Ship \(A\) is sailing towards north-east with velocity \(\vec{v}=30 \hat{i}+50 \hat{j}+50 \hat{i}~ \text{km/hr}\) where \(\hat{i}\) and \(\hat{j}\)$\stackrel{}{}$ north. Ship \(B\) is at a distance of \(80~\text{km}\) east and \(150~\text{km}\) north of Ship \(A\) and is sailing towards west at \(10~\text{km/hr}\). \(A\) will be at minimum distance from \(B\) in:
1. \(4.2~\text{hrs}\)
2. \(2.2~\text{hrs}\)
3. \(2.6~\text{hrs}\)
4. \(3.2~\text{hrs}\)

The stream of a river is flowing with a speed of \(2~\text{km/h}\). A swimmer can swim at a speed of \(4~\text{km/h}\). What should be the direction of the swimmer with respect to the flow of the river to cross the river straight?
1. \( 60^{\circ} \)
2. \( 90^{\circ} \)
3. \( 150^{\circ} \)
4. \( 120^{\circ}\)

When a car is at rest, its driver sees rain drops falling on it vertically. When driving the car with speed \(v,\) he sees that rain drops are coming at an angle \(60^\circ \) from the horizontal. On further increasing the speed of the car to \((1+\beta)v ,\) this angle changes to \(45^\circ.\) The value of \(\beta \) is close to:
1. \(0.41\)
2. \(0.50\)
3. \(0.37\)
4. \(0.73\)