Firstranker's choice
GATE 2015
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Q. 1 – Q. 25 carry one mark each.
Q.1 A satellite is moving in a circular orbit around the Earth. If T, V and E are its average kinetic, average potential and total energies, respectively, then which one of the following options is correct?
(A) V = -2T; E = -T
(B) V = -T; E = 0
(C) V = -T/2; E = T/2
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(D) V = -3T/2; E = -T/2
Q.2 The Pauli matrices for three spin-½ particles are ô1, ô2, and?3, respectively. The dimension of the Hilbert space required to define an operator Ô = 812 ×? is
Q.3 The mean kinetic energy of a nucleon in a nucleus of atomic weight A varies as A", where n is (upto two decimal places)
Q.4 Let L and be the angular and linear momentum operators, respectively, for a particle. The commutator [Lx,py] gives
(A) -ih p
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(B) 0
(C) ih p
(D) ih p
Q.5 The decay µe + y is forbidden, because it violates
(A) momentum and lepton number conservations
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(B) baryon and lepton number conservations
(C) angular momentum conservation
(D) lepton number conservation
Q.6 An operator for a spin-½ particle is given by A = ?· B, where B = (x + y), ? denotes Pauli matrices and is a constant. The eigenvalues of A are
(A) ± ? ? / v2
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(B) ±?
(C) 0, ?
(D) 0, - ? ?
Q.7 In an inertial frame S, two events A and B take place at (ct1 = 0, r1 = 0) and (ct1 =0, i1 =2y), respectively. The times at which these events take place in a frame S' moving with a velocity 0.6cy with respect to S are given by
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(A) ct'1 = 0; ct'1 = -3/2
(B) ct' = 0; ct' = 0
(C) ct' = 0; ct' = 3/2
(D) ct' = 0; ct' =1/2
Q.8 Given that the magnetic flux through the closed loop PQRSP is $. If [A·dl = p1 along PQR, the value of ? 1 di dl along PSR is
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(?) f-F
(?) F -f
(C) - F
(D) F
Q.9 If f(x) = e¯*² and g(x) = |x|e¯*², then
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(A) f andg are differentiable everywhere
(B) f is differentiable everywhere but g is not
(C) g is differentiable everywhere but f i
(D) g is discontinuous at x = 0
Q.10 In Bose-Einstein condensates, the particles
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(A) have strong interparticle attraction
(B) condense in real space
(C) have overlapping wavefuntions
(D) have large and positive chemical potential
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Q.11 Consider a system of N non-interacting spin-½ particles, each having a magnetic momentu, is in a magnetic field B = B 2. If E_is the total energy of the system, the number of accessible microstates O is given by
(?) O = N! / (N / 2-E/µB)! (N/2+E/µB)!
(?) O = (N / 2+E/µB)!
(C) O== (N / 2+E/µB)
(D) O = N! / (N/2+E/µB)!
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Q.12 For a black body radiation in a cavity, photons are created and annihilated freely as a result of emission and absorption by the walls of the cavity. This is because
(A) the chemical potential of the photons is zero
(B) photons obey Pauli exclusion principle
(C) photons are spin-1 particles
(D) the entropy of the photons is very large
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Q.13 Consider w= f(z) = u(x, y) + iv(x, y) to be an
(A) u(x, y) satisfies Laplace equation in D
(B) v(x, y) satisfies Laplace equation in D
(C) f(z)dz is depender on the choice of the contour between 21 and 22 in D
Q.14 The value of ft²(3t-6)dt is (upto one decimal place)
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Q.15 Which one of the following DOES NOT represent an exclusive OR operation for inputs A and B ?
(A) (A + B)AB
(B) AB + BA
(C) (A + B)(A + B)
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(D) (A + B)AB
Q.16 Consider a complex function f(z) = z(z++)cos (zn) Which one of the following statements is correct?
(A) f(z) has simple poles at z = 0 and z = – 1/2
(B) f(z) has a second order pole at z = – 1/2
(C) f(z) has infinite number of second order poles
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(D) f(z) has all simple poles
Q.17 In the given circuit, the voltage across the source resistor is 1 V. The drain voltage (in V) is
Q.18 A point charge is placed between two semi-infinite conducting plates which are inclined at an angle of 30° with respect to each other. The number of image charges is
Q.19 A beam of X-ray of intensity Io is incident normally on a metal sheet of thickness 2 mm. The intensity of the transmitted beam is 0.02510. The linear absorption coefficient of the metal sheet (in m¯¹) is (upto one decimal place)
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Q.20 The lattice parameters a, b, c of an orthorhombic crystal are related by a = 2b = 3c . In units of a, the interplanar separation between the (110) planes is (upto three decimal places)
Q.21 In a Hall effect experiment, the Hall voltage for an intrinsic semiconductor is negative. This is because (symbols carry usual meaning)
(A) n˜ p
(B) n>p
(C) µe > µa
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(D) me > m
Q.22 The space between two plates of a capacitor carrying charges +Q and –Q is filled with two different dielectric materials, as shown in the figure. Across the interface of the two dielectric materials, which one of the following statements is correct?
(A) E and D are continuous
(B) E is continuous and B is disconitnuous
(C) D is continuous and E is discontinuous
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) E and D are discontinuous
Q.23 The energy dependence of the density of states for a two dimensional non-relativistic electron gas is given by, g (E) = CE", where C is constant. The value of n is
Q.24 The dispersion relation for phonons in a one dimensional monatomic Bravais lattice with lattice spacing a and consisting of ions of masses M is given by, w(k) = 2C[1-cos(ka)], where @ is the frequency of oscillation, k is the wavevector and C is the spring constant. For the long wavelength modes (? > a), the ratio of the phase velocity to the group velocity is
Q.25 Four forces are given below in Cartesian and spherical polar coordinates.
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(i) F1 = K exp(-r² / R²)
(ii) F2 = K(x²y - y³2)
(iii) F = K(x²x+y³y)
(iv) F1 = K($/r) where K is a constant. Identify the correct option.
(A) (iii) and (iv) are conservative but (i) and (ii) are not
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(B) (i) and (ii) are conservative but (iii) and (iv) are not
(C) (ii) and (iii) are conservative but (i) and (iv) are not
(D) (i) and (iii) are conservative but (ii) and (iv) are not
Q. 26 – Q. 55 carry two marks each.
Q.26 Consider a system of eight non-interacting, identical quantum particles of spin-3/2 in a one dimensional box of length L. The minimum excitation energy of the system, in units of is p²² / 2mL2
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Q.27 The excitation wavelength of laser in a Raman effect experiment is 546 nm. If the Stokes' line is observed at 552 nm, then the wavenumber of the anti-Stokes' line (in cm¯¹) is
Q.28 The binding energy per molecule of NaCl (lattice parameter is 0.563 nm) is 7.95 eV. The repulsive term of the potential is of the form where K is a constant. The value of the Madelung constant is (upto three decimal places) (Electron charge e = -1.6W = -1.6 × 10-19 C; e? = 8.854 × 10-12 C2N-1m-2)
Q.29 Given that the Fermi energy of gold is 5.54 eV, the number density of electrons is × 1028 m-3 (upto one decimal place) (Mass of electron = 9.11 × 10-31 kg; h = 6.626 × 10-34J. s; 1 eV = 1.6 × 10-19J)
Q.30 The band gap of an intrinsic semiconductor is Eg = 0.72 eV and m1 = 6m. At 300 K, the Fermi level with respect to the edge of the valence band (in eV) is at (upto three decimal places) kB = 1.38 × 10-23JK-1
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Q.31 The number of permitted transitions from 2P3/2 ? 2S1/2 in the presence of a weak magnetic field is
Q.32 Which one of the following represents the electron occupancy for a superconductor in its normal and superconducting states?
(A)
(B)
(C)
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(D)
Q.33 A charge -q is distributed uniformly over a sphere, with a positive charge q at its center in (i). Also in (ii), a charge -q is distributed uniformly over an ellipsoid with a positive charge q at its center. With respect to the origin of the coordinate system, which one of the following statements is correct?
(A) The dipole moment is zero in both (i) and (ii)
(B) The dipole moment is non-zero in (i) but zero in (ii)
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(C) The dipole moment is zero in (i) but non-zero in (ii)
(D) The dipole moment is non-zero in both (i) and (ii)
Q.34 Consider the circuit shown in the figure, where RC = 1. For an input signal V1 shown below, choose the correct Vo from the options:
(A)
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(B)
(C)
(D)
Q.35 A long solenoid is embedded in a conducting medium and is insulated from the medium. If the current through the solenoid is increased at a constant rate, the induced current in the medium as a function of the radial distance r from the axis of the solenoid is proportional to
(A) r² inside the solenoid and = outside
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(B) r inside the solenoid and outside
(C) r² inside the solenoid and outside
(D) r inside the solenoid and outside
Q.36 In the simple current source shown in the figure, Q1 and Q2 are identical transistors with current gain ß = 100 and VBE = 0.7 V The current I, (in mA) is (upto two decimal places)
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Q.37 Match the phrases in Group I and Group II and identify the correct option.
Group I (P) Electron spin resonance (ESR) (Q) Nuclear magnetic resonance (NMR) (R) Transition between vibrational states of a molecule (S) Electronic transition
Group II (i) radio frequency (ii) visible range frequency (iii) microwave frequency (iv) far-infrared range
(A) (P-i), (Q-ii), (R-iii), (S-iv)
(B) (P-ii), (Q-i), (R-iv), (S-iii)
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(C) (P-iii), (Q-iv), (R-i), (S-ii)
(D) (P-iii), (Q-i), (R-iv), (S-ii)
Q.38 Consider the motion of the Sun with respect to the rotation of the Earth about its axis. If F and Fco denote the centrifugal and the Coriolis forces, respectively, acting on the Sun, then
(A) F is radially outward and Foo
(B) F is radially inward and Fco F= -2F
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(C) F is radially outward and Fco Fco=-2F
(D) F is radially outward and FC = 2F
Q.39 In a rigid-rotator of mass M, if the energy of the first excited state is 1 meV, then the fourth excited state energy (in meV) is
Q.40 A plane wave (x+iy)E, exp[i (kz – ot)] after passing through an optical element emerges as (x-iy)E, exp[i(kz – ot)], where k and are the wavevector and the angular frequency, respectively. The optical element is a
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(A) quarter wave plate
(B) half wave plate
(C) polarizer
(D) Faraday rotator
Q.41 The Lagrangian for a particle of mass m at a position moving with a velocity v is given by L = v² + Crv - V(r), where V(r) is a potential and C is a constant. If p is the canonical momentum, then its Hamiltonian is given by
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(A)(p+CF)²+V(r)
(B) (p-CF) +V(r)
(C) Pa+V(r)
(D) p²+c² r² + V(r)
Q.42 The Hamiltonian for a system of two particles of masses m1 and m2 at 7 and having velocities and v2 is given by_H =mv²+mv² one of the following statements is correct?
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(A) The total energy and total momentum conserved
(B) Only the total energy is conserved
(C) The total energy and the z - component of the total angular momentum are conserved
(D) The total energy and total angular momentum are conserved
Q.43 A particle of mass 0.01 kg falls freely in the earth's gravitational field with an initial velocity v (0) = 10 ms¯¹. If the air exerts a frictional force of the form, f = -kv, then for k = 0.05 Nm¯¹s, the velocity (in ms¯¹) at time t = 0.2 s is (use g = 10 ms¯² and e = 2.72) (upto two decimal places)
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Q.44 In the nuclear shell model, the potential is modeled as V V(r)==mw²r² - AL-S, ? > 0. The correct spin - parity and isospin assignments for the ground state of ¹³C is
(A) 1-/2;1/2
(B) 1+/2;1/2
(C) 3+/2;1/2
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(D) 3-/2;1/2
Q.45 A particle is confined in a box of length L as shown below.
If the potential Vo is treated as a perturbation, including the first order correction, the ground state energy is
(A) E = +V
(B) E =
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(C) E = +
Q.46 Suppose a linear harmonic oscillator of frequency @ and mass m is in the state at t = 0 where |%) and w1) are the ground and the first excited states, respectively. The value of (y|x|y) in the units of h / mo at t = 0 is
Q.47 A particle with rest mass M is at rest and decays into two particles of equal rest masses which move along the z axis. Their velocities are given by
(A) 1 = 2 = (0.8c) 2
(B) v1 = -2 = (0.8c) 2
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(C) 1 = -2 = (0.6c)2
(D) = (0.6c)2; v2 = (-0.8c)2
Q.48 In the given circuit, if the open loop gain_A = 10, the feedback configuration and the closed loop gain Af are
(A) series-shunt, Af = 9
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(B) series-series, Af = 10
(C) series-shunt, Af = 10
(D) shunt-shunt, Af = 10
Q.49 A function y(z) satisfies the ordinary differential equation y" += y' 2y = 0, where z m = 0,1,2,3,.... Consider the four statements P, Q, R, S as given below.
P: zm and z are linearly independent solutions for all values of m
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Q: z™ and z are linearly independent solutions for all values of m>0
R: In z and 1 are linearly independent solutions for m 0
S: z™ and ln z are linearly independent solutions for all values of m
The correct option for the combination of valid statements is
(A) P, R and S only
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(B) P and R only
(C) Q and R only
(D) R and S only
Q.50 The entropy of a gas containing N particles les enclosed in a volume V is given by S = NkB In where E is the total energy, a is a constant and k, is the Boltzmann constant. The chemical potential u of the system at a temperature 7 is given by
(A) µ = -kT In
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(?) µ = -kBT In
(C) µ = -kT In
(D) µ = -k&T In
Q.51 Let the Hamiltonian for two spin-½ particles of equal masses m, momenta p1 and p2 and positions and be H = P+ p²+=m² (r² + r²) + kô?¸·?2, where ?1 and ?2 denote the corresponding Pauli matrices, ha = 0.1eV and k = 0.2eV . If the ground state has net spin zero, then the energy (in eV) is
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Q.52 The average energy U of a one dimensional quantum oscillator of frequency @ and in contact with a heat bath at temperature T is given by
(A) U = -hwcoth - ß?
(B) U=hosinh ho
(C) U = = h tanh - Bho
(D) U ==h@coshß?
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Q.53 A monochromatic plane wave (wavelength = 600 nm) E, exp[i(kz – ot)] is incident normally on a diffraction grating giving rise to a plane wave E1 exp[-ot)] in the first order of diffraction. Here E,
Q.54 The Heaviside function is defined as H(t)= for t > 0 -1 fort
?
(A) /2
(B)
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(C) sin
?
(D) 0
Q.55 The atomic masses of 152 Eu, 152Sm, H and neutron are 151.921749, 151.919756, 1.007825 and 1.008665 in atomic mass units (amu), respectively. Using the above information, the Q-value of the reaction Eu + n ? 2Sm + p_is × 10-3 amu (upto three decimal places)
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