$\mathrm{If} {\sin }^{-1}\left(\mathrm{x}\right) - {\cos }^{-1}\left(\mathrm{x}\right) = \frac{\mathrm{\pi }}{6}, \mathrm{then} \mathrm{x} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\frac{1}{2}$

• $\frac{\sqrt{3}}{2}$

• $- \frac{1}{2}$

• $- \frac{\sqrt{3}}{2}$

${\sinh }^{-1}\left(2^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)$ is equal to

• $\log \left(3 + \sqrt{8}\right)$

• $\log \left(3 - \sqrt{8}\right)$

• $\log \left(2 + \sqrt{18}\right)$

• $\log \left(\sqrt{8} + \sqrt{27}\right)$

$\cos \left[{\cos }^{-1}\left(- \frac{1 }{7}\right) + {\sin }^{-1}\left(- \frac{1}{7}\right)\right] \mathrm{is} \mathrm{equal} \mathrm{to}$

• $- \frac{1}{3}$

• 0

• $\frac{1}{3}$

• $\frac{4}{9}$

$\mathrm{If} {\sin }^{-1}\left(\mathrm{x}\right) + {\sin }^{-1}\left(1 - \mathrm{x}\right) = {\cos }^{-1}\left(\mathrm{x}\right), \mathrm{then} \mathrm{x} \in \mathrm{to}$

• {1, 0}

• {- 1, 1}

• $\left\{0, \frac{1}{2}\right\}$

• {2, 0}

305.

${\sin }^{-1}\left(\frac{4}{5}\right) + 2{\tan }^{-1}\left(\frac{1}{3}\right)$ is equal to

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• 0

$2{\tanh }^{-1}\left(\frac{1}{2}\right)$ is equal to

• 0

• log(2)

• log(3)

• log(4)

The cartesian form of the polar equation $\mathrm{\theta } = {\tan }^{-1}\left(2\right) \mathrm{is}$

• x = 2y

• y = 2x

• x = 4y

• y = 4x

$$\begin{gathered} \mathrm{The} \mathrm{value} \mathrm{of} \mathrm{x}, \mathrm{where} \mathrm{x} > 0 \\ \mathrm{and} \tan \left({\sec }^{ - 1}\left(\frac{1}{\mathrm{x}}\right)\right) = \sin \left({\tan }^{-1}\left(2\right)\right) \mathrm{is} \end{gathered}$$

• $\sqrt{5}$

• $\frac{\sqrt{5}}{3}$

• 1

• $\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$

$\mathrm{If} {\sin }^{-1}\left(\frac{3}{\mathrm{x}}\right) + {\sin }^{-1}\left(\frac{4}{\mathrm{x}}\right) = \frac{\mathrm{\pi }}{2}, \mathrm{then} \mathrm{x} = ?$

• 3

• 5

• 7

• 11

${\cos }^{-1}\left(\frac{- 1}{2}\right) - 2{\sin }^{-1}\left(\frac{1}{2}\right) +3{\cos }^{-1}\left(\frac{- 1}{\sqrt{2}}\right) - 4{\tan }^{-1}\left(- 1\right) = ?$

• $\frac{19\mathrm{\pi }}{12}$

• $\frac{35\mathrm{\pi }}{12}$

• $\frac{47\mathrm{\pi }}{12}$

• $\frac{43\mathrm{\pi }}{12}$