CBSE Class 12 Mathematics Relations and Functions MCQs Set B

Question. The domain of y = x/√|x|-x is
(a) [0, ∞)
(b) (– ∞, 0)
(c) (– ∞, 0]
(d) [1, ∞)

Answer : B

Question. Let X = {– 1, 0, 1}, Y = {0, 2} and a function f : X → Y defined by y = 2x⁴, is
(a) one-one onto
(b) one-one into
(c) many-one onto
(d) many-one into

Answer : C

Question. A binary operation x on the set {0, 1, 2, 3, 4, 5} is defined as

the identity element is
(a) 0
(b) 1
(c) 2
(d) 3

Answer : A

Question. The mapping f : N → N given by f(n) = 1 + n², n ∈ N when N is the set of natural numbers, is
(a) one-one and onto
(b) onto but not one-one
(c) one-one but not onto
(d) neither one-one nor onto

Answer : C

Question. Let f : R → R be a function defined by f(x) = x³ + 4, then f is
(a) injective
(b) surjective
(c) bijective
(d) None of the options

Answer : C

Question. Which of the following functions from I to itself is a bijection?
(a) f(x) = x³
(b) f(x) = x + 2
(c) f (x) = 2x + 1
(d) f (x) = x² + x

Answer : B

Question. If f : R → R, f (x) = x³ + 2, then f –1 (x) is
(a) (x -1)^1/2
(b) x – 2
(c) (x – 2)^1/3
(d) (x – 2)1/2

Answer : C

Question. For real numbers x and y, we write x R y ⇔ x – y + √2 is an irrational number. Then, the relation R is
(a) Reflexive
(b) Symmetric
(c) Transitive
(d) None of the options

Answer : A

Question. Let f : R → R be defined by

Then f (– 1) + f (2) + f (4) is
(a) 9
(b) 14
(c) 5
(d) None of the options

Answer : A

Question. Let R be a relation on the set N be defined by {(x, y) l x, y ∈ N, 2x + y = 41}. Then, R is
(a) Reflexive
(b) Symmetric
(c) Transitive
(d) None of the options

Answer : D

Question. If the set A contains 7 elements and the set B contains 8 elements, then number of one-one and onto mappings from A to B is
(a) 24
(b) 120
(c) 0
(d) none of the options

Answer : C

Question. Let A = {1, 2, 3, 4}. Let R be the equivalence relation on A × A defined by (a, b) R (c, d) if a + d = b + c. Then the equivalence class [(1, 3)] is
(a) {(1, 3)}
(b) {(2, 4)}
(c) {(1, 8), (2, 4), (1, 4)}
(d) {(1, 3) (2, 4)}

Answer : D

Question. If f : R → R be defined by f(x) = 2/x, x ∀ R, then f is
(a) one-one
(b) onto
(c) bijective
(d) f is not defined

Answer : D

Question. Which of the following functions form Z into Z bijections?
(a) f (x) = x³ 
(b) f (x) = x + 2
(c) f (x) = 2x + 1
(d) f (x) = x² + 1

Answer : B

Question. If f : R → R be the function defined by f (x) = x³ + 5, then f^–1(x) is
(a) (x + 5)^1/3
(b) (x – 5)^1/3
(c) (5 – x)^1/3
(d) (5 – x)

Answer : B

Question. A function f : X → Y is said to be onto, if for every y ∈ Y, there exists an element x in X such that
(a) f(x) = y
(b) f(y) = x
(c) f(x) + y = 0
(d) f(y) + x = 0

Answer : A

Question. Let function f : R → R be defined by f (x) = 2x + sin x for x ∈ R , then f is
(a) one-one and onto
(b) one-one but NOT onto
(c) onto but NOT one-one
(d) neither one-one nor onto

Answer : A

Question. Let f : R → R be given by f(x) = tan x. Then f –1(1) is
(a) π/4
(b) {nπ + π/4 : n ∈ Z}
(c) does not exist
(d) None of the options

Answer : B
 

Case Based Questions

Consider the mapping f : A → B is defined by f(x) = x – 1/X -2 such that f is a bijection.

Based on the above information, answer the following questions:

Question. Domain of f is
(a) R – {2}
(b) R
(c) R – {1, 2}
(d) R – {0}

Answer : A

Question. Range of f is
(a) R
(b) R – {1}
(c) R – {0}
(d) R – {1, 2}

Answer : B

Question. If g : R – {2} → R – {1} is defined by g(x) = 2f(x) – 1, then g(x) in terms of x is

Answer : D

Question. The function g defined above, is
(a) One-one
(b) Many-one
(c) into
(d) None of these

Answer : A

Question. A function f(x) is said to be one-one if
(a) f(x₁) = f(x₂) ⇒ –x₁ = x₂
(b) f(–x₁) = f(–x₂) ⇒ –x₁ = x₂
(c) f(x₁) = f(x₂) ⇒ x₁ = x₂
(d) None of these

Answer : C

A general election of Lok Sabha is a gigantic exercise. About 911 million people were eligible to vote and voter turnout was about 67%, the highest ever

Let I be the set of all citizens of India who were eligible to exercise their voting right in general election held in 2019. A relation ‘R’ is defined on I as follows:
R = {(V₁, V₂)} : V₁, V₂ ∈ I and both use their voting right in general election — 2019}

Based on the above information answer the following:

Question. Two neighbours X and Y ∈ I. X exercised his voting right while Y did not cast her vote in general election — 2019. Which of the following is true?
(a) (X, Y) ∈ R
(b) (Y, X) ∈ R
(c) (X, X) ∉ R
(d) (X, Y) ∉ R

Answer : D

Question. Mr. ‘X’ and his wife ‘W’ both exercised their voting right in general election — 2019, which of the following is true?
(a) Both (X, W) and (W, X) ∈ R
(b) (X, W) ∈ R but (W, X) ∉ R
(c) Both (X, W) and (W, X) ∉ R
(d) (W, X) ∈ R but (X, W) ∉ R

Answer : A

Question. Three friends F₁, F₂ and F₃ exercised their voting right in general election — 2019, then which of the following is true?
(a) (F₁, F₂) ∈ R, (F₂, F₃) ∈ R and (F₁, F₃) ∈ R
(b) (F₁, F₂) ∈ R, (F₂, F₃) ∈ R and (F₁, F₃) ∉ R
(c) (F₁, F₂) ∈ R, (F₂, F₃) ∈ R but (F₁, F₃) ∉ R
(d) (F₁, F₂) ∉ R, (F₂, F₃) ∉ R and (F₁, F₃) ∉ R

Answer : A

Question. The above defined relation R is …………. .
(a) Symmetric and transitive but not reflexive
(b) Universal relation
(c) Equivalence relation
(d) Reflexive but not symmetric and transitive

Answer : C

Question. Mr. Shyam exercised his voting right in General Election — 2019, then Mr. Shyam is related to which of the following?
(a) All those eligible voters who cast their votes
(b) Family members of Mr. Shyam
(c) All citizens of India
(d) Eligible voters of India

Answer : A
 

Raji visited the Exhibition along with her family.
The Exhibition had a huge swing, which attracted many children. Raji found that the swing traced the path of a Parabola as given by y = x².

Answer the following questions using the above information:

Question. Let f : R → R be defined by f(x) = x² is ____
(a) Neither Surjective nor Injective
(b) Surjective
(c) Injective
(d) Bijective

Answer : A

Question. Let f : N → N be defined by f(x) = x² is _____
(a) Surjective but not Injective
(b) Surjective
(c) Injective
(d) Bijective

Answer : C

Question. Let f: {1, 2, 3,…} → {1, 4, 9, …} be defined by f(x) = x² is _______ .
(a) Bijective
(b) Surjective but not injective
(c) Injective but Surjective
(d) Neither Surjective nor Injective

Answer : A

Question. Let : N → R be defined by f(x) = x². Range of the function among the following is ______
(a) {1, 4, 9, 16, …}
(b) {1, 4, 8, 9, 10,…}
(c) {1, 4, 9, 15, 16,…}
(d) {1, 4, 8, 16,…}

Answer : A

Question. The function f : Z → Z defined by f(x) = x² is ________
(a) Neither Injective nor Surjective
(b) Injective
(c) Surjective
(d) Bijective

Answer : A
 

An organization conducted bike race under 2 different categories—boys and girls. Totally there was 250 participants. Among all of them finally three from Category 1 and two from Category 2 were selected for the final race. Ravi forms two sets B and G with these participants for this college project.
Let B = {b₁, b₂, b₃} and G = {g₁, g₂} where B
represents the set of boys selected and G the set of girls who were selected for the final race.

types of relations and functions.

Based on the above information answer the following:

Question. Ravi wishes to form all the relations possible from B to G. How many such relations are possible?
(a) 26
(b) 25
(c) 0
(d) 23

Answer : A

Question. Let R : B → B be defined by R = {(x, y) : x and y are students of same sex}, then this relation R is …………. .
(a) Equivalence
(b) Reflexive only
(c) Reflexive and symmetric but not transitive
(d) Reflexive and transitive but not symmetric

Answer : A

Question. Ravi wants to know among those relations, how many functions can be formed from B to G?
(a) 2²
(b) 2¹²
(c) 3²
(d) 2³

Answer : D

Question. Let R : B → G be defined by R = {(b₁, g₁), (b₂, g₂), (b₃, g₁)}, then R is ______ .
(a) Injective
(b) Surjective
(c) Neither Surjective nor Injective
(d) Surjective and Injective

Answer : B

Question. Ravi wants to find the number of injective functions from B to G. How many numbers of injective functions are possible?
(a) 0
(b) 2!
(c) 3!
(d) 0!

Answer : A