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Relations and Functions Set-1 MCQ Online Practice Test | Class 12 | Quiz Tweek

Relations and Functions Set-1

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

Relations and functions form the backbone of Class 12 Mathematics, linking algebraic ideas with graph behavior, inverse mappings, and domain–range reasoning. They are heavily tested in board exams for understanding “one-one/onto” properties and in competitive exams through function composition, injectivity–surjectivity conditions, and functional equations like $f(f(x))=x$ . Mastery here also strengthens your approach to many JEE/NEET-style questions involving monotonicity, inverses, and bijections.

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

(A)

$f$ is one-one but not onto.

(B)

$f$ is both one-one and onto.

(C)

$f$ is onto but not one-one.

(D)

$f$ is neither one-one nor onto.

Q2. Let $f:\mathbb{R}\to\mathbb{R}$ be given by $f(x)=\dfrac{x}{1+x^2}$ . Which of the following correctly describes injectivity and range of $f$ ?

(A)

$f$ is one-one and range is $\mathbb{R}$ .

(B)

$f$ is one-one and range is $\left[-\tfrac{1}{2},\tfrac{1}{2}\right]$ .

(C)

$f$ is not one-one and range is $\left(-\tfrac{1}{2},\tfrac{1}{2}\right)$ .

(D)

$f$ is not one-one and range is $\left[-\tfrac{1}{2},\tfrac{1}{2}\right]$ .

Q3. For which real values of $a$ is the function $f:\mathbb{R}\to\mathbb{R}$ defined by $f(x)=x^3+ax$ one-one on $\mathbb{R}$ ?

(A)

$a\ge 0$

(B)

$a>0$

(C)

$a\le 0$

(D)

all real $a$

Q4. Let $S=\{1,2,3,4,5,6\}$ . How many functions $f:S\to S$ satisfy $f(f(x))=x$ for every $x\in S$ ?

(A)

$64$

(B)

$90$

(C)

$76$

(D)

$120$

Q5. Let $f:\mathbb{R}\to\mathbb{R}$ be continuous and satisfy $f(f(x))=x$ for all $x\in\mathbb{R}$ . Which one of the following statements must necessarily be true?

(A)

$f$ is strictly increasing on $\mathbb{R}$ .

(B)

$f(x)=x$ for all $x\in\mathbb{R}$ .

(C)

$f$ has exactly one fixed point.

(D)

$f$ is bijective (one-one and onto).

Q6. 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 not injective.

(B)

$f$ is injective but not surjective.

(C)

$f$ is bijective.

(D)

$f$ is neither injective nor surjective.

Q7. Let $f:\mathbb{R}\setminus\{-2\}\to\mathbb{R}$ be defined by $f(x)=\dfrac{2x+3}{x+2}$ . Which statement is correct?

(A)

$f$ is injective and its range is $\mathbb{R}$ .

(B)

$f$ is not injective.

(C)

The range of $f$ is $\mathbb{R}\setminus\{-2\}$ .

(D)

$f$ is a bijection from $\mathbb{R}\setminus\{-2\}$ onto $\mathbb{R}\setminus\{2\}$ .

Q8. Let $f:\mathbb{R}\to\mathbb{R}$ be given by $f(x)=\dfrac{x}{1+|x|}$ . Which of the following correctly describes $f$ ?

(A)

$f$ is a bijection from $\mathbb{R}$ onto $(-1,1)$ .

(B)

$f$ is injective and its range is $(-1,0)\cup(0,1)$ .

(C)

$f$ is surjective onto $(-1,1)$ but not injective.

(D)

$f$ attains the maximum $1$ and minimum $-1$ .

Q9. Assertion (A): Let $f:A\to B$ and $g:B\to C$ be functions. If $g\circ f$ is bijective then $f$ is injective.
Reason (R): If $g\circ f$ is bijective then $g$ is surjective.
Choose the correct option:

(A)

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

(B)

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

(C)

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

(D)

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

Q10. Let $f:\mathbb{Z}_{12}\to\mathbb{Z}_{12}$ be defined by $f([x])=[ax+b]\pmod{12}$ with $a,b\in\{0,1,\dots,11\}$ . For how many ordered pairs $(a,b)$ is $f$ a bijection of $\mathbb{Z}_{12}$ ?

(A)

$12$

(B)

$24$

(C)

$48$

(D)

$144$

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