If ${\tan }^{-1}\left(\mathrm{x}\right) = \frac{\mathrm{\pi }}{4} - {\tan }^{-1}\left(\frac{1}{3}\right)$, then x is

• $\frac{1}{3}$

• $\frac{1}{2}$

• $\frac{1}{4}$

• $\frac{1}{6}$

$\mathrm{If} {\cos }^{-1}\left(\frac{\mathrm{y}}{\mathrm{b}}\right) = \mathrm{nlog}\left(\frac{\mathrm{x}}{\mathrm{n}}\right)$, then

• ${\mathrm{xy}}_1 = \mathrm{n}\sqrt{{\mathrm{b}}^2 - {\mathrm{y}}^2}$

• ${\mathrm{xy}}_1 + \mathrm{n}\sqrt{{\mathrm{b}}^2 - {\mathrm{y}}^2} = 0$

• ${\mathrm{y}}_1 = \mathrm{x}\sqrt{{\mathrm{b}}^2 - {\mathrm{y}}^2}$

• ${\mathrm{xy}}_1 - \sqrt{{\mathrm{b}}^2 - {\mathrm{y}}^2} = 0$

$2{\cos }^{-1}\left(\mathrm{x}\right) = {\sin }^{-1}\left(2\mathrm{x}\sqrt{1 - {\mathrm{x}}^2}\right)$ is valid for all values of x satisfying

• $- 1 \le \mathrm{x} \le 1$

• $0 \le \mathrm{x} \le 1$

• $\frac{1}{\sqrt{2}} \le \mathrm{x} \le 1$

• $0 \le \mathrm{x} \le \frac{1}{\sqrt{2}}$

If ${\mathrm{msin}}^{-1}\left(\mathrm{x}\right) = {\log }_{\mathrm{e}}\left(\mathrm{y}\right), \mathrm{then} \left(1 - {\mathrm{x}}^2\right)\mathrm{y}\text{'}\text{'} - \mathrm{xy}\text{'}$ is equal to

• m²y

• - m²y

• 2y

• - 2y

265.

If $\frac{3\mathrm{x} + 1}{\left(\mathrm{x} - 1\right)\left(\mathrm{x} + 3\right)} = \frac{\mathrm{A}}{\mathrm{x} - 1} + \frac{\mathrm{B}}{\mathrm{x} + 3}$, ${\sin }^{-1}\left(\frac{\mathrm{A}}{\mathrm{B}}\right)$ is equal to

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{4}$

The number of real solutions of the equation ${\tan }^{-1}\left(\sqrt{\mathrm{x}\left(\mathrm{x} + 1\right)}\right) + {\sin }^{-1}\left(\sqrt{{\mathrm{x}}^2 + \mathrm{x} + 1}\right) = \frac{\mathrm{\pi }}{2}$ is

• one

• four

• two

• infinitely many

If $\frac{{\left(\mathrm{x} + 1\right)}^2}{{\mathrm{x}}^3 + \mathrm{x}} = \frac{\mathrm{A}}{\mathrm{x}} + \frac{\mathrm{Bx} + \mathrm{C}}{{\mathrm{x}}^2 + 1}$, then ${\sin }^{-1}\left(\mathrm{A}\right) + {\tan }^{-1}\left(\mathrm{B}\right) + {\sec }^{-1}\left(\mathrm{C}\right)$ is equal to

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{6}$

• 0

• $\frac{5\mathrm{\pi }}{6}$

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

• $\frac{1}{5}$

• $\frac{- 2\sqrt{6}}{5}$

• $- \frac{1}{5}$

• $\frac{\sqrt{3}}{2}$

The value of ${\tan }^{-1}\left(\frac{\mathrm{x}}{\mathrm{y}}\right) - {\tan }^{-1}\left(\frac{\mathrm{x} - \mathrm{y}}{\mathrm{x} +\hspace{0.17em}\mathrm{y}}\right)$ (where, x, y > 0) is

• $\frac{\mathrm{\pi }}{4}$

• $- \frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• $- \frac{\mathrm{\pi }}{2}$

If y = (tan^-1(x)), then (x² + 1)²y₂ + 2x(x² + 1)y₁ is equal to

• 4

• 0

• 2

• 1