Ship \(A\) is travelling with a velocity of \(5~\text{km/h}\) due east. A second ship is heading $\mathrm{}$\(30^\circ\) east of north. What should be the speed of the second ship if it is to remain always due north with respect to the first ship?
1. \(10~\text{km/h}\)
2. \(9~\text{km/h}\)
3. \(8~\text{km/h}\)
4. \(7~\text{km/h}\)

A boat is moving with a velocity \(3\hat i+ 4\hat j\) with respect to ground. The water in the river is moving with a velocity \(-3\hat i- 4\hat j\) with respect to the ground. The relative velocity of the boat with respect to water is:
1. \(8\hat j\)
2. \(-6\hat i-8\hat j\)
3. \(6\hat i + 8\hat j\)
4. \(5\sqrt{2}\)

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

1. \(\dfrac{1}{5}~\text h\)

2. \(\dfrac{1}{25}~\text h\)

3. \(\dfrac{1}{15}~\text h\)

4. \(\dfrac{1}{20}~\text h\)

The speed of a boat is \(5\) km/hr in still water. It crosses a river of width \(1\) km along the shortest possible path in \(15\) minutes. The velocity of the river water is:
1. \(3\) km/hr
2. \(4\) km/hr
3. \(5\) km/hr
4. \(2\) km/hr

On a rainy day, a boy standing on the road finds that he needs to hold his umbrella at an angle of \(30^\circ\) with respect to the vertical to avoid getting drenched in the rain. He throws the umbrella and starts running at a speed of \(10\) km/h. He observes that the raindrops are falling vertically on his head. What is the speed of the raindrops with respect to the ground?
1. \(15\) km/h
2. \(20\) km/h
3. \(30\) km/h
4. \(35\) km/h