Q. One mole of an ideal monatomic gas undergoes the following four reversible processes:
Step $1$: It is first compressed adiabatically from volume $V_1$ to $1m^3$.
Step $2$: then expanded isothermally to volume $10m^3$.
Step $3$: then expanded adiabatically to volume $V_3$.
Step $4$: then compressed isothermally to volume $V_1$.
If the efficiency of the above cycle is $3/4$ then $V_1$ is,

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

Q. If some charge is given to a solid metallic sphere, the field inside remains zero and by Gauss's law all the charge resides on the surface. Suppose now that Coulomb's force between two charges varies as $1/r^3$. Then, for a charged solid metallic sphere

1. field inside will be zero and charge density inside will be zero
2. field inside will not be zero and charge density inside will not be zero
3. field inside will be zero and charge density inside will not be zero
4. field inside will not be zero and charge density inside will be zero

Q. A neutron star with magnetic moment of magnitude $m$ is spinning with angular velocity $\omega$ about its magnetic axis. The electromagnetic power $P$ radiated by it is given by ${\mu }_0^x{m}^y{\omega }^z{c}^u$, where ${\mu }_0$ and $c$ are the permeability and speed of light in free space, respectively. Then

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

Q. A square-shaped conducting wire loop of dimension $a$ moving parallel to the x-axis approaches a square region of size $b(a<b)$ where a uniform magnetic field $B$ exists pointing into the plane of the paper (see figure). As the loop passes through this region, the plot correctly depiciting its speed $\left(v\right)$ as a function of $x$ is

1. 
   
2. 
   
3. none of these
   
   
4. 
   

Q. Using dimensional analysis the resistivity in terms of fundamental constants $h,m_e,c,e,{\epsilon }_o$ can be expressed as.

1. $\frac{h}{{\epsilon }_0{m}_{e}c{e}^2}$
2. $\frac{{\epsilon }_0{m}_{e}c{e}^2}{h}$
3. $\frac{h^2}{m_{e}c{e}^2}$
4. $\frac{m_e{\epsilon }_0}{c{e}^2}$

Q. Force $\overrightarrow{F}$ applied on a body is written as $\overrightarrow{F}=(\hat{n}\cdot \hat{f})\hat{n}+\overrightarrow{G}$, where $\hat{n}$ is a unit vector. The vector $\overrightarrow{G}$ is equal to.

1. $\hat{n}\times \overrightarrow{F}$
2. $(\hat{n}\times \overrightarrow{F})\times \overrightarrow{F}/|\overrightarrow{F}|$
3. $(\hat{n}\times \overrightarrow{F})\times \hat{n}$
4. $\hat{n}\times (\hat{n}\times \overrightarrow{F})$

Q. The figure of a centimeter scale below shows a particular position of the Vernier calipers. In this position the value of $x$ shown in the figure is (figure is not to scale).

1. $0.20cm$
2. $3.65cm$
3. $4.15cm$
4. $0.03cm$

Q. A parallel beam of light is incident on a tank filled with water up to a height of $61.5mm$ as shown in the figure below. Ultrasonic waves of frequency $0.5MHz$ are sent along the length of the water column using a transducer placed at the top, and they form longitudinal standing waves in the water. Which of the schematic plots below best describes the intensity distribution of the light as seen on the screen? Take the speed of sound in water to be $1,500m/s$.

1. 
   
2. 
   
3. 
   
4. 
   

Q. An electron in an electron microscope with initial velocity $v_{0}i$ enters a region of a stray transverse electric field $E_{0}j$. The time taken for the change in its de Broglie wavelength from the initial value of $\lambda$ to $\frac{\lambda }{3}$ is proportional to.

1. $E_0$
2. $\frac{1}{E_0}$
3. $\frac{1}{\sqrt{E_0}}$
4. $\sqrt{E_0}$

Q. A transverse wave of frequency $500Hz$ and speed $100m/s$ is travelling in the positive x direction on a long string. At time $t=0s$ the displacements at $x=0.0m$ and at $x=0.25m$ are $0.0m$ and $0.02m$, respectively. The displacement at $x=0.2m$ at $t=5\times {10}^{-4}s$ is?

1. $-0.04$m
2. $-0.02$m
3. $0.04$m
4. $0.02$m

Q. Two bottles A and B have radii $R_A$ and $R_B$ and heights $h_A$ and $h_B$ respectively with $R_B=2R_A$ and $h_B=2h_A$. These are filled with hot water at ${60}^o$C. Consider that heat loss for the bottles takes place only from side surfaces. If the time the water takes to cool down to ${50}^o$C is $t_A$ and $t_B$ for bottles A and B, respectively, then $t_A$ and $t_B$ are best related as.

1. $t_B=2t_A$
2. $t_A=t_B$
3. $t_B=4t_A$
4. $t_B=t_A/2$

Q. On a pulley of mass M hangs a rope with two masses $m_1$ and $m_2(m_1>m_2)$ tied at the ends as shown in the figure. The pulley rotates without any friction, whereas the friction between the rope and the pulley is large enough to prevent any slipping. Which of the following plots best represents the difference between the tensions in the rope on the two sides of the pulley as a function of the mass of the pulley?

1. 
   
2. 
   
3. 
   
4. 
   

Q. A carrot looks orange in colour because of the $\beta$ carotene molecule in it. This means that the $\beta$ carotene molecule absorbs light of wavelengths.

1. Longer than $550$ nm
2. Shorter than $550$ nm
3. Shorter than $700$ nm
4. Longer than $700$ nm

Q. Two circularly shaped linear polarisers are placed coaxially. The transmission axis of the first polarizer is at ${30}^o$ from the vertical while the second one is at ${60}^o$, both in the clockwise sense. If an unpolarised beam of light of intensity $I=20$ $W/m^2$ is incident on this pair of polarisers, then the intensities $I_1$ and $I_2$ transmitted by the first and the second polarisers, respectively, will be close to.

1. $I_1=10.0W/m^2$ and $I_2=7.5W/m^2$
2. $I_1=15.0W/m^2$ and $I_2=0.0W/m^2$
3. $I_1=20W/m^2$ and $I_2=15W/m^2$
4. $I_1=10.0W/m^2$ and $I_2=8.6W/m^2$

Q. It was found that the refractive index of material of a certain prism varied as $1.5+0.004/{\lambda }^2$, where $\lambda$ is the wavelength of light used to measure the refractive index. The same material was then used to construct a thin prism of apex angle ${10}^o$. Angles of minimum deviation $\left({\delta }_m\right)$ of the prism were recorded for the sources with wavelength ${\lambda }_1$ and ${\lambda }_2$ respectively. Then.

1. ${\delta }_m\left({\lambda }_1\right)>{\delta }_m\left({\lambda }_2\right)$ if ${\lambda }_1>{\lambda }_2$
2. ${\delta }_m\left({\lambda }_1\right)<{\delta }_m\left({\lambda }_2\right)$ if ${\lambda }_1<{\lambda }_2$
3. ${\delta }_m\left({\lambda }_1\right)>{\delta }_m\left({\lambda }_2\right)$ if ${\lambda }_1<{\lambda }_2$
4. ${\delta }_m$ is the same in both the cases

Q. A star of mass $M$ (equal to the solar mass) with a planet (much smaller than the star) revolves around the star in a circular orbit. The velocity of the star with respect to the center of mass of the star-planet system is shown below:
The radius of the planet's orbit is closest to $(1A.U.=Earth-Sundistance)$.

1. $0.008A.U.$
2. $0.12A.U.$
3. $0.004A.U.$
4. $0.04A.U.$

Q. To calculate the size of a hydrogen anion using the Bohr model, we assume that its two electrons move in an orbit such that they are always on diametrically opposite sides of the nucleus. With each electron having the angular momentum $\hslash =h/2\pi$, and taking electron interaction into account the radius of the orbit in terms of the Bohr radius of hydrogen atom $a_B=\frac{4\pi {ϵ}_0{\hslash }^2}{m{e}^2}$ is

1. $\frac{2}{3}{a}_B$
2. $\frac{4}{3}{a}_B$
3. $a_B$
4. $\frac{3}{2}{a}_B$

Q. The magnitude of acceleration of the electron in the $n^{th}$ orbit of hydrogen atom is $a_H$ and that of singly ionized helium atom is $a_{He}$. The ratio $a_H:a_{He}$ is?

1. $1:8$
2. $1:4$
3. $1:2$
4. Dependent on n

Q. One end of a rod of length $L=1m$ is fixed to a point on the circumference of a wheel of radius $R=1/\sqrt{3}m$. The other end is sliding freely along a straight channel passing through the center $O$ of the wheel as shown in the figure below. The wheel is rotating with a constant angular velocity $\omega$ about $O$.
The speed of the sliding end $P$ when $\theta ={90}^{\circ }$ is

1. $\frac{\omega }{3}$
2. $\frac{2\omega }{3}$
3. $\frac{2\omega }{\sqrt{3}}$
4. $\frac{\omega }{\sqrt{3}}$

Q. Consider a bowl filled with water on which some black pepper powder have been sprinkled uniformly. Now a drop of liquid soap is added at the center of the surface of water. The picture of the surface immediately after this will look like.

1. 
   
2. 
   
3. 
   
4. 
   

Q. A positive charge q is placed at the center of a neutral hollow cylindrical conducting shell with its cross section as shown in the figure. Which one of the following figures correctly indicates the induced charge distribution on the conductor (ignore edge effects).

1. 
   
2. 
   
3. 
   
4. 
   

Q. A gas obeying the equation of state $PV=RT$ undergoes a hypothetical reversible process described by the equation, $P{V}^{5/3}exp(-\frac{PV}{E_0})=C_1$ where $C_1$ and $E_0$ are dimensioned constants. Then, for this process, the thermal compressibility at high temperature.

1. Approaches a constant value
2. Is proportional to $T$
3. Is proportional to $T^{1/2}$
4. Is proportional to $T^2$

Q. Two satellites $S_1$ and $S_2$ are revolving around a planet in the opposite sense in coplanar circular concentric orbits. At time $t=0$, the satellites are farthest apart. The periods of revolution of $S_1$ ad $S_2$ and $3$h and $24$h respectively. The radius of the orbit of $S_1$ is $3\times {10}^4$km. Then the orbital speed of $S_2$ as observed from.

1. The planet is $4\pi \times {10}^4$km $h^{-1}$ when $S_2$ is closest from $S_1$
2. The planet is $2\pi \times {10}^4$km $h^{-1}$ when $S_2$ is farthest from $S_1$
3. $S_1$ is $\pi \times {10}^4$km $h^{-1}$ when $S_2$ is closest from $S_1$
4. $S_1$ is $3\pi \times {10}^4$km $h^{-1}$ when $S_2$ is closest to $S_1$

Q. A bird sitting on a single high tension wire does not get electrocuted because:

1. The circuit is not complete
2. The bird feet has an insulating covering
3. Capacitance of the bird is too small and the line frequency is too small
4. None of the above

Q. A thin piece of thermal conductor of constant thermal conductivity insulated on the lateral sides connects two reservoirs which are maintained at temperatures $T_1$ and $T_2$ as shown. Assuming that the system is in steady state. which of the following plots best represents the dependence of the rate of change of entropy on the ratio of temperatures $\frac{T_1}{T_2}$.

1. 
   
2. 
   
3. 
   
4. 
   

Q. The number of gas molecules striking per second per square meter of the top surface of a table placed in a room at ${20}^o$C and $1$ atmospheric pressure is of the order of($k_B=1.4\times {10}^{-23}J/K$, and the average mass of an air molecule is $5\times {10}^{-27}$kg)

1. ${10}^{27}$
2. ${10}^{25}$
3. ${10}^{23}$
4. ${10}^{29}$

Q. A particle of mass m moves around the origin in a potential $\frac{1}{2}m{w}^2{r}^2$, where r is the distance from the origin. Applying the Bohr model in this case, the radius of the particle in its $n^{th}$ orbit in terms of $a=\sqrt{h/\left(2\pi m\omega \right)}$ is?

1. $an$
2. $a{n}^2$
3. $an\sqrt{n}$
4. $a\sqrt{n}$

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 a fixed axis $AB$. A bullet of mass $m(m<<M)$ and speed $v$ is shot horizontally at the face opposite to $ABCD$ at a height of $4a/3$ from the surface to impart the cube an angular speed $\omega$. It strikes the face and embeds in the cube. Then $\omega$ is close to (note: the moment of inertia of the cube about an axis perpendicular to the face and passing through the centre of mass is $2M{a}^2/3)$.

1. $Mv/ma$
2. $mv/Ma$
3. $Mv/2ma$
4. $mv/2Ma$

Q. Which of the following plots represents schematically the dependence of the time period of a pendulum if measured and plotted as a function of the amplitude of its oscillations?(Note: amplitude need not be small)

1. 
   
2. 
   
3. 
   
4. 
   

Q. A rectangular region of dimensions $w\times l(W\ll l)$ has a constant magnetic field into the plane of the paper as shown. On one side the region is bounded by a screen. On the other side positive ions of mass m and charge q are accelerated from rest and towards the screen by a parallel plate capacitor at constant potential difference $V<0$, and come out through a small hole in the upper plate. Which one of the following statements is correct regarding the charge on the ions that hit the screen?

1. Ions with $q>\frac{2|v|m}{B^2{w}^2}$ will hit the screen
2. Ions with $q<\frac{2|v|m}{B^2{w}^2}$ will hit the screen
3. Only ions with $q=\frac{2|v|m}{B^2{w}^2}$ will hit the screen
4. All ions will hit the screen