The direction ratios of the diagonals of a cube which joins the origin to the opposite corner are (when the 3 concurrent edges of the cube are coordinate axes)

• $\frac{2}{\sqrt{3}}, \frac{{\displaystyle 2}}{{\displaystyle \sqrt{3}}}, \frac{{\displaystyle 2}}{{\displaystyle \sqrt{3}}}$

• 1, 1, 1

• 2, - 2, 1

• 1, 2, 3

The equation of the plane in which the lines $\frac{\mathrm{x} - 5}{4} = \frac{\mathrm{y} - 7}{4} = \frac{\mathrm{z} + 3}{- 5} \mathrm{and} \frac{\mathrm{x} - 8}{7} = \frac{\mathrm{y} - 4}{1} = \frac{\mathrm{z} - 5}{3}$ lie, is

• 17x - 47y - 24z + 172 = 0

• 17x + 47y - 24z + 172 = 0

• 17x + 47y + 24z + 172 = 0

• 17x - 47y + 24z + 172 = 0

If the lines $\frac{\mathrm{x} - 1}{- 3} = \frac{\mathrm{y} - 2}{2\mathrm{k}} = \frac{\mathrm{z} - 3}{2}$ and $\frac{\mathrm{x} - 1}{3\mathrm{k}} = \frac{\mathrm{y} - 5}{1} = \frac{\mathrm{z} - 6}{- 5}$ are at right angles, then the value of k will be

• $- \frac{10}{7}$

• $- \frac{7}{10}$

• - 10

• 7

674.

The coordinates of the point where the line $\frac{\mathrm{x} - 6}{- 1} = \frac{\mathrm{y} + 1}{0} = \frac{\mathrm{z} + 3}{4}$ meets the plane x + y - z = 3, are

• (2, 1, 0)

• (7, - 1, 7)

• (1, 2, - 6)

• (5, - 1, 1)

The angle between the straight lines $\frac{\mathrm{x} + 1}{2} = \frac{\mathrm{y} - 2}{5} = \frac{\mathrm{z} + 3}{4}$ and $\frac{\mathrm{x} - 1}{1} = \frac{\mathrm{y} + 2}{2} = \frac{\mathrm{z} - 3}{- 3}$ is

• $45°$

• $30°$

• $60°$

• $90°$

The acute angle between the planes 2x - y + z = 6 and x + y + 2z = 3 is

• $45°$

• $60°$

• $30°$

• $90°$

The ratio in which the line joining the points (2, 4, 5) and (3, 5, - 4) is divided by the YZ-plane, is

• 2 : 3

• 3 : 2

• - 2 : 3

• 4 : - 3

A unit vector perpendicular to the plane of $\mathrm{a} = 2\hat{\mathrm{i}} - 6\hat{\mathrm{j}} - 3\hat{\mathrm{k}}$, $\mathrm{b} = 4\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - \hat{\mathrm{k}}$, is

• $\frac{4\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - \hat{\mathrm{k}}}{\sqrt{26}}$

• $\frac{2\hat{\mathrm{i}} - 6\hat{\mathrm{j}} - 6\hat{\mathrm{k}}}{7}$

• $\frac{3\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 6\hat{\mathrm{k}}}{7}$

• $\frac{2\hat{\mathrm{i}} - 3\hat{\mathrm{j}} - 6\hat{\mathrm{k}}}{7}$

If a = $\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}$, b = $2\hat{\mathrm{i}} - 4\hat{\mathrm{k}}$ and c = $\hat{\mathrm{i}} + \mathrm{\lambda }\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$ are coplanar, then the value of $\mathrm{\lambda }$, is

• 5/2

• 3/5

• 7/3

• None of these

The ratio rn which the XY- plane meets the line joining the points (- 3, 4, - 8)and (5, - 6, 4 ) is

• 2 : 3

• 2 : 1

• 4 : 5

• None of these