#### Online Mock Tests

#### Chapters

Chapter 2: Inverse Trigonometric Functions

Chapter 3: Matrices

Chapter 4: Determinants

Chapter 5: Continuity And Differentiability

Chapter 6: Application Of Derivatives

Chapter 7: Integrals

Chapter 8: Application Of Integrals

Chapter 9: Differential Equations

Chapter 10: Vector Algebra

Chapter 11: Three Dimensional Geometry

Chapter 12: Linear Programming

Chapter 13: Probability

## Chapter 1: Relations And Functions

### NCERT solutions for Mathematics Exemplar Class 12 Chapter 1 Relations And FunctionsSolved Examples [Pages 3 - 11]

#### Short Answer

Let A = {0, 1, 2, 3} and define a relation R on A as follows: R = {(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (2, 2), (3, 0), (3, 3)}. Is R reflexive? symmetric? transitive?

For the set A = {1, 2, 3}, define a relation R in the set A as follows: R = {(1, 1), (2, 2), (3, 3), (1, 3)}. Write the ordered pairs to be added to R to make it the smallest equivalence relation.

Let R be the equivalence relation in the set Z of integers given by R = {(a, b): 2 divides a – b}. Write the equivalence class [0]

Let the function f: R → R be defined by f(x) = 4x – 1, ∀ x ∈ R. Then, show that f is one-one.

If f = {(5, 2), (6, 3)}, g = {(2, 5), (3, 6)}, write f o g

Let f: R → R be the function defined by f(x) = 4x – 3 ∀ x ∈ R. Then write f^{–1}

Is the binary operation * defined on Z (set of integer) by m * n = m – n + mn ∀ m, n ∈ Z commutative?

If f = {(5, 2), (6, 3)} and g = {(2, 5), (3, 6)}, write the range of f and g

If A = {1, 2, 3} and f, g are relations corresponding to the subset of A × A indicated against them, which of f, g is a function? Why?

f = {(1, 3), (2, 3), (3, 2)}

g = {(1, 2), (1, 3), (3, 1)}

If A = {a, b, c, d} and f = {a, b), (b, d), (c, a), (d, c)}, show that f is one-one from A onto A. Find f^{–1}

In the set N of natural numbers, define the binary operation * by m * n = g.c.d (m, n), m, n ∈ N. Is the operation * commutative and associative?

#### Long Answer

In the set of natural numbers N, define a relation R as follows: ∀ n, m ∈ N, nRm if on division by 5 each of the integers n and m leaves the remainder less than 5, i.e. one of the numbers 0, 1, 2, 3 and 4. Show that R is equivalence relation. Also, obtain the pairwise disjoint subsets determined by R

Show that the function f: R → R defined by f(x) = `x/(x^2 + 1)`, ∀ ∈ + R , is neither one-one nor onto

Let f, g: R → R be two functions defined as f(x) = |x| + x and g(x) = x – x ∀ x ∈ R. Then, find f o g and g o f

Let R be the set of real numbers and f: R → R be the function defined by f(x) = 4x + 5. Show that f is invertible and find f^{–1}.

Let * be a binary operation defined on Q. Find which of the following binary operations are associative

a * b = a – b for a, b ∈ Q

Let * be a binary operation defined on Q. Find which of the following binary operations are associative

a * b = `"ab"/4` for a, b ∈ Q.

Let * be a binary operation defined on Q. Find which of the following binary operations are associative

a * b = a – b + ab for a, b ∈ Q

a * b = ab^{2} for a, b ∈ Q

#### Objective Type Questions Examples 17 to 25

Let R be a relation on the set N of natural numbers defined by nRm if n divides m. Then R is ______.

Reflexive and symmetric

Transitive and symmetric

Equivalence

Reflexive, transitive but not symmetric

Let L denote the set of all straight lines in a plane. Let a relation R be defined by lRm if and only if l is perpendicular to m ∀ l, m ∈ L. Then R is ______.

Reflexive

Symmetric

Transitive

None of these

Let N be the set of natural numbers and the function f: N → N be defined by f(n) = 2n + 3 ∀ n ∈ N. Then f is ______.

Surjective

Injective

Bijective

None of these

Set A has 3 elements and the set B has 4 elements. Then the number of injective mappings that can be defined from A to B is ______.

144

12

24

64

Let f: R → R be defined by f(x) = sin x and g: R → R be defined by g(x) = x 2 , then f o g is ______.

x

^{2}sin x(sin x)

^{2}sin x

^{2}`sinx/x^2`

Let f: R → R be defined by f(x) = 3x – 4. Then f^{–1}(x) is given by ______.

`(x + 4)/3`

`x/3 - 4`

3x + 4

None of these

Let f: R → R be defined by f(x) = x^{2} + 1. Then, pre-images of 17 and – 3, respectively, are ______.

φ, {4, – 4}

{3, – 3}, φ

{4, – 4}, φ

{4, – 4, {2, – 2}

For real numbers x and y, define xRy if and only if x – y + `sqrt(2)` is an irrational number. Then the relation R is ______.

Reflexive

Symmetric

Transitive

None of these

#### Fill in the blank Examples 25 to 30

Consider the set A = {1, 2, 3} and R be the smallest equivalence relation on A, then R = ______

The domain of the function f: R → R defined by f(x) = `sqrt(x^2 - 3x + 2)` is ______

Consider the set A containing n elements. Then, the total number of injective functions from A onto itself is ______

Let Z be the set of integers and R be the relation defined in Z such that aRb if a – b is divisible by 3. Then R partitions the set Z into ______ pairwise disjoint subsets

Let R be the set of real numbers and * be the binary operation defined on R as a * b = a + b – ab ∀ a, b ∈ R. Then, the identity element with respect to the binary operation * is ______.

#### State True or False for the statements in each of the Examples 30 to 34

Consider the set A = {1, 2, 3} and the relation R = {(1, 2), (1, 3)}. R is a transitive relation.

True

False

Let A be a finite set. Then, each injective function from A into itself is not surjective.

True

False

For sets A, B and C, let f: A → B, g: B → C be functions such that g o f is injective. Then both f and g are injective functions.

True

False

For sets A, B and C, let f: A → B, g: B → C be functions such that g o f is surjective. Then g is surjective.

True

False

Let N be the set of natural numbers. Then, the binary operation * in N defined as a * b = a + b, ∀ a, b ∈ N has identity element.

True

False

### NCERT solutions for Mathematics Exemplar Class 12 Chapter 1 Relations And FunctionsExercise [Pages 11 - 17]

#### Short Answer

Let A = {a, b, c} and the relation R be defined on A as follows:

R = {(a, a), (b, c), (a, b)}.

Then, write minimum number of ordered pairs to be added in R to make R reflexive and transitive

Let D be the domain of the real valued function f defined by f(x) = `sqrt(25 - x^2)`. Then, write D

Let f, g: R → R be defined by f(x) = 2x + 1 and g(x) = x^{2} – 2, ∀ x ∈ R, respectively. Then, find gof

Let f: R → R be the function defined by f(x) = 2x – 3 ∀ x ∈ R. write f^{–1}

If A = {a, b, c, d} and the function f = {(a, b), (b, d), (c, a), (d, c)}, write f^{–1}

If f: R → R is defined by f(x) = x^{2} – 3x + 2, write f(f (x))

Is g = {(1, 1), (2, 3), (3, 5), (4, 7)} a function? If g is described by g (x) = αx + β, then what value should be assigned to α and β

Are the following set of ordered pairs functions? If so, examine whether the mapping is injective or surjective.

{(x, y): x is a person, y is the mother of x}

Are the following set of ordered pairs functions? If so, examine whether the mapping is injective or surjective.

{(a, b): a is a person, b is an ancestor of a}

If the mappings f and g are given by f = {(1, 2), (3, 5), (4, 1)} and g = {(2, 3), (5, 1), (1, 3)}, write f o g.

Let C be the set of complex numbers. Prove that the mapping f: C → R given by f(z) = |z|, ∀ z ∈ C, is neither one-one nor onto.

Let the function f: R → R be defined by f(x) = cosx, ∀ x ∈ R. Show that f is neither one-one nor onto

Let X = {1, 2, 3}and Y = {4, 5}. Find whether the following subset of X ×Y are function from X to Y or not

f = {(1, 4), (1, 5), (2, 4), (3, 5)}

Let X = {1, 2, 3}and Y = {4, 5}. Find whether the following subset of X ×Y are function from X to Y or not

g = {(1, 4), (2, 4), (3, 4)}

Let X = {1, 2, 3}and Y = {4, 5}. Find whether the following subset of X ×Y are function from X to Y or not

h = {(1,4), (2, 5), (3, 5)}

k = {(1,4), (2, 5)}

If functions f: A → B and g: B → A satisfy gof = I_{A}, then show that f is one-one and g is onto

Let f: R → R be the function defined by f(x) = `1/(2 - cosx)` ∀ x ∈ R.Then, find the range of f

Let n be a fixed positive integer. Define a relation R in Z as follows: ∀ a, b ∈ Z, aRb if and only if a – b is divisible by n. Show that R is an equivalance relation

#### Long Answer

If A = {1, 2, 3, 4 }, define relations on A which have properties of being:

reflexive, transitive but not symmetric

If A = {1, 2, 3, 4 }, define relations on A which have properties of being:

symmetric but neither reflexive nor transitive

If A = {1, 2, 3, 4 }, define relations on A which have properties of being:

reflexive, symmetric and transitive

Let R be relation defined on the set of natural number N as follows:

R = {(x, y): x ∈N, y ∈N, 2x + y = 41}. Find the domain and range of the relation R. Also verify whether R is reflexive, symmetric and transitive

Given A = {2, 3, 4}, B = {2, 5, 6, 7}. Construct an example of the following:

an injective mapping from A to B

Given A = {2, 3, 4}, B = {2, 5, 6, 7}. Construct an example of the following:

a mapping from A to B which is not injective

Given A = {2, 3, 4}, B = {2, 5, 6, 7}. Construct an example of the following:

a mapping from B to A

Give an example of a map which is one-one but not onto

Give an example of a map which is not one-one but onto

Give an example of a map which is neither one-one nor onto

Let A = R – {3}, B = R – {1}. Let f: A → B be defined by f(x) = `(x - 2)/(x - 3)` ∀ x ∈ A . Then show that f is bijective

Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:

f(x) = `x/2`

Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:

g(x) = |x|

Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:

h(x) = x|x|

k(x) = x^{2}

The following defines a relation on N:

x is greater than y, x, y ∈ N

Determine which of the above relations are reflexive, symmetric and transitive.

The following defines a relation on N:

x + y = 10, x, y ∈ N

Determine which of the above relations are reflexive, symmetric and transitive.

The following defines a relation on N:

x y is square of an integer x, y ∈ N

Determine which of the above relations are reflexive, symmetric and transitive.

The following defines a relation on N:

x + 4y = 10 x, y ∈ N.

Determine which of the above relations are reflexive, symmetric and transitive.

Let A = {1, 2, 3, ... 9} and R be the relation in A × A defined by (a, b) R(c, d) if a + d = b + c for (a, b), (c, d) in A × A. Prove that R is an equivalence relation and also obtain the equivalent class [(2, 5)]

Using the definition, prove that the function f: A→ B is invertible if and only if f is both one-one and onto

Functions f , g: R → R are defined, respectively, by f(x) = x 2 + 3x + 1, g(x) = 2x – 3, find f o g

Functions f , g: R → R are defined, respectively, by f(x) = x 2 + 3x + 1, g(x) = 2x – 3, find g o f

Functions f , g: R → R are defined, respectively, by f(x) = x 2 + 3x + 1, g(x) = 2x – 3, find f o f

Functions f , g: R → R are defined, respectively, by f(x) = x 2 + 3x + 1, g(x) = 2x – 3, find g o g

Let * be the binary operation defined on Q. Find which of the following binary operations are commutative

a * b = a – b ∀ a, b ∈ Q

Let * be the binary operation defined on Q. Find which of the following binary operations are commutative

a * b = a^{2} + b^{2} ∀ a, b ∈ Q

Let * be the binary operation defined on Q. Find which of the following binary operations are commutative

a * b = a + ab ∀ a, b ∈ Q

a * b = (a – b)^{2} ∀ a, b ∈ Q

Let * be binary operation defined on R by a * b = 1 + ab, ∀ a, b ∈ R. Then the operation * is ______.

Commutative but not associative

Associative but not commutative

Neither commutative nor associative

Both commutative and associative

#### Objective Type Questions from 28 to 47

Let T be the set of all triangles in the Euclidean plane, and let a relation R on T be defined as aRb if a is congruent to b ∀ a, b ∈ T. Then R is ______.

Reflexive but not transitive

Transitive but not symmetric

Equivalence

None of these

Consider the non-empty set consisting of children in a family and a relation R defined as aRb if a is brother of b. Then R is ______.

Symmetric but not transitive

Transitive but not symmetric

Neither symmetric nor transitive

Both symmetric and transitive

The maximum number of equivalence relations on the set A = {1, 2, 3} are ______.

1

2

3

5

If a relation R on the set {1, 2, 3} be defined by R = {(1, 2)}, then R is ______.

Reflexive

Transitive

Symmetric

None of these

Let us define a relation R in R as aRb if a ≥ b. Then R is ______.

An equivalence relation

Reflexive, transitive but not symmetric

Symmetric, transitive but not reflexive

Neither transitive nor reflexive but symmetric

Let A = {1, 2, 3} and consider the relation R = {1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1,3)}. Then R is ______.

Reflexive but not symmetric

Reflexive but not transitive

Symmetric and transitive

Neither symmetric, nor transitive

The identity element for the binary operation * defined on Q ~ {0} as a * b = `"ab"/2` ∀ a, b ∈ Q ~ {0} is ______.

1

0

2

none of these

If the set A contains 5 elements and the set B contains 6 elements, then the number of one-one and onto mappings from A to B is ______.

720

120

0

none of these

Let A = {1, 2, 3, ...n} and B = {a, b}. Then the number of surjections from A into B is ______.

^{n}P_{2}2

^{n}– 22

^{n}– 1None of these

Let f: R → R be defined by f(x) = `1/x` ∀ x ∈ R. Then f is ______.

One-one

Onto

Bijective

F is not defined

Let f: R → R be defined by f(x) = 3x 2 – 5 and g: R → R by g(x) = `x/(x^2 + 1)` Then gof is ______.

`(3x^2 - 5)/(9x^4 - 30x^2 + 26)`

`(3x^2 - 5)/(9x^4 - 6x^2 + 26)`

`(3x^2)/(x^4 + 2x^2 - 4)`

`(3x^2)/(9x^4 + 30x^2 - 2`

Which of the following functions from Z into Z are bijections?

f(x) = x

^{3}f(x) = x + 2

f(x) = 2x + 1

f(x) = x

^{2}+ 1

Let f: R → R be the functions defined by f(x) = x^{3} + 5. Then f^{–1}(x) is ______.

`(x + 5)^(1/3)`

`(x - 5)^(1/3)`

`(5 - x)^(1/3)`

5 – x

Let f: A → B and g: B → C be the bijective functions. Then (g o f)^{–1} is ______.

f

^{–1}o g^{–1}f o g

g

^{–1}o f^{–1}g o f

Let f: R – `{3/5}` → R be defined by f(x) = `(3x + 2)/(5x - 3)`. Then ______.

f

^{–1}(x) = f(x)f

^{–1}(x) = – f(x)(f o f)x = – x

f

^{–1}(x) = `1/19` f(x)

Let f: [0, 1] → [0, 1] be defined by f(x) = `{{:(x",", "if" x "is rational"),(1 - x",", "if" x "is irrational"):}`. Then (f o f) x is ______.

Constant

1 + x

x

None of these

Let f: `[2, oo)` → R be the function defined by f(x) = x^{2} – 4x + 5, then the range of f is ______.

R

`[1, oo)`

`[4, oo)`

`[5, oo)`

Let f: N → R be the function defined by f(x) = `(2x - 1)/2` and g: Q → R be another function defined by g(x) = x + 2. Then (g o f) `3/2` is ______.

1

1

`7/2`

None of these

Let f: R → R be defined by f(x) = `{{:(2x",", x > 3),(x^2",", 1 < x ≤ 3),(3x",", x ≤ 1):}`. Then f(–1) + f(2) + f(4) is ______.

9

14

5

None of these

Let f: R → R be given by f(x) = tan x. Then f^{–1}(1) is ______.

`pi/4`

`{"n" pi + pi/4 : "n" ∈ "Z"}`

Does not exist

None of these

#### Fill in the blanks in the Exercise 48 to 52

Let the relation R be defined in N by aRb if 2a + 3b = 30. Then R = ______.

Let the relation R be defined on the set A = {1, 2, 3, 4, 5} by R = {(a, b) : |a^{2} – b^{2}| < 8. Then R is given by ______.

Let f = {(1, 2), (3, 5), (4, 1) and g = {(2, 3), (5, 1), (1, 3)}. Then g o f = ______ and f o g = ______.

Let f: R → R be defined by f(x) = `x/sqrt(1 + x^2)`. Then (f o f o f) (x) = ______.

If f(x) = (4 – (x – 7)^{3}}, then f^{–1}(x) = ______.

#### State True or False for the statement in the Exercise 53 to 63

Let R = {(3, 1), (1, 3), (3, 3)} be a relation defined on the set A = {1, 2, 3}. Then R is symmetric, transitive but not reflexive.

True

False

Let f: R → R be the function defined by f(x) = sin (3x+2) ∀ x ∈ R. Then f is invertible.

True

False

Every relation which is symmetric and transitive is also reflexive.

True

False

An integer m is said to be related to another integer n if m is a integral multiple of n. This relation in Z is reflexive, symmetric and transitive.

True

False

Let A = {0, 1} and N be the set of natural numbers. Then the mapping f: N → A defined by f(2n – 1) = 0, f(2n) = 1, ∀ n ∈ N, is onto.

True

False

The relation R on the set A = {1, 2, 3} defined as R = {{1, 1), (1, 2), (2, 1), (3, 3)} is reflexive, symmetric and transitive.

True

False

The composition of functions is commutative.

True

False

The composition of functions is associative.

True

False

Every function is invertible.

True

False

A binary operation on a set has always the identity element.

True

False

## Chapter 1: Relations And Functions

## NCERT solutions for Mathematics Exemplar Class 12 chapter 1 - Relations And Functions

NCERT solutions for Mathematics Exemplar Class 12 chapter 1 (Relations And Functions) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Exemplar Class 12 solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. NCERT textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Mathematics Exemplar Class 12 chapter 1 Relations And Functions are Composition of Functions and Invertible Function, Types of Functions, Types of Relations, Introduction of Relations and Functions, Concept of Binary Operations, Inverse of a Function.

Using NCERT Class 12 solutions Relations And Functions exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in NCERT Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 12 prefer NCERT Textbook Solutions to score more in exam.

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