Q. $n$ mole of a perfect gas undergoes a cyclic process ABCA (see figure) consisting of the following processes.
A $\to$ B: Isothermal expansion at temperature $T$ so that the volume is doubled from $V_1$ to $V_2=2V_1$ and pressure changes from $P_1$ to $P_2$
B $\to$ C: Isobaric compression at pressure $P_2$ to initial volume $V_1$.
C $\to$ A: Isochoric change leading to change of pressure from $P_2$ to $P_1$.
Total work done in the complete cycle ABCA is:

1. $0$
2. $nRT(\ln 2+\frac{1}{2})$
3. $nRT\ln 2$
4. $nRT(\ln 2-\frac{1}{2})$

Q. The focal length $f$ is related to the radius of curvature $r$ of the spherical convex mirror by:

1. $f=+\frac{1}{2}r$
2. $f=-r$
3. $f=-\frac{1}{2}r$
4. $f=r$

Q. In a Young's double slit experiment, the width of the one of the slit is three times the other slit. The amplitude of the light coming from a slit is proportional to the slit-width. Find the ratio of the maximum to the minimum intensity in the interference pattern.

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

Q. Two stars of masses $m$ and $2m$ at a distance $d$ rotate about their common center of mass in free space. The period of revolution is:

1. $\frac{1}{2\pi }\sqrt{\frac{d^3}{3Gm}}$
2. $2\pi \sqrt{\frac{d^3}{3Gm}}$
3. $\frac{1}{2\pi }\sqrt{\frac{3Gm}{d^3}}$
4. $2\pi \sqrt{\frac{3Gm}{d^3}}$

Q. A current through a wire depends on time as $i={\alpha }_{0}t+\beta t^2$ where ${\alpha }_0=20\text{A/s}$ and $\beta =8{\text{As}}^{-2}$. Find the charge crossed through a section of the wire in $15s$.

1. $2250C$
2. $11250C$
3. $2100C$
4. $260C$

Q. Moment of inertia (M.I) of four bodies, having same mass and radius are reported as;
$I_1$ = M.I of thin circular ring about its diameter.
$I_2$ = M.I of circular disc about an axis perpendicular to the disc and going through the centre,
$I_3$ = M.I of solid cylinder about its axis and
$I_4$ = M.I of solid sphere about its diameter
Then:

1. $I_1+I_3<I_2+I_4$
2. $I_1+I_2=I_3+\frac{5}{2}{I}_4$
3. $I_1=I_2=I_3>I_4$
4. $I_1=I_2=I_3<I_4$

Q. Given below are two statements:
Statement I: Two photons having equal linear momenta have equal wavelengths.
Statement- II: If the wavelength of photon is decreased, then the momentum and energy of a photon will also decrease.
In the light of the above statements, choose the correct answer from the options given below.

1. Both Statement I and Statement II are true.
2. Statement I is false but Statement II is true
3. Both Statement I and Statement II are false
4. Statement I is true but Statement II is false

Q. In the given figure, a mass M is attached to a horizontal spring which is fixed on one side to a rigid support. The spring constant of the spring is k. The mass oscillates on a frictionless surface with time period T and amplitude A. When the mass in equilibrium position, as shown in the figure, another mass m is gently fixed upon it. The new amplitude of oscillation will be:

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

Q. If $Y,K$ and $\eta$ are the values of Young's modulus, Bulk modulus and Modulus of Rigidity of any material respectively. Choose the correct relation for these parameters.

1. $Y=\frac{9K\eta }{3K-\eta }N/m^2$
2. $\eta =\frac{3YK}{9K+Y}N/m^2$
3. $Y=\frac{9K\eta }{2\eta -3K}N/m^2$
4. $K=\frac{Y\eta }{9\eta -3Y}N/m^2$

Q. In the given figure, the energy levels of hydrogen atom have been shown along with some transitions marked A, B, C, D and E. The transitions A , B and C respectively represent:

1. The ionization potential of hydrogen, second member of Balmer series and third member of Paschen series.
2. The first member of the Lyman series, third member of Balmer series and second member of Paschen series.
3. The series limit of Lyman series, third member of Balmer series and second member of Paschen series.
4. The series limit of Lyman series, second member of Balmer series and second member of Paschen series.

Q. Four identical particles of equal masses $1kg$ (equidistant from each other) are made to move along the circumference of a circle of radius $1m$ under the action of their own mutual gravitational attraction. The speed of each particle will be:

1. $\sqrt{\frac{G}{2}(1+2\sqrt{2})}$
2. $\sqrt{G(1+2\sqrt{2})}$
3. $\sqrt{\frac{G}{2}(2\sqrt{2}-1)}$
4. $\sqrt{\frac{(1+2\sqrt{2})G}{4}}$

Q. If the velocity-time graph has the shape AMB, what would be the shape of the corresponding acceleration-time graph ?

Q. Two equal capacitors are first connected in series and then in parallel. The ratio of the equivalent capacities in the two cases will be:

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

Q. If an emitter current is changed by $4\text{mA}$, the collector current changes by $3.5\text{mA}$. The value of $\beta$ will be:

1. $7$
2. $0.5$
3. $0.875$
4. $3.5$

Q. Match List-I with List-II:
List - I List-II
(a) Isothermal (i) Pressure constant
(b) Isochoric (ii) Temperature constant
(c) Adiabatic (iii) Volume constant
(d) Isobaric (iv) Heat content is constant
Choose the correct answer from the options given below:

1. (a) $\to$ (i), (b) $\to$ (iii) , (c) $\to$ (ii) , d $\to$ (iv)
2. (a) $\to$ (ii) , (b) $\to$ (iii) , (c) $\to$ (iv) , (d) $\to$ (i)
3. (a) $\to$ (ii), (b) $\to$ (iv) , (c) $\to$ (iii) , (d) $\to$ (i)
4. (a) $\to$ (iii) , (b) $\to$ (ii) , (c) $\to$ (i) , (d) $\to$ (iv)

Q. Each side of a box made of metal sheet in cubic shape is '$a$' at room temperature '$T$', the coefficient of linear expansion of the metal sheet is '$\alpha$'. The metal sheet is heated uniformly, by a small temperature $\Delta T$, so that its new temperature is $T+\Delta T$. Calculate the increase in the volume of the metal box.

1. $3a^3\alpha \Delta T$
2. $4a^3\alpha \Delta T$
3. $4\pi a^3\alpha \Delta T$
4. $\frac{4}{3}\pi a^3\alpha \Delta T$

Q. A cell $E$ of emf $6V$ and internal resistance $2\Omega$ is connected with another cell $E_2$ of emf $4V$ and internal resistance $8\Omega$ (as shown in the figure). The potential difference across points X and Y is:

1. $10.0V$
2. $3.6V$
3. $5.6V$
4. $2.0V$

Q. A cube of side '$a$' has point charges $+Q$ located at each of its vertices except at the origin where the charge is $-Q$. The electric field at the center of cube is:

1. $\frac{-Q}{3\sqrt{3}\pi {\epsilon }_0{a}^2}(\hat{x}+\hat{y}+\hat{z})$
2. $\frac{-2Q}{3\sqrt{3}\pi {\epsilon }_0{a}^2}(\hat{x}+\hat{y}+\hat{z})$
3. $\frac{2Q}{3\sqrt{3}\pi {\epsilon }_0{a}^2}(\hat{x}+\hat{y}+\hat{z})$
4. $\frac{Q}{3\sqrt{3}\pi {\epsilon }_0{a}^2}(\hat{x}+\hat{y}+\hat{z})$

Q. Consider two satellites $S_1$ and $S_2$ with periods of revolution $1$ hr. and $8$ hr. respectively revolving around a planet in circular orbits. The ratio of angular velocity of satellite $S_1$ to the angular velocity of satellite $S_2$ is:

1. $8:1$
2. $1:4$
3. $2:1$
4. $1:8$

Q. The work done by a gas molecule in an isolated system is given by, $W=\alpha {\beta }^2{e}^{-\frac{x^2}{\alpha kT}}$, where $x$ is the displacement, $k$ is the Boltzmann constant and $T$ is the temperature, $\alpha$ and $\beta$ are constants. Then the dimension of $\beta$ will be:

1. $\left[M{L}^2{T}^{-2}\right]$
2. $\left[ML{T}^{-2}\right]$
3. $\left[M^{2}L{T}^{-2}\right]$
4. $\left[M^{0}L{T}^0\right]$

Q. The coefficient of static friction between a wooden block of mass $0.5kg$ and a vertical rough wall is $0.2$. The magnitude of horizontal force that should be applied on the block to keep it adhere to the wall will be _________ N.
$[g=10m{s}^{-2}]$

Q. A resonance circuit having inductance and resistance $2\times {10}^{-4}H$ and $6.28\Omega$ respectively, oscillates at $10MHz$ frequency. The value of quality factor of this resonator is _________.
$[\pi =3.14]$

Q. A hydraulic press can lift $100kg$ when a mass '$m$' is placed on the smaller piston. It can lift ________ kg when the diameter of the larger piston is increased by $4$ times and that of the smaller piston is decreased by $4$ times keeping the same mass '$m$' on the smaller piston.

Q. An inclined plane is bent in such a way that the vertical cross-section is given by $y=\frac{x^2}{4}$ where $y$ is in vertical and $x$ in horizontal direction. If the upper surface of this curved plane is rough with coefficient of friction $\mu =0.5$, the maximum height in cm at which a stationary block will not slip downward is ___________ cm.
($x$ and $y$ are in meters)

Q. An electromagnetic wave of frequency $5\text{GHz}$, is travelling in a medium whose relative electric permittivity and relative magnetic permeability both are $2$. Its velocity in this medium is _________ $\times {10}^{7}m/s$.

Q. In connection with the circuit drawn below, the value of current flowing through $2k\Omega$ resistor is _________ $\times {10}^{-4}A$.

Q. An audio signal $v=20\sin 2\pi \left(1500t\right)$ amplitude modulates a carrier wave given by
$v_c=80\sin 2\pi (100,000t)$.
The value of percent modulation is _________.

Q. A ball with a speed of $9m/s$ collides with another identical ball at rest. After the collision, the direction of each ball makes an angle of ${30}^{\circ }$ with the original direction. The ratio of velocities of the balls after collision is $x:y$ is ___________.

Q. A common transistor radio set requires $12V$ (D.C) for its operation. The D.C source is constructed by using a transformer and a rectifier circuit, which are operated at $220V$ (A.C) on standard domestic A.C. supply. The number of turns of secondary coil are $24$, then the number of turns of primary are ___________.

Q. An unpolarized light beam is incident on the polarizer of a polarization experiment and the intensity of light beam emerging from the analyzer is measured as $100$ Lumens. Now, if the analyzer is rotated w.r.t. the horizontal axis (direction of light) by ${30}^{\circ }$ in clockwise direction, the intensity of emerging light will be __________ Lumens.