If $\overrightarrow{\mathrm{a}}$ is perpendicular to $\overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$, $\left|\overrightarrow{\mathrm{a}}\right| = 2, \left|\overrightarrow{\mathrm{b}}\right| = 3, \left|\overrightarrow{\mathrm{c}}\right| = 4$ and the angle between $\overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$ is $\frac{2\mathrm{\pi }}{3}$, then $\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$ is equal to

• $4\sqrt{3}$

• $6\sqrt{3}$

• $12\sqrt{3}$

• $18\sqrt{3}$

The vector $\hat{\mathrm{i}} + \mathrm{x}\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$ is rotated through an angle $\mathrm{\theta }$ and doubled in magnitude, then it becomes $4\hat{\mathrm{i}} + \left(4\mathrm{x} - 2\right)\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$. The value of x is

• $\left\{- \frac{2}{3}, 2\right\}$

• $\left\{\frac{1}{3}, 2\right\}$

• $\left\{\frac{2}{3}, 0\right\}$

• {2, 7}

A ball of mass 3 kg moving with a velocity of 3 m/s collides with another ball of mass 1 kg moving with velocity u in the opposite direction. If the first ball comes to rest after the impact and e  = $\frac{2}{7}$, then u is m/s, is

• $\frac{13}{3}$

• $\frac{17}{3}$

• $\frac{19}{3}$

• $\frac{23}{3}$

264.

Three forces ofmagnitude 30, 60 and P acting at a point are in equilibrium. If the angle between the first two is 60°, the value of P is

• $30\sqrt{7}$

• $30\sqrt{3}$

• $20\sqrt{6}$

• $25\sqrt{2}$

Let a and b be two equal vectors inclined at an angle $\mathrm{\theta }$, then $\mathrm{a} \sin \left(\frac{\mathrm{\theta }}{2}\right)$ is equal to

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

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

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

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

If $\overrightarrow{\mathrm{a}} = \left(1, - 1\right) \mathrm{and} \overrightarrow{\mathrm{b}} = \left(- 2, \mathrm{m}\right)$are two collinear vectors, then m is equal to

• 2

• 4

• 3

• 0

A ball falls from rest from top of a tower. If the ball reaches the foot of the tower in 3 s, then height of toweris (take g = 10m/s)

• 45 m

• 50 m

• 40 m

• None of these

A particle is thrown vertically upwards with a velocity of 490 cm/s. It will return to this position after

• 1 s

• 0.5 s

• 2 s

• None of these

The resultant oftwo forces P and 2P is P$\sqrt{3}$ If 1st force is doubled and reversed, the resultant of forces is

• 2P$\sqrt{3}$

• P$\sqrt{3}$

• 4P

• None of these

If $\mathrm{\theta }$ be the angle between the vectors $\overrightarrow{\mathrm{a}} = 2\hat{\mathrm{i}} + 2\hat{\mathrm{j}} - \hat{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{b}} = 6\hat{\mathrm{i}} - 3\hat{\mathrm{j}}+ 2\hat{\mathrm{k}}$, then

• $\cos \left(\mathrm{\theta }\right) = \frac{4}{21}$

• $\cos \left(\mathrm{\theta }\right) = \frac{3}{19}$

• $\cos \left(\mathrm{\theta }\right) = \frac{2}{19}$

• $\cos \left(\mathrm{\theta }\right) = \frac{5}{21}$