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1.

A ship A is moving westwards with a speed of 10 km ${\mathrm{h}}^{-1}$ and a ship B, 100 km south of A is moving northwards with a speed of 10 km ${\mathrm{h}}^{-1}$. The time after which the distance between them becomes the shortest, is:
1. 5 hr
2. $5\sqrt{2}$ hr
3. $10\sqrt{2}$ hr
4. 0 hr

2.

A particle is projected with a velocity u making an angle $\mathrm{\theta }$ with the horizontal. At any instant, its velocity v is at right angles to its initial velocity u; then v is:
1.  ucos$\mathrm{\theta }$
2.  utan$\mathrm{\theta }$
3.  ucot$\mathrm{\theta }$
4.  usec$\mathrm{\theta }$

3.

A particle moves in a circular path with decreasing speed. Choose the correct statement. 
(1) Angular momentum remains constant
(2) Acceleration ($\overrightarrow{a}$) is towards the center
(3) Particle moves in a spiral path with decreasing radius
(4) The direction of angular momentum remains constant

4.

A river is flowing from W to E with a speed of 5 m/min. A man can swim in still water with a velocity 10 m/min. In which direction should the man swim so as to take the shortest possible path to go to the south. 
1. 30° with downstream
2. 60° with downstream
3. 120° with downstream
4. South

5.

For a particle in a non-uniform accelerated circular motion 
1. Velocity is radial and acceleration is transverse only
2. Velocity is transverse and acceleration is radial only
3. Velocity is radial and acceleration has both radial and transverse components
4. Velocity is transverse and acceleration has both radial and transverse components

6.

A projectile is given an initial velocity of $\hat{i}+2\hat{j}$. The cartesian equation of its path is (g = 10 ${\mathrm{ms}}^{-2}$) 
1. $\mathrm{y}=2\mathrm{x}-5{\mathrm{x}}^2$
2. $\mathrm{y}=\mathrm{x}-5{\mathrm{x}}^2$
3. $4\mathrm{y}=2\mathrm{x}-5{\mathrm{x}}^2$
4. $\mathrm{y}=2\mathrm{x}-25{\mathrm{x}}^2$

7. A wheel is subjected to uniform angular acceleration about its axis. Initially, its angular velocity is zero. In the first \(2\) s, it rotates through an angle \(\theta_1\). In the next \(2\) s, it rotates through an additional angle \(\theta_2\). The ratio of \(\frac{\theta_2}{\theta_1}\) is: 
1. \(1\)
2. \(2\)
3. \(3\)
4. \(4\)

8.

A bomber plane moves horizontally with a speed of \(500~\text{m/s}\) and a bomb is released from it. The bomb strikes the ground in \(10~\text{s}\). The angle at which it strikes the ground will be: \((g= 10~\text{m/s}^2)\)
1. \(\tan ^{-1}\left(\frac{1}{5}\right )\)
2. \(\tan \left(\frac{1}{5}\right)\)
3. \(\tan ^{-1}(1)\)
4. \(\tan ^{-1}(5)\)

9.

A man standing on a road holds his umbrella at 30° with the vertical to keep the rain away. He throws the umbrella and starts running at 10 km/hr. He finds that raindrops are hitting his head vertically, the speed of raindrops with respect to the road will be:
1. 10 km/hr
2. 20 km/hr
3. 30 km/hr
4. 40 km/hr

10.

Which of the following sets of factors will affect the horizontal distance covered by an athlete in a long–jump event?
1. speed before he jumps and his weight
2. the direction in which he leaps and the initial speed
3. the force with which he pushes the ground and his speed
4. none of the above

11.

If the range of a gun that fires a shell with muzzle speed v is R, then the angle of elevation of the gun is 
(1) ${\cos }^{-1}\left(\frac{{\displaystyle v^2}}{{\displaystyle Rg}}\right)$
(2) ${\cos }^{-1}\left(\frac{{\displaystyle gR}}{{\displaystyle v^2}}\right)$
(3) $\frac{1}{2}\left(\frac{{\displaystyle v^2}}{{\displaystyle Rg}}\right)$
(4) $\frac{1}{2}{\sin }^{-1}\left(\frac{{\displaystyle gR}}{{\displaystyle v^2}}\right)$

12.

A particle moves so that its position vector is given by \(r=\cos \omega t \hat{x}+\sin \omega t \hat{y}\) where \(\omega\) is a constant. Based on the information given, which of the following is true?

1.	The velocity and acceleration, both are parallel to \(r.\)
2.	The velocity is perpendicular to \(r\) and acceleration is directed towards the origin.
3.	The velocity is not perpendicular to \(r\) and acceleration is directed away from the origin.
4.	The velocity and acceleration, both are perpendicular to \(r.\)

13.

A body is projected with a velocity \(u\) with an angle of projection \(\theta.\) The change in velocity after the time \((t)\) from the time of projection will be:

1.	\(gt\)	2.	\(\frac{1}{2}gt^2\)
3.	\(u\sin\theta\)	4.	\(u\cos\theta\)

14.

For a given velocity, a projectile has the same range of R for two angles of projection. If t₁ and t₂ are the times of flight in the two cases then:
(1) $t_1{t}_2\propto R^2$
(2) $t_1{t}_2\propto R$
(3) $t_1{t}_2\propto \frac{1}{R}$
(4) $t_1{t}_2\propto \frac{1}{R^2}$

15.

Two particles having position vectors \(\overrightarrow{r_{1}} = \left( 3 \hat{i} + 5 \hat{j}\right)\) metres and \(\overrightarrow{r_{2}} = \left(- 5 \hat{i} - 3 \hat{j} \right)\) metres are moving with velocities \(\overrightarrow{v}_{1} = \left( 4 \hat{i} + 3 \hat{j}\right)\)\(\text{m/s}\) and \(\overrightarrow{v}_{2} = \left(\alpha\hat{i} + 7 \hat{j} \right)\)\(\text{m/s}\). If they collide after \(2\) seconds, the value of \(\alpha\) is:

1.	\(2\)	2.	\(4\)
3.	\(6\)	4.	\(8\)