Q. A donor atom in a semiconductor has a loosely bound electron. The orbit of this electron is considerably affected by the semiconductor material but behaves in many ways like an electron orbiting a hydrogen nucleus. Given that the electrons has an effective mass of $0.07m_e$ (where $m_e$ is mass of the free electron) and the space in which it moves has a permittivity $13{ϵ}_0$, then the radius of the electron's lowermost energy orbit will be close to (The Bohr radius of the hydrogen atom is $0.53\stackrel{\circ }{A})$.

1. $100\stackrel{\circ }{A}$
2. $0.53\stackrel{\circ }{A}$
3. $243\stackrel{\circ }{A}$
4. $10\stackrel{\circ }{A}$

Q. The moments of inertia of a non-uniform circular disc (of mass M and radius R) about four mutually perpendicular tangents AB, BC CD, DA are $I_1,I_2,I_3$ and $I_4$ respectively (the square ABCD circumscribes the circle.) The distance of the center of mass of the disc from its geometrical center is given by.

1. $\frac{1}{4MR}\sqrt{(I_3-I_3{)}^2+(I_2-I_4{)}^2}$
2. $\frac{1}{12MR}\sqrt{(I_3-I_3{)}^2+(I_2-I_4{)}^2}$
3. $\frac{1}{3MR}\sqrt{(I_1-I_2{)}^2+(I_3-I_4{)}^2}$
4. $\frac{1}{2MR}\sqrt{(I_1-I_2{)}^2+(I_3-I_4{)}^2}$

Q. A small box resting on one edge of the table is struck in such a way that it slides off the other edge, $1m$ away, after $2$ seconds. The coefficient of kinetic friction between the box and the table.

1. Must be less than $0.05$
2. Must be exactly $0.5$
3. Must be exactly zero
4. Must be more than $0.05$

Q. A metallic ring of radius $a$ and resistance $R$ is held fixed with its axis along a spatially uniform magnetic field whose magnitude is $B_0\sin \left(\omega t\right)$. Neglect gravity. Then;

1. The current in the ring oscillates with a frequency of $2\omega$
2. The joule heating loss in the ring is proportional to $a^2$
3. The force per unit length on the ring will be proportional to $B_0^2$
4. The net force on the ring is non-zero

Q. An ac voltmeter connected between points $A$ and $B$ in the circuit below reads $36V$. If it is connected between $A$ and $C$, the reading is $39V$. The reading when it is connected between $B$ and $D$ is $25V$. What will the voltmeter read when it is connected between $A$ and $D$?) Assume that the voltmeter reads true rms voltage values and that the source generated a pure ac.)

1. $\sqrt{481}V$
2. $31V$
3. $61V$
4. $\sqrt{3361}V$

Q. Two identical objects each of radii $R$ and masses $m_1$ and $m_2$ are suspended using two strings of equal length $L$ as shown in the figure $(R<<L)$. The angle $\theta$ which mass $m_2$ makes with the vertical is approximately.

1. $\frac{m_{1}R}{(m_1+m_2)L}$
2. $\frac{2m_{2}R}{(m_1+m_2)L}$
3. $\frac{2m_{1}R}{(m_1+m_2)L}$
4. $\frac{m_{2}R}{(m_1+m_2)L}$

Q. A simple pendulum of length $l$ is made to oscillate with amplitude of $45$ degrees. The acceleration due to gravity is $g$. Let $T_0=2\pi \sqrt{l/g}$. The time period of oscillation of this pendulum will be.

1. $T_0$ irrespective of the amplitude
2. Slightly less than $T_0$
3. Slightly more then $T_0$
4. Dependent on whether it swings in a plane aligned with the north-south or east west directions

Q. A horizontal disk of moment of inertia $4.25kg-m^2$ with respect to its axis of symmetry is spinning counter clockwise at $15$ revolutions per second about its axis, as viewed from above. A second disk of moment of inertia $1.80kg-m^2$ with respect to its axis of symmetry is spinning clockwise at $25$ revolutions per second as viewed from above about the same axis and is dropped on top of the first disk.The two disks stick together and rotate as one about their axis of symmetry. The new angular velocity of the system as viewed from above is close to.

1. $18$ revolutions/second and clockwise
2. $18$ revolutions/second and counter clockwise
3. $3$ revolutions/second and clockwise
4. $3$ revolutions/second and counter clockwise

Q. A $160watt$ infrared source is radiating light of wavelength $50000\stackrel{\circ }{A}$ uniformly in all directions. The photon flux at a distance of $1.8m$ is of the order of.

1. $10m^2{s}^{-1}$
2. ${10}^{10}{m}^{-2}{s}^{-1}$
3. ${10}^{20}{m}^{-2}{s}^{-1}$
4. ${10}^{15}{m}^{-2}{s}^{-1}$

Q. In the circuit shown below, all the inductors (assumed ideal) and resistors are identical. The current through the resistance on the right is I after the key K has been switched on for a long time. The currents through the three resistors(in order, from left to right) immediately after the key is switched off are

1. 2 I downwards, I downwards and I downwards.
2. I downwards, I downwards and I downwards
3. 0, I downwards and I downwards
4. 2 I upwards, I downwards and I downwards

Q. An ideal gas undergoes a circular cycle centered at 4atm, 4 lit as shown in the diagram. The maximum temperature attained in this process is close to.

1. 30/R
2. 36/R
3. 24/R
4. 16/R

Q. A cubical vessel has opaque walls. An observer (dark circle in figure below) is located such that she can see only the wall CD but not the bottom. Nearly to what height should water be poured so that she can see an object placed at the bottom at a distance of 10 cm from the corner C ? (Refractive index of water is 1.33.

1. 10 cm
2. 16 cm
3. 45 cm
4. 27 cm

Q. An object is propelled vertically to a maximum height of 4R from the surface of a planet of radius R and mass M. The speed of object when it returns to the surface of the planet is.

1. $2\sqrt{\frac{2GM}{5R}}$
2. $\sqrt{\frac{GM}{2R}}$
3. $\frac{GM}{5R}$
4. $\sqrt{\frac{3GM}{2R}}$

Q. Electromagnetic waves emanating from a point A (in air) are incident on a rectangular block of material M and emerge from the other side as shown. The angles i and r are angles of incidence and refraction when the wave travels from air to the medium. Such paths for the rays are possible

1. If the material has a refractive index very nearly equal to zero
2. If the material has a refractive index less than zero
3. Only if the wave travels in M with a speed faster than the speed of light in vacuum.
4. Only with gamma rays with a wavelength smaller than atomic nuclei of the material

Q. The dimensions of the area $A$ of a black hole can be written in terms of the universal constant $G$, its mass $M$ and the speed of light $c$ as $A=G^{\alpha }{M}^{\beta }{c}^{\gamma }$. Here.

1. $\alpha =-2,\beta =-2$, and $\gamma =4$
2. $\alpha =2,\beta =2$, and $\gamma =-4$
3. $\alpha =3,\beta =3$, and $\gamma =-2$
4. $\alpha =-3,\beta =-3$, and $\gamma =2$

Q. Two small metal balls of different mass $m_1$ and $m_2$ are connected by strings of equal length to a fixed point. When the balls are given charges, the angles that the two strings make with the vertical are ${30}^o$ and ${60}^o$, respectively. The ratio $m_1/m_2$ is close to :

1. $0.58$
2. $3.0$
3. $1.7$
4. $2.0$

Q. A point charge $Q(=3\times {10}^{-12}C)$ rotates uniformly in a vertical circle of radius $R=1mm$. The axis of the circle is aligned along he magnetic axis of the earth. At what value of the angular speed $\omega$, the effective magnetic field at the center of the circle will be reduced to zero? (Horizontal component of Earth's magnetic field is $30$ micro Tesla).

1. ${10}^{11}rad/s$
2. ${10}^{9}rad/s$
3. ${10}^{13}rad/s$
4. ${10}^{7}rad/s$

Q. A closed bottle containing water at ${30}^{\circ }C$ is open on the surface of the moon. Then.

1. The water will boil
2. The water will freeze
3. The water will come out as a spherical ball
4. The water decompose into hydrogen and oxygen

Q. One end of a slack wire (Youngs modulus $Y$, length $L$ and cross -sectional area A) is clamped to a rigid wall and the other end to a block (mass m) which rests on a smooth horizontal plane. The block is set in motion with a speed v. What is the maximum distance the block will travel after the wire becomes taut?

1. $v\sqrt{\frac{mL}{AY}}$
2. $v\sqrt{\frac{2mL}{AY}}$
3. $v\sqrt{\frac{mL}{2AY}}$
4. $L\sqrt{\frac{mv}{AY}}$

Q. For a diode connected in parallel with a resistor, which is the most likely current(I) - voltage (V) characteristic?

1. 
   
2. 
   
3. 
   
4. 
   

Q. Two uniform plates of the same thickness and area but of different materials, one shaped like an isosceles triangle and the other shaped like a rectangle are joined together to form a composite body as shown in the figure. If the centre of mass of the composite body is located at the midpoint of their common side, the ratio between masses of the triangle to that of the rectangle is:

1. $4:3$
2. $3:4$
3. $1:1$
4. $2:1$

Q. Consider the regular array of vertical identical current carrying wires (with direction of current flow as indicated in the figure) protruding through a horizontal table. If we scatter some diamagnetic particles on the table, they are likely to accumulate.

1. Around regions such as A
2. Around regions such as B
3. Uniformly everywhere
4. In circular regions around individual wires such as C

Q. The intensity of sound during the festival season increased by $100$ times. This could imply a decibel level rise from

1. $20$ to $120dB$
2. $70$ to $72dB$
3. $100$ to $10000dB$
4. $80$ to $100dB$

Q. A horizontal steel railroad track has a length of 100 m when the temperature is ${25}^{o}C$. The track is constrained from expanding or bending. The stress on the track on a hot summer day, when the temperature is ${40}^{o}C$, is (Note: The linear coefficient of thermal expansion for steel is $1.1\times {10}^{-5}{/}^{o}C$ and the Youngs modulus of steel is $2\times {10}^{11}$ Pa)

1. $6.6\times {10}^{7}Pa$
2. $8.8\times {10}^{7}Pa$
3. $5.5\times {10}^{7}Pa$
4. $3.3\times {10}^{7}Pa$

Q. A boy is standing on top of a tower of height $85m$ and throws a ball in the vertically upward direction with a certain speed. If $5.25$ seconds later he hears the ball hitting the ground, then the speed with which the boy threw the ball is (take $g=10m/s^2$, the speed of sound in air $=340m/s)$:

1. $6m/s$
2. $8m/s$
3. $10m/s$
4. $12m/s$

Q. The distance between the vertex and the center of mass of a uniform solid planar circular segment of angular size $\theta$ and radius R is given by.

1. $\frac{4}{3}R\frac{sin\left(\theta /2\right)}{\theta }$
2. $R\frac{sin\left(\theta /2\right)}{\theta }$
3. $\frac{4}{3}Rcos\left(\frac{\left(\theta \right)}{2}\right)$
4. $\frac{2}{3}Rcos\left(\theta \right)$

Q. Carbon $-11$ decays to $boron-11$ according to the following formula.
${}_6^{11}C{\Rightarrow }_5^{11}B+e^{+}+v_e+0.96MeV$
Assume that positrons $\left(e^{+}\right)$ produced in the decay combine with free electrons in the atmosphere and annihilate each other almost immediately. Also assume that the neutrinos $\left(v_e\right)$ are massless and do not intersect with the environment. At $t=0$ we have $1\mu g$ of ${}_6^{12}C$. If the half-life of the decay process is $t_0$, the net energy produced between time $t=0$ and $t=2t_0$ will be nearly.

1. $8\times {10}^{18}MeV$
2. $8\times {10}^{16}MeV$
3. $4\times {10}^{8}MeV$
4. $4\times {10}^{16}MeV$

Q. The state of an ideal gas was changed isobarically. The graph depicts three such isobaric lines. Which of the following is true about the pressures of the gas?

1. $P_1=P_2=P_3$
2. $P_1>P_2>P_3$
3. $P_1<P_2<P_3$
4. $P_1/P_2=P_3/P_1$

Q. A wire bent in the shape of a regular $n$ polygonal loop carries a steady current $I$. Let $l$ be the perpendicular distance of a given segment and $R$ be the distance of a vertex both from the centre of the loop. The magnitude of the magnetic field at the centre of the loop is given by:

1. $\frac{n{\mu }_{0}I}{2\pi l}\sin \left(\pi /n\right)$
2. $\frac{n{\mu }_{0}I}{2\pi R}\sin \left(\pi /n\right)$
3. $\frac{n{\mu }_{0}I}{2\pi l}\cos \left(\pi /n\right)$
4. $\frac{n{\mu }_{0}I}{2\pi R}\cos \left(\pi /n\right)$

Q. A beam of monoenergetic electrons, which have been accelerated from rest by a potential $U$, is used to form an interference pattern in a Youngs Double Slit experiment. The electrons are now accelerated by potential $4U$. Then the fringe width.

1. Is half the original fringe width
2. Is twice the original fringe width
3. Remains the same
4. Is one-fourth the original fringe width