Q.1) A point charge q is placed symmetrically at a distance d from two perpendicularly placed grounded conducting infinite plates as shown in the figure. The net force on the charge (in units of1/4πε0) is
a) q28d2(22-1) away from the corner
b) q28d2(22-1)towards the corner
c) q222d2 towards the corner
d) 3q28d2 away from the corner
Ans: B
Q.2) If the electrostatic potential V(r,θ,ϕ) in a charge free region has the form V(r,θ,ϕ)=f(r)cosθ, then the functional form of f(r) (in the following a and b are constants) is.
a) ar2+br
b) ar+br2
c) ar+br
d) a ln rb
Ans: B
Q.3) Let four point charges q,-q2,q and –q/2 be placed at the vertices of a square of side a. Let another point charge –q be placed at the centre of the square (see the figure)
a) 1
b) 12
c) 14
d) 18
Ans: D
Q.4) The force between two long and parallel wires carrying currents I1 and I2 and separated by a distance D is proportional to
a) I1I2/D
b) (I1+I2)/D
c) (I1I2/D)2
d) I1I2/D2
Ans: A
Q.5) A horizontal metal disc rotates about the vertical axis in a uniform magnetic field pointing up as shown in the figure. A circuit is made by connecting one end A of a resistor to the center of the disc and the other B to its edge through a sliding contact. The current that flows through the resistor is
a) Zero
b) DC from A to B
c) DC from B to A
d) AC
Ans: C
Q.6) The electric field of an electromagnetic wave is given by
E→=E0cos[π(0.3x+0.4y-1000t)] k^
The associated magnetic field B→ is
a) 10-3E0cos[π(0.3x+0.4y-1000t)] k^
b) 10-4E0cos[π(0.3x+0.4y-1000t)](4i^-3j^)
c) E0cos[π(0.3x+0.4y-1000t)](0.3i^-0.4j^)
d) 102E0cos[π(0.3x+0.4y-1000t)](3i^-4j^)
Ans: B
Q.7) Let (V,A→) and V’,A’→ denote two sets of scalar and vector potentials, and ψ is a scalar function. Which of the following transformations leave the electric and magnetic fields (and hence Maxwell’s equations) unchanged?
a) A’→=A→+∇ψ and V’=V-∂ψ∂t
b) A’→=A→-∇ψ andV’=V+2∂ψ∂t
c) A’→=A→+∇ψ and V’=V+∂ψ∂t
d) A’→=A→-∇ψ and V’=V-∂ψ∂t
Ans: A
Q.8) The expression ∂2∂x12+∂2∂x22+∂2∂x32+∂2∂x42 1×12+x22+x32+∂42 is proportional to
a) δ(x1+x2+x3+x4)
b) δ(x1)δ(x2)δ(x3)δ(x4)
c) (x12+x22+x32+x42)-3/2
d) (x12+x22+x32+x42)-2
Ans: B
Q.9) If A→=i^̂yz+j^̂xz+k^ky,then the integral ∮C.A→∙dl→ (where C is along the perimeter of a rectangular area bounded by x-0,x-a and y-0,y-b) is
a) 12(a3+b3)
b) π(ab2+a2b)
c) π(a3+b3)
d) 0
Ans: D
Q.10) Consider an n×n(n>1) matrix A, in which Aij is the product of the indices i and j (namelyAij=ij) The matrix A
a) has one degenerate eigenvalue with degeneracy (n-1)
b) has two degenerate eigenvalues with degeneracies 2 and (n-2)
c) has one degenerate eigenvalue with degeneracy n
d) does not have any degenerate eigenvalue
Ans: A
Q.11) If A, and B C are non-zero Hermitian operators, which of the following relations must be false?
a) [A,B]=C
b) AB+BA=C
c) ABA=C
d) A+B=C
Ans: A
Q.12) Two different thermodynamic systems are described by the following equations of state: 1T1=3RN12U1and 1T2=5RN22U2where T1,2,N1,2 and U1,2 are respectively the temperature, the mole numbers and the internal energies of the two systems, and R is the gas constant. LetUtotdenote the total energy when these two systems are put in contact and attain thermal equilibrium. The ratio U1Utotis
a) 5N23N1+5N2
b) 3N13N1+5N2
c) N1N1+N2
d) N2N15N2
Ans: B
Q.13) The Fourier transform of the derivative of the Dirac function, namely δ'(x), is proportional to
a) 0
b) 1
c) sin k
d) ik
Ans: D
Q.14) A Carnot cycle operates as a heat engine between two bodies of equal heat capacity until their temperatures become equal. If the initial temperatures of the bodies are T1 and T2,respectively and T> 1 T2, then their common final temperature is
a) T12/T2
b) T22/T1
c) T1T2
d) 12T1+T2
Ans: C
Q.15) The entropy S of a thermodynamic system as a function of energy E is given by the following graph. The temperatures of the phases A, B and C denoted byTA,TB and TC respectively. Satisfy the following inequalities
a) TC>TB>TC
b) TA>TC>TB
c) TB>TC>TA
d) TB>TA>TC
Ans: C
Q.16) Given that the integral ∫0∞dxy2+x2)=π2y, the value of ∫0∞dx(y2+x2)2
a) πy3
b) π4y3
c) π8y3
d) π2y3
Ans: B
Q.17) Which of the following functions cannot be the real part of a complex analytic function of =x+iy?
a) x2y
b) x2-y2
c) x3-3xy2
d) 3x2y-y-y3
Ans: A
Q.18) Three sets of data A,and B C from an experiment, represented by x, and 0, are plotted on a log-log scale. Each of these are fitted with straight lines as shown in the figure
a) x,x and x2
b) -x2,x and 2x
c) 1×2,x and x2
d) 1x,x and x2
Ans: D
Q.19) A sample of Si has electron and hole mobilities of0.13 and 0.05m2/V-s respectively at300 K. It is doped with P and Al with doping densities of1.5×1021/m3 and 2.5×1021/m3respectively. The conductivity of the doped Si sample at 300 K is
a) 8Ω-1m-1
b) 32Ω-1m-1
c) 20.8Ω-1m-1
d) 83.2Ω-1m-1
Ans: A
Q.20) Two identical zener diodes are placed back-to-back in series and are connected to a variable DC power supply. The best representation of the I-V characteristics of the circuitis
a) 
b) 
c) 
d) 
Ans: D
Q.21) Consider the melting transition of ice into water at constant pressure. Which of the following thermodynamic quantities does not exhibit a discontinuous change across the phase transition?
a) Internal energy
b) Helmholtz free energy
c) Gibbs free energy
d) Entropy
Ans: C
Q.22) Consider the op-amp circuit shown in the figure. If the input is a sinusoidal wave Vi=5sin(1000t), then the amplitude of the output V0 is
a) 52
b) 5
c) 522
d) 52
Ans: C
Q.23) A system of N classical non-interacting particles, each of mass m, is at a temperature T and is confined by the external potential Vr=12Ar2(where A is a constant) in three dimensions. The internal energy of the system is
a) 3NkBT
b) 32NkBT
c) N2mA32kBT
d) NAmInkBTm
Ans: A
Q.24) Three identical spin -1/2 fermions are to be distributed in two non-degenerate distinct energy levels. The number of ways this can be done is
a) 8
b) 4
c) 3
d) 2
Ans: B
Q.25) If one of the inputs of a J-K flip flop is high and the other is low, then the outputs Q and Q
a) Oscillate between low and high in race around condition
b) Toggle and the circuit acts like a T flip flop
c) Are opposite to the inputs
d) Follow the inputs and the circuit acts like a R- S flip flop
Ans: D
Q.26) A 4-varible switching function is given by f=∑(5,7,8,10,13,15)+d(0,1,2), where d is the do-not-care-condition. The minimized form of f is sum of products (SOP) from is
a) A C+B D
b) A B+CD
c) AD+BC
d) B D+BD
Ans: D
Q.27) A particle is in the ground state of an infinite square well potential is given by, V(x)={0 for-a≤x≤a∞ otherwise
The probability to find the particle in the interval between –a 2and a 2 is
a) 12
b) 12+1π
c) 12-1π
d) 1π
Ans: B
Q.28) If Ψ(x)=Aexp(-x4) is the eigenfunction of a one dimensional Hamiltonian with eigenvalue E=0, the potential V(x) (in units where ℏ=2m=1) is
a) 12×2
b) 16×6
c) 16×2+12×2
d) 16×2-12×2
Ans: D
Q.29) A particle moves in a potential V=x2+y2+z22 . Which component(s) of the angular momentum is /are constant (s) of motion?
a) None
b) Lx,Ly and Lz
c) only Lx and Ly
d) only Lz
Ans: D
Q.30) A pendulum consists of a ring of mass M and radius R suspended by a massless rigid rod of length l attached to its rim. When the pendulum oscillates in the plane of the ring, the time period of oscillation is
a) 2πl+Rg
b) 2πg2R2+2Rl+l21/4
c) 2π2R2+2Rl+l2g(R+l)
d) 2πg2R2+2Rl+l21/4
Ans: C
Q.31) Consider a particle of mass m attached two identical springs each of length l and spring constant k (see the figure). The equilibrium configuration is the one where the springs are unstretched. There are no other external forces on the system. If the particle is given a small displacement along the x-axis , which of the describes the equation of motion for small oscillations?
a) mx¨+kx3l2=0
b) mx¨ +kx = 0
c) mx¨+2kx = 0
d) mx¨+kx3l=0
Ans: A
Q.32) The Hamiltonian of a relativistic particle of rest mass m and momentum p is given byH=p2+m2+V(x), in units in which the speed of light c=1. The corresponding Lagrangian is
a) L=m1+x2-V(x)
b) L=-m1-x2-V(x)
c) L=1+mx˙2-V(x)
d) L=12mx˙2-V(x)
Ans: B
Q.33) The motion of a particle of mass m in one dimension is described by the Hamiltonian H=p22m+12mω2×2+λx.What is the difference between the (quantized) energies of the first two levels? (In the following, 〈x〉 is the expectation value of x in the ground state)
a) ℏω-λ〈x〉
b) ℏω+λ〈x〉
c) ℏω-λ22mω2
d) ℏω
Ans: D
Q.34) The expectation value of the x- component of the orbital angular momentum Lx in the state Ψ=15[3Ψ2,1,-1+5Ψ2,1,0-11Ψ2,1,+1]
(where Ψnlm are the Eigen functions in usual notation), is
a) -ℏ1025(11-3)
b) 0
c) ℏ1025(11+3)
d) ℏ2
Ans: A
Q.35) Let Ψnlm denote the eigenfunctions of a Hamiltonian for a spherically symmetric potential V(r). The expectation value of Lz in the state Ψ=16[Ψ200+5Ψ210+10Ψ21-1+00Ψ211] is
a) -518 ℏ
b) 56ℏ
c) ℏ
d) 518 ℏ
Ans: D
Q.36) Let A,B and C be function of phase space variable (coordinates and momenta of a mechanical system). If , represent the poisson bracket, the value of A,B,C-A,B,Cis given by
a) 0
b) B,C,A
c) A,C,B
d) C,A,B
Ans: D
Q.37) A spin -12 particle is in the state χ=111(1+i3) in the eigenbasis of S2 and SzIf we measure Sz, the probabilities of getting +h2 and -h2 ,respectively are
a) 12 and 12
b) 211 and 911
c) 0 and 1
d) 111 and 311
Ans: B
Q.38) A perturbation Vpert=aL2 is added to the Hydrogen atom potential. The shift in the energy level of the 2P state, when the effects of spin are neglected up to second order in a is
a) 0
b) 2aℏ2+a2ℏ4
c) 2aℏ2
d) aℏ2+32a2ℏ4
Ans: C
Q.39) The spectroscopic symbol for the ground state of Al 13 is P1/ 2 2 . Under the action of a strong magnetic field (when L – S coupling can be neglected) the ground state energy level will split into
a) 3 levels
b) 4 levels
c) 5 levels
d) 6 levels
Ans: C
Q.40) A gas laser cavity has been designed to operate at λ=0.5μm with a cavity length of 1m .With this set-up, the frequency is found to be larger than the desired frequency by 100 Hz.The change in the effective length of the cavity required to retune the laser is
a) -0.334 ×10-12m
b) 0.334 ×10-12m
c) 0.167 ×10-12m
d) -0.167 ×10-12m
Ans: D
Q.41) A uniform linear monoatomic chain is modelled by a spring-masssystem of masses m separated by nearest neighbour distance a and spring constant mω02.The dispersion relation for this system is
a) ωk=2ω01-coska2
b) ωk=2ω0sin2ka2
c) ωk=2ω0sinka2
d) ωk=2ω0tanka2
Ans: C
Q.42) The energy of an electron in a band as a function of its wave vector k is given by Ek=E0-Bcoskxa+coskya+coskza, where E0,B and a are constant. The effective mass of the electron near the bottom of the band is
a) 2ℏ23βa2
b) ℏ23βa2
c) ℏ22βa2
d) ℏ2βa2
Ans: D
Q.43) The physical phenomenon that cannot be used for memory storage applications is
a) large variation in magnetoresistance as a function of applied magnetic field
b) variation in magnetization of a ferromagnet as a function of applied magnetic field
c) variation in polarization of a ferroelectric as a function of applied electric field
d) variation is resistance of a metal as a function of applied electric field
Ans: D
Q.44) A DC voltage V is applied across a Josephson junction between two superconductors with a phase differenceϕ0. If I0 and k are constants that depend on the properties of the junction, the current flowing through it has the form
a) I0sin2eVtℏ+ϕ0
b) kVsin2eVtℏ+ϕ0
c) kV sin ϕ0
d) I0sinϕ0+kV
Ans: A
Q.45) A child makes a random walk on a square lattice of constant a taking a step in the north, east, south or west directions with probabilities 0.255, 0.255, 0.245 and 0.245, respectively. After a large number of steps, N, the expected position of the child with respect to the starting point is at a distance
a) 2×10-2Na in the north-east direction
b) 2N ×10-2 a in the north-east direction
c) 22×10-2Na in the south-west direction
d) 0
Ans: A
Q.46) A particle is prepared in a stimultaneous eigenstate of L2L^2 and LzL_z. If ℓ(ℓ+1)ℏ2\ell(\ell + 1)\hbar^2 and mℏm\hbar are respectively the eigenvalues of L2L^2and LzL_z, then the expectation value ⟨Lx2⟩\langle L_x^2 \rangle of the particles in this state satisfies
a) ⟨Lx2⟩=0\langle L_x^2 \rangle = 0
b) 0≤⟨Lx2⟩≤ℓ2ℏ20 \leq \langle L_x^2 \rangle \leq \ell^2 \hbar^2
c) 0≤⟨Lx2⟩≤ℓ(ℓ+1)ℏ230 \leq \langle L_x^2 \rangle \leq \dfrac{\ell(\ell+1)\hbar^2}{3}
d) ℓℏ22≤⟨Lx2⟩≤ℓ(ℓ+1)ℏ23\dfrac{\ell \hbar^2}{2} \leq \langle L_x^2 \rangle \leq \dfrac{\ell(\ell+1)\hbar^2}{3}
Ans: D
Q.47) According to the shell model, the total angular momentum (in units of ℏ) and the parity of the ground state of the Li 37 nucleus is
a) 32With negative parity
b) 32With positive parity
c) 12With positive parity
d) 72With negative parity
Ans: A
Q.48) Let A and B be two vectors in three-dimensional Euclidean space. Under rotation, the tensor product Tij=AjBj
a) reduces to a direct sum of three 3-dimensional representations
b) is an irreducible 9-dimensional representation
c) reduces to a direct sum of a 1-dimensional, a 3-dimensional and a 5-dimensional inreducible representations
d) reduces to a direct sum of a 1-dimensional and an 8-dimensional irreducible representations
Ans: D
Q.49) The intrinsic electric dipole moment of a nucleus X ZA
a) increases with Z, but independent of A
b) decreases with Z, but independent of A
c) is always zero
d) increases with Z and A
Ans: C
Q.50) Consider the following ratios of the partial decay widths
R1=Γ(ρ+→π++π0)Γ(ρ−→π−+π0)and R2=Γ(Δ++→π++ρ)Γ(Δ−→π−+n)
If the effects of electromagnetic and weak interactions are neglected, then R1R_1 and R2R_2 are respectively
a) 1 and 2
b) 1 and 2
c) 2 and 1
d) 1 and 1
Ans: B
Q.51) Spherical particles of a given material of density pp are released from rest inside a liquid medium of lower density. The viscous drag force may be approximated by the Stoke’s law, i.e.,Fd=6πηRv,F_d = 6 \pi \eta R v, where η\eta is the viscosity of the medium, RR the radius of a particle and vv its instantaneous velocity. If τ(m)\tau(m) is the time taken by a particle of mass mm to reach half its terminal velocity, then the ratio τ(8m)/τ(m)\tau(8m) / \tau(m) is
a) 8
b) 1/8
c) 4
d) 1/4
Ans: C
Q.52) The speed v of the molecules of mass m of an ideal gas obeys Maxewll’s velocity distribution law at an equilibrium temperature T. Let (vx,vy,vz)denote the components of the velocity and kB the Boltzmann constant.The average value of (avx-βy)2,where a and β are constants , is
a) (a2-β2)kBT/m
b) (a2+β2)kBT/m
c) (a+β)2kBT/m
d) (a+β)2kBT/m
Ans: B
Q.53) Two monochromatic sources, L1and L2 emit light at 600 and 700 nm, respectively. If their frequency bandwidths are 10-1and 10-3GHz, respectively, then the ratio of linewidth of L1 to L2 is approximately
a) 100:1
b) 1:85
c) 75:1
d) 1:75
Ans: C
Q.54) A ring of mass m and radius R rolls (without slipping ) down an inclined plane starting from rest. If the centre of the ring is initially at a height h, the angular velocity when the ring reaches the base is
a) g/(h-R)tanθ
b) g/(h/R)
c) g/(h-R)/R2
d) 2g/(h-R)
Ans: C
Q.55) A loaded dice has the probabilities 121,221,321,421,521 and 621of turning up 1, 2, 3, 4, 5 and 6, respectively. If it is thrown twice, what is the probability that the sum of the members that turn up is even?
a) 144441
b) 225441
c) 221441
d) 220441
Ans: B
Q.56) What does the diagram below establish?
a) π>3
b) π≥22
c) π<4
d) π is closer to 3 and 4
Ans: D
Q.57) 366 players participate in a knock-out tournament. In each round all competing players pair together and play a match, the winner of each match moving to the next round. If at the end of a round there is an odd number of winners, the unpaired one moves to the next round without playing a match. What is the total number of matches played?
a) 366
b) 282
c) 365
d) 416
Ans: C
Q.58) A student buys a book from an online shop at 20% discount. His friend buys another copy of the same book in a book fair for Rs.192 paying 20% less than his friend. What is the full price of the book?
a) Rs. 275
b) Rs. 300
c) Rs. 320
d) Rs. 392
Ans: B
Q.59) Consider the sequence of order sets of natural numbers:
1,2,3,4,5,6,…
What is the last number in the 10th set?
a) 10
b) 19
c) 55
d) 67
Ans: C
Q.60) (25÷5+3–2 x 4)+(16×4–3)=
a) 61
b) 22
c) 41/24
d) 16
Ans: A
Q.61) What is the arithmetic mean of 11×2,12×3,13×4,14×5…..1100×101?
a) 0.01
b) 1101
c) 0.00111…
d) 149×50+150×512
Ans: B
Q.62) Three fishermen caught fishes and went to sleep. One of them woke up, took away one fish and 1/3nd of the remainder as his share, without others’ knowledge. Later, the three of them divided the remainder equally. How many fishes were caught?
a) 58
b) 19
c) 76
d) 88
Ans: B
Q.63) What is the perimeter of the given figure, where adjacent sides are at right angles to each other?
a) 20cm
b) 18cm
c) 21cm
d) cannot be determined
Ans: B
Q.64) What is the next number in the following sequence?
39, 42, 46, 50,…….
a) 52
b) 53
c) 54
d) 55
Ans: C
Q.65) There are 2 hills, A and B, in a region. If hills A is located N30˚E of hills B, what will be the direction of hills B when observed from hill A? (N30˚E means 30˚ from north towards east).
a) S30˚W
b) S60˚W
c) S30˚W
d) S60˚W
Ans: A
Q.66) A circle circumscribes identical, close-packed circles of unit diameter as shown in the figure. What is the total area of the shaded portion?
a) 2
b) 2π
c) 1⁄2
d) π⁄2
Ans: D
Q.67) One of the four A B, C and D committed a crime. A said, “I did it”, B said, “I didn’t”, C said, “B did it’, D said, ‘A did it”. Who is lying?
a) A
b) B
c) C
d) D
Ans: C
Q.68) A circle of radius 7 units lying in the fourth quadrant touches the x-axis at (10,0).The centre of the circle has coordinates
a) (7,7)
b) (-10,7)
c) (10.-7)
d) (7.-7)
Ans: C
Q.69) How many nine-digit positive integers are there, the sum of squares of whose digits is 2?
a) 8
b) 9
c) 10
d) 11
Ans: A
Q.70) If a+b+c+d+e=10 (all positive numbers), then the maximum value of a×b×c×d×e is
a) 12
b) 32
c) 48
d) 72
Ans: B
Q.71) A car is moving along a straight track. Its speed is changing with time as shown below. Which of the following statements is correct?
a) The speed is never zero.
b) The acceleration is zero once on the path.
c) The distance covered initially increases and then decreases.
d) The car comes back to its initial position once.
Ans: B
Q.72) A farmer gives 7 full, 7 half-full and 7 empty bottles of honey to his three sons and asks them to share these among themselves such that each of them gets the same amount of honey and the same number of bottles. In how many ways can this be done? (bottles camot be distinguished otherwise, they are sealed and cannot be broken).
a) 1
b) 2
c) 0
d) 3
Ans: B
Q.73) Every time a ball falls to ground it bounces back to half the height it fell from. A ball is dropped from a height of 1024 cm. The maximum height from the ground to which it can rise after the tenth bounce is
a) 102.4cm
b) 1.24 cm
c) 1cm
d) 2cm
Ans: C
Q.74) For real number s x andy,x2+(y-y)2=0.Then the value of x+y is
a) 0
b) 2
c) 2
d) 4
Ans: D
Q.75) A cylinder of radius 1 cm is broken into three piece. Which of the following MUST be true?
a) At least one piece has volume equal to 1cm3
b) At least two pieces have equal volumes
c) At least one piece has volume less than 1cm3
d) At least one piece has volume greater than 1cm3
Ans: D
