The equation of the plane through the line of intersection of the planes x - y + z + 3 = 0 and x + y + 22 + 1 = 0 and parallel to x-axis is

• 2y - z = 2

• 2y + z = 2

• 4y + z = 4

• y - 2z = 3

If 3p + 2q = i + j + k and 3p - 2q = i - j - k, then the angle between p and q is

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

   

The point of intersection of the straight line $\frac{\mathrm{x} - 2}{2} = \frac{\mathrm{y} - 1}{- 3} = \frac{\mathrm{z} + 2}{1}$ with the plane x + 3y - z + 1 = 0 is

• (3, - 1, 1)

• (- 5, 1, - 1)

• (2, 0, 3)

• (4, - 2, - 1)

If the lines $\frac{2\mathrm{x} - 1}{2} = \frac{3 - \mathrm{y}}{1} = \frac{\mathrm{z} - 1}{3}$ and $\frac{\mathrm{x} + 3}{2} = \frac{\mathrm{z} + 1}{\mathrm{p}} = \frac{\mathrm{y} + 2}{5}$ are perpendicular to each other, then p is equal to

• 1

• - 1

• 10

• - $\frac{7}{5}$

The point P(x, y, z) lies in the first octant and its distance from the origin is 12 units. If the position vector of P make 45° and 60° with the x-axis and y-axis respectively, then the coordinates of P are

• $\left(3\sqrt{3}, 6, 3\sqrt{2}\right)$

• $\left(4\sqrt{3}, 8, 4\sqrt{2}\right)$

• $\left(6\sqrt{2}, 6, 6\right)$

• $\left(6, 6, 6\sqrt{2}\right)$

The distance between the planes r . (i + 2j - 2k) + 5 = 0 and r . (2i + 4j - 4k) - 16 = 0 is

• 3

• $\frac{11}{3}$

• 13

• $\frac{13}{3}$

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

• $\frac{\mathrm{\pi }}{4}$

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

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

• $\frac{\mathrm{\pi }}{2}$