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Relations and Functions Class 12 MCQs Questions with Answers
Students are advised to solve the Relations and Functions Multiple Choice Questions of Class 12 Maths to know different concepts. Practicing the MCQ Questions on Relations and Functions Class 12 with answers will boost your confidence thereby helping you score well in the exam.
Explore numerous MCQ Questions of Relations and Functions Class 12 with answers provided with detailed solutions by looking below.
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) = x4. 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) x1/3
(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, Df = (-∞, 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) = ex and g(x) = log x, then gof is ……………….
Answer
Answer: x.
Hint:
gof (x) = g(f(x) = g (ex)
= log ex = 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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