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Relations and Functions Set-2 MCQ Online Practice Test | Class 12 | Quiz Tweek

Relations and Functions Set-2

Practice Important MCQs for Relations and Functions — Mathematics, Class 12.

In CBSE and competitive exams, “Relations and Functions” is a foundational chapter that builds key skills for understanding injective/surjective/bijective mappings, inverse functions, and function compositions. Many conceptual problems also appear in JEE/NEET and board exams in the form of checking injectivity/surjectivity, finding inverses, and using functional equations—so strong clarity here directly improves scoring accuracy.

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Q1. Let $f:\mathbb{R}\to\mathbb{R}$ be defined by $f(x)=x^3+x$ . Which of the following is true about $f$ ?

(A)

$f$ is one-one but not onto.

(B)

$f$ is bijective.

(C)

$f$ is onto but not one-one.

(D)

$f$ is neither injective nor surjective.

Q2. Let $f:\mathbb{R}\to(-1,1)$ be defined by $f(x)=\dfrac{x}{1+|x|}$ . Then which of the following is correct?

(A)

$f$ is injective but not surjective; its range is $[-1,1)$ .

(B)

$f$ is surjective but not injective; its range is $(-1,1)$ .

(C)

$f$ is bijective and $f^{-1}(y)=\dfrac{y}{1-|y|}$ for $y\in(-1,1)$ .

(D)

$f$ is bijective;

f^{-1}(y)=\begin{aligned}\dfrac{y}{1+y},&\quad y\ge 0 <br>\dfrac{y}{1-y},&\quad y<0\end{aligned}

Q3. Let $f:\mathbb{R}\setminus\{0\}\to\mathbb{R}$ be defined by $f(x)=ax+\dfrac{1}{x}$ , where $a\in\mathbb{R}$ . For which real values of $a$ is $f$ injective?

(A)

$a=0$ only

(B)

all $a\neq 0$

(C)

all real $a$

(D)

no real value of $a$

Q4. Assertion (A): Let $f:S\to S$ satisfy $f(f(x))=f(x)$ for all $x\in S$ . Then every element of $\operatorname{Im}(f)$ is a fixed point of $f$ , i.e. for every $y\in\operatorname{Im}(f)$ we have $f(y)=y$ .
Reason (R): If $y\in\operatorname{Im}(f)$ then there exists $x\in S$ with $y=f(x)$ . Hence $f(y)=f(f(x))=f(x)=y$ .
(opt1) Both (A) and (R) are true but (R) does not correctly explain (A).

(A)

Both (A) and (R) are true but (R) does not correctly explain (A).

(B)

(A) is true and (R) is false.

(C)

Both (A) and (R) are true and (R) correctly explains (A).

(D)

(A) is false but (R) is true.

Q5. Let $f:\mathbb{R}\to\mathbb{R}$ be defined by

f(x)=\begin{aligned}x,&\quad x\in\mathbb{Q} <br> x+1,&\quad x\notin\mathbb{Q}\end{aligned}

Which of the following statements is correct?

(A)

$f$ is injective but not surjective.

(B)

$f$ is bijective;

f^{-1}(y)=\begin{aligned}y,&\quad y\in\mathbb{Q} <br> y-1,&\quad y\notin\mathbb{Q}\end{aligned}

(C)

$f$ is surjective but not injective.

(D)

$f$ is neither injective nor surjective.

Q6. Let $f:\mathbb{R}\setminus\{1\}\to\mathbb{R}\setminus\{2\}$ be defined by $f(x)=\dfrac{2x+3}{x-1}$ . Then $f^{-1}(x)=$

(A)

$\,\dfrac{3x-2}{x+1},\quad x\neq -1$

(B)

$\,\dfrac{x-3}{x-2},\quad x\neq 2$

(C)

$\,\dfrac{x+3}{x-2},\quad x\neq 2$

(D)

$\,\dfrac{x+2}{2-x},\quad x\neq 2$

Q7. Let $f:\mathbb{R}\setminus\{0\}\to\mathbb{R}$ be defined by $f(x)=\dfrac{x+1}{x}$ . The number of real $x$ in the domain such that $f(f(x))=x$ is

(A)

$2$

(B)

$1$

(C)

$\infty$

(D)

$0$

Q8. Let $f:\mathbb{R}\to(-1,1)$ be defined by $f(x)=\dfrac{x}{1+|x|}$ . The inverse $f^{-1}:(-1,1)\to\mathbb{R}$ is given by

(A)

$f^{-1}(y)=\dfrac{y}{1-y}\quad\text{for all }y\in(-1,1)$

(B)

\begin{aligned} f^{-1}(y)=\dfrac{y}{1-y},&\quad y<0 <br> f^{-1}(y)=\dfrac{y}{1+y},&\quad y\ge 0 \end{aligned}

(C)

$f^{-1}(y)=\dfrac{y}{1+y}\quad\text{for all }y\in(-1,1)$

(D)

\begin{aligned} f^{-1}(y)=\dfrac{y}{1-y},&\quad y\ge 0 <br> f^{-1}(y)=\dfrac{y}{1+y},&\quad y<0 \end{aligned}

Q9. Consider functions $f:A\to B$ and $g:B\to C$ .
Assertion (A): If $g\circ f$ is bijective then both $f$ and $g$ are bijective.
Reason (R): If $g\circ f$ is bijective then $f$ is injective and $g$ is surjective.

(A)

Both (A) and (R) are true and (R) is a correct explanation of (A)

(B)

(A) is false and (R) is true

(C)

Both (A) and (R) are true but (R) is not a correct explanation of (A)

(D)

(A) is true and (R) is false

Q10. Let $\mathbb{Z}_{12}=\{[0],[1],\dots,[11]\}$ . For $a,b\in\{0,1,\dots,11\}$ define $f:\mathbb{Z}_{12}\to\mathbb{Z}_{12}$ by $f([x])=[ax+b]$ . The number of ordered pairs $(a,b)$ for which $f$ is a bijection is

(A)

$48$

(B)

$36$

(C)

$144$

(D)

$24$

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