$\mathrm{If} {\tan }^{-1}\left(\mathrm{x}\right) + {\tan }^{-1}\left(\mathrm{y}\right) + {\tan }^{-1}\left(\mathrm{z}\right) = \frac{\mathrm{\pi }}{2}, \mathrm{then} 1 - \mathrm{xy} - \mathrm{yz} - \mathrm{zx} = ?$

• 1

• 0

• - 1

• 2

$\mathrm{If} {\tanh }^{-1}\left(\mathrm{x}\right) = \mathrm{alog}\left(\frac{1 + \mathrm{x}}{1 - \mathrm{x}}\right), \left|\mathrm{x}\right| < 1, \mathrm{then} \mathrm{a} = ?$

• 1

• 2

• $\frac{1}{2}$

• $\frac{1}{4}$

${\left({\tan }^{-1}\left(\mathrm{x}\right)\right)}^2 + {\left({\cot }^{-1}\left(\mathrm{x}\right)\right)}^2 = \frac{5{\mathrm{\pi }}^2}{8} \Rightarrow \mathrm{x} =$

• - 1

• 1

• 0

• $\mathrm{\pi }\sqrt{\frac{5}{8}}$

$\mathrm{For} 0 < \mathrm{x} \le \mathrm{\pi }, {\sinh }^{-1}\left(\cot \left(\mathrm{x}\right)\right) = ?$

• $\log \left(\cot \left(\frac{\mathrm{x}}{2}\right)\right)$

• $\log \left(\tan \left(\frac{\mathrm{x}}{2}\right)\right)$

• $\log \left(1 + \cot \left(\mathrm{x}\right)\right)$

• $\log \left(1 + \tan \left(\mathrm{x}\right)\right)$

$$\begin{gathered} \mathrm{If} \frac{1}{2} \le \mathrm{x} \le 1, \mathrm{then} \\ {\cos }^{-1}\left(\mathrm{x}\right) + {\cos }^{-1}\left(\frac{\mathrm{x}}{2} + \frac{1}{2}\sqrt{3 - 3{\mathrm{x}}^2}\right) = ? \end{gathered}$$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{3}$

• $\mathrm{\pi }$

• 0

$\mathrm{If} {\cos }^{-1}\left(\frac{5}{13}\right) + {\cos }^{-1}\left(\frac{3}{5}\right) = {\cos }^{-1}\left(\mathrm{x}\right), \mathrm{then} \mathrm{x} = ?$

• $\frac{3}{63}$

• $\frac{- 36}{65}$

• $\frac{- 33}{65}$

• - 1

${\tanh }^{-1}\left(\frac{1}{2}\right) + {\coth }^{-1}\left(2\right) = ?$

• $\frac{1}{3}\log \left(3\right)$

• $\frac{1}{2}\log \left(6\right)$

• $\frac{1}{2}\log \left(12\right)$

• $\log \left(3\right)$

$$\begin{gathered} \mathrm{If} \mathrm{x} > 0, \mathrm{y} > 0, \mathrm{z} > 0, \mathrm{xy} + \mathrm{yz} + \mathrm{zx} < 1 \mathrm{and} \mathrm{if} {\tan }^{-1}\left(\mathrm{x}\right) + {\tan }^{-1}\left(\mathrm{y}\right) + {\tan }^{-1}\left(\mathrm{z}\right) = \mathrm{\pi }, \\ \mathrm{then} \mathrm{x} + \mathrm{y} +\hspace{0.17em}\mathrm{z} =? \end{gathered}$$

• 0

• xyz

• 3xyz

• $\sqrt{\mathrm{xyz}}$

189.

${\mathrm{sech}}^{-1}\left(\frac{1}{2}\right) - {\mathrm{cosec}}^{-1}\left(\frac{3}{4}\right) = ?$

• ${\log }_{\mathrm{e}}\left(3\left(2 + \sqrt{3}\right)\right)$

• ${\log }_{\mathrm{e}}\left(\frac{1 + \sqrt{3}}{3}\right)$

• ${\log }_{\mathrm{e}}\left(\frac{2 + \sqrt{3}}{3}\right)$

• ${\log }_{\mathrm{e}}\left(\frac{2 - \sqrt{3}}{3}\right)$

If ${\sin }^{-1}\left(\mathrm{x} - \frac{{\mathrm{x}}^2}{2} + \frac{{\mathrm{x}}^3}{4} - . ... \infty \right) + {\cos }^{-1}\left({\mathrm{x}}^2 - \frac{{\mathrm{x}}^4}{2} + \frac{{\mathrm{x}}^6}{4} - . ... \infty \right) = \frac{\mathrm{\pi }}{2}$ and 0 < x < $\sqrt{2}$, then x is equal to

• $\frac{1}{2}$

• 1

• $- \frac{1}{2}$

• - 1