31.

The area between the curves y = xe^x and y = xe^-x and the line x = 1, in sq unit, is :

• $2\left(\mathrm{e} + \frac{1}{\mathrm{e}}\right)$ sq unit

• 0 sq unit

• 2e sq unit

• $\frac{2}{\mathrm{e}}$ sq unit

If the tangent to the graph function y = f(x) makes angles $\frac{\mathrm{\pi }}{4} \mathrm{and} \frac{\mathrm{\pi }}{3}$ with the x-axis is at the point x = 2 and x = 4 respectively, the value of ${\int }_2^4\mathrm{f}\text{'}\left(\mathrm{x}\right)\mathrm{f}\text{'}\text{'}\left(\mathrm{x}\right)\mathrm{dx}$ :

• f(4) f(2)

• f(4)

• f(2)

• 1

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

• $\log \left(2\right)$

• $2\log \left(2\right)$

• ${\left(\log \left(2\right)\right)}^2$

• $\frac{1}{2}\log \left(2\right)$

The differential equation of all non-horizontal lines in a plane is :

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}$ = 0

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

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

• $\frac{{\mathrm{d}}^2\mathrm{x}}{{\mathrm{dy}}^2}$ = 0

The order and degree of the differential equation $\sqrt{\mathrm{y} + \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}} = \mathrm{x} + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ are :

• 2, 2

• 2, 1

• 1, 2

• 2, 3

The solution of 2(y + 3) - xy $\frac{\mathrm{dy}}{\mathrm{dx}}$ = 0 with y = - 2,when x = 1 is

• (y + 3) = x²

• x²(y + 3) = 1

• x⁴(y + 3) = 1

• x²(y + 3)³ = e^{y + 2}

Let f : R $\to$ R be a differentiable function and f(1) = 4. Then the value of $\underset{\mathrm{x}\to 1}{\lim }{\int }_4^{\mathrm{f}\left(\mathrm{x}\right)}\frac{2\mathrm{t}}{\mathrm{x} - 1}\mathrm{dt}$, if f'(1) = 2 is :

• 16

• 8

• 4

• 2

The solution of $\frac{\mathrm{dy}}{\mathrm{dx}}$ + y tan(x) = sec(x) is :

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

• $\mathrm{ytan}\left(\mathrm{x}\right) = \sec \left(\mathrm{x}\right) + \mathrm{c}$

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

• $\mathrm{xsec}\left(\mathrm{x}\right) = \tan \left(\mathrm{y}\right) + \mathrm{c}$

The solution of $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{ax} + \mathrm{h}}{\mathrm{by} + \mathrm{k}}$ represents a parabola, when :

• a = 0, b = 0

• a = 1, b = 2

• a = 0, b $\ne$ 0

• a = 2, b = 1

If $\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}} = \overrightarrow{0}, \left|\overrightarrow{\mathrm{a}}\right| = 3, \left|\overrightarrow{\mathrm{b}}\right| = 5 \mathrm{and} \left|\overrightarrow{\mathrm{c}}\right| = 7$, then the angle between $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ is :

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

• ${\cos }^{-1}\left(\frac{2}{225}\right)$

• $\frac{\mathrm{\pi }}{4}$