The position vector of the point where the line $\overrightarrow{\mathrm{r}} = \hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}} + \mathrm{t}\left(\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}\right)$ meets the plane $\overrightarrow{\mathrm{r}} . \left(\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}\right)$ = 5 is :

• $5\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}$

• $5\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - 3\hat{\mathrm{k}}$

• $2\hat{\mathrm{i}} + \hat{\mathrm{j}} + 2\hat{\mathrm{k}}$

• $5\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}$

If the distance of the point (1, 1, 1) from the origin is half its distance from the plane x + y + z+ k = 0, then k is equal to :

• ± 3

• ± 6

• - 3, 9

• 3, - 9

The angle between the line $\frac{\mathrm{x} - 3}{2} = \frac{\mathrm{y} - 1}{1} = \frac{\mathrm{x} + 4}{- 2}$ and the plane, x + y + z + 5 = 0 is :

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

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

• $\frac{\mathrm{\pi }}{4}$

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

The point of intersection of the line $\overrightarrow{\mathrm{r}} = 7\hat{\mathrm{i}} + 10\hat{\mathrm{j}} + 13\hat{\mathrm{k}} + \mathrm{s}\left(2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 4\hat{\mathrm{k}}\right)$ and $\overrightarrow{\mathrm{r}} = 3\hat{\mathrm{i}} + 5\hat{\mathrm{j}} + 7\hat{\mathrm{k}} + \mathrm{s}\left(\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}\right)$ is :

• $\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}$

• $2\hat{\mathrm{i}} - \hat{\mathrm{j}} + 4\hat{\mathrm{k}}$

• $\hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}$

• $\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}$

The interior angles of a polygon are AP. The smallest angle is 120° and the common difference is 5°. The number of sides of the polygon is :

• 9

• 10

• 16

• 5

The direction cosines of the line 4x - 4 = 1 - 3y = 2 - 1 are :

• $\frac{3}{\sqrt{56}}, \frac{{\displaystyle - 4}}{{\displaystyle \sqrt{56}}}, \frac{{\displaystyle 6}}{{\displaystyle \sqrt{56}}}$

• $\frac{3}{\sqrt{29}}, \frac{{\displaystyle - 4}}{{\displaystyle \sqrt{29}}}, \frac{{\displaystyle 6}}{{\displaystyle \sqrt{29}}}$

• $\frac{3}{\sqrt{61}}, \frac{{\displaystyle - 4}}{{\displaystyle \sqrt{61}}}, \frac{{\displaystyle 6}}{{\displaystyle \sqrt{61}}}$

• 4, - 3, 2

The centre of the sphere passing through the origin and through the intersection pomts of the plane$\frac{\mathrm{x}}{\mathrm{a}} + \frac{\mathrm{y}}{\mathrm{b}} + \frac{\mathrm{z}}{\mathrm{c}} = 1$ with axes is :

• $\left(\frac{\mathrm{a}}{2}, 0, 0\right)$

• $\left(0, \frac{\mathrm{a}}{2}, 0\right)$

• $\left(0, 0, \frac{\mathrm{a}}{2}\right)$

• $\left(\frac{\mathrm{a}}{2}, \frac{\mathrm{b}}{2}, \frac{\mathrm{c}}{2}\right)$

548.

The line 2x - y = 1 bisects angle between two lines. If equation of one line is y = x, then the equation of the other line is :

• 7x - y - 6 = 0

• x - 2y + 1 = 0

• 3x - 2y - 1 = 0

• x - 7y + 6 = 0

The angle between the lines $\sqrt{3}\mathrm{x} - \mathrm{y} - 2 = 0$ and x - $\sqrt{3}\mathrm{y} + 1 = 0$ is :

• 90°

• 60°

• 45°

• 30°

A force of magnitude $\sqrt{6}$ acting along the line joining the points A(2, - 1, 1) and B (3, 1, 2) displaces a particle from A to B. The work done by the force is :

• 6

• $6\sqrt{6}$

• $\sqrt{6}$

• 12