A photon and an electron, each of \(20 eV\) energy, move in free space. The ratio of linear momentum of electron \(p_e\) to that of photon \(p_{ph}\), \(\frac{p_e}{p_{ph}}\) is :
(Take speed of light \(= 3 \times 10^8 ms^{-1}\), charge of electron \(= -1.6 \times 10^{-19} C\) and mass of electron \(= 9 \times 10^{-31} kg\))

• (A) \(275\)
• (B) \(450\)
• (C) \(\frac{1}{250}\)
• (D) \(225\)

Water flows in a streamline motion through a horizontal pipe of circular cross-section as shown in the figure. The pressure difference of water between \(P\) and \(Q\) is \(15 Nm^{-2}\). The area of cross-section at \(P\) and \(Q\) are \(40 cm^2\) and \(20 cm^2\), respectively. The rate of flow of water through the pipe, in \(cm^3s^{-1}\), is :
[Take density of water \(= 1000 kg m^{-3}\)]

• (A) \(400\)
• (B) \(100\)
• (C) \(200\)
• (D) \(300\)

A thin horizontal disc is rotating about a vertical axis passing through its fixed centre \(O\). Its angular momentum is \(L_A\) and \(L_B\) computed about points \(A\) and \(B\), respectively, with \(OB = 2 \times OA\). The value of \(\frac{L_A}{L_B}\) is :

• (A) \(2\)
• (B) \(\frac{1}{4}\)
• (C) \(\frac{1}{2}\)
• (D) \(1\)

Consider a long solenoid of length \(l\) and radius \(r\). If \(n\) is the number of turns per unit length and \(\mu_0\) is the permeability of free space, the inductance of the solenoid is :

• (A) \(2\mu_0 \pi n^2 r^2 l\)
• (B) \(\mu_0 \pi n^2 r^2 l\)
• (C) \(\mu_0 n^2 r^2 l\)
• (D) \((\mu_0 / 2\pi) n^2 r^2 l\)

The temperature of a metallic sphere of radius \(R\) is increased by a small amount \(\Delta T\). If the linear coefficient of thermal expansion of the metal is \(\alpha\), the approximate increase in the volume of the sphere is :

• (A) \(6 \pi R^3 \alpha \Delta T\)
• (B) \(2 \pi R^3 \alpha \Delta T\)
• (C) \(3 \pi R^3 \alpha \Delta T\)
• (D) \(4 \pi R^3 \alpha \Delta T\)

Consider two circuits, (A) and (B), each having two resistors. One of them has a positive temperature coefficient of resistance, \(+\alpha\), while the other one has a negative temperature coefficient, \(-\alpha\), as shown in the figure. The current through these circuits are denoted by \(I_A\) and \(I_B\). At initial temperature, the resistance of the two resistors is \(R_0\). As the temperature is increased, the correct option that describes the variation of current in these circuits is :

• (A) both \(I_A\) and \(I_B\) remain constant
• (B) \(I_A\) remains constant while \(I_B\) increases
• (C) \(I_A\) decreases while \(I_B\) increases
• (D) \(I_A\) increases while \(I_B\) decreases

A beam of light falls on a metal surface such that photo-electrons are generated. If power of the light source starts to decrease linearly with time \(t\), then variation of the photocurrent \(I\) and magnitude of the stopping potential \(|V|\) with time is best represented by :

An ideal Zener diode with breakdown voltage of \(-3 V\) is reverse biased with a negative input voltage \(V_i = -5 V\). The magnitude of voltage difference between points \(B\) and \(A\) is :

• (A) \(0 V\)
• (B) \(3 V\)
• (C) \(2 V\)
• (D) \(1 V\)

In the measurement of viscosity of liquids using terminal velocity experiment, spherical balls of same radius but having different densities are used. The variation of the terminal velocity (\(v\)) with the ratio of density of spherical ball (\(\sigma\)) to density of the liquid (\(\rho\)), is best represented by :

Two planets \(P_1\) and \(P_2\) with equal mass have radii \(R_1\) and \(R_2\), respectively, where \(R_2 = \frac{R_1}{2}\). The escape speeds of \(P_1\) and \(P_2\) are \(v_1\) and \(v_2\), respectively. Then \(\frac{v_2}{v_1}\) is :

• (A) \(2\)
• (B) \(\frac{1}{\sqrt{2}}\)
• (C) \(1\)
• (D) \(\sqrt{2}\)

An ac voltage \(V = 220\sin(2\times 10^3 t)\) Volt is applied to a series LCR circuit. Then the current amplitude in this circuit is :
(Given : \(L = 10 mH\), \(C = 25\muF\), \(R = 100\ \Omega\))

• (A) \(22.0 A\)
• (B) \(2.2 A\)
• (C) \(5.5 A\)
• (D) \(11.0 A\)

Two identical inductors are connected in two different configurations P and Q, where a time varying current \(I(t)\) is flowing, as shown in the figure. The induced emf between points a and b for configuration P is \(E_P\) and that for configuration Q is \(E_Q\). The ratio \(E_P/E_Q\) is :
[Neglect the effect of mutual inductance.]

• (A) \(2\)
• (B) \(1/4\)
• (C) \(1/2\)
• (D) \(1\)

Three identical capacitors P, Q and S, each of the capacitance C, are connected to a battery of voltage V, as shown in the figure. If the energy stored in the capacitor P and total energy stored in the system are \(U_P\) and \(U_T\), respectively, then the ratio \(U_P/U_T\) is :

• (A) \(1/6\)
• (B) \(2/3\)
• (C) \(1/3\)
• (D) \(1/2\)

A conducting loop of finite resistance lies on the \(x - y\) plane. There is a constant magnetic field in the \(z\) direction. The area of the loop varies with time \(t\), as \(A = A_0(1 + \sin t)\) in appropriate units. The figure that correctly indicates the qualitative behaviour of the power \(P\) dissipated in the loop as a function of time is :

In an adiabatic expansion, the temperature of one mole of an ideal monatomic gas (\(\gamma = 5/3\)) decreases from \(60 K\) to \(50 K\). The work done by the gas in the process is :
[Take the universal gas constant as \(R = 8.3 J mol^{-1} K^{-1}\)]

• (A) \(166 J\)
• (B) \(41.5 J\)
• (C) \(83 J\)
• (D) \(124.5 J\)

Consider a particle moving along a straight line, whose position as a function of time is given by \(s(t) = \alpha t^2 - \beta t + \gamma\), where \(\alpha = 1 ms^{-2}\), \(\beta = 6 ms^{-1}\) and \(\gamma = 5 m\). The average speed of the particle, in \(ms^{-1}\), from \(t = 0\) to \(t = 6 s\) is :

• (A) \(0\)
• (B) \(12\)
• (C) \(6\)
• (D) \(3\)

The following table presents the part of the electromagnetic spectrum and their corresponding major applications.

{{Part of the electromagnetic spectrum}} & {{Applications}}

P & Microwave & I & For purifying the water

Q & UV rays & II & For warming the food

R & Gamma rays & III & For AM and FM communication systems

S & Radio wave & IV & For treating the Cancer cells

The correct option is :

• (A) P-II, Q-IV, R-III, S-I
• (B) P-I, Q-II, R-III, S-IV
• (C) P-I, Q-IV, R-II, S-III
• (D) P-II, Q-I, R-IV, S-III

An ideal gas is made of polyatomic molecules. Each of the molecules has three translational, three rotational and \(f\) number of vibrational modes. If the ratio of heat capacities \(C_P/C_V\) of the gas is \(8/7\), then the value of \(f\) is :

• (A) \(1\)
• (B) \(4\)
• (C) \(3\)
• (D) \(2\)

A unit positive point charge is taken slowly through an infinitesimally thin tube that is inside a charged dielectric sphere of radius \(R\), having uniform positive charge density \(\rho\), as shown in the figure. The initial and final positions of the charge are marked by A and B at distances \(2R\) and \(3R\) respectively, from the centre of the sphere. In this process, the magnitude of the total work done on the point charge is \(\frac{\rho R^2}{n \epsilon_0}\). The value of \(n\) is :
(\(\epsilon_0\) is the permittivity of vacuum)

• (A) \(18\)
• (B) \(2\)
• (C) \(6\)
• (D) \(9\)

A current \(I_0\) flows through a metallic circular loop of radius r as shown in the figure. Resistance of the segment ABC is half that of ADC. Magnitude of magnetic field at the center O of the loop is :

• (A) \(\frac{\mu_0 I_0}{2\pi r}\)
• (B) \(\frac{\mu_0 I_0}{12 r}\)
• (C) \(\frac{\mu_0 I_0}{4 r}\)
• (D) \(\frac{\mu_0 I_0}{2 r}\)

Bob B of mass m at rest is hanging vertically from the ceiling via a massless string of length 10 m, as shown in the figure. Point mass A of mass m travelling horizontally with speed 10 ms\(^{-1}\) hits bob B elastically. The bob B rises h meter after the collision. Taking the acceleration due to gravity \(g = 10 ms^{-2}\) and neglecting the size of the bob, the value of h is :

• (A) \(2.5\)
• (B) \(8\)
• (C) \(7\)
• (D) \(5\)

An electromagnetic wave travelling in a lossless dielectric medium having a dielectric constant, \(\epsilon_r = 9\), has the electric field, \(E_x = E_0 \sin (kz - 2\pi \times 10^6 t) Vm^{-1}\) where \(E_0\) is the amplitude and \(k\) is the wave vector. Among the following options, the incorrect choice is :

• (A) The direction of propagation of the electromagnetic wave is along +z.
• (B) The speed of the electromagnetic wave inside the medium is \(10^8 ms^{-1}\).
• (C) The wavelength of the electromagnetic wave inside the medium is \(300 m\).
• (D) The magnetic field is given by the relation \(B_y = \frac{E_0}{v} \sin (kz - 2\pi \times 10^6 t)\) where \(v\) is the speed of the electromagnetic wave inside the medium.

A particle of mass M moves along a horizontal x axis from \(x=0\) to \(x=L\). The coefficient of kinetic friction varies as a function of x as \(\mu_k(x) = \mu_0 - \alpha x\), where \(\mu_0, \alpha\) are constants of appropriate dimensions, so that \(\mu_k(L) = 0\). The total work done by the frictional force during the motion is \(n \mu_0 Mg L\), where g is the acceleration due to gravity. The value of n is :

• (A) \(1/2\)
• (B) \(1\)
• (C) \(1\)
• (D) \(1/3\)

Consider three media P, Q and R with refractive indices 1, 1.25, and 1.5, respectively. The medium Q having a thickness of 5 cm is placed between extended media P and R as shown in the figure. An object O is placed at the center of medium Q. If viewed from medium P near the normal direction, the apparent depth of O is \(h_1\). For similar observation from medium R, the apparent depth is \(h_2\). The value of \(|h_1 - h_2|\), in cm, is :

• (A) \(3\)
• (B) \(0\)
• (C) \(1\)
• (D) \(2\)

Consider a fixed uniformly charged insulating sphere with radius R and total charge +Q. A point charge -q (\(q \ll Q\)) with mass m is released from rest at a distance of 3R from the centre of the charged sphere. When the point charge reaches the surface of the sphere, its speed is :
(\(\epsilon_0\) is the permittivity of vacuum, neglect gravitational forces).

• (A) \(\sqrt{\frac{Qq}{4\pi\epsilon_0 m R}}\)
• (B) \(\sqrt{\frac{3Qq}{4\pi\epsilon_0 m R}}\)
• (C) \(\sqrt{\frac{2Qq}{3\pi\epsilon_0 m R}}\)
• (D) \(\sqrt{\frac{Qq}{3\pi\epsilon_0 m R}}\)

A car travels on a circular racetrack of radius 50 m, which is banked at an angle \(\theta\). If the car travels at a speed \(10 ms^{-1}\), then the wear and tear on its tyres is minimum. Taking the acceleration due to gravity to be \(10 ms^{-2}\), the value of \(\theta\) is :

• (A) \(\tan^{-1}(2\sqrt{5})\)
• (B) \(\tan^{-1}(1/5)\)
• (C) \(\tan^{-1}(2/5)\)
• (D) \(\tan^{-1}(\sqrt{3}/2)\)

A frictionless circular wire of unit radius is fixed on the horizontal plane. Two point particles of unit mass start moving simultaneously from point \(A(\theta = \pi/2)\) with identical uniform angular speeds in opposite directions, and meet again at point \(B(\theta = -\pi/2)\). During this time, which of the following figures schematically represent the magnitude of the total linear momentum P of the system, as a function of \(\theta\) ?

Three identical p-n junction diodes \(D_1\), \(D_2\) and \(D_3\) are connected across a battery as shown in the figure. If the width of the depletion regions of \(D_1\), \(D_2\) and \(D_3\) are \(W_1\), \(W_2\) and \(W_3\), respectively, then the correct option is :

• (A) \(W_2 > W_1 = W_3\)
• (B) \(W_1 > W_2 > W_3\)
• (C) \(W_3 = W_1 > W_2\)
• (D) \(W_3 > W_2 > W_1\)

The lens combination as shown in the figure, consists of two lenses, \(L_1\) and \(L_2\), of the focal lengths \(+10 cm\) and \(-10 cm\), respectively. The position of the image formed is :

• (A) \(60 cm\) to the right of the concave lens
• (B) \(20 cm\) to the left of the concave lens
• (C) \(60 cm\) to the left of the concave lens
• (D) \(30 cm\) to the right of the concave lens

A solid sphere A of radius R and mass M is attached at a point to a smaller solid sphere B of radius r \(<\) R and mass m \(<\) M. Assume that the line joining their centres lies along the horizontal. The moment of inertia of the system calculated about a vertical axis passing through the centre of A is \(I_A\) and that calculated about a vertical axis passing through the centre of B is \(I_B\). The difference \(I_A - I_B\) is :

• (A) \(0\)
• (B) \((m - M)(R + r)^2\)
• (C) \((m - M)(R + r)^2\) (duplicate option in original text, solved accurately below)
• (D) \((m - M)(R - r)^2\)

Consider that an electron is revolving in an excited state of Hydrogen atom with velocity \(\sqrt{25.6} \times 10^5 ms^{-1}\). The radius of the orbit is \(x \times 10^{-9} m\). The value of \(x\) is :
[Take the mass of electron to be \(9 \times 10^{-31} kg\), charge of electron \(= 1.6 \times 10^{-19} C\) and \(\frac{1}{4\pi\epsilon_0} = 9 \times 10^9 N m^2 C^{-2}\)]

• (A) \(1\)
• (B) \(4\)
• (C) \(3\)
• (D) \(2\)

The mean free path of molecules in an ideal gas A is half that of another ideal gas B. The diameter of the spherical molecules of gas A is twice the diameter of the molecules of B. If number densities of the gases A and B are \(n_A\) and \(n_B\), respectively, then the correct option is :

• (A) \(n_A = \frac{1}{2} n_B\)
• (B) \(n_A = n_B\)
• (C) \(n_A = 2 n_B\)
• (D) \(n_A = \frac{1}{4} n_B\)

A cylindrical cork of uniform density floats in a liquid of density \(\rho_1\). If the cork is depressed slightly and released, it oscillates harmonically with time period T. If the same cork floats in another liquid of density \(\rho_2\), then the similar oscillation has time period 2T. The value of \(\rho_2/\rho_1\) is :

• (A) \(1/4\)
• (B) \(4\)
• (C) \(2\)
• (D) \(1/2\)

For sound waves, if the number of nodes for the 5th harmonic of an open-ended pipe is n and that for the 9th harmonic of the same pipe with one of its ends closed is m, the ratio n/m is :

• (A) \(3/5\)
• (B) \(5/9\)
• (C) \(9/5\)
• (D) \(1\)

Consider the following nuclear reaction :
\(^{238}U \to ^{234}Th + ^4He\)
Take masses of \(^{238}U\), \(^{234}Th\) and \(^4He\) as \(238.050 u\), \(234.043 u\) and \(4.003 u\), respectively. The Q value for the reaction, in keV, is :
[Given : \(1 u = 931.5 MeV c^{-2}\)]

• (A) \(3740\)
• (B) \(3726\)
• (C) \(3730\)
• (D) \(3736\)

Which of the following measurements require 'index correction' ?

• (A) Measurement of speed of sound using resonance tube
• (B) Measurement of resistance of a wire using meter bridge
• (C) Measurement of gravitational acceleration using simple pendulum
• (D) Measurement of focal length of lenses using optical bench

In a solar system, the time-period of revolution of a planet tracing a circular orbit of radius R is proportional to :

• (A) \(R^3\)
• (B) \(R^{1/2}\)
• (C) \(R^{3/2}\)
• (D) \(R^2\)

Consider that \(\sigma\), \(k_B\), \(b\) represent Stefan-Boltzmann constant, Boltzmann constant and Wien's displacement law constant, respectively. The dimension of \(\sigma k_B^{-1} b\) is :

• (A) \([L^{-1} T^{-1} K^{-4}]\)
• (B) \([L^{-1} T^{-1} K^{-2}]\)
• (C) \([L^{-1} K^{-2}]\)
• (D) \([L^{-1} T^{-1} K^{-3}]\)

A ray of light with wavelength \(\lambda\) is incident on three different photo-electric cells namely 1, 2 and 3. The threshold wavelength of these photo-electric cells are \(\lambda_1\), \(\lambda_2\), and \(\lambda_3\), respectively and the magnitude of stopping potentials of these cells are \(V_1\), \(V_2\) and \(V_3\), respectively. The relation between \(\lambda\) and threshold wavelengths are \(\lambda_1 < \lambda\), \(\lambda_2 > \lambda\) and \(\lambda_3 \gg \lambda\). The correct option is :

• (A) \(V_1 < V_2\), \(V_3 = 0\)
• (B) \(V_1 = 0\), \(V_2 < V_3\)
• (C) \(V_1 = 0\), \(V_2 > V_3\)
• (D) \(V_1 > V_2\), \(V_3 = 0\)

One main scale division of a Vernier calliper is equal to 1 mm and the number of divisions on the Vernier scale is 10. When both the jaws touch each other, the Vernier scale shifts to the left of zero of the main scale in such a way that 4th Vernier division coincides with a division of the main scale. If this Vernier calliper measures the length of a wire to be 1 cm, the actual length of the wire is :

• (A) \(1.04 cm\)
• (B) \(0.60 cm\)
• (C) \(0.96 cm\)
• (D) \(1.00 cm\)

A point charge Q is placed inside a cavity within a solid isolated conducting sphere. Consider points A, B and C as shown in the figure, where the magnitudes of the electric fields are \(E_A\), \(E_B\) and \(E_C\), respectively. The points B and C are at the same distance from the center of the solid sphere. The correct option is :

• (A) \(E_A \neq 0\), \(E_B < E_C\)
• (B) \(E_A = 0\), \(E_B = E_C\)
• (C) \(E_A \neq 0\), \(E_B = E_C\)
• (D) \(E_A = 0\), \(E_B > E_C\)

In Geiger-Marsden experiment, the number of scattered \(\alpha\)-particles \(N(\theta)\) is plotted as a function of scattering angle \(\theta\). Which of the following options represents the correct plot ?

One mole of an ideal monatomic gas undergoes a cyclic process as shown in the figure. The total heat supplied to the gas is :

• (A) \(800 J\)
• (B) \(400 J\)
• (C) \(500 J\)
• (D) \(600 J\)

Two infinitely long parallel conducting wires A and B carry currents I and 2I, respectively, in the same direction. The wire A has uniform mass per unit length \(\lambda\) and lies on an insulated floor. The wire B is kept fixed at a height h above the floor. The minimum magnitude of h so that the wire A does not rise from the floor is :
[g is the acceleration due to gravity and \(\mu_0\) is the permeability of free space.]

• (A) \(\frac{4\mu_0 I^2}{\pi \lambda g}\)
• (B) \(\frac{\mu_0 I^2}{2\pi \lambda g}\)
• (C) \(\frac{\mu_0 I^2}{\pi \lambda g}\)
• (D) \(\frac{2\mu_0 I^2}{\pi \lambda g}\)

Consider a spring-mass simple harmonic oscillator in one dimension. The mass of the particle is m kg and the spring constant is k N/m. At a given instant, the extension of the spring is x meter and the speed of the particle is v m/s. On the x - v plane, if the graph of v as a function of x is a circle, then the correct option is :

• (A) \(k = \sqrt{m}\)
• (B) \(k = 1/m\)
• (C) \(k = m\)
• (D) \(k = m^2\)