${\int }_{- \mathrm{a}}^{\mathrm{a}}\mathrm{x}\sqrt{\left({\mathrm{a}}^2 - {\mathrm{x}}^2\right)}\mathrm{dx}$ is equal to

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{8}$

• 0

If y = ${\tan }^{-1}\left(\sqrt{\frac{1 - \cos \left(\mathrm{x}\right)}{1 + \cos \left(\mathrm{x}\right)}}\right)$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ will be

• $\sin \left(\mathrm{x}\right)\cos \left(\mathrm{x}\right)$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{1}{2}$

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

The value of $\underset{\mathrm{x}\to 1}{\lim }\left(1 - \mathrm{x}\right) . \tan \left(\frac{\mathrm{\pi x}}{2}\right)$ will be

• $\frac{\mathrm{\pi }}{2}$

• $\frac{2}{\mathrm{\pi }}$

• $2\mathrm{\pi }$

• $\mathrm{\pi }$

Let f(x) = $$\begin{gathered} \left\{\frac{{\mathrm{x}}^2 - 4\mathrm{x} + 3}{{\mathrm{x}}^2 + 2\mathrm{x} - 3}, \mathrm{x} \ne 1 \\ \mathrm{k} , \mathrm{x} = 1\right\} \end{gathered}$$ If f(x) is continuous at x = 1, then the value of k will be

• 1

• $\frac{1}{2}$

• - 1

• $- \frac{1}{2}$

Let f(x) = $$\begin{gathered} \left\{{\mathrm{x}}^{\mathrm{n}} . \sin \left(\frac{1}{\mathrm{x}}\right), \mathrm{x} \ne 0 \\ 0 , \mathrm{x} = 0\right\} \end{gathered}$$ Then, f(x) is differentiable at x = 0, if

• $\mathrm{n} \in \left(0, 1\right)$

• $\mathrm{n} \in \left(1, 2\right)$

• $\mathrm{n} \in \left(1, \infty \right)$

• $\mathrm{n} \in \left(- \infty , \infty \right)$

In which interval the function $\mathrm{f}\left(\mathrm{x}\right) = \sqrt{{\log }_{10}\left(\frac{5\mathrm{x} - {\mathrm{x}}^2}{4}\right)}$ is defined ?

• [1, 4]

• [0, 5)

• (0, 1)

• (- 1, $\infty$)

$\underset{\mathrm{x}\to 0}{\lim }\frac{\mathrm{x} . 2^{\mathrm{x}} - \mathrm{x}}{1 - \cos \left(\mathrm{x}\right)}$ equals

• $\log \left(2\right)$

• $\frac{1}{2}\log \left(2\right)$

• $2\log \left(2\right)$

• None of these

Let f(2) = 4 and f'(2)= 1. Then, $\underset{\mathrm{x}\to 2}{\lim }\frac{\mathrm{xf}\left(2\right) - 2\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{x} - 2}$ is given by

• 2

• - 2

• - 4

• 3

$\mathrm{If} \mathrm{y} = {\log }_{\sin \left(\mathrm{x}\right)}\left(\tan \left(\mathrm{x}\right)\right), \mathrm{then} {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}_{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}$ is equal to

• $\frac{4}{\log \left(2\right)}$

• $- 4\log \left(2\right)$

• $- \frac{4}{\log \left(2\right)}$

• None of these

If y = ${\log }_{{\mathrm{e}}^{\mathrm{x}}}{\left(\mathrm{x} - 2\right)}^2$ for $\mathrm{x} \ne 0$, 2, then y'(3) is equal to

• $\frac{1}{3}$

• $\frac{2}{3}$

• $\frac{4}{3}$

• None of these