Class XII Maths Relations and Functions MCQs (Difficulty-wise) — Answers & Explanations
Use this page if you want a clean revision flow: Easy (definitions), Medium (apply properties), and Hard (mixed reasoning). Each question has a short explanation, so you can quickly spot the exact definition you forgot.
For a timed mixed test, use the mixed page linked at the end.
Easy (Definitions First)
1) A relation $R$ on $A$ is symmetric if
A. $(a,a)\in R$ for all $a$
B. $(a,b)\in R\Rightarrow (b,a)\in R$
C. $(a,b)\in R$ and $(b,c)\in R\Rightarrow (a,c)\in R$
D. $R=\varnothing$
Correct Answer: B
Explanation: That is the definition of symmetric.
2) A relation is an equivalence relation if it is
A. reflexive, symmetric, transitive
B. reflexive, antisymmetric, transitive
C. symmetric, antisymmetric, transitive
D. only transitive
Correct Answer: A
3) If $f: \mathbb{R}\to\mathbb{R}$ is $f(x)=3x$, then $f$ is
A. one-one and onto
B. one-one but not onto
C. onto but not one-one
D. neither
Correct Answer: A
Explanation: Non-zero linear function on $\mathbb{R}$ is bijective.
4) The number of functions from a set with $m$ elements to a set with $n$ elements is
A. $m^n$
B. $n^m$
C. $m+n$
D. $m\cdot n$
Correct Answer: B
5) If $f:X\to Y$ has an inverse function $f^{-1}$ (as a function), then $f$ must be
A. onto only
B. one-one only
C. bijective
D. constant
Correct Answer: C
6) On $\mathbb{R}$, $aRb \iff a\le b$ is
A. symmetric
B. reflexive and transitive but not symmetric
C. not reflexive
D. an equivalence relation
Correct Answer: B
Revision Tip: For relation questions, write three mini-checks: reflexive? symmetric? transitive? Don’t guess.
Medium (Apply Properties)
7) On $A={1,2,3,4,5,6}$, define $xRy$ iff $x$ divides $y$. Then $R$ is
A. reflexive and transitive but not symmetric
B. symmetric and transitive but not reflexive
C. an equivalence relation
D. not transitive
Correct Answer: A
8) If $f:\mathbb{N}\to\mathbb{N}$ is $f(x)=2x$, then $f$ is
A. onto
B. one-one but not onto
C. onto but not one-one
D. neither
Correct Answer: B
Explanation: Odd naturals have no preimage.
9) If $X$ is a finite set and $f:X\to X$ is one-one, then $f$ is
A. necessarily onto
B. never onto
C. always constant
D. many-one
Correct Answer: A
10) The relation “is perpendicular to” on all lines in a plane is
A. reflexive
B. symmetric but not transitive
C. transitive
D. an equivalence relation
Correct Answer: B
11) If $f:X\to Y$ and $g:Y\to Z$, then $g\circ f$ maps
A. $Y\to X$
B. $X\to Z$
C. $Z\to X$
D. $X\to Y$
Correct Answer: B
12) If $R$ is an equivalence relation on $X$, then the equivalence classes of $R$
A. overlap
B. form a partition of $X$
C. never contain more than 2 elements
D. exist only for finite sets
Correct Answer: B
Hard (Mixed Reasoning)
13) If $f:\mathbb{R}\to\mathbb{R}$ is $f(x)=x^2$, then $f$ is
A. bijective
B. one-one only
C. onto only
D. neither one-one nor onto
Correct Answer: D
14) On ${1,2,3}$ define $(a,b)\in R$ iff $a\equiv b\pmod{2}$. Then the equivalence class of 1 is
A. ${1}$
B. ${2}$
C. ${1,3}$
D. ${1,2,3}$
Correct Answer: C
Explanation: 1 is congruent mod 2 to odd numbers.
15) If $|X|=3$ and $|Y|=2$, then number of onto functions $X\to Y$ is
A. 6
B. 8
C. 2
D. 0
Correct Answer: A
Explanation: Total functions $=2^3=8$. Not onto means all map to same element: 2 such functions. So onto $=8-2=6$.
16) If $f:X\to Y$ is bijective, then $f^{-1}:Y\to X$ is
A. not a function
B. a bijection
C. many-one
D. never onto
Correct Answer: B
Explanation: Inverse of a bijection is also a bijection.
17) A relation that is reflexive and transitive but not symmetric is called a
A. partial order
B. pre-order
C. symmetric order
D. universal relation
Correct Answer: B
Explanation: Preorder = reflexive + transitive. (Partial order additionally needs antisymmetry.)
18) If $A={1,2,3}$, number of relations on $A$ is
A. $2^3$
B. $2^6$
C. $2^9$
D. $3^2$
Correct Answer: C
Explanation: $|A\times A|=9$. Any subset of $A\times A$ is a relation.