If the line $\frac{\mathrm{x} - 1}{2} = \frac{\mathrm{y} +\hspace{0.17em}1}{3} = \frac{\mathrm{z} - 2}{4}$ meets the plane, x + 2y + 3z = 15 at a point P, then the distance of P from the origin is :

• $\frac{\sqrt{5}}{2}$

• $2\sqrt{5}$

• $\frac{9}{2}$

• $\frac{7}{2}$

12.

The solution of the differential equation $\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}} + 2\mathrm{y} = {\mathrm{x}}^2, (\mathrm{x} \ne 0)$ with y(1)  = 1, is :

• $\mathrm{y} = \frac{4}{5}{\mathrm{x}}^3 + \frac{1}{5{\mathrm{x}}^2}$

• $\mathrm{y} = \frac{3}{4}{\mathrm{x}}^3 + \frac{1}{4{\mathrm{x}}^2}$

• $\mathrm{y} = \frac{{\mathrm{x}}^2}{4} + \frac{3}{4{\mathrm{x}}^2}$

• $\mathrm{y} = \frac{{\mathrm{x}}^3}{5} + \frac{1}{5{\mathrm{x}}^2}$

The integral $\int {\sec }^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(\mathrm{x}\right){\csc }^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

• $- 3 {\cot }^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(\mathrm{x}\right) + \mathrm{C}$

• $- 3 {\tan }^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{C}$

• $3 {\tan }^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{C}$

• $- \frac{3}{4} {\tan }^{\raisebox{1ex}{$-4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{C}$

The area (in sq. units) of the region {9x, y) : x² $\le \mathrm{y} \le \mathrm{x}$ + 2} is :

• $\frac{9}{2}$

• $\frac{10}{3}$

• $\frac{13}{6}$

• $\frac{31}{6}$

A plane passing through the points (0, - 1, 0) and (0, 0, 1) and making an angle $\frac{\mathrm{\pi }}{4}$ with the plane y - z + 5 = 0, also passes through the point :

• $\left(\sqrt{2}, - 1, 4\right)$

• $\left(- \sqrt{2}, - 1, - 4\right)$

• $\left(- \sqrt{2}, 1, - 4\right)$

• $\left(\sqrt{2}, 1, 4\right)$