31.

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

$\int \frac{{\mathrm{e}}^{\mathrm{x}}}{\left(2 + {\mathrm{e}}^{\mathrm{x}}\right)\left({\mathrm{e}}^{\mathrm{x}} + 1\right)}\mathrm{dx}$ is equal to

• $\log \left(\frac{{\mathrm{e}}^{\mathrm{x}} + 1}{{\mathrm{e}}^{\mathrm{x}} + 2}\right) + \mathrm{c}$

• $\log \left(\frac{{\mathrm{e}}^{\mathrm{x}} + 2}{{\mathrm{e}}^{\mathrm{x}} + 1}\right) + \mathrm{c}$

• $\left(\frac{{\mathrm{e}}^{\mathrm{x}} + 1}{{\mathrm{e}}^{\mathrm{x}} + 2}\right) + \mathrm{c}$

• $\left(\frac{{\mathrm{e}}^{\mathrm{x}} + 2}{{\mathrm{e}}^{\mathrm{x}} + 1}\right) + \mathrm{c}$

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

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

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

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

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

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

• $\frac{\mathrm{P}}{2}$