PART-1
Q.1) Four equal point charges are kept fixed at the four vertices of a square. How many neutral points (i.e. points where the electric field vanishes) will be found inside the square?
- a) 1
- b) 4
- c) 5
- d) 7
Ans: C
Q.2) A static charge distribution gives rise to an electric field of the form E⃗=α(1-e-r/R)r^r2,where α and R are positive constants. The charge contained within a sphere of radius Centered at the origin is
- a) παε0eR2
- b) παε0e2R2
- c) 4παε0Re
- d) παε0R2e
Ans: D
Q.3) Consider a solenoid of radius R with n turns per unit length, in which a time dependent current I=I0 sin ωt (where ωR/c≪1) flows. The magnitude of the electric field at a perpendicular distance r<R from the axis of symmetry of the solenoid, is
- a) 0
- b) 12ωμ0nI0R2cos ωt
- c) 12ωμ0nI0rsin ωt
- d) 12ωμ0nI0rcos ωt
Ans: D
Q.4) Consider three polarizer’s P1,P2 and P3 placed along an axis as shown in the figure.
- a) 0
- b) I02
- c) I08sin22θ
- d) I04sin22θ
Ans: C
Q.5) In a young’s double-slit interference experiment, the slits are at a distance of 2L from each other and the screen is at a distance D from the slits. If a glass slab of refractive index μ and thickness d is placed in the path of one of the beams, the minimum value of d for the central fringe to be dark is
- a) λD(μ-1)D2+L2
- b) λD(μ-1)L
- c) λD(μ-1)
- d) λ2(μ-1)
Ans: D
Q.6) A constant electric current I in an infinitely long straight wire is suddenly switched on at t=0. The vector potential at a perpendicular distance r from the wire is given by A→=k^μ0I2π1n1rct+c2t2-r2. The electric field at a distance r(<ct) is
- a) 0
- b) μ0I2πt12(i^-j^)
- c) cμ0I2πc2t2-r212(i^+j^)
- d) cμ0I2πc2t2-r2k^
Ans: D
Q.7) The equation of the plane that is tangent to the surface xyz=8 at the point (1, 2, 4) is
- a) x+2y+4z=12
- b) 4x+2y+z=12
- c) x+4y+2=0
- d) x+y+z+=7
Ans: B
Q.8) A 3×3 matrix M has Tr[M]=6,Tr[M2]=26 and Tr[M3]=90 Which of the following can be a possible set of eigenvalues of M?
- a) 1,1,4
- b) -1,0,7
- c) -1,3,4
- d) 2,2,2
Ans: C
Q.9) A generic 3×3 real matrix A has eigenvalues 0, 1 and 6, and I is the 3 x 3 identity matrix. The quantity/quantities that cannot be determined from this information is/are the
- a) eigenvalues of (I+A)-1
- b) eigenvalues of (I+ATA)
- c) determinant of ATA
- d) rank of A
Ans: B
Q.10) Consider a Maxwellian distribution of the velocity of the molecules of an ideal gas. Let Vmp andVrms denote the most probable velocity and the root mean square velocity, respectively. The magnitude of the ratio Vmp/Vrms is.
- a) 1
- b) 2/3
- c) 23
- d) 3 2
Ans: C
Q.11) Let x1(t) and x2(t) be two linearly independent solutions of the differential equation d2xdt2+2dxdt+f(t)x=0 and let w(t)=x1(t)dx2(t)dx1-x2(t)dx1(t)dt. If w(0)=1, then w(1) is given by
- a) 1
- b) e2
- c) 1/e
- d) 1/e2
Ans: D
Q.12) The generating function f(x,t)=∑n=0∞ pn(x)tn for the Legendre polynomials pn(x) is .F(x,t)=(1-2xt+t2)-1⁄2The value of p3(-1) is
- a) 5/2
- b) 3/2
- c) +1
- d) -1
Ans: D
Q.13) The graph of the function f(x)=12n≤x≤2n+102n+1≤x≤2n+2where n=(0,1,2,……) is shown below. Its Laplace transform f~(s) is
- a) 1+e-ss
- b) 1-e-ss
- c) 1s1+e-s
- d) 1s1-e-s
Ans: C
Q.14) The internal energy E of a system is given by E=bS3VN,where b is a constant and other symbols have their usual meaning. The temperature of this system is equal to
- a) bS2VN
- b) 3bS2VN
- c) bS3V2N
- d) SN2
Ans: B
Q.15) A system of N non-interacting spin-1/2 particles is placed in an external magnetic fieldH.The behaviour of the entropy of the system as a function of energy is given by
- a)
- b)
- c)
- d)
Ans: A
Q.16) Let u(x,y)=x+12(x2-y2) be the real part of analytic function f(z) of the complex variable z=x+iy. The imaginary part of f(z) is
- a) y+xy
- b) xy
- c) y
- d) y2-x2
Ans: A
Q.17) The first few terms in the Taylor series expansion of the function f(x)=sin x around x=π4 are:
- a) 121+x-π4+12!x-π42+13!x-π43……
- b) 121+x-π4-12!x-π42-13!x-π43……
- c) x-π4-13!x-π43
- d) 121-x+x22!-x33!
Ans: B
Q.18) An unbiased dice is thrown three times successively. The probability that the number of dots on the uppermost surface add up to 16 is
- a) 116
- b) 136
- c) 1108
- d) 1216
Ans: B
Q.19) In the op-amp circuit shown in the figure below, the input voltage is 1V. The value of the output V0
- a) -33V
- b) -0.50V
- c) -1.00V
- d) -0.25V
Ans: C
Q.20) In the operational amplifier circuit below, the voltage at point A is
- a) 1.0V
- b) 0.5V
- c) 0V
- d) -5.0V
Ans: A
PART-2
Q.1) Gas molecules of mass m are confined in a cylinder of radius Rand height L (with R≫L) kept vertically in the Earth’s gravitational field. The average energy of the gas at low temperatures (such that mgL≫kBT) is given by
- a) NkBT/2
- b) 3NkBT
- c) 2NkBT
- d) 5NkBT/2
Ans: D
Q.2) A one-dimensional chain consists of a set of N rods each of length a. When stretched by a load, each rod can align either parallel or perpendicular to the length of the chain.The energy of a rod is-ε when perpendicular to it. When the chain is in thermal equilibrium at temperature T, its average length is.
- a) Na/2
- b) Na
- c) Na/1+e-2ε/kBT
- d) Na1+e-2ε/kBT
Ans: C
Q.3) A gas of N non-interacting particles is in thermal equilibrium at temperature T. Each particle can be in any of the possible non-degenerate states of energy 0,2ε and 4ε.The average energy per particle of the gas, when βε≪1,is
- a) 2ε
- b) 3ε
- c) 2ε/3
- d) ε
Ans: A
Q.4) The figure below shown a voltage regulator utilizing a Zener diode of breakdown voltage 5 V and a positive triangular wave input of amplitude 10 V. (a)
- a)
- b)
- c)
- d)
Ans: A
Q.5) If the number density of a free electron gas in three dimensions is increased eight times, its Fermi temperature will
- a) Increase by a factor of 4
- b) Decrease by a factor of 4
- c) Increase by a factor of 8
- d) Decrease by a factor of 8
Ans: A
Q.6) A counter consists of four-flops connected as shown in the figure:
- a) 1000
- b) 0001
- c) 0011
- d) 1100
Ans: B
Q.7) The pins 0, 1, 2 and 3 of part A of a microcontroller are connected with resistors to drive an LED at various intensities as shown in the figure. For VCC=4.2V and a voltage drop of 1.2 V across the LED, the range (maximum current) and resolution (step size) of the drive current are, respectively,
- a) 4.0 mA and 1.0 mA
- b) 15.0 mA and 1.0 mA
- c) 7.5 mA and 0.5 mA
- d) 4.0 mA and 0.5 mA
Ans: C
Q.8) An LED operates at 1.5 V and 5mA in forward bias. Assuming an 80% external efficiency of the LED, how many photons are emitted per second?
a) 5.0×106
b) 1.5×1016
c) 0.8×1016
d) 2.5×1016
Ans: D
Q.9) The wave function of a particle at time t=0 is given by |Ψ(0)⟩=12(|u1⟩+|u2⟩), where |u1⟩ and |u2⟩ are the normalized eigenstates with eigenvalues E1 and E2 respectively, (E2>E1), The shortest time after which |Ψ (t)〉 will become orthogonal to |Ψ (0)〉 is
a) -ℏπ2(E2-E1)
b) ℏπE2-E1
c) 2ℏπE2-E1
d) 2ℏπE2-E1
Ans: B
Q.10) A planet of mass m moves in the inverse square central force field of the sum of mass M. If the semi-major and semi-minor axes of the orbit are a and b, respectively the total energy of the planet is:
a) –GMma+b
b) –GMm1a+1b
c) -GMma1b-1a
d) –GMm a-b(a+b)2
Ans: A
Q.11) A double pendulum consists of two-point masses m attached by strings of length l as shown in the figure: The kinetic energy of the pendulum is
a) 12ml2[θ˙12+θ˙22]
b) 12ml2[2˙θ12+θ˙22+2θ˙1θ˙2cos (θ1-θ2)]
c) 12ml2[θ˙12+2θ22˙+2θ˙1θ˙2cos (θ1-θ2)]
d) 12ml2[2˙θ12+θ˙22+2θ˙1θ˙2cos (θ1+θ2)]
Ans: B
Q.12) A particle of mass m moves inside a bowl. If the surface of the bowl is given by the equation z=12a(x2+y2), where a is a constant, the Lagrangian of the particle is
a) 12m(r˙2+r2ϕ˙2-gar2)
b) 12m[(1+a2r2)r˙2+r2ϕ˙2]
c) 12m[r˙2+r2θ˙2+r2sin2θϕ˙2-gar2]
d) 12m[(1+a2r2)r˙2+r2ϕ˙2-gar2]
Ans: D
Q.13) Let | 0〉 and |1〉 denote the normalized eigenstates corresponding to the ground and first excited state of a one dimensional harmonic oscillator. The uncertainty ∆p in the state 12(|0〉+|1〉), is
a) ∆p=ℏmω/2
b) ∆p=ℏmω/2
c) ∆p=ℏmω
d) ∆p=2ℏmω
Ans: C
Q.14) The energy of the first excited quantum state of a particle in two-dimensional potential V(x,y)=12mω2(x2+4y2) is
a) 2ℏω
b) 3ℏω
c) 32ℏω
d) 52ℏω
Ans: D
Q.15) The potential of a diatomic molecule as a function of the distance r between the atoms is given by V(r)=ar6+br12. The value of the potential at equilibrium separation between the atoms is:
a) -4a2/b
b) -2a2/b
c) -a2/2b
d) -a2/4b
Ans: D
Q.16) Two particles of identical mass move in circular orbits under a central potentialV(r)=12 kr2.Let l1and l2 be the angular momenta and r1,r2 be the radii of the orbits respectively. If l1l2=2, the value of r1r2is:
a) 2
b) 1/2
c) 2
d) 1/2
Ans: A
Q.17) Consider a particle in a dimensional potential that satisfies V(x)=V(-x).Let |Ψ0⟩ and |Ψ1⟩ denote the ground and the first excited states, respectively, and let |Ψ⟩=α0 |Ψ0⟩+α0 |Ψ1⟩ be a normalized state with α0 and α1 being real constants. The expectation value 〈x〉 of the position operator x in the state |Ψ〉 is given by
a) α02⟨Ψ0│x│Ψ0⟩+α12⟨Ψ1│x│Ψ1⟩
b) α0α1⟨Ψ0│x│Ψ1⟩+⟨Ψ1│x│Ψ0⟩
c) α02+α12
d) 2α0α1
Ans: B
Q.18) A constant perturbation as shown in the figure below acts on a particle of mass m confined in an infinite potential well between 0 and L.
a) V02
b) 3V04
c) V04
d) 3V02
Ans: B
Q.19) The perturbation H’=bx4, where b is a constant, is added to the one dimensional harmonic oscillator potential V(x)=12mω2×2. Which of the following denotes the correction to the ground state energy to first order in b?
[Hint: The normalized ground state wave function of the one dimensional harmonic oscillator potential is Ψ0=(mωℏπ)1/4e-mωx2/2ℏ .You may use the following integral ∫-∞∞x2ne-ax2dx =a-n-12Г(n+12)].
a) 3bℏ24m2ω2
b) 3bℏ22m2ω2
c) 3bℏ22πm2ω2
d) 15bℏ24m2ω2
Ans: A
Q.20) A constant force F is applied to a relativistic particle of rest mass m. If the particle starts from rest at t=0, its speed after a time t is
a) Ftm
b) c tanhFtmc
c) c(1-e-Ft/mc)
d) FctF2t2+m2c2
Ans: D
Q.21) Given that the ground state energy of the hydrogen atom is -13.6 eV the ground state energy of positronium (which is a bound state of an electron and a positron is
,
a) + 6.8 eV
b) – 6.8 eV
c) – 13.6 eV
d) – 27.2 eV
Ans: B
Q.22) An atom of mass M can be excited to a state of mass (M+∆) by photon capture. The frequency of a photon which can cause this transition is
a) ∆c22h
b) ∆c2h
c) ∆2c22Mh
d) ∆2c22Mh(∆+2M)
Ans: D
Q.23) The ratio of intensities of the D1 and D2 lines of sodium at high temperature is
a) 1:1
b) 2:3
c) 1:3
d) 1:2
Ans: D
Q.24) A laser operating at 500 nm is used to excite a molecule. If the Stokes line is observed at 770 cm-1, the approximate positions of the Stokes and the anti-Stokes lines are
a) 481.5 nm and 520 nm
b) 481.5 nm and 500 nm
c) 500 nm and 520 nm
d) 500 nm and 600 nm
Ans: A
Q.25) The potential of a diatomic molecule as a function of the distance r between the atoms is given by Vr=-ar6+br12.The value of the potential at equilibrium separation between the atoms is:
a) -4a2/b
b) -2a2/b
c) -a2/2b
d) -a2/4b
Ans: D
PART-3
Q.1) The excitations of a three-dimensional solid are bosonic in nature with their frequency ω and wave-number k are related by ω∝k2 in the large wavelength limit. If the chemical potential is zero, the behaviour of the specific heat of the system at low temperature is proportional to
a) T1/2
b) T
c) T32
d) T3
Ans: C
Q.2) If the number density of a free electron gas in three dimensions is increased eight times, its Fermi temperature will
a) increase by a factor of 4
b) decrease by a factor of 4
c) increase by a factor of 8
d) ecrease by a factor of 8
Ans: A
Q.3) The difference in the Coulomb energy between the mirror nuclei Cr 2449 and Mn 2549 is 6.0 MeV. Assuming that the nuclei have a spherically symmetric charge distribution and that e2 is approximately1.0 MeV-fm, the radius of the Mn 2549 nucleus is
a) 4.9×10-13m
b) 4.9×10-15m
c) 5.1×10-13m
d) 5.1×10-15m
Ans: B
Q.4) According to the shell model the spin and parity of the two nuclei Sb 51125 and Sr 3889 are, respectively,
a) 52+ and 52+
b) 52+ and 72+
c) 72+ and 52+
d) 72+ and 72+
Ans: D
Q.5) An electron of energy 27 GeV collides with a proton of energy 820 GeV. The heaviest particle which can be produced in this collision has mass close to our.com
a) 300 GeV
b) 821 GeV
c) 850 GeV
d) 1127 GeV
Ans: A
Q.6) Assume that the free energy of a magnetic system has an expansion in the order parameter M of the form FM,T=aT-TC M2+bM4+cM6, with a, b and c>0. As the temperature is lowered below TC, the system undergoes a phase transition. The behaviour of the order parameter just below the transition, where T-TC is very small, is best described by
a) M∝TC-T-1/2
b) M∝TC-T1/2
c) M∝(TC-T)
d) M∝TC-T3
Ans: B
Q.7) The energy levels of electrons of mass ‘m’ and charge ‘e’ confined in an area A in the xy-plane with a uniform magnetic field B applied in the z-direction are given by En=n+12eBℏmc, n=0, 1, 2,… The degeneracy of each level is eBAℏc. The lowest level is completely filled the other are empty. The fermi energy ℏ2N2πmA, where N is the total number of electrons, is:
a) coincident with the n = 0 level
b) coincident with the n = 1 level
c) midway between then 0 and the n=1 levels
d) midway between then-1 and the n = 2 levels.
Ans: D
Q.8) An annulus of mass M made of a material of uniform density has inner and outer radii ‘a’ and ‘b’ respectively. Its principal moment of inertia along the axis of symmetry perpendicular to the plane of
a) 12Mb4+a4(b2-a2)
b) 12 M π b2-a2
c) 12M b2-a2
d) 12 M b2+a2
Ans: D
Q.9) The principal value of the real integral I=∫-3+3dxx2+3x+2 is:
a) 3π2
b) ℓn 25
c) ∞
d) 0
Ans: B
Q.10) The minimum energy of an electron (the rest mass of which is 0.5 MeV) that can emit Cherenkov radiation while passing through water (of refractive index 1.5) is approximately
a) 1.0 MeV
b) 3.0 MeV
c) 0.6 MeV
d) 0.5 MeV
Ans: C
Q.11) A heater and a thermocouple are used to measure and control temperature T of a sample at T0=250°C. A feedback circuit supplies power ‘P’ to the heater according to the equation P=P0+GT0-T-DdTdt with appropriately tuned values of the coefficients G and D. In order to maintain temperature stability in the presence of an external heat perturbation which causes small but rapid fluctuations of temperature, it is necessary to
a) decrease D
b) increase D
c) decrease G
d) increase G
Ans: A
Q.12) The trajectory on the zpz-plane (phase-space trajectory) of a ball bouncing perfectly elastically off a hard surface at == 0 is given by approximately by (neglect friction):
a) 
b) 
c) 
d) 
Ans: A
Q.13) According to the shell model the spin and parity of the two nuclei Sb 51225 and Sr 3889 are, respectively.
a) 52+and 52+
b) 52+and 72+
c) 72+and 52+
d) 72+and 72+
Ans: C
Q.14) The first few terms in the Taylor series expansion of the function f(x)=sin x around x=π4 are
a) 12 1+x-π4+12! x-π42+13! x-π43…
b) 12 1+x-π4-12! x-π42-13! x-π43…
c) x-π4-13! x-π43…
d) 12 1-x+x22!-X33!…
Ans: B
Q.15) A one-dimensional chain consists of a set of N rods each of length a. When stretched by a load, each rod can align either parallel or perpendicular to the length of the chain. The energy of a rod is -ε when aligned parallel to the length of the chain and is +ε when perpendicular to it. When the chain is in thermal equilibrium at temperature T, its average length is:
a) Na/2
b) Na
c) Na/(1+e-2ε/kBT)
d) Na/(1+e-2ε/kBT)
Ans: C
Q.16) If the hyperfine interaction in an atom is given by H=aS→c∙S→p, Where S→c and S→p denote the electron and proton spins, respectively, the splitting between the S1 3 and S0 1 state is
a) aℏ2/2
b) aℏ2
c) aℏ2/2
d) 2a ℏ2
Ans: B
Q.17) Monochromatic light of wavelength 660 nm and intensity 100 mW/cm2 falls on a solar cell of area 30 cm2. The conversion efficiency of the solar cell is 10%. If each converted photon results in an electron-hole pair, what is the maximum circuit current supplied by the solar cell? (Takeh=6.6×10-34J-s, c=3×108 m/s and e=1.6×10-19C).
a) 160 mA
b) 320 mA
c) 1600 mA
d) 3200 mA
Ans: A
Q.18) Consider three polarizers P1, P2 and P3 placed along an axis as shown in the Figure.P1 and P2
a) 0
b) I02
c) I08sin22θ
d) I04sin22θ
Ans: C
Q.19) A point particle of mass m carrying an electric charge q is attached to a spring of stiffness constant k.
A constant electric field E along the direction of the spring is switched on for a time interval T (where T<<m/k). Neglecting radiation loss, the amplitude of oscillation after the field is switched off is:
a) qE/k
b) qET2/m
c) qEm/Tk3/2
d) qET/mk
Ans: D
Q.20) In a Young’s double slit interference experiment, the slits are at a distance 2L from each other and the screen is at a distance D from the slits. If a glass slab of refractive index \mu and thickness d is placed in the path of one of the beams, the minimum value of d for the central fringe to be dark is
a) λDμ-1D2+L2
b) λD(μ-1)L
c) λ(μ-1)
d) λ2(μ-1)
Ans: D
Q.21) Consider a particle in a one dimensional potential that satisfies Vx=V-x. Let ψ0 and ψ1denote the ground and the first excited states, respectively, and let ψ=α0 ψ0+α1ψ1 be a normalized state with α0 and α1 being real constants. The expectation value x of the position operator x in the state ψ is given by
a) α02 ψ0|x|ψ0+α12 ψ1|x|ψ1
b) α0α1 [ψ0|x|ψ1+ψ1|x|ψ0]
c) α02+α12
d) 2α0α1
Ans: D
Q.22) A battery powers two circuits C1 and C2 as shown in t the figure
a) 7.0 mA
b) 7.5 mA
c) 5.0 mA
d) 10.5 mA
Ans: D
Q.23) There are two identical vessels of volume V each, one empty, end the other containing a block of wood of weight w. The vessels are then filled with water up to the brim. The two arrangements are shown as A and B in the figure. If the density of water is \rho and g is the acceleration due to gravity, Then
a) A and B have equal weights.
b) A is heavier than B by an amount w.
c) A is heavier than B by an amount V ρg-w.
d) B is heavier than A by an amount V ρg-w.
Ans: A
Q.24) If the father has blood group O and the mother has blood group AB, what are the possible blood groups of their children?
a) O, AB, A
b) A, B
c) A, O
d) B, AB
Ans: B
Q.25) Nucleus of P 32 and S 32, accelerated through the same potential difference enter a uniform, transverse magnetic field (Z = 15 for P and Z- 16 for S). As they emerge from the magnetic field.
a) both nucleus emerge undeflected
b) P 12 is deflected less than S 12
c) P 12 is deflected more than S 32
d) both are equally deflected
Ans: B
Q.26) A person chewing a bubble gum did not experience ear pain in a jet plane while landing whereas another person not chewing a gum had ear pain. The reason could be
a) chewing gum is a pain killer
b) chewing equilibrates pressure on both sides of the ear drum
c) chewing gum closes the ear drum
d) chewing distracts the person
Ans: B
Q.27) The reason why a lunar eclipse does not occur at every full moon is
a) The position of the sun is not favourable at all full moons.
b) The orbital planes of the moon and that of the earth are inclined to each other by a small angle.
c) the shape of the earth is not a perfect sphere.
d) The moon reflects only from one hemisphere.
Ans: B
Q.28) A boy throws a stone vertically upwards with certain unitial velocity. Which of the following graphs depicts the velocity as a function of time, if the acceleration due to gravity is assumed to be uniform and constant?
a) 
b) 
c) 
d) 
Ans: A
Q.29) A rigid uniform bar of a certain mass has two bobs of the same size, but with different densities \rho and 2\rho suspended identically from its ends.
a) 2d-d’
b) d>2d’
c) d=2d’
d) d<2d’
Ans: C
Q.30) There are two points A and A’ on the equator at longitudes 00 and 900 E, and two other points B and B’ on the same longitudes, respectively, but at latitude 600S. The distances (along the latitudes) between the points A, A’ and B, B’ are related by
a) AA’-BB’
b) AA’=2BB’
c) AA’=3BB’
d) AA’=2BB’
Ans: B
