The part of circle x² + y² = 9 in between y = 0 and y = 2 is revolved about y-axis. The volume of generating solid will be

• $\frac{46}{3}\mathrm{\pi }$

• $12\mathrm{\pi }$

• $16\mathrm{\pi }$

• None of these

12.

The solution of the differential equation xdy - ydx = $\left(\sqrt{{\mathrm{x}}^2 + {\mathrm{y}}^2}\right)\mathrm{dx}$ is

• $\mathrm{y} - \sqrt{{\mathrm{x}}^2 + {\mathrm{y}}^2} = {\mathrm{Cx}}^2$

• $\mathrm{y} + \sqrt{{\mathrm{x}}^2 + {\mathrm{y}}^2} = {\mathrm{Cx}}^2$

• $\mathrm{y} + \sqrt{{\mathrm{x}}^2 + {\mathrm{y}}^2} + {\mathrm{Cx}}^{2 }= 0$

• None of the above

The solution of $\frac{\mathrm{dy}}{\mathrm{dx}} = \cos \left(\mathrm{x}\right)\left(2 - \mathrm{ycsc}\left(\mathrm{x}\right)\right)$ where y = 2, when x = $\frac{\mathrm{\pi }}{2}$ is

• $\mathrm{y} = \sin \left(\mathrm{x}\right) + \csc \left(\mathrm{x}\right)$

• $\mathrm{y} = \tan \left(\frac{\mathrm{x}}{2}\right) + \cot \left(\frac{\mathrm{x}}{2}\right)$

• $\mathrm{y} = \frac{1}{\sqrt{2}}\sec \left(\frac{\mathrm{x}}{2}\right) + \sqrt{2}\cos \left(\frac{\mathrm{x}}{2}\right)$

• None of the above

The solution of the equation ${\sin }^{-1}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) = \mathrm{x} + \mathrm{y}$ is

• $\tan \left(\mathrm{x} + \mathrm{y}\right) + \sec \left(\mathrm{x} + \mathrm{y}\right) = \mathrm{x} + \mathrm{C}$

• $\tan \left(\mathrm{x} + \mathrm{y}\right) - \sec \left(\mathrm{x} + \mathrm{y}\right) = \mathrm{x} + \mathrm{C}$

• $\tan \left(\mathrm{x} + \mathrm{y}\right) - \sec \left(\mathrm{x} + \mathrm{y}\right) + \mathrm{x} + \mathrm{C} = 0$

• None of the above

The angle between two diagonals of a cube will be

• ${\sin }^{-1}\left(\frac{1}{3}\right)$

• ${\cos }^{-1}\left(\frac{1}{3}\right)$

• variable

• None of these

The lines $\frac{\mathrm{x} - \mathrm{a} + \mathrm{d}}{\mathrm{\alpha } - \mathrm{\delta }} = \frac{\mathrm{y} - \mathrm{a}}{\mathrm{\alpha }} = \frac{\mathrm{z} - \mathrm{a} - \mathrm{d}}{\mathrm{\alpha } + \mathrm{\delta }}$ and $\frac{\mathrm{x} - \mathrm{b} + \mathrm{c}}{\mathrm{\beta } - \mathrm{\tau }} = \frac{\mathrm{y} - \mathrm{b}}{\mathrm{\beta }} = \frac{\mathrm{z} - \mathrm{b} - \mathrm{c}}{\mathrm{\beta } + \mathrm{\tau }}$ are coplanar and then equation to the plane in which they lie, is

• x + y + z = 0

• x - y + z = 0

• x - 2y + z = 0

• x + y - 2z = 0

If ${\int }_0^{{\mathrm{t}}^2}\mathrm{x} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} = \frac{2}{5}{\mathrm{t}}^5,\mathrm{t} > 0, \mathrm{then} \mathrm{f}\left(\frac{4}{25}\right)$ is

• $\frac{2}{5}$

• $\frac{5}{2}$

• $- \frac{2}{5}$

• None of these

Three forces of magnitudes 1, 2 and 3 dynes meet in a point and act along diagonals of three adjacent faces of a cube. The resultant force is

• 114 dynes

• 6 dynes

• 5 dynes

• None of the above

The vectors AB = 3i + 5j + 4k and AC = 5i - 5j + 2k are side of a $∆$ABC. The length ofthe median through A is

• $\sqrt{13} \mathrm{units}$

• $2\sqrt{5} \mathrm{units}$

• 5 units

• 10 units

Let a = 2i + j + k, b = i + 2j - 1, and a unit vector c be caplanar. If c is perpendicular to a, then c is

• $\frac{1}{\sqrt{2}}\left(- \mathrm{j} + \mathrm{k}\right)$

• $\frac{1}{\sqrt{3}}\left(- \mathrm{i} - \mathrm{j} - \mathrm{k}\right)$

• $\frac{1}{\sqrt{5}}\left(\mathrm{i} - 2\mathrm{j}\right)$

• $\frac{1}{\sqrt{3}}\left(\mathrm{i} - \mathrm{j} - \mathrm{k}\right)$