Q. Three circularly shaped linear polarisers are placed coaxially. The transmission axis of the first polariser is at ${30}^o$, the second one is at ${60}^o$ and the third at ${90}^o$ to the vertical all in the clockwise sense. Each polariser additionally absorbs $10\%$ of the light. If a vertically polarised beam of light of intensity $I=100W/m^2$ is incident on this assembly of polarisers, then the final intensity of the transmitted light will be close to:

1. $20W/m^2$
2. $10W/m^2$
3. $30W/m^2$
4. $50W/m^2$

Q. A bomb explodes at time $t=0$ in a uniform, isotropic medium of density $\rho$ and releases energy E, generating a spherical blast wave. The radius R of this blast wave varies with time t as.

1. $t^{2/5}$
2. t
3. $t^{1/4}$
4. $t^{3/2}$

Q. A rope of mass $5kg$ is hanging between two supports as shown. The tension at the lowest point of the rope is close to (take $g=10m/s^2$).

1. $22N$
2. $44N$
3. $28N$
4. $14N$

Q. A small ring is rolling without slipping on the circumference of a large bowl as shown in the figure. The ring is moving down at $P_1$, comes down to the lower most point $P_2$ and is climbing up at $P_3$. Let $\overrightarrow{v}$CM denote the velocity of the centre of mass of the ring. Choose the correct statement regarding the frictional force on the ring.

1. It is opposite to $\overrightarrow{v}CM$ at the points $P_1,P_2$ and $P_3$
2. It is zero at the points $P_1,P_2$ and $P_3$
3. It is opposite to $\overrightarrow{v}CM$ at $P_1$, and in the same direction as $\overrightarrow{v}CM$ at $P_3$
4. It is in the same direction as $\overrightarrow{v}CM$ at $P_1$, and opposite to $\overrightarrow{v}CM$ at $P_3$.

Q. A massive black hole of mass m and radius R is spinning with angular velocity $\omega$. The power P radiated by it as gravitational waves is given by $P=G{c}^{-5}{m}^x{R}^y{\omega }^z$, where c and G are speed of light in free space, and the universal gravitational constant, respectively. Then-

1. $x=-1,y=2,z=4$
2. $x=1,y=1,z=4$
3. $x=-1,y=4,z=4$
4. $x=2,y=4,z=6$

Q. When light shines on a $p-n$ junction diode, the current $\left(I\right)vs$. voltage $\left(V\right)$ is observed as in the figure below:
In which quadrant(s) does the diode generate power, so that it can be used as a solar cell?

1. Quad $1$ and $3$ only
2. Quad $1$ only
3. Quad $4$ only
4. Quad $1$ and $4$ only

Q. The black shapes in the figure are closed surfaces. The electric field lines are in red. For which case the net flux through the surfaces is non-zero?

1. In all cases net flux is non-zero
2. Only (c) and (d)
3. Only (a) and (b)
4. Only (b), (c) and (d)

Q. A solid cube of wood of side $2a$ and mass $M$ is resting on a horizontal surface as shown in the figure. The cube is free to rotate about the fixed axis $AB$. A bullet of mass $m(<<M)$ and speed $v$ is shot horizontally at the face opposite to $ABCD$ at a height $h$ above the surface to impart the cube an angular speed ${\omega }_c$ so that the cube just topples over. Then ${\omega }_c$ is (note: the moment of inertia of the cube about an axis perpendicular to the face and passing through the centre of mass is $2M{a}^3/3)$.

1. $\sqrt{3gM/2ma}$
2. $\sqrt{3g/4h}$
3. $\sqrt{3g(\sqrt{2}-1)/2a}$
4. $\sqrt{3g(\sqrt{2}-1)/4a}$

Q. The distance s travelled by a particle in time t is $s=ut-\frac{1}{2}g{t}^2$. The initial velocity of the particle was measured to be $u=1.11\pm 0.01$ m/s and the time interval of the experimental was $t=1.01\pm 0.1s$. The acceleration was taken to be $g=9.8\pm 0.1m/s^2$. With these measurements, the student estimates the total distance travelled. How should the student report the result?

1. $1.121\pm 0.1m$
2. $3.8\pm 1.1m$
3. $1.12\pm 0.07m$
4. $1.1\pm 0.07m$

Q. Consider the following statements for air molecules in an air tight container.
(I) The average speed of molecules is larger than root mean square speed.
(II) Mean free path of molecules is larger than the mean distance between molecules.
(III) Mean free path of molecules increases with temperature.
(IV) The rms speed of nitrogen molecule is smaller than oxygen molecule. The true statements are.

1. Only II
2. II & III
3. II & IV
4. I, II & IV

Q. A uniform rope of total length l is at rest on a table with fraction f of its length hanging (see figure). If the coefficient of friction between the table and the chain is $\mu$ then which of the following relation is true:

1. $f=\mu$
2. $f=1/(1+\mu )$
3. $f=1/(1+\frac{1}{\mu })$
4. $f=1/(\mu +1/\mu )$

Q. One mole of an ideal monatomic gas undergoes the following four reversible processes:
$Step1$ - it is first compressed adiabatically from volume $8.0m^3$ to $1.0m^3$.
$Step2$ - then expanded isothermally at temperature $T_1$ to volume $10.0m^3$.
$Step3$ - then expanded adiabatically to volume $80.0m^3$.
$Step4$ - then compressed isothermally at temperature $T_2$ to volume $8.0m^3$.
Then $T_1/T_2$ is

1. $2$
2. $6$
3. $4$
4. $8$

Q. A star of mass M and radius R is made up of gases. The average gravitational pressure compressing the star due to gravitational pull of the gases making up the star depends on R as.

1. $1/R$
2. $1/R^6$
3. $1/R^2$
4. $1/R^4$

Q. Four identical beakers contain same amount of water as shown below. Beaker $a$ contains only water. A lead ball is held submerged in the beaker $b$ by string from above. A same sized plastic ball, say a table tennis $\left(TT\right)$ ball, is held submerged in beaker $c$ by a string attached to a stand from outside. Beaker $d$ contains same sized $TT$ ball which is held submerged from a string attached to the bottom of the beaker. These beakers (without stand) are placed on weighing pans and register readings $W_a,W_b,W_c$ and $W_d$ for $a,b,c$ and $d$, respectively. (Effects of the mass and volume of the stand and string are to be neglected.)

1. $W_a=W_b=W_c=W_d$
2. $W_b=W_c>W_d>W_a$
3. $W_b>W_c>W_d>W_a$
4. $W_b=W_c>W_a>W_d$

Q. Let the electrostatic field E at distance r from a point charge q not be an inverse square but, instead an inverse cubic, e.g., $\overrightarrow{E}=k\frac{q}{r^3}\hat{r}$. Here k is a constant. Consider the following two statements.
(i) Flux through a spherical surface enclosing the charge is $\Phi =q_{enclosed}/{ϵ}_0$
(ii) A charge placed inside uniformly charged shell will experience a force.
Choose the correct option.

1. Only (i) is valid
2. Only (ii) is valid
3. Both (i) and (ii) are invalid
4. Both (i) and (ii) are valid

Q. Some of the wavelengths observed in the emission spectrum of neutral hydrogen gas are: $912,1026,1216,3646,6563\stackrel{o}{A}$. If broad band light is passing through neutral hydrogen gas at room temperature, the wavelength that will not be absorbed strongly is?

1. $1026\stackrel{o}{A}$
2. $1216\stackrel{o}{A}$
3. $3646\stackrel{o}{A}$
4. $912\stackrel{o}{A}$

Q. A uniform thin wooden plank $AB$ of length $L$ and mass $M$ is kept on a table with its $B$ end slightly outside the edge of the table. When an impulse $J$ is given to the end $B$, the plank moves up with centre of mass rising a distance $h$ from the surface of the table. Then,

1. $h>9J^2/8M^{2}g$
2. $h=J^2/2M^{2}g$
3. $J^2/2M^{2}g<h<9J^2/8M^{2}g$
4. $h<J^2/2M^{2}g$

Q. A particle of charge q and mass m enters a region of a transverse electric field of $E_0\hat{j}$ with initial velocity $v_o\hat{i}$. The time taken for the change in the de Broglie wavelength of the charge from the initial value of ${\lambda }_0$ to ${\lambda }_0/3$ is proportional to.

1. $\frac{q}{m}$
2. $\frac{m}{q}$
3. $\sqrt{\frac{q}{m}}$
4. $\sqrt{\frac{m}{q}}$

Q. The emission series of hydrogen atom is given by $\frac{1}{\lambda }=R(\frac{1}{n_1^2}-\frac{1}{n_2^2})$ where $R$ is the Rydberg constant. For a transition from $n_2$ to $n_1$, the relative change $△\lambda /\lambda$ in the emission wavelength if hydrogen is replaced by deuterium (assume that the mass of proton and neutron are the same and approximately $2000$ times larger than that of electrons) is

1. $0.025$%
2. $0.005$%
3. $0.0025$%
4. $0.05$%

Q. Back surface of a glass (refractive index $n$ and thickness $t$) is polished to work as a mirror as shown below. A laser beam falls on it and is partially reflected and refracted at the air-glass interface and fully reflected at the mirror surface respectively. A pattern of discrete spots of light is observed on the screen.
The spacing between the spots on the screen will be

1. $\frac{t\sin 2\theta }{\sqrt{n^2-{\sin }^2\theta }}$
2. $\frac{2t\sin \theta }{\sqrt{n^2-{\sin }^2\theta }}$
3. $\frac{2t\tan \theta }{\sqrt{n^2-{\sin }^2\theta }}$
4. $\frac{2t\sin \theta }{\sqrt{1-\frac{{\sin }^2\theta }{n^2}}}$

Q. Consider the $R-L-C$ circuit given below. The circuit is driven by a $50HzAC$ source with peak voltage $220V$. If $R=400\Omega ,C=200\mu F$ and $L=6H$, the maximum current in the circuit is closest to

1. $0.55A$
2. $1.2A$
3. $0.120A$
4. $5.5A$

Q. Consider the following nuclear reactions.
I. ${}_7^{14}N+{}_2^{4}He\to {}_8^{17}O+X$
II. ${}_4^{9}Be+{}_2^{4}H\to {}_6^{12}He+Y$. Then.

1. X and Y are both neutrons
2. X and Y are both protons
3. X is a proton and Y is a neutron
4. X is a neutron and Y is a proton

Q. Consider a plane parallel beam of light incident on a plano-cylindrical lens as shown. Which of the following will you observe on a screen placed at the focal plane of the lens?

1. There will be a single bright line on the screen parallel to the x-axis
2. There will be a single bright spot on the screen
3. There will be a single bright line on the screen parallel to the y-axis
4. The screen will be uniformly illuminated

Q. One end of a rod of length L is fixed to a point on the circumference of a wheel of radius R. The other end is sliding freely along a straight channel passing through the centre O of the wheel as shown in the figure. The wheel is rotating with a constant angular velocity $\omega$ about O. Taking $T=\frac{2\pi }{P\omega }$ the motion of the rod is?

1. Simple harmonic with a period of $T/2$
2. Simple harmonic with a period of T
3. Not simple harmonic but periodic with a period of T
4. Not simple harmonic but periodic with a period of $T/2$

Q. A planet of radius $R_p$ is revolving around a star of radius $R^{\ast }$, which is at temperature $T^{\ast }$. The distance between the star and the planet is d. If the planet's temperature is $f{T}^{\ast }$, then f is proportional to.

1. $\sqrt{R^{\ast }/d}$
2. $R^{\ast }/d$
3. $(R^{\ast }/d{)}^4$
4. $R^{\ast }{R}_p/d^2$

Q. The n-side of the depletion layer of a p-n junction.

1. Always has same width as of the p-side
2. Is negatively charged
3. Has no bound charges
4. Is positively charged

Q. Consider the following statements regarding the photoelectric effect experiment:
(I) Photo electrons are emitted as soon as the metal is exposed to light.
(II) There is a minimum frequency below which no photo-current is observed.
(III) The stopping potential is proportional to the frequency of light.
(IV) The photo-current varies linearly with the intensity of the light.
Which of the above statements indicate that light consists of quanta (photons) with energy proportional to frequency?

1. I and III only
2. II and III only
3. I, II and III only
4. II, III and IV only

Q. A square-shaped wire loop of mass $m$, resistance $R$ and side a moving with speed $v_0$, parallel to the x-axis, enters a region of uniform magnetic field $B$, which is perpendicular to the plane of the loop. The speed of the loop changes with distance $x(x<a)$ in the field, as

1. $v_0-\frac{B^2{a}^2}{Rm}x$
2. $v_0-\frac{B^2{a}^2}{2Rm}x$
3. $v_0+\frac{B^{2}a}{Rm}x$
4. $v_0$

Q. A light beam travelling along the x axis with planar wavefront is incident on a medium of thickness t. In the region, where light is falling the refractive index can be taken to be varying such that $\frac{dn}{dy}>0$. The light beam on the other side of the medium will emerge.

1. Parallel to the x-axis
2. Bending downward
3. Bending upward
4. Split into two or more beams

Q. A closed pipe of length $300$cm contains some sand. A speaker is connected at one of its ends. The frequency of the speaker at which the sand will arrange itself in $20$ equidistant piles is close to(velocity of sound is $300$m/s).

1. $5$kHz
2. $100$kHz
3. $1$kHz
4. $10$kHz