91.

If r² = x² + y² + z² and ${\tan }^{-1}\left(\frac{\mathrm{yz}}{\mathrm{xr}}\right) + {\tan }^{-1}\left(\frac{\mathrm{xz}}{\mathrm{yr}}\right) = \frac{\mathrm{\pi }}{2} - {\tan }^{-1}\left(\mathrm{\varphi }\right)$, then

• $\mathrm{\varphi } = \frac{\mathrm{zr}}{\mathrm{xy}}$

• $\mathrm{\varphi } = \frac{\mathrm{xy}}{\mathrm{zr}}$

• $\mathrm{\varphi } = \frac{\mathrm{x} + \mathrm{y}}{\mathrm{zr}}$

• $\mathrm{\varphi } = \frac{\mathrm{yz}}{\mathrm{xr}} + \frac{\mathrm{xz}}{\mathrm{yr}}$

If $\mathrm{\alpha } = {\sin }^{-1}\left(\cos \left({\sin }^{-1}\left(\mathrm{x}\right)\right)\right)$ and $\mathrm{\beta } = {\cos }^{-1}\left(\sin \left({\cos }^{-1}\left(\mathrm{x}\right)\right)\right)$, then $\tan \left(\mathrm{\alpha }\right) . \tan \left(\mathrm{\beta }\right)$ is equal to

• 1

• - 1

• 2

• $\frac{1}{2}$

The value of $2{\tan }^{-1}\left(\csc \left({\tan }^{-1}\left(\mathrm{x}\right)\right) - \tan \left({\cot }^{-1}\left(\mathrm{x}\right)\right)\right)$ is

• ${\tan }^{-1}\left(\mathrm{x}\right)$

• tan(x)

• cot(x)

• ${\csc }^{-1}\left(\mathrm{x}\right)$

If ${\left({\tan }^{-1}\left(\mathrm{x}\right)\right)}^2 + {\left({\cot }^{-1}\left(\mathrm{x}\right)\right)}^2 = \frac{5{\mathrm{\pi }}^2}{8}$, then x is equal to

• - 1

• 1

• 0

• None of these

$\frac{{\mathrm{\alpha }}^3}{2}{\csc }^2\left(\frac{1}{2}{\tan }^{-1}\left(\frac{\mathrm{\alpha }}{\mathrm{\beta }}\right)\right) + \frac{{\displaystyle {\mathrm{\beta }}^3}}{{\displaystyle 2}}{\sec }^2\left(\frac{{\displaystyle 1}}{{\displaystyle 2}}\right)\left(\frac{1}{2}{\tan }^{-1}\left(\frac{{\displaystyle \mathrm{\beta }}}{{\displaystyle \mathrm{\alpha }}}\right)\right)$ is equal to

• $\left(\mathrm{\alpha } - \mathrm{\beta }\right)\left({\mathrm{\alpha }}^2 + {\mathrm{\beta }}^2\right)$

• $\left(\mathrm{\alpha } + \mathrm{\beta }\right)\left({\mathrm{\alpha }}^2 - {\mathrm{\beta }}^2\right)$

• $\left(\mathrm{\alpha } + \mathrm{\beta }\right)\left({\mathrm{\alpha }}^2 + {\mathrm{\beta }}^2\right)$

• None of the above

$\tan \left[\frac{\mathrm{\pi }}{4} + \frac{1}{2}{\cos }^{-1}\left(\frac{\mathrm{a}}{\mathrm{b}}\right)\right] + \tan \left[\frac{\mathrm{\pi }}{4} - \frac{1}{2}{\cos }^{-1}\left(\frac{\mathrm{a}}{\mathrm{b}}\right)\right]$ is equal to

• $\frac{2\mathrm{a}}{\mathrm{b}}$

• $\frac{2\mathrm{b}}{\mathrm{a}}$

• $\frac{\mathrm{a}}{\mathrm{b}}$

• $\frac{\mathrm{b}}{\mathrm{a}}$

The solution of equation 

${\sin }^{-1}\left(1 - \frac{\mathrm{x}}{2}\right) - 2{\sin }^{-1}\left(\frac{\mathrm{x}}{2}\right) = \frac{\mathrm{\pi }}{2}$ is

• x = 0

• $\mathrm{x} = \frac{1}{2}$

• $\mathrm{x} = 0 \mathrm{and} \frac{1}{2}$

• None of these

The value of $\sin \left[\frac{\mathrm{\pi }}{2} - {\sin }^{-1}\left(- \frac{\sqrt{3}}{2}\right)\right]$ is

• $\frac{1}{2}$

• $- \frac{1}{2}$

• 1

• - 1

If $\mathrm{a} \le {\sin }^{-1}\left(\mathrm{x}\right) + {\cos }^{-1}\left(\mathrm{x}\right) + {\tan }^{-1}\left(\mathrm{x}\right) \le \mathrm{b}$, then

• $\mathrm{a} = 0, \mathrm{b} = \mathrm{\pi }$

• $\mathrm{a} = 0, \mathrm{b} = \frac{\mathrm{\pi }}{2}$

• $\mathrm{a} = \frac{\mathrm{\pi }}{2}, \mathrm{b} = \mathrm{\pi }$

• None of these

If ${\sin }^{-1}\left(\mathrm{x}\right) + {\sin }^{-1}\left(\mathrm{y}\right) = \frac{2\mathrm{\pi }}{3}$ and ${\cos }^{-1}\left(\mathrm{x}\right) + {\cos }^{-1}\left(\mathrm{y}\right) = \frac{\mathrm{\pi }}{3}$. Then, (x, y) is equal to

• (0, 1)

• (1/2, 1)

• (1, 1/2)

• $\left(\sqrt{3}/2, 1\right)$