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

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

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

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

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

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

If $2{\sinh }^{-1}\left(\frac{\mathrm{a}}{\sqrt{1 - {\mathrm{a}}^2}}\right) = \log \left(\frac{1 + \mathrm{x}}{1 - \mathrm{x}}\right)$, then x is equal to

• a

• $\frac{1}{\mathrm{a}}$

• $\sqrt{1 - {\mathrm{a}}^2}$

• $\frac{1}{\sqrt{1 - {\mathrm{a}}^2}}$

$\mathrm{If} \mathrm{x} =\sin \left(2{\tan }^{-1}\left(2\right)\right) \mathrm{and} \mathrm{y} = \sin \left(\frac{1}{2}{\tan }^{-1}\left(\frac{4}{3}\right)\right), \mathrm{then}$

• $\mathrm{x} > \mathrm{y}$

• x = y

• x = 0 = y

• $\mathrm{x}< \mathrm{y}$

$\mathrm{If} \cosh \left(\mathrm{x}\right) = \frac{5}{4}, \mathrm{then} \cosh \left(3\mathrm{x}\right) = ?$

• $\frac{61}{16}$

• $\frac{63}{16}$

• $\frac{65}{16}$

• $\frac{61}{63}$

$\mathrm{In} \mathrm{a}∆\mathrm{ABC}, \mathrm{if} <\mathrm{A} = 90°, \mathrm{then} {\cos }^{-1}\left(\frac{\mathrm{R}}{{\mathrm{r}}_2 +\hspace{0.17em}{\mathrm{r}}_3}\right) = ?$

• $90°$

• $30°$

• $60°$

• $45°$

$\mathrm{If} \cot \left({\cos }^{-1}\left(\mathrm{x}\right)\right) = \sec \left\{{\tan }^{-1}\left(\frac{\mathrm{a}}{\sqrt{{\mathrm{b}}^2 - {\mathrm{a}}^2}}\right)\right\} : \mathrm{b} > \mathrm{a}, \mathrm{then} \mathrm{x} =$

• $\frac{\mathrm{b}}{\sqrt{2{\mathrm{b}}^2 - {\mathrm{a}}^2}}$

• $\frac{\sqrt{{\mathrm{b}}^2 - {\mathrm{a}}^2}}{\mathrm{ab}}$

• $\frac{\mathrm{a}}{\sqrt{2{\mathrm{b}}^2 - {\mathrm{a}}^2}}$

• $\frac{\sqrt{{\mathrm{b}}^2 - {\mathrm{a}}^2}}{\mathrm{a}}$

$2\mathrm{\pi } - \left[{\sin }^{-1}\left(\frac{4}{5}\right) + {\sin }^{-1}\left(\frac{5}{13}\right) + {\sin }^{-1}\left(\frac{16}{65}\right)\right] = ?$

• $\frac{\mathrm{\pi }}{2}$

• $\mathrm{\pi }$

• $\frac{3\mathrm{\pi }}{2}$

• $- \frac{\mathrm{\pi }}{2}$

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