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## Relations and Functions Class 12 MCQs Questions with Answers

Don’t forget to practice the multitude of MCQ Questions on Relations and Functions Class 12 with answers so you can apply your skills during the exam.

Question 1.

Let R be the relation in the set (1, 2, 3, 4}, given by:

R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}.

Then:

(a) R is reflexive and symmetric but not transitive

(b) R is reflexive and transitive but not symmetric

(c) R is symmetric and transitive but not reflexive

(d) R is an equivalence relation.

## Answer

Answer: (b) R is reflexive and transitive but not symmetric

Question 2.

Let R be the relation in the set N given by : R = {(a, b): a = b – 2, b > 6}. Then:

(a) (2, 4) ∈ R

(b) (3, 8) ∈ R

(c) (6, 8) ∈ R

(d) (8, 7) ∈ R.

## Answer

Answer: (c) (6, 8) ∈ R

Question 3.

Let A = {1, 2, 3}. Then number of relations containing {1, 2} and {1, 3}, which are reflexive and symmetric but not transitive is:

(a) 1

(b) 2

(c) 3

(d) 4.

## Answer

Answer: (a) 1

Question 4.

Let A = (1, 2, 3). Then the number of equivalence relations containing (1, 2) is

(a) 1

(b) 2

(c) 3

(d) 4.

## Answer

Answer: (b) 2

Question 5.

Let f: R → R be defined as f(x) = x^{4}. Then

(a) f is one-one onto

(b) f is many-one onto

(c) f is one-one but not onto

(d) f is neither one-one nor onto.

## Answer

Answer: (d) f is neither one-one nor onto.

Question 6.

Let f : R → R be defined as f(x) = 3x. Then

(a) f is one-one onto

(b) f is many-one onto

(c) f is one-one but not onto

(d) f is neither one-one nor onto.

## Answer

Answer: (a) f is one-one onto

Question 7.

If f: R → R be given by f(x) = (3 – x³)^{1/3}, then fof (x) is

(a) x^{1/3}

(b) x³

(c) x

(d) 3 – x³.

## Answer

Answer: (c) x

Question 8.

Let f: R – {-\(\frac { 4 }{3}\)} → R be a function defined as: f(x) = \(\frac { 4x }{3x+4}\), x ≠ –\(\frac { 4 }{3}\). The inverse of f is map g : Range f → R -{-\(\frac { 4 }{3}\)} given by

(a) g(y) = \(\frac { 3y }{3-4y}\)

(b) g(y) = \(\frac { 4y }{4-3y}\)

(c) g(y) = \(\frac { 4y }{3-4y}\)

(d) g(y) = \(\frac { 3y }{4-3y}\)

## Answer

Answer: (b) g(y) = \(\frac { 4y }{4-3y}\)

Question 9.

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

(a) Reflexive and symmetric

(b) Transitive and symmetric

(c) Equivalence

(d) Reflexive, transitive but not symmetric.

## Answer

Answer: (b) Transitive and symmetric

Question 10.

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:

(a) 144

(b) 12

(c) 24

(d) 64

## Answer

Answer: (c) 24

Question 11.

Let f: R → R be defined by f(x) = sin x and g : R → R be defined by g(x) = x², then fog is

(a) x² sin x

(b) (sin x)²

(c) sin x²

(d) \(\frac { sin x }{x^2}\)

## Answer

Answer: (c) sin x²

Question 12.

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

(a) ø, {4,-4}

(b) {3, -3}, ø

(c) {4, -4}, ø

(d) {4, -4}, {2,-2}.

## Answer

Answer: (c) {4, -4}, $

Question 13.

Let f: R → R be defined by

f(x)= \(\left\{\begin{array}{lr}

2 x ; & x>3 \\

x^{2} ; & 1<x<3 \\

3 x ; & x \leq 1

\end{array}\right.\)

(a) 9

(b) 14

(c) 5

(d) None of these.

## Answer

Answer: (a) 9

Question 14.

The domain of the function f (x) = \(\frac { 1 }{\sqrt{|x|-x}}\) is

(a) (-∞, ∞)

(b) (0, ∞)

(c) (-∞, 0)

(d) (-∞, ∞) – {0}.

## Answer

Answer: (c) (-∞, 0)

Hint:

f(x) = \(\frac { 1 }{\sqrt{|x|-x}}\)

f(x) is defined if |x| – x > 0

if |x| > x

if x < 0.

Hence, D_{f} = (-∞, 0).

Question 15.

If a ∈ R and the equation

-3(x – [x] )2 + 2 (x – [x]) + a² = 0,

where [x] denotes the greatest integer (≤ x) has no integral solution, then all possible values of a lie in the interval:

(a) (1, 2)

(b) (-2, -1)

(c) (-∞, -2) ∪(2, ∞)

(d) (-1, 0) ∪ (0, 1).

## Answer

Answer: (d) (-1, 0) ∪ (0, 1).

Hint:

Put x – [x] = t.

Then – 3t² + 2t + a² = 0

⇒ a² = 3t² – 2t.

For non-integral solutions, 0 < a² < 1.

Hence, as (- 1, 0) ∪ (0, 1).

Fill in the Blanks

Question 1.

Let A = {1, 2, 3}. Then the number of equivalence relations containing (1, 2) is ………………

## Answer

Answer: 2.

Question 2.

If A = {0, 1, 3}, then the number of relations on A is ………………..

## Answer

Answer: 9.

Question 3.

A bijective function is both ……………….. and ………………

## Answer

Answer: one-one, onto.

Question 4.

Let R be a relation defined on A = { 1, 2, 3) by R = {(1, 3), (3, 1), (2, 2)}, R is ………………..

## Answer

Answer: symmetric.

Question 5.

If f be the greatest integer function defined as f(x) = [x] and g be the modulus function defined as g(x) = |x|, then the value of gof (\(\frac { -5 }{4}\)) is ………………..

## Answer

Answer: 2.

Hint:

gof (\(\frac { -5 }{4}\)) = g (f(\(\frac { -5 }{4}\)))

= g(-2) = |-2| = 2.

Question 6.

If f : R → R is defined by 3x + 4, then f(f(x)) is …………….

## Answer

Answer: 9x + 16.

Hint:

f(f (x)) = f(3x + 4)

= 3f(x) + 4

= 3 (3x + 4) + 4

= 9x + 16.

Question 7.

If f(x) = e^{x} and g(x) = log x, then gof is ……………….

## Answer

Answer: x.

Hint:

gof (x) = g(f(x) = g (e^{x})

= log e^{x} = x log e = x(1) = x.

Question 8.

The domain of the function f (x) = \(\frac { x }{|x|}\) is ………………..

## Answer

Answer: R – {0}.

Hint:

f is defined for all x ∈ R except at x = 0.

Question 9.

Let f: R – {-\(\frac { 4 }{3}\)} → R be a function defined as: f(x) = \(\frac { 4x }{3x+4}\), x ≠ –\(\frac { 4 }{3}\)

Then inverse of f is map g : Range f → R – {-\(\frac { 4 }{3}\)} given by …………………

## Answer

Answer: g(y) = \(\frac { 4x }{3x+4}\)

Hint:

Let y = f(x)= \(\frac { 4x }{3x+4}\)

⇒ 3xy + 4y = 4x

⇒ (4 – 3y)x = 4y

⇒ x = \(\frac { 4y }{4-3y}\)

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