If g is the inverse of f and f'(x) = $\frac{1}{1 + {\mathrm{x}}^2}$, then g'(x) is equal to

• 1 + [g(x)]²

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

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

• None of these

If ${\int }_0^1{\tan }^{-1}\left(\mathrm{x}\right)\mathrm{dx} = \mathrm{p}$, then the value of ${\int }_0^1{\tan }^{-1}\left(\frac{1 - \mathrm{x}}{1 + \mathrm{x}}\right)$dx is

• $\frac{\mathrm{\pi }}{4} + \mathrm{p}$

• $\frac{\mathrm{\pi }}{4} - \mathrm{p}$

• 1 + p

• 1 - p

If f(x) = x , g(x) = sin(x), then $\int \mathrm{f}\left(\mathrm{g}\left(\mathrm{x}\right)\right)\mathrm{dx}$ is equal to 

• sin(x) + c

• - cos(x) + c

• $\frac{{\mathrm{x}}^2}{2} + \mathrm{c}$

• x sin(x) + c

The value of ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\log \left(\csc \left(\mathrm{x}\right)\right)d\mathrm{x}$ is

• $\frac{\mathrm{\pi }}{2}\log \left(2\right)$

• $\mathrm{\pi log}\left(2\right)$

• $- \frac{\mathrm{\pi }}{2}\log \left(2\right)$

• $2\mathrm{\pi log}\left(2\right)$

15.

If $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ are three non-coplanar vectors and $\overrightarrow{\mathrm{p}}, \overrightarrow{\mathrm{q}}, \overrightarrow{\mathrm{r}}$ are defined by the relations

$\overrightarrow{\mathrm{p}} = \frac{\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}}{\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]}, \overrightarrow{\mathrm{q}} = \frac{\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}}}{\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]} \mathrm{and} \overrightarrow{\mathrm{r}} = \frac{\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{a}}}{\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]}$

then $\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{p}} + \overrightarrow{\mathrm{b}} . \overrightarrow{\mathrm{q}} + \overrightarrow{\mathrm{c}} . \overrightarrow{\mathrm{r}}$ is equl to

• 0

• 1

• 2

• 3

The volume of a parallelopiped whose coterminous edges are  $2\overrightarrow{\mathrm{a}}, \overrightarrow{2\mathrm{b}}, 2\overrightarrow{\mathrm{c}}$ is

• $2\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$

• $4\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$

• $8\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$

• $\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$

The volume of the solid formed by rotating the area enclosed between the curve y² = 4x, x = 4 and x = 5 about x-axis is (in cubic units)

• $18\mathrm{\pi }$

• $36\mathrm{\pi }$

• $9\mathrm{\pi }$

• $24\mathrm{\pi }$

$\int {\mathrm{e}}^{\tan \left(\mathrm{x}\right)}\left({\sec }^2\left(\mathrm{x}\right) + {\sec }^3\left(\mathrm{x}\right)\sin \left(\mathrm{x}\right)\right)\mathrm{dx}$ is equal to

• $\sec \left(\mathrm{x}\right){\mathrm{e}}^{\tan \left(\mathrm{x}\right)} + \mathrm{c}$

• $\tan \left(\mathrm{x}\right){\mathrm{e}}^{\tan \left(\mathrm{x}\right)} + \mathrm{c}$

• ${\mathrm{e}}^{\tan \left(\mathrm{x}\right)} + \tan \left(\mathrm{x}\right) + \mathrm{c}$

• $\left(1 + \tan \left(\mathrm{x}\right)\right){\mathrm{e}}^{\tan \left(\mathrm{x}\right)} + \mathrm{c}$

$\int \frac{1}{16{\mathrm{x}}^2 + 9}\mathrm{dx}$ is equal to

• $\frac{1}{3}{\tan }^{-1}\left(\frac{4\mathrm{x}}{3}\right) + \mathrm{c}$

• $\frac{1}{4}{\tan }^{-1}\left(\frac{4\mathrm{x}}{3}\right) + \mathrm{c}$

• $\frac{1}{12}{\tan }^{-1}\left(\frac{4\mathrm{x}}{3}\right) + \mathrm{c}$

• $\frac{1}{12}{\tan }^{-1}\left(\frac{3\mathrm{x}}{4}\right) + \mathrm{c}$

The value of ${\int }_4^7\frac{{\left(11 - \mathrm{x}\right)}^2}{{\mathrm{x}}^2 + {\left(11 - \mathrm{x}\right)}^2}\mathrm{dx}$ is

• 1

• 1/2

• 3/2

• 0