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

Answer: A-I, B-IV, C-III, D-II. Solution: Standard principal value ranges: ^(-1): [0, ] (I); ^(-1): (-/2, /2) (IV); ^(-1): [-/2, /2] - \0\ (III); ^(-1): [0, ] - \/2\ (II). So A-I, B-IV, C-III, D-II.

Geometry

Trigonometry

Easy

Match List I with List II

List I (Functions) List II (Principal value branches)
A. $f(x) = \cos^{-1} x$ I. $[0, \pi]$
B. $f(x) = \tan^{-1} x$ II. $[0, \pi] - \left\{\frac{\pi}{2}\right\}$
C. $f(x) = \text{cosec}^{-1} x$ III. $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] - \{0\}$
D. $f(x) = \sec^{-1} x$ IV. $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$

Choose the correct answer from the options given below:

List I (Functions)	List II (Principal value branches)
A. $f(x) = \cos^{-1} x$	I. $[0, \pi]$
B. $f(x) = \tan^{-1} x$	II. $[0, \pi] - \left\{\frac{\pi}{2}\right\}$
C. $f(x) = \text{cosec}^{-1} x$	III. $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] - \{0\}$
D. $f(x) = \sec^{-1} x$	IV. $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$

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

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

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

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

✅ Correct Option: 2

Standard principal value ranges: $\cos^{-1}: [0, \pi]$ (I); $\tan^{-1}: (-\pi/2, \pi/2)$ (IV); $\csc^{-1}: [-\pi/2, \pi/2] - \{0\}$ (III); $\sec^{-1}: [0, \pi] - \{\pi/2\}$ (II). So A-I, B-IV, C-III, D-II.

2023: 23 May Shift 3

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

2026: 29 May Shift 2

Match the LIST-I with LIST-II

	LIST-I		LIST-II
A.	If $\cos^{-1}\alpha + \cos^{-1}\beta + \cos^{-1}\gamma = 3\pi$ then $\alpha(2\beta+\gamma) + \beta(2\gamma+\alpha) + \gamma(2\alpha+\beta)$ equals	I.	$1$
B.	The value of $\tan^2(\sec^{-1}4) + \cot^2(\text{cosec}^{-1}5)$ is	II.	$9$
C.	If $\sin\left(\cos^{-1}\dfrac{2}{5} + \sin^{-1}x\right) = 1$ then $10x$ equals	III.	$4$
D.	The length of the interval of domain of $\sin^{-1}2x$ is	IV.	$39$

Choose the correct answer from the options given below:

2022: 30 Aug Shift 1

The value of $\sin\left[2\cot^{-1}\left(\frac{-5}{12}\right)\right]$ is :

2026: 18 May Shift 2

Match the LIST-I with LIST-II

LIST-I Functions	LIST-II Principal value
A. $\sin^{-1}\left(\dfrac{1}{2}\right)$	I. $\dfrac{\pi}{4}$
B. $\sin^{-1}\left(-\dfrac{1}{\sqrt{2}}\right)$	II. $\dfrac{\pi}{6}$
C. $\tan^{-1}(1)$	III. $-\dfrac{\pi}{4}$
D. $\cos^{-1}\left(-\dfrac{1}{2}\right)$	IV. $\dfrac{2\pi}{3}$

Choose the correct answer from the options given below:

2023: 23 May Shift 3

The simplest form of $\tan^{-1}\left\{\frac{x}{\sqrt{a^2 - x^2}}\right\}$ is, where $-a < x < a$ .

2023: 30 May Shift 3

The maximum value of $(\sin x)(\cos x)$ is :

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