Select the incorrect statement:

1.	It is possible to have \(\left|\frac{{d} \overrightarrow{v}}{dt}\right| = 0 \) and \(\frac{{d}|\overrightarrow{v}|}{{dt}} \neq 0 \)
2.	It is possible to have\(\left|\frac{{d} \overrightarrow{{v}}}{{dt}}\right| \neq 0 \) and \(\frac{{d}|\overrightarrow{{v}}|}{dt}=0 .\)
3.	it is possible to have\(\left|\frac{{d} \overrightarrow{v}}{{dt}}\right|=0\) and \(\frac{{d}|\overrightarrow{{v}}|}{dt}=0 . \)
4.	It is possible to have \(\left|\frac{{d} \overrightarrow{{v}}}{{dt}}\right| \neq 0\) and \(\frac{{d} \overrightarrow{{v}}}{{dt}} \neq 0 \)

A ball is projected horizontally from a cliff 40 m high at a speed of 40 m/s and simultaneously a ball is dropped. If the time taken by the two balls to reach the ground are ${\mathrm{t}}_1<mspace\; width="1em"></mspace>\mathrm{and}<mspace\; width="1em"></mspace>{\mathrm{t}}_2$ respectively (taking air friction into consideration), then

(1) ${\mathrm{t}}_1<mspace\; width="1em"></mspace>><mspace\; width="1em"></mspace>{\mathrm{t}}_2$

(2) ${\mathrm{t}}_1<mspace\; width="1em"></mspace><<mspace\; width="1em"></mspace>{\mathrm{t}}_2$

(3) ${\mathrm{t}}_1<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>{\mathrm{t}}_2$

(4) Depending on air friction ${\mathrm{t}}_1$ maybe less or more than ${\mathrm{t}}_2$

Six friends stand at the six vertices of a regular hexagonal park. Each friend always maintains a direction towards the friend at the next corner with a constant speed of \(2~\text{m/s}.\) They eventually meet after \(60~\text{s}.\) What is the length of each side of the hexagon?
1. \(120~\text{m}\)
2. \(30~\text{m}\)
3. \(240~\text{m}\)
4. \(60~\text{m}\)

Which of the following is an appropriate expression for the radius of curvature of a projectile at the highest point? (R $\to$ Range of projectile)

(1) $\frac{\mathrm{R}}{2}<mspace\; width="1em"></mspace>\cot <mspace\; width="1em"></mspace>\mathrm{\theta }$

(2) $\frac{\mathrm{R}}{2}<mspace\; width="1em"></mspace>\tan <mspace\; width="1em"></mspace>\mathrm{\theta }$

(3) 2R cot $\mathrm{\theta }$

(4) 2R tan $\mathrm{\theta }$

An aeroplane flies \(400\) m north and then \(300\) m west and then flies \(1200\) m upwards. Its net displacement is:

A ball is projected horizontally from the top of wall A of height 80 m with speed 10 m/s. At what distance from the wall ball hits the ground?

(1) 20 m

(2) 50 m

(3) 60 m

(4) 40 m

A person is rotating in a horizontal circle of radius 7 m. Find the period of rotation for which the acceleration is equal to 3g.

(1) 2.6 s

(2) 2.8 s

(3) 3.1 s

(4) 4.3 s

Which of the following is the angle between velocity and acceleration of a body in uniform circular motion?
1. \(30^\circ\)
2. \(45^\circ\)
3. \(60^\circ\)
4. \(90^\circ\)

An object of mass m is projected from the ground with a momentum \(p\) at such an angle that its maximum height is \(\frac{1}{4}\)th of its horizontal range. Its minimum kinetic energy in its path will be:

A body is executing uniform circular motion in a circle of radius 100 m at speed 10 m/s. Acceleration of the body is :

(1) Zero

(2) 2 $\mathrm{m}/{\mathrm{s}}^2$

(3) 1 $\mathrm{m}/{\mathrm{s}}^2$

(4) 5 $\mathrm{m}/{\mathrm{s}}^2$