31.

The value of $\frac{\mathrm{d}}{\mathrm{dx}}\left(\sin \left({\tan }^{-1}\left({\mathrm{e}}^{\mathrm{x}}\right)\right)\right)$ at x = 0  is

• 0

• $- \sqrt{2}$

• $\frac{- 1}{2\sqrt{2}}$

• $- \frac{1}{\sqrt{2}}$

A cylindrical tank of radius 2 m is being filled with rice at the rate of 314 cubic m/h. The depth ofthe rice is increasing at the rate of

• 25 cubic m/h

• 0.25 cubic m/h

• 1 cubic m/h

• $\frac{3}{4}$ cubic m/h

The distance of the point on the curve x² = 2y, which is nearest to the pont (0, 5) is

• 3

• 4

• $2\sqrt{2}$

• None of these

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\sin \left(2\mathrm{x}\right){\tan }^{-1}\left(\sin \left(\mathrm{x}\right)\right)\mathrm{dx}$ =

• 1

• 0

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{2} - 1$

The order and degree of the differential equation ${\left(\mathrm{y}\text{'}\right)}^4 = \sqrt{1 - \mathrm{y}\text{'}\text{'}\text{'}}$ are respectively.

• 3, 4

• 1, 2

• 3, 2

• 3, 1

The differential equation of the family of circles having centre on Y-axis and radius 4 is

• $\left({\mathrm{x}}^2 - 4\right){\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2 + {\mathrm{x}}^2 = 0$

• $\left({\mathrm{x}}^2 - 9\right){\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2 + {\mathrm{x}}^2 = 0$

• $\left({\mathrm{x}}^2 - 9\right)\frac{\mathrm{dy}}{\mathrm{dx}} + {\mathrm{x}}^2 = 0$

• $\left({\mathrm{x}}^2 - 16\right){\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2 + {\mathrm{x}}^2 = 0$

The solution of the differential equation $\left(1 + {\mathrm{y}}^2\right) + \left(\mathrm{x} - {\mathrm{e}}^{{\tan }^{-1}\left(\mathrm{y}\right)}\right)\frac{\mathrm{dy}}{\mathrm{dx}} = 0$ is given by

• $\mathrm{x} - 2 = {\mathrm{ke}}^{{\tan }^{-1}\left(\mathrm{y}\right)}$

• $2{\mathrm{xe}}^{{\tan }^{-1}\left(\mathrm{y}\right)} = {\mathrm{e}}^{2{\tan }^{-1}\left(\mathrm{y}\right)} + \mathrm{k}$

• ${\mathrm{xe}}^{{\tan }^{-1}\left(\mathrm{y}\right)} = {\tan }^{-1}\left(y\right) + \mathrm{k}$

• ${\mathrm{xe}}^{{\tan }^{-1}\left(\mathrm{y}\right)} = {\mathrm{e}}^{{\tan }^{-1}\left(\mathrm{y}\right)} + \mathrm{k}$

The integrating factor of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} + \frac{\mathrm{y}}{\mathrm{x}} = {\mathrm{x}}^3 - 3$ will be

• x

• log(x)

• - x

• e^x

Dividing the interval (1, 2) into four equal parts and using Simpson's rule, the value of ${\int }_1^2\frac{\mathrm{dx}}{\mathrm{x}}$ will be

• 0.6932

• 0.6753

• 0.6692

• 0.7132

Taking four subintervals, the value of ${\int }_0^1\frac{\mathrm{dx}}{1 + \mathrm{x}}$ by usin trapezoidal rule will be

• 0.6870

• 0.6677

• 0.6970

• 0.5970