Mathematics : Three Dimensional Geometry

The equation of the plane through intersection of planes x + 2y + 3z = 4 and 2x + y - z = - 5 and perpendicular to the plane 5x + 3y + 6z = - 8 is

• 23x + 14y - 9z = - 8

• 51x + 15y - 50z = - 173

• 7x - 2y + 3z = - 81

• None of the above

If l, m, n are the direction cosines of a line, then the maximum value of lmn is

• $\frac{1}{2\sqrt{3}}$

• $\frac{1}{5\sqrt{3}}$

• $\frac{1}{\sqrt{3}}$

• None of the above

If the shortest distance between the lines $\frac{\mathrm{x} - 1}{2} = \frac{\mathrm{y} - 2}{3} = \frac{\mathrm{z} - 3}{4}$ and $\frac{\mathrm{x} - 2}{3} = \frac{\mathrm{y} - 4}{4} = \frac{\mathrm{z} - 5}{5}$ is d, then [d], where [.] is the greatest integer function, is equal to

• 0

• 1

• 2

• 3

The angle between the planes 3x-  4y + 5z = 0 and 2x - y - 2z = 5 is

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

• None of these

The distance between the lines $\frac{\mathrm{x} - 1}{3} = \frac{\mathrm{y} + 2}{- 2} = \frac{\mathrm{z} - 1}{2}$ and the plane 2x + 2y - z = 6 is

• 9

• 3

• 2

• 1

The cosine of the angle between any two diagonals of a cube is

• $\frac{1}{3}$

• $\frac{2}{3}$

• $- \frac{2}{3}$

• $\frac{1}{2}$

ABCD is a parallelogram, with AC, BD as diagonals, then AC - BD is equal to

• 4AB

• AB

• 3AB

• 2AB

If $\mathrm{\theta }$ is the angle between a and b and $\left|\mathrm{a} \times \mathrm{b}\right| = \left|\mathrm{a} . \mathrm{b}\right|$, then $\mathrm{\theta }$ is equal to

• 0

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{4}$

a, b, c, d are coplanar vectors, then (a x b) x (c x d) is equal to

• 0

• 1

• a

• b

If the foot of the perpendicular from (0, 0, 0) to the plane is (1, 2, 2), then the equation ofthe plane is

• - x + 2y + 8z - 9 = 0

• x + 2y + 2z - 9 = 0

• x + y + z - 5 = 0

• x + 2y - 3z + 1 = 0