If a and b are positive numbers such that a > b, then the minimum value of $\mathrm{asec}\left(\mathrm{\theta }\right) - \mathrm{btan}\left(0 < \mathrm{\theta } < \frac{\mathrm{\pi }}{2}\right)$ is

• $\frac{1}{\sqrt{{\mathrm{a}}^2 - {\mathrm{b}}^2}}$

• $\frac{1}{\sqrt{{\mathrm{a}}^2 + {\mathrm{b}}^2}}$

• $\sqrt{{\mathrm{a}}^2 + {\mathrm{b}}^2}$

• $\sqrt{{\mathrm{a}}^2 - {\mathrm{b}}^2}$

If $- \frac{\mathrm{\pi }}{2} < {\sin }^{-1}\left(\mathrm{x}\right) < \frac{\mathrm{\pi }}{2}$, then $\tan \left({\sin }^{-1}\left(\mathrm{x}\right)\right)$ is equal to

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

• $\frac{\mathrm{x}}{1 +\hspace{0.17em}{\mathrm{x}}^2}$

• $\frac{\mathrm{x}}{\sqrt{1 - {\mathrm{x}}^2}}$

• $\frac{1}{\sqrt{1 - {\mathrm{x}}^2}}$

The number of discontinuities of the greatest integer function f (x) = [x], $\mathrm{x} \in \left(- \frac{7}{2}, 100\right)$ is equal to

• 104

• 100

• 102

• 103

If f(x) = $$\begin{gathered} \left\{\frac{3\sin \left(\mathrm{\pi x}\right)}{5\mathrm{x}}, \mathrm{x} \ne 0 \\ 2\mathrm{k}, \mathrm{x} = 0\right\} \end{gathered}$$ is continuous at x = 0, then the value of k is equal to

• $\frac{3\mathrm{\pi }}{10}$

• $\frac{3\mathrm{\pi }}{5}$

• $\frac{\mathrm{\pi }}{10}$

• $\frac{3\mathrm{\pi }}{2}$

If a function f satisfies f {f(x)} = x + 1 for all real values of x and if f (0) = $\frac{1}{2}$, then f(1) is equal to

• $\frac{1}{2}$

• 1

• $\frac{3}{2}$

• 2

If $\mathrm{y} = \sin \left[{\cos }^{-1}\left\{\sin \left({\cos }^{-1}\left(\mathrm{x}\right)\right)\right\}\right], \mathrm{then} \frac{\mathrm{dy}}{\mathrm{dx}}$ at $\mathrm{x} = \frac{1}{2}$ is equal to

• 0

• - 1

• $\frac{2}{\sqrt{3}}$

• 1

If x² + y² = $\mathrm{t} - \frac{1}{\mathrm{t}}$ and x⁴ + y⁴ = ${\mathrm{t}}^2 + \frac{1}{{\mathrm{t}}^2}$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$

• $\frac{1}{{\mathrm{x}}^2{\mathrm{y}}^3}$

• $\frac{1}{{\mathrm{xy}}^3}$

• $\frac{1}{{\mathrm{x}}^2{\mathrm{y}}^2}$

• $\frac{1}{{\mathrm{x}}^3\mathrm{y}}$

If $\mathrm{y} = {\sec }^{-1}\left[\csc \left(\mathrm{x}\right)\right] + {\csc }^{-1}\left[\sec \left(\mathrm{x}\right)\right] + {\sin }^{-1}\left[\cos \left(\mathrm{x}\right)\right] + {\cos }^{-1}\left[\sin \left(\mathrm{x}\right)\right]$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• 0

• 2

• - 2

• - 4

19.

If $\mathrm{y} = {\mathrm{e}}^{\mathrm{x}} . {\mathrm{e}}^{{\mathrm{x}}^2} . {\mathrm{e}}^{{\mathrm{x}}^3} . ... {\mathrm{e}}^{{\mathrm{x}}^{\mathrm{n}}} ...,$ for 0 < x < 1, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ at x = $\frac{1}{2}$ is

• e

• 4e

• 2e

• 3e

The derivative of ${\tan }^{-1}\left(\frac{2\mathrm{x}}{1 - {\mathrm{x}}^2}\right)$ with repsect to ${\cos }^{-1}\left(\sqrt{1 - {\mathrm{x}}^2}\right)$ is

• $\frac{\sqrt{1 - {\mathrm{x}}^2}}{1 + {\mathrm{x}}^2}$

• $\frac{1}{\sqrt{1 - {\mathrm{x}}^2}}$

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

• $\frac{2\sqrt{1 - {\mathrm{x}}^2}}{1 + {\mathrm{x}}^2}$