$\int \mathrm{xlog}\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

• $\frac{{\mathrm{x}}^2}{4}\left(2\log \left(\mathrm{x}\right) - 1\right) + \mathrm{c}$

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

• $\frac{{\mathrm{x}}^2}{4}\left(2\log \left(\mathrm{x}\right) + 1\right) + \mathrm{c}$

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

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

• 0

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{4}$

• $\mathrm{\pi }$

23.

If $2\overrightarrow{\mathrm{a}} + 3\overrightarrow{\mathrm{b}} - 5\overrightarrow{\mathrm{c}} = \overrightarrow{0}$, then ratio in which $\overrightarrow{\mathrm{c}}$ divides $\overrightarrow{\mathrm{AB}}$ is

• 3 : 2 internally

• 3 : 2 externally

• 2 : 3 internally

• 2 : 3 externally

Volume of the parallelopiped having vertices at 0 = (0, 0, 0), A = (2, - 2, 1), B = (5, - 4, 4) and C = (1, - 2, 4) is

• 5 cu unit

• 10 cu unit

• 15 cu unit

• 20 cu unit