CBSE Class 12 Mathematics Relations and Functions VBQs Set 02

Question. 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: (d) Reflexive, transitive but not symmetric.

Question. If the set A contains 7 elements and set B contains 10 elements, then the number of one-one functions from A to B is :
(a) \( ^{10}C_7 \)
(b) \( ^{10}C_7 \times 7! \)
(c) \( 7^{10} \)
(d) \( 10^7 \)
Answer: (b) \( ^{10}C_7 \times 7! \)

Question. Let N be the set of natural numbers and the function f : N → N be defined by f(n) = 2n + 3 \( \forall n \in N \). Then f is
(a) surjective
(b) injective
(c) bijective
(d) none of the options
Answer: (b) injective

Question. 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: (c) 24

Question. If A = {a, b, c} and B = {4, 5, 6}, then number of functions from A to B is :
(a) 9
(b) 27
(c) 18
(d) 81
Answer: (b) 27

Question. Let f : R → R be defined by f(x) = \( x^2 + 1 \). Then, pre images of 17 and – 3, respectively, are:
(a) \( \phi \), {4, – 4}
(b) {3, – 3}, \( \phi \)
(c) {4, – 4}, \( \phi \)
(d) {4, – 4}, {2, – 2}
Answer: (c) {4, – 4}, \( \phi \)

Question. 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:
(a) reflexive
(b) symmetric
(c) transitive
(d) none of the options
Answer: (a) reflexive

Question. Let A = {2, 3, 6}. Which of the following relations on A are reflexive?
(a) R = {(2, 2), (3, 3), (6, 6)}
(b) R = {(2, 2), (3, 3), (3, 6), (6, 3)}
(c) R = {(2, 2), (3, 6), (2, 6)}
(d) None of the options
Answer: (a) R = {(2, 2), (3, 3), (6, 6)}

Question. Let R be the relation on N defined by R = {(x, y): x + 2y = 8}. Then, the domain of R is:
(a) {2, 4, 6, 8}
(b) {2, 4, 8}
(c) {2, 4, 6}
(d) {1, 2, 3, 4}
Answer: (c) {2, 4, 6}

Question. Let f : R → R be defined as \( f(x) = \begin{cases} 2x, & \text{if } x > 3 \\ x^2, & \text{if } 1 < x \le 3 \\ 3x, & \text{if } x \le 1 \end{cases} \).
Then f(– 1) + f(2) + f(4) =
(a) 9
(b) 14
(c) 5
(d) None of the options
Answer: (a) 9

Question. The relation R in the set of natural numbers N defined as R = {(x, y) : y = x + 5 and x < 4} is :
(a) reflexive
(b) symmetric
(c) transitive
(d) None of the options
Answer: (c) transitive

Question. 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)}
Then, the ordered pair to be added to R to make it the smallest equivalence relation is : 
(a) (1, 3)
(b) (3, 1)
(c) (2, 1)
(d) (1, 2)
Answer: (b) (3, 1)

Question. If A = {x ∈ Z : 0 ≤ x ≤ 12} and R is the relation in A given by R = {(a, b) : a = b). Then, the set of all elements related to 1 is :
(a) {1, 2}
(b) {2, 3}
(c) {1}
(d) {2}
Answer: (c) {1}

Question. f : X → Y is onto, if and only if :
(a) range of f = Y
(b) range of f ≠ Y
(c) range of f < Y
(d) range of f ≥ Y
Answer: (a) range of f = Y

Question. The number of all one-one functions from set A = {1, 2, 3} to itself is :
(a) 2
(b) 6
(c) 3
(d) 1
Answer: (b) 6

Question. Let A = {1, 2, 3, …, n} and B = {a, b}. Then the number of surjections from A into B is :
(a) \( ^nP_2 \)
(b) \( 2^n – 2 \)
(c) \( 2^n – 1 \)
(d) None of the options
Answer: (b) \( 2^n – 2 \)

Question. 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 :
(a) 720
(b) 120
(c) 0
(d) None of the options
Answer: (c) 0

Question. The greatest integer function f : R → R, given by f(x) = [x] is :
(a) one-one
(b) onto
(c) both one-one and onto
(d) neither one-one nor onto
Answer: (d) neither one-one nor onto

Question. Set A has 3 elements and the set B has 4 element then the total number of injective mapping :
(a) 144
(b) 12
(c) 24
(d) 64
Answer: (c) 24

Question. The relation of the relation R = {(x, x^2) : x is a prime number less than 13} :
(a) {2, 3, 5, 7}
(b) {4, 9, 25, 49, 121}
(c) {2, 3, 5, 7, 11}
(d) {1, 4, 9, 25, 49, 121}
Answer: (b) {4, 9, 25, 49, 121}

Question. Let f : R → R be defined by \( f(x) = \frac{x^2 - 8}{x^2 + 2} \), then f is :
(a) One-one but not onto
(b) One-one and onto
(c) Onto but not one-one
(d) Neither one-one nor onto
Answer: (d) Neither one-one nor onto

Question. Let R be the relation on the set A = {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (3, 3), (2, 3), (1, 3)}. 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: (b) R is reflexive and transitive but not symmetric

Question. R is a relation from {11, 12, 13} to {8, 10, 12} defined by y = x – 3 then \( R^{-1} \) is :
(a) {(8, 11), (10, 13)}
(b) {(11, 8), (13, 10)}
(c) {(10, 13), (8, 11), (8, 10)}
(d) None of the options
Answer: (a) {(8, 11), (10, 13)}

Question. Which one of the following is an identity relation?
(a) (1, 2), (2, 3), (1, 3)
(b) (5, 5), (4, 4), (2, 2)
(c) (1, 3), (3, 1), (2, 3)
(d) None of the options
Answer: (b) (5, 5), (4, 4), (2, 2)

Question. 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 for all a, b ∈ T, then R is :
(a) Reflexive but not symmetric
(b) Transitive but not symmetric
(c) Equivalence
(d) Neither symmetric nor transitive
Answer: (c) Equivalence

Question. One-one, onto function is also called :
(a) Injective function
(b) Surjective function
(c) Bijective function
(d) All of the options
Answer: (c) Bijective function

Question. The relation R on R defined by R = {(a, b) : \( a \le b^3 \)} is :
(a) Reflexive
(b) Symmetric
(c) Transitive
(d) None of the options
Answer: (d) None of the options

Question. The maximum number of equivalence relations on the set A = {1, 2, 3} are: 
(a) 1
(b) 2
(c) 3
(d) 5
Answer: (d) 5

Question. If a relation R on the set {1, 2, 3} be defined by R = {(1, 2)} then R is : 
(a) reflexive
(b) transitive
(c) symmetric
(d) none of the options
Answer: (d) none of the options

Question. Let us define a relation R in R as aRb if a ≥ b. Then R is : 
(a) an equivalence relation
(b) reflexive, transitive but not symmetric
(c) symmetric, transitive but not reflexive
(d) neither transitive nor reflexive but symmetric
Answer: (b) reflexive, transitive but not symmetric

Question. Let A = {1, 2, 3} and consider the relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)}
Then R is : 
(a) reflexive but not symmetric
(b) reflexive but not transitive
(c) symmetric and transitive
(d) neither symmetric nor transitive
Answer: (a) reflexive but not symmetric

Question. 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)}. Choose the correct answer.
(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: (b) R is reflexive and transitive but not symmetric

Question. Let f : R → R be defined as \( f(x) = x^4 \). Choose the correct answer : 
(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: (d) f is neither one-one nor onto