Let x₁ and x₂ be solutions of the equation ${\sin }^{-1}\left({\mathrm{x}}^2 - 3\mathrm{x} + \frac{5}{2}\right) = \frac{\mathrm{\pi }}{6}$. Then, the value of x₁² + x₂² is :

• 4

• 5

• $\frac{5}{2}$

• 6

If $\mathrm{\alpha } = {\cos }^{-1}\left(\frac{3}{5}\right), \mathrm{\beta } = {\tan }^{-1}\left(\frac{1}{3}\right)$, where $0 < \mathrm{\alpha }, \mathrm{\beta } < \frac{\mathrm{\pi }}{2}$, then $\mathrm{\alpha } - \mathrm{\beta }$ is equal to :

• ${\tan }^{-1}\left(\frac{9}{5\sqrt{10}}\right)$

• ${\tan }^{-1}\left(\frac{9}{14}\right)$

• ${\sin }^{-1}\left(\frac{9}{5\sqrt{10}}\right)$

• ${\cos }^{-1}\left(\frac{9}{5\sqrt{10}}\right)$

If ${\cos }^{-1}\left(\mathrm{x}\right) - {\cos }^{-1}\left(\frac{\mathrm{y}}{2}\right) = \mathrm{\alpha }$, where $- 1 \le \mathrm{x} \le 1$, $- 2 \le \mathrm{y} \le 2, \mathrm{x} \le \frac{\mathrm{y}}{2}$ then for all x, y, 4x² - 4xy$\cos \left(\mathrm{\alpha }\right)$ + y² is equal to

• $4{\sin }^2\left(\mathrm{\alpha }\right)$

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

• $4{\cos }^2\left(\mathrm{\alpha }\right) +\hspace{0.17em}2{\mathrm{x}}^2{\mathrm{y}}^2$

• $4{\sin }^2\left(\mathrm{\alpha }\right) - 2{\mathrm{x}}^2{\mathrm{y}}^2$

y = $\log \left(\tan \left(\raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)\right) + {\sin }^{-1}\left(\cos \left(\mathrm{x}\right)\right)$, then dy/dx is

• csc(x) - 1

• cs(x)

• csc(x) + 1

• x

If $2{\tan }^{-1}\left(\cos \left(\mathrm{x}\right)\right) = {\tan }^{-1}\left(2\csc \left(\mathrm{x}\right)\right)$, then $\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)$ is equal to

• $2\sqrt{2}$

• $\sqrt{2}$

• $\frac{1}{\sqrt{2}}$

• $\frac{1}{2}$

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

• $\frac{4}{5}$

• $- \frac{4}{5}$

• $\frac{3}{5}$

• 0

If $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ are roots of the equation x² + 5$\left|\mathrm{x}\right|$ - 6 = 0, then the value of $\left|{\tan }^{-1}\left(\mathrm{\alpha }\right) - {\tan }^{-1}\left(\mathrm{\beta }\right)\right|$ is

• $\frac{\mathrm{\pi }}{2}$

• 0

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{4}$

The value of ${\cos }^{-1}\left(\cot \left(\frac{\mathrm{\pi }}{2}\right)\right) + {\cos }^{-1}\left(\sin \left(\frac{2\mathrm{\pi }}{3}\right)\right)$ is

• $\frac{2\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

• $\mathrm{\pi }$

89.

The range of x for which the formula $3{\sin }^{-1}\left(\mathrm{x}\right) = {\sin }^{-1}\left[\mathrm{x}\left(3 - 4{\mathrm{x}}^2\right)\right]$ hold is

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

• $- \frac{1}{4} \le \mathrm{x} \le \frac{2}{3}$

• $- \frac{1}{3} \le \mathrm{x} \le 1$

• $- \frac{2}{3} \le \mathrm{x} \le \frac{2}{3}$

The value of $\tan \left[\frac{1}{2}{\cos }^{-1}\left(\frac{2}{3}\right)\right]$ is

• $\frac{1}{\sqrt{5}}$

• $\sqrt{\frac{3}{10}}$

• $\sqrt{\frac{5}{2}}$

• $1 - \sqrt{\frac{5}{2}}$