PHY-MATH

Functions CBSE Important questions - Class 12 maths Relations and Functions

2021-04-23T09:09:00.004-07:00

What are Functions ? - Part 2 / Class 12 maths

2021-04-20T08:57:00.002-07:00

What are Functions – II ?

Hello everyone,

So, in the previous post I explained the meaning of Functions. If you missed the post, just click on the following link

Now, we’ll see Functions in more details:

A function is a process (or a relation) that associates each element $x$ of a set $X$ , the DOMAIN of the function, to a single element $y$ of another set $Y$ (possibly the same set), the CODOMAIN of the function.

Now you know why it is called “FUNCTION” ?

Because it does something, it’s a process.

I some times see FUNCTIONS as some kind of machine in which when we put some value of x of set X ( DOMAIN) and in return it gives a single value y of set Y ( CODOMAIN)

This was just a general presentation friends now let us see some real functions too.

So as in Real life we have different gadgets to fulfill our daily needs, similarly to fulfill our mathematical needs there are many types of functions. Let us discuss one such function

So, after this much study of modulus function, we can also represent it like this

F = {(0,0), (-1,1), (1,1), (-2,2) , (2,2) , (-3,3), (3,3),……. }

Note: Range elements are those elements of codomain which are associated with the elements of Domain, In General Domain elements are denoted by x and elements of Range are denoted by y

Can we draw a graph for it?

If we take terms before “ ,” as values of x ( Domain element)

If we take terms after “ ,” as values of y ( Range element)

See at the left side of the image. All the points of the function

f = {(0,0), (-1,1), (1,1), (-2,2) , (2,2) , (-3,3), (3,3),…….}

O D A E B F C

are marked on the graph paper and after joining those points we got “ V “ shaped graph.

Similarly there are more functions:

Greatest integer function [x] , signum function etc.

So, dear students we have studied function in a new prospective and soon we'll be studying types of functions and composition.

For this post I guess this much is enough 😊😊

Bie

Relations - CBSE Important questions - Class 12 chapter 1

2021-04-20T08:26:00.009-07:00

Study hard.

For any doubts or suggestions comment below👍

What are Functions ? - Class 12 maths chapter 1 - Relations and Functions

2021-04-17T22:57:00.006-07:00

WHAT ARE FUNCTIONS?

So in the previous post, we learnt about “ RELATIONS”. If you missed those posts kindly follow the link, as it is pre requisite knowledge to study functions.

CLICK THIS LINK

WHAT ARE RELATIONS?

Dear friends

“ FUNCTIONS” are special type of “RELATIONS”

In a Relation we associate elements of set “A” (DOMAIN) with the elements of set “B” ( CO DOMAIN) following a fixed rule.

R = { (0,0), ( 1, -1) , (1, 1) , ( 4, -2) , ( 4, 2) , ( 9, -3) , ( 9, 3) }

See how 1 associated with -1 and 1 in red colour.

4 with -2 and 2 in blue and 9 with -3 and 3 in green.

R = { (0,0), (-1,1), (1,1), (-2,4) , ( 2,4), ( -3,9) , ( 3,9), ( -4,16), (4,16) }

See how -1 and 1 are associated with 1 in yellow colour

-2 and 2 with 4 in green ; -3 and 3 with 9 in red and -4 and 4 with 16 in black

Now, we’ll take just one more example and then we can understand what functions are

See how each -2 is connected with -1 only, -1 with 0 only, 0 with 1 only and so on.

Now see, in the e.g.1 first diagram we have associations like this

In e.g.2 second diagram , association is like this

In e.g.3 third diagram , association is like this

“Relations” in which elements have association of type 2 and type 3 are termed as “Functions”

In other words

The RELATIONS in which each element of the DOMAIN are related with one and only one element in the CODOMAIN are termed as FUNCTIONS.

See in e.g. 1

“ 4” ( element of DOMAIN) is related to two elements -2 and 2 ( elements of CODOMAIN). Similarly , 1 is associated with -1 and 1

Therefore this relation is not a function.

Let us understand it more with one question:

Question

Which of the following relation is a function?

A = { (1,a), (2,b), (3,c), (1,d)} B = { (1,a), (1,b), (2,c), (2,d), (3,c), (3,d)}

C = { (1,a) , (2,a), (3,a),(4,b)} D = { (1,a), (2, a), (3,a), (1,b)}

Take some time and try to solve it using diagram

Not a function because 1 is associated with two elements “a” and “d”

Not a function because 1 is associated with two elements “a” and “b”

2 is associated with two elements “c” and “d”

3 is associated with two elements “c” and “d”

It is a function1 is associated with only a

2 is associated with only a

3 is associated with only a

4 is associated with inly b

NOTE : elements of CODOMAIN can associate with any number of elements of domain but restriction is only on DOMAIN elements. Each domain element should associate with one and only one element of the CODOMAIN.

Not a function because 1 is associated with two elements “a” and “b”

So, in this post I guess this much knowledge is enough.

In the next post we’ll see Functions with a different prospective and you’ll come to know more about functions.

click on the link below to see functions with a different linse.

What are functions ?- Part 2

Till then take care, Keep studying Maths and stay blessed.😄😄🙋🙋

RELATIONS - TYPES OF RELATIONS( PART 2) - CLASS 12 MATHEMATICS

2021-04-17T07:15:00.004-07:00

Types of Relations

Dear friends, in the previous blog we learnt about the meaning of "RELATION" and meaning of "REFLEXIVE RELATION". If you have missed click below

Relations - post 1

Now we'll continue with the topic 😊😊

Symmetric relations:

I wish this relation were symmetric.

( poor zoo zoo).

Let us define a relation in the set of human beings to understand the meaning of symmetric relations

e.g. A is related to B if A and B live in the same locality, where A and B are human beings.

So, Mr. Mehra is related to Mr. Sharma as both live in the same locality.

So Mr. Mehra can tell Mr. Sharma ,” I am related to you.”

Can Mr. Sharma say the same words to Mr. Mehra?

Or in other words, is Mr. Sharma related to Mr. Mehra with the same relation?

The answer is YES.

Because if “ A “ lives in the neighborhood of “B” then definitely “B” also lives in the neighborhood of “A”

Friends, Such relations are called symmetric relations.

So, we can generalize it like this:

If a is related to b with a fixed relation and b is also related to a with same relation, then we’ll say the relation is symmetric.

( Now you know why poor zoo zoo’s relation should be symmetric)

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Let us discuss one more example

A is related to B if A is father of B, where A and B are human beings.

Now think on this..

If Mr. Raj is father of Nikhil , can we say that Nikhil is related to Mr. Raj with the same relation?

The answer is NO.

Nikhil is related to Mr. Raj with the relation “ is son of “

It means Nikhil does not have the same relation with Mr. Raj.

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Ok now let us go to the world of numbers.

So,

R = { (a,b) : 2a + b = 0 } in the set of Real numbers.

Read it like this, a is related to b such that 2a + b = 0

Here, 1 is related to -2

As, 2 (1) + ( -2) = 0 is true

( see the relation above i.e. a is related to b if 2a + b=0)

Can we say -2 is also related to 1?

Let us see : 2 ( -2) + 1 = -4 + 1 = -3 .

So it is not equal to 0.

Therefore – 2 is no related to 1

So, the conclusion is that 1 is related to -2 but -2 is not related to 1

Therefore, it is a relation which is not symmetric.

Generalizing the definition:

So, in simple words if element “a” is related to element “b” and then element “b” is also related to “a” with the same relation, then the relation is called symmetric relation.

Transitive relations:

So, we are at the last type of relations now. “ Transitive relations”.

Now consider an relation in the set of human beings denoted by H.

R = { (a,b) : a is brother of b , a,b }

Read as a is related to b if a is brother of b , where a and b “ belong to “ set H

( a and b are Human beings)

So, Here Sonu (a) is related to Monu (b) since Sonu is Monu’s brother

Similarly, Monu (b) is related to Gonu (c) since Monu is Gonu’s brother

In place of names we can always call them “a” , “b” and “c”

Now “a” is related to “b” and “b” is related to ”c”.

Now, a question for you

Is “a” related to “c” with the same relation

( or in other words )

( Is Sonu, Gonu’s brother ?)

The answer is “Yes”.

I guess even you answered correctly.

So, such relations in which if “a” is related to “b” and “b” is related to “c”, it implies that “a” is related to “c” then relation is called transitive relation.

Okay, one question , is the following relation also transitive?

In the set of all the lines in a 2 dimensional plane

A line “a” is related to “b” if line ”a” is perpendicular to line “b”

Answer in the comments:

So, friends now we have understood three types of relations. Now we can define “EQUIVALENCE RELATIONS” .

A relation which is “reflexive, “symmetric” and “transitive” is termed as an equivalence relation.

So by this last definition we come to the end of this topic.

FOR NUMERICALS CLICK ON THE LINK

RELATIONS IMPORTANT QUSTIONS

See you later. Stay blessed.😊🙋

What are relations?. Types of Relations Class XII - maths chapter 1

2021-04-17T05:23:00.004-07:00

What are relations?

Before explaining what the meaning of relations is, we’ll recall the meaning of cartesian product.

So if we have two sets

A = { a, b, c}

B = { x, y}

Then randomly we can associate each element of A with each element of B

So, Cartesian product is

A X B = { (a, x). (a, y), ( b, x ) ( b, y), (c, x) , (c, y)}

Now if we set a rule on the basis of which we associate each element of set A with element(s) of set B then that ordered pair will be called relation. Same thing we do in real life.

e.g.1 if a man gets married to a woman then that man will be called the husband of the woman. So , a man will be related to a woman with the relation of “ is a husband” if and only if he has married that woman.

Mr. Alex is husband of Mrs. Sophia since he married her

e.g.2 “Alphabets” of set A are related to “word(s)” of set B if the word starts with the letter selected from set A

We can write this relation as

R = { (a, apple), (b, ball), (b, box) , (c, cat), ( c, car) }

Generalizing the definition:

So, in simple words if element “a” of set A gets associated with element “b” of set B on the basis of a rule then we’ll say that “a” is related to “b”.

TYPES OF RELATIONS

Now, within a few seconds he again measured his weight and no surprises it is still 100 kg.

Is Mr. A related to himself with the above relation?

The answer is YES.

As in the statement : A is related to B if A’s weight is equal to B’s weight, where A and B are human beings.

If we replace B by A the statement makes sense and is meaningful

A is related to A if A’s weight is equal to A’s weight, where A and A are human beings.

So, If a relation is such that , that each and every element of the given set is related with itself then the relation will termed as Reflexive relation.

Now think , can A be 7 cm taller than himself?

The answer is NO.

No one can be taller than himself.

So A is not related to A in this case.

So in this case relation is not reflexive

Generalizing the definition:

So, in simple words if element “a” of set A is related to “a” for each and every “a” belongs to “a” then the relation is reflexive.

I guess already the blog is too long . Let us discuss the other two types of relations in the next blog.

PART - 2 RELATIONS ( SYMMETRIC AND TRANSITIVE RELATIONS)

Keep studying maths and stay blessed.

Mathematics case study questions for class 10 pdf

2021-04-13T19:45:00.001-07:00

Dear friends,

As we all know, this year in class 10th maths exam (standard maths(041) and badic maths(241)), 4 questions based on case study will be there. Each question will have a word problem depicting a real life situation, on the basis of which there will be five parts. Students need to do any 4 parts for which they'll be awarded 4 marks.

Many of the students are facing problems in such questions. To deal with such questions, Students require good understanding of the concept and also must have the habit and capability to apply the knowledge in different situations.

To practice such questions I am sharing a link with you. Just click on the link and download the pdf containing such questions.

cbse class 10 mathematics case study questions

Enjoy learning , stay blessed😃

5 marks Important questions ( Three Dimensional Geometry)

2021-04-13T07:39:00.003-07:00

Dear friends, practice these questions again and again to ensure good marks in the upcoming CBSE exams. We'll be posting more study material ( chapter vise and marks vise ) here to help you score good. May the almighty bless us all with the will and strength to work harder everyday.

(Leave your comments for suggestions and to request notes on specific topics. We'll make sure to reply each one of you.)