If the angle $\mathrm{\theta }$ between the line $\frac{\mathrm{x} + 1}{1} = \frac{\mathrm{y} - 1}{2} = \frac{\mathrm{z} - 2}{2}$ and the plane 2x - y + $\sqrt{\mathrm{p}}$z + 4 = 0 is such that $\sin \left(\mathrm{\theta }\right) = \frac{1}{3}$, then the value of p is

• 0

• $\frac{1}{3}$

• $\frac{2}{3}$

• $\frac{5}{3}$

The ratio in which the plane y - 1 = 0 divides the straight line joining (1, - 1, 3) and (- 2, 5, 4) is

• 1 : 2

• 3 : 1

• 5 : 2

• 1 : 3

Equation of the line passing through i + j - 3k and perpendicular to the plane 2x - 4y + 3z + 5 = 0 is

• $\frac{\mathrm{x} - 1}{2} = \frac{1 - \mathrm{y}}{- 4} = \frac{\mathrm{z} - 3}{3}$

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

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

• $\frac{\mathrm{x} - 1}{- 2} = \frac{1 - \mathrm{y}}{- 4} = \frac{\mathrm{z} - 3}{3}$

The angle between the straight lines $\mathrm{x} - 1 = \frac{2\mathrm{y} + 3}{3} = \frac{\mathrm{z} +\hspace{0.17em}5}{2}$ and x = 3r + 2; y = - 2r - 1; z = 2, where r is a parameter, is

• $\frac{\mathrm{\pi }}{4}$

• ${\cos }^{-1}\left(\frac{- 3}{\sqrt{182}}\right)$

• ${\sin }^{-1}\left(\frac{- 3}{\sqrt{182}}\right)$

• $\frac{\mathrm{\pi }}{2}$

485.

Equation of the line through the point (2, 3, 1) and parallel to the line of intersection of the planes x - 2y - z + 5 = 0 and x + y + 3z = 6 is

• $\frac{\mathrm{x} - 2}{- 5} = \frac{\mathrm{y} - 3}{- 4} = \frac{\mathrm{z} - 1}{3}$

• $\frac{\mathrm{x} - 2}{5} = \frac{\mathrm{y} - 3}{- 4} = \frac{\mathrm{z} - 1}{3}$

• $\frac{\mathrm{x} - 2}{5} = \frac{\mathrm{y} - 3}{- 4} = \frac{\mathrm{z} - 1}{3}$

• $\frac{\mathrm{x} - 2}{4} = \frac{\mathrm{y} - 3}{4} = \frac{\mathrm{z} - 1}{2}$

The angle between a normal to the plane 2x - y + 2z - 1 = 0 and the Z-axis is

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

• ${\sin }^{-1}\left(\frac{2}{3}\right)$

• ${\cos }^{-1}\left(\frac{2}{3}\right)$

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

Foot of the perpendicular drawn from the origin to the plane 2x - 3y + 4z = 29 is

• (5, - 1, 4)

• (7, - 1, 3)

• (5, - 2, 3)

• (2, - 3, 4)

The distance between the X-axis and the point (3, 12, 5) is

• 3

• 13

• 14

• 12

The angle between the lines 2x = 3 y = - z and 6x = - y = - 4z is

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

The projection of the line segment joining (2, 0, - 3) and (5, - 1, 2) on a straight line whose direction ratios are 2, 4, 4, is

• $\frac{11}{6}$

• $\frac{10}{3}$

• $\frac{13}{3}$

• $\frac{11}{3}$