If ${\sin }^{-1}\left(\frac{5}{\mathrm{x}}\right) + {\sin }^{-1}\left(\frac{12}{\mathrm{x}}\right) = \frac{\mathrm{\pi }}{2}$, then x is equal to :

• $\frac{7}{13}$

• $\frac{4}{3}$

• 13

• $\frac{13}{7}$

The sum of the series ${\tan }^{-1}\left(\frac{1}{1 + 1 + 1^2}\right) + {\tan }^{-1}\left(\frac{1}{2 + 2 + 2^2}\right) + {\tan }^{-1}\left(\frac{1}{3 + 3 + 3^2}\right) + ... \infty$ is equal to :

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{6}$

The avlue of $\sec \left[{\tan }^{-1}\left(\frac{\mathrm{b} + \mathrm{a}}{\mathrm{b} - \mathrm{a}}\right) - {\tan }^{-1}\left(\frac{\mathrm{a}}{\mathrm{b}}\right)\right]$ is :

• 2

• $\sqrt{2}$

• 4

• 1

The regression coefficient of y on x is 2/3 and that of x on y is 4/3. The acute angle between the two regression lines is tan^-1(k), where k is equal to :

• $\frac{1}{9}$

• $\frac{2}{9}$

• $\frac{1}{18}$

• $\frac{1}{3}$

75.

If $\sin \left({\sin }^{-1}\left(\frac{1}{5}\right) + {\cos }^{-1}\left(\mathrm{x}\right)\right) = 1$, then the value of x is :

• - 1

• $\frac{2}{5}$

• $\frac{1}{3}$

• $\frac{1}{5}$

$\sin \left[{\sin }^{-1}\left(\frac{1}{5}\right)\right]$ is equal to :

• $\frac{71}{125}$

• $\frac{74}{125}$

• $\frac{3}{5}$

• $\frac{1}{2}$

${\cos }^{-1}\left(\frac{3 + 5\cos \left(\mathrm{x}\right)}{5 + 3\cos \left(\mathrm{x}\right)}\right)$ is equal to :

• ${\tan }^{-1}\left(\frac{1}{2}\tan \left(\frac{\mathrm{x}}{2}\right)\right)$

• $2{\tan }^{-1}\left(2\tan \left(\frac{\mathrm{x}}{2}\right)\right)$

• $\frac{1}{2}{\tan }^{-1}\left(2\tan \left(\frac{\mathrm{x}}{2}\right)\right)$

• $2{\tan }^{-1}\left(\frac{1}{2}\tan \left(\frac{\mathrm{x}}{2}\right)\right)$

If $\angle \mathrm{A} = 90°$ in the triangle ABC, then ${\tan }^{-1}\left(\frac{\mathrm{c}}{\mathrm{a} +\hspace{0.17em}\mathrm{b}}\right) + {\tan }^{-1}\left(\frac{\mathrm{b}}{\mathrm{a} +\hspace{0.17em}\mathrm{c}}\right)$ is equal to :

• 0

• 1

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{6}$

The domain of sin^-1(log₃(x)) is :

• [- 1, 1]

• [0, 1]

• [0, $\infty$]

• $\left[\frac{1}{3}, 3\right]$

The value of $\cos \left[2{\tan }^{-1}\left(- 7\right)\right]$ is :

• $\frac{49}{50}$

• $- \frac{49}{50}$

• $\frac{24}{25}$

• $- \frac{24}{25}$