A cat is situated at point \(A\) (\(0,3,4\)) and a rat is situated at point \(B\) (\(5,3,-8\)$\mathrm{}$). The cat is free to move but the rat is always at rest. The minimum distance travelled by the cat to catch the rat is:
1. 5 unit
2. 12 unit
3. 13 unit
4. 17 unit

Two bullets are fired simultaneously horizontally and at different speeds from the same place. Which bullet will hit the ground first? (Air resistance is neglected)

A bus is going to the North at a speed of 30 kmph. It makes a 90° left turn without changing the speed. The change in the velocity of the bus is:

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. 1/8 h

2. 1/4 h

3. 1/2 h

4. 1 h

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 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:

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

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

The position vector of a particle is $\overrightarrow{\mathrm{r}}$ $=$ $\left(\mathrm{a}\right)$ $\sin$ $\mathrm{\omega t}\hat{\mathrm{i}}$ $+$ $\left(\mathrm{a}\right)$ $\cos$ $\mathrm{\omega t}\hat{\mathrm{j}}$. The velocity of the particle is: