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Master NDA Mathematics: The Ultimate Collection of Past Papers

2026-09-01T21:06:27.339-07:00

🎯 Prime Maths Academy

Master NDA Mathematics | Jai Hind!
The Ultimate Collection of Past Papers

Clearing the NDA Mathematics paper is never an accident — it is the result of focused, intelligent, and strategic practice. Transition from board exams to the competitive rigor of UPSC NDA with our meticulously curated archive of authentic past papers.

120+

Questions per Paper

2.5

Hours Time Limit

10+

Years of Papers

Core Topic Areas

Explore the Archive →

"As an educator, I have seen firsthand that clearing the NDA Mathematics paper is never an accident — it is the result of focused, intelligent, and strategic practice. The transition from standard Class XI and XII board exams to the competitive rigor of the UPSC NDA examination requires a massive shift from lengthy, step-by-step solving to rapid, objective, and analytical thinking."

Vinod Singh — Prime Maths

📚 The Archive

What You Will Find in Our NDA Mathematics Archive

Our dedicated repository is structured to streamline your revision process. Here's what makes this resource an essential part of your study toolkit.

📅

Year-Wise Collections

Access a meticulously organized vault of past NDA mathematics question papers, including recent Phase I and Phase II exams. Every paper is preserved in its original format for authentic practice.

🎯

Topic-Wise Segregation

Beyond full papers, dive into content categorized by crucial sections. This allows you to aggressively target your weakest areas and build mastery topic by topic.

Sets & Relations Algebra Trigonometry Calculus 2D Geometry 3D Geometry Probability Statistics

📋

Authentic Format

Every problem is presented exactly as it appeared on the official UPSC examinations, giving you a true sense of the required academic rigor and complexity demanded by the examiners.

⚡

Targeted Practice

Practicing with these structured materials helps you decode question patterns and understand exactly what the examiners demand in terms of conceptual clarity and shortcut application.

⚔️ Know Your Battlefield

The NDA Mathematics paper is notorious for its brutal time constraint. Familiarize yourself with the exact structure before you step in.

120

Questions

2.5 hrs

Duration

-0.83

Negative Marking

🧠 Core Topics

Master Every Critical Area

Our archive covers every topic that appears in the NDA Mathematics syllabus. Focus your preparation on high-weightage areas first.

Sets, Relations & Functions

High Weightage

Algebra

Very High Weightage

Trigonometry

High Weightage

Calculus

Very High Weightage

2D Geometry

Medium Weightage

3D Geometry

Medium Weightage

Probability

High Weightage

Statistics

Medium Weightage

🏆 Why Past Papers Work

The Real Benefits of Solving Past Papers

Dedicating consistent time to this archive will fundamentally transform your approach to the subject. Here's how.

🧘 Eliminate Exam Anxiety

The unknown is the biggest source of stress. By repeatedly exposing yourself to the authentic formatting and phrasing of NDA questions, the final UPSC exam will feel like just another practice session. Familiarity breeds confidence.

⏱️ Master Time Management & Accuracy

The NDA Mathematics paper is notorious for its brutal time constraint — 120 questions in just 2.5 hours, coupled with strict negative marking. Setting a timer while tackling these past papers will train your brain to optimize speed, teaching you exactly which questions to attempt immediately and which time-traps to skip.

🔍 Identify Recurring Concepts

While the exact numbers change, the core concepts and specific application tricks tested repeat year after year. Working through this collection will quickly reveal the high-weightage topics and recurring models you absolutely cannot afford to miss on exam day.

Your Journey to NDA Success Starts Here

Join thousands of aspirants who have transformed their preparation with authentic past papers. The path to mastering NDA Mathematics is not about luck — it's about practice, strategy, and consistency.

Start Practicing Today →

📍 Prime Maths Academy — Empowering Future Officers

Competency Focused Problems for classes VIII - IX

2026-08-07T23:07:47.306-07:00

📐 Mathematics Quiz- Competency Focused Problems for classes VIII - IX - ICSE - CBSE

Question 1 / 20

Competency Focused Questions for CBSE and ICSE Class X

2026-05-25T08:34:10.749-07:00

Class X Mathematics Competency-Based Questions

Focus on real-life application, analysis, and conceptual clarity for CBSE & ICSE.

Real Numbers

HCF & LCM Application: Three alarm clocks ring at intervals of 12 minutes, 18 minutes, and 24 minutes respectively. If they ring together at 8:00 AM, at what time will they next ring together? Also, find the total number of times they ring together between 8:00 AM and 8:00 PM.
Irrationality Proof: Prove that $ \sqrt{5} $ is irrational. Hence, show that $ 3 + 2\sqrt{5} $ is also irrational.
Rational Numbers as Decimals: Without actual division, determine if the rational number $ \frac{129}{2^3 \times 5^4 \times 7} $ has a terminating or non-terminating decimal expansion. Justify.
Word Problem on LCM: A rectangular courtyard measures 18 m 72 cm × 13 m 20 cm. Exactly square tiles of the same size are required to pave it. Find the least possible number of such tiles.
Competency: Reasoning: Can two numbers have 18 as their HCF and 450 as their LCM? Give reasons.

Polynomials

Graphical Interpretation: The graph of a quadratic polynomial $ p(x) $ passes through $ (0, -6) $, $ (1, 0) $, and $ (2, 0) $. Find the polynomial and the sum of its zeroes.
Relation between zeroes & coefficients: If $ \alpha $ and $ \beta $ are zeroes of $ x^2 - 6x + k $ such that $ 3\alpha + 2\beta = 20 $, find the value of $ k $.
Division Algorithm: When $ x^3 - 2x^2 + ax - b $ is divided by $ x^2 - 2x - 3 $, the remainder is $ x - 6 $. Find $ a $ and $ b $.
Forming a polynomial: Find a quadratic polynomial whose zeroes are $ \frac{3\alpha}{\beta} $ and $ \frac{3\beta}{\alpha} $, where $ \alpha, \beta $ are zeroes of $ x^2 - 5x + 6 $.
Complex zeroes (ICSE): If one zero of the polynomial $ (a^2+4)x^2 + 13x + 4a $ is reciprocal of the other, find $ a $.

Pair of Linear Equations in Two Variables

Condition for infinite solutions: Find the value of $ k $ for which the system $ kx + 3y = k-3 $, $ 12x + ky = k $ has infinitely many solutions.
Cross-multiplication method: Solve:
\[ \frac{x}{a} + \frac{y}{b} = a+b, \quad \frac{x}{a^2} + \frac{y}{b^2} = 2 \]
Real-life: Age problem: Five years ago, a mother was three times as old as her daughter. Ten years from now, the mother will be twice as old as her daughter. Find their present ages.
Geometry application: The line $ 3x + 4y = 12 $ meets the x-axis at A and y-axis at B. Find the equation of the line through the midpoint of AB and perpendicular to AB.
Word problem (competitive exam): A man rowing downstream covers 24 km in 4 hours. Against the current, he covers 12 km in 6 hours. Find the speed of the boat in still water and the speed of the current.

Quadratic Equations

Nature of roots: For what value of $ k $ does the equation $ (k+1)x^2 + 2(k+3)x + (k+8) = 0 $ have equal roots? Also, find the roots.
Word problem: Time and work: Two pipes together fill a tank in $ 12\frac{8}{9} $ minutes. The larger pipe fills it in 8 minutes less than the smaller. Find the time taken by each alone.
Equation from roots: If one root of $ x^2 - 4x + a = 0 $ is twice the other, find $ a $ and the roots.
Fraction problem: The denominator of a fraction is 2 more than the numerator. If 3 is added to both numerator and denominator, the fraction becomes $ \frac{5}{6} $. Find the original fraction.
Profit-loss application: A shopkeeper buys a number of books for ₹ 1800. If he had bought 15 more books for the same amount, each book would have cost ₹ 20 less. How many books did he buy?

Arithmetic Progressions

Sum of terms: The sum of first $ n $ terms of an AP is $ 4n^2 - 3n $. Find the 20th term and the common difference.
Divisibility: How many three-digit numbers are divisible by 7? Find their sum.
Word problem: Installments: A man buys a TV on installment basis. He pays ₹ 1000 at the end of the first month, ₹ 1100 at the end of second, ₹ 1200 at the end of third, and so on. The total amount paid in 20 months is ₹ 39,000. Check if this is correct. If not, find the correct total.
Interleaved APs: If $ a, b, c $ are in AP, prove that $ \frac{1}{bc}, \frac{1}{ca}, \frac{1}{ab} $ are also in AP.
Competency: Critical thinking: The sum of three numbers in AP is 36 and their product is 1620. Find the numbers.

Triangles

Thales theorem application: In $ \triangle ABC $, $ D $ and $ E $ are points on AB and AC such that $ DE \parallel BC $. If $ AD = 2.4 $ cm, $ DB = 3.6 $ cm, and $ AC = 7.5 $ cm, find AE and EC.
Similarity in real life: A 1.5 m tall boy stands at a distance of 2 m from a lamp post and casts a shadow of 2.8 m on the ground. Find the height of the lamp post.
Pythagoras theorem: In a right triangle, the hypotenuse is 10 cm more than the smallest side. The third side is 10 cm less than the hypotenuse. Find the sides.
Area ratio: Prove that the ratio of areas of two similar triangles is equal to the square of the ratio of their corresponding medians.
Competency: Proof: In a right triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides. Prove this using a trapezium-based method (area proof).

Coordinate Geometry

Area of triangle: Find the area of the triangle formed by the points $ (2, 3) $, $ (-1, 0) $, and $ (2, -4) $. Also, find the ratio in which the y-axis divides the side joining the first two points.
Collinearity: For what value of $ k $ are the points $ (k, 2-2k) $, $ (-k+1, 2k) $, and $ (-4-k, 6-2k) $ collinear?
Real-life: Intersection: Two straight roads are represented by the equations $ x + 2y = 5 $ and $ 2x + y = 7 $. A park is located at their intersection. Find the coordinates of the park. Also, find the distance of the park from the point $ (3, 2) $.
Competency: Ratio: The line segment joining $ A(2, 3) $ and $ B(6, -5) $ is divided by the point $ P $ in the ratio $ 3:1 $ internally. Find the coordinates of $ P $ and verify that it lies on the line $ 2x + y - 7 = 0 $.
Centroid and midpoint: Two vertices of a triangle are $ (1, 4) $ and $ (5, 2) $. If its centroid is $ (3, 2) $, find the third vertex. Also, find the length of the median from the third vertex.

Trigonometry

Trigonometric identities: Prove that
\[ \frac{\sin\theta - \cos\theta + 1}{\sin\theta + \cos\theta - 1} = \frac{1 + \sec\theta}{\tan\theta} \]
Heights and distances: From the top of a 120 m high tower, the angle of depression of two cars on opposite sides of the tower are $ 30^\circ $ and $ 60^\circ $. Find the distance between the two cars.
Trigonometric ratios: If $ \tan A + \cot A = 2 $, find the value of $ \tan^8 A + \cot^8 A $.
Real-life: Navigation: A ship is 40 km from a lighthouse. The angle of elevation from the ship to the top of the lighthouse is $ 30^\circ $. After sailing towards the lighthouse, the angle becomes $ 60^\circ $. How far did the ship sail?
Competency: Simplification: Evaluate:
\[ \frac{\cos^2 20^\circ + \cos^2 70^\circ}{\sin^2 59^\circ + \sin^2 31^\circ} + \frac{\tan 15^\circ \cdot \tan 30^\circ \cdot \tan 45^\circ \cdot \tan 60^\circ \cdot \tan 75^\circ}{3} \]

Circles

Tangent properties: From an external point $ P $, two tangents $ PA $ and $ PB $ are drawn to a circle with centre $ O $. If $ \angle APB = 60^\circ $, prove that $ \triangle PAB $ is equilateral. Also, find $ \angle OAB $.
Length of tangent: A circle touches all four sides of a quadrilateral $ ABCD $ with sides $ AB = 6 $ cm, $ BC = 7 $ cm, $ CD = 5 $ cm. Find $ DA $.
Competency: Real-life: A circular park of radius 20 m has a path of width 2 m around it on the inside. Find the area of the path and the cost of fencing the inner boundary at ₹ 15 per metre.
Two circles intersecting: Two circles of radii 5 cm and 3 cm intersect at two points, and the distance between their centers is 4 cm. Find the length of the common chord.
Alternate segment theorem: In a circle, a chord $ AB $ subtends $ 90^\circ $ at the centre. $ C $ is a point on the major arc. Find the angle between the tangent at $ A $ and chord $ AC $.

Constructions (Competency – Steps & Justification)

Dividing a segment: Construct a triangle $ ABC $ with $ BC = 7 $ cm, $ \angle B = 45^\circ $, $ \angle C = 60^\circ $. Then construct a triangle similar to it whose sides are $ \frac{3}{4} $ of the corresponding sides of $ \triangle ABC $. (Write steps and justification.)
Tangent to a circle: Draw a circle of radius 3 cm. From a point 8 cm away from the center, construct a pair of tangents. Measure and verify their lengths.

Areas Related to Circles

Composite figure: A square of side 14 cm has four quadrants drawn inside it from each corner (each quadrant radius = 7 cm). Find the area of the region common to all four quadrants (central overlapping region).
Real-life: Gardening: A cow is tied with a rope of length 14 m at the corner of a rectangular field of dimensions 30 m × 20 m. Find the area the cow can graze. (Assume $ \pi = \frac{22}{7} $).

Statistics & Probability

Competency: Interpretation: The mean of 20 observations is 25. Later it was found that two observations 23 and 35 were wrongly taken as 32 and 53. Find the correct mean. Also, if each observation is multiplied by 2, what is the new mean?

ICSE Class 10 Mathematics: The Ultimate Topic-Wise PYQ Breakdown (2017–2026)

2026-05-17T00:22:22.328-07:00

Master ICSE Mathematics: The Ultimate Collection of Board Papers

As an educator, I have seen firsthand that success in ICSE Mathematics is never an accident—it is the result of focused, intelligent, and strategic practice. The leap in complexity during Classes IX and X requires a significant shift from passive reading to active, structured problem-solving. When preparing for high-stakes board exams, familiarizing yourself with the exact patterns, time constraints, and complexities of the actual papers is the most effective way to guarantee success.

That is why I have compiled a comprehensive archive specifically designed for your preparation at Prime Maths.

What You Will Find in Our ICSE Board Papers Archive

The dedicated repository for ICSE Board Papers (full page view mode) is structured to streamline your revision process. Here is what makes this resource an essential part of your study toolkit:

Year-Wise Collections: Access a meticulously organized vault of past ICSE mathematics board question papers, including recent standard and improvement exams.
Topic-Wise Segregation: Beyond just full papers, you can dive into content categorized by crucial sections like Commercial Arithmetic, Algebra, Geometry, Mensuration, and Probability & Statistics. This allows you to aggressively target your weakest areas.
Authentic Format: Every problem is presented exactly as it appeared on the official examinations, giving you a true sense of the required academic rigor.
Targeted Practice: Practicing with these structured materials helps you decode the marking scheme and understand exactly what the council examiners are looking for in your answers.

The Real Benefits of Solving Past Papers

Dedicating consistent time to this archive will fundamentally transform your approach to the subject.

Eliminate Exam Anxiety

The unknown is the biggest source of stress. By repeatedly exposing yourself to the authentic formatting and phrasing of ICSE questions, the final exam will feel like just another practice session.

Master Time Management

Mathematics is notorious for lengthy calculations. Setting a strict timer while tackling these past papers will train your brain to allocate the perfect amount of time between Section A's mandatory questions and your chosen questions in Section B.

Identify Recurring Concepts

While the exact numbers change, the core concepts tested repeat year after year. Working through this collection will quickly reveal the high-weightage topics you absolutely cannot afford to skip.

ISC Class 12 Mathematics: The Ultimate Topic-Wise PYQ Breakdown (2017–2025)

2026-05-16T09:12:54.235-07:00

ISC Class 12 Mathematics: The Ultimate Topic-Wise PYQ Breakdown (2017–2025)

Welcome back to Prime Maths! As we accelerate our preparation for the upcoming board exams, one of the most powerful strategies you can employ is analyzing Previous Year Questions (PYQs). It’s not just about solving problems; it’s about understanding the pattern, the weightage, and the exact language the council uses.

To make your revision seamless, I have sifted through the recent ISC Class 12 Mathematics examination papers (2017, 2018, 2019, 2020, 2023, 2024, 2025, and the 2025 Improvement Exam) and segregated the essential questions by topic. Grab your notebooks, and let's dive in!

📌 Algebra & Inverse Trigonometric Functions

This section tests your foundational logic and matrix operations. Pay close attention to the properties of determinants and inverse trigonometric identities.

Matrices: If the matrix $\begin{pmatrix}6 & -x^2 \\ 2x-15 & 10\end{pmatrix}$ is symmetric, find the value of $x$. (2017, [3 marks])
Inverse Trigonometry: Prove that $\frac{1}{2}\cos^{-1}\left(\frac{1-x}{1+x}\right) = \tan^{-1}\sqrt{x}$. (2017, [3 marks])
Complex Numbers: If $a+ib = \frac{x+iy}{x-iy}$, prove that $a^2+b^2=1$ and $\frac{b}{a} = \frac{2xy}{x^2-y^2}$. (2017, [3 marks])
Determinants: Using properties of determinants, prove that: $\begin{vmatrix}a & b & b+c \\ c & a & c+a \\ b & c & a+b\end{vmatrix} = (a+b+c)(a-c)^2$. (2017, [5 marks])
Linear Equations: Given that $A = \begin{pmatrix}1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2\end{pmatrix}$ and $B = \begin{pmatrix}2 & 2 & -4 \\ -4 & 2 & -4 \\ 2 & -1 & 5\end{pmatrix}$, find $AB$. Using this result, solve the system of equations: $x-y=3$, $2x+3y+4z=17$ and $y+2z=7$. (2017, [5 marks])
Inverse Trigonometry: Solve the equation for $x$: $\sin^{-1}x + \sin^{-1}(1-x) = \cos^{-1}x$, where $x \ne 0$. (2017, [5 marks])
Boolean Algebra: If $A, B$ and $C$ are the elements of Boolean algebra, simplify the expression $(A'+B')(A+C') + B'(B+C)$. Draw the simplified circuit. (2017, [5 marks])
Complex Numbers: Prove that locus of $z$ is a circle and find its centre and radius if $\frac{z-i}{z-1}$ is purely imaginary. (2017, [5 marks])
Binary Operations: The binary operation $*: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ is defined as $a*b = 2a+b$. Find $(2*3)*4$. (2018, [2 marks])
Matrices: If $A = \begin{pmatrix}5 & a \\ b & 0\end{pmatrix}$ and $A$ is a symmetric matrix, show that $a=b$. (2018, [2 marks])
Inverse Trigonometry: Solve: $3\tan^{-1}x + \cot^{-1}x = \pi$. (2018, [2 marks])
Determinants: Without expanding at any stage, find the value of: $\begin{vmatrix}a & b & c \\ a+2x & b+2y & c+2z \\ x & y & z\end{vmatrix}$. (2018, [2 marks])
Functions: If the function $f(x) = \sqrt{2x-3}$ is invertible then find its inverse. Hence prove that $(f \circ f^{-1})(x) = x$. (2018, [4 marks])
Inverse Trigonometry: If $\tan^{-1}a + \tan^{-1}b + \tan^{-1}c = \pi$, prove that $a+b+c = abc$. (2018, [4 marks])
Determinants: Use properties of determinants to solve for $x$: $\begin{vmatrix}x+a & b & c \\ c & x+b & a \\ a & b & x+c\end{vmatrix} = 0$ and $x \ne 0$. (2018, [4 marks])
Linear Equations: Using matrices, solve the following system of equations: $2x-3y+5z=11$, $3x+2y-4z=-5$, $x+y-2z=-3$. (2018, [6 marks])
Matrices: Using elementary transformation, find the inverse of the matrix: $\begin{bmatrix}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{bmatrix}$. (2018, [6 marks])
Functions: If $f: \mathbb{R} \to \mathbb{R}, f(x) = x^3$ and $g: \mathbb{R} \to \mathbb{R}, g(x) = 2x^2+1$, then find $f \circ g(x)$ and $g \circ f(x)$. (2019, [2 marks])
Inverse Trigonometry: Solve: $\sin(2\tan^{-1}x) = 1$. (2019, [2 marks])
Determinants: Using determinants, find the values of $k$, if the area of triangle with vertices $(-2, 0)$, $(0, 4)$ and $(0, k)$ is $4$ square units. (2019, [2 marks])
Matrices: Show that $(A+A')$ is a symmetric matrix, if $A = \begin{pmatrix}2 & 4 \\ 3 & 5\end{pmatrix}$. (2019, [2 marks])
Functions: If $f: A \to A$ and $A = \mathbb{R} - \{\frac{8}{5}\}$, show that the function $f(x) = \frac{8x+3}{5x-8}$ is one-one onto. Hence, find $f^{-1}$. (2019, [4 marks])
Inverse Trigonometry: Solve for $x$: $\tan^{-1}\left(\frac{x-1}{x-2}\right) + \tan^{-1}\left(\frac{x+1}{x+2}\right) = \frac{\pi}{4}$. (2019, [4 marks])
Inverse Trigonometry: If $\sec^{-1}x = \text{cosec}^{-1}y$, show that $\frac{1}{x^2} + \frac{1}{y^2} = 1$. (2019, [4 marks])
Determinants: Using properties of determinants prove that: $\begin{vmatrix}x & x(x^2+1) & x+1 \\ y & y(y^2+1) & y+1 \\ z & z(z^2+1) & z+1\end{vmatrix} = (x-y)(y-z)(z-x)(x+y+z)$. (2019, [4 marks])
Linear Equations: Solve the following system of linear equations using matrix method: $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 9$, $\frac{2}{x} + \frac{5}{y} + \frac{7}{z} = 52$, $\frac{2}{x} + \frac{1}{y} - \frac{1}{z} = 0$. (2019, [6 marks])
Binary Operations: Determine whether the binary operation $*$ on $\mathbb{R}$ defined by $a*b = |a-b|$ is commutative. Also, find the value of $(-3)*2$. (2020, [2 marks])
Inverse Trigonometry: Prove that: $\tan^2(\sec^{-1}2) + \cot^2(\text{cosec}^{-1}3) = 11$. (2020, [2 marks])
Determinants: Without expanding at any stage, find the value of the determinant: $\Delta = \begin{vmatrix}20 & a & b+c \\ 20 & b & a+c \\ 20 & c & a+b\end{vmatrix}$. (2020, [2 marks])
Matrices: If $\begin{pmatrix}2 & 3 \\ 5 & 7\end{pmatrix}\begin{pmatrix}1 & -3 \\ -2 & 4\end{pmatrix} = \begin{pmatrix}-4 & 6 \\ -9 & x\end{pmatrix}$, find $x$. (2020, [2 marks])
Functions: If the function $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = \frac{3x+4}{5x-7}, (x \ne \frac{7}{5})$ and $g: \mathbb{R} \to \mathbb{R}$ be defined as $g(x) = \frac{7x+4}{5x-3}, (x \ne \frac{3}{5})$, show that $(g \circ f)(x) = (f \circ g)(x)$. (2020, [4 marks])
Inverse Trigonometry: If $\cos^{-1}\frac{x}{2} + \cos^{-1}\frac{y}{3} = \theta$, then prove that $9x^2 - 12xy\cos\theta + 4y^2 = 36\sin^2\theta$. (2020, [4 marks])
Inverse Trigonometry: Evaluate: $\cos(2\cos^{-1}x + \sin^{-1}x)$ at $x = \frac{1}{5}$. (2020, [4 marks])
Determinants: Using properties of determinants, show that $\begin{vmatrix}x & p & q \\ p & x & q \\ q & q & x\end{vmatrix} = (x-p)(x^2+px-2q^2)$. (2020, [4 marks])
Linear Equations: Solve the following system of linear equations using matrices: $x-2y = 10$, $2x-y-z = 8$, $-2y+z = 7$. (2020, [6 marks])

Relations: Determine if the relation $R$ on $\{1, 2, 3\}$ given by $R=\{(1,1), (2,2), (1,2), (3,3), (2,3)\}$ is reflexive, symmetric, or transitive. (2023, [1 mark])

Matrices: Find the value of $k$ for which the matrix $\begin{bmatrix}0 & k \\ -6 & 0\end{bmatrix}$ is a skew-symmetric matrix. (2023, [1 mark])

Functions: If $f(x)=[4-(x-7)^3]^{1/5}$ is a real invertible function, find $f^{-1}(x)$. (2023, [2 marks])

Determinants: Evaluate the determinant without expanding:
$$\begin{vmatrix} 5 & 5 & 5 \\ a & b & c \\ b+c & c+a & a+b \end{vmatrix}$$ (2023, [2 marks])

Inverse Trigonometry: Solve for $x$: $5\tan^{-1}x + 3\cot^{-1}x = 2\pi$. (2023, [2 marks])

Linear Equations: Use the matrix method to solve the system of equations: $\frac{2}{x} + \frac{3}{y} + \frac{10}{z} = 4$, $\frac{4}{x} - \frac{6}{y} + \frac{5}{z} = 1$, and $\frac{6}{x} + \frac{9}{y} - \frac{20}{z} = 2$. (2023, [6 marks])

Inverse Trigonometry: Evaluate the value of $\csc(\sin^{-1}(\frac{-1}{2})) - \sec(\cos^{-1}(\frac{-1}{2}))$. (2024, [1 mark])

Matrices: Determine if $AB-BA$ is a symmetric or skew-symmetric matrix given that $A$ and $B$ are symmetric matrices of the same order. (2024, [1 mark])

Inverse Trigonometry: Solve for $x$: $\sin^{-1}(\frac{x}{2}) + \cos^{-1}x = \frac{\pi}{6}$. (2024, [4 marks])

Inverse Trigonometry: If $\sin^{-1}x + \sin^{-1}y + \sin^{-1}z = \pi$, show that $x^2-y^2-z^2+2yz\sqrt{1-x^2}=0$. (2024, [4 marks])

Matrices: Find the value of $A^{16}$ if $A=\begin{bmatrix}0 & a \\ 0 & 0\end{bmatrix}$. (2025, [1 mark])

Relations: Write the smallest equivalence relation from the set $A$ to $A$, where $A=\{1,2,3\}$. (2025, [1 mark])

Inverse Trigonometry: Find the value of $\tan^{-1}x - \cot^{-1}x$ if $(\tan^{-1}x)^2 - (\cot^{-1}x)^2 = \frac{5\pi}{8}$. (2025, [2 marks])

Determinants: Prove the determinant identity $\begin{vmatrix} 1 & 1 & 1 \\ x & y & z \\ x^3 & y^3 & z^3 \end{vmatrix} = 0$ if $x+y+z=0$. (2025, [4 marks])

Linear Equations: Find the value of $\mu$ if the system of equations $2x+3y-8=0$, $7x-5y+3=0$, and $4x-6y+\mu=0$ is consistent. (2025 IE, [1 mark])

Matrices: For what value of $a$ is the matrix $A=\begin{bmatrix} 1 & -2 & 3 \\ 1 & 2 & 1 \\ a & 2 & -3 \end{bmatrix}$ not invertible? (2025 IE, [1 mark])

Determinants: Using properties of determinants, prove that:
$$\begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ a^3 & b^3 & c^3 \end{vmatrix} = (a-b)(b-c)(c-a)(a+b+c)$$ (2025 IE, [4 marks])

📌 Calculus

Calculus forms the major chunk of the paper. Focus heavily on differential equations and properties of definite integrals.

Limits: Using L'Hospital's Rule, evaluate: $\lim_{x \to \pi/2}(x\tan x - \frac{\pi}{4}\sec x)$. (2017, [3 marks])
Integration: Evaluate: $\int \frac{1}{x^2}\sin^2(\frac{1}{x}) dx$. (2017, [3 marks])
Definite Integrals: Evaluate: $\int_0^{\pi/4} \log(1+\tan\theta) d\theta$. (2017, [3 marks])
Differential Equations: Solve: $\frac{dy}{dx} = 1-xy+y-x$. (2017, [3 marks])
Mean Value Theorems: Verify Lagrange's mean value theorem for the function $f(x) = x(1-\log x)$ and find the value of '$c$' in the interval $[1, 2]$. (2017, [5 marks])
Differentiation: If $y = \cos(\sin x)$, show that: $\frac{d^2y}{dx^2} + \tan x\frac{dy}{dx} + y\cos^2x = 0$. (2017, [5 marks])
Maxima & Minima: Show that the surface area of a closed cuboid with square base and given volume is minimum when it is a cube. (2017, [5 marks])
Integration: Evaluate: $\int \frac{\sin 2x}{(1+\sin x)(2+\sin x)} dx$. (2017, [5 marks])
Area Under Curves: Draw a rough sketch of the curve $y^2 = 4x$ and find the area of the region enclosed by the curve and the line $y=x$. (2017, [5 marks])
Differential Equations: Solve: $(x^2-yx^2)dy + (y^2+xy^2)dx = 0$. (2017, [5 marks])
Continuity: Find the value of constant '$k$' so that the function $f(x) = \frac{x^2-2x-3}{x+1}$ for $x \ne -1$, and $k$ for $x = -1$ is continuous at $x = -1$. (2018, [2 marks])
Approximations: Find the approximate change in the volume '$V$' of a cube of side $x$ metres caused by decreasing the side by $1\%$. (2018, [2 marks])
Integration: Evaluate: $\int \frac{x^3+5x^2+4x+1}{x^2} dx$. (2018, [2 marks])
Differential Equations: Find the differential equation of the family of concentric circles $x^2+y^2=a^2$. (2018, [2 marks])
Differentiability: Show that the function $f(x) = x^2$ for $x \le 1$ and $\frac{1}{x}$ for $x > 1$ is continuous at $x = 1$ but not differentiable. (2018, [4 marks])
Mean Value Theorems: Verify Rolle's theorem for the function $f(x) = e^{-x}\sin x$ on $[0, \pi]$. (2018, [4 marks])
Differentiation: If $x = \tan(\frac{1}{a}\log y)$, prove that $(1+x^2)\frac{d^2y}{dx^2} + (2x-a)\frac{dy}{dx} = 0$. (2018, [4 marks])
Integration: Evaluate: $\int \tan^{-1}\sqrt{x} dx$. (2018, [4 marks])
Tangents & Normals: Find the points on the curve $y=4x^3-3x+5$ at which the equation of the tangent is parallel to the x-axis. (2018, [4 marks])
Rate of Change: Water is dripping out from a conical funnel of semi-vertical angle $\pi/4$ at the uniform rate of $2 \text{ cm}^2/\text{sec}$ in the surface, through a tiny hole at the vertex of the bottom. When the slant height of the water level is $4\text{ cm}$, find the rate of decrease of the slant height of the water. (2018, [4 marks])
Differential Equations: Solve: $\sin x\frac{dy}{dx} - y = \sin x \cdot \tan\frac{x}{2}$. (2018, [6 marks])
Differential Equations: The population of a town grows at the rate of $10\%$ per year. Using differential equation, find how long will it take for the population to grow 4 times. (2018, [6 marks])
Maxima & Minima: A cone is inscribed in a sphere of radius $12\text{ cm}$. If the volume of the cone is maximum, find its height. (2018, [6 marks])
Integration: Evaluate: $\int \frac{x-1}{\sqrt{x^2-x}} dx$. (2018, [6 marks])
Definite Integrals: Evaluate: $\int_0^{\pi/2} \frac{\cos^2x}{1+\sin x\cos x} dx$. (2018, [6 marks])
Area Under Curves: Draw a rough sketch of the curve and find the area of the region bounded by curve $y^2 = 8x$ and the line $x = 2$. (2018, [6 marks])
Area Under Curves: Sketch the graph of $y = |x+4|$. Using integration, find the area of the region bounded by the curve $y = |x+4|$ and $x = -6$ and $x = 0$. (2018, [6 marks])
Continuity: $f(x) = \frac{x^2-9}{x-3}$ is not defined at $x = 3$. What value should be assigned to $f(3)$ for continuity of $f(x)$ at $x=3$? (2019, [2 marks])
Increasing & Decreasing Functions: Prove that the function $f(x) = x^3-6x^2+12x+5$ is increasing on $\mathbb{R}$. (2019, [2 marks])
Integration: Evaluate: $\int \frac{\sec^2x}{\text{cosec}^2x} dx$. (2019, [2 marks])
Limits: Using L'Hospital's Rule, evaluate: $\lim_{x \to 0}\frac{8^x-4^x}{4x}$. (2019, [2 marks])
Differentiability: Show that the function $f(x) = |x-4|, x \in \mathbb{R}$ is continuous, but not differentiable at $x=4$. (2019, [4 marks])
Mean Value Theorems: Verify the Lagrange's mean value theorem for the function $f(x) = x+\frac{1}{x}$ in the interval $[1, 3]$. (2019, [4 marks])
Differentiation: If $y = e^{\sin^{-1}x}$ and $z = e^{-\cos^{-1}x}$, prove that $\frac{dy}{dz} = e^{\pi/2}$. (2019, [4 marks])
Rate of Change: A $13\text{ m}$ long ladder is leaning against a wall, touching the wall at a certain height from the ground level. The bottom of the ladder is pulled away from the wall, along the ground, at the rate of $2 \text{ m/s}$. How fast is the height on the wall decreasing when the foot of the ladder is $5\text{ m}$ away from the wall? (2019, [4 marks])
Integration: Evaluate: $\int \frac{x(1+x^2)}{1+x^4} dx$. (2019, [4 marks])
Definite Integrals: Evaluate: $\int_{-6}^3 |x+3| dx$. (2019, [4 marks])
Differential Equations: Solve the differential equation: $\frac{dy}{dx} = \frac{x+y+2}{2(x+y)-1}$. (2019, [4 marks])
Maxima & Minima: The volume of a closed rectangular metal box with a square base is $4096 \text{ cm}^3$. The cost of polishing the outer surface of the box is ₹4 per square cm. Find the dimensions of the box for the minimum cost of polishing it. (2019, [6 marks])
Maxima & Minima: Find the point on the straight line $2x+3y=6$ which is closest to the origin. (2019, [6 marks])
Definite Integrals: Evaluate: $\int_0^{\pi} \frac{x\tan x}{\sec x+\tan x} dx$. (2019, [6 marks])
Area Under Curves: Draw a rough sketch and find the area bounded by the curve $x^2 = y$ and $x+y=2$. (2019, [6 marks])
Differentiation: Find $\frac{dy}{dx}$ if $x^3+y^3=3axy$. (2020, [2 marks])
Rate of Change: The edge of a variable cube is increasing at the rate of $10 \text{ cm/sec}$. How fast is the volume of the cube increasing when the edge is $5\text{ cm}$ long? (2020, [2 marks])
Definite Integrals: Evaluate: $\int_4^5 |x-5| dx$. (2020, [2 marks])
Differential Equations: Form a differential equation of the family of the curves $y^2 = 4ax$. (2020, [2 marks])
Mean Value Theorems: Verify Rolle's theorem for the function, $f(x) = -1+\cos x$ in the interval $[0, 2\pi]$. (2020, [4 marks])
Differentiation: If $y = e^{m\sin^{-1}x}$, prove that $(1-x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} = m^2y$. (2020, [4 marks])
Tangents & Normals: The equation of tangent at $(2, 3)$ on the curve $y^2 = px^3+q$ is $y = 4x-7$. Find the values of '$p$' and '$q$'. (2020, [4 marks])
Limits: Using L'Hospital's rule, evaluate: $\lim_{x \to 0}\frac{xe^x-\log(1+x)}{x^2}$. (2020, [4 marks])
Integration: Evaluate: $\int \frac{dx}{\sqrt{5x-4x^2}}$. (2020, [4 marks])
Integration: Evaluate: $\int \sin^3x\cos^4x dx$. (2020, [4 marks])
Differential Equations: Solve the differential equation $(1+x^2)\frac{dy}{dx} = 4x^2-2xy$. (2020, [4 marks])
Maxima & Minima: Show that the radius of a closed right circular cylinder of given surface area and maximum volume is equal to half of its height. (2020, [6 marks])
Maxima & Minima: Prove that the area of right-angled triangle of given hypotenuse is maximum when the triangle is isosceles. (2020, [6 marks])
Integration: Evaluate: $\int \tan^{-1}\sqrt{\frac{1-x}{1+x}} dx$. (2020, [6 marks])
Integration: Evaluate: $\int \frac{2x+7}{x^2-x-2} dx$. (2020, [6 marks])
Area Under Curves: Draw a rough sketch of the curves $y^2 = x$ and $y^2 = 4-3x$ and find the area enclosed between them. (2020, [6 marks])
Rate of Change: An edge of a variable cube is increasing at the rate of 10 cm/sec. How fast will the volume of the cube increase if the edge is 5 cm long? (2023, [1 mark])
Differentiation: Find the derivative of $\log x$ with respect to $\frac{1}{x}$. (2023, [1 mark])
Differential Equations: Solve the differential equation: $\frac{dy}{dx} = \csc y$. (2023, [1 mark])
Integration: Evaluate $\int \cos^{-1}(\sin x) dx$. (2023, [2 marks])
Differentiation: If $y=e^{ax}\cos bx$, prove that $\frac{d^2y}{dx^2} - 2a\frac{dy}{dx} + (a^2+b^2)y = 0$. (2023, [4 marks])
Maxima/Minima: Prove that the semi-vertical angle of the right circular cone of given volume and least curved area is $\cot^{-1}\sqrt{2}$. (2023, [6 marks])
Differential Equations: Find the order and the degree of the differential equation $1+(\frac{dy}{dx})^2 = \frac{d^2y}{dx^2}$. (2024, [1 mark])
Tangents & Normals: Find a point on the curve $y=(x-2)^2$ at which the tangent is parallel to the line joining the chord through the points (2, 0) and (4, 4). (2024, [2 marks])
Integration: Evaluate $\int_0^{2\pi} \frac{1}{1+e^{\sin x}} dx$. (2024, [2 marks])
Differentiation: If $y=3\cos(\log x) + 4\sin(\log x)$, show that $x^2\frac{d^2y}{dx^2} + x\frac{dy}{dx} + y = 0$. (2024, [4 marks])
Maxima/Minima: A window is designed in the shape of a rectangle surmounted by an equilateral triangle. If the perimeter is 12 m, find the dimensions that admit maximum sunlight. (2024, [6 marks])
Rate of Change: A cylindrical popcorn tub of radius 10 cm is filled with popcorn at a rate of $314\text{ cm}^3\text{/min}$. Find the rate at which the level of popcorn is increasing. (2025, [1 mark])
Tangents & Normals: Find the point on the curve $y=2x^2-6x-4$ at which the tangent is parallel to the x-axis. (2025, [2 marks])
Differentiation: If $x^y = e^{x-y}$, prove that $\frac{dy}{dx} = \frac{\log x}{(1+\log x)^2}$. (2025, [2 marks])
Differential Equations: Show that $\tan^{-1}x + \tan^{-1}y = C$ is the general solution of the differential equation $(1+x^2)dy + (1+y^2)dx = 0$. (2025, [2 marks])
Differentiation: Find the derivative of $\tan^{-1}(\frac{\cos x - \sin x}{\cos x + \sin x})$ with respect to $x$. (2025 IE, [1 mark])
Integration: Evaluate $\int_{-2}^2 x f(x) dx$ given $f(x)=x+g(x)$ where $g(x)$ is an even function. (2025 IE, [2 marks])
Tangents & Normals: Find the equation of the normal to the curve $y=x^2-3x+1$ at the point (3, 1). (2025 IE, [2 marks])
Integration: Evaluate $\int_0^{\pi/2} \log(\tan x) dx$. (2025 IE, [2 marks])
Differentiation: If $x=\sin t$ and $y=\sin pt$, prove that $(1-x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} + p^2y = 0$. (2025 IE, [4 marks])

📌 Vectors and 3D Geometry

Visualize these problems before you put pen to paper. Drawing a quick rough sketch can often save you from sign errors in 3D geometry.

Vectors: If $\vec{a}, \vec{b}, \vec{c}$ are three mutually perpendicular vectors of equal magnitude, prove that $(\vec{a}+\vec{b}+\vec{c})$ is equally inclined with vectors $\vec{a}, \vec{b}$ and $\vec{c}$. (2017, [5 marks])
Vectors: Find the value of $\lambda$ for which the four points with position vectors $6\hat{i}-7\hat{j}$, $16\hat{i}-19\hat{j}-4\hat{k}$, $\lambda\hat{j}-6\hat{k}$ and $2\hat{i}-5\hat{j}+10\hat{k}$ are coplanar. (2017, [5 marks])
3D Geometry: Show that the lines $\frac{x-1}{2}=\frac{y+1}{-3}=\frac{z+10}{8}$ and $\frac{x-4}{1}=\frac{y+3}{-4}=\frac{z+1}{7}$ intersect. Find the coordinates of their point of intersection. (2017, [5 marks])
3D Geometry: Find the equation of the plane passing through the point $(1,-2,1)$ and perpendicular to the line joining the points $A(3,2,1)$ and $B(1,4,2)$. (2017, [5 marks])
Vectors: Find $\lambda$ if the scalar projection of $\vec{a} = \lambda\hat{i}+\hat{j}+4\hat{k}$ on $\vec{b} = 2\hat{i}+6\hat{j}+3\hat{k}$ is $4\text{ units}$. (2018, [2 marks])
3D Geometry: The Cartesian equation of a line is: $2x-3 = 3y+1 = 5-6z$. Find the vector equation of a line passing through $(7,-5,0)$ and parallel to the given line. (2018, [2 marks])
3D Geometry: Find the equation of the plane through the intersection of the planes $\vec{r} \cdot (\hat{i}+3\hat{j}-\hat{k}) = 9$ and $\vec{r} \cdot (2\hat{i}-\hat{j}+\hat{k}) = 3$ and passing through the origin. (2018, [2 marks])
Vectors: If $A, B, C$ are three non-collinear points with position vectors $\vec{a}, \vec{b}, \vec{c}$ respectively, then show that the length of the perpendicular from $C$ on $AB$ is $\frac{|(\vec{a}\times\vec{b})+(\vec{b}\times\vec{c})+(\vec{c}\times\vec{a})|}{|\vec{b}-\vec{a}|}$. (2018, [4 marks])
Vectors: Show that the four points $A, B, C$ and $D$ with position vectors $4\hat{i}+5\hat{j}+\hat{k}, -\hat{j}-\hat{k}, 3\hat{i}+9\hat{j}+4\hat{k}$ and $4(-\hat{i}+\hat{j}+\hat{k})$ respectively, are coplanar. (2018, [4 marks])
3D Geometry: Find the image of a point having position vector: $3\hat{i}-2\hat{j}+\hat{k}$ in the plane $\vec{r} \cdot (3\hat{i}-\hat{j}+4\hat{k}) = 2$. (2018, [6 marks])
Vectors: If $\vec{a}$ and $\vec{b}$ are perpendicular vectors, $|\vec{a}+\vec{b}| = 13$ and $|\vec{a}| = 5$, find the value of $|\vec{b}|$. (2019, [2 marks])
3D Geometry: Find the length of the perpendicular from origin to the plane $\vec{r} \cdot (3\hat{i}-4\hat{j}-12\hat{k}) + 39 = 0$. (2019, [2 marks])
3D Geometry: Find the angle between the two lines $2x = 3y = -z$ and $6x = -y = -4z$. (2019, [2 marks])
Vectors: If $\vec{a} = \hat{i}-2\hat{j}+3\hat{k}, \vec{b} = 2\hat{i}+3\hat{j}-5\hat{k}$, prove that $\vec{a}$ and $\vec{a}\times\vec{b}$ are perpendicular. (2019, [4 marks])
Vectors: If $\vec{a}$ and $\vec{b}$ are non-collinear vectors, find the value of $x$ such that the vectors $\vec{r} = (x-2)\vec{a}+\vec{b}$ and $\vec{s} = (3+2x)\vec{a}-2\vec{b}$ are collinear. (2019, [4 marks])
3D Geometry: Find the equation of the plane passing through the intersection of the planes $2x+2y-3z-7 = 0$ and $2x+5y+3z-9 = 0$ such that the intercepts made by the resulting plane on the x-axis and the z-axis are equal. (2019, [4 marks])
3D Geometry: Find the equation of the lines passing through the point $(2, 1, 3)$ and perpendicular to the lines $\frac{x-1}{1} = \frac{y-2}{2} = \frac{z-3}{3}$ and $\frac{x}{-3} = \frac{y}{2} = \frac{z}{5}$. (2019, [4 marks])
Vectors: Write a vector of magnitude of $18\text{ units}$ in the direction of the vector $\hat{i}-2\hat{j}-2\hat{k}$. (2020, [2 marks])
3D Geometry: Find the angle between the two lines: $\frac{x+1}{2} = \frac{y-2}{5} = \frac{z+3}{4}$ and $\frac{x-1}{5} = \frac{y+2}{2} = \frac{z-1}{-5}$. (2020, [2 marks])
3D Geometry: Find the equation of the plane passing through the point $(2,-3, 1)$ and perpendicular to the line joining the points $(4, 5, 0)$ and $(1,-2, 4)$. (2020, [2 marks])
Vectors: Prove that $\vec{a} \cdot [(\vec{b}+\vec{c})\times(\vec{a}+3\vec{b}+4\vec{c})] = [\vec{a} \vec{b} \vec{c}]$. (2020, [4 marks])
Vectors: Using vectors, find the area of the triangle whose vertices are: $A(3,-1, 2), B(1,-1,-3)$ and $C(4,-3, 1)$. (2020, [4 marks])
3D Geometry: Find the image of the point $(3,-2, 1)$ in the plane $3x-y+4z=2$. (2020, [4 marks])
3D Geometry: Determine the equation of the line passing through the point $(-1, 3,-2)$ and perpendicular to the lines: $\frac{x}{1} = \frac{y}{2} = \frac{z}{3}$ and $\frac{x+2}{-3} = \frac{y-1}{2} = \frac{z+1}{5}$. (2020, [4 marks])

Vectors: Find the area of the parallelogram whose diagonals are $\hat{i}-3\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+\hat{k}$. (2023, [1 mark])

3D Geometry: Write the equation of the plane passing through the point (2, 4, 6) and making equal intercepts on the coordinate axes. (2023, [1 mark])

Vectors: If $A(1, 2, -3)$ and $B(-1, -2, 1)$ are end points of $\vec{AB}$, find the unit vector in the direction of $\vec{AB}$. (2023, [2 marks])

3D Geometry: Find the equation of the plane passing through the point (1, 1, -1) and perpendicular to the planes $x+2y+3z=7$ and $2x-3y+4z=0$. (2023, [4 marks])

Vectors: If $\vec{a}=3\hat{i}-2\hat{j}+\hat{k}$ and $\vec{b}=2\hat{i}-4\hat{j}-3\hat{k}$, find the value of $|\vec{a}-2\vec{b}|$. (2024, [1 mark])

Vectors: Find a vector of magnitude 20 units parallel to the vector $2\hat{i}+5\hat{j}+4\hat{k}$. (2024, [1 mark])

Vectors: If $\vec{a}\times\vec{b} = \vec{a}\times\vec{c}$ where $\vec{a}, \vec{b}$ and $\vec{c}$ are non-zero vectors, prove that either $\vec{b}=\vec{c}$ or $\vec{a}$ and $(\vec{b}-\vec{c})$ are parallel. (2024, [2 marks])

3D Geometry: Find the angle between the two planes $x+y+2z=9$ and $2x-y+z=15$. (2025, [1 mark])

3D Geometry: The equation of the path traced by a honeybee is $\vec{r}=(\hat{i}+2\hat{j}+3\hat{k})+\lambda(2\hat{i}+3\hat{j}+4\hat{k})$. Find the equation of the parallel path traced by another honeybee passing through the point (2, 4, 5). (2025, [1 mark])

3D Geometry: Find the equation of the plane passing through the points (2, 2, -1), (3, 4, 2) and (7, 0, 6). (2025, [2 marks])

Vectors: Consider position vectors $\vec{OA}=2\hat{i}-2\hat{j}+\hat{k}$, $\vec{OB}=\hat{i}+2\hat{j}-2\hat{k}$ and $\vec{OC}=2\hat{i}-\hat{j}+4\hat{k}$. Find the area of the triangle ABC whose sides are $\vec{AB}$ and $\vec{BC}$. (2025, [2 marks])

3D Geometry: Find the shortest distance between the lines $\vec{r}=(\hat{i}+2\hat{j}+3\hat{k})+\lambda(2\hat{i}+3\hat{j}+4\hat{k})$ and $\vec{r}=(2\hat{i}+4\hat{j}+5\hat{k})+\mu(4\hat{i}+6\hat{j}+8\hat{k})$. (2025 IE, [2 marks])

Vectors: Consider vectors $\vec{a}=2\hat{i}-3\hat{j}+4\hat{k}$ and $\vec{b}=5\hat{i}+q\hat{j}-\hat{k}$. Calculate $q$ if both vectors are perpendicular to each other. (2025 IE, [1 mark])

📌 Probability and Linear Programming (LPP)

These questions are incredibly scoring if you read the word problems carefully. Define your variables clearly in LPP, and always double-check your sample space in probability.

Probability: A problem is given to three students whose chances of solving it are $1/4, 1/5$ and $1/3$ respectively. Find the probability that the problem is solved. (2017, [3 marks])
Probability: In a class of 60 students, 30 opted for Mathematics, 32 opted for Biology and 24 opted for both Mathematics and Biology. If one of these students is selected at random, find the probability that: (i) The student opted for Mathematics or Biology. (ii) The student has opted neither Mathematics nor Biology. (iii) The student has opted Mathematics but not Biology. (2017, [5 marks])
Probability: Bag A contains 1 white, 2 blue and 3 red balls. Bag B contains 3 white, 3 blue and 2 red balls. Bag C contains 2 white, 3 blue and 4 red balls. One bag is selected at random and then two balls are drawn from the selected bag. Find the probability that the balls drawn are white and red. (2017, [5 marks])
Probability (Bayes' Theorem): A fair die is rolled. If face 1 turns up, a ball is drawn from Bag A. If face 2 or 3 turns up, a ball is drawn from Bag B. If face 4 or 5 or 6 turns up, a ball is drawn from Bag C. Bag A contains 3 red and 2 white balls, Bag B contains 3 red and 4 white balls and Bag C contains 4 red and 5 white balls. The die is rolled, a Bag is picked up and a ball is drawn. If the drawn ball is red, what is the probability that it is drawn from Bag B? (2017, [5 marks])
Probability Distribution: An urn contains 25 balls of which 10 balls are red and the remaining green. A ball is drawn at random from the urn, the colour is noted and the ball is replaced. If 6 balls are drawn in this way, find the probability that: (i) All the balls are red. (ii) Not more than 2 balls are green. (iii) Number of red balls and green balls are equal. (2017, [5 marks])
LPP Formulation: A farmer has a supply of chemical fertilizer of type A which contains $10\%$ nitrogen and $6\%$ phosphoric acid and of type B which contains $5\%$ nitrogen and $10\%$ phosphoric acid. After soil test, it is found that at least $7\text{ kg}$ of nitrogen and same quantity of phosphoric acid is required for a good crop. The fertilizer of type A costs $₹5.00\text{ per kg}$ and the type B costs $₹8.00\text{ per kg}$. Using Linear programming, find how many kilograms of each type of the fertilizer should be bought to meet the requirement and for the cost to be minimum. Find the feasible region in the graph. (2017, [5 marks])
Probability: If $A$ and $B$ are events such that $P(A) = 1/2, P(B) = 1/3$ and $P(A \cap B) = 1/4$ then find $P(A/B)$ and $P(B/A)$. (2018, [2 marks])
Probability: In a race, the probabilities of $A$ and $B$ winning the race are $1/3$ and $1/6$ respectively. Find the probability of neither of them winning the race. (2018, [2 marks])
Probability: $A$ speaks truth in $60\%$ of the cases, while $B$ in $40\%$ of the cases. In what percent of cases are they likely to contradict each other in stating the same fact? (2018, [4 marks])
Probability Distribution: From a lot of 6 items containing 2 defective items, a sample of 4 items are drawn at random. Let the random variable $X$ denote the number of defective items in the sample. If the sample is drawn without replacement, find: (a) The probability distribution of X (b) Mean of X (c) Variance of X. (2018, [6 marks])
LPP Formulation: A manufacturing company makes two types of teaching aids $A$ and $B$ of Mathematics for Class X... The company makes a profit of ₹80 on each piece of type $A$ and ₹120 on each piece of type $B$. How many pieces of type $A$ and type $B$ should be manufactured per week to get a maximum profit? Formulate this as Linear Programming Problem and solve it. Identify the feasible region from the rough sketch. (2018, [6 marks])
Probability: Two balls are drawn from an urn containing 3 white, 5 red and 2 black balls, one by one without replacement. What is the probability that at least one ball is red? (2019, [2 marks])
Probability: If events $A$ and $B$ are independent, such that $P(A) = 3/5, P(B) = 2/3$, find $P(A \cup B)$. (2019, [2 marks])
Probability: Bag A contains 4 white balls and 3 black balls, while Bag B contains 3 white balls and 5 black balls. Two balls are drawn from Bag A and placed in Bag B. Then, what is the probability of drawing a white ball from Bag B? (2019, [4 marks])
Probability (Bayes' Theorem): Given three identical Boxes A, B and C... A person chooses a Box at random and takes out a coin. If the coin drawn is of silver, find the probability that it has been drawn from the Box which has the remaining two coins also of silver. (2019, [6 marks])
Probability Distribution: Determine the binomial distribution where mean is 9 and standard deviation is 3/2. Also, find the probability of obtaining at most one success. (2019, [6 marks])
LPP Formulation: A carpenter has 90, 80 and 50 running feet respectively of teak wood, plywood and rosewood... Formulate the above as a Linear Programming Problem and solve it, indicating clearly the feasible region in the graph. (2019, [6 marks])
Probability: A bag contains 5 white, 7 red and 4 black balls. If four balls are drawn one by one with replacement, what is the probability that none is white? (2020, [2 marks])
Probability: Let $A$ and $B$ be two events such that $P(A) = 1/2, P(B) = p$ and $P(A \cup B) = 3/5$, find '$p$' if $A$ and $B$ are independent events. (2020, [2 marks])
Probability: Three persons A, B and C shoot to hit a target. Their probabilities of hitting the target are 5/6, 4/5 and 3/4 respectively. Find the probability that: (i) Exactly two persons hit the target. (ii) At least one person hits the target. (2020, [4 marks])
Probability Distribution: The probability that a bulb produced in a factory will fuse after 150 days of use is 0.05. Find the probability that out of 5 such bulbs: (i) None will fuse... (ii) Not more than one will fuse... (iii) More than one will fuse... (iv) At least one will fuse after 150 days of use. (2020, [6 marks])
LPP Formulation: A company uses three machines to manufacture two types of shirts, half sleeves and full sleeves. None of the machines can be in operation for more than 40 hours per week... How many of each type of shirts should be made per week to maximize the company's profit? (2020, [6 marks])
Probability: A bag contains 19 tickets, numbered from 1 to 19. Two tickets are drawn randomly in succession with replacement. Find the probability that both the tickets drawn are even numbers. (2023, [1 mark])
Probability: The probability of event A occurring is $\frac{1}{3}$ and event B occurring is $\frac{1}{2}$. If A and B are independent, find the probability of neither A nor B occurring. (2023, [2 marks])
Probability Distribution: A box contains 30 fruits, out of which 10 are rotten. Two fruits are selected at random one by one without replacement. Find the probability distribution and mean of the number of unspoiled fruits. (2023, [6 marks])
LPP: Solve the Linear Programming Problem graphically: Maximise $z=5x+2y$ subject to $x-2y\le 2$, $3x+2y\le 12$, $-3x+2y\le 3$, $x\ge 0, y\ge 0$. (2023, [6 marks])
Probability: Evaluate $P(A\cup B)$ if $2P(A)=P(B)=\frac{5}{13}$ and $P(A|B)=\frac{2}{5}$. (2024, [2 marks])
LPP Formulation: Aman has ₹1500 to purchase rice and wheat. Each sack of rice and wheat costs ₹180 and ₹120 respectively. He can store a maximum of 10 bags and will earn a profit of ₹11 per bag of rice and ₹9 per bag of wheat. Formulate a Linear Programming Problem to maximise profit. (2024, [4 marks])
Probability: Three critics review a book. Odds in favour of the book are 5:2, 4:3 and 3:4 respectively. Find the probability that all critics are in favour of the book. (2025, [1 mark])
Probability: Pia, Sia and Dia displayed 15, 5 and 10 of their paintings respectively in an art exhibition. A person bought three paintings. Find the probability that he bought one painting from each of them. (2025, [2 marks])
LPP Formulation: Two different types of books have to be stacked. Type 1 weighs 1 kg with a thickness of 6 cm. Type 2 weighs 1.5 kg with a thickness of 4 cm. The 96 cm long shelf holds a maximum of 21 kg. Formulate an LPP to maximize the number of books. (2025, [4 marks])
Probability: In a class of 50 students, 25 study English, 10 study History and 10 study both. Find the probability that a randomly selected student studies either English or History. (2025 IE, [1 mark])
Probability Distribution: In a school committee of 30 teachers, 20 never commit errors. Two teachers are selected at random. Let $X$ be the number of selected teachers who never make an error. Find the probability distribution and its mean. (2025 IE, [4 marks])
LPP: A cooperative society has 50 hectares to grow crops X and Y (profits ₹10500 and ₹9000 per hectare). A liquid herbicide constraint limits usage to 20 litres and 10 litres per hectare respectively, with a maximum total of 800 litres. Formulate the LPP to maximize profit and solve graphically. (2025 IE, [4 marks])

A Final Word of Advice:

Don't just read these questions—solve them. Write out the steps completely as if you were sitting in the exam hall. If you get stuck on any of these, let me know in the comments below, and we can tackle the solutions together in an upcoming post.

Keep practicing, and stay focused!

Quick Revision Guide | CBSE | ICSE | X

2026-05-16T03:31:12.716-07:00

Master Your Class 10 Board Exams: The Ultimate Quick Revision Guide by Prime Maths

As a Class 10 student, the countdown to Board exams can feel incredibly overwhelming. Mathematics, in particular, demands a fine balance between understanding core concepts and executing formulas flawlessly under time constraints. When exam day approaches, scrolling through massive textbooks is the last thing you want to do.

This is exactly where the Prime Maths - Quick Revision - X portal comes to the rescue. Curated specifically for Grade 10 students looking for high-yield, streamlined study resources, this page acts as a digital hub for your last-minute preparation needs.

Let’s dive into what makes this platform a must-bookmark resource for every Class 10 math student.

What is the "Quick Revision - X" Portal?

The Quick Revision - X page is a centralized repository hosted by Prime Maths, designed to eliminate textbook clutter. Instead of spending hours compiling your own notes or searching the internet for reliable worksheets, this platform provides direct access to targeted study material.

At the heart of the page is the Embedded Files section. Here, students can find curated PDFs, formulas sheets, and chapter summaries that condense extensive chapters into bite-sized, high-retention revision guides.

Key Features Every Class 10 Student Needs

While the platform is incredibly clean and straightforward to navigate, its utility extends far beyond standard notes. By exploring the page and its surrounding ecosystem, students gain access to a full suite of mathematical tools:

Direct Access to Embedded Revision Files: Skip the search. The platform serves as a direct pipeline to downloadable material perfect for offline viewing, printing, or active recall sessions right before your test.
Seamless Transition to School Algebra: Need a quick refresher on quadratic equations, progressions, or linear inequations? The sidebar gives you instant access to the School Algebra sub-section, ensuring you can bridge your fundamental gaps without leaving the hub.
Interactive Homework Verification Tools: One of the standout features of the broader Prime Maths platform is its dedicated suite of mathematical calculators. When practicing statistics, you don't have to wonder if your calculations are right. You can jump directly to their specialized tools, including:
- Mean Calculator
- Median, Mode & Percentile Calculator

These interactive calculators allow you to check your work instantly, which is an invaluable asset when self-studying for boards.

How to Use This Page for Maximum Results

To squeeze the most value out of the Prime Maths Quick Revision hub, try integrating it into your daily study routine using these three steps:

Step 1: The Formula Blitz: Before attempting any sample paper, visit the Embedded Files section to quickly review the formulas for that day's topics (such as Trigonometry, Mensuration, or Coordinate Geometry).
Step 2: Active Problem Solving: Utilize their School Algebra and revision sheets to solve problems without looking at the solutions first.
Step 3: Verify and Refine: Use the integrated Mean and Median calculators to instantly verify complex statistical problems. Finding your error in seconds saves precious time that you can redirect toward weaker topics.

The Verdict

Success in Class 10 Mathematics isn't about studying harder; it’s about studying smarter. By organizing essential revision files and linking them alongside practical calculation tools, Prime Maths provides a frictionless environment for high-scoring students.

Don't wait until the night before the exam. Bookmark the Prime Maths Quick Revision - X page today, download your core revision sheets, and take control of your board preparation!

Happy Studying!

Mastering Algebra for Classes VIII–X

2026-05-14T18:39:19.933-07:00

Prime Maths

Direct Problem Bank Access

Mastering Algebra for Classes VIII to X:
The Ultimate Practice Guide for CBSE, ICSE & State Boards

Algebra is often the point in a student's mathematical journey where numbers give way to letters, and concrete arithmetic transitions into abstract logic. For students in Classes VIII to X, building a rock-solid foundation in algebra is not just about passing the next test—it is about developing the critical problem-solving skills required for higher secondary mathematics and future competitive exams.

Here at Prime Maths, we understand that mastering math requires more than just reading through theorems; it demands consistent, structured practice. That is why we have compiled a comprehensive, categorized algebraic problem bank designed specifically to bridge the gap between foundational classroom learning and advanced mathematical proficiency.

The Utility of a Structured Problem Bank

When tackling algebra, jumping straight into complex word problems or quadratic equations without mastering the basics can leave students frustrated. Our problem set is meticulously categorized to ensure a smooth, progressive learning curve:

Key Benefits of Extensive Algebra Practice

Tailored for Board Success

The curriculum requirements for classes VIII to X across CBSE, ICSE, and State Boards are rigorous. This problem set aligns perfectly with these syllabi, ensuring that whether a student is facing a standard board exam or a more conceptual competitive paper, they are fully prepared.

Bridges the Gap to Competitive Math

Standard textbook exercises often stop just as the problems get interesting. This curated list pushes boundaries, taking students from standard textbook applications to the nuanced proofs and rational equations often found in Olympiads or foundation courses.

Develops Algorithmic Thinking

By working through categorized problems, students naturally develop algorithmic thinking. They learn to break down a daunting complex fraction or a multi-step simultaneous equation into smaller, manageable, and logical steps.

Eliminates "Silly Mistakes"

Algebraic errors usually stem from a lack of focus on signs (like a dropped negative) or basic arithmetic slips. The repetitive, targeted practice offered in the earlier sections builds muscle memory, drastically reducing calculation errors in high-stakes exams.

Mathematics is not a spectator sport. The only way to truly understand algebra is to roll up your sleeves and solve problems.

Whether you are struggling to factorize a quadratic equation or looking to perfect your graphical solutions for simultaneous equations, this structured approach will help you build confidence step-by-step.

Access the Complete Prime Maths Algebra Problem Bank

Direct link: https://sites.google.com/view/primemaths/math-ix-x/school-algebra

📘 Grab a notebook, pick a category, and start solving. Consistent practice today will pave the way for a perfect score tomorrow!

Vinod Singh (Mathematics Educator, Prime Maths)

M.Sc. Pure Mathematics (Calcutta University, First Class) B.Sc. Mathematics (St. Xavier's College Kolkata, First Class) Contact: +91-9038126497

Passionate about bridging foundational gaps and creating rigorous problem banks that empower students to excel in board exams and competitive mathematics. The Algebra Mastery Series is designed to help students transition from foundational concepts to advanced problem-solving fluency.

Comprehensive Curriculum Aligned with NEP 2020 24/7 Access to Problem Bank

Verified Resource | Prime Maths

Perfect for self-study & revision • Designed for CBSE, ICSE, and major State Boards

Similarity | CBSE | ICSE | Problem Solving

2026-05-10T03:42:00.000-07:00

Geometry – Similarity & Area | ICSE/CBSE Practice

📐 Class X – Geometry

Similarity, Area & Proportions

A structured problem set for ICSE / CBSE board practice

📅 Updated 2026 📖 9 Problems ⏱️ ~60 min practice 🏷️ Triangles • Parallelograms • Trapezium

📚 Quick Recap – Geometry Essentials

1. Similarity of Triangles

Two triangles are similar if their corresponding angles are equal and corresponding sides are proportional. The main criteria are:

AA (Angle‑Angle): Two angles of one triangle equal to two angles of another.
SAS (Side‑Angle‑Side): One angle equal and the sides including it are proportional.
SSS (Side‑Side‑Side): All three sides are proportional.

2. Basic Proportionality Theorem (Thales)

If a line is drawn parallel to one side of a triangle intersecting the other two sides, it divides those sides in the same ratio.

3. Mid‑Point Theorem

The line segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.

4. Properties of Parallelograms

Opposite sides are equal and parallel.
Diagonals bisect each other.
Opposite angles are equal.

5. Area Ratios of Similar Triangles

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

            💡 Pro‑tip: Always look for parallel lines – they are the key to setting up similarity and proportional segments.
        

📝 Problem 1 / 9

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Rational and Irrational Numbers | CBSE | Grade 8 |

2026-05-09T23:27:00.000-07:00

📐 Grade 8 Mathematics

Rational Numbers

A complete interactive practice guide with step-by-step solutions

📅 Updated 2026 📖 17 Problems ⏱️ ~45 min read 🏷️ Number Systems

📚 Understanding Rational & Irrational Numbers

1. What is a Rational Number?

A rational number is any number that can be expressed in the form $ \frac{p}{q} $, where $ p $ and $ q $ are integers and $ q \neq 0 $. In other words, a rational number is a ratio of two integers.

Examples: $ \frac{1}{2}, -\frac{3}{4}, 5 \;(=\frac{5}{1}), 0 \;(=\frac{0}{1}), 0.75 \;(=\frac{3}{4}) $
The set of all rational numbers is denoted by $ \mathbb{Q} $.

2. What is an Irrational Number?

An irrational number is a number that cannot be expressed as a ratio of two integers. Their decimal expansions are non-terminating and non-repeating.

Examples: $ \pi, e, \sqrt{2}, \sqrt{3}, \sqrt{5} $
The set of irrational numbers is not closed under addition or multiplication (e.g., $ \sqrt{2} + (-\sqrt{2}) = 0 $, which is rational).

            🔑 Key Insight: Every rational number has a decimal expansion that either
            terminates (like \( \frac{1}{4} = 0.25 \)) or repeats
            (like \( \frac{1}{3} = 0.\overline{3} \)). Irrational numbers never terminate and never repeat.
        

3. Closure Property

A set is said to be closed under an operation if applying that operation to any two elements of the set always produces an element that also belongs to the same set.

Formally: A set $ S $ is closed under operation $ * $ if for all $ a, b \in S $, we have $ a * b \in S $.

➕ Addition

$ \mathbb{N}, \mathbb{W}, \mathbb{Z}, \mathbb{Q}, \mathbb{R} $ are all closed under addition.

✖️ Multiplication

$ \mathbb{N}, \mathbb{W}, \mathbb{Z}, \mathbb{Q}, \mathbb{R} $ are all closed under multiplication.

➖ Subtraction

$ \mathbb{Z}, \mathbb{Q}, \mathbb{R} $ are closed. $ \mathbb{N} $ and $ \mathbb{W} $ are not closed.

➗ Division

Only $ \mathbb{Q} $ and $ \mathbb{R} $ are closed (excluding division by zero). $ \mathbb{N}, \mathbb{W}, \mathbb{Z} $ are not closed.

4. Identity Elements

Additive Identity: $ 0 $ — because $ a + 0 = 0 + a = a $ for any $ a $.
Multiplicative Identity: $ 1 $ — because $ a \times 1 = 1 \times a = a $ for any $ a $.

5. Inverse Elements

Additive Inverse: For any $ a $, the number $ -a $ such that $ a + (-a) = 0 $.
Multiplicative Inverse: For any $ a \neq 0 $, the number $ \frac{1}{a} $ such that $ a \times \frac{1}{a} = 1 $. Zero has no multiplicative inverse.

6. Commutative & Associative Properties

Commutative: $ a * b = b * a $ (order doesn't matter).
Associative: $ (a * b) * c = a * (b * c) $ (grouping doesn't matter).
Addition and multiplication are both commutative and associative on $ \mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R} $.
Subtraction and division are neither commutative nor associative.

📝 Problem 1 / 17

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ICSE Class 10 Results 2027: Expected Dates, Past Trends, and How to Check

2026-04-27T17:30:00.000-07:00

The wait is almost over for lakhs of ICSE Class 10 students! As we step into the final days of April, anticipation is running high. The Council for the Indian School Certificate Examinations (CISCE) is expected to announce the much-awaited ICSE results anytime between April 28 and May 7, 2027.

It is completely normal to feel a mix of excitement and anxiety right now. However, staying informed and prepared is the best way to handle the pre-result jitters. Here is a breakdown of everything you need to know about the upcoming results, including historical trends and exactly where to find your scorecards.

📅 A Look at Previous Years' Trends

While CISCE has not yet officially confirmed the exact date and time for 2026, looking at the patterns from recent years gives us a highly reliable estimate. The current expected window of late April to early May aligns perfectly with the council's recent track record:

2026: Results were announced on April 30.
2025: Results were announced on April 30.
2024: Results were announced on May 6.
2023: Results were announced on May 14.

Based on this timeline, an official declaration in the coming days is right on schedule.

💻 Where and How to Check Your Results

When the results go live, the official websites often experience a massive surge in traffic. It is best to have your admit card ready beforehand and know exactly where to navigate.

Official CISCE Portals:

cisce.org results.cisce.org

Step-by-Step Guide to Accessing Your Score:

Visit either of the official portals linked above.
Select "ICSE" from the course drop-down menu.
Enter your Unique ID and Index Number exactly as they appear on your board admit card.
Type in the captcha code displayed on the screen for verification.
Click "Submit" to view your scorecard.
Download or print a digital copy of your result immediately for your personal records until your school distributes the physical certificates.

🔍 What to Do After the Results

Keep in mind that these results, while an important academic milestone, are just one stepping stone in your journey. If your scores are not exactly what you hoped for, you will have official avenues to address them:

Rechecking: CISCE typically opens a short window for recheck applications immediately after the results are declared.
Improvement Exams: If you want another chance to improve your academic standing, CISCE generally schedules improvement exams for the month of July.

Bridging the Gap: Foundation Mathematics for Competitive Excellence | Classes IX & X CBSE ICSE

2026-04-17T23:50:00.000-07:00

Bridging the Gap: Foundation Mathematics for Competitive Excellence

Overview

This problem set is meticulously curated for students in Classes IX and X (CBSE/ICSE) who are ready to step beyond the standard textbook curriculum. While the regular syllabus equips you with mathematical tools, these questions train you in the art of mathematical thinking. Designed in alignment with the competency-based approach of the new NCF, this module shifts the focus from procedural calculation to logical deduction, pattern recognition, and critical analysis.

Whether your goal is to tackle the Mathematics Olympiads, prepare for the rigorous foundations of IIT-JEE, or simply elevate your problem-solving skills, these challenges will test your conceptual depth.

Thematic Breakdown & Core Concepts

1. The Power of Patterns (Cyclicity & Last Digits)

Target Questions: 3, 4, 6, 7, 10

The Objective: In school, you learn to calculate exact values. In competitive mathematics, you are often asked to find the behavior of numbers that are too massive to compute (like $3^{80}$ or $2009^{2009}$).

Skills Developed: These questions introduce the foundational concepts of modular arithmetic and the cyclicity of unit digits. Students learn to observe repetitive patterns, extrapolate rules, and apply them to complex exponents and factorials.

2. Advanced Algebraic Reasoning & Exponents

Target Questions: 1, 2, 5

The Objective: Moving beyond basic index laws, these problems require multi-step logical framing. For instance, proving that a number is never divisible by 3 demands a solid grasp of mathematical proofs and parity.

Skills Developed: Students will enhance their ability to manipulate nested exponents (power towers) and use algebraic identities to prove divisibility rules, a staple skill for ISI and CMI entrance exams.

3. Number Theory & Combinatorial Thinking

Target Questions: 8, 9, 11

The Objective: Questions involving perfect cubes, sums of squares, and counting divisors require an intimate understanding of prime factorization.

Skills Developed: This section sharpens combinatorial logic. Instead of manually counting, students learn to use prime factorization as a blueprint to determine the number of divisors (Question 11) or to analyze the boundaries of perfect cubes within a massive range (Question 8).

Why This Matters for Your Development

NCF Alignment: Emphasizes analytical thinking over rote memorization. You aren't just applying formulas; you are building them.
Stamina & Resilience: Problems like evaluating $9! + 3^{9966}$ teach you not to be intimidated by the scale of a problem, breaking it down into manageable, logical pieces.
Competitive Edge: The transition from Class X to Class XI mathematics is notoriously steep. Mastering these number theory and algebra concepts now builds a robust foundation, making advanced calculus and discrete mathematics much more intuitive later on.

Answers: $\quad $ 1. $a^2$ $\quad $ 2. 1$\quad $ 3. 1$\quad $ 4. 9$\quad $ 5. (Proof required)$\quad $ 6. 8$\quad $ 7. 3$\quad $ 8. 10,000$\quad $ 9. (B) 1997$\quad $ 10. 9$\quad $ 11. 16$\quad $

1. If $a^3 = 1$ and $x = a^{2009^{2009^{2009}}}$, find the simplest value of $x$.

2. Find the remainder when $2009^{2009^{2009}}$ is divided by 2.

3. Find the remainder when $3^{80}$ is divided by 10.

4. Find the last digit of $9! + 3^{9966}$.

5. Show that $16^n$ is never divisible by 3 for any natural number $n$.

6. Find the last digit of $4^{2n} + 2$, where $n = 2026$.

7. Find the digit in the units place of the integer $1! + 2! + 3! + \dots + 99!$ (where $n! = 1 \times 2 \times 3 \times \dots \times n$).

8. Find the number of perfect cubes from 1 to $10^{12}$.

9. Which of the following numbers can be expressed as the sum of the squares of two integers?
* (A) 1995
* (B) 1997
* (C) 2003

10. Find the last digit of $(2137)^{754}$.

11. Find the number of divisors of 1000.

✍️ Vinod Singh 📞 9038126497

🧠 Mathematics educator | Prime Maths

Can You Solve These 20 Tricky Math Problems for Classes IX and X CBSE ICSE

2026-04-10T06:43:00.000-07:00

📚 20 Mathematical Challenges – CBSE, ICSE

Welcome! This slider shows two problems side by side on each page. Use Previous / Next buttons (or left/right arrow keys) to navigate through the 10 slides. Each problem has its own “Show hint” button – click to reveal a subtle nudge.

✨ Covers rational proofs, surd simplifications, Diophantine equations, and more. Suitable for grades 9–12 and competitive exams. Featuring 20 hand-picked problems covering everything from rational, irrational numbers to algebra and number theory. Whether you are prepping for a competitive exam or just love a good brain-teaser, this is for you. Suitable for students of IX and X of CBSE and ICSE. If these problems feel a bit intimidating, don't worry—they are designed to be! While standard high school mathematics focuses heavily on rote memorization and applying standard formulas to straightforward questions, this problem set bridges the gap between the regular classroom and competitive mathematics (like the AMC, math Olympiads, or advanced entrance exams). Instead of just asking you to "solve for x," these questions require creative algebraic manipulation, pattern recognition, and proof-based logical reasoning. They test whether you can combine multiple mathematical concepts to find an elegant solution, rather than just grinding through a standard algorithm. Key Mathematical Areas Covered This set of 20 problems targets several advanced domains: Advanced Algebra & Identities: Moving beyond basic factoring to use conditional identities (like the sum of cubes) and symmetric functions. Number Theory: Exploring the properties of prime numbers, divisibility rules, and solving basic Diophantine equations (finding integer solutions to polynomial equations). Complex Radicals & Surds: Tackling nested square roots, cube roots of binomial surds, and rationalizing multi-term denominators. Telescoping Series: Recognizing patterns in sequences that cancel each other out to reveal a clean, simple answer. Introductory Complex Numbers & Logarithms: Understanding the cyclical nature of imaginary numbers and manipulating logarithmic bases.

Slide 1 / 10

Integral Calculus for JEE Main & Advanced: Practice Problems

2026-04-02T04:01:00.000-07:00

Relations and Functions Advanced Problems for JEE Mains IIT Advanced Indian Statistical Institute and WBJEE

2026-02-22T07:10:00.000-08:00

Circles for ICSE and CBSE | 25 Competency Problems

2026-02-17T01:58:00.000-08:00

Class 11 Math Advanced Problem Set | Full Practice Set with Answer Key

2026-02-14T22:08:00.000-08:00

📐 Prime Maths · Practice Problem Sheet XI

Class XI · Vinod Singh — CBSE, ISC, HS, JEE Mains & WBJEE.
One question at a time, with persistent answers. This practice set covers all essential chapters to help you test your preparation level and speed. Includes questions on Sets, Relations and Functions , Trigonometric Functions , Complex Numbers , Linear Inequalities , Permutations & Combinations , Coordinate Geometry, and Limits & Derivatives. This Class 11 Maths exercise is perfect for students of CBSE, ISC, WBCHSE and other State Boards looking for high-quality practice material to score better in their upcoming board and entrance exams.

📚 CBSE📘 ISC🏫 HS⚡ JEE Mains🧠 WBJEE

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Madhyamik Mathematics Board Paper 2026 | Full Solution | English Version

2026-02-14T02:43:07.364-08:00

🔢 Madhyamik 2026 Board Question Paper - Mathematics

Prime Maths · Vinod Singh 9038126497

📌 1. Choose the correct option in each case from the following questions: [ 1 x 6 = 6 ]

Q1(i). If a principal becomes double in 10 years, the rate of simple interest per annum is:

📘 view stepwise solution

Step 1 – define variables: Let principal $P$. Amount after 10 years $A = 2P$.
Step 2 – interest earned: $SI = A - P = 2P - P = P$.
Step 3 – apply simple interest formula: $SI = \dfrac{P \times R \times T}{100}$ with $T = 10$.
\[ P = \frac{P \times R \times 10}{100} \]
Step 4 – cancel $P$ (since $P \neq 0$): $1 = \dfrac{R \times 10}{100}$
Step 5 – solve for $R$: $1 = \dfrac{R}{10} \implies R = 10\%$.
✅ correct option: (b) 10%

Q1(ii). Condition of two roots of $ax^2+bx+c=0\;(a>0)$ equal in magnitude but opposite in sign:

$b=0,c=0$
$b=0,c>0$
$b=0,c<0$
$b>0,c=0$

📘 view stepwise solution

Step 1 – let the roots be $\alpha$ and $-\alpha$.
Step 2 – sum of roots: $\alpha + (-\alpha) = 0 = -\dfrac{b}{a} \implies b = 0$ (since $a\neq0$).
Step 3 – product of roots: $\alpha \cdot (-\alpha) = -\alpha^2 = \dfrac{c}{a}$.
Step 4 – sign analysis: $\alpha^2 > 0$ (real roots), so $-\alpha^2 < 0$. Hence $\dfrac{c}{a} < 0$.
Step 5 – given $a>0$, therefore $c < 0$.
✅ correct option: (c) $b=0,\;c<0$

Q1(iii). If average of $6,7,x,y,16$ is $9$. Then:

$x+y=21$
$x+y=16$
$x-y=21$
$x-y=19$

📘 view stepwise solution

Step 1 – average formula: $\dfrac{6+7+x+y+16}{5} = 9$.
Step 2 – sum of known numbers: $6+7+16 = 29$. So $\dfrac{29+x+y}{5} = 9$.
Step 3 – multiply by 5: $29+x+y = 45$.
Step 4 – solve: $x+y = 45-29 = 16$.
✅ correct option: (b) $x+y=16$

Q1(iv). Arc of length $121$ cm of a circle makes $77^\circ$ angle at the centre of the circle, then the radius of the circle will be. Radius = ?

110 cm
100 cm
90 cm
70 cm

📘 view stepwise solution

Step 1 – arc length formula: $L = 2\pi r \times \dfrac{\theta}{360^\circ}$. Use $\pi = \dfrac{22}{7}$.
Step 2 – substitute: $121 = 2 \cdot \dfrac{22}{7} \cdot r \cdot \dfrac{77}{360}$.
Step 3 – simplify inside: $2\cdot\dfrac{22}{7} = \dfrac{44}{7}$. Then $\dfrac{44}{7} \times \dfrac{77}{360} = \dfrac{44 \times 77}{7 \times 360}$.
Step 4 – cancel 77 with 7: $\dfrac{77}{7} = 11$, so expression becomes $\dfrac{44 \times 11}{360} = \dfrac{484}{360}$.
Step 5 – equation: $121 = r \cdot \dfrac{484}{360}$.
Step 6 – solve for $r$: $r = \dfrac{121 \times 360}{484}$. Note $484 = 4 \times 121$. Thus $r = \dfrac{360}{4} = 90$ cm.
✅ correct option: (c) 90 cm

Q1(v). Length of a side of a cube be $'a'$ unit and length of the diagonal be $'d'$ unit then the relation of a and d will be:

$\sqrt{2}a = d$
$\sqrt{3}a = d$
$a = \sqrt{3}d$
$a = \sqrt{2}d$

📘 view stepwise solution

Step 1 – space diagonal of a cube: $d = \sqrt{a^2 + a^2 + a^2}$.
Step 2 – simplify: $d = \sqrt{3a^2} = a\sqrt{3}$.
Step 3 – rewrite: $\sqrt{3}\,a = d$.
✅ correct option: (b) $\sqrt{3}a = d$

Q1(vi). $ABCD$ be a cyclic quadrilateral whose centre is $O$. $BC$ is extended upto $E$. If $\angle DCE = 96°$ then the value of $\angle BOD$ will be:

42°
84°
142°
168°

📘 view stepwise solution

Step 1 – linear pair: $\angle BCD + \angle DCE = 180^\circ$ (straight line BCE). So $\angle BCD = 180^\circ - 96^\circ = 84^\circ$.
Step 2 – central angle theorem: Angle at centre ($\angle BOD$) is twice the angle at circumference subtended by the same arc BAD. Here $\angle BCD$ subtends arc BAD.
Step 3 – calculate: $\angle BOD = 2 \times \angle BCD = 2 \times 84^\circ = 168^\circ$.
✅ correct option: (d) 168°

📝 2. Fill in the blanks (any five) [ 1 x 5 = 5]

(i) If the ratio of principal and yearly amount be 8:9, then yearly rate of interest is _________.

🔍 detailed solution

Let $P = 8k$, $A = 9k$. Interest for 1 year $= A-P = k$. Simple interest formula: $k = \dfrac{8k \times R \times 1}{100}$. Cancel $k$: $1 = \dfrac{8R}{100} \Rightarrow R = \dfrac{100}{8} = 12.5\%$.
$\Rightarrow$ $12\frac{1}{2}\%$.

(ii) Conjugate surd of $\sqrt{3} - 5$ is ______.

🔍 detailed solution

Write as $-5 + \sqrt{3}$. Conjugate: change sign of the irrational part $\rightarrow -5 - \sqrt{3}$.

(iii) Two tangents at the end point of a diameter of a circle are mutually _________.

🔍 detailed solution

Parallel (radius ⟂ tangent, both radii are collinear, so tangents are perpendicular to the same line → parallel).

(iv) If $x = a\sec\theta,\; y = b\cot\theta$, then $\dfrac{x^2}{a^2} - \dfrac{b^2}{y^2} =$ ______.

🔍 detailed solution

$\dfrac{x}{a} = \sec\theta,\; \dfrac{y}{b} = \cot\theta \Rightarrow \dfrac{b}{y} = \tan\theta$. Then $\dfrac{x^2}{a^2} - \dfrac{b^2}{y^2} = \sec^2\theta - \tan^2\theta = 1$.

(v) The radius of a solid hemisphere is $3r$ unit, the area of total surface is _________.

🔍 detailed solution

TSA of hemisphere (solid) = $3\pi R^2$. Here $R = 3r$ → $3\pi (3r)^2 = 3\pi \cdot 9r^2 = 27\pi r^2$ sq units.

(vi) The frequency of $1, 2, 3, 4, 5$ are respectively $1, 2, 3, 4, f$ and their arithmetic mean is $4$ then value of $f$ is _________.

🔍 detailed solution

$\sum fx = 1\cdot1 + 2\cdot2 + 3\cdot3 + 4\cdot4 + 5f = 1+4+9+16+5f = 30+5f$.
$\sum f = 1+2+3+4+f = 10+f$. Mean = $\frac{30+5f}{10+f} = 4$.
Cross multiply: $30+5f = 40+4f \Rightarrow f = 10$.

🔎 3. True or False (any five) [ 1 x 5 = 5 ]

(i) $\sin^2\theta = (\sin\theta)^2$ for $0° \theta<90°$.

explanation

True – conventional notation: $\sin^2\theta$ means $(\sin\theta)^2$.

(ii) The length of a side of a largest cube be $4\sqrt{2}$ cm inscribed in a sphere of radius 4 cm.

explanation

False. Sphere diameter = 8 cm = space diagonal of cube $\Rightarrow \sqrt{3}\,a = 8 \Rightarrow a = 8/\sqrt{3} \approx 4.618$ cm. $4\sqrt{2}\approx5.657$ cm (that would be face diagonal).

(iii) The angle in the segment of a circle which is greater than semicircle is an obtuse angle.

explanation

False. Major segment ( > semicircle) contains an acute angle; obtuse angle lies in minor segment.

(iv) If the Arithmetic mean of $x-3, x-1, 7, x, 2x-1, 3x-5$ be $7.5$, then their median will be $3$.

explanation

Sum = $(x-3)+(x-1)+7+x+(2x-1)+(3x-5) = 8x -3$. Mean = $\frac{8x-3}{6}=7.5 \Rightarrow 8x-3=45 \Rightarrow x=6$.
Data: 3,5,7,6,11,13 → sorted 3,5,6,7,11,13 → median = $\frac{6+7}{2}=6.5 \neq 3$. False.

(v) If $x \propto 1/y$ then $(xy)^{10}$ is a constant.

explanation

True. $xy = k$ (constant) ⇒ $(xy)^{10}=k^{10}$ constant.

(vi) In a business, the ratio of capital of Raju and Asif is $5:4$. If Raju got Rs. 80 of total profit then Asif got Rs. 100.

explanation

False. Profit ratio = capital ratio = 5:4. So Asif = $\frac{4}{5}\times 80 = 64$.

✍️ 4. Answer any ten (2 marks each) [ 2 x 10 = 20]

(i) A and B started a business by investing Rs. 15,000 and Rs. 45,000 respectively. After 6 months B got a profit of Rs. 3,030, what is the profit of A?

stepwise solution

Capital ratio A:B = 15000:45000 = 1:3. Profits shared in same ratio (no time adjustment as period same).
Let A's profit = $x$, then B's profit = $3x$. Given $3x = 3030 \Rightarrow x = 1010$. So A gets ₹1010.

(ii) In $\triangle ABC$, line parallel to BC intersects AB, AC at P, Q. AP=4 cm, QC=9 cm, PB=AQ. Find PB.

stepwise solution

By Basic Proportionality Theorem: $\dfrac{AP}{PB} = \dfrac{AQ}{QC}$. Set $PB = AQ = x$. Then $\dfrac{4}{x} = \dfrac{x}{9} \Rightarrow x^2 = 36 \Rightarrow x = 6$ (positive). PB = 6 cm.

(iii) Two Chords AB and CD are equidistant from centre O. If ∠AOB = 60° and CD = 6 cm, find radius.

stepwise solution

Equidistant chords ⇒ equal lengths: AB = CD = 6 cm. In ΔAOB, OA=OB (radii), ∠AOB=60° ⇒ triangle is equilateral (isosceles with vertex 60° ⇒ base angles each 60°). Hence radius = OA = AB = 6 cm.

(iv) If $\tanθ + \cotθ = 2$ then find $\tan^7θ + \cot^7θ$.

stepwise solution

$\tanθ + \frac{1}{\tanθ}=2$. Multiply by $\tanθ$: $\tan^2θ + 1 = 2\tanθ \Rightarrow \tan^2θ -2\tanθ +1=0 \Rightarrow (\tanθ-1)^2=0 \Rightarrow \tanθ=1$. So $\cotθ=1$. Then $\tan^7θ+\cot^7θ = 1+1 = 2$.

(v) If $x$ and $y$ are positive real numbers then $\sec \theta = \frac{x}{y}$ is correct or not? Give reason.

stepwise solution

No. For real θ between $0^{\circ}$ and $90^{\circ}$, $0 \leq \cos \theta \leq 1$. As $\sec \theta = \frac{x}{y}$ which means $\cos \theta = \frac{y}{x}$ and for $y > x $, $ \cos \theta > 1 $ which is not possible.

(vi) For two right circular cylinders, if ratio of their heights be $1:2$ and ratio of the circumference of the base be $3:4$ then find the ratio of their volume.

stepwise solution

Let $h_1:h_2 = 1:2$. Circumference $2πr_1 : 2πr_2 = 3:4 \Rightarrow r_1:r_2 = 3:4$. Volume = $πr^2h$. Ratio = $\left(\frac{r_1}{r_2}\right)^2 \times \frac{h_1}{h_2} = \left(\frac{3}{4}\right)^2 \times \frac{1}{2} = \frac{9}{16}\times\frac12 = \frac{9}{32}$.

(vii) Arithmetic mean of $x_1,x_2,......,x_n$ is $\bar x$. Prove that $\sum (x_i-\bar{x})^2 = \sum x_i^2 - n\bar{x}^2 $.

stepwise proof

$\sum (x_i-\bar{x})^2 = \sum (x_i^2 - 2x_i\bar{x} + \bar{x}^2) = \sum x_i^2 - 2\bar{x}\sum x_i + \sum \bar{x}^2$.
Now $\sum x_i = n\bar{x}$ and $\sum \bar{x}^2 = n\bar{x}^2$. Hence expression = $\sum x_i^2 - 2\bar{x}(n\bar{x}) + n\bar{x}^2 = \sum x_i^2 - 2n\bar{x}^2 + n\bar{x}^2 = \sum x_i^2 - n\bar{x}^2$.

(viii)If the rate of interest increases from 5.5% to 6% then the yearly interest increases by ₹49.50. Find the capital.

stepwise solution

Increase in rate = $0.5\%$. This increase applies to the whole capital: $0.5\% \times \text{capital} = 49.50$. So $\frac{0.5}{100} \times C = 49.50 \Rightarrow C = 49.50 \times \frac{100}{0.5} = 49.50 \times 200 = 9900$.

(ix)If the sum of the roots of the equation $x^2-4x = K(x-1)-5$ is 7. Find the value of $K$.

stepwise solution

Expand RHS: $Kx - K -5$. Bring all terms: $x^2 -4x - Kx + K +5 =0 \Rightarrow x^2 -(4+K)x + (K+5)=0$. Sum of roots = $4+K$. Given $4+K = 7 \Rightarrow K=3$.

(x) If $(a+b):\sqrt{ab}=2:1$ then find $a:b$.

stepwise solution

$\frac{a+b}{\sqrt{ab}} = 2 \Rightarrow a+b = 2\sqrt{ab}$. Square both sides: $(a+b)^2 = 4ab \Rightarrow a^2+2ab+b^2 = 4ab \Rightarrow a^2-2ab+b^2=0 \Rightarrow (a-b)^2=0 \Rightarrow a=b$. Ratio $1:1$.

(xi) If the radius of the sphere is increased by 50% find the percentage increase of the volume.

stepwise solution

New radius = $1.5r$. Volume ∝ $r^3$. New volume = $(1.5)^3 = 3.375$ times original. Increase = $3.375-1 = 2.375$ → $237.5\%$.

(xii) $ABCD$ is a cyclic quadrilateral. If $AD=AB$, ∠DAC=60°, ∠BDC=50° then find the ∠ACD.

stepwise solution

1. Angles in same segment: $\angle BAC = \angle BDC = 50^\circ$ (subtended by arc BC).
2. So $\angle DAB = \angle DAC + \angle CAB = 60^\circ+50^\circ=110^\circ$.
3. In $\triangle ADB$, $AD=AB$ ⇒ base angles equal: $\angle ABD = \angle ADB = \frac{180^\circ-110^\circ}{2}=35^\circ$.
4. Again, $\angle ACD = \angle ABD$ (angles subtended by arc AD). Hence $\angle ACD = 35^\circ$.

📊 5. Arithmetic (answer any one)

(i) If the rate of compound interest be 4% in the 1st year and 5% in the 2nd year, then find the interest of Rs. 25,000 for two years.

stepwise solution

Amount = $25000 \times \left(1+\frac{4}{100}\right) \times \left(1+\frac{5}{100}\right) = 25000 \times 1.04 \times 1.05$.
$1.04\times1.05 = 1.092$. So $A = 25000\times1.092 = 27300$. Interest = $27300-25000 = 2300$.

(ii) Three friends invest Rs. 4,800, Rs. 6,600 and Rs. 9,600. respectively in a business. 1^st person received 1/8 th of the profit as salary for looking after the business and the remaining profit was distributed among them in ratio of their capitals. If after one year 1^st person received Rs. 780 find the amount recieved by other two.

stepwise solution

Capitals ratio = 48:66:96 = 8:11:16 (sum 35). Let total profit = $P$. Salary = $P/8$. Remaining = $7P/8$ divided in ratio 8:11:16.
1st's extra = $\frac{8}{35}\times\frac{7P}{8} = \frac{P}{5}$. So 1st total = $\frac{P}{8} + \frac{P}{5} = \frac{5P+8P}{40} = \frac{13P}{40}$. Given $\frac{13P}{40}=780 \Rightarrow P = 2400$.
Remaining after salary = $2400 - 2400/8 = 2400-300 = 2100$.
2nd gets $\frac{11}{35}\times2100 = 660$; 3rd gets $\frac{16}{35}\times2100 = 960$.

🧩 6. Algebra – quadratic eq. (any one)

(i) Solve: $b(c-a)x^2 + c(a-b)x + a(b-c)=0$.

stepwise solution

Check sum of coefficients: $b(c-a)+c(a-b)+a(b-c) = bc - ab + ac - bc + ab - ac = 0$. Hence one root is $1$.
Product of roots = $\frac{a(b-c)}{b(c-a)}$. Thus other root = $\frac{a(b-c)}{b(c-a)}$.
Alternatively: $b(c-a)x^2 + c(a-b)x + a(b-c)=0 $ $\implies b(c-a)x^2 -(b(c-a)+a(b-c))x + a(b-c)=0$ $\implies b(c-a)x^2 -b(c-a)x-a(b-c)x + a(b-c)=0$ $\implies b(c-a)x(x-1)-a(b-c)(x-1)=0$ Now complete!

(ii) Digit in ten's place of a two digit number is less by 3 than the digit in the unit place. Product of digits is less than the number by 15. Find the number.

stepwise solution

Let units = $y$, tens = $x = y-3$. Number = $10x+y = 10(y-3)+y = 11y-30$.
Product $xy = (y-3)y = y^2-3y$. Given $xy = (11y-30) - 15$ → $y^2-3y = 11y-45$.
Rearr: $y^2 -14y +45=0$ ⇒ $(y-5)(y-9)=0$ ⇒ $y=5$ or $9$. Then numbers: $25$ or $69$.

⚡ 7. Algebra – variation / surds (any one)

(i) If $(x^3+y^3)\propto (x^3-y^3)$, then prove that $(x^2+y^2)\propto xy$.

stepwise proof

$\frac{x^3+y^3}{x^3-y^3}=k$ (constant). Apply componendo-dividendo: $\frac{(x^3+y^3)+(x^3-y^3)}{(x^3+y^3)-(x^3-y^3)} = \frac{k+1}{k-1}$ ⇒ $\frac{2x^3}{2y^3} = \frac{k+1}{k-1}$ ⇒ $\left(\frac{x}{y}\right)^3 = \text{constant}$. Hence $\frac{x}{y}=c$ (constant). Then $\frac{x^2+y^2}{xy} = \frac{c^2+1}{c}$ = constant ⇒ $(x^2+y^2)\propto xy$.

(ii) If $x(2-\sqrt3)=y(2+\sqrt3)=1$ then find the value of $3x^2-5xy+3y^2$.

stepwise solution

$x = \frac{1}{2-\sqrt3}$. Rationalise: multiply numerator and denominator by $2+\sqrt3$ ⇒ $x = 2+\sqrt3$.
$y = \frac{1}{2+\sqrt3} = 2-\sqrt3$. Then $x+y=4,\; xy=1$.
$3x^2-5xy+3y^2 = 3(x^2+y^2)-5$. Now $x^2+y^2 = (x+y)^2 - 2xy = 16-2=14$. So expression = $3\times14-5 = 42-5=37$.

⚖️ 8. Ratio & proportion (any one)

(i) If $\frac{a+b-c}{a+b}=\frac{b+c-a}{b+c}=\frac{c+a-b}{c+a}$ and $a+b+c \neq 0$, then prove that $a=b=c$.

stepwise proof

Subtract each fraction from 1: $1-\frac{a+b-c}{a+b} = \frac{c}{a+b}$; similarly $\frac{a}{b+c}$ and $\frac{b}{c+a}$. So $\frac{c}{a+b}=\frac{a}{b+c}=\frac{b}{c+a}$. Add 1 to each: $\frac{a+b+c}{a+b}=\frac{a+b+c}{b+c}=\frac{a+b+c}{c+a}$. Assuming $a+b+c\neq0$, denominators equal ⇒ $a+b=b+c=c+a$ ⇒ $a=b=c$.

(ii) If $x=\frac{8ab}{a+b}$, then find the value of $\frac{x+4a}{x-4a}+\frac{x+4b}{x-4b}$.

stepwise solution

$\frac{x}{4a} = \frac{2b}{a+b}$. Apply componendo-dividendo: $\frac{x+4a}{x-4a} = \frac{2b+(a+b)}{2b-(a+b)} = \frac{a+3b}{b-a}$.
Similarly $\frac{x}{4b} = \frac{2a}{a+b}$ ⇒ $\frac{x+4b}{x-4b} = \frac{2a+(a+b)}{2a-(a+b)} = \frac{3a+b}{a-b} = -\frac{3a+b}{b-a}$.
Sum = $\frac{a+3b - (3a+b)}{b-a} = \frac{-2a+2b}{b-a} = 2$.

🧠 For more problems keep visiting Prime Maths

📐 9. Geometry theorems (any one)

(i) Prove that the angle subtended by an arc at the centre of a circle is double the angle subtended by the same arc at any point on the remaining part of the circle.

📘 view stepwise proof

Given: Circle with centre $O$. Arc $AB$ subtends $\angle AOB$ at the centre and $\angle ACB$ at a point $C$ on the remaining part.
To prove: $\angle AOB = 2\angle ACB$.
Construction: Join $CO$ and extend it to point $D$.
Proof:

In $\triangle AOC$, $OA = OC$ (radii) ⇒ $\angle OAC = \angle OCA$.
Exterior angle $\angle AOD = \angle OAC + \angle OCA = 2\angle OCA$.
In $\triangle BOC$, $OB = OC$ ⇒ $\angle OBC = \angle OCB$.
Exterior angle $\angle BOD = \angle OBC + \angle OCB = 2\angle OCB$.
Adding: $\angle AOD + \angle BOD = 2(\angle OCA + \angle OCB)$.
But $\angle AOD + \angle BOD = \angle AOB$ and $\angle OCA + \angle OCB = \angle ACB$.
Hence $\angle AOB = 2\angle ACB$. ∎

(ii) If two circles touch each other, prove that the point of contact lies on the straight line joining their centres.

📘 view stepwise proof

Given: Two circles with centres $A$ and $B$ touching at point $P$.
To prove: $A$, $P$, $B$ are collinear.
Construction: Draw a common tangent $ST$ at $P$.
Proof:

Radius $AP$ is perpendicular to tangent $ST$ ⇒ $\angle APT = 90^\circ$.
Radius $BP$ is perpendicular to tangent $ST$ ⇒ $\angle BPT = 90^\circ$.
Thus $\angle APT + \angle BPT = 180^\circ$.
Angles $\angle APT$ and $\angle BPT$ form a linear pair on line $ST$.
Therefore, $AP$ and $BP$ are along the same straight line ⇒ $A$, $P$, $B$ are collinear. ∎

🔧 10. Geometry applications (any one)

(i) In an isosceles $\triangle ABC$, $\angle B = 90^\circ$. The bisector of $\angle BAC$ meets $BC$ at $D$. Prove that $CD^2 = 2\,BD^2$.

📘 view stepwise solution

Given: $AB = BC$ (isosceles right‑angled at $B$). $AD$ bisects $\angle BAC$.
To prove: $CD^2 = 2 BD^2$.
Proof:

Let $AB = BC = a$. Then hypotenuse $AC = \sqrt{a^2 + a^2} = a\sqrt{2}$.
By angle‑bisector theorem in $\triangle ABC$: \[ \frac{BD}{CD} = \frac{AB}{AC} = \frac{a}{a\sqrt{2}} = \frac{1}{\sqrt{2}}. \]
Hence $CD = \sqrt{2}\, BD$.
Squaring both sides: $CD^2 = 2 BD^2$. ∎

(ii) $O$ is any point inside rectangle $ABCD$. Prove that $OA^2 + OC^2 = OB^2 + OD^2$.

📘 view stepwise solution

Construction: Through $O$ draw lines parallel to the sides, meeting $AB, BC, CD, DA$ at $P, Q, R, S$ respectively.
Proof:

By Pythagoras in right triangles: \[ \begin{aligned} OA^2 &= AP^2 + OP^2,\\ OC^2 &= CQ^2 + OQ^2,\\ OB^2 &= BP^2 + OP^2,\\ OD^2 &= DQ^2 + OQ^2. \end{aligned} \]
Because of the parallel lines, $AP = BQ$ and $CQ = PD$.
Adding the first two: $OA^2 + OC^2 = AP^2 + OP^2 + CQ^2 + OQ^2$.
Adding the last two: $OB^2 + OD^2 = BP^2 + OP^2 + DQ^2 + OQ^2$.
But $AP^2 = BQ^2$ and $CQ^2 = DQ^2$. Hence both sums are equal. ∎

🛠️ 11. Construction (any one)

(i) Construct the circumcircle of $\triangle ABC$ where $BC = 6\ \text{cm}$, $\angle ABC = 60^\circ$ and $AB = 8\ \text{cm}$. (Write the main steps.)

📘 view construction steps

Draw a line segment $BC = 6\ \text{cm}$.
At $B$, construct $\angle CBA = 60^\circ$ and cut $AB = 8\ \text{cm}$ from the ray.
Join $AC$ to complete $\triangle ABC$.
Draw the perpendicular bisectors of sides $AB$ and $BC$. Let them intersect at $O$.
With centre $O$ and radius $OA$ (or $OB$ or $OC$), draw the circle – this is the required circumcircle.

(ii) Construct a square equal in area to an equilateral triangle of side $6\ \text{cm}$. (Write the main steps.)

📘 view construction steps

Draw an equilateral triangle $ABC$ with side $6\ \text{cm}$.
Compute (or measure) its altitude: $h = \sqrt{6^2 - 3^2} = \sqrt{36-9} = \sqrt{27} = 3\sqrt{3}\ \text{cm}$.
Area of triangle = $\frac{1}{2}\times 6 \times 3\sqrt{3} = 9\sqrt{3}\ \text{cm}^2$.
Construct a rectangle with length $6\ \text{cm}$ and width $3\sqrt{3}\ \text{cm}$ (using a right triangle to obtain $\sqrt{3}$).
Now construct a square with area equal to that rectangle using the mean‑proportional method:
- Extend the rectangle’s length $L$ by its width $W$.
- Draw a semicircle on the total segment as diameter.
- Erect a perpendicular at the junction; its length to the circle gives the side of the square.

📐 12. Trigonometry (any two)

(i) If the ratio of three angles of a triangle is $2:3:4$ then determine the circular value of the greatest angle.

📘 view stepwise solution

Step 1: Let the angles be $2x,\ 3x,\ 4x$. Sum = $2x+3x+4x = 9x$.
Step 2: In a triangle, $9x = 180^\circ$ ⇒ $x = 20^\circ$.
Step 3: Greatest angle = $4x = 80^\circ$.
Step 4: Convert to radians: $80^\circ \times \frac{\pi}{180} = \frac{4\pi}{9}$.
✅ Greatest angle = $\frac{4\pi}{9}$ radians

(ii) If $\tan\theta = \frac{4}{3}$, find $\sin\theta + \cos\theta$.

📘 view stepwise solution

Step 1: Construct a right triangle with opposite = $4$, adjacent = $3$. Hypotenuse = $\sqrt{4^2+3^2} = \sqrt{16+9}=5$.
Step 2: $\sin\theta = \frac{\text{opp}}{\text{hyp}} = \frac{4}{5}$, $\cos\theta = \frac{\text{adj}}{\text{hyp}} = \frac{3}{5}$.
Step 3: $\sin\theta + \cos\theta = \frac{4}{5} + \frac{3}{5} = \frac{7}{5}$.
✅ $\sin\theta + \cos\theta = \frac{7}{5}$

(iii) If $A$ and $B$ are complementary angles, prove that $(\sin A + \cos B)^2 = 1 + 2\sin A \sin B$.

📘 view stepwise proof

Given: $A + B = 90^\circ$. Hence $\cos B = \sin A$ and $\sin B = \cos A$.
Proof: \[ \begin{aligned} \text{LHS} &= (\sin A + \cos B)^2 \\ &= (\sin A + \sin A)^2 \quad (\because \cos B = \sin A)\\ &= (2\sin A)^2 = 4\sin^2 A. \end{aligned} \] Wrong Problem!!. Usually the intended statement is: $(\sin A + \sin B)^2 = 1 + 2\sin A \sin B$ because $\sin B = \cos A$. Let's verify the standard form: \[ (\sin A + \sin B)^2 = \sin^2 A + \sin^2 B + 2\sin A\sin B. \] Since $\sin^2 A + \sin^2 B = \sin^2 A + \cos^2 A = 1$, we get $1 + 2\sin A\sin B$.
If the problem statement uses $\cos B$ instead of $\sin B$, then note $\cos B = \sin A$ and the expression becomes $(\sin A + \sin A)^2 = 4\sin^2 A$, which is not generally equal to $1+2\sin A\sin B$.
Assuming the intended identity uses $\sin B$ (common in complementary angle problems):
\[ (\sin A + \sin B)^2 = 1 + 2\sin A \sin B \quad\text{(proved above).} \] ✅ Full marks on attempt.

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📏 13. Height and Distance (any one)

(i) From the roof of a building, the angles of depression of the top and foot of a lamp post are $30^\circ$ and $\theta^\circ$ respectively. The ratio of the heights of the building and the lamp post is $3:2$, then find the value of $\theta$.

📘 view stepwise solution

Step 1 – assign variables: Let height of building $H = 3x$, height of lamp post $h = 2x$.
Step 2 – vertical difference: The top of the lamp is $H-h = 3x - 2x = x$ below the roof.
Step 3 – horizontal distance $d$: From roof, angle of depression $30^\circ$ to top of lamp ⇒ $\tan 30^\circ = \frac{x}{d}$.
\[ \frac{1}{\sqrt{3}} = \frac{x}{d} \quad\Rightarrow\quad d = x\sqrt{3}. \] Step 4 – angle $\theta$ to foot of lamp: From roof, foot is at depth $3x$ below, same horizontal distance $d$.
\[ \tan\theta = \frac{3x}{d} = \frac{3x}{x\sqrt{3}} = \sqrt{3}. \] Step 5 – solve for $\theta$: $\tan\theta = \sqrt{3} \Rightarrow \theta = 60^\circ$.
✅ $\theta = 60^\circ$

(ii) From the foot of a Tilla the angle of elevation of the top is $45^\circ$. By moving $100$ m towards the Tilla along a slope of $30^\circ$, the angle of elevation of the top becomes$60^\circ$. Find the height of the Tilla.

📘 view stepwise solution

Step 1 – Main triangle: The distance of the point from the foot of the Tilla is $h$, since $\tan 45^{\circ}=1=\frac{h}{distance}$
Step 2 – movement along slope: From the starting point walk $100$ m at $30^\circ$ upward slope (towards the Tilla). The displacement components: Horizontal distance $ = 100\cos30^\circ = 100 \times \frac{\sqrt3}{2} = 50\sqrt3$, Vertical distance $= 100\sin30^\circ = 50$. So the new position is at a horizontal distance of $h-50\sqrt{3}$ from the Tilla and its vertical distance from the top of the Tilla is $h-50$
Step 3 – elevation from $B$: Angle of elevation to top of Tilla is $60^\circ$. \[ \tan60^\circ = \frac{h-50}{h - 50\sqrt3}. \] Step 4 – solve for $h$: $\tan60^\circ = \sqrt3$: \[ \sqrt3 = \frac{h-50}{h - 50\sqrt3} \implies \sqrt3\,(h - 50\sqrt3) = h - 50 \implies \sqrt3\,h - 150 = h - 50. \] \[ (\sqrt3 - 1)h = 100 \quad\Rightarrow\quad h = \frac{100}{\sqrt3 - 1}. \] Step 5 – rationalise: Multiply numerator and denominator by $\sqrt3 + 1$: \[ h = \frac{100(\sqrt3+1)}{3-1} = 50(\sqrt3 + 1)\ \text{m}. \] ✅ Height $= 50(\sqrt3 + 1)$ m

🧊 14. Mensuration (any two)

(i) The ratio of the length, breadth and height of a solid rectangular parallelopiped is $4:3:2$ and area of the whole surface is $468\ \text{cm}^2$. Find the volume of the parallelopiped.

📘 view stepwise solution

Step 1 – let dimensions: Let length $l = 4x$, breadth $b = 3x$, height $h = 2x$.
Step 2 – total surface area (TSA): \[ TSA = 2(lb + bh + hl) = 2\bigl(4x\cdot3x + 3x\cdot2x + 2x\cdot4x\bigr) = 2(12x^2 + 6x^2 + 8x^2) = 2 \times 26x^2 = 52x^2. \] Step 3 – given TSA = 468: \[ 52x^2 = 468 \quad\Rightarrow\quad x^2 = 9 \quad\Rightarrow\quad x = 3. \] Step 4 – actual dimensions: $l = 12\ \text{cm},\ b = 9\ \text{cm},\ h = 6\ \text{cm}$.
Step 5 – volume: \[ V = l \times b \times h = 12 \times 9 \times 6 = 648\ \text{cm}^3. \] ✅ Volume = $648\ \text{cm}^3$

(ii) The internal and external radius of a hollow cylinder of height $20\ \text{cm}$ are $4\ \text{cm}$ and $5\ \text{cm}$ respectively. By meltingthis cylinder a solid cone of height equal to one‑third the height of the cylinder is formed. Find the diameter of the base of the cone.

📘 view stepwise solution

Step 1 – volume of hollow cylinder: \[ V_{\text{cyl}} = \pi h_{\text{cyl}} (R^2 - r^2) = \pi \times 20 \times (5^2 - 4^2) = 20\pi (25-16) = 20\pi \times 9 = 180\pi\ \text{cm}^3. \] Step 2 – height of cone: $h_{\text{cone}} = \frac{1}{3}\times 20 = \frac{20}{3}\ \text{cm}$.
Step 3 – volume of cone: $V_{\text{cone}} = \frac13 \pi r_{\text{cone}}^2 h_{\text{cone}}$. Equate volumes: \[ \frac13 \pi r_{\text{cone}}^2 \times \frac{20}{3} = 180\pi. \] Cancel $\pi$ and simplify: \[ \frac{20}{9} r_{\text{cone}}^2 = 180 \quad\Rightarrow\quad r_{\text{cone}}^2 = 180 \times \frac{9}{20} = 81. \] Step 4 – radius and diameter: $r_{\text{cone}} = 9\ \text{cm}$ (positive). Diameter = $2\times 9 = 18\ \text{cm}$.
✅ Base diameter = $18\ \text{cm}$

(iii) A hemispherical bowl of internal radius $9 \text{cm}$ is full of water. How many cylindrical bottles each of diameter $3\ \text{cm}$ and height $4\ \text{cm}$ are required to fill this water?

📘 view stepwise solution

Step 1 – volume of hemispherical bowl: \[ V_{\text{bowl}} = \frac23 \pi R^3 = \frac23 \pi (9)^3 = \frac23 \pi \times 729 = 486\pi\ \text{cm}^3. \] Step 2 – volume of one cylindrical bottle: radius $r = \frac{3}{2} = 1.5\ \text{cm}$, height $h = 4\ \text{cm}$. \[ V_{\text{bottle}} = \pi r^2 h = \pi (1.5)^2 \times 4 = \pi \times 2.25 \times 4 = 9\pi\ \text{cm}^3. \] Step 3 – number of bottles: \[ n = \frac{V_{\text{bowl}}}{V_{\text{bottle}}} = \frac{486\pi}{9\pi} = 54. \] ✅ 54 bottles are required

📊 15. Statistics (any two)

(i) Find the arithmetic mean for the following frequency distribution:

Class	5-14	15-24	25-34	35-44	45-54	55-64
Frequency	3	6	18	20	10	3

📘 view stepwise solution

Step 1 – find midpoints (class marks): \[ x_1 = \frac{5+14}{2} = 9.5,\; x_2 = 19.5,\; x_3 = 29.5,\; x_4 = 39.5,\; x_5 = 49.5,\; x_6 = 59.5. \] Step 2 – compute $f_i x_i$: \[ \begin{aligned} f_1x_1 &= 3 \times 9.5 = 28.5,\\ f_2x_2 &= 6 \times 19.5 = 117,\\ f_3x_3 &= 18 \times 29.5 = 531,\\ f_4x_4 &= 20 \times 39.5 = 790,\\ f_5x_5 &= 10 \times 49.5 = 495,\\ f_6x_6 &= 3 \times 59.5 = 178.5. \end{aligned} \] Step 3 – total frequency and sum of products: \[ \sum f = 3+6+18+20+10+3 = 60, \] \[ \sum fx = 28.5 + 117 + 531 + 790 + 495 + 178.5 = 2140. \] Step 4 – arithmetic mean: \[ \bar{x} = \frac{\sum fx}{\sum f} = \frac{2140}{60} \approx 35.666\ldots \approx 35.67. \] ✅ Mean $\approx 35.67$

(ii) For the following frequency distribution, construct a 'greater than' cumulative frequency table and draw the ogive (more than type).

Class	100-120	120-140	140-160	160-180	180-200
Frequency	8	14	10	12	4

📘 view stepwise solution

Step 1 – compute total frequency: \[ N = 8 + 14 + 10 + 12 + 4 = 48. \] Step 2 – obtain 'greater than' cumulative frequencies:

More than 100 : $48$
More than 120 : $48 - 8 = 40$
More than 140 : $40 - 14 = 26$
More than 160 : $26 - 10 = 16$
More than 180 : $16 - 12 = 4$
More than 200 : $4 - 4 = 0$ (optional)

Step 3 – points to plot for the ogive: \[ (100,\,48),\quad (120,\,40),\quad (140,\,26),\quad (160,\,16),\quad (180,\,4). \] Step 4 – drawing the ogive: On a graph sheet, take the upper limits (or the lower limits for more‑than) on the x‑axis and cumulative frequencies on the y‑axis. Plot the points and join them with a smooth freehand curve. The resulting graph is the 'more than' ogive.
✅ Table and points ready for plotting.

(iii) Find the mode of the following frequency distribution:

Marks obtained	less than 10	less than 20	less than 30	less than 40	less than 50	less than 60
Number of students	8	15	29	42	60	70

📘 view stepwise solution

Step 1 – convert to ordinary frequency distribution: The given cumulative frequencies are for "less than" limits. Subtract successive cumulatives to obtain class frequencies.

$0-10$ : $8$
$10-20$: $15 - 8 = 7$
$20-30$: $29 - 15 = 14$
$30-40$: $42 - 29 = 13$ (this is $f_0$)
$40-50$: $60 - 42 = 18$ (this is $f_1$ – modal class)
$50-60$: $70 - 60 = 10$ (this is $f_2$)

Step 2 – identify modal class: The highest frequency is $18$ in the class $40-50$. So modal class = $40-50$ with \[ l = 40,\quad f_1 = 18,\quad f_0 = 13,\quad f_2 = 10,\quad h = 10. \] Step 3 – apply mode formula: \[ \text{Mode} = l + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h = 40 + \frac{18 - 13}{2\times18 - 13 - 10} \times 10. \] Step 4 – simplify: \[ 2f_1 - f_0 - f_2 = 36 - 13 - 10 = 13,\quad f_1-f_0 = 5. \] \[ \text{Mode} = 40 + \frac{5}{13} \times 10 = 40 + \frac{50}{13} \approx 40 + 3.846 = 43.846. \] ✅ Mode $\approx 43.85$ (rounded to two decimals)

📈 Prime Maths

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Class X Mathematics Practice Test Paper

Test Examination - Practice Test Paper - Prime Maths

Class X - Mathematics

Examination Overview

This comprehensive mathematics examination covers key concepts from the Class X curriculum including algebra, geometry, trigonometry, mensuration, statistics, and commercial mathematics. The paper is designed to test conceptual understanding, problem-solving skills, and application of mathematical principles in real-world scenarios.

Duration: 3 Hours

Total Marks: 90

Class: X (Secondary)

Year: 2023

No Calculator

Use of calculators is not permitted in this examination.

Geometrical Tools

Geometrical instruments are required for construction problems.

Comprehensive Coverage

Covers all major topics from the Class X mathematics syllabus.

General Instructions

All questions are compulsory unless specified otherwise.
Marks for each question are indicated in brackets.
Use of calculators is not permitted.
Draw neat diagrams wherever required.
For fill in the blanks and true/false questions, answer any five.
Show all working steps for full marks.
Write answers clearly and legibly.

Section 1: Multiple Choice Questions

[1 × 6 = 6 Marks]

Choose the correct answer in each of the following:

i In a partnership business, the ratio of profits received by two friends is $\frac{1}{2} : \frac{1}{3}$. The ratio of their capitals is:

(a) 2 : 3

(b) 3 : 2

(d) 5 : 3

ii If $p+q=\sqrt{13}$ and $p-q=\sqrt{5}$, then the value of $pq$ is:

(a) 2

(b) 18

(d) 8

iii O is the center of a circle and AB is a chord. ABCD is a cyclic quadrilateral. If $\angle ABC = 65^\circ$ and $\angle DAC = 40^\circ$, then the value of $\angle BCD$ is:

(a) 75°

(b) 105°

(d) 80°

iv If $\tan \alpha + \cot \alpha = 2$, then the value of $\tan^{13} \alpha + \cot^{13} \alpha$ is:

(a) 13

(b) 2

(d) 0

v Two cubes each with side $2\sqrt{6}$ cm are placed side by side to form a cuboid. The length of the diagonal of the cuboid is:

(a) 10 cm

(b) 6 cm

(d) 12 cm

vi If the mean of $x_1, x_2, x_3, \dots, x_{10}$ is 20, then the mean of $x_1 + 4$, $x_2 + 4$, $x_3 + 4$, $\dots$, $x_{10} + 4$ is:

(a) 20

(b) 24

(d) 10

Section 2: Fill in the Blanks

[1 × 5 = 5 Marks]

Answer any five of the following:

i If the annual compound interest rate is $r\%$ and the principal in the first year is $P$ rupees, then the principal in the second year is ______.

ii $7\sqrt{11}$ is a ______ number.

iii If the radius of a sphere is $r$ and its volume is $v$, then $v \propto$ ______.

iv Two triangles are similar if their corresponding sides are ______.

v If the opposite angles of a quadrilateral are supplementary, then the vertices of the quadrilateral are ______.

vi If the length, breadth, and height of a cuboid are equal, then the name of that solid is ______.

Section 3: True or False

[1 × 5 = 5 Marks]

Answer any five of the following:

i In compound interest, if the interest rates for the first, second, and third years are $r_1\%$, $r_2\%$, and $r_3\%$ respectively, then the amount for $P$ rupees after 3 years is \[P\left(1 + \frac{r_1}{100}\right)\left(1 + \frac{r_2}{100}\right)\left(1 + \frac{r_3}{100}\right) \text{ rupees.}\]

ii The values of cos 36° and sin 54° are equal.

iii Only one tangent can be drawn to a circle from an external point.

iv The compound ratio of $2ab : c^2$, $bc : a^2$, and $ca : 2b^2$ is 1 : 1.

v If the numerical values of the surface area and volume of a sphere are equal, then the radius is 3 units.

vi The mode of the numbers 5, 2, 4, 3, 5, 2, 5, 2, 5, 2 is 2.

Section 4: Short Answer Questions

[2 × 10 = 20 Marks]

Answer any ten of the following:

vii The radii of two circles are 8 cm and 3 cm. The distance between their centers is 13 cm. What is the length of the direct common tangent of the circles?

viii What is the radian measure of the angle swept by the hour hand of a clock in 1 hour?

ix If $\tan 4\theta \tan 6\theta = 1$ and $6\theta$ is positive acute, determine the value of $\theta$.

x The height of a right circular cylinder is 12 cm and its volume is 100π cubic cm. Determine the radius of the base of the cylinder.

xi If the ratio of the surface areas of two spheres is 1 : 4, determine the ratio of their volumes.

xii If $u_1 = \frac{x_1 - 35}{10}$, $\sum f_1 u_1 = 30$, and $\sum f_1 = 60$, then determine the value of $\bar{x}$.

Section 5: Compound Interest or Partnership

[5 Marks]

Answer any one of the following:

i Determine the compound interest on ₹40,000 for 3 years at 5% per annum compound interest.

ii Rajib started a business with a capital of ₹3,750. After 6 months, Sayan joined the business with a capital of ₹15,000. At the end of the year, the profit was ₹6,900. Determine each one's share of the profit.

Section 6: Algebra

[3 Marks]

Answer any one of the following:

i Solve: $\frac{x}{x+1} + \frac{x+1}{x} = 2 \frac{1}{12}$, where $x \neq 0, -1$

ii A train travels 200 km at a uniform speed. If the speed of the train is increased by 5 km/h, it takes 2 hours less to cover the same distance. Determine the speed of the train.

Section 7: Algebra and Proportion

[3 Marks]

Answer any one of the following:

i If $x = \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}}$ and $y = \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}}$, then find the value of $x^2 - xy + y^2$.

ii If $x \propto y$ and $y \propto z$, prove that $(x^2 + y^2 + z^2) \propto (xy + yz + zx)$.

Section 8: Proportions

[3 Marks]

Answer any one of the following:

i If $a, b, c, d$ are in continued proportion, show that \[(b^2 + d^2) : (b^2 - d^2) = (ab + cd) : (ab - cd)\]

ii If $\frac{a+b-c}{a+b} = \frac{b+c-a}{b+c} = \frac{c+a-b}{c+a}$ and $a+b+c \neq 0$, then prove that $a = b = c$.

Section 9: Geometry Proofs

[5 Marks]

Answer any one of the following:

i Prove that all angles in the same segment of a circle are equal.

ii Prove that if a perpendicular is drawn from the right angle vertex of a right triangle to the hypotenuse, then the two triangles formed on either side of the perpendicular are similar to the original triangle and to each other.

Section 10: Geometry

[3 Marks]

Answer any one of the following:

i O is the center of a circle and QR is a chord. Tangents drawn at Q and R intersect at P. If QM is a diameter of the circle, prove that $\frac{QR}{QM} = 2$.

ii In triangle ABC, $\angle C = 90^\circ$. If CD is the median, prove that $BC^2 = CD^2 + 3AD^2$.

Section 11: Construction

[5 Marks]

Answer any one of the following:

i Construct a triangle ABC with BC = 7 cm, AB = 5 cm, and AC = 6 cm. Draw the circumcircle of the triangle. (Only construction traces are required)

ii Determine geometrically the mean proportional between 4 cm and 3 cm. (Only construction traces are required)

Section 12: Trigonometry

[3 × 2 = 6 Marks]

Answer any two of the following:

i In $\triangle ABC$, $\angle C = 90^\circ$. If $BC = m$ and $AC = n$, show that \[m \sin A + n \sin B = \sqrt{m^2 + n^2}\]

ii Find the value: \[ \frac{4}{3} \cot^2 30^\circ + 3 \sin^2 60^\circ - 2 \cos^2 60^\circ - \frac{3}{4} \tan^2 30^\circ \]

iii If $\angle P + \angle Q = 90^\circ$, then show that \[ \sqrt{\frac{\sin P}{\cos Q}} - \sin P \cos Q = \cos^2 P \]

Section 13: Height and Distance

[5 Marks]

Answer any one of the following:

i The width of a river is 600 m. Two boats start from the same bank of the river to go to the opposite bank. If the first boat makes an angle of 30° with the bank and the second boat makes an angle of 90° with the direction of the first boat to reach the opposite bank, what will be the distance between the two boats after they reach the opposite bank?

ii On the roof of a three-story building, there is a flagpole of length 3.6 m. From a point on the road, the angles of elevation of the top and bottom of the flagpole are 50° and 45° respectively. What is the height of the building? (Take tan 50° = 1.2)

Section 14: Mensuration

[4 × 2 = 8 Marks]

Answer any two of the following:

i The ratio of length, breadth, and height of a solid cuboid is 4:3:2 and its total surface area is 468 sq. cm. Find the volume of the cuboid.

ii The radius of the base of a solid cylindrical rod is 3.2 dm. The rod is melted to form 21 solid spheres. If the radius of each sphere is 8 cm, find the length of the rod.

iii The slant height of a solid right circular cone is 7 cm and its total surface area is 147.84 sq. cm. Find the radius of the base of the cone.

Section 15: Statistics

[4 × 2 = 8 Marks]

Answer any two of the following:

i Calculate the arithmetic mean from the following frequency distribution:

Class Limit	10-20	20-30	30-40	40-50	50-60	60-70
Frequency	10	16	20	30	13	11

ii Find the median of the data below:

Class Limit	1-5	6-10	11-15	16-20	21-25	26-30	31-35
No. of Workers	2	4	6	8	10	5	4

iii Determine the mode of the following frequency distribution:

Class Limit	0-5	5-10	10-15	15-20	20-25	25-30	30-35	35-40
Frequency	2	6	10	16	22	11	8	5

Madhyamik Practice Test Paper

2026-01-15T02:42:00.000-08:00

Madhyamik Test Examination Practice Paper - Mathematics

Prime Maths | Vinod Singh

Madhyamik Test Examination – Practice Paper | Subject: MATHEMATICS

Time: 3 Hours 15 Minutes

Full Marks: 90

Type: Free Practice Test Paper

General Instructions:

The figures in the margin indicate full marks.
Use of calculators is not permitted.
Graph papers will be supplied if required.

1. Multiple Choice Questions [1 × 6 = 6]

Select the correct answer in each of the following cases:

iIf the mean of a frequency distribution is 8.1, $\sum f_i x_i = 132 + 5k$, and $\sum f_i = 20$, then the value of $k$ is:

(a) 4

(b) 2

(d) 6

iiIf $\cos\theta + \sec\theta = 2$, then the value of $\cos^5\theta + \sin^5\theta$ is:

(a) -1

(b) 1

(d) -2

iiiThe volumes of two right circular cones are equal. If the radii of their circular bases are in the ratio 1:2, their heights are in the ratio:

(a) 2:1

(b) 3:1

(d) 8:1

ivIn the right-angled triangle ABC, $\angle A$ is a right angle, AB = 12 cm, AC = 5 cm, and BC = 13 cm. AD is perpendicular to BC. The length of AD is:

(a) $\frac{65}{12}$ cm

(b) $\frac{60}{13}$ cm

(d) $\frac{5}{15}$ cm

vX and Y are two points on sides PQ and PR of triangle PQR respectively. If PX = 2 cm, XQ = 3.5 cm, YR = 7 cm, and PY = 4.25 cm, then:

(a) XY || QR

(b) XY = QR

(d) None of these

viIf $x \propto y$, then:

(a) $x^{420} \propto y^{202}$

(b) $x^{2025} \propto y^{3025}$

(d) $x^{2025} \propto y^{2025}$

2. Fill in the blanks (any five) [1 × 5 = 5]

iIf the difference between the roots of the equation $x^2 - px + q = 0$ is 1, then $p^2 - 4q = $ __________.

iiIf $x = 2 - \sqrt{3}$ and $y = 4\sqrt{2}-2$, then $(x + y)^4 = $ __________.

iiiThe tangents drawn at the two ends of a diameter of a circle are __________ to each other.

ivThe ratio of simple interest on a fixed amount of money at the same annual rate for 4 years and 8 years is __________.

vIf the compound ratio of 2:3, 4:5, and 6:$x$ is 2:5, then the value of $x = $ __________.

viA tangent drawn from an external point M to a circle with center O touches the circle at point N. If ON = 9 cm and MO = 15 cm, then the length of MN = __________.

3. Write True or False (any five) [1 × 5 = 5]

iA starts a business with ₹10,000, and 6 months later B invests ₹20,000. At the end of the year, their profit shares will be equal.

iiThe ratio of the volumes of a right circular cone and a right circular cylinder with the same base and same height is 1:3.

iiiThe arithmetic mean and mode of the numbers 4, 3, 2, 5, 3, 4, 5, 1, 7, 3, 2, 1 are equal.

ivIn trapezium ABCD, AB || CD; diagonals AC and BD intersect at point O, then $AO \cdot OD = BO \cdot OC$.

vAB is a diameter of a circle with center O. P is any point on the circumference. If $\angle POA = 120^\circ$, then $\angle PBO + \angle POA = 140^\circ$.

viThe current price of a mobile phone is ₹20,000. If the price depreciates by 15% every year, the price of the mobile phone after 2 years will be ₹14,450.

4. Answer the following questions (any ten) [2 × 10 = 20]

iIf the simple interest on ₹8,000 for 2 years is ₹720, find the annual rate of interest.

iiIf the compound interest on a sum of money for 2 years is ₹512 and the simple interest for the same time and at the same rate is ₹500, what is the rate of interest?

iiiIn a partnership business, the ratio of the capitals of three persons is 3 : 5 : 8. If the profit of the first person is ₹60 less than the profit of the third person, what was the total profit in the business?

ivIf 2 is a root of the equations $x^2 + bx + 12 = 0$ and $x^2 + bx + q = 0$, find the value of $q$.

vIf $a+b=2\sqrt{5}$ and $a-b=\sqrt{2}$, what is the value of $a^2 + b^2$?

viThe length of each of two parallel chords AB and CD is 16 cm. If the radius of the circle is 10 cm, find the distance between the two chords.

viiIn a circle, two chords PQ and PR are perpendicular to each other. If the length of the radius of the circle is $r$ cm, find the length of the chord QR.

viiiIf $(4a+5b):(4c+5d) = (4a-5b):(4c-5d)$, prove that a, b, c, and d are in proportion.

ixIf the volume of a right circular cone is $V$, base area is $A$, and height is $H$, find the value of $\frac{AH}{V}$.

xIf $\theta = 30^\circ$, what is the value of $\sin 2\theta + \cos 2\theta$?

xiFind the median and arithmetic mean of the numbers: 2, 2, 3, 4, 6, 5, 5, 8, 9, 10, 11.

xiiThe length, breadth, and height of a rectangular cuboid room are 5 m, 4 m, and 3 m respectively. Calculate the length of the longest rod that can be kept in that room.

5. Answer any one question [5 × 1 = 5]

iA person borrowed ₹500 at 6% simple interest per annum, and exactly two years after the first loan, borrowed a second loan of ₹900 at 4% simple interest per annum. How many years after taking the first loan will the interest on these two loans be equal, and what will be the total interest on the two loans at that time?

iiA and B started a joint business with ₹6,200 and ₹10,000 respectively. They decided that A would receive 20% of the profit for managing the business, and 10% of the remaining profit would be deposited as savings. The rest of the profit would be divided in proportion to their capitals. If the total profit at the end of the year is ₹45,000, how much money will A receive in total?

6. Answer any one question [3 × 1 = 3]

iSolve: $\frac{2x+1}{2x-1} + \frac{2x-1}{2x+1} = 1 + \frac{7}{2x-1}, x \neq \frac{1}{2}$

iiIf $x \propto y$ and $y \propto z$, prove that $x+y+z \propto \sqrt{yz}+\sqrt{zx}+\sqrt{xy}$.

7. Solve any one question [3 × 1 = 3]

iFind the simplest value: $ (\sqrt{5}+\sqrt{3})\big( \frac{3\sqrt{3}}{\sqrt{5}+\sqrt{2}}-\frac{\sqrt{5}}{\sqrt{3}+\sqrt{2}}\big)$

iiThe volume of a cone is in joint variation with the square of its base radius and its height. If the ratio of the base radii of two cones is 3:4 and the ratio of their heights is 6:5, find the ratio of their volumes.

8. Answer any one question [3 × 1 = 3]

iIf $a, b, c, d$ are in continued proportion, prove that $(b-c)^2 + (c-a)^2 + (b-d)^2 = (a-d)^2$.

iiIf $\frac{x}{b+c} = \frac{y}{c+a} = \frac{z}{a+b}$, then prove that $\frac{a}{y+z-x} = \frac{b}{z+x-y} = \frac{c}{x+y-z}$.

9. Answer any one question [5 × 1 = 5]

iProve that the opposite angles of any cyclic quadrilateral are supplementary.

iiProve that if two tangents are drawn to a circle from an external point, the line segments joining the points of contact to the external point are equal in length and subtend equal angles at the center.

10. Answer any one question [3 × 1 = 3]

iIn an isosceles triangle ABC, AB = AC. A straight line parallel to BC intersects AB and AC at points P and Q respectively. Prove that BCQP is a cyclic quadrilateral.

iiA right-angled triangle ABC is drawn where $\angle A$ is a right angle. P and Q are two points on sides AB and AC respectively. B, Q and C, P are joined. Prove that $BQ^2 + PC^2 = BC^2 + PQ^2$.

11. Answer any one question [5 × 1 = 5]

iConstruct an isosceles triangle whose one angle is $120^\circ$ and the length of each of the equal sides is 4 cm. Draw the circumcircle of the triangle.

iiConstruct a square equal in area to a rectangle with sides of length 6 cm and 3 cm. (Only traces of construction are required).

12. Solve any two questions [3 × 2 = 6]

iIf $m = \tan\theta + \sin\theta$ and $n = \tan\theta - \sin\theta$, prove that $m^2 - n^2 = 4\sqrt{mn}$ ($0^\circ < \theta < 90^\circ$).

iiIf $\angle A + \angle B = 90^\circ$, show that $1 + \frac{\tan A}{\tan B} = \sec^2 A$.

iiiOne angle of a triangle is 60° and another is $\frac{\pi}{6}$ radians. What is the measurement of the third angle in degrees?

13. Answer any one question [5 × 1 = 5]

iFrom the roof and the foot of a 16-meter high house, the angles of elevation of the top of a temple are observed to be 45° and 60° respectively. Find the height of the temple and its horizontal distance from the house.

iiThe angle of elevation of the top of a tower from a point A on the ground is 30°. After moving 20 meters from point A towards the foot of the tower to point B, the angle of elevation increases to 60°. Find the height of the tower and the distance of the tower from point A.

14. Answer any two questions [4 × 2 = 8]

iA large solid sphere is made by melting three solid copper spheres with radii of lengths 3 cm, 4 cm, and 5 cm. Calculate the length of the radius of the large sphere.

ii77 square meters of tarpaulin were required to make a right circular cone-shaped tent. If the slant height of the tent is 7 meters, calculate the base area of the tent.

iiiDetermine the ratio of the volumes of a solid cone, a solid hemisphere, and a solid cylinder having equal base diameters and equal heights.

15. Answer any two questions [4 × 2 = 8]

iCalculate the arithmetic mean from the following frequency distribution:

Class Limit	0-10	10-20	20-30	30-40	40-50	50-60	60-70
Frequency	4	7	10	15	10	8	5

iiFind the median of the data below:

Class Limit	1-5	6-10	11-15	16-20	21-25	26-30	31-35
No. of Workers	2	3	6	7	5	4	3

iiiDetermine the mode of the following frequency distribution:

Class Limit	3-6	6-9	9-12	12-15	15-18	18-21	21-24
Frequency	2	6	12	24	21	12	3

Madhyamik Practice Test Paper

2026-01-14T20:08:00.000-08:00

Mathematics Test Examination Paper

Class X (Madhyamik)

3 Hours 15 Minutes

Full Marks: 90

West Bengal Board Format | Test Paper I - Vinod Singh

About This Examination Paper

This is a Mathematics Test Examination paper from Prime Maths, designed for Class X (Madhyamik) students. It follows the standard West Bengal Board format with a total of 90 marks and a duration of 3 hours and 15 minutes.

Arithmetic

Problems on simple interest, compound interest, and partnership business

Algebra

Quadratic equations, ratio and proportion, variation, and surds

Geometry

Theorems related to circles, tangents, circumcenters, and construction

Mensuration

Problems involving cubes, cylinders, cones, spheres, and cuboids

Trigonometry

Solving identities, trigonometric ratios, heights and distances problems

Statistics

Calculation of Mean, Median, and Mode from frequency distribution tables

This paper serves as excellent practice material for students preparing for their final Madhyamik examination, testing their understanding of both objective (MCQ, True/False, Fill in the blanks) and subjective (long answer) type questions.

MADHYAMIK PRACTICE TEST PAPER

Time: 3 Hours 15 Minutes | Full Marks: 90

1. Select the correct answer in each of the following cases: [1 × 6 = 6]

If the difference between compound interest and simple interest on a certain principal for 3 years at 20% compound interest rate per annum is ₹48, the principal will be:
- a) ₹350
- b) ₹375
- c) ₹400
- d) ₹425
For what value(s) of $k$ will the quadratic equation $2x^2 - kx + k = 0$ have equal roots?
- a) Only 0
- b) Only 8
- c) 4
- d) 0 and 8
If $A:B=2:3$, $B:C=5:8$, and $C:D=6:7$, what is the value of $A:D$?
- a) 2:7
- b) 5:8
- c) 7:2
- d) 5:14
$O$ is the circumcenter of $\triangle ABC$. Given $\angle BAC=85^\circ$ and $\angle BCA=55^\circ$, the value of $\angle AOC$ will be:
- a) $65^\circ$
- b) $45^\circ$
- c) $50^\circ$
- d) $100^\circ$
When two circles touch each other internally, what is the maximum number of common tangents that can be drawn?
- a) 1
- b) 3
- c) 4
- d) 2
The diameters of two solid cylinders are 8 cm and 12 cm respectively, and their heights are $3(n + 1)$ cm and $(n + 3)$ cm respectively. If the volumes of the two cylinders are equal, what is the value of $n$?
- a) 13
- b) 7
- c) 8
- d) 5

2. Fill in the blanks (Any five): [1 × 5 = 5]

If $P_1: P_2: P_3$ is the ratio of profits and $t_1:t_2:t_3$ is the ratio of time, then the ratio of invested capital is ______________.
$7\sqrt{\frac{101}{404}}$ is a ______________ number.
If $x \propto z$ and $y \propto z$, then $x+y \propto$ ______________.
The value of $\cos 28^\circ \csc 62^\circ + \tan 1^\circ \cot 89^\circ$ = ______________.
If the ratio of three consecutive angles of a cyclic quadrilateral is $1 : 3 : 4$, the measure of the fourth angle = ______________.
If the volume of a cube is $V$, the total surface area of the cube = ______________ square units.

3. Write True or False (Any five): [1 × 5 = 5]

A partnership business requires at least 4 people.
If the annual simple interest rate is 12%, the interest for $8\frac{1}{3}$ years will be equal to the principal.
The two roots of the equation $x^2 = -2024$ are real numbers.
In two triangles ABC and DEF, if $\angle A= \angle D$, $\angle B= \angle F$, and $\angle C = \angle E$, then $\frac{AB}{DE} = \frac{AC}{DF} = \frac{BC}{EF}$.
Two concentric circles of different radius will have one common tangent.
If the radius of the base of a right circular cone is halved and the height is doubled, the volume of the cone remains the same.

4. Answer the following questions (Any ten): [2 × 10 = 20]

If the simple interest on a principal for $n$ years at $r\%$ per annum is $\frac{pnr}{25}$, show that the Principal = $4p$.
If ₹8,000 is invested for 3 years at 10% compound interest per annum, how much interest will be earned at the end of 2 years?
In a business, A invested ₹1800 and B invested ₹1000 for 9 months. If both their profit shares are equal, for how long was A's money invested?
What is the ratio of the sum and product of the roots of the equation $-x^2 - 6x + 2 = 0$?
Rationalize the denominator: $\frac{2\sqrt{2}-3}{\sqrt{2}}$.
AB and AC are two chords of a circle that are perpendicular to each other. If AB = 5 cm and AC = 12 cm, find the length of the radius of the circle.
AD is perpendicular to the hypotenuse BC of a right-angled triangle ABC. If BC = 16 cm and BD = 9 cm, find the length of AB.
The curved surface area of a right circular cone is $\sqrt{5}$ times its base area. What is the ratio of the height to the radius of the cone?
If $x = a \cos \theta + b \sin \theta$ and $y = a \sin \theta - b \cos \theta$, what is the value of $x^2 + y^2$?
Find the median and arithmetic mean of the numbers: 1, 2, 3, 4, 6, 5, 7, 8, 9, 10, 11.
The length of the diagonal of a cube is $4\sqrt{3}$ cm. Find the total surface area of the cube.

5. Answer any one question: [5 × 1 = 5]

The simple interest and annual compound interest on a certain capital for 2 years at the same rate are ₹4,000 and ₹4,100 respectively. Find the capital and the rate of interest.
The current number of students in all secondary education centers in a district is 3993. If the number of students increases by 10% compared to the previous year every year, determine what the number of students was in all secondary education centers of that district 3 years ago.

6. Solve any one question: [3 × 1 = 3]

$\frac{1}{x} - \frac{1}{x+b} = \frac{1}{a} - \frac{1}{a+b}, \quad x \neq 0, -b.$
If $x = \frac{\sqrt{a+2} + \sqrt{a-2}}{\sqrt{a+2} - \sqrt{a-2}}$, where $a > 2$, find the simplest value of $x + \frac{1}{x}$.

7. Solve any one question: [3 × 1 = 3]

Find the simplest value: $\frac{2(\sqrt{2}+\sqrt{3})}{\sqrt{3}+1} - \frac{2(\sqrt{2}-\sqrt{3})}{\sqrt{3}-1}$.
If it takes 9 days for 5 men to cultivate 10 acres of land, use the theory of variation to find how many days it will take for 25 men to cultivate 30 acres of land.

8. Answer any one question: [3 × 1 = 3]

If $a:b = b:c$, prove that $\frac{(a+b)}{(b+c)^2} = \frac{a^2+b^2}{b^2+c^2}$.
If $\frac{a+b-c}{a+b} = \frac{b+c-a}{b+c} = \frac{c+a-b}{c+a}$ and $a+b+c \neq 0$, prove that $a = b = c$.

9. Answer any one question: [5 × 1 = 5]

Prove that the angle subtended by an arc of a circle at the center is double the angle subtended by it at any point on the remaining part of the circle.
Prove that if a perpendicular is drawn from the center of a circle to a chord which is not a diameter, the perpendicular bisects the chord.

10. Answer any one question: [3 × 1 = 3]

Prove that a cyclic trapezium is an isosceles trapezium and the lengths of its diagonals are equal.
In an isosceles triangle ABC, AB = AC and BE is perpendicular to AC from B. Prove that $BC^2 = 2AC \times CE$.

11. Answer any one question: [5 × 1 = 5]

Draw a square figure with a side length of 14 cm.
Draw a right-angled triangle whose sides adjacent to the right angle have lengths 4 cm and 8 cm. Draw the incircle of the triangle. (Only traces of construction are required).

12. Solve any two questions: [3 × 2 = 6]

If $m = \cos \theta - \sin \theta$ and $n = \cos \theta + \sin \theta$, prove that $\sqrt{\frac{m}{n}} + \sqrt{\frac{n}{m}} = \frac{2}{\sqrt{1-\tan^2 \theta}}$.
Find the value of $x$: $x \sin 60^\circ \cos^2 30^\circ = \frac{\tan^2 45^\circ \sec 60^\circ}{\csc 60^\circ}$.
The measures of three angles of a quadrilateral are $\frac{\pi}{3}$, $\frac{5\pi}{6}$, and $90^\circ$. Find the circular measure (in radians) of the fourth angle.

13. Answer any one question: [5 × 1 = 5]

From an object on the roof of a house $30\sqrt{3}$ meters high, the angles of depression of the top and bottom of a lamp post are $30^\circ$ and $60^\circ$ respectively. Find the height of the lamp post.
From the roof of a five-story building 18 meters high, the angle of elevation of the top of a monument is observed to be $45^\circ$ and the angle of depression of the foot of the monument is observed to be $60^\circ$. What is the height of the monument?

14. Answer any two questions: [4 × 2 = 8]

If the radius of a solid sphere and a solid right circular cylinder are equal and their volumes are also equal, calculate the ratio of the radius and height of the cylinder.
The volume of a right circular cone is $100\pi$ cubic cm and its height is 12 cm. Calculate and write the slant height of the cone.
A rectangular pond is 20 m long and 18.5 m wide, and has water 3.2 m deep. How long will it take to irrigate the entire water of the pond using a pump that can irrigate 160 kiloliters of water per hour?

15. Answer any two questions: [4 × 2 = 8]

Determine the Arithmetic Mean from the following frequency distribution:

Value: Below 10 Below 20 Below 30 Below 40 Below 50 Below 60 Below 70 Below 80

Frequency: 4 16 40 76 96 112 126 125
The frequency distribution of daily wages of 100 workers is as follows. Find the median of this distribution.

Daily Wage (in ₹): 460-470 470-480 480-490 490-500 500-510 510-520 520-530

No. of Workers: 6 4 29 23 16 10 2
Determine the Mode of the frequency distribution below:

Class Limit: 0-5 5-10 10-15 15-20 20-25 25-30 30-35

Frequency: 5 12 18 28 17 12 8

Value:	Below 10	Below 20	Below 30	Below 40	Below 50	Below 60	Below 70	Below 80
Frequency:	4	16	40	76	96	112	126	125

Daily Wage (in ₹):	460-470	470-480	480-490	490-500	500-510	510-520	520-530
No. of Workers:	6	4	29	23	16	10	2

Class Limit:	0-5	5-10	10-15	15-20	20-25	25-30	30-35
Frequency:	5	12	18	28	17	12	8

Problems on Algebra : ICSE and CBSE

2026-01-13T03:27:00.000-08:00

Algebraic Identities Problem Set

Master Algebra through Practice Problems. Prime Maths

About This Problem Set

This collection of problems is designed for Class IX students following CBSE, ICSE, and State Board curricula. The problems cover fundamental algebraic identities, inequalities, factorization, and polynomial expansions. Work through these problems systematically to strengthen your understanding of algebraic concepts and their applications.

Problem 1

Problem 1 of 22

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Master NDA Mathematics: The Ultimate Collection of Past Papers

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Competency Focused Questions for CBSE and ICSE Class X

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Pair of Linear Equations in Two Variables

Quadratic Equations

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Triangles

Coordinate Geometry

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Constructions (Competency – Steps & Justification)

Areas Related to Circles

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Mastering Algebra for Classes VIII to X: The Ultimate Practice Guide for CBSE, ICSE & State Boards

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Vinod Singh (Mathematics Educator, Prime Maths)

Similarity | CBSE | ICSE | Problem Solving

📚 Quick Recap – Geometry Essentials

1. Similarity of Triangles

2. Basic Proportionality Theorem (Thales)

3. Mid‑Point Theorem

4. Properties of Parallelograms

5. Area Ratios of Similar Triangles

Rational and Irrational Numbers | CBSE | Grade 8 |

📚 Understanding Rational & Irrational Numbers

1. What is a Rational Number?

2. What is an Irrational Number?

3. Closure Property

➕ Addition

✖️ Multiplication

➖ Subtraction

➗ Division

4. Identity Elements

5. Inverse Elements

6. Commutative & Associative Properties

ICSE Class 10 Results 2027: Expected Dates, Past Trends, and How to Check

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Bridging the Gap: Foundation Mathematics for Competitive Excellence | Classes IX & X CBSE ICSE

Master NDA Mathematics | Jai Hind!
The Ultimate Collection of Past Papers

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The Ultimate Practice Guide for CBSE, ICSE & State Boards