The angle between the lines $\frac{\mathrm{x} - 2}{3} = \frac{\mathrm{y} + 1}{- 2}; \mathrm{z}= 2$ and $\frac{\mathrm{x} - 1}{1} = \frac{2\mathrm{y} + 3}{3}; \frac{\mathrm{z} +\hspace{0.17em}5}{2}$ is

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{4}$

The angle between planes 2x - y + z = 6 and x + y + 2z = 8 is

• 30°

• 60°

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

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

Equation of a plane passing through (- 1, 1, 1) and (1, - 1, 1) and perpendicular to x + 2y + 2z = 5 is

• 2x + 3y - 3z + 3 = 0

• x + y + 3z - 5 = 0

• 2x+ 2y - 3z + 3 = 0

• x + y + z - 3 = 0

The position vectors of three non-collinear points A, Band C are a, b and c, respectively. The perpendicular distance of point C from the straight line AB is

• $\frac{\left|\mathrm{b} \times \mathrm{c}\right|}{\left|\mathrm{b} - \mathrm{c}\right|}$

• $\frac{\left|\mathrm{a} \times \mathrm{b}\right|}{\left|\mathrm{b} - \mathrm{a}\right|}$

• $\frac{\left|\mathrm{c} \times \mathrm{a}\right|}{\left|\mathrm{c} - \mathrm{a}\right|}$

• $\frac{\left|\mathrm{b} \times \mathrm{c} + \mathrm{c} \times \mathrm{a} + \mathrm{a} \times \mathrm{b}\right|}{\left|\mathrm{b} - \mathrm{a}\right|}$

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

709.

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