation', and the secretary modifies the allotment register accordingly.
Arrangement Relating to Splitting of Allotment Letters:
An allottee may request the company to split their allotment into smaller denominations to sell a portion of the shares. The secretary issues split allotment letters after obtaining Board approval.
Filing Return of Allotment:
Within 30 days of the allotment of shares, the company secretary must file a document called the 'Return of Allotment' with the Registrar of Companies (ROC). It contains details like the names and addresses of allottees, the number of shares allotted, and the amount paid.
Preparation of Register of Members and Issue of Share Certificates:
The secretary enters the details of the allottees in the 'Register of Members'. Finally, within two months from the date of allotment, the company must issue and dispatch the Share Certificates to the shareholders. In the case of a Demat account, the shares are directly credited to the respective accounts.

HSC 12th Secretarial Practice (SP) Board Question Paper March 2026

HSC 12th OCM June 2026 Board Paper Solution (J-283) | Organisation of Commerce and Management

aaminomtex@gmail.com (Amin Buhari) — Fri, 23 Oct 2026 10:44:04 +0530

12th OCM June 2026 Board Question Paper with Full Solution | Maharashtra Board HSC Commerce

Here is the complete 12th OCM June 2026 Board Question Paper with Full Solution for Maharashtra State Board HSC Commerce students. All objective questions, concept explanations, case studies, distinguish between tables, and long answers are prepared strictly according to the official board textbook and marking scheme.

Q. 1. (A) Select the correct options and rewrite the sentences:

Organisation function is important for execution of plan which have been prepared by (a) top level management.
For online transaction (a) registration is required.
The term 'Ethics' is derived from the (c) Greek word 'ethos'.
Perishable goods such as vegetables, fruits, milk products, etc. are sold in (a) very short period market.
Air transport is (b) the costliest mode of transport.

Q. 1. (B) Give one word / phrase / term for the following statements:

It is an intellectual process of logical thinking and rational decision making: Planning
The transaction which is done with the help of internet: Online transaction / E-commerce / E-business
The right of consumer which is about safety and protection to his life and health: Right to Safety
The market for commodities which are produced in one country and sold in another countries: International Market
An art of exchanging ideas, facts, information, etc. from one person or entity to another: Communication

Q. 1. (C) Correct the underlined word and rewrite the sentences:

Esprit de corps means unity is strength.
Controlling is a function of comparing the actual performance with predetermined performance.
Moral values are based on perceptions of right or wrong.
In India, consumers are widely scattered and not united.
Packaging helps in avoiding breakage, damage and destruction of the product.

Q. 1. (D) Select the correct options and complete the following table:

Sr. No.	Group A	Group B
(i)	Central Bank	Apex bank of the country
(ii)	Warehouse	Time and place utility
(iii)	Superset of e-commerce	E-business
(iv)	Timely payment of taxes	Responsibility towards Government
(v)	2019	Consumer Protection Act

Q. 2. Explain the following terms / concepts (Any FOUR):

(i) Time study

Time study is a technique of scientific management developed by F.W. Taylor. It refers to the process of observing and recording the time taken by an average worker to complete a specific task under normal conditions. The main objective of time study is to determine the standard time required to perform a well-defined job, which helps in determining a fair day's work and fixing proper wage rates for employees.

(ii) Directing

Directing is a core managerial function that involves instructing, guiding, communicating, inspiring, and motivating employees to achieve organizational goals. It is the phase of the management process that initiates action, bridging the gap between planning and actual execution. Directing is considered the life-spark of an enterprise as it brings the human effort into motion.

(iii) Warehousing

Warehousing refers to the systematic and scientific process of storing and preserving goods from the time of their production or purchase until their actual consumption or sale. It creates 'time utility' by bridging the time gap between production and consumption. Warehousing ensures a continuous and steady supply of goods in the market and protects them from spoilage, theft, and damage.

(iv) District Commission

The District Commission is a consumer dispute redressal agency established at the district level by the State Government under the Consumer Protection Act, 2019. It consists of a President and not less than two members. It entertains consumer complaints where the value of the goods or services paid as consideration does not exceed Rs. 1 Crore (or up to Rs. 50 Lakhs as per the revised rules of 2021).

(v) Courier Service

Courier service is a specialized, fast, and secure facility for delivering parcels, important documents, and letters from one place to another. These services are generally operated by private companies and provide door-to-door delivery. While they are relatively more expensive than standard postal services, they offer advantages like speed, reliability, and real-time tracking.

(vi) Staffing

Staffing is the managerial function of executing the human resource plan by acquiring, developing, retaining, and fairly compensating the right people for the right jobs. It involves processes such as recruitment, selection, placement, training, and performance appraisal. The primary aim of staffing is to build and maintain a competent and satisfied workforce to ensure the smooth functioning of the organization.

Q. 3. Study the following case / situation and express your opinion (Any TWO):

(i) Case Study: Mr. Sharan

Situation: Mr. Sharan is a successful manufacturer. He is having production units at various locations. He has a large stock of raw material and finished goods. He is worried about safeguarding goods from any unwanted financial loss. He also requires to transfer raw material and finished goods from one unit to another but does not have any facility for that. He also requires funds for business expansion.

(a) Name the service which will help him to safeguard goods from any damage.
Ans: Warehousing is the service that will help Mr. Sharan safeguard his large stock of raw material and finished goods from physical damage, spoilage, and theft. Additionally, to safeguard specifically against unwanted financial loss caused by unforeseen risks or damages, he should avail of Insurance services.
(b) Which service will help him to remove difficulty of place?
Ans: Transport (Transportation) service will help him remove the difficulty of place by transferring raw material and finished goods from one production unit to another.
(c) From which service sector will Mr. Sharan get financial support?
Ans: Mr. Sharan will get financial support for his business expansion from the Banking (or Financial) service sector.

(ii) Case Study: Ajay and Jyoti

Situation: Ajay purchases some gift articles online from www.flipkart.com. At the same time Jyoti purchases gift from e-bay.com.

(a) Which website is related to C2C?
Ans: The website e-bay.com is related to Consumer-to-Consumer (C2C) e-business, as it facilitates transactions directly between consumers through online auctions and listings.
(b) Which website is related to B2C?
Ans: The website www.flipkart.com is related to Business-to-Consumer (B2C) e-business, where a business sells products directly to end customers.
(c) Which first step in online transaction does Ajay need to follow?
Ans: Registration is the first step in an online transaction that Ajay needs to follow before shopping on the website.

(iii) Case Study: Sanjyot Co. Ltd.

Situation: Sanjyot Co. Ltd. is providing facilities for their female staff like day care centre for kids and work from home facility. Even management takes their suggestions while taking the decisions though they are members of trade union.

(a) By doing this activity they are following social responsibility towards which interest groups?
Ans: By doing this activity, Sanjyot Co. Ltd. is following social responsibility towards its Employees (Workers).
(b) Write any one facility provided by the company to their employees.
Ans: The company provides a day care centre for kids (or work from home facility) to its female employees.
(c) What kind of responsibility employer follows in above case?
Ans: The employer is following Social Responsibility (specifically towards employees by promoting employee welfare, recognizing trade union rights, and encouraging workers' participation in management).

Q. 4. Distinguish between the following (Any THREE):

(i) Road Transport and Rail Transport

Sr. No.	Point of Distinction	Road Transport	Rail Transport
1.	Meaning	Road transport refers to the movement of passengers and goods on public roads using vehicles like trucks, buses, cars, bullock carts, etc.	Rail transport refers to the movement of passengers and goods on railway tracks using trains.
2.	Speed	Road transport has relatively limited speed due to road conditions, traffic congestion, and speed limits.	Rail transport operates at high speed as it runs on dedicated tracks without traffic obstruction.
3.	Carrying Capacity	It has a limited carrying capacity and is suitable for carrying smaller quantities of goods.	It has a huge carrying capacity and is ideal for transporting heavy and bulky goods in large quantities.
4.	Cost	It requires less capital investment in terms of construction and maintenance of vehicles and roads.	It requires heavy capital investment for laying tracks, purchasing engines, wagons, and maintaining stations.
5.	Distance	It is recommended and economical for short and medium distances.	It is recommended and economical for both short and long distances, especially cross-country routes.
6.	Door-to-Door Service	It provides convenient door-to-door service directly from the sender to the receiver.	It does not provide door-to-door service; goods and passengers must reach and leave from fixed railway stations.

(ii) Co-ordinating and Controlling

Sr. No.	Point of Distinction	Co-ordinating	Controlling
1.	Meaning	Co-ordinating refers to the orderly arrangement of group efforts to provide unity of action in the pursuit of common goals.	Controlling is the process of comparing actual performance with predetermined standards and taking corrective action if required.
2.	Objective	Its main objective is to ensure harmony of efforts among employees and smooth functioning of the organization.	Its main objective is to ensure that organizational goals are achieved strictly according to the formulated plans.
3.	Area of Function	It involves the integration of efforts of top, middle, and lower levels as well as various departments.	It involves setting standards, measuring actual performance, finding deviations, and taking corrective measures.
4.	Factors	In co-ordinating function, internal and human factors are harmonized to create team spirit.	In controlling function, both internal and external factors are considered for taking corrective action.
5.	Order	It is an important element of organizing and follows planning and organizing functions.	It is the end function of the management process and follows all other functions.
6.	Resources	It is strictly related to human resources and their collective efforts.	It is related to all organizational resources (human, physical, and financial) to achieve business goals.

(iii) State Commission and National Commission

Sr. No.	Point of Distinction	State Commission	National Commission
1.	Meaning	A consumer dispute redressal forum at the State level established by the State Government is known as the State Commission.	A consumer dispute redressal forum at the National level established by the Central Government is known as the National Commission.
2.	President	A person who is or has been a Judge of a High Court shall be appointed as its President by the State Government.	A person who is or has been a Judge of the Supreme Court shall be appointed as its President by the Central Government.
3.	Membership	It shall have a President and not less than four other members, or such number of members as may be prescribed.	It shall have a President and not less than four other members, or such number of members as may be prescribed.
4.	Area Covered	It covers the geographical area of a particular State.	It covers the entire territory of India.
5.	Monetary Jurisdiction	It entertains complaints where the value of goods or services paid as consideration exceeds Rs. 1 crore but does not exceed Rs. 10 crore (As per revised rules: exceeds Rs. 50 lakhs up to Rs. 2 crore).	It entertains complaints where the value of goods or services paid as consideration exceeds Rs. 10 crore (As per revised rules: exceeds Rs. 2 crore).
6.	Appeal	An appeal against the order of the State Commission can be made to the National Commission within 30 days.	An appeal against the order of the National Commission can be made to the Supreme Court within 30 days.

(iv) Life Insurance and Marine Insurance

Sr. No.	Point of Distinction	Life Insurance	Marine Insurance
1.	Meaning	A contract whereby the insurer undertakes to pay a certain sum of money either on death or on maturity (whichever is earlier) for a consideration (premium).	A contract whereby the insurer undertakes to compensate the insured against losses to the ship or cargo caused by perils of the sea.
2.	Subject Matter	Human life is the subject matter of life insurance.	Ship, cargo (goods), and freight are the subject matter of marine insurance.
3.	Insurable Interest	Insurable interest must exist at the time of taking the policy.	Insurable interest must exist at the time of the loss/damage.
4.	Period / Tenure	The policy can be issued for any number of years, usually for a long term, or until the death of the assured.	The policy is generally issued for a short period (up to one year) or for a specific voyage.
5.	Principle of Indemnity	It is not applicable to life insurance as the value of human life cannot be estimated in monetary terms.	It is strictly applicable as the insurer compensates only for the actual financial loss incurred due to sea perils.
6.	Surrender of Policy	The policy can be surrendered before the expiry of the term, subject to certain conditions.	Marine insurance policies cannot be surrendered before expiry.

Q. 5. Answer in brief (Any TWO):

(i) Explain any four points of significance of principles of management.

Ans: The principles of management are fundamental truths that act as guidelines for managerial decision-making and action. The significance of principles of management can be explained as follows:

Provides Useful Insight to Managers: The principles of management help managers to understand the organization, its situations, and problems in a better way. They improve the manager's understanding of how an organization works and guide them in finding practical solutions to various business situations, thereby reducing wastage of resources.
Helpful in Efficient Utilization of Resources: In every organization, two types of resources are used: physical resources (material, machines, money) and human resources (manpower). Management principles help in maintaining a proper balance between these resources, avoiding wastage, and putting them to optimum use to achieve maximum productivity at minimum cost.
Scientific Decisions: In business organizations, a scientific decision means a systematic and balanced decision. Management principles train managers to handle critical situations tactfully and make objective, realistic, and logical decisions based on facts than relying on trial and error or personal bias.
Understanding Social Responsibility: Management principles are not merely focused on handling organizational resources efficiently, but they also guide management in understanding the social responsibility of the business. This prompts organizations to focus on providing quality products at reasonable prices, avoiding artificial monopolistic situations, ensuring fair remuneration, and maintaining healthy working environments.
Encourages Research and Development (R&D): Principles of management are dynamic in nature and evolve continuously along with changes in the business environment. They enable managers to adopt new trends at the workplace and encourage continuous research and development to discover new techniques in production, finance, marketing, and human resources.

(ii) Explain any four rights of consumers.

Ans: The Consumer Protection Act, 2019 protects the interests of consumers by granting them several fundamental rights. Any four rights of consumers are explained below:

Right to Safety: This right protects consumers against products, production processes, and services which are hazardous to their health or life. It includes concern for consumers' long-term interests as well as their immediate requirements. According to this right, consumers must receive complete safety and protection in relation to medicines, electrical appliances, food items, automobiles, etc.
Right to Information: According to this right, the consumer should be provided with adequate and accurate information about all aspects of goods and services. This includes details such as price, name of the manufacturer, contents used, batch number, date of manufacture, date of expiry, user manual, and safety instructions. This right enables the consumer to select the right product and make an informed choice.
Right to Choose: The choices available to Indian consumers across the basket of goods and services have multiplied significantly. According to this right, a consumer should be given full freedom to select an article as per their requirement, liking, and purchasing capacity. The seller cannot compel or force the consumer to buy a particular product or create a monopoly to restrict choice.
Right to be Heard: Every business organization should listen to and solve the legitimate complaints of consumers. According to this right, consumers have the opportunity to voice their grievances and complaints to the consumer forum. Consumers also give suggestions to the manufacturer or trader on certain matters such as quality, quantity, price, packaging, etc. Nowadays, consumers can file online complaints through portals and mobile applications.
Right to Redress: Along with the right to represent, the consumer also has the right to get claims and complaints settled against manufacturers or sellers due to unfair trade practices. This right enables the consumer to demand repair, replacement, compensation, or refund for defective products or poor services through the three-tier consumer dispute redressal machinery (District, State, and National Commissions).

(iii) Explain types of market on the basis of Area covered and on the basis of Time.

Ans: Markets can be classified into different types based on geographical area covered and time duration:

A. Types of Market on the Basis of Area Covered:

Local Market: The market for the commodities which are sold within the local geographical limits of a region is known as a local market. It generally deals in perishable and semi-durable goods like vegetables, fruits, milk products, etc., where local buyers and sellers transact directly.
National Market: The market for the commodities which are sold throughout the country is known as a national market. It covers the domestic area within the political boundaries of a nation, and trade is governed by the domestic rules and regulations of that specific country.
International Market: The market for the commodities which are produced in one country and sold in other countries is known as an international market. It involves import and export transactions across national borders and is regulated by international trade policies.

B. Types of Market on the Basis of Time:

Very Short Period Market: This type of market has a very short time existence, typically lasting for a few hours or for a day at a particular time and place. In this type of market, perishable goods such as vegetables, fruits, milk products, fish, etc. are sold. Since supply cannot be increased instantly, the price is primarily determined by demand.
Short Period Market: This type of market exists for a short period, such as weekly markets, festival markets, or markets during fairs. Perishable goods and semi-durable goods are sold in this market, where supply can be adjusted to a limited extent using existing stock.
Long Period Market: This type of market exists for a long duration. In this market, durable commodities which are generally non-perishable in nature are sold. There is ample time to adjust the supply of goods according to changes in market demand.

Q. 6. Justify the following statements (Any TWO):

(i) Planning is the first function of management.

Justification:

Planning is the basic and foundational function of every management process. Every manager begins their managerial activity with planning before performing any other function.
Planning means deciding in advance what to do, how to do it, when to do it, and who is to do it. It involves setting clear organizational objectives and formulating policies, strategies, and programmes to achieve them.
Without setting goals and preparing a logical plan of action, no other functions of management—such as organizing, staffing, directing, co-ordinating, and controlling—can be initiated or executed effectively.
For instance, organizing resources or recruiting staff is impossible unless the management first plans the scale of operations and organizational requirements. Similarly, controlling cannot exist without the predetermined standards set during the planning stage.
Thus, it is rightly justified that planning is the first function of management as it provides the base for all other managerial functions.

(ii) Consumer organisations and NGOs play an important role in consumer protection.

Justification:

Non-Government Organisations (NGOs) and consumer associations are voluntary, non-profit organizations established specifically to safeguard and promote the interests of consumers against exploitation by unscrupulous traders.
They organize various consumer education campaigns, workshops, and training programmes to create widespread awareness among consumers regarding their legal rights and responsibilities.
They publish periodicals, journals, and brochures to keep consumers well-informed about market trends, product quality, comparative prices, and the malpractices of dishonest sellers.
They provide free legal advice and assistance to aggrieved consumers to help them seek timely legal remedies and relief. They also conduct independent testing of food items and household products in laboratories to detect adulteration and substandard quality.
Furthermore, they file Public Interest Litigations (PILs) in courts on crucial consumer issues on behalf of the general public. Hence, consumer organisations and NGOs play a vital role in strengthening the consumer protection movement.

(iii) E-business allows user to work across the globe in any field.

Justification:

E-business (Electronic Business) refers to conducting all kinds of business activities and commercial transactions through the internet and electronic platforms.
Because the internet is truly without boundaries, e-business completely eliminates geographical barriers and physical distance between buyers and sellers.
It allows a businessperson or user sitting in any corner of the world to connect, communicate, and transact with customers, suppliers, and business partners globally on a 24x7x365 basis.
On the one hand, it gives sellers access to a massive global market at very low setup and operational costs; on the other hand, it enables buyers from anywhere in the world to select and purchase goods and services across various fields (such as retail, banking, education, IT, consulting, healthcare, etc.).
Therefore, it is justified that e-business offers global reach and allows users to work and trade across the globe in any field.

(iv) Role of media has major influence on business organisation.

Justification:

Media refers to the various means of mass communication—including print media (newspapers, magazines), electronic media (television, radio), and digital/social media—that reach a vast audience very quickly.
In modern times, media acts as a vigilant watchdog of society. It is highly active and plays a prominent role in exposing unfair trade practices, black marketing, hoarding, misleading advertisements, and consumer exploitation by business organizations.
Media can easily influence the opinions, attitudes, and purchasing decisions of the general public. Negative publicity or reporting about a company's unethical practices or poor-quality products can instantly tarnish its goodwill, brand image, and market reputation.
Conversely, positive coverage by the media regarding a company's quality standards, ethical practices, and corporate social responsibility (CSR) initiatives helps build immense public trust, brand loyalty, and market prestige.
Since media has the power to make or break the reputation of an enterprise within a very short span of time, its role has a major influence on business organizations.

Q. 7. Attempt the following (Any TWO):

(i) Explain any five Henry Fayol's principles of management.

Ans: Henry Fayol (1841–1925), a French mining engineer who rose to become a chief managing director, is regarded as the 'Father of Modern Management'. In his famous book 'General and Industrial Administration', he proposed 14 fundamental principles of management. Any five of these principles are explained below:

Principle of Division of Work: According to this principle, the total work in an organization should be divided into smaller sub-parts such as technical, financial, commercial, security, accounting, and managerial operations. Each sub-part is then assigned to employees according to their qualities, skills, and capabilities. Performing the same task repeatedly helps build expertise, improves efficiency, and leads to specialization in the organization.
Principle of Authority and Responsibility: Authority is the right to take decisions and give orders to get things done from subordinates. Responsibility is the obligation to perform the assigned task properly within the given time. According to Fayol, authority and responsibility always go hand in hand. If a manager is given the responsibility to complete a task, they must also be given adequate authority to make the necessary decisions. Similarly, having authority without accountability can lead to the misuse of power.
Principle of Discipline: According to Fayol, discipline is the most essential thing in any organization. Employees must obey and respect the rules, regulations, and policies that govern the enterprise. Good discipline helps to achieve organizational goals smoothly. It is the result of effective leadership, a clear understanding between management and workers regarding organizational rules, and the judicious application of penalties.
Principle of Unity of Command: Each member of an organization should receive orders from only one superior at a time. This principle helps in managing conflicts and solving disputes among people in the organization. If an employee receives commands from more than one boss simultaneously, they will get confused about whose orders to follow first, leading to a breakdown in authority and efficiency.
Principle of Unity of Direction: This principle states that 'there should be one head and one plan' for every group of activities having the same objective. Each group in the organization should work together towards the same target under the guidance of a single manager using a single uniform plan. This ensures coordination of efforts and avoids duplication of work.
Principle of Esprit de Corps (Team Work): Henry Fayol has given great emphasis on team work. 'Esprit de corps' is a French phrase which means 'Unity is Strength'. Running any organization is a group activity, and human resources are a valuable asset. If all employees work as a united team with mutual trust and understanding, difficulties can be solved quickly and organizational goals can be achieved easily.

(ii) Explain the characteristics of an Entrepreneur.

Ans: An entrepreneur is a person who starts a business and is willing to risk loss in order to make money. Entrepreneurs have certain special qualities and features that set them apart. The key characteristics of an entrepreneur are explained below:

Intellectual Capabilities: An entrepreneur is a creative thinker who has reasonably high intelligence compared to a normal businessman. They possess the ability to analyze critical business situations accurately and make wise, logical decisions at the right time.
Future Vision: A successful entrepreneur has great foresight about future market conditions. They constantly observe changes in the external business environment, predict upcoming customer trends, and take proactive timely actions to capture new opportunities.
Hard Work: An entrepreneur is always ready to work hard and put in long hours, especially when starting a new venture. They have to handle various core areas independently and cannot afford to stop until the business becomes stable and successful.
Technical Knowledge: An entrepreneur possesses good technical knowledge about the products or services their business offers, as well as the processes of production. They also keep updating themselves with the latest technological advancements in their industry.
Communication Skills: An entrepreneur needs to communicate effectively with different people like customers, suppliers, creditors, employees, and financiers. There must be proper understanding between the sender and the receiver of the message to build strong business relationships and get work done smoothly.
Highly Optimistic: An entrepreneur always thinks positively and maintains a hopeful attitude even during tough times or market downturns. This positive mindset gives them the confidence to overcome business failures and keep moving forward toward their goals.
Risk-bearing Capacity: Successful entrepreneurs take calculated risks. Every business faces uncertainties and potential losses. An entrepreneur is ready to face these challenges and looks for new, unexplored avenues by weighing risks against potential rewards.
Self-Confidence: An entrepreneur has strong self-confidence and believes in their own abilities to achieve predetermined goals. They are self-motivated and do not get easily discouraged by obstacles along the way.

(iii) Explain the importance of marketing to the society.

Ans: Marketing is not only essential for business firms to earn profits, but it also plays a vital role in the welfare and development of society as a whole. The importance of marketing to society is explained below:

Increase in Standard of Living: The main objective of a marketing department is to supply goods and services to satisfy customer needs. Paul Mazur defined marketing as "the delivery of standard of living to the society." Marketing helps to identify the needs of customers and takes the initiative to provide quality goods at cheaper prices. By introducing new and better varieties of products and services, it helps consumers live a more comfortable life and directly improves their standard of living.
Provides Employment: Modern marketing is a total system that covers almost all functions of an organization such as buying, selling, financing, transport, warehousing, risk-bearing, research, and development. Running this vast system requires a massive workforce, which creates millions of direct and indirect job opportunities for people in society.
Decreases Distribution Costs: Marketing activities help to provide cheap goods and services to society. Effective utilization of channels of distribution can help in reducing the overall cost and prices of products. When distribution costs are reduced through efficient marketing, the benefit is passed on to consumers in the form of lower product prices, ensuring timely availability of goods.
Consumer Awareness: Marketing helps society by informing and educating consumers. Through advertising and sales promotion, it gives consumers detailed information about new products available in the market, their uses, prices, and features. It also creates awareness regarding how different products can satisfy their specific requirements and makes it easier for them to make smart buying choices.
Increase in National Income: Every economy revolves around three basic activities: marketing, production, and finance. Scientifically organized marketing activities help in the economic development of the country. Effective marketing of products and services leads to higher industrialization, creates more jobs, and boosts consumer spending, which directly increases the national income of the country.
Managing Consumer Expectations: Marketing research helps organizations understand the changing needs and expectations of consumers. Businesses use this feedback to develop or modify products accordingly. Moreover, marketing monitors quality standards and curbs malpractices through government regulations and consumer forums, ensuring that societal expectations for fair trade are met.

Q. 8. Answer the following question in detail (Any ONE):

(i) What is marketing? Explain different functions of marketing.

Meaning of Marketing:

Marketing is a key managerial function in every business organization. In simple words, marketing is the sum total of all activities that are involved in the flow of goods and services from the point of production to the point of ultimate consumption. It starts long before production begins (by identifying customer needs through market research) and continues even after the sale has been made (through after-sales service and customer feedback).

Definition of Marketing:

According to Philip Kotler, "Marketing is a societal process by which individuals and groups obtain what they need and want through creating, offering and freely exchanging products and services of value with others."

Functions of Marketing:

Success of a business depends largely on how effectively its marketing functions are performed. The different functions of marketing are explained in detail below:

Marketing Research: Effective marketing is only possible when a business clearly identifies the needs and wants of its consumers. To gather this information, marketers systematically collect and analyze data from consumers regarding what they want to buy, when they buy it, in what quantity, and at what price. This systematic process of collecting and analyzing market information to make crucial decisions is known as marketing research.
Buying and Assembling: It involves collecting raw materials from different sources at a single central place for production. This function is crucial because the quality and final cost of the finished goods directly depend upon the quality and purchase price of the raw materials bought and assembled.
Market Planning: After assessing the need for marketing through research, the businessman needs to chalk out a clear, practical plan and strategy to achieve the business objectives. Market planning is the process of organizing and defining the marketing aims of a company and gathering strategies and tactics to achieve them efficiently.
Product Development: Product development and design play an important role in the selling of the product. A business needs to develop a product that suits the specific needs of the consumer. Product design includes decisions related to quality, standards, shape, design, packing, colour, etc. Consumers always prefer better and attractively designed products, which gives a competitive advantage to the business.
Standardisation and Grading: Standardisation means determining standards related to the process, size, quality, design, weight, colour, etc. of the product. It helps in ensuring uniformity in the quality of the product and builds customer loyalty. Grading is the process of classification of products according to similar characteristics and/or quality. Grading is done on the basis of features like size, shape, and quality (typically in the case of agricultural products like wheat, rice, potatoes, etc.).
Packaging and Labelling: Package and label create the first impression on the consumer about the product. Attractive packaging can help to make a product successful. Packaging means designing the package for the product to avoid breakage, damage, and destruction of the goods during transit and storage. Packing material includes bottles, containers, plastic bags, tin, wooden boxes, etc. Label is a slip which is found on the product and provides all the information regarding the product and its producer. This slip is called a label and its process is called labelling.
Branding: Every businessman wants to have a special identity in the market for his product. Branding is a process of giving a special identity to a product through a unique brand name to differentiate it from the competitor's products. In simple words, giving a distinct name to one's product is called branding. Registered brands are known as Trademarks, which cannot be copied by others.
Customer Support Service: Customer is the king of the market; hence, a business needs to take necessary steps for the satisfaction of the customer. A business needs to take every possible effort to provide support services to the customer to gain their loyalty for repeat sales. Support services include pre-sales service, consumer helpline, after-sales service, technical assistance, product demo, etc.
Pricing of Product: Pricing is one of the most important and challenging functions of marketing. Many times, the price of a product decides the success or failure of that product, especially in a highly competitive market. While determining the price of a product, a businessman needs to consider critical factors like cost of production, desired profit margin, price of the competitor's product, demand for the product, market conditions, etc.
Promotional Channels: Promotional channels is the process of informing the consumers about the products, their features, uses, prices, etc. and encouraging them to buy these products. Advertising, Personal selling, Publicity, and Sales Promotion are some of the important tools of promotion. A business uses a combination of all or some of these four methods for promotion as per the need of the business.
Distribution: Distribution is the set of activities which is concerned with efficient movement of finished goods from the place of production to the consumer. It includes transportation, warehousing, material handling, inventory control, order processing, market forecasting, packaging, plant and warehouse location, and customer service.
Transportation: Transportation means physical movement of goods from the place of production to the place of consumption. It includes transportation of raw materials as well as finished goods. Analysis of geographical boundaries, nature of goods, cost, target market, speed, reliability, frequency, safety, etc. helps in deciding modes of transport (road, rail, air, water). Transportation creates 'place utility'.
Warehousing: There is a time-lag between the purchase or production of goods and their actual sale. It is important to store the goods at a safe place during this time-lag to prevent spoilage or damage. Any negligence during this period may damage the stock. Warehouse helps to maintain a smooth flow of goods, stabilizes prices in the market, and creates 'time utility'.

(ii) What is Bank? Explain primary functions of commercial banks.

Meaning of Bank:

The term 'Bank' comes from the French word 'Banco' which means a 'bench'. In earlier days, moneylenders used to display coins of different currencies in big heaps or on benches or tables for the purpose of lending or exchanging. A bank is a financial institution which deals with deposits and advances and other related services. Bank provides various services related to money or financial requirements of consumers and businesses.

Definition of Bank:

According to The Indian Banking Regulation Act, 1949, "Banking company means any company which transacts the business of banking in India." Further, banking has been defined as "Accepting for the purpose of lending or investment of deposits of money from the public, repayable on demand or otherwise, and withdrawable by cheque, draft, order or otherwise."

Primary Functions of Commercial Banks:

The primary functions of commercial banks are known as core banking functions. They are divided into two main categories: (A) Accepting Deposits and (B) Granting Loans and Advances.

(A) Accepting Deposits:

Commercial banks collect deposits from the individuals and organizations. Deposits can be classified into two types: Demand Deposits and Time Deposits.

1. Time Deposits: Time deposits are called as time deposits because they are repaid to the customers after the expiry of decided time. Time deposits are divided into two types:

(a) Fixed Deposit: Fixed deposit account is an account where fixed amount is kept for a fixed period of time bearing fixed interest rate. Rate of interest is more as compared to saving bank account and varies with the deposit period. Normally, withdrawal is not permitted before maturity date. However, depositor can withdraw amount before maturity date for which bank will reduce the interest rate. For the amount deposited in this account, a Fixed Deposit Receipt (FDR) is issued by the bank. Against this receipt, loan can be taken from the bank.
(b) Recurring Deposit: It is operated by salaried persons and businessmen having regular income. A certain fixed sum of money is deposited into the account every month for a specified period of time. Withdrawal of accumulated amount along with interest is paid after the maturity date. Rate of interest is higher which is similar to fixed deposit account. Separate passbook is provided to know the position of RD account.

2. Demand Deposits: Demand deposits are those which are repaid to customers whenever they demand. That means, money can be withdrawn as per the wish of the customer through withdrawal slips, cheques, ATM cards, online transfer, etc. Demand deposits are divided into two types:

(a) Saving Account: It is generally operated by those who earn regular or fixed income such as salary or wages. The main aim of this deposit account is to encourage habit of savings among people. These deposit accounts are meant for the purpose of maximum savings. There are restrictions on withdrawal limits from these accounts. These accounts carry low interest rates. Interest is credited monthly, quarterly, half-yearly and yearly basis on this account. Passbook facility, balance on SMS, account statement, etc. are provided to account holders to ascertain financial position. For saving account holders some banks provide separate facility of flexi deposit. This facility combines the advantages of saving account and fixed deposit account.
(b) Current Account: This account is operated by business firms and other commercial organizations such as hospitals, educational institutions, etc. who have regular banking transactions. In this account there is no restriction on deposits and withdrawals of amounts. No interest is paid by the bank on this account. Overdraft facility is available for this account. For current account, banks provide statement of account every month.

(B) Granting Loans and Advances:

Commercial banks grant loans and advances to business firms and others who are in need of bank funds. The loans and advances are provided for longer period of time from 1 year and more. Advances are provided for shorter period from 4 months to 1 year. The advances are in the form of cash credit, overdraft and discounting of bills, etc.

1. Loans: Commercial banks provide loan to businessman and others. The borrowers can use the entire amount sanctioned or can withdraw in installments. Interest is charged on the amount sanctioned. The loans are as follows:

(a) Short Term Loans: Short term loans are for a period up to 1 year to meet working capital requirements of the borrower.
(b) Medium Term Loans: These loans are for a period of 1 year to 5 years to meet working capital as well as fixed capital requirements of the borrower.
(c) Long Term Loans: These loans are for a period of 5 years or more to meet long term capital requirements of the borrower.

2. Advances: Advances are small term fund provided to businessman to satisfy different financial requirements of the business. Advances are as follows:

(a) Cash Credit (CC): The cash credit advances are provided to current account and saving account holders. It provides working capital for longer period of time. Interest rate is higher on CC. Separate CC account has to be maintained by the borrower. Under this facility, borrower can withdraw funds repeatedly up to a stipulated credit limit against the security of tangible assets. Interest is charged only on the actual amount withdrawn and not on the total sanctioned limit.
(b) Overdraft (OD): This facility is offered to current account holders to meet their working capital requirements. The period can vary from 15 to 60 days. Interest is charged on actual amount withdrawn. No separate account is maintained, and entries are shown in the current account. It is a temporary arrangement for a short period.
(c) Discounting of Bills of Exchange: The drawer of bill of exchange (or beneficiary) can discount the bill with bank and obtain an advance. On the maturity of the bill, the bank will recover the amount from the drawee.

HSC 12th OCM June 2026 Original Board Question Paper (Code J-283)

Below are the scanned pages of the official Maharashtra State Board HSC 12th Standard Organisation of Commerce and Management (OCM) June 2026 Board Question Paper (Subject Code: J-283) for reference and practice:

Page 1 of 5: HSC 12th OCM June 2026 Board Question Paper (Q. 1. A)

Page 2 of 5: HSC 12th OCM June 2026 Board Question Paper (Q. 1. B & Q. 1. C)

Page 3 of 5: HSC 12th OCM June 2026 Board Question Paper (Q. 1. D, Q. 2 & Q. 3)

Page 4 of 5: HSC 12th OCM June 2026 Board Question Paper (Q. 3, Q. 4 & Q. 5)

Page 5 of 5: HSC 12th OCM June 2026 Board Question Paper (Q. 6, Q. 7 & Q. 8)

SSC 10th Algebra (Maths-1) Board Paper Solution March 2026 | Maharashtra Board

aaminomtex@gmail.com (Amin) — Thu, 22 Oct 2026 23:58:14 +0530

SSC 10th Algebra (Mathematics Part - I) March 2026 Board Paper Solution

Subject: Mathematics (71) Algebra - Part I (E) | Paper Code: N 919 | Time: 2 Hours | Max. Marks: 40

Q. 1. (A) Choose the correct answer and write the alphabet: [4 Marks]

(i) Rate of GST on essential commodities is ............

(A) 5%
(B) 12%
(C) 0%
(D) 18%

Answer: (C) 0%

(ii) Out of the following equations which one is not a quadratic equation?

(A) x² + 4x = 11 + x²
(B) x² = 4x
(C) 5x² = 90
(D) 2x - x² = x² + 5

Answer: (A) x² + 4x = 11 + x²

Explanation: Simplifying x² + 4x = 11 + x² gives 4x - 11 = 0, which is a linear equation as the highest index of the variable x is 1.

(iii) To draw graph of 4x + 5y = 19, find the value of y when x = 1.

(A) 4
(B) 3
(C) 2
(D) -3

Answer: (B) 3

Explanation: Substituting x = 1 in 4x + 5y = 19:
4(1) + 5y = 19
4 + 5y = 19
5y = 19 - 4
5y = 15
y = 15 / 5 = 3

(iv) Sum of first five multiples of 3 is ............

(A) 45
(B) 55
(C) 15
(D) 75

Answer: (A) 45

Explanation: The first five multiples of 3 are 3, 6, 9, 12, and 15.
Sum = 3 + 6 + 9 + 12 + 15 = 45.

Q. 1. (B) Solve the following subquestions: [4 Marks]

(i) On certain article if rate of CGST is 9%, then what is the rate of GST?

Solution:

Given: Rate of CGST = 9%

We know that, Rate of GST = 2 × Rate of CGST

Rate of GST = 2 × 9% = 18%

Ans: The rate of GST is 18%.

(ii) For simultaneous equations in variables x and y, if D_x = -42 and D = -3, then find the value of x.

Solution:

Given: D_x = -42 and D = -3

By Cramer's Rule,

x = D_x / D

x = -42 / -3

x = 14

Ans: The value of x is 14.

(iii) When two coins are tossed simultaneously, write the sample space 'S'.

Solution:

When two coins are tossed simultaneously, the sample space 'S' is:

S = {HH, HT, TH, TT}

(iv) Write the following quadratic equation in the form ax² + bx + c = 0: 2y = 10 - y²

Solution:

Given quadratic equation: 2y = 10 - y²

Rearranging the terms in standard form ax² + bx + c = 0:

y² + 2y - 10 = 0

Q. 2. (A) Complete and write any two activities from the following: [4 Marks]

(i) If one die is rolled, then find the probability of the event, that the number on the upper face is prime.

Solution:

'S' is the sample space.

S = {1, 2, 3, 4, 5, 6}

∴ n(S) = [ 6 ]

Event A : Prime number on the upper face.

A = {2, 3, 5}

∴ n(A) = [ 3 ]

∴ P(A) = n(A) / [ n(S) ]

∴ P(A) = [ 3 ] / 6

∴ P(A) = 1 / 2

(ii) Write the correct number in the given boxes from the given A.P.; 70, 60, 50, 40, .............

Solution:

Here t₁ = [ 70 ], t₂ = 60, t₃ = [ 50 ]

∴ a = [ 70 ], d = [ -10 ]

(iii) Complete the following table to draw graph of the equation: x + y = 3

Solution:

x	3	[ -2 ]
y	[ 0 ]	5
(x, y)	[ (3, 0) ]	[ (-2, 5) ]

Q. 2. (B) Solve any four subquestions from the following: [8 Marks]

(i) If two coins are tossed simultaneously, find the probability of the event, "getting at least one head."

Solution:

Let 'S' be the sample space when two coins are tossed simultaneously.

S = {HH, HT, TH, TT}

∴ n(S) = 4

Let Event A be the event of getting at least one head.

A = {HH, HT, TH}

∴ n(A) = 3

P(A) = n(A) / n(S)

P(A) = 3 / 4

Ans: The probability of getting at least one head is 3/4.

(ii) Solve the following simultaneous equations: x + y = 4 ; 2x - y = 2

Solution:

x + y = 4 — (Equation 1)

2x - y = 2 — (Equation 2)

Adding Equation 1 and Equation 2:

(x + y) + (2x - y) = 4 + 2

3x = 6

x = 6 / 3

x = 2

Substitute x = 2 in Equation 1:

2 + y = 4

y = 4 - 2

y = 2

Ans: (x, y) = (2, 2) is the solution of the given simultaneous equations.

(iii) Solve the following quadratic equation by factorisation method: x² + x - 20 = 0

Solution:

x² + x - 20 = 0

x² + 5x - 4x - 20 = 0

x(x + 5) - 4(x + 5) = 0

(x + 5)(x - 4) = 0

∴ x + 5 = 0 or x - 4 = 0

∴ x = -5 or x = 4

Ans: The roots of the given quadratic equation are -5 and 4.

(iv) Given A.P. is 12, 16, 20, 24, ................. Find the 24th term of this progression.

Solution:

Given A.P.: 12, 16, 20, 24, ...

Here, first term (a) = 12

Common difference (d) = 16 - 12 = 4

We know that,

t_n = a + (n - 1)d

Substitute n = 24, a = 12, and d = 4:

t₂₄ = 12 + (24 - 1) × 4

t₂₄ = 12 + 23 × 4

t₂₄ = 12 + 92

t₂₄ = 104

Ans: The 24th term of the given progression is 104.

(v) If ∑x_if_i = 1265 and N = 50, then find the mean of the given data.

Solution:

Given: ∑x_if_i = 1265 and N = ∑f_i = 50

Mean (x̄) = ∑x_if_i / N

Mean (x̄) = 1265 / 50

Mean (x̄) = 25.3

Ans: The mean of the given data is 25.3.

Q. 3. (A) Complete and write any one activity from the following: [3 Marks]

(i) Shreekar bought a Laptop with 10% discount on printed price. The printed price of that Laptop was ₹ 50,000. 18% GST was charged on discounted price. Find the amount of CGST and SGST. What amount did Shreekar pay?

Solution:

Discount = 10% of 50,000 = [ ₹ 5,000 ]

Taxable value of Laptop = 50,000 - [ 5,000 ] = ₹ 45,000

Rate of GST = [ 18% ]

∴ Rate of CGST = 9%

CGST = 9% of 45,000

= [ 9 / 100 ] × 45,000

= [ ₹ 4,050 ]

SGST = ₹ 4,050

Total amount of Laptop = 45,000 + 4,050 + 4,050 = ₹ [ 53,100 ]

(ii) One of the roots of the quadratic equation 5m² + 2m + k = 0 is -7/5. Complete the following activity to find the value of 'k'.

Solution:

-7/5 is a root of quadratic equation 5m² + 2m + k = 0.

∴ put m = -7/5 in the equation.

5 × [ -7/5 ]² + 2 × [ -7/5 ] + k = 0

[ 49/5 ] + [ -14/5 ] + k = 0

[ 7 ] + k = 0

k = [ -7 ]

Q. 3. (B) Solve any two subquestions of the following: [6 Marks]

(i) Two numbers differ by 3. The sum of twice the smaller number and thrice the greater number is 19. Find the numbers.

Solution:

Let the greater number be x and the smaller number be y.

According to the first condition, the two numbers differ by 3:

x - y = 3 — (Equation 1)

According to the second condition, the sum of twice the smaller number and thrice the greater number is 19:

3x + 2y = 19 — (Equation 2)

Multiply Equation 1 by 2:

2x - 2y = 6 — (Equation 3)

Add Equation 2 and Equation 3:

(3x + 2y) + (2x - 2y) = 19 + 6

5x = 25

x = 25 / 5

x = 5

Substitute x = 5 in Equation 1:

5 - y = 3

y = 5 - 3

y = 2

Ans: The required numbers are 5 and 2.

(ii) Sachin invested some amount in a National Saving Certificate Scheme. In the first year he invested ₹ 5,000, in the second year ₹ 7,000, in the third year ₹ 9,000 and so on. Find the total amount that he invested in 12 years.

Solution:

The amounts invested by Sachin each year are ₹ 5,000, ₹ 7,000, ₹ 9,000, ...

Since the common difference between consecutive investments is constant (₹ 2,000), the sequence forms an Arithmetic Progression (A.P.).

Here:

First term (a) = 5000
Common difference (d) = 7000 - 5000 = 2000
Number of years (n) = 12

We know that the sum of the first n terms of an A.P. is:

S_n = (n / 2) × [2a + (n - 1)d]

Substitute n = 12, a = 5000, and d = 2000:

S₁₂ = (12 / 2) × [2(5000) + (12 - 1) × 2000]

S₁₂ = 6 × [10000 + 11 × 2000]

S₁₂ = 6 × [10000 + 22000]

S₁₂ = 6 × 32000

S₁₂ = 1,92,000

Ans: The total amount invested by Sachin in 12 years is ₹ 1,92,000.

(iii) There are six cards in a box, each bearing a number from 0 to 5. Find the probability of each of the following events, that: (1) The number on the card drawn is a natural number. (2) The number on the card drawn is a whole number.

Solution:

Sample space 'S' for six cards bearing numbers from 0 to 5:

S = {0, 1, 2, 3, 4, 5}

∴ n(S) = 6

(1) Let Event A be the event that the number on the card drawn is a natural number.

Natural numbers in the sample space are 1, 2, 3, 4, 5.

A = {1, 2, 3, 4, 5}

∴ n(A) = 5

P(A) = n(A) / n(S)

P(A) = 5 / 6

(2) Let Event B be the event that the number on the card drawn is a whole number.

Whole numbers in the sample space are 0, 1, 2, 3, 4, 5.

B = {0, 1, 2, 3, 4, 5}

∴ n(B) = 6

P(B) = n(B) / n(S) = 6 / 6

P(B) = 1

Ans: (1) Probability that the card bears a natural number is 5/6.
(2) Probability that the card bears a whole number is 1.

(iv) Draw a histogram of the following data:

Height of Students (cm)	No. of Students
135—140	4
140—145	12
145—150	16
150—155	8

Solution (Steps to construct the Histogram on Graph Paper):

Since the given classes (135—140, 140—145, 145—150, 150—155) are already continuous, no class adjustment is required.
Draw two perpendicular axes: the horizontal axis (X-axis) and the vertical axis (Y-axis) intersecting at the origin O.
On the X-axis, represent the Height of Students (in cm). Use a krink mark (zig-zag line) between the origin O and 135 because the class intervals start from 135 instead of 0.
On the Y-axis, represent the No. of Students (Frequency).
Choose a suitable scale:
- X-axis: 1 cm = 5 cm (Height)
- Y-axis: 1 cm = 2 students
Draw adjacent rectangles over each class interval with heights corresponding to their respective frequencies:
- For class 135—140: draw a rectangle of height 4 units.
- For class 140—145: draw a rectangle of height 12 units.
- For class 145—150: draw a rectangle of height 16 units.
- For class 150—155: draw a rectangle of height 8 units.

Q. 4. Solve any two subquestions of the following: [8 Marks]

(i) The sum of two roots of a quadratic equation is 7 and sum of their cubes is 91, find the equation.

Solution:

Let α and β be the two roots of the required quadratic equation.

According to the given conditions:

α + β = 7 — (1)

α³ + β³ = 91 — (2)

We know the algebraic identity:

α³ + β³ = (α + β)³ - 3αβ(α + β)

Substitute the values from (1) and (2) into the identity:

91 = (7)³ - 3αβ(7)

91 = 343 - 21αβ

21αβ = 343 - 91

21αβ = 252

αβ = 252 / 21

αβ = 12

A quadratic equation having roots α and β is given by:

x² - (α + β)x + αβ = 0

Substitute α + β = 7 and αβ = 12:

x² - 7x + 12 = 0

Ans: The required quadratic equation is x² - 7x + 12 = 0.

(ii) Smt. Tejaswini purchased 100 shares of FV ₹ 80 when the MV is ₹ 900. She paid brokerage at the rate of 0.5% and 18% GST on brokerage. Find the following:
(1) Net amount paid for 100 shares.
(2) Brokerage paid on sum invested.
(3) GST paid on brokerage.
(4) Total amount paid for 100 shares.

Solution:

Given:

Number of shares = 100
Face Value (FV) = ₹ 80
Market Value (MV) = ₹ 900
Rate of Brokerage = 0.5%
Rate of GST on brokerage = 18%

(1) Net amount paid for 100 shares (Sum Invested):

Sum Invested = Number of shares × Market Value (MV)

Sum Invested = 100 × 900

Net amount paid for 100 shares = ₹ 90,000

(2) Brokerage paid on sum invested:

Brokerage = 0.5% of Sum Invested

Brokerage = (0.5 / 100) × 90,000

Brokerage = 0.5 × 900

Brokerage paid = ₹ 450

(3) GST paid on brokerage:

GST = 18% of Brokerage

GST = (18 / 100) × 450

GST = 18 × 4.5

GST paid on brokerage = ₹ 81

(4) Total amount paid for 100 shares:

Total amount paid = Sum Invested + Brokerage + GST

Total amount paid = 90,000 + 450 + 81

Total amount paid for 100 shares = ₹ 90,531

Ans:

Net amount paid for 100 shares = ₹ 90,000
Brokerage paid on sum invested = ₹ 450
GST paid on brokerage = ₹ 81
Total amount paid for 100 shares = ₹ 90,531

(iii) The following table shows the funds collected by 105 students for flood affected people. Find the median of the data.

Fund (Rupees)	No. of Students
10—20	10
20—30	37
30—40	15
40—50	20
50—60	7
60—70	8
70—80	8

Solution:

Let us prepare the cumulative frequency table (less than type):

Class Interval (Fund in ₹)	Frequency (No. of Students) (f)	Cumulative Frequency (Less than type) (c.f.)
10—20	10	10
20—30	37	10 + 37 = 47 (c.f.)
30—40 (Median Class)	15 (f)	47 + 15 = 62
40—50	20	62 + 20 = 82
50—60	7	82 + 7 = 89
60—70	8	89 + 8 = 97
70—80	8	97 + 8 = 105
Total	N = ∑f = 105

Here, N = 105

∴ N / 2 = 105 / 2 = 52.5

The cumulative frequency just greater than or equal to 52.5 is 62, which corresponds to the class interval 30—40.

∴ Median Class = 30—40

Lower boundary of the median class (L) = 30
Frequency of the median class (f) = 15
Cumulative frequency of the class preceding the median class (c.f.) = 47
Class width (h) = 40 - 30 = 10

Formula for Median:

Median = L + [ (N/2 - c.f.) / f ] × h

Substitute the values into the formula:

Median = 30 + [ (52.5 - 47) / 15 ] × 10

Median = 30 + (5.5 / 15) × 10

Median = 30 + (55 / 15)

Median = 30 + (11 / 3)

Median = 30 + 3.67

Median = 33.67

Ans: The median of the funds collected is ₹ 33.67.

Q. 5. Solve any one subquestion of the following: [3 Marks]

(i) A survey was conducted in the school. Student using different modes of transport is given below:

Mode of Transport	Percentage of Students
Bicycle	35
Bus	a
Railway	2a
Rickshaw	20

Find the value of a and draw the Pie diagram.

Solution:

Step 1: Find the value of 'a'

The sum of all percentages is always 100.

35 + a + 2a + 20 = 100

55 + 3a = 100

3a = 100 - 55

3a = 45

a = 45 / 3

a = 15

Percentage of students using Bus = a = 15%
Percentage of students using Railway = 2a = 2 × 15 = 30%

Step 2: Calculate the measure of central angle (θ) for each mode of transport

Formula: Measure of Central Angle (θ) = (Percentage of Students / 100) × 360°

Mode of Transport	Percentage of Students	Measure of Central Angle (θ)
Bicycle	35%	(35 / 100) × 360° = 126°
Bus	15%	(15 / 100) × 360° = 54°
Railway	30%	(30 / 100) × 360° = 108°
Rickshaw	20%	(20 / 100) × 360° = 72°
Total	100%	360°

Step 3: Pie Diagram

Ans: The value of a = 15.

(ii) There are some students in group A and group B. If one student goes from group A to group B, then the number of students in both groups is same. But one student goes from group B to group A, then the number of students in group A becomes twice of group B. Find the number of students in each group.

Solution:

Let the number of students in Group A be x and the number of students in Group B be y.

Case 1: If one student goes from Group A to Group B:

Remaining students in Group A = x - 1
New number of students in Group B = y + 1

According to the first condition, the number of students in both groups becomes the same:

x - 1 = y + 1

x - y = 1 + 1

x - y = 2 — (Equation 1)

Case 2: If one student goes from Group B to Group A:

New number of students in Group A = x + 1
Remaining students in Group B = y - 1

According to the second condition, the number of students in Group A becomes twice that of Group B:

x + 1 = 2(y - 1)

x + 1 = 2y - 2

x - 2y = -2 - 1

x - 2y = -3 — (Equation 2)

Subtracting Equation 2 from Equation 1:

(x - y) - (x - 2y) = 2 - (-3)

x - y - x + 2y = 2 + 3

y = 5

Substitute y = 5 in Equation 1:

x - 5 = 2

x = 2 + 5

x = 7

Ans: The number of students in Group A is 7 and the number of students in Group B is 5.

HSC 12th Economics March 2026 Board Paper Solution | Maharashtra Board

aaminomtex@gmail.com (Amin) — Thu, 22 Oct 2026 13:05:10 +0530

HSC 12th Economics March 2026 Board Paper Solution

Q. 1. (A) Complete the correlation:

Micro economics : Slicing method :: Macro economics : Lumping method
Price rises : Expansion of supply :: Price falls : Contraction of supply
Salt : Essential good :: Diamond : Prestige good / Luxurious good
P0 : Base year price :: P1 : Current year price
Money market : Short term funds :: Capital market : Long term funds

Q. 1. (B) Give economic terms:

Capacity of a commodity to satisfy human want: Utility
A branch of economics which does microscopic study of the economy: Micro Economics
Cost incurred on fixed factors: Fixed Cost (Total Fixed Cost)
The cost incurred by the firm to promote sales: Selling Cost
A budget in which estimated revenue and expenditure of the government are equal: Balanced Budget

Q. 1. (C) Find the odd word:

Types of utility: Place utility, Form utility, Time utility, Commodity utility
Odd word: Commodity utility
Exceptions to the Law of supply: Giffen's goods, Supply of labour, Perishable goods, Rare goods
Odd word: Giffen's goods
Selling cost: Free gifts, Patents, Hoardings, Window display
Odd word: Patents
Determinants of demand: Income, Cost of production, Size of population, Price
Odd word: Cost of production
Types of Index Numbers: Selection Index Number, Price Index number, Value Index Number, Quantity Index Number
Odd word: Selection Index Number

Q. 1. (D) Choose the correct option:

Demand curve is parallel to 'Y'-axis
Answer: (4) only a (Perfectly inelastic demand)
Revenue per unit of output sold
Answer: (3) only c (Average revenue)
Statements that are incorrect in relation to index numbers
Answer: (2) a, b (Index number is a geographical tool, Index numbers measure changes in the air pressure)
Obligatory functions of the government
Answer: (3) only b (Maintaining internal law and order)
Examples of competitive demand are
Answer: (2) a, c (Tea - Coffee, Sugar - Jaggery)

Q. 2. (A) Identify and explain the following concepts (Any THREE):

Shabana paid wages to workers in her factory.
Identified Concept: Wages (Reward for Labour / Factor payment)

Explanation: Labour is an active human factor of production. Wages are the financial reward or remuneration paid to labourers for their physical and mental contribution to the production process.
India purchased petroleum from Iran.
Identified Concept: Import Trade

Explanation: Import trade refers to the purchase of goods and services by one country from another country. It involves the inflow of goods from a foreign country into the home country.
Gopal collected information about output of rice, wheat, cotton and jowar produced in 2016 and 2020 and derived Index Number.
Identified Concept: Quantity Index Number

Explanation: A quantity index number measures the changes in the level of output or physical volume of production of goods in the economy over a specific period of time.
40% fall in price of a commodity leads to 40% rise in quantity demanded.
Identified Concept: Unitary Elastic Demand (Ed = 1)

Explanation: Unitary elastic demand occurs when the percentage change in the quantity demanded is exactly equal to the percentage change in the price of the commodity.
Tina deposited a lumpsum amount of ₹ 50,000 in the bank for a period of one year.
Identified Concept: Fixed Deposit (Time Deposit)

Explanation: A fixed deposit refers to a lump sum amount deposited by a customer in a bank for a specific, predetermined period of time. It earns a higher and fixed rate of interest compared to other deposits.

Q. 2. (B) Distinguish between (Any THREE):

1. Micro Economics and Macro Economics

Points of Difference	Micro Economics	Macro Economics
Meaning	Micro economics is that branch of economics which studies the behaviour of individual economic units such as an individual consumer, firm, or industry.	Macro economics is that branch of economics which studies the behaviour of aggregates of the economy as a whole, such as total employment, national income, etc.
Scope	The scope of micro economics is narrow, as it deals only with individual economic variables.	The scope of macro economics is much wider, as it deals with overall macroeconomic variables.
Tools	Individual demand and individual supply are the main tools of micro economics.	Aggregate demand and aggregate supply are the main tools of macro economics.
Alternative Name	It is also known as Price Theory.	It is also known as Income and Employment Theory.

2. Increase in demand and Expansion of demand

Points of Difference	Increase in Demand	Expansion of Demand
Meaning	It refers to a rise in quantity demanded due to favourable changes in other factors, while the price remains constant.	It refers to a rise in quantity demanded strictly due to a fall in price alone, while other factors remain constant.
Cause	It is caused by changes in factors like increase in income, favourable changes in tastes and preferences, etc.	It is caused solely by a decrease in the price of the given commodity.
Effect on Demand Curve	The original demand curve shifts entirely to the right (outward shift) to form a new demand curve.	There is a downward movement along the same existing demand curve.
Concept Base	It is a part of 'Changes in Demand'.	It is a part of 'Variations in Demand'.

3. Average cost and Average revenue

Points of Difference	Average Cost (AC)	Average Revenue (AR)
Meaning	Average cost refers to the cost of production incurred per unit of output produced.	Average revenue refers to the revenue generated per unit of output sold.
Formula	AC = Total Cost (TC) / Total Quantity (TQ)	AR = Total Revenue (TR) / Total Quantity (TQ)
Significance	It helps a firm determine the minimum price required to cover the production expenses of a single unit.	It basically represents the price of the commodity, as price per unit is equal to average revenue.
Calculation Example	If Total Cost is ₹ 500 for 10 units, then AC = 500 / 10 = ₹ 50 per unit.	If Total Revenue is ₹ 800 from 10 units, then AR = 800 / 10 = ₹ 80 per unit.

4. Simple Index Number and Weighted Index Number

Points of Difference	Simple Index Number	Weighted Index Number
Meaning	In a simple index number, every commodity is given equal importance while constructing the index.	In a weighted index number, different commodities are given different weights based on their relative importance in the group.
Method of Construction	The method of constructing a simple index number is relatively easier and straightforward.	The method of constructing a weighted index number is relatively complex due to the assignment of weights.
Accuracy and Reliability	It is considered less accurate as it ignores the real-life relative importance of different commodities.	It is highly accurate and reliable as it reflects the true importance of various commodities through weights.
Use of Quantities	Quantities are usually not used as weights in the simple method.	Quantities of the base year or current year (e.g., Laspeyres' or Paasche's method) are explicitly used as weights.

5. Direct tax and Indirect tax

Points of Difference	Direct Tax	Indirect Tax
Meaning	It is a tax which is paid directly by the same person on whom it is legally imposed. The impact and incidence are on the same person.	It is a tax imposed on one person (like a producer/seller) but paid partly or wholly by another person (consumer). The impact and incidence are on different persons.
Burden / Shiftability	The burden of a direct tax cannot be shifted to any other person.	The burden of an indirect tax can easily be shifted to the final consumers in the form of higher prices.
Nature	Direct taxes are generally progressive in nature, meaning the tax rate increases as income increases.	Indirect taxes are mostly proportional or regressive in nature, affecting all income groups uniformly.
Examples	Income Tax, Wealth Tax, Corporate Tax, etc.	Goods and Services Tax (GST), Custom Duty, Excise Duty, etc.

Q. 3. Answer the following (Any THREE):

1. Explain any four features of utility.

Utility is the capacity of a commodity to satisfy human wants. Any four features of utility are as follows:

Relative Concept: Utility is related to time and place. It changes from time to time and place to place. For example, woollen clothes have more utility in winter than in summer, and sand has more utility at a construction site than on a sea shore.
Subjective Concept: Utility is a psychological concept and differs from person to person. It depends on the individual's likes, dislikes, nature, habits, and profession. For example, a stethoscope has utility for a doctor but not for a layman.
Ethically Neutral Concept: The concept of utility has no ethical consideration. A commodity should satisfy any want of a person, regardless of whether it is good or bad, moral or immoral. For example, a knife has utility to cut vegetables for a housewife and can also be used to harm someone by a killer.
Utility differs from Usefulness: Utility is the want-satisfying power of a commodity, whereas usefulness indicates value in use or benefit derived from it. A commodity may have utility but not usefulness. For example, liquor has utility for a drunkard but it is not useful as it harms health.

2. Explain the Ratio method of measuring elasticity of demand.

The Ratio method, also known as the Percentage method or Arithmetic method, was developed by Prof. Alfred Marshall. In this method, price elasticity of demand is measured by dividing the percentage change in quantity demanded by the percentage change in price.

Formula:

Ed = Percentage change in Quantity Demanded / Percentage change in Price

Ed = % ∆Q / % ∆P

Where:

∆Q = Change in quantity demanded (New Quantity - Original Quantity)
∆P = Change in price (New Price - Original Price)
Q = Original quantity demanded
P = Original price

Mathematically, it can be expressed as: Ed = (∆Q / Q) × (P / ∆P)

Depending on the numerical value obtained, elasticity can be classified as unitary elastic (Ed = 1), relatively elastic (Ed > 1), or relatively inelastic (Ed < 1).

3. Explain any four sources of non-tax revenue of the government.

Non-tax revenue refers to the public revenue received by the government from sources other than taxes. Any four sources are as follows:

Fees: A fee is a charge imposed by the government for certain specific services rendered to the citizens. It is not paid voluntarily but is compulsory for those who utilize the service. Examples include education fees, registration fees, etc.
Fines and Penalties: The government imposes fines and penalties on individuals who violate the laws of the country. The main objective of imposing fines is not to earn revenue but to deter people from breaking the laws. For example, fines for violating traffic rules.
Prices of Public Goods and Services: The government sells various goods and services to the public through public sector enterprises. The revenue generated from the sale of these goods (e.g., railway fares, postal charges, toll taxes) is a major source of non-tax revenue.
Gifts, Grants and Donations: The government receives financial assistance in the form of grants and gifts from foreign governments and international organizations (like the IMF or World Bank). Citizens also provide donations during times of natural calamities or emergencies.

4. Explain the concept of foreign trade and its types.

Concept of Foreign Trade: Foreign trade refers to the buying and selling of goods and services across the political and geographical boundaries of different countries. It is also known as international trade or external trade.

Types of Foreign Trade:

Import Trade: It refers to the purchase of goods and services by one country from another country. It involves the inflow of goods into the home country from foreign countries. For example, India purchasing petroleum from Iraq.
Export Trade: It refers to the sale of goods and services by one country to another country. It involves the outflow of goods from the home country to foreign countries. For example, India selling tea to the USA.
Entrepot Trade: It refers to the purchase of goods and services from one country and then selling them to another country after some processing or repacking. In this trade, goods are imported not for domestic consumption but for the purpose of exporting them to a third country. For example, Japan importing raw materials from India and exporting the finished electronic goods to Europe.

5. Calculate Price Index Number from the given data:

Given Data and Calculation Table:

Commodity	Base Year Price (p₀) in ₹	Current Year Price (p₁) in ₹
A	6	18
B	16	8
C	24	28
D	4	6
Total	Σp₀ = 50	Σp₁ = 60

Formula for Price Index Number (P₀₁):

P₀₁ = (Σp₁ / Σp₀) × 100

Where:

Σp₁ = Sum total of current year prices = 60
Σp₀ = Sum total of base year prices = 50

Calculation:

P₀₁ = (60 / 50) × 100

P₀₁ = 1.2 × 100

P₀₁ = 120

Answer: The calculated Price Index Number is 120.

Q. 4. State with reasons whether you agree or disagree with the following statements (Any THREE):

(i) Perfect competition means monopolistic competition.

Answer: No, I disagree with this statement.

Reasons:

Perfect competition and monopolistic competition are two distinct and entirely different market structures.
Perfect competition is an ideal and theoretical market concept where there are a large number of buyers and sellers selling a perfectly homogeneous (identical) product at a single uniform price.
On the other hand, monopolistic competition is a realistic market structure where there are many sellers, but they sell closely related and differentiated products (e.g., different brands of soaps).
In perfect competition, sellers are pure price takers and there is an absence of selling cost. In monopolistic competition, sellers have partial control over the price and incur heavy selling costs (like advertisement) to promote their products.

(ii) Scope of micro economics includes only the theory of product pricing.

Answer: No, I disagree with this statement.

Reasons:

The scope of micro economics is not limited merely to the theory of product pricing; its scope is much wider.
Apart from product pricing, the scope of micro economics includes the Theory of Factor Pricing, which explains how the rewards for the factors of production (rent for land, wages for labour, interest for capital, and profit for entrepreneur) are determined.
Furthermore, it also encompasses the Theory of Economic Welfare, which deals with efficiency in the allocation of resources. This involves efficiency in production, efficiency in consumption, and overall economic efficiency.
Therefore, micro economics is a comprehensive study that covers product pricing, factor pricing, and economic welfare.

(iii) There are many features of National Income.

Answer: Yes, I agree with this statement.

Reasons:

National income is a vital macroeconomic concept that studies the aggregate economy rather than individual economic units.
It is a flow concept because it shows the flow of goods and services produced in an economy during a specific period of time, generally a financial year.
It represents the money value of goods and services, as only those goods and services exchanged for money are included in its calculation.
It strictly accounts for the value of final goods and services only. The value of intermediate goods or raw materials is excluded to avoid the error of double counting.
It focuses on current production, meaning goods produced in the previous years are not included in the current year's national income.

(iv) Public expenditure includes only revenue expenditure.

Answer: No, I disagree with this statement.

Reasons:

Public expenditure refers to the expenses incurred by the public authorities (Central, State, and Local governments) for the protection of their citizens, for satisfying their collective needs, and for promoting economic and social welfare.
While public expenditure does include revenue expenditure (which is incurred for the regular, day-to-day running of government departments and administrative services), it is not restricted to it.
It also includes capital expenditure, which is the huge investment made for the economic progress and development of the country, such as building dams, roads, and hospitals.
Additionally, it encompasses developmental expenditure (productive expenses on health, education, and industry) and non-developmental expenditure (unproductive expenses like war costs and administrative overheads).

(v) The concept of total utility and marginal utility is the same.

Answer: No, I disagree with this statement.

Reasons:

Total utility and marginal utility are closely interrelated but are distinctly different concepts in consumer behaviour.
Total Utility (TU) refers to the aggregate or total amount of utility derived by a consumer from the consumption of all available units of a given commodity. It is the sum of all marginal utilities.
Marginal Utility (MU) refers to the additional utility derived by a consumer from the consumption of an extra or additional unit of that commodity.
As a consumer goes on consuming successive units of a commodity, Total Utility goes on increasing at a diminishing rate (until the point of maximum satisfaction), whereas Marginal Utility goes on diminishing continuously, and can even become zero or negative.

Q. 5. Study the following table, figure, passage and answer the questions given below it (Any TWO):

(i) Observe the following table and answer the questions given below it:

(a) Complete the above supply schedule.

Answer: The completed supply schedule is as follows, showing a direct relationship between price and quantity supplied (an increase of ₹5 in price leads to an increase of 50 units in supply):

Price in ₹	Qty. supplied in units
10	200
15	250
20	300
25	350
30	400
35	450
40	500

(b) Draw a diagram for the above supply schedule.

Answer: (For the exam, students must draw the graph based on the table)

X-axis: Represents Quantity Supplied in units (Scale: 1 cm = 50 units, starting from 200 to 500).
Y-axis: Represents Price in ₹ (Scale: 1 cm = ₹ 5, starting from 10 to 40).
Curve: Plot the points (10,200), (15,250), (20,300), (25,350), (30,400), (35,450), and (40,500). Join these points to form a straight line 'SS' sloping upwards from left to right, indicating a positive relationship between price and quantity supplied.

(ii) Observe the following figures and answer the questions given below them:

(1) Identify and explain the degree of elasticity of demand in figure 'A'.

Answer:

Identification: Perfectly Inelastic Demand (Ed = 0).
Explanation: When a percentage change in price has absolutely no effect on the quantity demanded of a commodity, it is called perfectly inelastic demand. In Figure 'A', the demand curve is a vertical straight line parallel to the Y-axis, showing that demand remains perfectly constant regardless of changes in price.

(2) Identify and explain the degree of elasticity of demand in figure 'B'.

Answer:

Identification: Relatively Elastic Demand (Ed > 1).
Explanation: When a percentage change in price leads to a more than proportionate percentage change in quantity demanded, the demand is said to be relatively elastic. In Figure 'B', a small change in price (from P to P1) leads to a larger change in quantity demanded (from Q to Q1). The resulting demand curve is flatter in shape.

(iii) Study the following passage and answer the questions given below it:

(1) What is Digital marketing?

Answer: Digital marketing, also called online marketing, refers to marketing campaigns that appear on a computer, phone, tablet, or other devices. It involves connecting with probable customers by using the internet and other digital communication media such as social media, web-based advertising, and email.

(2) What percentage of Americans are online more than once?

Answer: According to the passage, 43% of Americans go online more than once a day.

(3) Write your opinion from the above passage.

Answer: In my opinion, digital marketing is a highly effective and essential strategy in today's digital era. With a vast majority of the population constantly online, it offers businesses an incredibly convenient way to reach a larger, targeted audience. Unlike traditional marketing, digital channels like social media and search engines allow businesses to directly connect with interested individuals, build a strong brand presence, and ultimately deliver a highly personalized and superior customer experience.

Q. 6. Answer the following questions in detail (Any TWO):

(i) State and explain the law of demand with exceptions.

Introduction: The Law of Demand is one of the important and basic laws of consumption. It was introduced by Prof. Alfred Marshall in his book, "Principles of Economics", published in 1890. The law explains the functional relationship between price and quantity demanded.

Statement of the Law: According to Prof. Alfred Marshall, "Other things being equal, higher the price of a commodity, smaller is the quantity demanded and lower the price of a commodity, larger is the quantity demanded." In simple words, other factors remaining constant, if the price of a commodity rises, demand for it falls, and when the price of a commodity falls, demand for the commodity rises. Thus, there is an inverse relationship between price and quantity demanded.

Symbolic Representation: D_x = f (P_x)
[Where D = Demand for a commodity, x = Commodity, f = function, P_x = Price of a commodity x]

Exceptions to the Law of Demand: There are certain situations where the law of demand does not operate. In these cases, a rise in price leads to a rise in demand, and a fall in price leads to a fall in demand. The demand curve slopes upwards from left to right. The exceptions are:

Giffen's Paradox: Inferior goods or low-quality goods are those whose demand does not rise even if their price falls. Sir Robert Giffen observed this behaviour in England in relation to bread. When the price of bread declined, people did not buy more because of an increase in their real income or purchasing power. They preferred to buy superior goods like meat. This is known as Giffen's paradox.
Prestige Goods: Expensive goods like diamond, gold, etc., are status symbols. So rich people buy more of it, even when their prices are high, to display their wealth.
Speculation: The law of demand does not hold true when people expect prices to rise still further. In this case, although the prices have risen today, consumers will demand more in anticipation of further rise in price. For example, prices of essential goods rise before a festival, yet people go on purchasing more.
Price Illusion: Consumers have an illusion that high-priced goods are of a better quality. Therefore, the demand for such goods tends to increase with a rise in their prices. For example, branded products.
Ignorance: Sometimes, due to ignorance, people may buy more of a commodity at a higher price. This may happen when the consumer is unaware of the price of that commodity at other places.
Habitual Goods: Due to habit of consumption, certain goods like tea, coffee, cigarettes, etc., are purchased in required quantities even at higher prices.

(ii) Explain the practical difficulties involved in the measurement of national income.

In practice, a number of difficulties arise in the collection of required statistical data for estimating national income. The practical or statistical difficulties are as follows:

Problem of Double Counting: The greatest difficulty in calculating national income is of double counting. It arises from the failure to distinguish properly between a final and an intermediate product. For example, flour used by a bakery is an intermediate product, but it is a final product when used by a household. Doing so might lead to counting the value of the same product more than once.
Existence of Non-monetized Sector: In India, especially in rural areas, there exists a non-monetized sector. Agriculture is mostly for self-consumption. Therefore, a major part of the agricultural produce does not enter the market and is exchanged for other goods and services (barter system). This makes national income estimation difficult.
Inadequate and Unreliable Data: Adequate and reliable data are not available in respect of agriculture and allied activities. Besides this, data relating to unearned incomes, consumption, and investment expenditure of rural and urban population are not available. This does not reveal the exact size of national income.
Depreciation: The calculation of depreciation on capital goods is another practical difficulty. Depreciation refers to wear and tear of capital assets, due to their use in the process of production. There are no uniform, common, or accepted standard rates of depreciation applicable to the various capital assets.
Capital Gains or Losses: Capital gains or capital losses, which accrue to the property owners by increases or decreases in the market value of their capital assets or changes in demand, are not included in the national income because these changes do not result from current economic activities.
Illiteracy and Ignorance: Due to ignorance and illiteracy, small producers do not keep an account of their production. Hence, they cannot give information about the quantity or value of their output.
Difficulties in the Classification of Working Population: In India, working population is not clearly defined. For instance, farmers in India are not engaged in agriculture round the year. Obviously, in the off-season, they engage themselves in alternative occupations. In such a case, it is very difficult to identify their incomes from a particular occupation.
Valuation of Inventories: Raw materials, intermediate goods, semi-finished, and finished products in the stock of the producers are known as inventories. Any mistake in measuring the value of inventory will distort the value of the final production of the producer.

(iii) Explain the functions of Reserve Bank of India.

Introduction: The Reserve Bank of India (RBI) is the central bank of the country. It was established on April 1, 1935, under the RBI Act of 1934, and was nationalised on January 1, 1949. It acts as the apex monetary authority.

Functions of the Reserve Bank of India:

Issue of Currency Notes: RBI has the sole right to issue currency notes of all denominations, except one rupee note and coins (which are issued by the Ministry of Finance). As per the Minimum Reserve System of 1957, RBI is required to maintain a minimum reserve of ₹ 200 crores (comprising gold worth ₹ 115 crores and foreign securities worth ₹ 85 crores) to back the currency issuance.
Banker to the Government: RBI acts as a banker, agent, and advisor to the Government of India and State Governments. It transacts the business of both the Central and State Governments. It accepts money and makes payments on behalf of the government and undertakes the management of public debt.
Banker's Bank: RBI exercises statutory control over all commercial banks in India. All scheduled banks are required to maintain a certain minimum cash reserve with the RBI. It also provides financial assistance to banks in the form of discounting of eligible bills. RBI is known as the 'Lender of the Last Resort' for commercial banks in times of financial crisis.
Custodian of Foreign Exchange Reserves: RBI acts as a custodian of the country's foreign exchange reserves. It has to maintain the official rate of exchange of the rupee and ensure its stability. RBI also undertakes to buy and sell the currencies of all the member countries of the International Monetary Fund (IMF).
Controller of Credit: As a supreme banking authority, the RBI has the power to influence the volume of credit created by commercial banks. It controls credit through quantitative methods (like Bank Rate, Open Market Operations, Cash Reserve Ratio, Statutory Liquidity Ratio) and qualitative methods (like margin requirements, moral suasion, direct action).
Collection and Publication of Data: RBI collects and compiles statistical information related to banking and other financial sectors of the economy. It publishes reports and bulletins to inform the public about economic changes.
Promotional and Developmental Functions: RBI performs various developmental functions such as extending banking services to semi-urban and rural areas, providing security to depositors, developing specialized institutions for agricultural and industrial finance, and promoting financial inclusion.

Maharashtra HSC Board 2026 Book Keeping & Accountancy Question Paper with Solutions

aaminomtex@gmail.com (Amin) — Thu, 22 Oct 2026 11:34:24 +0530

HSC Board 2026 Book Keeping & Accountancy Question Paper Solution

Q. 1. All objective questions are compulsory:

(A) Complete the table:

(1) Cost of goods sold: = Opening Stock + Purchases + Direct Expenses - Closing Stock
(2) Expenditure: 27,000 (Calculated as Income 25,000 + Deficit 2,000)
(3) Due Date: 25.01.2022 (23.12.2021 + 1 month = 23.01.2022. Adding 3 grace days gives 26.01.2022. Since January 26 is a public holiday, the due date is the preceding working day)
(4) Third Party Liability: 1,00,000 (Creditors 40,000 + Bills Payable 60,000)
(5) Sales: 2,00,000 (Purchases 1,80,000 + Gross Profit 20,000)

(B) Select the correct options and rewrite the statements:

The profit or loss from revaluation on retirement of partner is shared by (b) All the partners.
Profit and Loss Suspense Account is shown in the new balance sheet on (c) Assets side.
Not for Profit Organisation is also called (a) Service organisation.
A (d) Goodwill is an intangible asset.
(a) Revaluation Account is credited when unrecorded asset is brought into business.

(C) Write a word/term/phrase which can substitute each of the following statements:

Debit balance of revaluation account: Loss on Revaluation (or Revaluation Loss)
It is a damaged software, cracked, but nearly fully functional: Pirated Software (or Cracked Software)
Average Profit - Normal Profit = Super Profit
An account which records only revenue items in case of Not for Profit Concern: Income and Expenditure Account
Partnership Agreement in written form: Partnership Deed

(D) Complete the sentences:

When face value of the share is 100 and issued price is 110 then it is said that the shares are issued at Premium.
A person who draws the bill of exchange is known as Drawer.
Dissolution expenses are credited to Cash / Bank account.
Credit side of Receipts and Payments Account shows cash Payments.
Expenses which are paid before due date are called as Prepaid Expenses.

Q. 2. Admission of a Partner (Mohit, Rohit and Sujit)

Journal Entries in the books of the firm

Date	Particulars	Debit (₹)	Credit (₹)
2021 April 1	General Reserve A/c ...Dr. To Mohit's Capital A/c To Rohit's Capital A/c (Being General Reserve distributed among old partners in their old profit sharing ratio i.e., 1:1)	20,000	10,000 10,000
April 1	Profit and Loss Adjustment A/c ...Dr. To Stock A/c To Plant and Machinery A/c To R.D.D. A/c (Being decrease in the value of assets and provision created for doubtful debts)	49,200	16,000 28,000 5,200
April 1	Land and Building A/c ...Dr. To Profit and Loss Adjustment A/c (Being value of Land and Building appreciated)	55,000	55,000
April 1	Profit and Loss Adjustment A/c ...Dr. To Mohit's Capital A/c To Rohit's Capital A/c (Being profit on revaluation transferred to old partners' capital accounts in 1:1 ratio)	5,800	2,900 2,900
April 1	Cash / Bank A/c ...Dr. To Sujit's Capital A/c (Being cash brought in by Sujit as his capital)	2,00,000	2,00,000
April 1	Cash / Bank A/c ...Dr. To Goodwill A/c (Being Sujit's share of goodwill brought in cash: 4,00,000 × 1/5)	80,000	80,000
April 1	Goodwill A/c ...Dr. To Mohit's Capital A/c To Rohit's Capital A/c (Being goodwill distributed among sacrificing partners in their sacrificing ratio i.e., 1:1)	80,000	40,000 40,000
	Total	4,90,000	4,90,000

Profit and Loss Adjustment Account (Revaluation Account)

Dr. Particulars	Amount (₹)	Cr. Particulars	Amount (₹)
To Stock A/c	16,000	By Land & Building A/c	55,000
To Plant & Machinery A/c	28,000
To R.D.D. A/c	5,200
To Profit transferred to Capital A/cs: Mohit: 2,900 Rohit: 2,900	5,800
Total	55,000	Total	55,000

Q. 3. Bills of Exchange (In the books of Daulat)

Journal Entries

No.	Particulars	Debit (₹)	Credit (₹)
(1)	Bank A/c ...Dr. Discount A/c ...Dr. To Bills Receivable A/c (Being Sarita's acceptance discounted with the bank)	19,800 200	20,000
(2)	Cash / Bank A/c ...Dr. Rebate on Bill Discounted A/c ...Dr. To Bills Receivable A/c (Being Siddhi's acceptance retired 1 month before due date at 12% p.a. rebate: 15,000 × 12% × 1/12 = 150)	14,850 150	15,000
(3)	Bhumika's A/c ...Dr. To Bills Receivable A/c To Discount Received A/c (Being Prachiti's acceptance endorsed to Bhumika in full settlement)	10,000	9,750 250
(4)	Cash / Bank A/c ...Dr. Bills Receivable A/c ...Dr. To Sales A/c (Being goods sold to Madhura at 10% trade discount, half amount received in cash and bill drawn for the balance)	9,000 9,000	18,000
	Cash / Bank A/c ...Dr. To Bills Receivable A/c (Being Madhura's acceptance met on the due date)	9,000	9,000

Q. 3. (OR) Dissolution of Partnership Firm (Rajeev, Sanjeev and Mahesh)

Realisation Account

Dr. Particulars	Amount (₹)	Cr. Particulars	Amount (₹)
To Sundry Assets A/c: Machinery: 2,00,000 Investment: 96,000 Debtors: 2,20,000 Stock: 80,000	5,96,000	By Sundry Liabilities A/c: Creditors: 1,60,000 Bills Payable: 56,000 By R.D.D. A/c	2,16,000 12,000
To Bank A/c (Liabilities Paid): Creditors: 1,60,000 Bills Payable: 56,000	2,16,000	By Bank A/c (Assets Realised): Machinery: 72,000 Investment: 1,80,000 Stock: 84,000 Debtors: 1,80,000 Goodwill: 96,000	6,12,000
To Bank A/c (Dissolution Exp)	12,000
To Profit transferred to Capital A/cs: Rajeev (2/5): 6,400 Sanjeev (2/5): 6,400 Mahesh (1/5): 3,200	16,000
Total	8,40,000	Total	8,40,000

Partners' Capital Accounts

Dr. Particulars	Rajeev (₹)	Sanjeev (₹)	Mahesh (₹)	Cr. Particulars	Rajeev (₹)	Sanjeev (₹)	Mahesh (₹)
To Bank A/c (Final Settlement)	2,56,000	96,000	88,000	By Balance b/d	2,40,000	80,000	80,000
				By Gen. Reserve A/c	9,600	9,600	4,800
				By Realisation A/c (Profit)	6,400	6,400	3,200
Total	2,56,000	96,000	88,000	Total	2,56,000	96,000	88,000

Bank Account

Dr. Particulars	Amount (₹)	Cr. Particulars	Amount (₹)
To Balance b/d	56,000	By Realisation A/c (Liabilities)	2,16,000
To Realisation A/c (Assets Realised)	6,12,000	By Realisation A/c (Expenses)	12,000
		By Rajeev's Capital A/c	2,56,000
		By Sanjeev's Capital A/c	96,000
		By Mahesh's Capital A/c	88,000
Total	6,68,000	Total	6,68,000

Q. 4. Death of a Partner (Mahendra, Jitendra and Hitendra)

In the Books of the Firm

Hitendra's Capital Account

Dr. Particulars	Amount (₹)	Cr. Particulars	Amount (₹)
To Drawings A/c	30,000	By Balance b/d	2,40,000
To Interest on Drawings A/c	2,000	By General Reserve A/c (96,000 × 1/6)	16,000
To Hitendra's Executors' Loan A/c (Bal. Fig.)	2,95,000	By Profit & Loss Adjustment A/c (Profit)	1,000
		By Goodwill A/c	60,000
		By Profit & Loss Suspense A/c	10,000
Total	3,27,000	Total	3,27,000

Working Notes:

(i) Calculation of Hitendra's share in Goodwill:

Total Profit of last 5 years = 1,80,000 + 1,50,000 + 2,00,000 + 1,20,000 + 2,50,000 = ₹ 9,00,000
Average Profit = Total Profit / Number of Years = 9,00,000 / 5 = ₹ 1,80,000
Goodwill of the Firm = Average Profit × 2 = 1,80,000 × 2 = ₹ 3,60,000
Hitendra's Share of Goodwill = Firm's Goodwill × Hitendra's Profit Sharing Ratio (1/6) = 3,60,000 × 1/6 = ₹ 60,000

(ii) Calculation of Hitendra's share in Profit (up to date of death):

Estimated Profit for the year 2022-23 = ₹ 1,20,000
Period from 1st April 2022 to 1st October 2022 = 6 months
Hitendra's Share of Profit = Estimated Profit × Hitendra's Ratio × (Months alive / 12)
Hitendra's Share of Profit = 1,20,000 × 1/6 × 6/12 = ₹ 10,000

(iii) Calculation of Profit/Loss on Revaluation:

Decrease in Machinery value (85,000 - 80,000) = ₹ 5,000 (Loss)
Decrease in Creditors value (80,000 - 69,000) = ₹ 11,000 (Profit)
Net Profit on Revaluation = 11,000 - 5,000 = ₹ 6,000
Hitendra's Share of Revaluation Profit = 6,000 × 1/6 = ₹ 1,000

Q. 4. (OR) Importance of Computerised Accounting System

A computerised accounting system is a software program that helps businesses manage their finances and accounting processes. Its importance can be stated as follows:

Speed and Efficiency: Accounting software processes data, performs calculations, and generates reports much faster than manual bookkeeping, saving valuable time.
Accuracy: It heavily minimizes mathematical and typographical errors commonly found in manual accounting, ensuring highly accurate financial records.
Automation: Routine tasks such as updating ledgers, calculating balances, and generating financial statements (Trial Balance, P&L Account, Balance Sheet) are done automatically once the initial journal entries are passed.
Up-to-date Information: The system updates records in real-time, providing instant access to current financial data for prompt management decision-making.
Security and Backup: Computerised systems offer enhanced data security through passwords and user permissions, and they allow easy digital backups to prevent data loss.
Storage and Retrieval: Large volumes of financial data can be stored compactly on digital drives, and specific records can be retrieved instantly using search functions.

Q. 5. Issue of Shares (Pavitra Co. Ltd.)

Working Note:

Number of Equity Shares Issued: Total Capital ₹ 20,00,000 / Face Value ₹ 100 = 20,000 Shares.
Application Money: 20,000 × ₹ 20 = ₹ 4,00,000.
Allotment Money: 20,000 × ₹ 40 = ₹ 8,00,000 (Includes ₹ 10 premium i.e., Capital ₹ 6,00,000 and Premium ₹ 2,00,000).
First & Final Call Money: 20,000 × ₹ 60 = ₹ 12,00,000 (Includes ₹ 10 premium i.e., Capital ₹ 10,00,000 and Premium ₹ 2,00,000).

Journal Entries

Date	Particulars	Debit (₹)	Credit (₹)
1.	Bank A/c ...Dr. To Equity Share Application A/c (Being application money received on 20,000 equity shares @ ₹ 20 per share)	4,00,000	4,00,000
2.	Equity Share Application A/c ...Dr. To Equity Share Capital A/c (Being application money transferred to share capital account)	4,00,000	4,00,000
3.	Equity Share Allotment A/c ...Dr. To Equity Share Capital A/c To Securities Premium A/c (Being allotment money made due on 20,000 shares @ ₹ 40 per share including premium of ₹ 10)	8,00,000	6,00,000 2,00,000
4.	Bank A/c ...Dr. To Equity Share Allotment A/c (Being allotment money received)	8,00,000	8,00,000
5.	Equity Share First and Final Call A/c ...Dr. To Equity Share Capital A/c To Securities Premium A/c (Being first and final call money made due on 20,000 shares @ ₹ 60 per share including premium of ₹ 10)	12,00,000	10,00,000 2,00,000
6.	Bank A/c ...Dr. To Equity Share First and Final Call A/c (Being first and final call money received)	12,00,000	12,00,000
	Total	48,00,000	48,00,000

Q. 5. (OR) Analysis of Financial Statements (Ratios)

1. Gross Profit Ratio

Formula: (Gross Profit / Net Sales) × 100
Calculation: (6,00,000 / 24,00,000) × 100 = 25%

2. Operating Ratio

Formula: [(Cost of Goods Sold + Operating Expenses) / Net Sales] × 100
Cost of Goods Sold (COGS): Sales - Gross Profit = 24,00,000 - 6,00,000 = ₹ 18,00,000
Operating Expenses: Office Expenses + Selling Expenses = 2,50,000 + 2,00,000 = ₹ 4,50,000
Calculation: [(18,00,000 + 4,50,000) / 24,00,000] × 100 = (22,50,000 / 24,00,000) × 100 = 93.75%

3. Net Profit Ratio

Formula: (Net Profit / Net Sales) × 100
Calculation: (90,000 / 24,00,000) × 100 = 3.75%

4. Current Ratio

Formula: Current Assets / Current Liabilities
Current Assets: Debtors (1,20,000) + Stock (60,000) + Loose Tools (20,000) = ₹ 2,00,000
Current Liabilities: Creditors (60,000) + Bills Payable (40,000) + Bank Overdraft (20,000) = ₹ 1,20,000
Calculation: 2,00,000 / 1,20,000 = 1.67 : 1 (or 5:3)

Q. 6. Not for Profit Concern (Samarth Vyayamshala, Chandgad)

Income and Expenditure Account for the year ended 31st March 2022

Expenditure	Amount (₹)	Amount (₹)	Income	Amount (₹)	Amount (₹)
To Salaries		18,000	By Subscriptions	54,000
To Entertainment Expenses		7,440	Add: Outstanding for current year	21,000	75,000
To Sundry Expenses		3,900	By Donations		18,000
To Electricity Charges		3,600	By Receipts from Entertainment		16,200
To Rent		2,100	By Interest		1,200
To Printing and Stationery		2,400	By Entrance Fees	18,600
To Postage		9,600	Less: 50% Capitalised	9,300	9,300
To Depreciation on:
Building	7,500
Furniture	4,500	12,000
To Surplus (Excess of Income over Expenditure)		60,660
Total		1,19,700	Total		1,19,700

Balance Sheet as on 31st March 2022

Liabilities	Amount (₹)	Amount (₹)	Assets	Amount (₹)	Amount (₹)
Capital Fund	2,10,000		Building	1,50,000
Add: 50% Entrance Fees	9,300		Less: 5% Depreciation	7,500	1,42,500
Add: Surplus	60,660	2,79,960	Furniture	45,000
Subscription received in advance (2022-23)		1,230	Less: 10% Depreciation	4,500	40,500
			Investment		45,000
			Fixed Deposit		11,700
			Outstanding Subscription (2021-22)		21,000
			Cash in Hand		20,490
Total		2,81,190	Total		2,81,190

Working Notes:

Calculation of Outstanding Subscription:
Total members = 1,500
Yearly subscription per member = ₹ 50
Total expected subscription for 2021-22 = 1,500 × 50 = ₹ 75,000
Subscription actually received for 2021-22 = ₹ 54,000
Outstanding Subscription = 75,000 - 54,000 = ₹ 21,000
Calculation of Depreciation:
Depreciation on Building (5% on 1,50,000) = 1,50,000 × 5/100 = ₹ 7,500
Depreciation on Furniture (10% on 45,000) = 45,000 × 10/100 = ₹ 4,500
Entrance Fees:
Total Entrance Fees = ₹ 18,600
50% to be capitalised = 18,600 × 50/100 = ₹ 9,300 (Added to Capital Fund)
Remaining 50% = ₹ 9,300 (Treated as Income)

Q. 7. Final Accounts of Partnership Firm (Snehal and Deepali)

Trading and Profit & Loss Account for the year ended 31st March 2021

Dr. Particulars	Amount (₹)	Amount (₹)	Cr. Particulars	Amount (₹)	Amount (₹)
To Opening Stock		1,79,000	By Sales	2,96,000
To Purchases	96,000		Less: Sales Return	16,000	2,80,000
Less: Purchase Return	12,000	84,000
To Wages and Salary	15,200		By Closing Stock		1,76,000
Add: Outstanding Wages	1,600	16,800
To Gross Profit c/d		1,76,200
Total		4,56,000	Total		4,56,000

To Carriage Outward		2,600	By Gross Profit b/d		1,76,200
To General Expenses		5,000
To Office Rent		10,000
To Advertisement (For 2 Years)	8,000
Less: Prepaid (For 1 Year)	4,000	4,000
To Salaries		9,000
To R.D.D. (New - 5% on Debtors)	6,200
Less: R.D.D. (Old)	2,400	3,800
To Depreciation on:
Motor Van (5%)	4,000
Plant and Machinery (10%)	2,760	6,760
To Net Profit transferred to Capital A/c (2:1):
Snehal	90,027
Deepali	45,013	1,35,040
Total		1,76,200	Total		1,76,200

Balance Sheet as on 31st March 2021

Liabilities	Amount (₹)	Amount (₹)	Assets	Amount (₹)	Amount (₹)
Capital Accounts:			Freehold Property		46,000
Snehal's Capital	1,60,000		Plant and Machinery	27,600
Add: Net Profit	90,027	2,50,027	Less: 10% Depreciation	2,760	24,840
Deepali's Capital	80,000		Motor Van	80,000
Add: Net Profit	45,013	1,25,013	Less: 5% Depreciation	4,000	76,000
Sundry Creditors		68,000	Sundry Debtors	1,24,000
Outstanding Wages		1,600	Less: 5% R.D.D.	6,200	1,17,800
			Closing Stock		1,76,000
			Prepaid Advertisement		4,000
Total		4,44,640	Total		4,44,640

Working Notes:

Profit Sharing Ratio: Profits and losses are shared in the ratio of their capital. Capital balances are ₹ 1,60,000 (Snehal) and ₹ 80,000 (Deepali). Therefore, the profit sharing ratio is 1,60,000 : 80,000 i.e., 2 : 1.
Net Profit Distribution: Total Net Profit = ₹ 1,35,040.
Snehal's Share: 1,35,040 × (2/3) = ₹ 90,026.67 (rounded off to ₹ 90,027).
Deepali's Share: 1,35,040 × (1/3) = ₹ 45,013.33 (rounded off to ₹ 45,013).
Hidden Adjustment (Advertisement): Advertisement is paid for 2 years (₹ 8,000). Assuming it is effective from the beginning of the current accounting year, the current year's expense is ₹ 4,000 and the remaining ₹ 4,000 is treated as Prepaid Advertisement (shown on the Asset side).
R.D.D. (Reserve for Doubtful Debts): New R.D.D is calculated as 5% on Debtors (1,24,000 × 5/100) = ₹ 6,200. Since the Old R.D.D given in the Trial Balance (Cr. Side) is ₹ 2,400, the net amount debited to the Profit and Loss Account is (6,200 - 2,400) = ₹ 3,800.

HSC Board 2026 Book Keeping & Accountancy Question Paper Images

Maharashtra Board HSC 12th English (J-101) March 2026 Complete Paper Solution

aaminomtex@gmail.com (Amin) — Wed, 21 Oct 2026 23:41:04 +0530

Section - I : Prose

(Reading for Comprehension, Language Study, Summary and Mind Mapping)

Q. 1. (A) Read the extract and complete the activities given below:

At length Soapy reached one of the avenues to the east where the glitter and turmoil was but faint. He dragged himself toward Madison Square, for the homing instinct survives even when the home is a park bench.

But, on an unusually quiet corner, Soapy came to a standstill. Here was an old church, quaint and rambling and gabled. Through one violet-stained window a soft light glowed, where, no doubt, the organist loitered over the keys, making sure of his mastery of the coming Sabbath Anthem. For there drifted out to Soapy’s ears sweet music that caught and held him transfixed against the convolutions of the iron fence.

The moon was above, full and radiant; vehicles and pedestrians were few; sparrows twittered sleepily in the eaves or a little while the scene might have been a country churchyard. And the anthem that the organist played cemented Soapy to the iron fence, for he had known it well in the days when his life contained such things as mothers and roses and ambitions and friends and immaculate thoughts and collars.

The conjunction of Soapy’s receptive state of mind and the influences about the old church brought a sudden and wonderful change in his soul. He viewed with rising horror the pit into which he had tumbled, the degraded days, unworthy desires, dead hopes, wrecked faculties and base motives that made up his existence.

And also in a moment his heart responded thrillingly to this strange mood. A strong impulse moved him to battle with his desperate fate. He would pull himself out of the mire and would make a man of himself again; he would conquer the evil that had enslaved him. There was time; he was young yet; he would resurrect his old eager ambitions and pursue them without faltering. Those solemn but sweet organ notes had set up a revolution in him. Tomorrow he would go into the roaring downtown district and find work. A fur importer had once offered him a place as driver. He would be somebody in the world. He would .....

Soapy felt a hand laid on his arm. He looked quickly around into the impassive face of a policeman.

“What are you doin’ here?” asked the officer.

“Nothin’,” said Soapy.

“Then come along,” said the policeman.

“Three months on the island,” said the Magistrate in the Police Court the next morning.

A1. Complete the given diagram expressing Soapy’s journey:

Solution:

Title / Heading: Soapy’s Journey (The Cop and the Anthem)
Branch 1: Listens to Sabbath Anthem
Branch 2: Sweet organ music from church
Branch 3: Transformation after listening
Branch 4: Resolves to lead a respectable life

A2. Soapy’s earlier life was much different from his present life. Highlight the difference in the given table:

Solution:

Sr. No.	Earlier Life	Present Life
(1)	Days of hope, mothers, and roses	Degraded days
(2)	Clean thoughts and high ideals / Good reputation in society	Unworthy desires and dead hopes / Homeless and miserable
(3)	Ambitious	Hopeless and desperate

A3. Complete the given web highlighting the change in Soapy after listening to the sweet and solemn organ notes:

Solution:

Effect of Organ Notes on Soapy:

A sudden and wonderful change took place in his soul.
He remembered his past life, mothers, roses, and friends.
He resolved to fight his desperate fate and pull himself out of the mire.
He decided to seek work and make a man of himself again.

A4. Write about any of your decisions which made you repent later.

Solution:

One decision that I deeply repented later was choosing to procrastinate on a major science project until the night before the deadline. I was overconfident and assumed I could easily complete the work in a few hours, choosing instead to spend the weekend watching a series. However, I severely underestimated the research and formatting required, which led to a sleepless night and a poorly constructed final submission. The low grade I received was a harsh wake-up call, making me regret my poor time management and teaching me the lasting value of discipline over temporary entertainment.

A5. Language Study

1. A fur importer had offered him a place as driver.
(Choose the correct alternative to get past continuous tense as the answer of the given sentence.)
(a) A fur importer had been offering him a place as driver.
(b) A fur importer will be offering him a place as driver.
(c) A fur importer was offering him a place as driver.
(d) A fur importer has been offering him a place as driver.

Correct Answer: (c) A fur importer was offering him a place as driver.

2. He would be somebody in the world.
(Choose the correct alternative from the given sentences showing ‘certainty’.)
(a) He must be somebody in the world.
(b) He will be somebody in the world.
(c) He should be somebody in the world.
(d) He might be somebody in the world.

Correct Answer: (a) He must be somebody in the world.

A6. Vocabulary: Find the antonym of the following words from the extract:

(i) defeat
(ii) moving

Solution:

(i) defeat — success
(ii) moving — still

B. Non-Textual Grammar

B1. Do as directed

1. Gitesh is one of the best football players in the college.
(Choose the correct alternative from the given options showing the correct transformation in Positive Degree and rewrite)

(A) No other football player in the college is as good as Gitesh.
(B) Very few football players in the college are as better as Gitesh.
(C) No other football player in the college is as best as Gitesh.
(D) Very few football players in the college are as good as Gitesh.

Correct Answer: (D) Very few football players in the college are as good as Gitesh.

2. No sooner does Rohit Sharma enter the ground than he gets a warm welcome.
(Choose the correct alternative from the given options showing the correct use of ‘as soon as’ and rewrite)

(A) As soon as Rohit Sharma had entered the ground, he gets a warm welcome.
(B) As soon as Rohit Sharma have entered the ground, he gets a warm welcome.
(C) As soon as Rohit Sharma enters the ground, he gets a warm welcome.
(D) As soon as Rohit Sharma will enter the ground, he gets a warm welcome.

Correct Answer: (C) As soon as Rohit Sharma enters the ground, he gets a warm welcome.

3. Swarali said to her mother, “Did you like my sketch?”
(Choose the correct alternative from the given options showing the ‘indirect speech’ form of the given sentence.)

(a) Swarali told her mother that she liked her sketch.
(b) Swarali told her mother if she liked her sketch.
(c) Swarali asked her mother that she has liked her sketch.
(d) Swarali asked her mother if she had liked her sketch.

Correct Answer: (d) Swarali asked her mother if she had liked her sketch.

Q. 2. (A) Read the extract and complete the activities given below:

Family can be classified into two types: joint family and nuclear family.

In the joint family along with parents and children other members like grandparents, uncles, aunts, cousins, etc. are included. In such family, socialization of children is quicker. Tasks are shared. Responsibilities are shared.

Establishment of emotional bonds are common leading to mental security. Members learn adjustment. But sometimes there are conflicts and misunderstandings among family members. Children do not get complete attention from their parents.

On the contrary nuclear family consists of parents and their children. So we see flexibility in parental roles. Parents share their responsibilities. Parent child relationships are very intimate. Full attention is paid on children’s personality development. But children lack in making adjustments. Children get too much attention from their parents and so they become demanding and selfish.

Parenting styles do influence children’s behaviour. The best known research on parenting style is Diana Baumrind’s early studies of pre-school children and their parents. She proposed three parenting styles : permissive, authoritarian and authoritative.

Permissive parents are inconsistent in their behaviour. So children of such parents become dependent, immature.

Authoritarian parents are rigid and punitive in nature. Children of such parents become unfriendly, unsocial and uninvolved. Parents provide food and shelter to their children but they neglect them and become emotionally detached. That leads to indifferent, rejecting behaviour on the part of the children.

Authoritative parents encourage independence, they set limits and goals, they are firm in their behaviour. This kind of parenting style makes children self-reliable, independent and develop social skills.

A1. State whether the following statements are True or False. Correct the false statements and rewrite:

(a) In the nuclear family, the members do not have close interaction among themselves.
Answer: False
Correction: In a nuclear family, the members have close interaction among themselves.

(b) There are more conflicts and misunderstandings among family members in joint family.
Answer: True

(c) Flexibility of the parents has worst effect in children to make adjustments in nuclear family.
Answer: False
Correction: Flexibility of the parents has a positive effect on children to make adjustments in a nuclear family.

(d) The child’s physical, emotional and social development begins in school.
Answer: False
Correction: The child’s physical, emotional and social development begins at home.

A2. Pick out the sentences from the extract which indicate the strict behaviour of parents affecting the children.

Solution:
The sentences that indicate strict parental behaviour are:
(i) "Authoritarian parents are rigid and punitive in nature."
(ii) "Children of such parents become unfriendly, unsocial and uninvolved."
(iii) "Parents provide food and shelter to their children but they neglect them and become emotionally detached."
(iv) "That leads to indifferent, rejecting behaviour on the part of the children."

A3. Give reasons for the following and rewrite:

(a) In nuclear family children become demanding and selfish because ............
Solution: In nuclear family children become demanding and selfish because they receive excessive attention and do not learn to share or adjust with others.

(b) Children of permissive parents become dependent and immature because ............
Solution: Children of permissive parents become dependent and immature because their parents do not set proper rules or guide them towards responsibility.

A4. Family plays an important role in developing your social behaviour. Write your thoughts on the given statement.

Solution:
Our family is our first school, and it strongly shapes how we behave in society. From childhood, we watch and copy our parents. By interacting with family members daily, we learn important social skills like sharing, showing respect, and talking politely to others. A supportive family helps us build confidence and teaches us how to make good friends and handle problems in the outside world.

A5. Language Study

(1) Children do not get complete attention from their parents.
(Identify the correct passive voice of the above sentence from the given options and rewrite.)
(A) Complete attention do not get from their parents by the children.
(B) Complete attention from their parents does not get by the children.
(C) Complete attention from their parents is not got by the children.
(D) Complete attention from their parents is not received by the children.

Correct Answer: (D) Complete attention from their parents is not received by the children.

(2) Authoritarian parents are rigid and punitive in nature.
(Identify the correct Wh-question to get an underlined answer from the given options and rewrite. *Note: Underlined part is "rigid and punitive"*.)
(a) What are authoritarian parents in nature?
(b) What authoritarian parents are in nature?
(c) How the authoritarian parents in nature are?
(d) How are the authoritarian parents in nature?

Correct Answer: (d) How are the authoritarian parents in nature?

A6. Find the synonyms of the words given below from the extract:

(i) impress
(ii) close

Solution:
(i) impress — influence
(ii) close — intimate / near

(B) Write a ‘Summary’ of the above extract by using the following points –

[Types of families — role of the parents — development of personality — socialization of behaviour]

Solution:

Summary: The Role of Family and Parenting in Child Development

Families are primarily classified into joint and nuclear families. In joint families, socialization happens quickly as children learn adjustment through shared responsibilities among many members. In contrast, nuclear families provide full parental attention, though children may sometimes lack adjustment skills and become demanding. A child's personality development is heavily influenced by the role of the parents and their parenting style. While permissive and authoritarian parenting can lead to children becoming dependent or unsocial, authoritative parents who set clear goals and encourage independence successfully raise self-reliant children with healthy social skills.

(C) Mind Mapping: Prepare a mind map on ‘Competitive Exams’ using your ideas/thoughts/concepts to illustrate.

Solution:

Central Topic: Competitive Exams

Types of Exams
- Civil Services (UPSC, MPSC)
- Medical & Engineering (NEET, JEE)
- Banking & Defense (IBPS, NDA)
- Management (CAT, MBA-CET)
Essential Qualities Needed
- Dedication and Consistency
- Focus and Concentration
- Patience and Perseverance
- Self-Discipline
Preparation Strategies
- Effective Time Management
- Solving Mock Tests and Past Papers
- Choosing the Right Study Material
- Making Short Notes and Mind Maps
Challenges and Realities
- High Stress and Pressure
- Intense Competition
- Vast Syllabus
- Fear of Failure
Ultimate Goals / Outcomes
- Securing a Bright Future
- Career Stability
- Gaining Immense Knowledge
- Achieving Personal Dreams

(Note: In your actual answer sheet, draw circles or boxes for each of these main points and connect them to the central topic "Competitive Exams" using lines or arrows to create a visual web diagram).

Section - II : Poetry

(Poetry and Appreciation)

Q. 3. (A) Read the extract and complete the activities given below:

[Extract from the poem]

Home again, I see him drinking weak tea,

Eating a stale chapati, reading a book.

He goes into the toilet to contemplate

Man’s estrangement from a man-made world.

Coming out he trembles at the sink,

The cold water running over his brown hands,

A few droplets cling to the greying hair on his wrists.

His sullen children have often refused to share

Jokes and secrets with him.

He will now go to sleep

Listening to the static on the radio, dreaming

Of his ancestors and grandchildren, thinking

Of nomads entering a subcontinent through a narrow pass.

A1. Select and write the two statements from the given list which justify the theme of the poem:

(a) The father has problems in performing daily activities.
(b) The routine of the father is explained in detail.
(c) Isolation from the society makes the father introvert.
(d) The father is lost in his own thoughts.

Correct Answer: (c) and (d)

Solution:
The two statements that justify the theme of the poem are:
(c) Isolation from the society makes the father introvert.
(d) The father is lost in his own thoughts.

A2. Complete the given points about the problems and difficulties faced by the father.

(a) Food : ______________________
(b) Response of the children : ______________________

Solution:

a) Food: He has to consume unappetizing, poor-quality food, specifically "drinking weak tea" and "eating a stale chapati".
(b) Response of the children: They are emotionally distant and unresponsive; his "sullen children have often refused to share / Jokes and secrets with him".

A3. Write about the way you respond to your father when he comes home from his work.

Solution:
When my father comes home, my first priority is to help him leave his work stress behind. I usually greet him warmly and make him a hot cup of tea, giving him some quiet time to freshen up and relax before asking him anything. Once he is settled, we sit together and I ask him how his day went, while sharing small updates about my own studies. I always make a conscious effort to share lighthearted moments with him, ensuring he feels deeply valued, connected, and cherished after a long day of hard work.

A4. Identify and explain the figure of speech in the following line:

‘Man’s estrangement from a man-made world’

Solution:

Figure of Speech: Alliteration

Explanation: The consonant sound 'm' is repeated in the words 'Man’s', 'man', and 'made' to create a poetic or musical effect.

Alternative answer you can also use:

Figure of Speech: Paradox (or Irony)

Explanation: The line presents a contradictory and ironic situation where a human being feels isolated and disconnected from the very world that humans themselves have created.

A5. Compose a poem of your own in about 2 to 4 lines on the topic ‘Family’.

Solution:

Here are three different options for the poem on 'Family'. You can pick the one you find easiest to remember:

Option 1 (Focus on unity and strength):

Roots that hold us firm and deep,
Promises we always keep.
A shelter from the stormy weather,
A family always stands together.

Option 2 (Focus on warmth and home):

A house is built with bricks and stone,
But family makes it feel like home.
Through every laugh and every tear,
They are the ones who stay right here.

Option 3 (Focus on guidance and love):

Like stars that guide us through the night,
A family makes the darkness bright.
No matter where we choose to roam,
Their endless love will guide us home.

(B) Read the extract and write the appreciation of the poem.

[Extract from the poem "Indian Weavers"]

Weavers, weaving at break of day,

Why do you weave a garment so gay ? ......

Blue as the wing of a halcyon wild,

We weave the robes of a new-born child.

Weavers, weaving at fall of night,

Why do you weave a garment so bright ? ........

Like the plumes of a peacock, purple and green.

We weave the marriage-veils of a queen.

Weavers, weaving solemn and still,

What do you weave in the moonlight chill ......

White as a feather and white as a cloud,

We weave a dead man’s funeral shroud.

Solution:

Appreciation of the Poem: "Indian Weavers"

1. Title and Poet: The poem "Indian Weavers" is a beautiful and thought-provoking piece written by the celebrated Indian poet, Sarojini Naidu, also known as the Nightingale of India.

2. Theme and Central Idea: The central theme of the poem is the continuous cycle of human life—birth, adulthood, and death. The poet masterfully connects these three fundamental stages of life with the weavers' continuous work across three different times of the day (morning, evening, and night) and matches them with specific symbolic colours.

3. Poetic Devices: The poet relies heavily on vivid visual imagery and colours. She makes brilliant use of Simile to draw comparisons, such as "Blue as the wing of a halcyon wild" and "White as a feather". Alliteration is also used effectively to create a musical effect, as seen in the repetition of consonant sounds in "Weavers, weaving". The poem follows a simple and rhythmic AABB rhyme scheme.

4. Language and Style: The language is simple, musical, and highly symbolic. A unique feature of the poem's style is its conversational Q&A format; the poet asks a question in the first two lines of each stanza, and the weavers answer in the next two. The tone shifts subtly to match each stage of life: hopeful in the morning, celebratory at dusk, and solemn and still at night.

5. Message and Personal Opinion: The poem imparts the philosophical message that the journey of human life is inevitable and fleeting, moving from the joy of birth to the sorrow of death. I find this poem deeply moving because it beautifully uses a simple, traditional Indian craft—weaving—to explain the deepest universal truth of human existence.

Section - III : Writing Skills

Q. 4. Complete the activities as per the instructions given below:

(A) Attempt ‘Any One’ of the following activities:

(I) Drafting a virtual message:

Mr. Avinash has been invited to attend a birthday programme. His son Rohit is at college and will reach home late by evening. Mr. Avinash has to convey the same as he has to leave early and would not be able to meet his son Rohit. Draft a ‘virtual message’ in about 50 words that he would write to his son Rohit.

Solution:

Here is a drafted virtual message you can use for your exam. In your answer sheet, remember to draw a box around the message.

MESSAGE

Date: 22 October 2026
Time: 04:00 PM

Dear Rohit,

I am leaving early to attend a birthday programme. Since you will be reaching home late from college this evening, we won't be able to meet before I leave. I have kept your dinner in the fridge. Please warm it up and eat.

Dad

(II) Statement of Purpose:

Imagine, after your HSC Exam, you wish to pursue a ‘Bachelor’s degree in Performing Arts,’ with Classical Vocal Music in a reputed university in India or abroad. Prepare a ‘Statement of Purpose’ (SOP), in about 150 words which will help you to secure an admission in your dream university.

Solution:

Statement of Purpose

Music has always been more than a hobby for me; it is my primary medium of expression and a profound spiritual anchor. Having recently completed my Higher Secondary Certificate (HSC) examinations, I am eager to transition my lifelong passion into a dedicated academic pursuit. I am writing to express my strong interest in securing admission to your esteemed university to pursue a Bachelor’s degree in Performing Arts, specializing in Classical Vocal Music.

Over the past six years, I have trained rigorously in classical vocals, participating in numerous state-level competitions and college cultural fests. This foundation has instilled in me a deep appreciation for rhythm, melody, and the rich heritage of classical ragas. However, to evolve into a professional performing artist and an ambassador of this timeless art form, I require the structured academic environment and expert mentorship that your prestigious institution provides.

I am particularly drawn to your university's diverse curriculum, distinguished faculty, and state-of-the-art acoustic facilities. I am confident that my dedication, self-discipline, and profound love for music make me a highly deserving candidate for your esteemed program, and I look forward to contributing meaningfully to your artistic community.

(III) Group Discussion:

Imagine your class has attended a guest lecture on ‘positive thinking.’ Write a group discussion in the form of dialogues associated with this guest lecture, held among 3/4 student participants (150 words).

Solution:

Group Discussion: Positive Thinking

Rahul: Hi everyone! Today's guest lecture on 'Positive Thinking' was truly eye-opening, wasn't it? The speaker's point about how our mindset shapes our reality really stuck with me.

Priya: Absolutely, Rahul. I loved when the speaker clarified that positive thinking isn't about ignoring life's problems, but rather approaching them with a solution-oriented mindset. It completely changes how we view our daily challenges.

Amit: I completely agree. As students, we face so much academic stress and competition. Applying positive thinking can actually help us manage exam pressure instead of just panicking. I found the 'gratitude journal' technique she shared to be very practical.

Neha: That’s a great point, Amit. Cultivating a positive attitude is a daily habit, not a one-time fix. If we start our day by focusing on our strengths rather than our fears, it naturally boosts our confidence and productivity.

Rahul: Well said, guys. It seems we all agree that adopting this positive mindset will not only help us in our current studies but also in our future careers. Let's make a conscious effort to implement these habits!

(B) Attempt ‘Any One’ of the following activities:

(I) E-mail Writing:

Draft an ‘E-mail’ to a director of a company, requesting him/her to help, support and facilitate educational resources, teaching materials and A/V aids by some funds and sponsorship. Specify the benefits and results about your requirements (150 words).

Solution:

Here is a draft of the email you can use for your exam. In your answer sheet, draw a box around it to make it look like an authentic email interface.

To: director.csr@techsolutions.com
Subject: Request for Sponsorship of Educational Resources and A/V Aids

Respected Sir/Madam,

I am writing to you on behalf of XYZ Junior College to request your esteemed company's support in upgrading our educational infrastructure. We are currently seeking financial sponsorship to procure modern teaching materials, reference books, and Audio-Visual (A/V) aids for our classrooms.

Currently, our students lack access to digital learning tools. By sponsoring these A/V aids and educational resources, you will directly facilitate a highly interactive and engaging learning environment. The integration of visual learning will significantly enhance concept clarity, boost student retention, and ultimately lead to improved academic results. Furthermore, it will empower our students with the technological exposure necessary for their future careers.

We highly admire your company’s commitment to social causes. We would be honored to display your company’s logo as our official sponsor on the newly equipped digital classrooms.

We look forward to a positive response.

Sincerely,

XYZ
Student Representative
XYZ Junior College

(II) Report Writing:

Imagine your class attended a lecture on ‘Self Defence and Safety’, delivered by an expert speaker. Write a report on the details of this event organised at your college in about 150 words.

Solution:

Seminar on 'Self-Defense and Safety' Empowers Students

By XYZ, Student Reporter

Mumbai, 22 October: XYZ Junior College organized an insightful guest lecture on 'Self Defense and Safety' for the students on 21st October in the college auditorium. The highly engaging session was conducted by Mr. Arun Sharma, a renowned martial arts expert and personal safety instructor.

The primary objective of the event was to equip students with practical skills and situational awareness to protect themselves in unforeseen emergencies. Mr. Sharma demonstrated several basic yet effective physical self-defense techniques, emphasizing how to use everyday items like bags, pens, and keys for protection. He also shared valuable tips on digital safety and the critical importance of remaining calm during a crisis.

The interactive session concluded with a lively Q&A round where students enthusiastically cleared their doubts. The college Principal delivered the vote of thanks, bringing the event to a close. The lecture was a resounding success, leaving the students feeling significantly more confident, alert, and empowered regarding their personal safety.

(III) Interview:

Imagine you have to conduct an interview of an ‘Ideal Teacher.’ With the help of the format, draft questions on the given fields. (Do not change the sequence of the questions.)

Solution:

Name of the interviewee: Mr. Ramesh Patil

Field / Reputation: Education / State Award-winning Ideal Teacher

Date / Venue / Time: 22nd October 2026 / College Auditorium / 10:00 AM

Duration of Interview: 30 Minutes

Questions:

Questions based on:

(1) Dream
What was your childhood dream, and did you always know you wanted to become a teacher?

(2) Mission / Vision
What is your ultimate vision for your students when they step out of your classroom into the real world?

(3) Objective
What are the core objectives you strive to achieve through your unique teaching methodology?

(4) Early Preparation
How did you prepare yourself in the early stages of your life to build such a successful career in education?

(5) Support
Who has been your greatest pillar of support throughout your challenging journey as an educator?

(6) Recognition
How did you feel when you first started receiving widespread recognition from both students and parents for your dedication?

(7) Awards / Felicitations
You recently received the 'Ideal Teacher Award'; how do such felicitations impact your motivation to teach?

(8) Aim / Destination
What is your ultimate aim or final destination that you wish to reach in the field of education before you retire?

(C) Attempt ‘Any One’ of the following activities:

(I) Speech:

Imagine you have to deliver a speech on ‘Making Vacation Meaningful’ at a competition in your college. Draft a speech in about 150 words on the given topic.

Solution:

Speech: Making Vacation Meaningful

Respected Principal, honorable judges, teachers, and my dear friends, a very warm good morning to you all. Today, I stand before you to share my thoughts on the topic, 'Making Vacation Meaningful.'

When we hear the word 'vacation,' our minds immediately jump to sleeping late, binge-watching shows, and endless scrolling on social media. While taking time to relax is absolutely essential to recharge our exhausted minds, completely wasting weeks of free time is an opportunity lost.

A meaningful vacation is about striking a healthy balance. It is the perfect time to step out of our comfort zones and invest in ourselves without the pressure of exams. We can use this time to learn a new language, pick up a musical instrument, read books outside our syllabus, or volunteer for a social cause in our community. Even dedicating time to help our parents with daily chores and responsibilities can be incredibly rewarding.

Let us treat our upcoming holidays not just as an escape from our studies, but as a golden chance for self-improvement, discovery, and personal growth.

Thank you all for listening so patiently.

(II) Compering:

Imagine, you are given the responsibility to ‘compere’ a programme by your college authorities. You need to prepare your script on the programme titled, ‘Read Literature, Enjoy Literature.’ Draft the script to decide the flow of the programme. (150 words)

Solution:

1. Prayer / Welcome Song: Good morning, respected Principal, distinguished guests, teachers, and my dear friends! Welcome to our annual Literary Fest. Let us begin this auspicious event with a peaceful prayer song performed by our college choir group to evoke blessings for the day.

2. Introduction of the Event: Literature is the mirror of life and a gateway to infinite worlds. Today’s event, titled ‘Read Literature, Enjoy Literature,’ aims to celebrate the joy of reading and inspire a deep appreciation for poetry, drama, and prose among young minds.

3. Introduction of the Guests: We are deeply honored to have renowned author Dr. A. P. Sharma with us as our Chief Guest, along with our esteemed Principal, Dr. Mehta. A warm welcome to both of you, sir!

4. Felicitations: I now request our honorable Principal to felicitate our Chief Guest with a floral bouquet and a token of love on behalf of our college.

5. Activities on Literature Fest: Today’s festival is packed with exciting activities! We have poetry recitation, a book review competition, character enactments, and an interactive Q&A session with our guest speaker.

6. Conclusion: As we draw to the close of this enriching morning, I express my sincere gratitude to everyone for making this literary gathering a resounding success. Keep reading and keep enjoying literature! Have a wonderful day ahead.

(III) Expansion of Idea:

Expand the following idea with the help of the points given below (150 words):
‘Self discipline is self-caring’

Solution:

Expansion of Idea: Self-Discipline is Self-Caring

We often mistake self-care for instant gratification or short-term comfort, but true self-care lies in practicing self-discipline. Self-discipline is the highest form of self-love because it prioritizes our long-term success and peace over temporary distractions.

To cultivate self-discipline, one needs persistence and determination to stay on the right path when motivation fades. It demands perseverance and hard work to push through difficulties and build healthy habits. Deep devotion and commitment ensure that we remain faithful to our promises and personal growth. Furthermore, maintaining a clear focus on our target or goal gives our life a strong direction and purpose. Finally, self-discipline fosters a spirit of self-learning, enabling us to continuously improve ourselves without relying on external pressure.

By choosing long-term well-being over fleeting pleasures, we truly respect our potential. Ultimately, disciplining oneself is the most genuine act of self-care.

(D) Attempt ‘Any One’ of the following activities:

(I) Review Writing:

You have recently watched a documentary on ‘Wildlife in India.’ Write a review on the same with the help of the following points: Factual information, story, characters; Emotional impact; Place, event, people involved; Interview, footage, visual elements.

Solution:

Review: Documentary on 'Wildlife in India'

1. Factual Information, Story, and Characters: The recently released documentary 'Wildlife in India' offers a fascinating exploration of the nation's rich and diverse ecosystems. Rather than following a scripted storyline, the narrative centers around real-life characters: frontline forest guards, local indigenous communities, and iconic wildlife species like the Bengal tiger, Asiatic lion, and Indian elephant. It presents rich factual details about habitat preservation, biodiversity loss, and ongoing conservation challenges.

2. Visual Elements, Footage, and Interviews: The documentary stands out for its breathtaking visual cinematography. High-definition camera footage, aerial drone shots, and intimate macro-lenses bring dense rainforests, arid deserts, and snowy Himalayan peaks vividly to life. The narrative is further enriched by insightful interviews with wildlife biologists, local trackers, and forest officers who share first-hand experiences about tracking and protecting these magnificent animals.

3. Setting and Emotional Impact: Filmed across iconic locations such as Jim Corbett National Park, Kaziranga, and Gir National Park, the film visually captures both the harsh realities of human-animal conflict and the triumphant returns of endangered species. The emotional impact is profound—it evokes deep awe for nature’s beauty while striking a urgent, poignant chord regarding human responsibility toward conservation.

Wildlife in India is an inspiring, visually stunning masterpiece that every nature lover and student must watch.

(II) Blog Writing:

Write a ‘Blog’ on ‘Proper Study Habits’ with the help of the following points (100–150 words): Importance; Self discipline; Titles / Quotes; Planning and organisation; Benefits and results.

Solution:

Mastering the Mind: The Power of Proper Study Habits

"Success is the sum of small efforts, repeated day in and day out."

Developing proper study habits is the ultimate foundation for academic success. Many students struggle not because they lack intelligence, but because they lack a structured approach to learning.

The core of effective studying lies in planning and organisation. Creating a clear daily timetable, breaking down complex topics into smaller goals, and organizing notes ahead of time eliminates last-minute stress. However, even the best plan fails without self-discipline. Consistent daily effort, minimizing digital distractions, and staying focused during study hours build long-term retention.

The benefits and results of cultivating good study habits are life-changing. Beyond securing top exam grades, proper study routines boost self-confidence, reduce academic anxiety, and teach time management skills that last a lifetime.

Start building your ideal routine today—small habits lead to big achievements!

(III) Appeal Writing:

Prepare an ‘Appeal’ on the topic ‘From Smart Phones to Smart Habits’, with the help of the following points (100–150 words): Engagement in smart phone and its effect; Cultivating some good habits and interests; Effect on mind and body; Healthy lifestyle.

Solution:

APPEAL FOR A BETTER FUTURE!

"FROM SMART PHONES TO SMART HABITS"

Are you losing your valuable time endlessly scrolling through screens?

★ THE REALITY:

Excessive smartphone engagement leads to reduced attention spans, sleep disturbances, physical lethargy, and mental stress. Don't let a screen control your life and well-being!

★ MAKE THE SMART SWITCH:

Swap screen time for real-time habits:

• Cultivate hobbies like reading, painting, or learning a musical instrument.

• Engage in outdoor sports, yoga, and regular physical exercises.

• Spend meaningful face-to-face quality time with family and friends.

★ THE OUTCOME:

A healthy lifestyle refreshes your mind, energizes your body, reduces anxiety, and unlocks your true potential.

DISCONNECT TO RECONNECT WITH REAL LIFE!

Join the Movement for a Healthier, Smarter You!

Issued in Public Interest by: Youth Club, ABC Junior College

Section - IV : Literary Genre - Novel

Q. 5. (A) Complete the activities given below as per the instructions:

(I) Match the columns:

Column ‘A’	Column ‘B’
(1) Dominant literary genre	(a) 2000 years
(2) History of Novel	(b) World’s first novelist
(3) Lady Murasaki Shikibu	(c) 20th century
(4) Modern Topics	(d) Novel

Solution:

Column ‘A’	Column ‘B’
(1) Dominant literary genre	(d) Novel
(2) History of Novel	(a) 2000 years
(3) Lady Murasaki Shikibu	(b) World’s first novelist
(4) Modern Topics	(c) 20th century

(II) Choose the correct alternatives and rewrite the sentences:

Solution:

(a) 18th century people had extreme interest in romances and tragedies.

(b) Fantastic fiction is for general entertainment in pulp magazines.

(d) Setting is the background in which the story takes place.

(B) Answer in about 50 words to the questions given below:

(I) Write the character sketch of Mr. Florian.

Solution:

Character Sketch of Mr. Florian (To Sir, With Love)

Mr. Florian is the headmaster of Greenslade School in London's East End. He is portrayed as a visionary, progressive, and open-minded educator who rejects traditional, harsh methods of discipline. Instead, he believes in treating students with respect, equality, and freedom, trusting them to take responsibility for their own behavior despite their difficult backgrounds.

During the Students' Council half-yearly report, Mr. Florian delivers the opening and closing addresses, listening to his students with immense pride and interest. His democratic leadership, patience, and unwavering belief in human potential create a supportive environment, making him an inspiring and transformative figure for both students and staff.

(II) Miss Joseph focuses on ‘the importance of international and interracial cooperation.’ Justify.

Solution:

Justification:

In Chapter 17 of To Sir, With Love, Miss Joseph speaks on behalf of the senior class during the Students' Council report. She strongly emphasizes international and interracial cooperation by explaining that no nation or race can exist in isolation.

To support her point, she presents how different countries are economically and socially interdependent for essential resources, trade, and cultural growth. She highlights that prejudice and racial discrimination only breed conflict, whereas global challenges like poverty and war require united efforts. Through her speech, Miss Joseph reflects Mr. Braithwaite's teachings, demonstrating that equality, mutual respect, and brotherhood across boundaries are essential for human progress and world peace.

(C) Answer in about 50 words to the questions given below:

(I) ‘Passepartout blamed himself for his master’s misfortune.’ Explain.

Solution:

Passepartout blames himself because he discovered Detective Fix’s true identity in Hong Kong but kept it a secret from Phileas Fogg. He realized too late that sharing his travel plans with Fix allowed the detective to track and arrest Fogg at Liverpool. Passepartout felt his silence and carelessness directly caused the arrest, making Fogg lose crucial time and miss winning the bet.

(II) Describe, ‘Winning heart over money,’ as the central idea of the novel, ‘Around the World in Eighty Days.’

Solution:

In Around the World in Eighty Days, Phileas Fogg initially sets out to win a 20,000-pound wager. However, throughout the journey, he selflessly spends his fortune to save Aouda and rescue Passepartout. Though he believes he lost the money, he gains Aouda's love and loyal friendship. Ultimately, the novel shows that winning love and human affection is infinitely more rewarding than monetary wealth.

(D) Answer in about 50 words to the questions given below:

(I) According to Miss Morstan, her case was not ‘A singular case’. Explain.

Solution:

When Sherlock Holmes describes her situation as "singular" (unusual), Miss Morstan clarifies that it is not entirely unique. She explains that her employer, Mrs. Cecil Forrester, had experienced a similar puzzling mystery years ago, which Holmes successfully solved. Therefore, Miss Morstan felt her case of her father's disappearance and anonymous pearls was not without precedent.

(II) Prove: Although Holmes appeared reserved, he worked on the case during their journey.

Solution:

During the cab journey to the Lyceum Theatre, Sherlock Holmes remained silent and reserved, but his mind was actively at work. He closely monitored their route through the foggy London streets, calling out locations like Rochester Row and Vauxhall Bridge Road. By cross-referencing these places with his map, he actively traced their destination in real time, proving his continuous dedication to the case.

Original Question Paper Images

Maharashtra Board SSC & HSC Time Table 2027: Exam Dates Announced (PDF Soon)

aaminomtex@gmail.com (Amin Buhari) — Wed, 21 Oct 2026 22:03:08 +0530

Maharashtra Board SSC & HSC Exam Dates 2027: Schedule Out, Time Table PDF Coming Soon!

Hello Students! If you are appearing for the Maharashtra State Board Class 10 (SSC) or Class 12 (HSC) exams in 2027, we have an important update for your preparation.

The Maharashtra State Board of Secondary and Higher Secondary Education (MSBSHSE) has officially announced the start and end dates for your upcoming board exams. While the detailed subject-wise Time Table PDF is not released yet, you can now start planning your revision strategy based on these confirmed dates.

🚨 Important Update: When Will the PDF Time Table Be Released?

Right now, only the primary exam date windows are out. The board will release the full subject-wise date sheet very soon. Don't worry about searching multiple websites! As soon as the official Time Table PDF is released, we will activate the direct download link right here on this page.

📌 Bookmark this page on omtexclasses.com and check back regularly. The download link will appear below the moment it is officially announced!

[Link Will Be Activated Here: Download SSC & HSC Time Table 2027 PDF]

Maharashtra Board Exam 2027: Official Date Window

Here are the officially announced date ranges for both your practical and theory examinations. Use this to create your study timetable:

Exam Event	HSC (Class 12) Dates 2027	SSC (Class 10) Dates 2027
Practical / Oral Exams	January 20 to February 6, 2027	January 28 to February 13, 2027
Theory Exams Start Date	February 8, 2027	February 15, 2027
Theory Exams End Date	March 16, 2027	March 11, 2027
Total Duration	37 Days	25 Days

Exam Timings & Shifts

Just like previous years, the board exams will be conducted in two shifts. Make sure you know which shift your specific subject falls under once the final timetable is out:

Morning Shift: 11:00 AM to 2:00 PM
Evening Shift: 3:00 PM to 6:00 PM

Quick Tips for Your Board Exam Preparation

Start Now: You know your exams start in mid-February. Outline your remaining syllabus and set weekly targets today.
Focus on Practicals: Practical exams start as early as January 20th for HSC. Keep your journals, assignments, and projects completed and signed on time.
Stay Connected: Keep visiting omtexclasses.com. We will provide the exact date sheet, important updates, and study materials as soon as they are available.

Best of luck with your preparations, students! Start studying hard, and check back soon for your complete PDF schedule.

\(p\)	\(q\)	\(\sim p\)	\(\sim q\)	\(p \wedge q\)	\(\sim p \vee \sim q\)	\((p \wedge q) \wedge (\sim p \vee \sim q)\)
T	T	F	F	T	F	F
T	F	F	T	F	T	F
F	T	T	F	F	T	F
F	F	T	T	F	T	F

\(x_i\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(6\)	Total
\(p_i\)	\(\frac{1}{6}\)	\(\frac{1}{6}\)	\(\frac{1}{6}\)	\(\frac{1}{6}\)	\(\frac{1}{6}\)	\(\frac{1}{6}\)	\(1\)
\(x_i p_i\)	\(\frac{1}{6}\)	\(\frac{2}{6}\)	\(\frac{3}{6}\)	\(\frac{4}{6}\)	\(\frac{5}{6}\)	\(\frac{6}{6}\)	\(\frac{21}{6}\)
\(x_i^2 p_i\)	\(\frac{1}{6}\)	\(\frac{4}{6}\)	\(\frac{9}{6}\)	\(\frac{16}{6}\)	\(\frac{25}{6}\)	\(\frac{36}{6}\)	\(\frac{91}{6}\)

Inequality	Boundary Equation	Points on Coordinate Axes	Origin Test \((0, 0)\)	Feasible Half-Plane
\(5x + y \ge 10\)	\(5x + y = 10\)	\(A(0, 10)\), \(C(2, 0)\)	\(0 \ge 10\) (False)	Non-origin side
\(x + y \ge 6\)	\(x + y = 6\)	\(D(0, 6)\), \(B(6, 0)\)	\(0 \ge 6\) (False)	Non-origin side

Corner Point	Value of Objective Function \(z = 5x + 2y\)
\(A(0, 10)\)	\(z(A) = 5(0) + 2(10) = 20\)
\(P(1, 5)\)	\(z(P) = 5(1) + 2(5) = 5 + 10 = \mathbf{15}\) (Minimum)
\(B(6, 0)\)	\(z(B) = 5(6) + 2(0) = 30\)

Commodities	Quantity in 2000 Base Year \((q_0)\)	Quantity in 2001 Current Year \((q_1)\)
A	20	45
B	40	70
C	60	105
D	20	60
Total	\(\sum q_0 = 140\)	\(\sum q_1 = 280\)

Lanthanoids	Actinoids
1. Except promethium (\(\ce{Pm}\)), all lanthanoids are non-radioactive.	1. All actinoids are radioactive elements.
2. Besides \(+3\), they show a maximum oxidation state of \(+4\) (and occasionally \(+2\)).	2. Besides \(+3\), they exhibit higher oxidation states such as \(+4, +5, +6,\) and \(+7\).
3. They have a relatively lesser tendency to form coordination complexes.	3. They have a much greater tendency to form coordination complexes.