51.

If $\tan \left[3{\tan }^{-1}\left(\frac{1}{5}\right), - \frac{\mathrm{\pi }}{4}\right]$ is equal to

• $- \frac{13}{46}$

• $- \frac{11}{46}$

• $- \frac{7}{46}$

• $- \frac{9}{46}$

If ${\tan }^{-1}\left(- \mathrm{x}\right) + {\cos }^{-1}\left(\frac{- 1}{2}\right) = \frac{\mathrm{\pi }}{2}$, then the value of x is

• $\sqrt{3}$

• $\frac{- 1}{\sqrt{3}}$

• $\frac{1}{\sqrt{3}}$

• $- \sqrt{3}$

If ab > - 1, bc > - 1 and ca > - 1, then ${\cot }^{-1}\left(\frac{\mathrm{ab} + 1}{\mathrm{a} - \mathrm{b}}\right) + {\cot }^{-1}\left(\frac{\mathrm{bc} + 1}{\mathrm{b} - \mathrm{c}}\right) + {\cot }^{-1}\left(\frac{\mathrm{ca} + 1}{\mathrm{c} - \mathrm{a}}\right)$ is

• - 1

• ${\cot }^{-1}\left(\mathrm{a} +\hspace{0.17em}\mathrm{b} +\hspace{0.17em}\mathrm{c}\right)$

• ${\cot }^{-1}\left(\mathrm{abc}\right)$

• 0

The value of ${\sin }^{-1}\left(\frac{2\sqrt{2}}{3}\right) + {\sin }^{-1}\left(\frac{1}{3}\right)$ is equal to

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$

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

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

If ab < 1 and ${\cos }^{-1}\left(\frac{1 - {\mathrm{a}}^2}{1 + {\mathrm{a}}^2}\right) + {\cos }^{-1}\left(\frac{1 - {\mathrm{b}}^2}{1 + {\mathrm{b}}^2}\right) = 2{\tan }^{-1}\left(\mathrm{x}\right)$, then x is equal to

• $\frac{\mathrm{a}}{1 + \mathrm{ab}}$

• $\frac{\mathrm{a}}{1 - \mathrm{ab}}$

• $\frac{\mathrm{a} +\hspace{0.17em}\mathrm{b}}{1 + \mathrm{ab}}$

• $\frac{\mathrm{a} +\hspace{0.17em}\mathrm{b}}{1 - \mathrm{ab}}$

${\cos }^{-1}\left[\cos \left(\frac{7\mathrm{\pi }}{5}\right)\right]$ is equal to

• $\frac{3\mathrm{\pi }}{5}$

• $\frac{2\mathrm{\pi }}{5}$

• $\frac{- 7\mathrm{\pi }}{5}$

• $\frac{7\mathrm{\pi }}{5}$

The value of ${\sec }^2\left({\tan }^{-1}\left(3\right)\right) + \csc \left({\cot }^{-1}\left(2\right)\right)$ is equal to

• 5

• 13

• 15

• 23

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

• ${\tan }^{-1}\left(\frac{16}{13}\right)$

• ${\tan }^{-1}\left(\frac{17}{23}\right)$

• $\frac{\mathrm{\pi }}{4}$

• 0

If ${\cos }^{-1}\left(\mathrm{x}\right) + {\cos }^{-1}\left(\mathrm{y}\right) = \frac{2\mathrm{\pi }}{7}$, then the value of ${\sin }^{-1}\left(\mathrm{x}\right) + {\sin }^{-1}\left(\mathrm{y}\right)$ is

• $\frac{4\mathrm{\pi }}{7}$

• $\frac{3\mathrm{\pi }}{7}$

• $\frac{2\mathrm{\pi }}{7}$

• $\frac{5\mathrm{\pi }}{7}$

If ${\cos }^{-1}\left(\mathrm{x}\right) > {\sin }^{-1}\left(\mathrm{x}\right)$, then x lies in the interval

• $(\frac{1}{2}, 1]$

• (0, 1]

• $[- 1, \frac{1}{\sqrt{2}})$

• [- 1, 1]