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1.

A body starts moving from rest on a horizontal ground such that the position vector of the body with respect to its starting point is given by \(r= 2 t\hat{i}+3t^2\hat j\). The equation of the trajectory of the body is:
1. \(y =1.5x\)
2. \(y =0.75x^2\)
3. \(y =1.5x^2\)
4. \(y =0.45x^2\)

2. For any arbitrary motion in space, which of the following relations is true?

1.	\(\vec{v}_{\text {avg }}=\frac{1}{2}\left[\vec{v}\left(t_1\right)+\vec{v}\left(t_2\right)\right]\)
2.	\(\vec{v}(t)=\vec{v}(0)+\vec{a} t\)
3.	\(\vec{r}({t})=\vec{r}(0)+\vec{v}(0){t}+\frac{1}{2} \vec{a}{t}^2\)
4.	\(\vec{v}_{\text {avg }}=\frac{\left[\vec{r}\left(t_2\right)-\vec{r}\left(t_1\right)\right]}{\left(t_2-t_1\right)}\)

3.

A particle moves in space such that:
\(x=2t^3+3t+4;~y=t^2+4t-1;~z=2\sin\pi t\)
where \(x,~y,~z\)  are measured in meters and \(t\) in seconds. The acceleration of the particle at \(t=3\) seconds will be:

1.	\(36 \hat{i}+2 \hat{j}+\hat{k} \) ms-2
2.	\(36 \hat{i}+2 \hat{j}+\pi \hat{k} \) ms-2
3.	\(36 \hat{i}+2 \hat{j} \) ms-2
4.	\(12 \hat{i}+2 \hat{j} \) ms-2

4.

A particle starts moving with constant acceleration with initial velocity (\(\hat{\mathrm{i}}+5\hat{\mathrm{j}}\)$\stackrel{}{\mathrm{}}$) m/s. After \(4\) seconds, its velocity becomes (\(3\hat{\mathrm{i}}-2\hat{\mathrm{j}}\)) m/s. The magnitude of its displacement in 4 seconds is:
1. \(5\) m
2. \(10\) m
3. \(15\) m
4. \(20\) m

5.

A river is flowing from east to west at a speed of \(5\) m/min. A man on south bank of river, capable of swimming \(10\) m/min in still water, wants to swim across the river in shortest time. He should swim:

1.	Due north
2.	Due north-east
3.	Due north-east with double the speed of the river
4.	None of the above

6.

In the given figure, \(a=15\) m/s² represents the total acceleration of a particle moving in the clockwise direction in a circle of radius \(R=2.5\) m at a given instant of time. The speed of the particle is:
           
1. \(4.5\) m/s
2. \(5.0\) m/s
3. \(5.7\) m/s
4. \(6.2\) m/s

7.

Two bullets are fired horizontally and simultaneously towards each other from the rooftops of two buildings (building being \(100~\text{m}\) apart and being of the same height of \(200~\text{m}\)) with the same velocity of \(25~\text{m/s}.\) When and where will the two bullets collide?
\((g = 10~\text{m/s}^2)\)

1.	After \(2~\text{s}\) at a height of \(180~\text{m}\)
2.	After \(2~\text{s}\) at a height of \(20~\text{m}\)
3.	After \(4~\text{s}\) at a height of \(120~\text{m}\)
4.	They will not collide.

8.

A car starts from rest and accelerates at \(5~\text{m/s}^{2}.\) At \(t=4~\text{s}\), a ball is dropped out of a window by a person sitting in the car. What is the velocity and acceleration of the ball at \(t=6~\text{s}?\) 
(Take \(g=10~\text{m/s}^2\))

1.	\(20\sqrt{2}~\text{m/s}, 0~\text{m/s}^2\)	2.	\(20\sqrt{2}~\text{m/s}, 10~\text{m/s}^2\)
3.	\(20~\text{m/s}, 5~\text{m/s}^2\)	4.	\(20~\text{m/s}, 0~\text{m/s}^2\)

9. For a smoothly running analog clock, the ratio of the number of rotations made in a day by the hour hand to the second hand, respectively, is:

1.	\(24 :1\)	2.	\(1:720\)
3.	\(1:60\)	4.	\(2:5\)

10. A bullet is fired from a gun at the speed of \(280~\text{ms}^{-1}\) in the direction \(30^\circ\) above the horizontal. The maximum height attained by the bullet is:\(\left(g=9.8~\text{m/s}^{2}, \sin30^{\circ}=0.5\right)\)

1.	\(3000~\text{m}\)	2.	\(2800~\text{m}\)
3.	\(2000~\text{m}\)	4.	\(1000~\text{m}\)

11.

A boy runs on a circular track of radius \(R\) (in km) with a speed of \(\dfrac{πR}{2}\) km/h in the clockwise direction for \(3\) h and then with \(πR\) km/h in the anticlockwise direction for \(1\) h. The magnitude of his displacement will be: 
1. \(\dfrac{πR}{2}\)
2. \(\dfrac{R}{\sqrt{2}}\)
3. \(\dfrac{3πR}{2}\)
4. \(\sqrt{2}R\)

12.

In a two-dimensional motion, the instantaneous speed of a particle remains constant at a positive value \(v_0.\) Which of the following statements must always be true?

1.	The particle has zero acceleration.
2.	The particle’s acceleration is increasing.
3.	The particle’s acceleration always lies in the plane of motion.
4.	The particle necessarily moves in a uniform circular path.

13. Given below are two statements: 

Assertion (A):	The maximum height of a projectile is always \(25\)% of the maximum range.
Reason (R):	For maximum height, the projectile should be projected at \(90^\circ.\)

1.	Both (A) and (R) are True and (R) is the correct explanation of (A).
2.	Both (A) and (R) are True but (R) is not the correct explanation of (A).
3.	(A) is True and (R) is False.
4.	(A) is False and (R) is True.

14.

A particle moves with constant angular velocity in a circle. During the motion its:

1.	Energy is conserved
2.	Momentum is conserved
3.	Energy and momentum both are conserved
4.	None of the above is conserved

15.

Six particles situated at the corners of a regular hexagon of side \(a\) move at constant speed \(v\). Each particle maintains a direction towards the particle at the next. The time which the particles will take to meet each other is:
1. \(\frac{2 a}{v}~\text{sec}\)
2. \(\frac{a}{v}~\text{sec}\)
3. \(\frac{2 a}{3v}~\text{sec}\)
4. \(\frac{3 a}{v}~\text{sec}\)