Algebra

Relations & Functions

Medium

core

If $f(x) = \sin^{-1}\left(\dfrac{2^{x+1}}{1+4^x}\right)$ , then

Match List-I with List-II

List-I List-II
Value of $x$ $\dfrac{1}{2}f(x)$
(A) $x = 0$ (I) $\tan^{-1}\sqrt{2}$
(B) $x = \dfrac{1}{2}$ (II) $\tan^{-1}4$
(C) $x = 1$ (III) $\dfrac{\pi}{4}$
(D) $x = 2$ (IV) $\tan^{-1}2$

Choose the correct answer from the options given below:

List-I	List-II
Value of $x$	$\dfrac{1}{2}f(x)$
(A) $x = 0$	(I) $\tan^{-1}\sqrt{2}$
(B) $x = \dfrac{1}{2}$	(II) $\tan^{-1}4$
(C) $x = 1$	(III) $\dfrac{\pi}{4}$
(D) $x = 2$	(IV) $\tan^{-1}2$

(A) - (III), (B) - (I), (C) - (IV), (D) - (II)

(A) - (III), (B) - (II), (C) - (IV), (D) - (I)

(A) - (I), (B) - (II), (C) - (III), (D) - (IV)

(A) - (IV), (B) - (III), (C) - (II), (D) - (I)

✅ Correct Option: 1

Put $t=2^x$ . Then $\frac{2^{x+1}}{1+4^x} = \frac{2t}{1+t^2} = \sin(2\tan^{-1}t)$ , so $f(x) = 2\tan^{-1}(2^x)$ and $\frac{1}{2}f(x) = \tan^{-1}(2^x)$ .

Hence $x=0\to\tan^{-1}1=\frac{\pi}{4}$ , $x=\frac{1}{2}\to\tan^{-1}\sqrt{2}$ , $x=1\to\tan^{-1}2$ , $x=2\to\tan^{-1}4$ .

2025: 13 May Shift 2

Let L be the set of all lines in a plane and R be the relation on set L defined by $R = \{(L_1, L_2): L_1 \perp L_2\}$ Then R is

(A) an equivalence Relation

(B) a symmetric Relation

(D) a reflexive Relation

Choose the correct answer from the options given below:

2026: 12 May Shift 2

Let $A$ be set of all real numbers and $R$ be a relation on $A$ defined by $R=\{(a,b): 1+ab>0\}$ , then relation $R$ is

2025: 14 May Shift 2

Let a relation R = {(a, b) : a is a factor of b, a, b $\in$ N}. Then, R is ______.

2022: 23 Aug Shift 1

Let $R : \mathbb{R} \to \mathbb{R}$ , where $\mathbb{R}$ is the set of real numbers and R be a relation. An element $(x, y) \in R$ if $x + y - \sqrt{2}$ belongs to the set of irrational numbers. Then the relation R is :

2025: 3 June Shift 2

The relation R on the set of real numbers defined by $R = \{(a, b): a \leq b^2\}$ is

2026: 19 May Shift 2

Match the LIST-I with LIST-II

[Where R and Z denotes set of real numbers and set of integers respectively]

	LIST-I Function		LIST-II Nature
A.	$f: R \to R, f(x) = \sin^2 x + \cos^2 x$	I.	Bijective
B.	$f: Z \to Z, f(x) = x^3$	II.	Surjective but not injective
C.	$f: R \to R, f(x) = x^3 - x$	III.	Injective but not surjective
D.	$f: R \to R, f(x) = 3 - 4x$	IV.	Neither injective nor surjective

Choose the correct answer from the options given below:

Solution

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