The plane $\overrightarrow{\mathrm{r}} = \mathrm{s}\left(\hat{\mathrm{i}} + \hat{\mathrm{j}} - 4\hat{\mathrm{k}}\right)$ $+ \mathrm{t}\left(3\hat{\mathrm{i}} +4 \hat{\mathrm{j}} - 4\hat{\mathrm{k}}\right)$ + $\left(1 - \mathrm{t}\right)\left(2\hat{\mathrm{i}} - 7\hat{\mathrm{j}} - 3\hat{\mathrm{k}}\right)$ is parallel to the line

• $\overrightarrow{\mathrm{r}} = \left(- \hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}\right) + \mathrm{t}\left(- \hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 4\hat{\mathrm{k}}\right)$

• $\overrightarrow{\mathrm{r}} = \left(- \hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}\right) + \mathrm{t}\left( \hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 4\hat{\mathrm{k}}\right)$

• $\overrightarrow{\mathrm{r}} = \left(\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}\right) + \mathrm{t}\left(- \hat{\mathrm{i}} - 4\hat{\mathrm{j}} + 7\hat{\mathrm{k}}\right)$

• $\overrightarrow{\mathrm{r}} = \left(- \hat{\mathrm{i}} + \hat{\mathrm{j}} - 3\hat{\mathrm{k}}\right) + \mathrm{t}\left(2\hat{\mathrm{i}} + 6\hat{\mathrm{j}} - 8\hat{\mathrm{k}}\right)$

The distance between the line $\overrightarrow{\mathrm{r}} = \left(2\hat{\mathrm{i}} + 2\hat{\mathrm{j}} - \hat{\mathrm{k}}\right) + \mathrm{\lambda }\left(2\hat{\mathrm{i}} + \hat{\mathrm{j}} - 2\hat{\mathrm{k}}\right)$ and the plane $\overrightarrow{\mathrm{r}} . \left(\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 2\hat{\mathrm{k}}\right)$ = 10 is equal to

• 5

• 4

• 3

• 2

Equation of the plane passing through t intersection of the planes x + y + z = 6 and 2x + 3y + 4z + 5 = 0 and the point (1, 1, 1)

• 20x + 23y + 26z - 69 = 0

• 31x + 45y + 49z + 52 = 0

• 8x + 5y + 2z - 69 = 0

• 4x + 5y + 6z - 7 = 0

The equation of the plane containing the line $\frac{\mathrm{x} - 1}{2}$ = $\frac{\mathrm{y} + 1}{- 1}$ = $\frac{\mathrm{z}}{3}$ and $\frac{\mathrm{x}}{2}$ = $\frac{\mathrm{y} - 2}{- 1}$ = $\frac{\mathrm{z} + 1}{3}$ is

• 8x - y + 5z - 8 = 0

• 8x + y - 5z - 7 = 0

• x - 8y + 3z + 6 = 0

• 8x + y - 5z + 7 = 0

The angle between the curves, y = x and y² - x = 0 at the point (1, 1) is

• $\frac{\mathrm{\pi }}{2}$

• ${\tan }^{-1}\left(\frac{4}{3}\right)$

• $\frac{\mathrm{\pi }}{3}$

• ${\tan }^{-1}\left(\frac{3}{4}\right)$

If the distance between (2, 3) and (- 5, 2) is equal to the distance between (x, 2) and (1, 3), then the values of x are

• - 6, 8

• 6, 8

• - 8, 6

• - 7, 7

The vertices of a triangle are A(3, 7), B (3, 4) and C (5, 4). The equation of the bisector of the angle ABC is

• y = x + 1

• y = x - 1

• y = 3x - 5

• y = x

If the angle between a and c is 25°, the angle between b and c is 65° and a + b = c, then the angle between a and b is

• 40°

• 115°

• 25°

• 90°

459.

The projection of the vector 2i + aj - k on the vector i - 2j + k is $\frac{- 5}{\sqrt{6}}$. Then, the value of a is equal to

• 1

• 2

• - 2

• 3

A unit vector in the XOY-plane that makes an angle 30° with the vector i + j and makes an angle 60° with i - j is

• $\frac{1}{4}\left[\left(\sqrt{6} + \sqrt{2}\right)\mathrm{i} - \left(\sqrt{6} - \sqrt{2}\right)\mathrm{j}\right]$

• $\frac{1}{2}\left[\left(\sqrt{6} - \sqrt{2}\right)\mathrm{i} + \left(\sqrt{6} + \sqrt{2}\right)\mathrm{j}\right]$

• $\frac{1}{4}\left[\left(\sqrt{6} - \sqrt{2}\right)\mathrm{i} + \left(\sqrt{6} + \sqrt{2}\right)\mathrm{j}\right]$

• $\frac{1}{4}\left[\left(\sqrt{6} + \sqrt{2}\right)\mathrm{i} + \left(\sqrt{6} - \sqrt{2}\right)\mathrm{j}\right]$