If $\int \frac{\sqrt{\mathrm{x}}}{\mathrm{x} + 1}\mathrm{dx} = \mathrm{A}\sqrt{\mathrm{x}} + {\mathrm{Btan}}^{-1}\left(\sqrt{\mathrm{x}}\right) + \mathrm{c}$, then :

• A = 1, B = 1

• A = 1, B = 2

• A = 2, B = 2

• A = 2, B = - 2

$\int \frac{{\mathrm{x}}^3\sin \left[{\tan }^{-1}\left({\mathrm{x}}^4\right)\right]}{1 + {\mathrm{x}}^8}\mathrm{dx}$ is equal to :

• $\frac{1}{4}\cos \left[{\tan }^{-1}\left({\mathrm{x}}^4\right)\right] + \mathrm{c}$

• $\frac{1}{4}\sin \left[{\tan }^{-1}\left({\mathrm{x}}^4\right)\right] + \mathrm{c}$

• $- \frac{1}{4}\cos \left[{\tan }^{-1}\left({\mathrm{x}}^4\right)\right] + \mathrm{c}$

• $\frac{1}{4}{\sec }^{-1}\left[{\tan }^{-1}\left({\mathrm{x}}^4\right)\right] + \mathrm{c}$

23.

I_n = ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{\tan }^{\mathrm{n}}\left(\mathrm{x}\right)\mathrm{dx}$, then $\underset{\mathrm{n}\to \infty }{\lim }\mathrm{n}\left[{\mathrm{I}}_{\mathrm{n}} + {\mathrm{I}}_{\mathrm{n} +\hspace{0.17em}2}\right]$ equals :

• $\frac{1}{2}$

• 1

• $\infty$

• zero

A bag contains 3 black, 3 white and 2 red balls. One by one, three balls are drawn without replacement. The probabilitythat the third ball is red is equal to :

• $\frac{2}{56}$

• $\frac{3}{56}$

• $\frac{1}{56}$

• $\frac{14}{56}$

The angle between the line $\frac{\mathrm{x} - 3}{2} = \frac{\mathrm{y} - 1}{1} = \frac{\mathrm{x} + 4}{- 2}$ and the plane, x + y + z + 5 = 0 is :

• ${\sin }^{-1}\left(\frac{2}{\sqrt{3}}\right)$

• ${\sin }^{-1}\left(\frac{1}{\sqrt{3}}\right)$

• $\frac{\mathrm{\pi }}{4}$

• ${\sin }^{-1}\left(\frac{1}{3\sqrt{3}}\right)$

A vector perpendicular to $2\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}$ and coplanar with $\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + \hat{\mathrm{k}}$ and $\hat{\mathrm{i}} + \hat{\mathrm{j}} + 2\hat{\mathrm{k}}$ is :

• $5\left(\hat{\mathrm{j}} - \hat{\mathrm{k}}\right)$

• $\hat{\mathrm{i}} + 7\hat{\mathrm{j}} - \hat{\mathrm{k}}$

• $5\left(\hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$

• $2\hat{\mathrm{i}} - \hat{\mathrm{j}} - \hat{\mathrm{k}}$

If $\overrightarrow{\mathrm{a}} = 2\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + \mathrm{p}\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}} = \overrightarrow{\mathrm{a}} = 4\hat{\mathrm{i}} + 2\hat{\mathrm{j}} - 2\hat{\mathrm{k}}$, then p is :

• 0

• - 1

• 1

• 2

Let $\overrightarrow{\mathrm{a}} = \hat{\mathrm{i}} - \hat{\mathrm{j}}, \overrightarrow{\mathrm{b}} = \hat{\mathrm{j}} - \hat{\mathrm{k}}, \overrightarrow{\mathrm{c}} = \hat{\mathrm{k}} - \hat{\mathrm{i}}$. If $\overrightarrow{\mathrm{d}}$ is a unit vector such that $\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{d}} = 0 = \left[\overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}} \overrightarrow{\mathrm{d}}\right]$, then $\overrightarrow{\mathrm{d}}$ is (are) :

• $\pm \frac{\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}}{\sqrt{3}}$

• $\pm \frac{\hat{\mathrm{i}} + \hat{\mathrm{j}} - 2\hat{\mathrm{k}}}{\sqrt{6}}$

• $\pm \frac{\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}}{\sqrt{3}}$

• $\pm \hat{\mathrm{k}}$

If $\int \mathrm{xf}\left(\mathrm{x}\right)\mathrm{dx} = \frac{\mathrm{f}\left(\mathrm{x}\right)}{2}$, then f(x) is equal to :

• e^x

• e^{- x}

• log(x)

• $\frac{{\mathrm{e}}^{{\mathrm{x}}^2}}{2}$

${\int }_0^2\left[{\mathrm{x}}^2\right]\mathrm{dx}$ is :

• $2 - \sqrt{2}$

• $2 + \sqrt{2}$

• $\sqrt{2} - 1$

• $- \sqrt{2} - \sqrt{3} + 5$