CBSE Class 12 Mathematics Relations and Functions MCQs

Multiple Choice Questions (MCQs)

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 , a,b∈ T.Then R is
a) equivalence
b) reflexive but not transitive
c) transitive but not symmetric
d) none of these
Answer : A

Question : The maximum number of equivalence relations on the set A = {2, 3, 4} are
(a) 1
(b) 27
(c) 3
(d) 5
Answer : D

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 : 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 these
Answer : B

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 : If f : A → B and g : B → C be the bijective functions, then (gof)–1 is
(a) f^–1og^–1
(b) fog
(c) g^–1of^–1
(d) gof
Answer : A

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 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 these
Answer : C

Question : If f : R – {3/5} → R be defined by f (x) = 3x + 2/5x - 3 then
(a) f^–1(x) = f (x)
(b) f^–1(x) = –f (x)
(c) fof (x) = –x
(d) f^–1(x) = 1/19 f (x)
Answer : A

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 : 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 (gof) 3/2 is
(a) 1
(b) – 1
(c) 7/2
(d) 3
Answer : D

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

Question :  If f: R→ R given by f(x) =(3 − x³)^1/3, find f0f(x)
a) x
b) (3- x³)
c) x³
d) None of these
Answer : A

Question : Let A = {1,2,3}. The number of equivalence relations containing (1,2) is
a) 2
b) 3
c) 4
d) None of these
Answer : A

Question :  Let f:R→R defined by f(x) = x⁴. Choose the correct answer
a) f is neither one-one nor onto
b) f is oneone but not onto
c) f is many one onto
d) None of these
Answer : A

Question : Let f:R→R defined by f(x) = 3x. Choose the correct answer
a) ƒ 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 :  A

Question :  If A = {1,2,3}, B = {4,6,9} and R is a relation from A to B defined by ‘ x is smaller than y’. The range of R is
a) {4,6,9}
b) {1}
c) none of these
d) {1, 4,6,9}
Answer : A

Question : The relation R = { (1,1),(2,2),(3,3)} on {1,2,3} is
a) an equivalence relation
b) transitive only
c) reflexive only
d) None of these
Answer : A

Question : 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
a) reflexive but not symmetric
b) symmetric and transitive
c) reflexive but not transitive
d) neither symmetric nor transitive
Answer : A

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

Case Based Questions

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. Ravi decides to explore these sets for various types of relations and functions.

Based on the above information answer the following:

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 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

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 :  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 → 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

CASE STUDY

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}

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,F2) ∈R but (F3,F3) ∉ R
d. (F1,F2 ) ∉ R, (F2,F3) ∉ R and (F1,F3) ∉ R
Answer : A

Question : Two neighbors 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 : 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.’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 : 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

CASE STUDY

Consider the mapping f : A → B is defined by f(x) = 𝑥 − 1/𝑥 – 2such that f is a bijection. Based on the above information, answer the following questions:

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

Question : Domain off is
(a) R – {2}
(b) R
(c) R – {1, 2}
(d) R – {0}
Answer : A

Question : If g : R – {2} → R – {1} is defined by g(x) = 2f(x) – 1, then g(x) in terms of x is
(a) 𝑥 + 2/𝑥
(b) 𝑥 + 1/𝑥 – 2
(c) 𝑥 − 2/𝑥
(d) 𝑥/𝑥 – 2
Answer : D

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

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