11.

$\cos \left(2{\cos }^{-1}\left(\frac{1}{5}\right) + {\sin }^{-1}\left(\frac{1}{5}\right)\right)$ 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 $\mathrm{x} + \mathrm{y} = {\tan }^{-1}\left(\mathrm{y}\right) \mathrm{and} \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} = \mathrm{f}\left(\mathrm{y}\right)\frac{\mathrm{dy}}{\mathrm{dx}}$, then f(y) is equal to

• $\frac{- 2}{{\mathrm{y}}^3}$

• $\frac{2}{{\mathrm{y}}^3}$

• $\frac{1}{\mathrm{y}}$

• $\frac{- 1}{\mathrm{y}}$

If f(x) = $$\begin{gathered} \left\{2\mathrm{a} - \mathrm{x} \mathrm{when} - \mathrm{a} < \mathrm{x} <\hspace{0.17em}\mathrm{a} \\ 3\mathrm{x} - 2\mathrm{a} \mathrm{when} \mathrm{a} \le \mathrm{x}\right\} \end{gathered}$$. Then, which of the following is true ?

• f(x) is not differentiable at x = a

• f(x) is discontinuous at x = a

• f(x) is continuous for all x < a

• f(x) is differentiable for all x $\ge$ a

If f(x) = ${\cos }^{-1}\left[\frac{1}{\sqrt{13}}\left(2\cos \left(\mathrm{x}\right) - 3\sin \left(\mathrm{x}\right)\right)\right]$. Then f'(0.5) is equal to

• 0.5

• 1

• 0

• - 1

If f(x) is a function such that f"(a) + f'(a) = 0 and g(x) =[f(x)]² + [f'(x)]² and g(3) = 8, then g(8) is equal to

• 0

• 3

• 5

• 8

If f(a) = f'(x) + f"(x) + f'"(x) + ... and f(0) = 1, then f(x) is equal to

• e^x/2

• e^x

• e^2x

• e^4x

The function f(x) = $\frac{\mathrm{x}}{3} + \frac{3}{\mathrm{x}}$ decreases in the interval

• (- 3, 3)

• $\left(- \infty , 3\right)$

• $\left(3, \infty \right)$

• (- 9, 9)

If ${\sin }^{-1}\left(\mathrm{a}\right)$ is the acute angle between the curves : x² + y² = 4x and x² + y² = 8 at (2, 2), then a is equal to

• 1

• 0

• $\frac{1}{\sqrt{2}}$

• $\frac{\sqrt{3}}{2}$

$\int \frac{{\cos }^{\mathrm{n} - 1}\left(\mathrm{x}\right)}{{\sin }^{\mathrm{n} + 1}\left(\mathrm{x}\right)}\mathrm{dx} \left(\mathrm{where}, \mathrm{n} \ne 0\right)$ is equal to

• $\frac{{\cot }^{\mathrm{n}}\left(\mathrm{x}\right)}{\mathrm{n}} + \mathrm{C}$

• $\frac{- {\cot }^{\mathrm{n} - 1}\left(\mathrm{x}\right)}{\mathrm{n} - 1} + \mathrm{C}$

• $\frac{- {\cot }^{\mathrm{n}}\left(\mathrm{x}\right)}{\mathrm{n}} + \mathrm{C}$

• $\frac{{\cot }^{\mathrm{n} - 1}\left(\mathrm{x}\right)}{\mathrm{n} - 1} + \mathrm{C}$