Q. A screw gauge gives the following reading when used to measure the diameter of a wire.
Main scale reading : 0 mm
Circular scale reading : 52 divisions
Given that 1 mm on main scale corresponds to 100 divisions of the circular scale. The diameter of wire from the above data is :

1. $0.52cm$
2. $0.052cm$
3. $0.026cm$
4. $0.005cm$

Q. A pulley of radius $2$ $m$ is rotated about its axis by a force $F=(20t-5t^2)$ Newton (where $t$ is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is $10$ $kg{m}^2$, the number of rotations made by the pulley before its direction of motion is reversed, is:

1. less than 3
2. more than 9
3. more than 3 but less than 6
4. more than 6 but less than 9

Q. A car is fitted with a convex side-view mirror of focal length $20cm$. A second car $2.8m$ behind the first car is overtaking the first car at a relative speed of $15m/s$. The speed of the image of the second car as seen in the mirror of the first one is :

1. $15m/s$
2. $\frac{1}{10}m/s$
3. $\frac{1}{15}m/s$
4. $10m/s$

Q. Two particles are executing simple harmonic motion of the same amplitude $A$ and frequency $co$ along the $x$-axis. Their mean position is separated by distance $X_0(X_0>A)$. If the maximum separation between them is $(X_0+A)$, the phase difference between their motion is

1. $\frac{\pi }{2}$
2. $\frac{\pi }{3}$
3. $\frac{\pi }{4}$
4. $\frac{\pi }{6}$

Q. A resistor $R$ and $2\mu F$ capacitor in series is connected through a switch to $200$ $V$ direct supply. Across the capacitor is a neon bulb that lights up at $120$ $V$. Calculate the value of $R$ make the bulb light up $5$ $s$ after the switch has been closed $({\log }_{10}2.5=0.4).$

1. $1.3\times {10}^4\Omega$
2. $1.7\times {10}^6\Omega$
3. $2.7\times {10}^6\Omega$
4. $3.3\times {10}^7\Omega$

Q. A boat is moving due east in a region where the earth's magnetic field is $5.0\times {10}^{-5}N{A}^{-1}{m}^{-1}$ due north and horizontal. The boat carries a vertical aerial $2$ $m$ long. If the speed of the boat is $1.50$ $m{s}^{-1}$, the magnitude of the induced emf in the wire of aerial is:

1. $0.75$ $mV$
2. $1$ $mV$
3. $0.15$ $mV$
4. $0.50$ $mV$

Q. The half life of a radioactive substance is $20$ minutes. The approximate time interval $(t_2-t_1)$ between the time $t_1$when $\frac{1}{3}$ of it had decayed and $t_2$ when $\frac{2}{3}$ of it had decayed is :

1. $7min$
2. $14min$
3. $20min$
4. $28min$

Q. A mass $m$ hangs with the help of a string wrapped around a pulley on a frictionless bearing.The pulley has mass $m$ and radius $R$. Assuming pulley to be a perfect uniform circular disc, the acceleration of the mass $m$, if the string does not slip on the pulley, is :

1. $\frac{3}{2}g$
2. $g$
3. $\frac{2}{3}g$
4. $\frac{g}{3}$

Q. The question has a paragraph followed by two statements, Statement 1 and Statement 2. Of the given four alternatives after the statements, choose the one that best describes the statements.
A thin air film is formed by putting the convex surface of a plane convex lens over a plane glass plate. With monochromatic light, this film gives an interference pattern due to light reflected from the top (convex) surface and the bottom (glass plate) surface of the film.
Statement 1 : When light reflects from the air-glass plate interface, the reflected wave suffers a phase change of $\pi$.
Statement 2 : The center of the interference pattern is dark.

1. Statement 1 is True, Statement 2 is False.
2. Statement 1 is True, Statement 2 is True; Statement 2 is a correct explanation for Statement 1.
3. Statement 1 is True, Statement 2 is True; Statement 2 is not the correct explanation for Statement 1.
4. Statement 1 is False, Statement 2 is True.

Q. If a wire is stretched to make it $0.1\%$ longer, its resistance will

1. increase by $0.05\%$
2. increase by $0.2\%$
3. decrease by $0.2\%$
4. decrease by $0.05\%$.

Q. $100$ $g$ of water is heated from ${30}^{o}C$ to ${50}^{o}C$. Ignoring the slight expansion of the water, the change in its internal energy is (specific heat of water is $4184$ $J/Kg/K$)

1. $84$ $kJ$
2. $8.4$ $kJ$
3. $4.2$ $kJ$
4. $2.1$ $kJ$

Q. A Carnot engine operating between temperatures $T_1$ and $T_2$ has efficiency $\frac{1}{6}$. When $T_2$ is lowered by $62$ $K$, it's efficiency increases to $\frac{1}{3}$. Then $T_1$ and $T_2$ are, respectively :

1. $372$ $K$ and $330$ $K$
2. $310$ $K$ and $248$ $K$
3. $330$ $K$ and $268$ $K$
4. $372$ $K$ and $310$ $K$

Q. Work done in increasing the size of a soap bubble from a radius of $3cm$ to $5cm$ is nearly (Surface tension of soap solution $=0.03$ $N{m}^{-1}$):

1. $2\pi$ $mJ$
2. $4\pi$ $mJ$
3. $0.2\pi$ $mJ$
4. $0.4\pi$ $mJ$

Q. Two bodies of masses $m$ and $4m$ are placed at a distance $r.$ The gravitational potential at a point on the line joining them where the gravitational field is zero is :

1. $zero$
2. $-\frac{4Gm}{r}$
3. $-\frac{6Gm}{r}$
4. $-\frac{9Gm}{r}$.

Q. Water is flowing continuously from a tap having an internal diameter $8\times {10}^{-3}m$. The water velocity as it leaves the tap is $0.4m{s}^{-1}$. The diameter of the water stream at a distance $2\times {10}^{-1}m$ below the tap is close to

1. $5.0\times {10}^{-3}m$
2. $7.5\times {10}^{-3}m$
3. $9.6\times {10}^{-3}m$
4. $3.6\times {10}^{-3}m$.

Q. A water fountain on the ground sprinkles water all around it. If the speed of water coming out of the fountain is $v$, the total area around the fountain that gets wet is :

1. $\frac{\pi v^4}{2g^2}$
2. $\pi \frac{v^2}{g^2}$.
3. $\pi \frac{v^4}{g^2}$
4. $\pi \frac{v^2}{g}$

Q. A mass $M$ attached to a horizontal spring, executes SHM with amplitude $A_1$. When the mass $M$ passes through its mean position then a smaller mass $m$ is placed over it and both of them move together with amplitude $A_2$. The ratio of $\left(\frac{A_1}{A_2}\right)$ is:

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

Q. A thermally insulated vessel contains an ideal gas of molecular mass $M$ and ratio of specific heats $\gamma$ . It is moving with speed $v$ and is suddenly brought to rest. Assuming no heat is lost to the surroundings, its temperature increases by

1. $\frac{(\gamma -1)}{2(\gamma +2)R}$ $M{v}^2$
2. $\frac{(\gamma -1)}{2\gamma R}$ $M{v}^2$
3. $\frac{\gamma M{v}^2}{2R}$
4. $\frac{(\gamma -1)}{2R}$ $M{v}^2$.

Q. Two identical charged spheres suspended from a common point by two massless strings of length $l$ are initially a distance $d(d<<\ell )$ apart because of their mutual repulsion. The charge begins to leak from both the spheres at a constant rate. As a result the charges approach each other with a velocity $v$. Then as a function of distance $x$ between them:

1. $v\infty x^{-1/2}$
2. $v\infty x^{-1}$
3. $v\infty x^{1/2}$
4. $v\infty x$

Q. A fully charged capacitor $C$ with initial charge $q_0$ is connected to a coil of self- inductance $L$ at $t=0$. The time at which, the energy is stored equally between the electric and the magnetic fields is

1. $\sqrt{LC}$
2. $\frac{\pi }{4}\sqrt{LC}$
3. $\pi \sqrt{LC}$
4. $2\pi \sqrt{LC}$

Q. A thin horizontal circular disc is rotating about a vertical axis passing through its center. An insect is at rest at a point near the rim of the disc. The insect now moves along a diameter of the disc of reach its other end. During the journey of the insect, the angular speed of the disc:

1. continuously decreases
2. remains unchanged
3. continuously increases
4. first increases and then decreases

Q. Three perfect gases at absolute temperatures $T_1,T_2$ and $T_3$ are mixed. The masses of molecules are $m_1,m_2$ and $m_3$ and the number of molecules are $n_1,n_2$ and $n_3$ respectively. Assuming no loss of energy, the final temperature of the mixture is:

1. $\frac{(T_1+T_2+T_3)}{3}$
2. $\frac{n_1{T}_l+n_2{T}_2+n_3{T}_3}{n_1+n_2+n_3}$
3. $\frac{n_1{T}_1^2+n_2{T}_2^2+n_3{T}_3^3}{n_1{T}_1+n_2{T}_2+n_3{T}_3}$
4. $\frac{n_1^2{T}_1^2+n_2^2{T}_2^2+n_3{T}_3^3}{n_1{T}_1+n_2{T}_2+n_3{T}_3}$

Q. The electrostatic potential inside a charged spherical ball is given by $\varphi =a{r}^2+b$ where $r$ is the distance from the centre; $a,b$ are constants. Then the charge density inside the ball is :

1. $-6a{ϵ}_{0}r$
2. $-24\pi a{ϵ}_{0}r$
3. $-24\pi a{ϵ}_0$
4. $-6a{ϵ}_0$

Q. Energy required for the electron excitation in $L{i}^{+}$ from the first to the third Bohr orbit is

1. $36.3$ $eV$
2. $108.8$ $eV$
3. $122.4$ $eV$
4. $12.1$ $eV$

Q. Direction : The question has Statement 1 and Statement 2. Of the four choices given after the statements, choose the one that describes the two statements.
Statement 1 : Sky wave signals are used for long distance radio communication. These signals are in general, less stable than ground wave signals.
Statement 2 : The state of ionosphere varies from hour to hour, day to day and season to season.

1. Statement 1 is True, Statement 2 is False.
2. Statement 1 is False, Statement 2 is True.
3. Statement 1 is True, Statement 2 is True; Statement 2 is a correct explanation for Statement 1.
4. Statement 1 is True, Statement 2 is True; Statement 2 is not the correct explanation for Statement 1.

Q. Let the $x-y$ plane be the boundary between two transparent media. Medium 1 in $z\ge 0$ has a refractive index of $\sqrt{2}$ and medium 2 with $z<0$ has a refractive index of $\sqrt{3}$. A ray of light in medium 1 given by the vector $\overrightarrow{A}=6\sqrt{3}\hat{i}+8\sqrt{3}\hat{j}+10\hat{k}$ is incident on the plane of separation. The angle of refraction in medium 2 is

1. ${30}^o$
2. ${45}^o$
3. ${60}^o$
4. ${75}^o$

Q. Direction : The question has Statement 1 and Statement 2. Of the four choices given after the statements, choose the one that describes the two statements.
Statement 1 : A metallic surface is irradiated by a monochromatic light of frequency $v>v_0$ (the threshold frequency). The maximum kinetic energy and the stopping potential are $K_{max}$ and $V_0$ respectively. If the frequency incident on the surface is doubled, both the $K_{max}$ and $V_0$ are also doubled.
Statement 2 : The maximum kinetic energy and the stopping potential of photoelectrons emitted from a surface are linearly dependent on the frequency of incident light.

1. Statement 1 is True, Statement 2 is False.
2. Statement 1 is True, Statement 2 is True; Statement 2 is a correct explanation for Statement 1.
3. Statement 1 is False, Statement 2 is True.
4. Statement 1 is True, Statement 2 is True; Statement 2 is not the correct explanation for Statement 1.

Q. The transverse displacement $y(x,t)$ of a wave on a string is given by $y(x,t)=e^{-(a{x}^2+b{t}^2+2\sqrt{ab}xt\}}$. This represents a :

1. wave moving in $+x$ direction with speed $\sqrt{\frac{a}{b}}$
2. standing wave of frequency $\sqrt{b}$
3. wave moving in $-x$ direction with speed $\sqrt{\frac{b}{a}}$
4. standing wave of frequency $\frac{1}{\sqrt{b}}$

Q. An object, moving with a speed of $6.25m{s}^{-1}$, is decelerated at a rate given by $\frac{dv}{dt}=-2.5\sqrt{v}$ where $v$ is the instantaneous speed. The time taken by the object, to come to rest, would be

1. $1s$
2. $2s$
3. $4s$
4. $8s$

Q. A current $I$ flows in a infinitely long wire with cross section in the form of a semicircular ring of radius R. The magnitude of the magnetic induction at its axis is

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