11.

Which one of the following is true?

• $\sin \left({\cos }^{-1}\left(\mathrm{x}\right)\right) = \cos \left({\sin }^{-1}\left(\mathrm{x}\right)\right)$

• $\sec \left({\tan }^{-1}\left(\mathrm{x}\right)\right) = \tan \left({\sec }^{-1}\left(\mathrm{x}\right)\right)$

• $\cos \left({\tan }^{-1}\left(\mathrm{x}\right)\right) = \tan \left({\cos }^{-1}\left(\mathrm{x}\right)\right)$

• $\tan \left({\sin }^{-1}\left(\mathrm{x}\right)\right) = \sin \left({\tan }^{-1}\left(\mathrm{x}\right)\right)$

If ${\tan }^{-1}\left(\mathrm{a}\right) + {\tan }^{-1}\left(\mathrm{b}\right) = {\sin }^{-1}\left(1\right) - {\tan }^{-1}\left(\mathrm{c}\right)$, then

• a + b + c = abc

• ab + bc + ca = abc

• $\frac{1}{\mathrm{a}} + \frac{1}{\mathrm{b}} + \frac{1}{\mathrm{c}} - \frac{1}{\mathrm{abc}} = 0$

• ab + bc + ca = a + b + c

If f(x) = $$\begin{gathered} \left\{\frac{2^{\mathrm{x}} - 1}{\sqrt{1 + \mathrm{x}} - 1}, - 1 \le \mathrm{x} < \infty , \mathrm{x} \ne 0 \\ \mathrm{k}, \mathrm{x} = 0\right\} \end{gathered}$$ is continuous everywhere, then k is equal to:

• $\frac{1}{2}\log \left(2\right)$

• $\log \left(4\right)$

• $\log \left(8\right)$

• $\log \left(2\right)$

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

• $\frac{\mathrm{x}}{\mathrm{y}}$

• $- \frac{\mathrm{x}}{\mathrm{y}}$

• $\frac{\mathrm{y}}{\mathrm{x}}$

• $- \frac{\mathrm{y}}{\mathrm{x}}$

The maximum value of $\frac{\log \left(\mathrm{x}\right)}{\mathrm{x}}$ is equal to :

• $\frac{2}{\mathrm{e}}$

• $\frac{1}{\mathrm{e}}$

• e

• 1

${\tan }^{-1}\left(\frac{\mathrm{m}}{\mathrm{n}}\right) - {\tan }^{-1}\left(\frac{\mathrm{m} - \mathrm{n}}{\mathrm{m} + \mathrm{n}}\right)$ is equal to :

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{8}$

Let S be the set of all real numbers. Then the relation R = {(a, b): 1 + ab > 0} on S is :

• reflexive and symmetric but not transitive

• reflexive and transitive but not symmetric

• symmetric and transitive but not reflexive

• reflexive, transitive and symmetric

For the principal value branch of the graph of the function $\mathrm{y} = {\sin }^{-1}\left(\mathrm{x}\right), - 1 \le \mathrm{x} \le 1$, which among the following is a true statement ?

• graph is symmetric about the x-axis

• graph is symmetric about the y-axis

• graph is not continuous

• the line x = 1 is a tangent

Let $\mathrm{f}\left(\mathrm{x} + \frac{1}{\mathrm{x}}\right) = {\mathrm{x}}^2 + \frac{1}{{\mathrm{x}}^2}, \mathrm{x} \ne 0$, then f(x) equals to :

• x²

• x² - 1

• x² - 2

• x² + 1

If g(x) = min (x, x) where x is a real number, then :

• g(x) is an increasing function

• g(x) is a decreasing function

• g(x) is a constant function

• g(x) is a continuous function except at x = 0