Let $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$ be vectors with magnitudes 3, 4 and 5 respecnvely and $\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}} = \overrightarrow{0}$, then the values of $\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{b}} .\overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{c}} . \overrightarrow{\mathrm{a}}$ is

• 47

• 25

• 50

• - 25

A particle is moving in a straight line with uniform velocity. Its acceleration is

• positive

• negative

• zero

• none of these

If a particle is projected with a velocity 49 m/s making an angle 60° with the horizontal, its time of flight is

• $10\sqrt{3}$

• $5\sqrt{3}$

• $\sqrt{3}\mathrm{s}$

• none of these

If two equal components (P) of a force (F) are acting at an angle 120°, then F is

• P

• $\mathrm{P}\sqrt{2}$

• 2P

• $\frac{\mathrm{P}}{2}$

The value of $\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}} \overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}\right]$ is :

• $\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$

• 0

• $2\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$

• $\overrightarrow{\mathrm{a}} \times \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}\right)$

The area of the triangle having vertices as $\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$, $- 2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - \hat{\mathrm{k}}$, $4\hat{\mathrm{i}} - 7\hat{\mathrm{j}} + 7\hat{\mathrm{k}}$ is :

• 36 sq unit

• 0  sq unit

• 39 sq unit

• 11 sq unit

The figure formed by the four points $\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}$, $2\hat{\mathrm{i}} + 3\hat{\mathrm{j}}$, $5\hat{\mathrm{j}} - \hat{\mathrm{k}} \mathrm{and} \hat{\mathrm{k}} - \hat{\mathrm{j}}$ is :

• trapezium

• rectangle

• parallelogram

• None of these

278.

The unit vector perpendicular to $\hat{\mathrm{i}} - \hat{\mathrm{j}}$ and coplanar with $\hat{\mathrm{i}} + 2\hat{\mathrm{j}}$ and $2\hat{\mathrm{i}} + 3\hat{\mathrm{j}}$ is :

• $\frac{2\hat{\mathrm{i}} - 5\hat{\mathrm{j}}}{\sqrt{29}}$

• $2\hat{\mathrm{i}} + 5\hat{\mathrm{j}}$

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

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

${\left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right)}^2 + {\left(\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}}\right)}^2$ is equal to :

• ${\overrightarrow{\mathrm{a}}}^2 {\overrightarrow{\mathrm{b}}}^2$

• ${\overrightarrow{\mathrm{a}}}^2 + {\overrightarrow{\mathrm{b}}}^2$

• 1

• $2\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}}$

If ${\left|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right|}^2 +\hspace{0.17em}{\left|\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}}\right|}^2 = 144 \mathrm{and} \left|\overrightarrow{\mathrm{a}}\right| = 4, \mathrm{then} \left|\overrightarrow{\mathrm{b}}\right|$ is equal to

• 12

• 3

• 8

• 4