If A(- 1, 3, 2),B (2, 3, 5) and C(3, 5, - 2) are vertices of a $∆$ABC, then angles of $∆\mathrm{ABC}$ are

• $\angle \mathrm{A} = 90°, \angle \mathrm{B} = 30°, \angle \mathrm{C} = 60°$

• $\angle \mathrm{A} = \angle \mathrm{B} = \angle \mathrm{C} = 90°$

• $\angle \mathrm{A} = \angle \mathrm{B} = 45°, \angle \mathrm{C} = 90°$

• None of the above

If a, band care three non-coplanar vectors, then [a x b b x c c x a] is equal to

• [a b c]³

• [a b c]²

• 0

• None of these

Image point of (1, 3, 4) in the plane 2x - y + z + 3 = 0 will be

• (3, 5, 2)

• (3, 5, - 2)

• (- 3, 5, 2)

• None of these

Distance of the point (2, 3, 4) from the plane 3x - 6 y + 2z + 11 = 0 is

• 0

• 1

• 2

• 3

The three lines of a triangle are given by (x² - y²)(2x + 3y - 6) = 0. If the point (- 2, $\mathrm{\lambda }$) lies inside and ($\mathrm{\mu }$, 1) lies outside the triangle, then

• $\mathrm{\lambda } \in \left(1, \frac{10}{3}\right), \mathrm{\mu } \in \left(- 3, 5\right)$

• $\mathrm{\lambda } \in \left(2, \frac{10}{3}\right), \mathrm{\mu } \in \left(- 1, 1\right)$

• $\mathrm{\lambda } \in \left(- 1, \frac{9}{2}\right), \mathrm{\mu } \in \left(- 2, \frac{10}{3}\right)$

• None of the above

A variable plane is at a constant distance p from the origin O and meets the axes at A, B and C. The locus of the centroid of the tetrahedron OABC is

• $\frac{1}{{\mathrm{x}}^2} + \frac{1}{{\mathrm{y}}^2} + \frac{1}{{\mathrm{z}}^2} = \frac{1}{{\mathrm{p}}^2}$

• $\frac{1}{{\mathrm{x}}^2} + \frac{1}{{\mathrm{y}}^2} + \frac{1}{{\mathrm{z}}^2} = \frac{16}{{\mathrm{p}}^2}$

• x² + y² + z² = 16p²

• x² + y² + z² = p²

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

378.

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