The speed of water in a river is 4 km/h and a man can swim at 5 km/h. The minimum time taken by the man to cross the river of width 200 m is:

1.  $\frac{1}{5} \mathrm{h}$

2.  $\frac{1}{25} \mathrm{h}$

3.  $\frac{1}{15} \mathrm{h}$

4.  $\frac{1}{20} \mathrm{h}$

A particle is moving on a circular path of radius \(R.\) When the particle moves from point \(A\) to \(B\) (angle \( \theta\)), the ratio of the distance to that of the magnitude of the displacement will be:

1. $\frac{\mathrm{\theta }}{\sin {\displaystyle \frac{\mathrm{\theta }}{2}}}$

2. $\frac{\mathrm{\theta }}{2\sin {\displaystyle \frac{\mathrm{\theta }}{2}}}$

3. $\frac{\mathrm{\theta }}{2\cos {\displaystyle \frac{\mathrm{\theta }}{2}}}$

4. $\frac{\mathrm{\theta }}{{\displaystyle \cos \frac{\mathrm{\theta }}{2}}}$

Two particles move from A to C and A to D on a circle of radius R and diameter AB. If the time taken by both particles are the same, then the ratio of magnitudes of their average velocities is:
                                
1. 2

2.  $2\sqrt{3}$

3.  $\sqrt{3}$

4.  $\frac{\sqrt{3}}{2}$

A particle moves on the curve ${\mathrm{}}^{}$\(x^2 = 2y\)$\mathrm{}$. The angle of its velocity vector with the x-axis at the point \(\left(1, \frac{1}{2}\right )\) will be:

A particle is moving along a curve. Select the correct statement.

A car is moving along east at 10 m/s and a bus is moving along north at 10 m/s. The velocity of the car with respect to the bus is along:

A particle starts moving from the origin in the XY plane and its velocity after time \(t\) is given by \(\overrightarrow{\mathrm{v}}=4 \hat{\mathrm{i}}+2 \mathrm{t} \hat{\mathrm{j}}\). The trajectory of the particle is correctly shown in the figure:

1.		2.
3.		4.

A particle is moving in the XY plane such that \(x = \left(t^2 -2t\right)\) m, and \(y = \left(2t^2-t\right)\) m, then:

It is raining at 20 m/s in still air. Now a wind starts blowing with speed 10 m/s in the north direction. If a cyclist starts moving at 10 m/s in the south direction, then the apparent velocity of rain with respect to a cyclist will be

1. 20 m/s

2. $20\sqrt{2}$ m/s

3. $10\sqrt{5}$ m/s

4. 30 m/s

River of width 500 m is flowing at a speed of 10 m/s. A swimmer can swim at a speed of 10 m/s in still water. If he starts swimming at an angle of 120° with the flow direction, then the distance he travels along the river while crossing the river is:

1. 250 m

2. $500\sqrt{3}$ m

3. $\frac{500}{\sqrt{3}}$ m

4. 500 m