21.

The value of $\int \frac{\mathrm{dx}}{\sqrt{2\mathrm{x} - {\mathrm{x}}^2}}$ is

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

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

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

• $- \sqrt{2\mathrm{x} - {\mathrm{x}}^2} + \mathrm{c}$

Which of the following functions is a solution of the differential equation ?

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

• y = 2x - 4

• y = 2x² - 4

• y = 2

• y = 2x

y = ae^mx + be^{- mx} satisfies which of the following differential equations

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

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

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

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

Solution of the differential equation $\frac{\mathrm{dx}}{\mathrm{x}} + \frac{\mathrm{dy}}{\mathrm{y}} = 0$ is

• $\frac{1}{\mathrm{x}} + \frac{1}{\mathrm{y}} = \mathrm{c}$

• log(x)log(y) = c

• xy = c

• x + y = c

The value of $\int \mathrm{xlog}\left({\mathrm{x}}^3\right)\mathrm{dx}$ is

• $\frac{1}{16}\left(4{\mathrm{x}}^4\log \left(\mathrm{x}\right) - {\mathrm{x}}^4 + \mathrm{c}\right)$

• $\frac{1}{8}\left({\mathrm{x}}^4\log \left(\mathrm{x}\right) - 4{\mathrm{x}}^4 + \mathrm{c}\right)$

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

• $\frac{1}{16}\left(4{\mathrm{x}}^4\log \left(\mathrm{x}\right) + {\mathrm{x}}^4 + \mathrm{c}\right)$

The value of ${\int }_0^{\mathrm{\pi }}\frac{\mathrm{dx}}{5 + 3\cos \left(\mathrm{x}\right)}$ is

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{8}$

• $\frac{\mathrm{\pi }}{2}$

• zero

If the vectors $3\hat{\mathrm{i}} + \hat{\mathrm{j}} - 2\hat{\mathrm{k}}$, $\hat{\mathrm{i}} + 2\hat{\mathrm{j}} - 3\hat{\mathrm{k}}$, $3\hat{\mathrm{i}} + \mathrm{\lambda }\hat{\mathrm{j}} + 5\hat{\mathrm{k}}$ are co-planar, the value of $\mathrm{\lambda }$ is

• - 4

• 4

• 8

• - 8

The value of ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\log \left(\tan \left(\mathrm{x}\right)\right)\mathrm{dx}$ is

• - 1

• $\frac{1}{2}$

• zero

• 1

The value of ${\int }_{- \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\cos \left(\mathrm{x}\right)\left[\log \left(\frac{1 - \mathrm{x}}{1 + \mathrm{x}}\right)\right]\mathrm{dx}$ is

• 2e^1/2

• 1

• e^1/2

• zero

The area of the figure bounded by the curves y = cos(x) and y = sin(x) and the coordinate x = 0 and x = $\frac{\mathrm{\pi }}{4}$

• $\sqrt{2} + 1$

• $\sqrt{2} - 1$

• $\frac{1}{\sqrt{2}}$

• $\frac{\sqrt{2} - 1}{\sqrt{2}}$