Q. The total energy of a black body radiation source is collected for five minutes and used to heat water. The temperature of the water increases from ${10.0}^{o}C$ to ${11.0}^{o}C$. The absolute temperature of the black body is doubled and its surface area halved and the experiment repeated for the same time. Which of the following statements would be most nearly correct ?

1. The temperature of the water would increase from ${10.0}^{o}C$ to a final temperature of ${12}^{o}C$
2. The temperature of the water would increase from ${10.0}^{o}C$ to a final temperature of ${11}^{o}C$
3. The temperature of the water would increase from ${10.0}^{o}C$ to a final temperature of ${18}^{o}C$
4. The temperature of the water would increase from ${10.0}^{o}C$ to a final temperature of ${14}^{o}C$

Q. An ideal monatomic gas expands to twice its volume. If the process is isothermal, the magnitude of work done by the gas is $W_i$. If the process is adiabatic, the magnitude of work done by the gas is $W_a$. Which of the following is true?

1. $W_i=W_a>0$
2. $W_i>W_a=0$
3. $W_i>W_a>0$
4. $W_a=W_i=0$

Q. A steady current $I$ flows through a wire of radius $r$, length $L$ and resistivity $\rho$. The current produces heat in the wire. The rate of heat loss in a wire is proportional to its surface area. The steady temperature of the wire is independent of

1. $I$
2. $L$
3. $r$
4. $\rho$

Q. As shown in the figure below, a cube is formed with ten identical resistances R (thick lines) and two honing wires (dotted lines) along the arms AC and BD.
A Resistance between point A and B is.

1. R/2
2. 5R/6
3. R
4. 3R/4

Q. The capacitor of capacitance $C$ in the circuit shown is fully charged initially. Resistance is $R$.
After the switch $S$ is closed, the time taken to reduce the stored energy in the capacitor to half its initial value is

1. $\frac{RC}{2}$
2. $2RC\ln 2$
3. $RC\ln 2$
4. $\frac{RC\ln 2}{2}$

Q. A stream of photons having energy 3 eV each impinges on a potassium surface. The work function of potassium is 2.3 eV. The emerging photo-electrons are slowed down by a copper plate placed 5 mm away. If the potential difference between the two metal plates is 1 V, the maximum distance the electrons can move away from the potassium surface before being turned back is.

1. 2.5 mm
2. 5.0 mm
3. 3.5 mm
4. 1.5 mm

Q. A small asteroid is orbiting around the sun in a circular orbit of radius $r_0$ with speed $V_o$. A rocket is launched from the asteroid with speed $V=\alpha V_o$, where V is the speed relative to the sun. The highest value of $\alpha$ for which the rocket will remain bound to the solar system is (ignoring gravity due to the asteroid and effects of other planets)

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

Q. A body is executing simple harmonic motion of amplitude $a$ and period $T$ about the equilibrium position $x=0$. Large numbers of snapshots are taken at random of this body in motion. The probability of the body being found in a very small interval $x$ to $x+\left|dx\right|$ is highest at

1. $x=\pm a/2$
2. $x=\pm a$
3. $x=0$
4. $x=\pm a/\sqrt{2}$

Q. On a bright sunny day, a diver of height $h$ stands at the bottom of a lake of depth $H$. Looking upward, he can see objects outside the lake in a circular region of radius $R$. Beyond this circle he see the images of objects lying on the floor of the lake. If refractive index of water is $4/3$, then the value of $R$ is :

1. $3(H-h)/\sqrt{5/3}$
2. $(H-h)/\sqrt{7/3}$
3. $3h\sqrt{7}$
4. $3(H-h)/\sqrt{7}$

Q. A radioactive nucleus A has a single decay mode with half life ${\tau }_A$. Another radioactive nucleus B has two decay modes 1 and 2. If decay mode 2 were absent, the half life of B would have been${\tau }_A/2$. If decay mode 1 were absent, the half life of B would have been 3${\tau }_A$. If the actual half life of B is ${\tau }_B$, then the ratio ${\tau }_A/{\tau }_B$ is :

1. $7/3$
2. $3/7$
3. $7/2$
4. $1$

Q. A ball of mass $m$ suspended from a rigid support by an inextensible massless string is released from a height $h$ above its lowest point. At its lowest point it collides elastically with a block of mass $2m$ at rest on a frictionless surface. Neglect the dimensions of the ball and the block. After the collision the ball rises to a maximum height of :

1. $h/2$
2. $h/8$
3. $h/3$
4. $h/9$

Q. A liquid drop placed on a horizontal plane has a near spherical shape (slightly flattened due to gravity). Let $R$ be the radius of its largest horizontal section. A small disturbance causes the drop to vibrate with frequency $v$ about its equilibrium shape. By dimensional analysis the ratio $\frac{v}{\sqrt{\frac{\sigma }{\rho R^3}}}$ can be (Here $\sigma$ is surface tension, $\rho$ is density, $g$ is acceleration due to gravity, and $k$ is an arbitrary dimensionless constant).

1. $\frac{k\rho }{g\sigma }$
2. $\frac{k\rho g{R}^2}{\sigma }$
3. $\frac{k\rho R^3}{g\sigma }$
4. $\frac{k\rho R^2}{g\sigma }$

Q. A simple pendulum is released from rest at the horizontally stretched position. When the string makes an angle $\theta$ with the vertical, then the angle $\alpha$, which the acceleration vector of the bob makes with the string is equal to

1. $ta{n}^{-1}\left(2tan\theta \right)$
2. $ta{n}^{-1}\left(\frac{tan\theta }{2}\right)$
3. $\pi /2$
4. $0$

Q. An unpolarized beam of light of intensity $I_0$ passes through two linear polarizers making an angle of ${30}^o$ with respect to each other. The emergent beam will have an intensity

1. $\frac{3I_0}{4}$
2. $\frac{\sqrt{3}{I}_0}{4}$
3. $\frac{3I_0}{8}$
4. $\frac{I_0}{8}$

Q. Seven identical coins are rigidly arranged on a flat table in the pattern shown below so that each coin touches its neighbours. Each coin is a thin disc of mass $m$ and radius $r$. Note that the moment of inertia of an individual coin about an axis passing through center and perpendicular to the plane of the coin is $\frac{m{r}^2}{2}$.
The moment of inertia of the system of seven coins about an axis that passes through the point $P$ (the center of the coin positioned directly to the right of the central coin) and perpendicular to the plane of the coins is

1. $\frac{127}{2}m{r}^2$
2. $\frac{111}{2}m{r}^2$
3. $55m{r}^2$
4. $\frac{55}{2}m{r}^2$

Q. Consider three concentric metallic spheres A, B and C of radii a, b, c respectively where a < b < c. A and B are connected whereas C is grounded. The potential of the middle sphere B is raised to V then the charge on the sphere C is

1. $-4\pi {\epsilon }_{0}V\frac{ac}{c-a}$
2. $-4\pi {\epsilon }_{0}V\frac{bc}{c-b}$
3. $+4\pi {\epsilon }_{0}V\frac{bc}{c-b}$
4. $zero$

Q. A conducting rod of mass $m$ and length $l$ is free to move without friction on two parallel long conducting rails, as shown below. There is a resistance $R$ across the rails. In the entire space around, there is a uniform magnetic field $B$ normal to the plane of the rod and rails. The rod is given an impulsive velocity $v_0$. Finally, the initial energy $\frac{1}{2}m{v}_0^2$

1. Will enable rod to continue to move with velocity $v_0$ since the rails are frictionless
2. Will be converted into the work done against the magnetic field
3. Will be converted fully into heat energy in the resistor
4. Will be converted fully into magnetic energy due to induced current

Q. Two identical bodies are made of a material for which the heat capacity increases with temperature. One of these is held at a temperature of $100$ while the other one is kept at $0$. If the two are brought into contact, then, assuming no heat loss to the environment, the final temperature that they will reach is :

1. More than $50$
2. $0$
3. $50$
4. Less than $50$

Q. Two blocks (1 and 2) of equal mass m are connected by an ideal string (see figure below) over a frictionless pulley. The blocks arc attached to the ground by springs having spring constants $k_1$ and $k_2$ such that $k_1>k_2$.
Initially, both springs are unstretched. The block 1 is slowly pulled down a distance x and released. Just after the release the possible values of the magnitudes of the acceleration of the blocks $a_1$ and $a_2$ can be.

1. either $(a_1=a_2=\frac{(k_1+k_2)x}{2m})$ or
   $(a_1=\frac{k_{1}x}{m}-gand{a}_2=\frac{k_{2}x}{m}+g)$
2. $(a_1=a_2=\frac{(k_1+k_2)x}{2m})$ only
3. either $(a_1=a_2=\frac{(k_1-k_2)x}{2m})$ or
   $(a_1=a_2\frac{\left(k_1{k}_2\right)x}{(k_1+k_2)m}-g)$
4. $(a_1=a_2=\frac{(k_1-k_2)x}{2m})$ only

Q. A parent nucleus $X$ is decaying into daughter nucleus $Y$ which in turn decays to $Z$. The half lives of $X$ and $Y$ are $40000$ years and $20$ years respectively. In a certain sample, it is found that the number of $Y$ nuclei hardly changes with time. If the number of $X$ nuclei in the sample is $4\times {10}^{20}$, the number of $Y$ nuclei present in it is

1. $4\times {10}^{23}$
2. $2\times {10}^{20}$
3. $2\times {10}^{17}$
4. $4\times {10}^{20}$

Q. A standing wave in a pipe with a length $L=1.2m$ is described by $y(x,t)=y_{0}sin\left[\right(2\pi /L\left)x\right]sin\left[\right(2\left(\pi /L\right)x+\pi /4)$]. Based on above information, which one of the following statement is incorrect.
( Speed of sound in air is $300m{s}^{-1}$).

1. The pipe is closed at both ends
2. The wavelength of the wave could be $1.2m$
3. There could be a node at $x=0$ and antinode at $x=\frac{L}{2}$
4. The frequency of the fundamental mode of vibrations is $137.5Hz$

Q. A loop carrying current $I$ has the shape of a regular polygon of $n$ sides. If $R$ is the distance from the center to any vertex, then the magnitude of the magnetic induction vector $\overrightarrow{B}$ at the center of the loop is:

1. $n\frac{{\mu }_{0}I}{2\pi R}\tan \frac{\pi }{n}$
2. $\frac{{\mu }_{0}I}{\pi R}\tan \frac{\pi }{n}$
3. $n\frac{{\mu }_{0}I}{2\pi R}\tan \frac{2\pi }{n}$
4. $\frac{{\mu }_{0}I}{2R}$

Q. A cylindrical steel rod of length $0.10m$ and thermal conductivity $50W{m}^{-1}{K}^{-1}$ is welded end to end to copper rod of thermal conductivity $400W{m}^{-1}{K}^{-1}$ and of the same area of cross section but $0.20m$ long. The free end of the steel rod is maintained at $100$ K and that of the copper rod at $0$ K. Assuming that the rods are perfectly insulated from the surrounding, the temperature at the junction of the two rods is :

1. $50$ K
2. $30$ K
3. $20$ K
4. $40$ K

Q. The ratio of the speed of sound to the average speed of an air molecule at $300K$ and $1$ atmospheric pressure is close to

1. $1$
2. $\sqrt{1/300}$
3. $\sqrt{300}$
4. $300$

Q. A particle is acted upon by a force given by $F=-\alpha x^3-\beta x^4$ where $\alpha$ and $\beta$ are positive constants. At the point $x=0$, the particle is

1. In unstable equilibrium
2. In stable equilibrium
3. In neutral equilibrium
4. Not in equilibrium

Q. In a Young's double slit experiment the intensity of light at each slit is $I_0$. Interference pattern is observed along a direction parallel to the line $S_1{S}_2$ on screen $S$. The minimum, maximum, and the intensity averaged over the entire screen are respectively

1. $0,4I_0,2I_0$
2. $I_0,2I_0,3I_0/2$
3. $0,4I_0,I_0$
4. $0,2I_0,I_0$

Q. The potential energy of a point particle is given by the expression $V\left(x\right)=-\alpha x+\beta \sin \left(x/\gamma \right)$. A dimensionless combination of the constants $\alpha ,\beta$ and $\gamma$ is

1. $\alpha /\beta \gamma$
2. $\gamma /\alpha \beta$
3. $\alpha \gamma /\beta$
4. ${\alpha }^2/\beta \gamma$

Q. A particle released from rest is falling through a thick fluid under gravity. The fluid exerts a resistive force on the particle proportional to the square of its speed. Which one of the following graphs best depicts the variation of its speed $v$ with time $t$?

1. 
   
2. 
   
3. 
   
4. 
   

Q. A planet orbits in an elliptical path of eccentricity $e$ around a massive star considered fixed at one of the foci. The point in space where it is closest to the star is denoted by $P$ and the point where it is farthest is denoted by $A$. Let $v_P$ and $v_A$ be the respective speeds at $P$ and $A$. Then

1. $\frac{V_P}{V_A}=\frac{1+e}{1-e}$
2. $\frac{V_P}{V_A}=1$
3. $\frac{V_P}{V_A}=\frac{1+e^2}{1-e}$
4. $\frac{V_P}{V_A}=\frac{1+e^2}{1-e^2}$

Q. In one model of the electron, the electron of mass $m_e$ is thought to be a uniformly charged shell of radius $R$ and total charge $e$, whose electrostatic energy $E$ is equivalent to its mass $m_e$ via Einstein's mass energy relation $E=m_e{c}^2$. In this model, $R$ is approximately ($m_e=9.1\times {10}^{-31}kg$, $c=3\times {10}^{8}m{s}^{-1}$, $1/4\pi {\epsilon }_0=9\times {10}^{9}Farads{m}^{-1}$, magnitude of the electron charge $=1.6\times {10}^{-19}C$)

1. $1.4\times {10}^{-15}m$
2. $2.8\times {10}^{-35}m$
3. $2\times {10}^{-13}m$
4. $5.3\times {10}^{-11}m$