1.

The value of k which makes

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) =\left\{ \sin \left(\mathrm{x}\right) \mathrm{x} \ne 0 \\ \mathrm{k} \mathrm{x} = 0\right\} \mathrm{continuous} \mathrm{at} \mathrm{x} = 0, \mathrm{is} \end{gathered}$$

• 2

• 1

•  - 1

• 0

What is the derivative of sec²(tan^-1(x)) with respect to x ?

• 2x

• x² + 1

• x + 1

• x²

$\mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = \mathrm{IxI} + \mathrm{Ix} - 1\mathrm{I}, \mathrm{then} \mathrm{which} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{is} \mathrm{correct} ?$

• $\mathrm{f}\left(\mathrm{x}\right) \mathrm{is} \mathrm{continuos} \mathrm{at} \mathrm{x} = 0 \mathrm{and} \mathrm{x} = 1$

• $\mathrm{f}\left(\mathrm{x}\right) \mathrm{is} \mathrm{continuous} \mathrm{at} \mathrm{x} = 0 \mathrm{but} \mathrm{not} \mathrm{at} \mathrm{x} = 1$

• f(x) is continuous at x = 1 but not at x = 0

• f(x) is neither continuous at x = 0 nor at x = 1

Consider the function $$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left\{{\mathrm{x}}^2\ln \left(\mathrm{x}\right) \mathrm{x} \ne 0 \\ 0 \mathrm{x} = 0\right\}. \mathrm{What} \mathrm{is} \mathrm{f}\text{'}\left(0\right) = ? \end{gathered}$$

• 0

• 1

•  - 1

• It does not exist

$$\begin{gathered} \mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = \frac{{\mathrm{x}}^2 - 9}{{\mathrm{x}}^2 - 2\mathrm{x} - 3}, \mathrm{x}\ne 3 \mathrm{is} \mathrm{continuous} \mathrm{at} \mathrm{x} = 3, \\ \mathrm{then} \mathrm{which} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{is} \mathrm{correct} ? \end{gathered}$$

• $\mathrm{f}\left(3\right) = 0$

• $\mathrm{f}\left(3\right) = 1.5$

• $\mathrm{f}\left(3\right) = 3$

• $\mathrm{f}\left(3\right) = - 1.5$

$$\begin{gathered} \mathrm{Consider} \mathrm{the} \mathrm{function} \\ \mathrm{f}\left(\mathrm{x}\right) = \left\{\frac{\sin \left(2\mathrm{x}\right)}{5\mathrm{x}} \mathrm{if} \mathrm{x} \ne 0 \\ \frac{2}{15} \mathrm{if} \mathrm{x} = 0\right\} \end{gathered}$$

Which one of the following is correct in respect of the function ?

• It is not continuous at x = 0

• It is continuous at every x

• It is not continuous at x = $\mathrm{\pi }$

• It is continuous at x = 0

$\mathrm{If} \mathrm{y} = {\tan }^{-1}\left(\frac{5 - 2\tan \left(\sqrt{\mathrm{x}}\right)}{2 + 5\tan \left(\sqrt{\mathrm{x}}\right)}\right), \mathrm{then} \mathrm{what} \mathrm{is} \frac{\mathrm{dy}}{\mathrm{dx}} = ?$

• $- \frac{1}{2\sqrt{\mathrm{x}}}$

• 1

•  - 1

• $\frac{1}{2\sqrt{\mathrm{x}}}$

Let f(x) be defined as follows:

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) =\left\{ 2\mathrm{x} + 1, - 3 < \mathrm{x} < - 2 \\ \mathrm{x} - 1, - 2 \le \mathrm{x} \le 0 \\ \mathrm{x}\hspace{0.17em}+ 2, 0 \le \mathrm{x} < 1\right\} \end{gathered}$$

Which one of the following statements is correct in respect of the above function ?

• It is discontinuous at x = - 2 but continuous at every other point.

• It is continuous only in the interval (- 3, - 2).

• It is discontinuous at x = 0 but continuous at every other point

• It is discontinuous at every point.

What is the derivative of ${\log }_{10}\left(5{\mathrm{x}}^2 + 3\right)$ with respect to x ?

• $\frac{{\mathrm{xlog}}_{10}\left(\mathrm{e}\right)}{5{\mathrm{x}}^2 + 3}$

• $\frac{2{\mathrm{xlog}}_{10}\left(\mathrm{e}\right)}{5{\mathrm{x}}^2 + 3}$

• $\frac{10{\mathrm{xlog}}_{10}\left(\mathrm{e}\right)}{5{\mathrm{x}}^2 + 3}$

• $\frac{10{\mathrm{xlog}}_{\mathrm{e}}\left(10\right)}{5{\mathrm{x}}^2 + 3}$

Suppose the function f(x) = x'', n $\ne$ 0 is differentiable for all x. Then n can be any element of the interval

• $\left(1, \infty \right)$

• $\left(0, \infty \right)$

• $\left(\frac{1}{2}, \infty \right)$

• None of the above