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

• $\frac{11}{6}$

• $\frac{10}{3}$

• $\frac{13}{3}$

• $\frac{11}{3}$

The angle between the straight line $\mathrm{r} = \left(\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + \hat{\mathrm{k}}\right) + \left(\hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$ and the plane $\mathrm{r} . \left(2\hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$ = 4 is

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

• ${\sin }^{-1}\left(\frac{\sqrt{2}}{6}\right)$

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

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

If a straight line makes angles $\mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma }$ with the coordinate axes, then $\frac{1 - {\tan }^2\left(\mathrm{\alpha }\right)}{1 - {\tan }^2\left(\mathrm{\alpha }\right)} + \frac{1}{\sec \left(2\mathrm{\beta }\right)} - 2{\sin }^2\left(\mathrm{\gamma }\right)$ is equal to

• - 1

• 1

• - 2

• 2

The equation of the plane which bisects the· line segment joining the points (3, 2, 6) and (5, 4, 8) and is perpendicular to the same line segment, is

• x + y + z = 16

• x + y + z = 10

• x + y + z = 12

• x + y + z = 14

The foot of the perpendicular from the point (1, 6, 3) to the line $\frac{\mathrm{x}}{1} = \frac{\mathrm{y} - 1}{2} = \frac{\mathrm{z} - 2}{3}$ is

• (1, 3, 5)

• (- 1, - 1, - 1)

• (2, 5, 8)

• (- 2, - 3, - 4)