What are the direction cosines of z-axis ?

• $< 1, 1, 1 >$

• $< 1, 0, 0 >$

• $< 0, 1, 0 >$

• $< 0, 0, 1 >$

What is the equation of the plane passing through the points (- 2, 6, - 6), (- 3, 10, - 9) and (- 5, 0, - 6) ?

• 2x - y - 2z = 2

• 2x + y + 3z = 3

• x + y + z = 6

• x - y - z = 3

What is the equation to the sphere whose centre is at ( - 2. 3, 4) and radius is 6 units ?

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 + {\mathrm{z}}^2 + 4\mathrm{x} - 6\mathrm{y} - 8\mathrm{z} = 7$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 + {\mathrm{z}}^2 + 4\mathrm{x} - 6\mathrm{y} - 8\mathrm{z} = 4$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 + {\mathrm{z}}^2 + 4\mathrm{x} - 6\mathrm{y} - 8\mathrm{z} = 7$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 + {\mathrm{z}}^2 + 4\mathrm{x} + 6\mathrm{y} + 8\mathrm{z} = 4$

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

• 1 unit

• 2 units

• 3 units

• 4 units

$\mathrm{The} \mathrm{line} \frac{\mathrm{x} - 1}{2} = \frac{\mathrm{y} - 2}{3} = \frac{\mathrm{z} - 3}{4} \mathrm{is} \mathrm{given} \mathrm{by}$

• x + y + z = 6, x + 2y - 3z = - 4

• 3x + 2y - 3z = 0, 3x - 6y + 3z = - 2

• 3x + 2y - 3z = - 2, 3x - 6y + 3z = - 2

• 3x + 2y - 3z = -2, 3x - 6y + 3z = 0

$$\begin{gathered} \mathrm{Consider} \mathrm{the} \mathrm{following} \mathrm{statements}: \\ 1. \mathrm{The} \mathrm{angle} \mathrm{between} \mathrm{the} \mathrm{planes} \\ 2\mathrm{x} - \mathrm{y} + \mathrm{z} = 1 \mathrm{and} \mathrm{x} + \mathrm{y} + 2\mathrm{z} = 3 \mathrm{is} \frac{\mathrm{\pi }}{3} \\ 2. \mathrm{The}. \mathrm{distance} \mathrm{between} \mathrm{the} \mathrm{planes} \\ 6\mathrm{x} - 3\mathrm{y} + 6\mathrm{z} + 2 = 0 \mathrm{and} 2\mathrm{x} - \mathrm{y} + 2\mathrm{z} + 4 = 0 \mathrm{is} \frac{10}{9} \\ \mathrm{Which} \mathrm{of} \mathrm{the} \mathrm{above} \mathrm{statements} \mathrm{is}/\mathrm{are} \mathrm{correct} ? \end{gathered}$$

• 1 only

• 2 only 

• both 1 and 2

• neither 1 nor 2

7.

The incentre of the triangle with vertices A(1, $\sqrt{3}$), B(0, 0) and C(2, 0) is

• $\left(1, \frac{\sqrt{3}}{2}\right)$

• $\left(\frac{2}{3}, \frac{1}{\sqrt{3}}\right)$

• $\left(\frac{2}{3}, \frac{\sqrt{3}}{2}\right)$

• $\left(1, \frac{1}{\sqrt{3}}\right)$

The angle of elevation of a stationary cloud from a point 25 m above a lake is 15° and the angle of depression of its image in the lake is 45°. The height of the cloud above the lake level is :

• 25m

• 25$\sqrt{3}$

• 50m

• $50\sqrt{3}\mathrm{m}$

The point of intersection of the line joining the points ( - 3, 4, - 8) and (5, - 6, 4) with the XY-plane is

• $\left(\frac{7}{3}, - \frac{8}{3}, 0\right)$

• $\left( - \frac{7}{3}, - \frac{8}{3}, 0\right)$

• $\left( - \frac{7}{3}, \frac{8}{3}, 0\right)$

• $\left(\frac{7}{3}, \frac{8}{3}, 0\right)$

If the angle between the lines whose direction ratios are (2, - 1, 2) and (x, 3, 5) is ,then the smaller value of x is :

• 52

• 4

• 2

• 1