Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| Function | Domain | ||
| (A) | \(\cos^{-1}\left(\displaystyle\frac{x}{2}\right)\) | (I) | \([-2, -1]\) |
| (B) | \(\sin^{-1}(x - 1)\) | (II) | \(\mathrm{R} - (-1, 1)\) |
| (C) | \(\sec^{-1} x\) | (III) | \([-2, 2]\) |
| (D) | \(\sin^{-1}(2x + 3)\) | (IV) | \([0, 2]\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A)-(III), (B)-(IV), (C)-(II), (D)-(I)
\(\sin^{-1} t\) and \(\cos^{-1} t\) need the argument in \([-1, 1]\); \(\sec^{-1} t\) needs \(t \le -1\) or \(t \ge 1\).
(A) \(\cos^{-1}\left(\frac{x}{2}\right)\)
This matches (III) \([-2, 2]\):
\( \displaystyle -1 \le \frac{x}{2} \le 1 \)
\(-2 \le x \le 2\)
(B) \(\sin^{-1}(x - 1)\)
This matches (IV) \([0, 2]\):
\(-1 \le x - 1 \le 1\)
\(0 \le x \le 2\)
(C) \(\sec^{-1} x\)
This matches (II) \(\mathrm{R} - (-1, 1)\): the secant never takes a value strictly between \(-1\) and 1, since \(|\sec\theta| = \frac{1}{|\cos\theta|} \ge 1\).
(D) \(\sin^{-1}(2x + 3)\)
This matches (I) \([-2, -1]\):
\(-1 \le 2x + 3 \le 1\)
\(-4 \le 2x \le -2\)
\(-2 \le x \le -1\)