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1.

A particle moves at a constant speed along the circumference of a circle with a radius \(R,\) subject to a central fictitious force \(F\) that is inversely proportional to \(R^3.\) Its time period of revolution will be given by:

1.	\( T \propto R^2 \)	2.	\( T \propto R^{\frac{3}{2}} \)
3.	\( T \propto R^{\frac{5}{2}} \)	4.	\(T \propto R^{\frac{4}{3}} \)

2.

The trajectory of a projectile in a vertical plane is \(y=\alpha x-\beta x^2\), where \(\alpha\) and \(\beta\) are constants and \(x\) & \(y\) are respectively the horizontal and vertical distances of the projectile from the point of projection. The angle of projection \(\theta\) will be:
1. \( \tan ^{-1} \alpha \)
2. \( \tan ^{-1} \beta \)
3. \( \tan ^{-1} 2 \beta \)
4. \( \tan ^{-1} 2 \alpha\)

3. The following diagram represents three vectors of equal magnitudes in a plane.
             
What arrow best represents the direction of the sum of three vectors?
1. \(\rightarrow\)
2. \(\leftarrow\)
3. \(\uparrow\)
4. \(\downarrow\)

4. Two particles move uniformly along the periphery of a circle, one completing a revolution in \(1~\text s\) while the other does it in \(3~\text s.\) If they start from the same point and move in opposite directions, they will meet, for the first time, in:

1.	less than \(1~\text s\)	2.	\(2~\text s\)
3.	\(3~\text s\)	4.	more than \(3~\text s\)

5. A simple pendulum consists of a bob of mass \(m\) attached to a suspension point \(O\) by means of a light string of length \(l.\) It is set oscillating, as shown in the figure. The tension in the string, when the bob is at its lowest point, is (i.e. at \(B\)):

1. \(mg\)
2. \(\dfrac{mg}{2}\)
3. less than \(mg\)
4. greater than \(mg\)

6. A car is going up a slight slope decelerating at \(0.1\) m/s². It comes to a stop after going for \(5\) s. What was its initial velocity? 
1. \(0.02~\text {m/s}\)
2. \(0.25~\text {m/s}\)
3. \(0.5~\text{m/s}\)
4. \(1.0~\text{m/s}\)

7.

A particle moves around a circle with a unique uniform speed in each revolution. After the first revolution and during the \(2\)^nd revolution: its speed doubles; and during the \(3\)^rd revolution, its speed becomes \(3\) times the initial speed and so on. The time for the \(1\)^st revolution is \(12\) sec. The average time per revolution, for the first four revolutions, is:
1. \(4.8\) s
2. \(9.6\) s
3. \(6.25\) s
4. \(6\) s

8.

Planck's constant (\(h\)), speed of light in the vacuum (\(c\)), and Newton's gravitational constant (\(G\)) are the three fundamental constants. Which of the following combinations of these has the dimension of length?

1.	\(\dfrac{\sqrt{hG}}{c^{3/2}}\)	2.	\(\dfrac{\sqrt{hG}}{c^{5/2}}\)
3.	\(\dfrac{\sqrt{hG}}{G}\)	4.	\(\dfrac{\sqrt{Gc}}{h^{3/2}}\)

9.

If vectors \(\overrightarrow{{A}}=\cos \omega t \hat{{i}}+\sin \omega t \hat{j}\) and \(\overrightarrow{{B}}=\cos \left(\frac{\omega t}{2}\right)\hat{{i}}+\sin \left(\frac{\omega t}{2}\right) \hat{j}\) are functions of time. Then, at what value of \(t\) are they orthogonal to one another?

1.	\(t = \frac{\pi}{4\omega}\)	2.	\(t = \frac{\pi}{2\omega}\)
3.	\(t = \frac{\pi}{\omega}\)	4.	\(t = 0\)

10. A girl of mass \(45~\text{kg}\) stands on a weighing machine \((A) \) which is placed on top of a second weighing machine \((B).\) The weighing machines each weigh \(5~\text{kg}.\) Assume that the readings of the weighing machines can be seen easily. The readings of \(A\) and \(B\) are: (take \(g=10~\text{m/s}^2,\) if required)

1.	\(45~\text{kg},~50~\text{kg}\)	2.	\(50~\text{kg},~55~\text{kg}\)
3.	\(47.5~\text{kg},~52.5~\text{kg}\)	4.	\(45~\text{kg},~45~\text{kg}\)

11. The angle of \(1'\) (minute of an arc) in radian is nearly equal to:
1. \(2.91 \times 10^{-4}~\text{rad} \)
2. \(4.85 \times 10^{-4}~\text{rad} \)
3. \(4.80 \times 10^{-6} ~\text{rad} \)
4. \(1.75 \times 10^{-2}~\text{rad} \)

12. With what minimum acceleration can a fireman slide down a rope whose breaking strength is \(\frac{2}{5}^{\text{th}}\) of his weight?
1. \(1~g\)
2. \(0.4~g\)
3. \(0.6~g\)
4. \(0.8~g\)

13.

The magnitude of the vector 
\(\widehat i+\widehat i\times\widehat j+(\widehat i\times\widehat j)\times\widehat i+((\widehat i\times\widehat j)\times\widehat i)\times\widehat j\):
1. \(1\)
2. \(\sqrt2\)
3. \(\sqrt3\)
4. \(2\)

14.

A stone is thrown vertically downwards with an initial velocity of \(40\) m/s from the top of a building. If it reaches the ground with a velocity of \(60\) m/s, then the height of the building is: (take \(g=10\) m/s²)

1.	\(120\) m	2.	\(140\) m
3.	\(80\) m	4.	\(100\) m

15.

A mass is projected from the ground with a certain velocity, at an angle with the horizontal. Neglecting air resistance, we can say that its speed will be:

1.	constant during the flight
2.	decreasing continuously during the flight
3.	minimum at the highest point
4.	maximum at the highest point

16. The acceleration-time graph of a particle is shown in the figure. What is the velocity of the particle at \(t= 8~\text{s}\) if the initial velocity of the particle is \(3~\text{m/s}\)?
        
1. \(4~\text{m/s}\)
2. \(5~\text{m/s}\)
3. \(6~\text{m/s}\)
4. \(7~\text{m/s}\)

17. The error in the measurement of distance is \(2\%,\) while that in time is  \(1\%.\) The error in speed, computed from the above, is:

1.	\(2\%+1\%\)	2.	\(2\%-1\%\)
3.	\(\dfrac{2\%+1\%}{2}\)	4.	\(\dfrac{2\%-1\%}{2}\)

18. Match Column-I with Column-II and choose the correct option.

Column-I		Column-II
I.	Torque	(a)	\([M^0LT^{-2}]\)
II.	Stress	(b)	\([ML^{-1}T^{-1}]\)
III.	Coefficient of viscosity	(c)	\([ML^{-1}T^{-2}]\)
IV.	Gravitational potential gradient	(d)	\([ML^2T^{-2}]\)

Choose the correct option from the given ones:

1.	I → (a), II → (c), III → (b), IV → (d)
2.	I → (d), II → (b), III → (c), IV → (a)
3.	I → (d), II → (c), III → (b), IV → (a)
4.	I → (a), II → (c), III → (d), IV → (b)

19. A force \(F\) is acting at an angle of \(60^\circ,\) on a block of mass \(m\) resting on a smooth horizontal plane. If \(F=mg,\) then the acceleration of the block will be: (\(g\) is the acceleration due to gravity)
       

1.	\(g,\) at \(60^\circ\) above the horizontal
2.	\(g,\) at \(60^\circ\) below the horizontal
3.	\(\dfrac g2\) along the horizontal
4.	\(\dfrac {\sqrt3g}{2}\) along the horizontal

20. If the magnitude of the sum of two vectors is equal to the magnitude of the difference of the two vectors, then the angle between these vectors is:
1. \(0^\circ\)
2. \(45^\circ\)
3. \(90^\circ\)
4. \(180^\circ\)