The change in the value of $g$ at a height '$h$' above the surface of the earth is the same as at a depth '$d$' below the surface of earth. When both '$d$' and '$h$' are much smaller than the radius of earth, then which one of the following is correct?
(1) $d = \frac{h}{2}$
(2) $\mathrm{d} = \frac{3\mathrm{h}}{2}$
(3) $d = 2\mathrm{h}$
(4) $d = h$

A particle of mass 10 g is kept on the surface of a uniform sphere of mass 100 kg and radius 10 cm. Find the work to be done against the gravitational force between them to take the particle far away from the sphere (you may take $\mathrm{G} = 6.67 \times 10^{-11} \mathrm{Nm}^2/\mathrm{kg}^2$)
(1) $13.34 \times 10^{-10} \mathrm{~J}$
(2) $3.33 \times 10^{-10} \mathrm{~J}$
(3) $6.67 \times 10^{-9} \mathrm{~J}$
(4) $6.67 \times 10^{-10} \mathrm{~J}$

The function $\sin^2(\omega t)$ represents
(1) a periodic, but not simple harmonic motion with a period $2\pi/\omega$
(2) a periodic, but not simple harmonic motion with a period $\pi/\omega$
(3) a simple harmonic motion with a period $2\pi/\omega$
(4) a simple harmonic motion with a period $\pi/\omega$

Two simple harmonic motions are represented by the equation $\mathrm{y}_1 = 0.1\sin\left(100\pi t + \frac{\pi}{3}\right)$ and $y_2 = 0.1\cos\pi t$. The phase difference of the velocity of particle 1 w.r.t. the velocity of the particle 2 is
(1) $-\pi/6$
(2) $\pi/3$
(3) $-\pi/3$
(4) $\pi/6$

If a simple harmonic motion is represented by $\frac{d^2x}{dt^2} + \alpha x = 0$, its time period is
(1) $\frac{2\pi}{\alpha}$
(2) $\frac{2\pi}{\sqrt{\alpha}}$
(3) $2\pi\alpha$
(4) $2\pi\sqrt{\alpha}$

A fish looking up through the water sees the outside world contained in a circular horizon. If the refractive index of water is $4/3$ and the fish is 12 cm below the surface, the radius of this circle in cm is
(1) $36\sqrt{7}$
(2) $36/\sqrt{7}$
(3) $36\sqrt{5}$
(4) $4\sqrt{5}$

A thin glass (refractive index 1.5) lens has optical power of $-5\,D$ in air. Its optical power in a liquid medium with refractive index 1.6 will be
(1) 1 D
(2) $-$1 D
(3) 25 D
(4) None of these

A Young's double slit experiment uses a monochromatic source. The shape of the interference fringes formed on a screen is
(1) hyperbola
(2) circle
(3) straight line
(4) parabola

Two point white dots are 1 mm apart on a black paper. They are viewed by eye of pupil diameter 3 mm. Approximately, what is the maximum distance at which these dots can be resolved by the eye? [Take wavelength of light $= 500\,\mathrm{nm}$]
(1) 5 m
(2) 1 m
(3) 6 m
(4) 3 m

When an unpolarized light of intensity $I_0$ is incident on a polarizing sheet, the intensity of the light which does not get transmitted is
(1) $\frac{1}{2}\mathrm{I}_0$
(2) $\frac{1}{4}\mathrm{I}_0$
(3) zero
(4) $I_0$

If $\mathrm{I}_0$ is the intensity of the principal maximum in the single slit diffraction pattern, then what will be its intensity when the slit width is doubled?
(1) $2\mathrm{I}_0$
(2) $4I_0$
(3) $I_0$
(4) $I_0/2$

A photocell is illuminated by a small bright source placed 1 m away. When the same source of light is placed $\frac{1}{2}\mathrm{~m}$ away, the number of electrons emitted by photo cathode would
(1) decrease by a factor of 4
(2) increase by a factor of 4
(3) decrease by a factor of 2
(4) increase by a factor of 2

If the kinetic energy of a free electron doubles. Its deBroglie wavelength changes by the factor
(1) $\frac{1}{2}$
(2) 2
(3) $\frac{1}{\sqrt{2}}$
(4) $\sqrt{2}$

The intensity of gamma radiation from a given source is I. On passing through 36 mm of lead, it is reduced to $\frac{I}{8}$. The thickness of lead which will reduce the intensity to $\frac{I}{2}$ will be
(1) 6 mm
(2) 9 mm
(3) 18 mm
(4) 12 mm

Starting with a sample of pure ${}^{66}\mathrm{Cu}$, $7/8$ of it decays into Zn in 15 minutes. The corresponding half-life is
(1) 10 minutes
(2) 15 minutes
(3) 5 minutes
(4) $7\frac{1}{2}$ minutes

If radius of ${}_{13}^{27}\mathrm{Al}$ nucleus is estimated to be 3.6 Fermi then the radius ${}_{52}^{125}\mathrm{Te}$ nucleus be nearly
(1) 6 fermi
(2) 8 fermi
(3) 4 fermi
(4) 5 fermi

A nuclear transformation is denoted by $X(n,\alpha){}_{3}^{7}\mathrm{Li}$. Which of the following is the nucleus of element $X$?
(1) ${}^{12}\mathrm{C}_6$
(2) ${}_{5}^{10}\mathrm{B}$
(3) ${}_{5}^{9}\mathrm{B}$
(4) ${}_{4}^{11}\mathrm{Be}$

A particle located at $x = 0$ at time $t = 0$, starts moving along the positive $x$-direction with a velocity '$v$' that varies as $v = \alpha \sqrt{x}$. The displacement of the particle varies with time as
(1) $t^{3}$
(2) $t^{2}$
(3) $t$
(4) $t^{1/2}$

A body falling from rest under gravity passes a certain point $P$. It was at a distance of 400 m from $P$, $4$ s prior to passing through $P$. If $g = 10$ m/s$^2$, then the height above the point P from where the body began to fall is
(1) 720 m
(2) 900 m
(3) 320 m
(4) 680 m

A mass of M kg is suspended by a weightless string. The horizontal force that is required to displace it until the string makes an angle of $45^{\circ}$ with the initial vertical direction is
(1) $Mg(\sqrt{2} - 1)$
(2) $Mg(\sqrt{2} + 1)$
(3) $Mg\sqrt{2}$
(4) $\frac{Mg}{\sqrt{2}}$

A player caught a cricket ball of mass 150 g moving at a rate of $20$ m/s. If the catching process is completed in 0.1 s, the force of the blow exerted by the ball on the hand of the player is equal to
(1) 300 N
(2) 150 N
(3) 3 N
(4) 30 N

A particle of mass 100 g is thrown vertically upwards with a speed of $5$ m/s. The work done by the force of gravity during the time the particle goes up is
(1) 0.5 J
(2) $-0.5$ J
(3) $-1.25$ J
(4) 1.25 J

A ball of mass 0.2 kg is thrown vertically upwards by applying a force by hand. If the hand moves 0.2 m while applying the force and the ball goes upto 2 m height further, find the magnitude of the force. Consider $g = 10$ m/s$^2$
(1) 22 N
(2) 4 N
(3) 16 N
(4) 20 N

The potential energy of a 1 kg particle free to move along the $x$-axis is given by $$V(x) = \left(\frac{x^4}{4} - \frac{x^2}{2}\right) \text{J}$$ The total mechanical energy of the particle is 2 J. Then, the maximum speed (in m/s) is
(1) 2
(2) $3/\sqrt{2}$
(3) $\sqrt{2}$
(4) $1/\sqrt{2}$

A bomb of mass 16 kg at rest explodes into two pieces of masses of 4 kg and 12 kg. The velocity of the 12 kg mass is $4$ ms$^{-1}$. The kinetic energy of the other mass is
(1) 96 J
(2) 144 J
(3) 288 J
(4) 192 J