# NCERT solutions for Mathematics Exemplar Class 12 chapter 1 - Relations And Functions [Latest edition]

#### Chapters ## Chapter 1: Relations And Functions

Solved ExamplesExercise
Solved Examples [Pages 3 - 11]

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

Solved Examples | Q 1 | Page 3

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?

Solved Examples | Q 2 | Page 3

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.

Solved Examples | Q 3 | Page 3

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 

Solved Examples | Q 4 | Page 3

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

Solved Examples | Q 5 | Page 3

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

Solved Examples | Q 6 | Page 4

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

Solved Examples | Q 7 | Page 4

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

Solved Examples | Q 8 | Page 4

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

Solved Examples | Q 9 | Page 4

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)}

Solved Examples | Q 10 | Page 4

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

Solved Examples | Q 11 | Page 4

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?

Solved Examples | Q 12 | Page 5

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

Solved Examples | Q 13 | Page 5

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

Solved Examples | Q 14 | Page 6

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

Solved Examples | Q 15 | Page 6

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.

Solved Examples | Q 16. (i) | Page 7

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

Solved Examples | Q 16. (ii) | Page 7

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.

Solved Examples | Q 16. (iii) | Page 7

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

Solved Examples | Q 16. (iv) | Page 7

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

a * b = ab2 for a, b ∈ Q

#### Objective Type Questions Examples 17 to 25

Solved Examples | Q 17 | Page 8

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

Solved Examples | Q 18 | Page 8

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

Solved Examples | Q 19 | Page 8

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

Solved Examples | Q 20 | Page 8

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

Solved Examples | Q 21 | Page 9

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 ______.

• x2 sin x

• (sin x)2

• sin x2

• sinx/x^2

Solved Examples | Q 22 | Page 9

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

Solved Examples | Q 23 | Page 9

Let f: R → R be defined by f(x) = x2 + 1. Then, pre-images of 17 and – 3, respectively, are ______.

• φ, {4, – 4}

• {3, – 3}, φ

• {4, – 4}, φ

• {4, – 4, {2, – 2}

Solved Examples | Q 24 | Page 9

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

Solved Examples | Q 25 | Page 9

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

Solved Examples | Q 26 | Page 10

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

Solved Examples | Q 27 | Page 10

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

Solved Examples | Q 28 | Page 10

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

Solved Examples | Q 29 | Page 10

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

Solved Examples | Q 30 | Page 10

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

• True

• False

Solved Examples | Q 31 | Page 10

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

• True

• False

Solved Examples | Q 32 | Page 10

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

Solved Examples | Q 33 | Page 10

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

Solved Examples | Q 34 | Page 11

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

Exercise [Pages 11 - 17]

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

Exercise | Q 1 | Page 11

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

Exercise | Q 2 | Page 11

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

Exercise | Q 3 | Page 11

Let f, g: R → R be defined by f(x) = 2x + 1 and g(x) = x2 – 2, ∀ x ∈ R, respectively. Then, find gof

Exercise | Q 4 | Page 11

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

Exercise | Q 5 | Page 11

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

Exercise | Q 6 | Page 11

If f: R → R is defined by f(x) = x2 – 3x + 2, write f(f (x))

Exercise | Q 7 | Page 11

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

Exercise | Q 8. (i) | Page 11

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}

Exercise | Q 8. (ii) | Page 11

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}

Exercise | Q 9 | Page 11

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.

Exercise | Q 10 | Page 11

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.

Exercise | Q 11 | Page 11

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

Exercise | Q 12. (i) | Page 11

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)}

Exercise | Q 12. (ii) | Page 11

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)}

Exercise | Q 12. (iii) | Page 11

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)}

Exercise | Q 12. (iv) | Page 11

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

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

Exercise | Q 13 | Page 11

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

Exercise | Q 14 | Page 12

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

Exercise | Q 15 | Page 12

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

Exercise | Q 16. (a) | Page 12

If A = {1, 2, 3, 4 }, define relations on A which have properties of being:
reflexive, transitive but not symmetric

Exercise | Q 16. (b) | Page 12

If A = {1, 2, 3, 4 }, define relations on A which have properties of being:
symmetric but neither reflexive nor transitive

Exercise | Q 16. (c) | Page 12

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

Exercise | Q 17 | Page 12

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

Exercise | Q 18. (a) | Page 12

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

Exercise | Q 1. (b) | Page 12

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

Exercise | Q 18. (c) | Page 12

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

Exercise | Q 19. (i) | Page 12

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

Exercise | Q 19. (ii) | Page 12

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

Exercise | Q 19. (iii) | Page 12

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

Exercise | Q 20 | Page 12

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

Exercise | Q 21. (i) | Page 12

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

f(x) = x/2

Exercise | Q 21. (ii) | Page 12

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

g(x) = |x|

Exercise | Q 21. (iii) | Page 12

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

h(x) = x|x|

Exercise | Q 21. (iv) | Page 12

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

k(x) = x2

Exercise | Q 22. (i) | Page 12

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.

Exercise | Q 22. (ii) | Page 12

The following defines a relation on N:
x + y = 10, x, y ∈ N
Determine which of the above relations are reflexive, symmetric and transitive.

Exercise | Q 22. (iii) | Page 12

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.

Exercise | Q 22. (iv) | Page 12

The following defines a relation on N:
x + 4y = 10 x, y ∈ N.
Determine which of the above relations are reflexive, symmetric and transitive.

Exercise | Q 23 | Page 13

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)]

Exercise | Q 24 | Page 13

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

Exercise | Q 25. (i) | Page 13

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

Exercise | Q 25. (ii) | Page 13

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

Exercise | Q 25. (iii) | Page 13

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

Exercise | Q 25. (iv) | Page 13

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

Exercise | Q 26. (i) | Page 13

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

a * b = a – b ∀ a, b ∈ Q

Exercise | Q 26. (ii) | Page 13

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

a * b = a2 + b2 ∀ a, b ∈ Q

Exercise | Q 26. (iii) | Page 13

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

a * b = a + ab ∀ a, b ∈ Q

Exercise | Q 26. (iv) | Page 13

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

a * b = (a – b)2 ∀ a, b ∈ Q

Exercise | Q 27 | Page 13

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

Exercise | Q 28 | Page 13

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

Exercise | Q 29 | Page 13

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

Exercise | Q 30 | Page 14

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

• 1

• 2

• 3

• 5

Exercise | Q 31 | Page 14

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

Exercise | Q 32 | Page 14

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

Exercise | Q 33 | Page 14

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

Exercise | Q 34 | Page 14

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

Exercise | Q 35 | Page 14

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

Exercise | Q 36 | Page 14

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

• nP2

• 2n – 2

• 2n – 1

• None of these

Exercise | Q 37 | Page 15

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

• One-one

• Onto

• Bijective

• F is not defined

Exercise | Q 38 | Page 15

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

Exercise | Q 39 | Page 15

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

• f(x) = x3

• f(x) = x + 2

• f(x) = 2x + 1

• f(x) = x2 + 1

Exercise | Q 40 | Page 15

Let f: R → R be the functions defined by f(x) = x3 + 5. Then f–1(x) is ______.

• (x + 5)^(1/3)

• (x - 5)^(1/3)

• (5 - x)^(1/3)

• 5 – x

Exercise | Q 41 | Page 15

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

Exercise | Q 42 | Page 15

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)

Exercise | Q 43 | Page 15

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

Exercise | Q 44 | Page 16

Let f: [2, oo) → R be the function defined by f(x) = x2 – 4x + 5, then the range of f is ______.

• R

• [1, oo)

• [4, oo)

• [5, oo)

Exercise | Q 45 | Page 16

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

Exercise | Q 46 | Page 16

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

Exercise | Q 47 | Page 16

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

Exercise | Q 48 | Page 16

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

Exercise | Q 49 | Page 16

Let the relation R be defined on the set A = {1, 2, 3, 4, 5} by R = {(a, b) : |a2 – b2| < 8. Then R is given by ______.

Exercise | Q 50 | Page 16

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

Exercise | Q 51 | Page 17

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

Exercise | Q 52 | Page 17

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

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

Exercise | Q 53 | Page 17

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

Exercise | Q 54 | Page 17

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

• True

• False

Exercise | Q 55 | Page 17

Every relation which is symmetric and transitive is also reflexive.

• True

• False

Exercise | Q 56 | Page 17

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

Exercise | Q 57 | Page 17

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

Exercise | Q 58 | Page 17

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

Exercise | Q 59 | Page 17

The composition of functions is commutative.

• True

• False

Exercise | Q 60 | Page 17

The composition of functions is associative.

• True

• False

Exercise | Q 61 | Page 17

Every function is invertible.

• True

• False

Exercise | Q 62 | Page 17

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

• True

• False

## Chapter 1: Relations And Functions

Solved ExamplesExercise ## 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.

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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.

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