If P = (0, 1, 2), Q =(4, - 2, 1), O =(0, 0, 0), then $\angle$POQ is equal to

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{3}$

A variable plane is at a constant distance h from the origin and meets the coordinate axes in A, B, C. Locus of centroid of ABC is

• x² + y² + z² = h^{- 2}

• x² + y² + z² = 4h^{- 2}

• x² + y² + z² = 16h²

• $\frac{1}{{\mathrm{x}}^2} + \frac{1}{{\mathrm{y}}^2} + \frac{1}{{\mathrm{z}}^2} = \frac{9}{{\mathrm{h}}^2}$

The acute angle between the two lines whose direction ratios are given by l + m - n = 0 and  ${\mathrm{l}}^2 + {\mathrm{m}}^2 + {\mathrm{n}}^2 = 0, \mathrm{is}$

• 0

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

The direction ratios of normal to the plane passing through (0, 0, 1), (0, 1, 2) and (1, 0, 3) are

• (2, 1, - 1)

• (1, 0, 1)

• (0, 0, - 1)

• (1, 0, 0)

If P = (0, 1, 0), Q = (0, 1, 0), then the projection of PQ on the plane x + y + z = 3 is

• 2

• $\sqrt{2}$

• 3

• $\sqrt{3}$

In the space the equation by + cz + d = 0 represents a plane perpendicular to the

• YOZ-plane

• ZOX-plane

• XOY-plane

• None of these

A plane x passes through the point (1, 1, 1). If b, c, a are the direction ratios of a normal to the plane, where a, b, c (a < b < c) are the factors of 2001, then the equation of plane is 

• 29x + 31y + 3z = 63

• 23x + 29y - 29z = 23

• 23x + 29y + 3z = 55

• 31x + 37y + 3z = 71

728.

If the plane 7x + 11y + 13z = 3003 meets the co-ordinate axes in A, B, C, then the centroid of the $∆\mathrm{ABC}$ is

• (143, 91, 77)

• (143, 77, 91)

• (91, 143, 77)

• (143, 66, 91)

If a,b, c are three non-coplanar vectors, then the vector equation $\overrightarrow{\mathrm{r}} = \left(1 - \mathrm{p} - \mathrm{q}\right)\overrightarrow{\mathrm{a}} + \mathrm{p}\overrightarrow{\mathrm{b}} + \mathrm{q}\overrightarrow{\mathrm{c}}$ represents a :

• straight line

• plane

• plane apssing through origin

• sphere

If a,b, c are three vectors such that $\overrightarrow{\mathrm{a}} = \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}$ and the angle between $\overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$ is $\frac{\mathrm{\pi }}{2}$, then

• a² = b² + c²

• b² = c² + a²

• c² = a² + b²

• 2a² - b² = c²