21.

$\int \frac{\left(\mathrm{x} - 1\right){\mathrm{e}}^{\mathrm{x}}}{{\left(\mathrm{x} +\hspace{0.17em}1\right)}^3}\mathrm{dx}$ is equal to

• $\frac{{\mathrm{e}}^{\mathrm{x}}}{\left(\mathrm{x} +\hspace{0.17em}1\right)} +\hspace{0.17em}\mathrm{C}$

• $\frac{{\mathrm{e}}^{\mathrm{x}}}{{\left(\mathrm{x} +\hspace{0.17em}1\right)}^2} +\hspace{0.17em}\mathrm{C}$

• $\frac{{\mathrm{e}}^{\mathrm{x}}}{{\left(\mathrm{x} +\hspace{0.17em}1\right)}^3} +\hspace{0.17em}\mathrm{C}$

• $\frac{{\mathrm{xe}}^{\mathrm{x}}}{\left(\mathrm{x} +\hspace{0.17em}1\right)} +\hspace{0.17em}\mathrm{C}$

If I₁ = ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\mathrm{xsin}\left(\mathrm{x}\right)\mathrm{dx}$, I₂ = ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\mathrm{xcos}\left(\mathrm{x}\right)\mathrm{dx}$, then whi ch one f the followin is true ?

• I₁ = I₂

• I₁ + I₂ = 0

• ${\mathrm{I}}_1 = \frac{\mathrm{\pi }}{2} . {\mathrm{I}}_2$

• ${\mathrm{I}}_1 + {\mathrm{I}}_2 = \frac{\mathrm{\pi }}{2}$

The value of ${\int }_{- 1}^2\frac{\left|\mathrm{x}\right|}{\mathrm{x}}\mathrm{dx}$, is

• 0

• 1

• 2

• 3

${\int }_0^{\mathrm{\pi }}\frac{{\cos }^4\left(\mathrm{x}\right)}{{\cos }^4\left(\mathrm{x}\right) + {\sin }^4\left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{8}$

• $\mathrm{\pi }$

The area bounded by the curve y = $\sin \left(\frac{\mathrm{x}}{3}\right)$, x-axis and lines x = 0 and z = 3$\mathrm{\pi }$ is

• 9

• 0

• 6

• 3

The general solution of the differential equation $\sqrt{1 - {\mathrm{x}}^2{\mathrm{y}}^2} . \mathrm{dx} = \mathrm{y} . \mathrm{dx} + \mathrm{x} . \mathrm{dy}$ is

• $\sin \left(\mathrm{xy}\right) = \mathrm{x} +\hspace{0.17em}\mathrm{C}$

• ${\sin }^{-1}\left(\mathrm{xy}\right) + \mathrm{x} = \mathrm{C}$

• $\sin \left(\mathrm{x} +\hspace{0.17em}\mathrm{C}\right) = \mathrm{xy}$

• $\sin \left(\mathrm{xy}\right) + \mathrm{x} = \mathrm{C}$

If m and n are order and degree of the differential equation ${\left(\mathrm{y}\text{'}\text{'}\right)}^5 + \frac{{\left(\mathrm{y}\text{'}\text{'}\right)}^3}{\mathrm{y}\text{'}\text{'}\text{'}} + \mathrm{y}\text{'}\text{'}\text{'} = \sin \left(\mathrm{x}\right)$, then

• m = 3, n = 5

• m = 3, n = 1

• m = 3, n = 3

• m = 3, n = 2

If a = (1,2, 3), b = (2, - 1, 1), c = (3, 2, 1) and $\mathrm{a} \times \left(\mathrm{b} \times \mathrm{c}\right) = \mathrm{\alpha a} + \mathrm{\beta b} + \mathrm{\gamma c}$, then

• $\mathrm{\alpha } = 1, \mathrm{\beta } = 10, \mathrm{\gamma } = 3$

• $\mathrm{\alpha } = 0, \mathrm{\beta } = 10, \mathrm{\gamma } = - 3$

• $\mathrm{\alpha } + \mathrm{\beta } + \mathrm{\gamma } = 8$

• $\mathrm{\alpha } = \mathrm{\beta } = \mathrm{\gamma } = 0$

If $\mathrm{a} \perp \mathrm{b} \mathrm{and} \left(\mathrm{a} + \mathrm{b}\right) \perp \left(\mathrm{a} + \mathrm{mb}\right)$, then m is equal to

• - 1

• 1

• $\frac{- {\left|\mathrm{a}\right|}^2}{{\left|\mathrm{b}\right|}^2}$

• 0

If a, b and c are unit vectors such that a + b + c = 0, then a . b + b . c + c . a is equal to

• $\frac{3}{2}$

• $- \frac{3}{2}$

• $\frac{2}{3}$

• $\frac{1}{2}$