$\sin \left[{\sin }^{-1}\left(\frac{1}{5}\right)\right]$ 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 value of f at x = 0 so that function $\mathrm{f}\left(\mathrm{x}\right) = \frac{2^{\mathrm{x}} - 2^{- \mathrm{x}}}{\mathrm{x}}, \mathrm{x} \ne 0$.  is continuous at x = 0, is :

• 0

• log(2)

• 4

• log(4)

15.

The domain of sin^-1(log₃(x)) is :

• [- 1, 1]

• [0, 1]

• [0, $\infty$]

• $\left[\frac{1}{3}, 3\right]$

If y = a^x . b^{2x - 1}, then $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}$ is :

• ${\mathrm{y}}^2 . \log \left({\mathrm{ab}}^2\right)$

• $\mathrm{y} . \log \left({\mathrm{ab}}^2\right)$

• y²

• $\mathrm{y} . {\left(\log \left({\mathrm{ab}}^2\right)\right)}^2$

The value of $\frac{\mathrm{d}}{\mathrm{dx}}\left[{\tan }^{-1}\left(\frac{\sqrt{\mathrm{x}}\left(3 - \mathrm{x}\right)}{1 - 3\mathrm{x}}\right)\right]$ is

• $\frac{1}{2\left(1 + \mathrm{x}\right)\sqrt{\mathrm{x}}}$

• $\frac{3}{2\left(1 + \mathrm{x}\right)\sqrt{\mathrm{x}}}$

• $\frac{2}{\left(1 + \mathrm{x}\right)\sqrt{\mathrm{x}}}$

• $\frac{3}{2\left(1 + \mathrm{x}\right)\sqrt{\mathrm{x}}}$

The derivative of y = (1 - x)(2 - x) ... (n - x) at x = 1 is equal to :

• 0

• (- 1)(n - 1)!

• n! - 1

• (- 1)^{n - 1}(n - 1)!

Let f(x + y) = f(x)f(y) and f(x) = 1 + sin(3x) g(x), where g(x) is continuous, then f'(x) is : 

• f(x)g(0)

• 3g(0)

• f(x)$\cos \left(3\mathrm{x}\right)$

• 3f(x)g(0)

Let f be continuous on [1, 5] and differentiable in (1, 5). If f (1) = - 3 and f'(x) $\ge$ 9 for all x $\in$ (1, 5), then

• $\mathrm{f}\left(5\right) \ge 33$

• $\mathrm{f}\left(5\right) \ge 36$

• $\mathrm{f}\left(5\right) \le 36$

• $\mathrm{f}\left(5\right) \ge 9$