21.

The integrating factor of the differential equation  is :

• x

• in x

• 0

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\mathrm{xsin}}^2\left(\mathrm{x}\right){\cos }^2\left(\mathrm{x}\right)\mathrm{dx}$ is equal to :

• $\frac{{\mathrm{\pi }}^2}{32}$

• $\frac{{\mathrm{\pi }}^2}{16}$

• $\frac{\mathrm{\pi }}{32}$

• None of these

Angle between the vectors $\sqrt{3}\left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right)$ and $\overrightarrow{\mathrm{b}} - \left(\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}}\right)\overrightarrow{\mathrm{a}}$ is :

• $\frac{\mathrm{\pi }}{2}$

• 0

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

If $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ are any three coplanar unit vectors, then

• $\overrightarrow{\mathrm{a}} . \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}\right) = 1$

• $\overrightarrow{\mathrm{a}} . \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}\right)$ = 3

• $\overrightarrow{\mathrm{a}} . \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}\right) = 0$

• $\left(\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}}\right) . \overrightarrow{\mathrm{b}} = 1$

If a force $\overrightarrow{\mathrm{F}} = 3\hat{\mathrm{i}} + 2\hat{\mathrm{j}} - 4\hat{\mathrm{k}}$ is acting at the point P(1, - 1, 2), then the moment of $\overrightarrow{\mathrm{F}}$ about the point Q(2, - 1, 3) is :

• $\sqrt{57}$

• $\sqrt{39}$

• 12

• 17

A vector of magnitude 5 and perpendicular to  ($\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + \hat{\mathrm{k}}$) and $\left(2\hat{\mathrm{i}} + \hat{\mathrm{j}} - 3\hat{\mathrm{k}}\right)$ :

• $\frac{5\sqrt{3}}{3}\left(\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$

• $\frac{5\sqrt{3}}{3}\left(\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}\right)$

• $\frac{5\sqrt{3}}{3}\left(\hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$

• $\frac{5\sqrt{3}}{3}\left(- \hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$

${\int }_{- \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\frac{\mathrm{xsin}\left(\mathrm{x}\right)}{{\cos }^2\left(\mathrm{x}\right)}\mathrm{dx}$ is :

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

• $\frac{4\mathrm{\pi }}{3} - 2\log \left(\tan \left(\frac{5\mathrm{\pi }}{12}\right)\right)$

• $\frac{4\mathrm{\pi }}{3} + \log \left(\tan \left(\frac{5\mathrm{\pi }}{12}\right)\right)$

• None of these

The angle between the lines 2x = 3y = - z and 6x = - y = -  4z is :

• 90°

• 0°

• 30°

• 45°