21.

 is equal to

• $\frac{1}{3}{\tan }^{-1}\left(\frac{\mathrm{x} + 2}{3}\right) + \mathrm{c}$

• $\log \left(2\mathrm{x} +\hspace{0.17em}4\right) +\hspace{0.17em}\mathrm{c}$

• $\frac{2\mathrm{x} +\hspace{0.17em}4}{{\left({\mathrm{x}}^2 + 4\mathrm{x} + 13\right)}^2} + \mathrm{c}$

The value of ${\int }_2^3\frac{\mathrm{x} + 1}{{\mathrm{x}}^2\left(\mathrm{x} - 1\right)}\mathrm{dx}$ is

• $\log \left(\frac{16}{9}\right) + \frac{1}{6}$

• $\log \left(\frac{16}{9}\right) - \frac{1}{6}$

• $2\log \left(2\right) - \frac{1}{6}$

• $\log \left(\frac{4}{3}\right) - \frac{1}{6}$

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\left(\cos \left(\mathrm{x}\right) - \sin \left(\mathrm{x}\right)\right)\mathrm{dx} + {\int }_{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}^{\raisebox{1ex}{$5\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\left(\sin \left(\mathrm{x}\right) - \cos \left(\mathrm{x}\right)\right)\mathrm{dx} + {\int }_{2\mathrm{\pi }}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\left(\cos \left(\mathrm{x}\right) - \sin \left(\mathrm{x}\right)\right)\mathrm{dx}$

is equal to

• $\sqrt{2} - 2$

• $2\sqrt{2} - 2$

• $3\sqrt{2} - 2$

• $4\sqrt{2} - 2$

If the planes x + 2y + kz = 0 and 2x + y - 2z = 0, are at right angles, then the value of k is

• 2

• - 2

• $\frac{1}{2}$

• $- \frac{1}{2}$

The ratio in which the line joining (2, 4, 5), (3, 5, - 4) is divided by the yz-plane is

• 2 : 3

• 3 : 2

• - 2 : 3

• 4 : - 3

$\int \frac{{\mathrm{a}}^{{\displaystyle \raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{\sqrt{{\mathrm{a}}^{- \mathrm{x}} - {\mathrm{a}}^{\mathrm{x}}}}\mathrm{dx}$ is equal to

• $\frac{1}{\log \left(\mathrm{a}\right)}{\sin }^{-1}\left({\mathrm{a}}^{\mathrm{x}}\right) + \mathrm{c}$

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

• $2\sqrt{{\mathrm{a}}^{- \mathrm{x}} - {\mathrm{a}}^{\mathrm{x}}} + \mathrm{c}$

• $\log \left({\mathrm{a}}^{\mathrm{x}} - 1\right) + \mathrm{c}$

The value of ${\int }_0^1\frac{{\mathrm{x}}^4 + 1}{{\mathrm{x}}^2 +\hspace{0.17em}1}d\mathrm{x}$ is

• $\frac{1}{6}\left(3 - 4\mathrm{\pi }\right)$

• $\frac{1}{6}\left(3\mathrm{\pi } + 4\right)$

• $\frac{1}{6}\left(3 + 4\mathrm{\pi }\right)$

• $\frac{1}{6}\left(3\mathrm{\pi } - 4\right)$

The solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} = \mathrm{y} \tan \left(\mathrm{x}\right) - 2 \sin \left(\mathrm{x}\right)$, is

• $\mathrm{y} \sin \left(\mathrm{x}\right) = \mathrm{c} + \sin \left(2\mathrm{x}\right)$

• $\mathrm{y} \cos \left(\mathrm{x}\right) = \mathrm{c} + \frac{1}{2}\sin \left(2\mathrm{x}\right)$

• $\mathrm{y} \cos \left(\mathrm{x}\right) = \mathrm{c} - \sin \left(2\mathrm{x}\right)$

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

If $\overrightarrow{\mathrm{a}} = \left(1, - 1\right) \mathrm{and} \overrightarrow{\mathrm{b}} = \left(- 2, \mathrm{m}\right)$are two collinear vectors, then m is equal to

• 2

• 4

• 3

• 0

A ball falls from rest from top of a tower. If the ball reaches the foot of the tower in 3 s, then height of toweris (take g = 10m/s)

• 45 m

• 50 m

• 40 m

• None of these