If $\cos \left(\mathrm{\alpha }\right), \cos \left(\mathrm{\beta }\right), \cos \left(\mathrm{\gamma }\right)$ are the direction cosines of a vector a, then $\cos \left(2\mathrm{\alpha }\right) + \cos \left(2\mathrm{\beta }\right) + \cos \left(2\mathrm{\gamma }\right)$ is equal to

• 2

• 3

• - 1

• 0

642.

If a and b are unit vectors, then angle between a and b for $\sqrt{3}\mathrm{a} - \mathrm{b}$ to be unit vectori

• 45°

• 60°

• 90°

• 30°

The plane 2x - 3y + 6z - 11 = 0 makes an angle sin^-1($\mathrm{\alpha }$) with X-axis, the value of $\mathrm{\alpha }$ is equal to

• $\frac{\sqrt{2}}{3}$

• $\frac{2}{7}$

• $\frac{\sqrt{3}}{2}$

• $\frac{3}{7}$

The perpendicular distance of the point P(6, 7, 8) from XY-plane is

• 6

• 7

• 5

• 8

Reflexion of the point $\left(\mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma }\right)$ in XY-plane is

• $\left(0, 0, \mathrm{\gamma }\right)$

• $\left(- \mathrm{\alpha }, - \mathrm{\beta }, \mathrm{\gamma }\right)$

• $\left(\mathrm{\alpha }, \mathrm{\beta }, - \mathrm{\gamma }\right)$

• $\left(\mathrm{\alpha }, \mathrm{\beta }, 0\right)$

The distance of the point (- 2, 4, - 5) from the line $\frac{\mathrm{x} + 3}{3} = \frac{\mathrm{y} - 4}{5} = \frac{\mathrm{z} + 8}{6}$ is

• $\sqrt{\frac{37}{10}}$

• $\frac{37}{10}$

• $\frac{\sqrt{37}}{10}$

• $\frac{37}{\sqrt{10}}$

The equation of straight line passing through the point (a, b, c) and parallel to Z-axis, is

• $\frac{\mathrm{x} - \mathrm{a}}{1} = \frac{\mathrm{y} - \mathrm{b}}{1} = \frac{\mathrm{z} - \mathrm{c}}{0}$

• $\frac{\mathrm{x} - \mathrm{a}}{0} = \frac{\mathrm{y} - \mathrm{b}}{1} = \frac{\mathrm{z} - \mathrm{c}}{1}$

• $\frac{\mathrm{x} - \mathrm{a}}{1} = \frac{\mathrm{y} - \mathrm{b}}{0} = \frac{\mathrm{z} - \mathrm{c}}{0}$

• $\frac{\mathrm{x} - \mathrm{a}}{0} = \frac{\mathrm{y} - \mathrm{b}}{0} = \frac{\mathrm{z} - \mathrm{c}}{1}$

If the equation of the locus of a point equidistant from the points (a₁, b₁) and (a₂, b₂) is (a₁ - a₂)r + (b₁ - b₂)y + c = 0, then the value of 'c' is

• $\frac{1}{2}\left({{\mathrm{a}}_2}^2 + {{\mathrm{b}}_2}^2 - {{\mathrm{a}}_1}^2 - {{\mathrm{b}}_1}^2\right)$

• ${{\mathrm{a}}_1}^2 - {{\mathrm{a}}_2}^2 + {{\mathrm{b}}_1}^2 - {{\mathrm{b}}_2}^2$

• $\frac{1}{2}\left({{\mathrm{a}}_1}^2 + {{\mathrm{a}}_2}^2 + {{\mathrm{b}}_1}^2 + {{\mathrm{b}}_2}^2\right)$

• $\sqrt{{{\mathrm{a}}_1}^2 + {{\mathrm{b}}_1}^2 - {{\mathrm{a}}_2}^2 - {{\mathrm{b}}_2}^2}$

A tetrahedron has vertices at 0(0, 0, 0), A(1, 2, 1), B(2, 1, 3) and C(- 1, 1, 2). Then, the angle between the faces OAB and ABC will be

• ${\cos }^{-1}\left(\frac{19}{35}\right)$

• ${\cos }^{-1}\left(\frac{17}{31}\right)$

• $30°$

• $90°$

Distance between parallel planes 2x - 2y + z + 3 = 0 and 4x - 4y + 2z + 5 = 0, is

• $\frac{1}{2}$

• $\frac{1}{3}$

• $\frac{1}{4}$

• $\frac{1}{6}$