GATE PH 2015 Question Paper & Solutions

Quantitative Aptitude

Choose the appropriate word/phrase, out of the four options given below, to complete the following sentence: Apparent lifelessness (A) harbours (B) leads to dormant life. (C) supports (D) affects

Verbal Aptitude

Fill in the blank with the correct idiom/phrase. That boy from the town was a (A) dog out of herd (C) fish out of water in the sleepy village. (B) sheep from the heap (D) bird from the flock

Quantitative Aptitude

Choose the statement where underlined word is used correctly. (A) When the teacher eludes to different authors, he is being elusive. (B) When the thief keeps eluding the police, he is being elusive. (C) Matters that are difficult to understand, identify or remember are allusive. (D) Mirages can be allusive, but a better way to express them is illusory.

Quantitative Aptitude

Tanya is older than Eric. Cliff is older than Tanya. Enic is older than Chff. If the first two statements are true, then the third statement is: (A) True (B) False (C) Uncertain (D) Data insufficient

Quantitative Aptitude

Question Type : MCO Five teams have to compete in a league, with every team playing every other team exactly once, before going to the next round. How many matches will have to be held to complete the league round of matches? (A) 20 (B) 10 (C) 8 (D) 5

Quantitative Aptitude

Select the appropriate option in place of underlined part of the sentence. Increased productivity necessary reflects greater efforts made by the employees. (A) Increase in productivity necessary (B) Increase productivity is necessary (C) Increase in productivity necessarily D) No improvement required

Quantitative Aptitude

Question Type : MCO Given below are two statements followed by two conclusions. Assuming these statements to be true, decide which one logically follows. Statements: I. No manager is a leader. II. All leaders are executives. Conclusions: I. No manager is an executive. II. No executive is a manager. (A) Only conclusion I follows. (B) Only conclusion II follows. (C) Neither conclusion I nor II follows. (D) Both conclusions I and II follow.

Quantitative Aptitude

In the given figure angle Q is a right angle, PS:QS = 3: 1, RT:QT = 5:2 and PU: UR = 1:1. If area of triangle QTS is 20 cm', then the area of triangle PQR in cm' is. R P Q

Quantitative Aptitude

Right triangle PQR is to be constructed in the xy - plane so that the right angle is at P and line PR is parallel to the x-axis. The x and y coordinates of P, Q. and R are to be integers that satisfy the inequalities: -4 ≤ x≤ 5 and 6 ≤ y ≤ 16. How many different triangles could be constructed with these properties? (A) 110 (B) 1,100 (C) 9,900 (D) 10,000

Quantitative Aptitude

A coin is tossed thrice. Let X be the event that head occurs in each of the first two tosses. Let Y be the event that a tail occurs on the third toss. Let Z be the event that two tails occur in three tosses. Based on the above information, which one of the following statements is TRUE? (A) X and Y are not independent (B) Y and Z are dependent (C) Y and Z are independent (D) X and Z are independent

Lagrangian & Hamiltonian Dynamics

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 (A) V = - 2T; E = -T (C) V = -T/2; E = T/2

Schrodinger equation & Wave Mechanics

The Pauli matrices for three spin-½ particles are ô,, 02, and o3, respectively. The dimension of the Hilbert space required to define an operator Ô=õ, \cdotô, xô, is

Nuclear Models & Decays

The mean kinetic energy of a nucleon in a nucleus of atomic weight A varies as A", where n is (upto two decimal places)

Schrodinger equation & Wave Mechanics

Let L and p be the angular and linear momentum operators, respectively, for a particle. The commutator [Lx, P,] gives (B) 0 (C) it px (D) itip= (A) - it p=

Lagrangian & Hamiltonian Dynamics

The decay it → e + y is forbidden, because it violates (A) momentum and lepton number conservations (B) baryon and lepton number conservations (C) angular momentum conservation (D) lepton number conservation

Linear Algebra & Vector Calculus

An operator for a spin-½ particle is given by A= 2ỡ - B, where B= matrices and 2 is a constant. The eigenvalues of A are (A) + 2B/V2 (B) $^B (С) 0, Л В - (*+ý), o denotes Pauli (D) 0, - 1 B

Complex analysis

In an inertial frame S, two events A and B take place at (ct, = 0, 7, = 0) and (ct = 0, 7g = 2ý), respectively. The times at which these events take place in a frame S' moving with a velocity 0.6c with respect to S are given by (A) ct, =0; ct = -3/2 (B) ct, =0; Ct =0 (C) ct, = 0; ct = 3/2 (D) ct, = 0; ct =1/2

Complex analysis

vaine of ( - di slang sRis (А) Ф-Ф, (В) Ф -ф

Complex analysis

Question Type : MCO If f(x) = e-** and g (x) = xe-* , then (A) fandg are differentiable everywhere (B) fis differentiable everywhere but g is not (C) g is differentiable everywhere but f is not (D) g is discontinuous at x = 0 (C) - Ф

Electrostatics and Magnetostatics

R (D) Ф In Bose-Einstein condensates, the particles (A) have strong interparticle attraction (B) condense in real space (C) have overlapping wavefuntions (D) have large and positive chemical potential

Electrostatics and Magnetostatics

Consider a system of N non-interacting spin-½ particles, each having a magnetic moment p, is in a magnetic field B = B ?. If E is the total energy of the system, the number of accessible microstates S is given by N! (A) Q= (B) 2= N+- 4B ) N! (D) 0=

Electrostatics and Magnetostatics

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

Complex analysis

Consider w = f (z) = u(x, y) + iv(x, y) to be an analytic function in a domain D. Which one of the following options is NOT correct? (A) u(x, y) satisfies Laplace equation in D (B) v(x, y) satisfies Laplace equation in D 10) (e) d: is dependent on the choice of the confour between , and 2 in 20 (D) f(z) can be Taylor expanded in D

Complex analysis

The value of J? t2 8(3t-6)dt is (upto one decimal place)

Complex analysis

Which one of the following DOES NOT represent an exclusive OR operation for imputs A and B ? (A) (A + B)AB (C) (A + B)(A + B) (B) AB + BA (D) (A + B)AB

Complex analysis

Question Type : MCO Consider a complex function f(z) = 2(*+*)cos (2x) Which one of the following statements is correct? (A) f(z) has simple poles at z = 0 and z = - ½/ (B) Az) has a second order pole at z = - ½2 (C) f(z) has infinite number of second order poles (D) f(z) has all simple poles

Complex analysis

In the given circuit, the voltage across the source resistor is 1 V. The drain voltage (in V) is 25 V 5k2 2 M2 500 92

Electrostatics and Magnetostatics

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

Complex analysis

A beam of X-ray of intensity lo is incident normally on a metal sheet of thickness 2 mm. The intensity of the transmitted beam is 0.0251. The linear absorption coefficient of the metal sheet (in m') is . (upto one decimal place)

Crystal Structure & Lattice Dynamics

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)

Crystal Structure & Lattice Dynamics

In a Hall effect experiment, the Hall voltage for an intrinsic semiconductor is negative. This is because (symbols carry usual meaning) (A) no P (B) n>p (C) Me > 144 (D) me > my

Electrostatics and Magnetostatics

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? +Q -Q (A) E and D are continuous (B) E is continuous and D is disconitnuous (C) D is continuous and E is discontinuous (D) E and D are discontinuous

Complex analysis

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

Crystal Structure & Lattice Dynamics

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 o t) = i [1 - cos(ka), where a is the frequency of oscillation, k is the wavevector and C is the spring constant. For the long wavelength modes (1 >> a), the ratio of the phase velocity to the group velocity is

Complex analysis

Question Type : MCO Four forces are given below in Cartesian and spherical polar coordinates. F=Kexp(-7/R')f (ii) F, = K(xy-y°₴) (iv) F, = K(ф/r) where K is a constant. Identify the correct option. (A) (i11) and (iv) are conservative but (i) and (i1) are not (B) (1) and (11) are conservative but (111) and (iv) are not (C) ii) and (ini) are conservative but (i) and (iv) are not (D) (i) and (iii) are conservative but (i1) and (iv) are not

Complex analysis

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 2mE

Atomic Spectra & Multi-Electron Atoms

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") is

Electrostatics and Magnetostatics

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 yo, where K is a constant. The value of the Madelung constant 15 _ _ (upto three decimal places) (Electron charge e = -1.6 × 10-19 C; E0 = 8.854 × 10-12 C2N-4m=2)

Statistical Ensembles & Partition Functions

Given that the Fermi energy of gold is 5.54 eV, the number density of electrons × 1028 m-3 (upto one decimal place) (Mass of electron = 9.11 × 10-31 kg; h = 6.626 × 10-34].s; 1 eV = 1.6 × 10- 191)

Statistical Ensembles & Partition Functions

The band gap of an intrinsic semiconductor is E, = 0.72 eV and my = 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

Electrostatics and Magnetostatics

The number of permitted transitions from ?P3/2 → 3S12 in the presence of a weak magnetic field is

Crystal Structure & Lattice Dynamics

Which one of the following represents the electron occupancy for a superconductor in its normal and superconducting states? (A) (B) Normal state Superconducting state Normal state Superconducting state f(E) f(E) f(E) f(E) E . E E E (C) (D) Normal state Superconducting state Normal state Superconducting state f(E) f(E) f(E) f(E) E E E E

Electrostatics and Magnetostatics

A charge -q 15 distributed unformly over a sphere, with a positive charge q at its center in (1) Also in (11), a charge -q is distributed uniformly over an ellipsod 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? AX AX Y (i) (ii) (A) The dipole moment is zero in both (i) and (i1) (B) The dipole moment is non-zero in (i) but zero in (ii) (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)

Complex analysis

Consider the circuit shown in the figure, where RC = 1. For an input signal Vi shown below, choose the correct V. from the options: R Vi o- 1 o Vo R$ 1 2 3 (A) (B) 2 -1 (C) Vo T 0.1 (D) VoT 1 3 -0.1

Electromagnetic Waves in Media

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) p? inside the solenoid and = outside (B) r inside the solenoid and z outside (C) p? inside the solenoid and outside D) r inside the solenoid and ½ outside

Complex analysis

In the simple current source shown in the figure, Qi and Q are identical transistors with current gain B = 100 and VE = 0.7 V 5k23 The current I. (in mA) is (upto two decimal places)

Atomic Spectra & Multi-Electron Atoms

Match the phrases in Group I and Group II and identify the correct option. Group I Group I (P) Electron spin resonance (ESR) (Q) Nuclear magnetic resonance (NMR) (R) Transition between vibrational states of a molecule (S) Electronic transition 1) radio frequency (i1) visible range frequency (111) microwave frequency (1v) far-infrared range (A) (P-i), (Q-ii), (R-iii), (S-iv) (B) (P-ii), (Q-i), (R-iv), (S-iii) (C) (P-iii), (Q-iv), (R-i), (S-ii) (D) (P-iii), (Q-i), (R-iv), (S-ii)

Lagrangian & Hamiltonian Dynamics

Consider the motion of the Sun with respect to the rotation of the Earth about its axis. If F. and Fc. denote the centrifugal and the Coriolis forces, respectively, acting on the Sun, then (A) F is radially outward and Fco= Fo (B) F is radially inward and Fco= - 2F. (C) F is radially outward and Fco= - 2F. (D) F, is radially outward and Fco= 2F.

Complex analysis

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 _

Electromagnetic Waves in Media

A plane wave (* + iý)E, expli(kz - cot) after passing through an optical element emerges as (*- iỷ)E, exp (kz- cot), where k and co are the wavevector and the angular frequency, respectively. The optical element is a (A) quarter wave plate (B) half wave plate (C) polarizer (D) Faraday rotator

Lagrangian & Hamiltonian Dynamics

Question Type : MCO The Lagrangian for a particle of mass m at a position 7 moving with a velocity v is given by L= =D° + CFV -V), where V(v) is a potential and C is a constant. If P. is the canonical momentum, then its Hamiltonian is given by 1 (A) 2m -(P. + C7) + V(r) (B) 1 _(P. -CF) +V(r) 2m (D) 2m

Lagrangian & Hamiltonian Dynamics

The Hamiltonian for a system of two particles of masses mi and mz at f and F having velocities Tand ½, is given by H = _my + one of the following statements is correct? 77\cdot(4x%), where Cis a constant. Which (4-7) (A) The total energy and total momentum are 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

Complex analysis

A particle of mass 0.01 kg falls freely in the earth's gravitational field with an initial velocity v (0) = 10 ms'. If the air exerts a frictional force of the form, f = -kv, then fork = 0.05 Nm 1s, the velocity (in ms) at time t = 0.2 s is (upto two decimal places) (use g = 10 ms-2 and e = 2.72)

Electrostatics and Magnetostatics

Question Type : MCO In the nuclear shell model, the potential is modeled as 1(r) == mor'-AL-S, 2>0. The correct spin - parity and isospin assignments for the ground state of 3C' is (A) 1 2 -1 2 1+ (B) - 2 2 (C) 3+ 2 1 2

Electrostatics and Magnetostatics

A particle is confined in a box of length L as shown below. Vo 1 L/2 If the potential Vo is treated as a perturbation, including the first order correction, the ground state energy 1s tỉ T (A) E= (C) E =- 2mE +- 4 (B) E= 2mE? tỉ Tỉ (D) E= 2ml + 2

Schrodinger equation & Wave Mechanics

Suppose a linear harmonic oscillator of frequency c and mass m is in the state at t = 0 where (Vo) and (41) are the ground and the first excited states, respectively. The value of (y xy) in the units of Vmo at t = 0 15

Nuclear Models & Decays

3 A particle with rest mass M is at rest and decays into two particles of equal rest masses -M 10 which move along the z axis. Their velocities are given by (A) = V= (0.80) = (C) V, = -V, = (0.6c) ₴ (B) V, = -V, = (0.8c) ₴ (D) = (0.6c) 2; V, = (-0.8c) ₴

Complex analysis

Question Type : MCO In the given circuit, if the open loop gain A = 105 , the feedback configuration and the closed loop gain Af are Vio 0 Vo 9 kg 1 kn $ 1 (A) series-shunt, Af = 9 (C) series-shunt, Af = 10 (B) series-series, Ag = 10 (D) shunt-shunt, Af = 10

Complex analysis

A function (2) satisfies the ordinary diferential equation y° + - * = 0, where m = 0,1,2,3,.... Consider the four statements P, Q, R, S as given below. P: z'' and z*™ are linearly independent solutions for all values of m Q: z'' and z'' are linearly independent solutions for all values of m>o R: In z and 1 are linearly independent solutions for m = 0 S: z'' and In 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 (B) P and R only (C) Q and R only (D) R and S only

Electrostatics and Magnetostatics

The entropy of a gas containing N particles enclosed in a volume V is given by aVE/ S = Nkg In (w% Where is the total energy, a is a constant and kg is the Boltzmann constant. The chemical potential u of the system at a temperature T is given by (A) M = -KI aVEl win. (B) #= -kgT aVEZ (C) M=-k I In (D) M=-k I mavEy

Lagrangian & Hamiltonian Dynamics

Let the Hamiltonian for two spin-/ particles of equal masses m, momenta p, and P, and where o, and o, denote the corresponding Pauli matrices, to = 0.1eV and k = 0.2eV . If the ground state has net spin zero, then the energy (in eV) is

Statistical Ensembles & Partition Functions

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 (B) U=-rosinh ¿Bro) (0) U=natanh - Bro

Complex analysis

A monochromatic plane wave (wavelength = 600 nm) E, expli(kz - ct)] is incident normally on a diffraction grating giving rise to a plane wave E, exp ilk, \cdotF- cot) in the first order of diffraction. Here E, < Eo and k, The period (in um) of the diffraction grating is (upto one decimal place)

Fourier and Laplace transforms

f+1 fort> 0 The Heaviside function is defined as H (t) = 1-1 fort <0 - 21/c. The Fourier transform of - 1(1+1/2) - 17(1-1/2) is and its Fourier transform is given by (A) (0) sin(;)

Complex analysis

The atomic masses of 53 Eu, 62Sm, }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 63 Eu +n → 625m+ p is _ × 103 amu (upto three decimal places)