Class XII Maths Relations and Functions MCQs (Mixed) — Answers & Explanations
In Class XII, Relations and Functions questions often look “theoretical”, but they are very scoring if your definitions are sharp: reflexive/symmetric/transitive, equivalence classes, one-one/onto, composition, and inverse.
This mixed practice set is designed like a quick self-test: each MCQ has one correct option and a short explanation that links directly to the definition or property being used.
If you prefer a structured Easy → Medium → Hard set, use the difficulty-wise page linked at the end.
MCQs (Mixed Practice)
1) A relation $R$ on a set $A$ is reflexive if
A. $(a,b)\in R\Rightarrow (b,a)\in R$
B. $(a,a)\in R$ for every $a\in A$
C. $(a,b)\in R$ and $(b,c)\in R\Rightarrow (a,c)\in R$
D. $R=A\times A$
Correct Answer: B
Explanation: Reflexive means every element relates to itself.
2) On $\mathbb{R}$, the relation $aRb \iff a\le b$ is
A. reflexive and symmetric
B. symmetric and transitive
C. reflexive and transitive but not symmetric
D. an equivalence relation
Correct Answer: C
Explanation: $a\le a$ so reflexive; and $a\le b, b\le c \Rightarrow a\le c$ so transitive; not symmetric.
3) Relation “is perpendicular to” on the set of all lines in a plane is
A. reflexive
B. symmetric but not transitive
C. transitive
D. an equivalence relation
Correct Answer: B
Explanation: If $L_1\perp L_2$ then $L_2\perp L_1$ (symmetric). It’s not transitive.
4) A relation is an equivalence relation if it is
A. reflexive, symmetric, transitive
B. reflexive, antisymmetric, transitive
C. symmetric only
D. transitive only
Correct Answer: A
Explanation: That’s the definition.
5) 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 one-one nor onto
Correct Answer: A
Explanation: Non-zero linear function on $\mathbb{R}$ is bijective.
Revision Tip: For onto, always ask: “Given any $y$ in codomain, can I solve $f(x)=y$ for some $x$ in domain?”
6) The function $f: \mathbb{R}\to\mathbb{R}$, $f(x)=x^2$ is
A. one-one and onto
B. one-one but not onto
C. onto but not one-one
D. neither one-one nor onto
Correct Answer: D
Explanation: Not one-one ($f(1)=f(-1)$) and not onto $\mathbb{R}$ (no negative outputs).
7) If $A$ has $n$ elements, the number of equivalence relations on $A$ equals
A. $2^{n^2}$
B. $n!$
C. the number of partitions of $A$
D. $n^n$
Correct Answer: C
Explanation: Equivalence relations ↔ partitions of the set.
8) If $f:X\to Y$ is one-one and $X$ is finite, then
A. $|X|<|Y|$ always
B. $|X|\le |Y|$
C. $|X|>|Y|$
D. $|X|=|Y|$ always
Correct Answer: B
Explanation: Injective mapping can’t “fit” more elements into fewer images.
9) If $f: X\to X$ is one-one and $X$ is a finite set, then $f$ is
A. necessarily onto
B. necessarily constant
C. necessarily many-one
D. never onto
Correct Answer: A
Explanation: On a finite set, injective implies surjective.
10) On $A={1,2,3,4,5,6}$, define $xRy$ if $x$ divides $y$. Then $R$ is
A. reflexive and transitive but not symmetric
B. symmetric and transitive but not reflexive
C. an equivalence relation
D. neither reflexive nor transitive
Correct Answer: A
Explanation: $x|x$ (reflexive), divisibility is transitive, not symmetric.
11) A function $f:X\to Y$ has an inverse function $f^{-1}$ (as a function) if and only if $f$ is
A. onto
B. one-one
C. bijective
D. constant
Correct Answer: C
Explanation: Inverse exists as a function only for bijections.
12) If $f:X\to Y$ and $g:Y\to Z$, then $g\circ f$ is defined from
A. $Y$ to $X$
B. $X$ to $Z$
C. $Z$ to $X$
D. $X$ to $Y$
Correct Answer: B
Explanation: Composition takes input from the first function’s domain and outputs in the second’s codomain.
13) Let $R$ be the relation on ${1,2,3}$ defined by $(a,b)\in R$ iff $a\equiv b\pmod{2}$. Then $R$ is
A. not reflexive
B. not symmetric
C. an equivalence relation
D. not transitive
Correct Answer: C
Explanation: Congruence mod 2 is reflexive, symmetric, transitive.
14) The number of functions from a set with 3 elements to a set with 2 elements is
A. $2^3$
B. $3^2$
C. $2\cdot 3$
D. $3!$
Correct Answer: A
Explanation: If $|X|=m$, $|Y|=n$, then number of functions $X\to Y$ is $n^m$.
15) 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 one-one nor onto
Correct Answer: B
Explanation: Injective (different inputs give different outputs), but odd numbers have no preimage.