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. \(\frac{1}{5}~\text{h}\)
2. \(\frac{1}{25}~\text{h}\)
3. \(\frac{1}{15}~\text{h}\)
4. \(\frac{1}{20}~\text{h}\)

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. \(\dfrac{\theta}{\sin\frac{\theta}{2}}\)
2. \(\dfrac{\theta}{2\sin\frac{\theta}{2}}\)
3. \(\dfrac{\theta}{2\cos\frac{\theta}{2}}\)
4. \(\dfrac{\theta}{\cos\frac{\theta}{2}}\)

Two particles move from \(A\) to \(C\) and \(A\) to \(D\) on a circle of radius \(R\) and the diameter \(AB.\) If the time taken by both particles is the same, then the ratio of magnitudes of their average velocities is:
                                
1. \(2\)
2. \(2\sqrt{3}\)
3. \(\sqrt{3}\)
4. \(\dfrac{\sqrt{3}}{2}\)

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 \(\overrightarrow{{v}}=4 \hat{{i}}+2 {t} \hat{{j}}\). 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 \(x = \left(t^2 -2t\right)~\text m,\) and \(y = \left(2t^2-t\right)~\text m,\) 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 \(120^{\circ}\) with the flow direction, then the distance he travels along the river while crossing the river is:
1. \(250~\text{m}\)
2. \(500\sqrt{3}~\text{m}\)
3. \(\frac{500}{\sqrt{3}}~\text{m}\)
4. \(500~\text{m}\)