31.

The solution of $\frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{P}\left(\mathrm{x}\right)\mathrm{y} = 0$ is

• $\mathrm{y} = {\mathrm{ce}}^{\int \mathrm{Pdx}}$

• $\mathrm{y} = {\mathrm{ce}}^{- \int \mathrm{Pdx}}$

• $\mathrm{x} = {\mathrm{ce}}^{- \int \mathrm{Pdy}}$

• $\mathrm{x} = {\mathrm{ce}}^{\int \mathrm{Pdy}}$

The differential equation of the family of lines passing through the origin is :

• $\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{y} = 0$

• $\mathrm{x} + \frac{\mathrm{dy}}{\mathrm{dx}} = 0$

• $\frac{\mathrm{dy}}{\mathrm{dx}} = \mathrm{y}$

• $\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}} - \mathrm{y} = 0$

The solution of $\frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{y} = {\mathrm{e}}^{- \mathrm{x}}$; y(0) = 0 is :

• y = e^{- x}(x - 1)

• y = xe^{- x}

• y = xe^{- x} + 1

• y = (x + 1)e^-x

${\int }_{- 2}^2\left|1 - {\mathrm{x}}^2\right|\mathrm{dx}$ is equal to :

• 4

• 2

• - 2

• 0

$\int \frac{\sqrt{\tan \left(\mathrm{x}\right)}}{\sin \left(\mathrm{x}\right)\cos \left(\mathrm{x}\right)}\mathrm{dx}$ is equal to :

• $2\tan \left(\mathrm{x}\right) + \mathrm{c}$

• $\sqrt{\cot \left(\mathrm{x}\right)} + \mathrm{c}$

• $2\sqrt{\tan \left(\mathrm{x}\right)} + \mathrm{c}$

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

$\int \frac{\mathrm{dx}}{{\mathrm{x}}^2 + 4\mathrm{x} + 13}$ is equal to :

• $\log \left({\mathrm{x}}^2 + 4\mathrm{x} + 130\right) + \mathrm{c}$

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

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

• $\frac{1}{{\mathrm{x}}^2 + 4\mathrm{x} + 13} + \mathrm{c}$

${\int }_0^1\frac{\mathrm{d}}{\mathrm{dx}}\left[{\sin }^{-1}\left(\frac{2\mathrm{x}}{1 + {\mathrm{x}}^2}\right)\right]\mathrm{dx}$ is equal to :

• 0

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{4}$

${\int }_0^{\frac{\mathrm{\pi }}{2}}\frac{\sin \left(\mathrm{x}\right)}{\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$

The area bounded by the parabolas y² = 4ax and x² = 4ay is :

• $\frac{8{\mathrm{a}}^3}{3} \mathrm{sq} \mathrm{unit}$

• $\frac{16{\mathrm{a}}^2}{3} \mathrm{sq} \mathrm{unit}$

• $\frac{32{\mathrm{a}}^2}{3} \mathrm{sq} \mathrm{unit}$

• $\frac{64{\mathrm{a}}^2}{3} \mathrm{sq} \mathrm{unit}$

The area of the regior $\left\{\left(\mathrm{x}, \mathrm{y}\right) : {\mathrm{x}}^2 + {\mathrm{y}}^2 \le 1 \le \mathrm{x} + \mathrm{y}\right\}$ is :

• $\frac{{\mathrm{\pi }}^2}{5} \mathrm{sq} \mathrm{unit}$

• $\frac{{\mathrm{\pi }}^2}{2} \mathrm{sq} \mathrm{unit}$

• $\frac{{\mathrm{\pi }}^2}{3} \mathrm{sq} \mathrm{unit}$

• $\left(\frac{\mathrm{\pi }}{4} - \frac{1}{2}\right) \mathrm{sq} \mathrm{unit}$