A person observes the top of a tower from a point A on the ground. The elevation of the tower from this point is 60°. He moves 60 min the direction perpendicular to the line joining A and base of the tower. The angle of elevation of the tower from this point is 45°.Then, the height of the tower (in metres) is

• $60\sqrt{\frac{3}{2}}$

• $60\sqrt{2}$

• $60\sqrt{\frac{2}{3}}$

• $60\sqrt{3}$

The direction ratios of the two lines AB and AC are 1, - 1, - 1 and 2, - 1, 1. The direction ratios of the normal to the plane ABC are

• 2, 3, -1

• 2, 2, 1

• 3, 2, - 1

• - 1, 2, 3

A plane passing through(- 1, 2, 3) and whose normal makes equal angles with the coordinate axes is

• x + y + z + 4 = 0

• x - y + z + 4 = 0

• x + y + z - 4 = 0

• x + y + z = 0

A variable plane passes through a fixed point (1, 2, 3). Then, the foot of the perpendicular from the origin to the plane lies on

• a circle

• a sphere

• an ellipse

• a parabola

The angle between the lines $\hat{\mathrm{r}} = \left(2\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + \hat{\mathrm{k}}\right) + \mathrm{\lambda }\left(\hat{\mathrm{i}} + 4\hat{\mathrm{j}} + 3\hat{\mathrm{k}}\right)$ and $\hat{\mathrm{r}} = \left(\hat{\mathrm{i}} - \hat{\mathrm{j}} + 2\hat{\mathrm{k}}\right) + \mathrm{\mu }\left(\hat{\mathrm{i}} + 2\hat{\mathrm{j}} - 3\hat{\mathrm{k}}\right)$ is

• $\frac{\mathrm{\pi }}{2}$

• ${\cos }^{-1}\left(\frac{9}{\sqrt{91}}\right)$

• ${\cos }^{-1}\left(\frac{7}{\sqrt{84}}\right)$

• $\frac{\mathrm{\pi }}{3}$

The locus of the centroid of the triangle with vertices at (acos(θ), asin(θ)), (bsin(θ), - bcos(θ)) and (1, 0) is (here, θ is a parameter)

• ${\left(3\mathrm{x} + 1\right)}^2 + 9{\mathrm{y}}^2 = {\mathrm{a}}^2 + {\mathrm{b}}^2$

• ${\left(3\mathrm{x} - 1\right)}^2 + 9{\mathrm{y}}^2 = {\mathrm{a}}^2 - {\mathrm{b}}^2$

• ${\left(3\mathrm{x} - 1\right)}^2 + 9{\mathrm{y}}^2 = {\mathrm{a}}^2 + {\mathrm{b}}^2$

• ${\left(3\mathrm{x} + 1\right)}^2 + 9{\mathrm{y}}^2 = {\mathrm{a}}^2 - {\mathrm{b}}^2$

If (2, - 1, 2) and (K, 3, 5) are the triads of direction ratios of two lines and the angle between them is 45°, then the value of K is

• 2

• 3

• 4

• 6

The length of perpendicular from the origin to the plane which makes intercepts $\frac{1}{3}, \frac{1}{4} \mathrm{and} \frac{1}{5}$ respectively on the coordinate axes is

• $\frac{1}{5\sqrt{2}}$

• $\frac{1}{10}$

• $5\sqrt{2}$

• 5

If the plane 56x + 4y + 9z = 2016 meets the coordinate axes in A, B, C, then the centroid of the $∆$ABC is

• $\left(12, 168, \frac{224}{3}\right)$

• (12, 168, 224)

• (12, 168, 112)

• $\left(12, -168, \frac{224}{3}\right)$

The equation of the plane through (4,4,0) and perpendicular to the planes 2x + y + 2z + 3 = 0 and 3x + 3y + 2z - 8 = 0

• 4x + 3y + 3z = 28

• 4x - 2y - 3z = 8

• 4x + 2y + 3z = 24

• 4x +2y - 3z = 24