The number of solutions of the equation tan(x) + sec(x) = 2cos(x) and cos(x) $\ne$ 0 lying in the interval ($0, \mathrm{\pi }$) is :

• 2

• 1

• 0

• 3

The angle between $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ is $\frac{5\mathrm{\pi }}{6}$ and the projection of $\overrightarrow{\mathrm{a}}$ in the direction of $\overrightarrow{\mathrm{b}}$ is $\frac{- 6}{\sqrt{3}}$, then $\left|\overrightarrow{\mathrm{a}}\right|$ is equal to :

• 6

• $\frac{\sqrt{3}}{2}$

• 12

• 4

A unit vector in the plane of $\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + \hat{\mathrm{k}}$ and $\hat{\mathrm{i}} + \hat{\mathrm{j}} + 2\hat{\mathrm{k}}$ and perpendicular yo $2\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}$ is :

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

• $\frac{\hat{\mathrm{i}} + \hat{\mathrm{j}}}{\sqrt{2}}$

• $\frac{\hat{\mathrm{j}} + \hat{\mathrm{k}}}{\sqrt{2}}$

• $\frac{\hat{\mathrm{j}} - \hat{\mathrm{k}}}{\sqrt{2}}$

The equation of the plane through the point (2, - 1, - 3) and parallel to the lines $\frac{\mathrm{x} - 1}{3} = \frac{\mathrm{y} + 2}{2} = \frac{\mathrm{z}}{- 4}$ and $\frac{\mathrm{x}}{2} = \frac{\mathrm{y} - 1}{- 3} = \frac{\mathrm{z} - 2}{2}$ is :

• 8x + 14y + 13z + 37 = 0

• 8x - 14y + 13z + 37 = 0

• 8x + 14y - 13z + 37 = 0

• 8x + 14y + 13z - 37 = 0

If for a plane, the intercepts on the co-ordinate axes are 8, 4, 4, then the length of the perpendicular from the origin on to the plane is :

• $\frac{8}{3}$

• $\frac{3}{8}$

• 3

• $\frac{4}{3}$

If a plane meets the co-ordinate axes at A, B and C such that the centroid of the triangle is (1, 2, 4), then the equation of the plane is:

• x + 2y + 4z = 12

• 4x + 2y + z = 12

• x + 2y + 4z = 3

• 4x + 2y + z = 3

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 :

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

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

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

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