If $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ be three unit vectors such that $\overrightarrow{\mathrm{a}} \times \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}\right) = \frac{1}{2}\overrightarrow{\mathrm{b}}$, $\overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$ being non-parallel. If ${\mathrm{\theta }}_1$ is the angle between $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ and ${\mathrm{\theta }}_2$ is the angle between $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{c}}$, then

• ${\mathrm{\theta }}_1 = \frac{\mathrm{\pi }}{6}, {\mathrm{\theta }}_2 = \frac{\mathrm{\pi }}{3}$

• ${\mathrm{\theta }}_1 = \frac{\mathrm{\pi }}{3}, {\mathrm{\theta }}_2 = \frac{\mathrm{\pi }}{6}$

• ${\mathrm{\theta }}_1 = \frac{\mathrm{\pi }}{2}, {\mathrm{\theta }}_2 = \frac{\mathrm{\pi }}{3}$

• ${\mathrm{\theta }}_1 = \frac{\mathrm{\pi }}{3}, {\mathrm{\theta }}_2 = \frac{\mathrm{\pi }}{2}$

The ${\overrightarrow{\mathrm{r}}}^2 - \overrightarrow{\mathrm{r}} . \overrightarrow{\mathrm{c}} + \mathrm{h} = 0$, $\left|\overrightarrow{\mathrm{c}}\right| > \sqrt{\mathrm{h}}$, represents

• circle

• ellipse

• cone

• sphere

$\overrightarrow{\mathrm{a}} = \hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}} = 2\hat{\mathrm{i}} + 4\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$ are one of the sides and medians respectively, of a triangle through the same vertex, then area of the triangle is

• $\frac{1}{2}\sqrt{83}$

• $\sqrt{83}$

• $\frac{1}{2}\sqrt{85}$

• $\sqrt{86}$

If $\overrightarrow{\mathrm{b}}$ is a unit vector, then $\left(\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}}\right)\overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{b}}\left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right)$ is :

• ${\left|\overrightarrow{\mathrm{a}}\right|}^2 \overrightarrow{\mathrm{b}}$

• $\left|\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}}\right|\overrightarrow{\mathrm{a}}$

• $\overrightarrow{\mathrm{a}}$

• $\overrightarrow{\mathrm{b}}$

65.

The position vectors of the points A and B with respect to O are 2i + 2j + k and 2i + 4j+ 4k. The length of the internal bisector of $\angle$BOA of $∆$AOB is

• $\frac{\sqrt{136}}{9}$

• $\frac{\sqrt{136}}{3}$

• $\frac{20}{3}$

• $\frac{\sqrt{217}}{9}$

Given (a x b) x (c x d) = 5c x 6d, then the value of (a . b) x (a + c + 2d) is

• 7

• 16

• - 1

• 4

Let a, b and c be non-zero vectors such that $\left(\mathrm{a} \times \mathrm{b}\right) \times \mathrm{c} = \frac{- 1}{4}\left|\mathrm{b}\right|\left|\mathrm{c}\right|\mathrm{a}$.  If $\mathrm{\theta }$ the acute angle between the vectors b and c, then the angle between a and c is equal to

• $\frac{2\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

A vector of magnitude 12 unit perpendicular to the plane containing the vectors 4i + 6j - k and 3i + 8j + k is

• - 8i + 4j + 8k

• 8i + 4j + 8k

• 8i - 4j + 8k

• 8i - 4j - 8k

Forces of magnitudes 3 and 4 unit acting alon 6i + 2j + 3k and 3i - 2j + 6k, respectively act on a particle and displace it from (2, 2,- 1) to (4, 3, 1). The work done is

• $\frac{124}{7}$

• $\frac{120}{7}$

• $\frac{125}{7}$

• $\frac{121}{7}$

If ABCD be a parallelogram and M be the point of intersection of the diagonals. If O is any point, then OA + OB + OC + OD is

• 3OM

• 4OM

• OM

• 2OM