Match List-I with List-II, List-I (Functions), List-II (Range, principal value), (A) ^(-1)x, (I) [0,]-\()/2\, (B) ^(-1)x, (II) [-()/2,()/2]-\0\, (C) cosec^(-1)x, (III) [-()/2,()/2], (D) ^(-1)x, (IV) (-()/2,()/2), Choose the correct answer from the options given below: Option 1: (A) - (IV), (B) - (III), (C) - (I), (D) - (II) Option 2: (A) - (III), (B) - (IV), (C) - (II), (D) - (I) Option 3: (A) - (I), (B) - (II), (C) - (III), (D) - (IV) Option 4: (A) - (III), (B) - (I), (C) - (II), (D) - (IV)

Answer: (A) - (III), (B) - (IV), (C) - (II), (D) - (I). Solution: Principal value branches: ^(-1)x has the closed interval [-()/2,()/2]; ^(-1)x has the open interval; cosec^(-1)x is the sine branch with 0 removed; ^(-1)x is [0,] with ()/2 removed.

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Match List-I with List-II

List-I (Functions) List-II (Range, principal value)
(A) $\sin^{-1}x$ (I) $[0,\pi]-\left\{\dfrac{\pi}{2}\right\}$
(B) $\tan^{-1}x$ (II) $\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]-\{0\}$
(C) $cosec^{-1}x$ (III) $\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]$
(D) $\sec^{-1}x$ (IV) $\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$

Choose the correct answer from the options given below:

List-I (Functions)	List-II (Range, principal value)
(A) $\sin^{-1}x$	(I) $[0,\pi]-\left\{\dfrac{\pi}{2}\right\}$
(B) $\tan^{-1}x$	(II) $\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]-\{0\}$
(C) $cosec^{-1}x$	(III) $\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]$
(D) $\sec^{-1}x$	(IV) $\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$

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

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

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

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

✅ Correct Option: 2

Principal value branches: $\sin^{-1}x$ has the closed interval $\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]$ ; $\tan^{-1}x$ has the open interval; $cosec^{-1}x$ is the sine branch with $0$ removed; $\sec^{-1}x$ is $[0,\pi]$ with $\dfrac{\pi}{2}$ removed.

2026: 22 May Shift 1

Match List-I with List-II

List-I (Function)	List-II (Principal value of y)
(A) $y=\sin^{-1}\left(-\frac{1}{2}\right)$	(I) $\frac{3\pi}{4}$
(B) $y=\cot^{-1}\left(-\frac{1}{\sqrt3}\right)$	(II) $-\frac{\pi}{4}$
(C) $y=cosec^{-1}(-\sqrt2)$	(III) $-\frac{\pi}{6}$
(D) $y=\cos^{-1}\left(-\frac{1}{\sqrt2}\right)$	(IV) $\frac{2\pi}{3}$

Choose the correct answer from the options given below:

2022: 17 Aug Shift 2

The value of $\sin\left[\frac{\pi}{2} - \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right]$ is :

2026: 15 May Shift 1

The value of $\tan\left(\dfrac{1}{2}\cos^{-1}\dfrac{\sqrt5}{3}\right)$ is equal to

2023: 23 May Shift 3

The principal value of $\cot^{-1}\left(-\frac{1}{\sqrt{3}}\right)$

2025: 22 May Shift 2

The value of $\cot\left(\cos^{-1}\frac{7}{25}\right)$ is

2023: 30 May Shift 3

Let $a \leq \tan^{-1} x + \cot^{-1} x + \sin^{-1} x \leq b$ . If $\alpha$ and $\beta$ denote the minimum and maximum possible values of a and b respectively, then :

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