Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| Expression | Principal value | ||
| (A) | \(\cot^{-1}\left(\sqrt{3}\right)\) | (I) | \(\displaystyle\frac{\pi}{3}\) |
| (B) | \(\sec^{-1}(2)\) | (II) | \(\displaystyle\frac{5\pi}{6}\) |
| (C) | \(\mathrm{cosec}^{-1}\left(-\displaystyle\frac{2}{\sqrt{3}}\right)\) | (III) | \(-\displaystyle\frac{\pi}{3}\) |
| (D) | \(\cos^{-1}\left(-\displaystyle\frac{\sqrt{3}}{2}\right)\) | (IV) | \(\displaystyle\frac{\pi}{6}\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A)-(IV), (B)-(I), (C)-(III), (D)-(II)
Each value must lie in the principal value branch: \((0, \pi)\) for \(\cot^{-1}\), \([0, \pi] - \left\{\frac{\pi}{2}\right\}\) for \(\sec^{-1}\), \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] - \{0\}\) for \(\mathrm{cosec}^{-1}\) and \([0, \pi]\) for \(\cos^{-1}\).
(A) \(\cot^{-1}\left(\sqrt{3}\right)\)
This matches (IV) \(\frac{\pi}{6}\):
\( \displaystyle \cot\frac{\pi}{6} = \sqrt{3} \)
(B) \(\sec^{-1}(2)\)
This matches (I) \(\frac{\pi}{3}\):
\( \displaystyle \sec\frac{\pi}{3} = \frac{1}{\cos\frac{\pi}{3}} = 2 \)
(C) \(\mathrm{cosec}^{-1}\left(-\frac{2}{\sqrt{3}}\right)\)
This matches (III) \(-\frac{\pi}{3}\), a negative angle because the number is negative:
\( \displaystyle \mathrm{cosec}\left(-\frac{\pi}{3}\right) = \frac{1}{\sin\left(-\frac{\pi}{3}\right)} = -\frac{2}{\sqrt{3}} \)
(D) \(\cos^{-1}\left(-\frac{\sqrt{3}}{2}\right)\)
This matches (II) \(\frac{5\pi}{6}\), the second-quadrant angle in \([0, \pi]\):
\( \displaystyle \cos\frac{5\pi}{6} = \cos\left(\pi - \frac{\pi}{6}\right) = -\frac{\sqrt{3}}{2} \)