A man is walking on a horizontal road with a speed of \(4\) km/h. Suddenly, the rain starts vertically downwards with a speed of \(7\) km/h. The magnitude of the relative velocity of the rain with respect to the man is:
1. \(\sqrt{33}\) km/h
2. \(\sqrt{65}\) km/h
3. \(8\) km/h
4. \(4\) 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 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. \(5\sqrt{2}\)

Raindrops are falling with speed \(v\) vertically downwards and a man is running on a horizontal road with speed \(u.\) The magnitude of the velocity of the raindrops with respect to the man is:
1. \(v-u\)
2. \(v+u\)
3. \(\sqrt{{v}^2 + {u}^2 \over 2}\)
4. \(\sqrt{{v}^2 + {u}^2}\)

A person, reaches a point directly opposite on the other bank of a flowing river, while swimming at a speed of \(5\) m/s at an angle of \(120^\circ\) with the flow. The speed of the flow must be:
1. \(2.5\) m/s
2. \(3\) m/s
3. \(4\) m/s
4. \(1.5\) m/s

A man can row a boat with a speed of \(10\) kmph in still water. The river flows at \(6\) kmph. If he crosses the river from one bank to the other along the shortest possible path, time taken to cross the river of width \(1\) km is:
1. \(\frac{1}{8}~\text{h}\)
2. \(\frac{1}{4}~\text{h}\)
3. \(\frac{1}{2}~\text{h}\)
4. \(1~\text{h}\)

Two particles are separated by a horizontal distance \(x\) as shown in the figure. They are projected at the same time as shown in the figure with different initial speeds. The time after which the horizontal distance between them becomes zero will be:

Two boys are standing at the ends \(A\) and \(B\) of the ground where \(AB =a.\) The boy at \(B\) starts running in a direction perpendicular to \(AB\) with velocity \(v_1.\) The boy at \(A\) starts running simultaneously with velocity \(v\) and catches the other boy in a time \(t,\) where \(t\) is:

Two men \(P\) & \(Q\) are standing at corners \(A\) & \(B\) of square \(ABCD\) of side \(8\) m. They start moving along the track with constant speed \(2\) m/s and \(10\) m/s respectively. The time when they will meet for the first time, is equal to:
     
1. \(2\) sec
2. \(3\) sec
3. \(1\) sec
4. \(6\) sec