GATE PH 2014 Question Paper & Solutions

Quantitative & Analytical Aptitude

The unit vector perpendicular to the surface 32 2 2= + +z y x at the point ( )1 , 1 , 1is (A) 3 ˆˆ ˆ z y x- + (B) 3 ˆˆ ˆ z y x- - (C) 3 ˆˆ ˆ z y x+ - (D) 3 ˆˆ ˆ z y x+ +

Quantitative & Analytical Aptitude

Which one of the following quantities is invariant under Lorentz transformation? (A) Charge density (B) Charge (C) Current (D) Electric field

Quantitative & Analytical Aptitude

The number of normal Zeeman splitting components of 1P → 1D transition is (A) 3 (B) 4 (C) 8 (D) 9

Quantitative & Analytical Aptitude

If the half -life of an elementary particle moving with speed 0.9 c in the laboratory frame is 810 5-× s, then the proper half-life is __________ × 10-8 s. (cc = 3 × 108 mm/ss)

Quantitative & Analytical Aptitude

An unpolarized light wave is incident from air on a glass surface at the Brewster angle. The angle between the reflected and the refracted wave is (A) 00 (B) 045 (C) 090 (D) 0120

Quantitative & Analytical Aptitude

Two masses m and 3m are attached to the two ends of a massless spring with force constant K. If m = 100 g and K = 0.3 N/m, then the natural angular frequency of oscillation is ___________ Hz.

Quantitative & Analytical Aptitude

The electric field of a uniform plane wave propagating in a dielectric, non-conducting medium is given by, ( )z txE ππ 4 . 0 10 6cos10ˆ 7 - ×=  V/m. The phase velocity of the wave is ________________ × 108 m/s.

Quantitative & Analytical Aptitude

The matrix     - - += 1 1 1 1 3 1 i iA is (A) orthogonal (B) symmetric (C) anti-symmetric (D) unitary

Quantitative & Analytical Aptitude

The recoil momentum of an atom is Ap when it emits an infrared photon of wavelength 1500 nm, and it is Bp when it emits a photon of visible wavelength 500 nm. The ratio B A p p is (A) 1 : 1 (B) 3 : 1 (C) 3 : 1 (D) 2 : 3

Quantitative & Analytical Aptitude

For a gas under isothermal conditions, its pressure P varies with volume V as 3 / 5-∝ V P . The bulk modulus B is proportional to (A) 2 / 1-V (B) 3 / 2-V (C) 5 / 3-V (D) 3 / 5-V

Nuclear & Particle Physics

Which one of the following high energy processes is allowed by conservation laws? (A) 0 0Λ + Λ → +p p (B) n p+ → +- 0π π (C) ee p nν+ + →- (D) γ µ+ →+ +e PH 1/10 PH (GATE 2014) 1 PHYSICS-PH

Mathematical Physics

The length element ds of an arc is given by, ( ) ( ) 2 122212 32 ) ( dx dxdx dx ds+ + = .The metric tensor ijg is

Quantum Mechanics

The ground state and the first excited state wave functions of a one dimensional infinite potential well are ψ1 and ψ2, respectively. When two spin-up electrons are placed in this potential, which one of the following, with x 1 and x2 denoting the position of the two electrons, correctly represents the space part of the ground state wave function of the system? (A) [ ]) ( ) ( ) ( ) ( 2 1 2 2 2 1 1 2 1 1x x x xψ ψ ψ ψ- (B) [ ]) ( ) ( ) ( ) ( 2 1 1 2 2 1 2 2 1 1x x x xψ ψ ψ ψ+ (C) [ ]) ( ) ( ) ( ) ( 2 1 2 2 2 1 1 2 1 1x x x xψ ψ ψ ψ+ (D) [ ]) ( ) ( ) ( ) ( 2 1 1 2 2 1 2 2 1 1x x x xψ ψ ψ ψ-

Electromagnetic Theory

If the vector potential ,ˆ3ˆ2ˆ z z y y x x A- + =α  satisfies the Coulomb gauge, the value of the constant α is ___________________ .

Quantum Mechanics

At a given temperature, T, the average energy per particle of a non -interacting gas of two-dimensional classical harmonic oscillators is ________________ kBT (kB is the Boltzmann constant).

Nuclear & Particle Physics

Which one of the following is a fermion? (A) α particle (B) 4Be7 nucleus (C) Hydrogen atom (D) Deuteron

Nuclear & Particle Physics

Which one of the following three-quark states (qqq), denoted by X, CAN NOT be a possible baryon? The corresponding electric charge is indicated in the superscript. (A) X + + (B) X + (C) X - (D) X - -

Classical Mechanics

The Hamilton's canonical equations of motion in terms of Poisson Brackets are

Nuclear & Particle Physics

The Miller indices of a plane passing through the three points having coordinates (0,0,1), (1,0,0), � 1 2 , 1 2 , 1 4� are (A) (212) (B) (111) (C) (121) (D) (211) PH 2/10 PH (GATE 2014) 1 PHYSICS-PH

Nuclear & Particle Physics

The plot of specific heat versus temperature across the superconducting transition temperature (TTCC ) is most appropriately represented by (A) (B) (C) (D)

Quantum Mechanics

If LL�⃗ is the orbital angular momentum and SS⃗ is the spin angular momentum, then LL�⃗. SS⃗ does NOT commute with (A) SSzz (B) LL2 (C) SS2 (D) �LL�⃗ + SS⃗� 2 T Cp TTCC Cp T TTCC Cp T TTCC Cp T TTCC PH 3/10 PH (GATE 2014) 1 PHYSICS-PH

Solid State Physics

The energy, 𝜀𝜀kk for band electrons as a function of the wave vector, kk in the first Brillouin zone (- 𝜋𝜋 aa⁄ ≤ kk ≤ 𝜋𝜋 aa⁄ ) of a one dimensional monatomic lattice is shown as (aa is lattice constant) The variation of the group velocity vvkk is most appropriately represented by (A) (B) (C) (D)

Nuclear & Particle Physics

For a free electron gas in two dimensions, the variation of the density of states, NN(EE) as a function of energy E, is best represented by (A) (B) (C) (D) N(E) E N(E) E N(E) E N(E) E O π/a -π/a kk 𝜀𝜀kk vvkk π/a -π/a O k vvkk π/a -π/a O k π/a -π/a O k vvkk π/a -π/a O k vvkk PH 4/10 PH (GATE 2014) 1 PHYSICS-PH

Electronics

The input given to an ideal OP-AMP integrator circuit is The correct output of the integrator circuit is (A) (B) (C) (D)

Electronics

The minimum number of flip-flops required to construct a mod-75 counter is _______. V t t0 V0 V t V0 t0 V t V0 t0 V t V0 t0 V t t0 V0 PH 5/10 PH (GATE 2014) 1 PHYSICS-PH Q. 26 - Q. 55 carry two marks each.

Nuclear & Particle Physics

A bead of mass m can slide without friction along a massless rod kept at 45 0 with the verti cal as shown in the figure. The rod is rotating about the vertical axis with a constant angular speed ω. At any instant, r is the distance of the bead from the origin. The momentum conjugate to r is (A) r m (B) r m 2 1 (C) r m 2 1 (D) r m2

Quantum Mechanics

An electron in the ground state of the hydrogen atom has the wave function Ѱ(rr⃗) = 1 �𝜋𝜋aa0 3 ee-�rr aa0� � where aa0 is constant. The expectation value of the operator 2 2ˆ r z Q- =, where θcosr z= is (Hint: ∫ ee-𝛼𝛼 rr rrnn ddrr = 𝛤𝛤(nn) 𝛼𝛼 nn +1 = (nn-1)! 𝛼𝛼 nn +1 ∞ 0 ) (A) -aa0 2/2 (B) -aa0 2 (C) -3aa0 2/2 (D) -2aa0 2

Atomic & Molecular Physics

For Nickel , the number density is 8 × 1023 aaaaaammss/ccmm3 and electronic configuration is 1s22s22p63s23p63d84s2. T he value of the saturation magnetization of Nickel in its ferromagnetic state is _________________× 109 AA/mm. (Given the value of Bohr magneton µBB = 9.21 × 10-21 AAmm2)

Nuclear & Particle Physics

A particle of mass m is in a potential given by ,3) ( 3 2 0 r ar r ar V + - = where a and r0 are positive constants. When disturbed slightly from its stable equilibrium position, it undergoes a simple harmonic oscillation. The time period of oscillation is (A) a mr 22 3 0π (B) a mr3 02π (C) a mr3 022π (D) a mr3 04π

Nuclear & Particle Physics

The donor concentration in a sample of n -type silicon is increased by a factor of 100. The shift in the position of the Fermi level at 300K, assuming the sample to be non degenerate is __________ meV. (kkBB TT = 25 mmeemm at 300 KK)

Nuclear & Particle Physics

A particle of mass mm is subjected to a potential, mm(xx, yy) = 1 2 mm𝜔𝜔2(xx2 + yy2), -∞ ≤ xx ≤ ∞, -∞ ≤ yy ≤ ∞ The state with energy 4ℏ𝜔𝜔 is gg-fold degenerate. The value of gg is ___________ . r 450 ω zẑ xx� m PH 6/10 PH (GATE 2014) 1 PHYSICS-PH

Quantum Mechanics

A hydrogen atom is in the state Ψ = � 8 21 ψ200 - �3 7 ψ310 + � 4 21 ψ321, where nn, ll, mm in ψnnllmm denote the principal, orbital and magnetic quantum numbers , respectively. If LL�⃗ is the angular momentum operator, the average value of LL2 is ______________ ħ2.

Quantum Mechanics

A planet of mass m moves in a circular orbit of radius r0 in the gravitational potential mm(rr) = - kk rr , where k is a positive constant. The orbital angular momentum of the planet is (A) kmr02 (B) kmr02 (C) kmr0 (D) kmr0

Atomic & Molecular Physics

The moment of inertia of a rigid diatomic molecule A is 6 times that of another rigid diatomic molecule B. If the rotational energies of the two molecules are equal, then the corresponding values of the rotational quantum numbers JA and JB are (A) JA = 2, JB = 1 (B) JA = 3, JB = 1 (C) JA = 5, JB = 0 (D) JA = 6, JB = 1

Nuclear & Particle Physics

The value of the integral dze z C z∫ +1 2 , where C is the circle 4=z , is (A) iπ2 (B) i22π (C) i34π (D) i24π

Electromagnetic Theory

A ray of light inside Region 1 in the xy- plane is incident at the semicircular boundary that carries no free charges. The electric field at the point ( )0 , /4P r π in plane polar coordinates is 1 73ˆˆrE ee ϕ= -  , where ˆˆ andree ϕ are the unit vectors. The em erging ray in R egion 2 has the electric field 2E  parallel to x- axis. If 12 and εε are the dielectric constants of Region 1 and Region 2 respectively, then 2 1 ε ε is ____________________________ PH 7/10 PH (GATE 2014) 1 PHYSICS-PH

Mathematical Physics

The solution of the differential equation 02 2 = -yt d y d , subject to the boundary conditions ( ) 1 0=y and ( ) 0= ∞y , is (A) cos t + sin t (B) cosh t + sinh t (C) cos t - sin t (D) cosh t - sinh t

Classical Mechanics

Given that the linear transformation of a generalized coordinate q and the corresponding momentum p, Q = q + 4ap P = q + 2p is canonical, the value of the constant a is ___________ .

Electromagnetic Theory

The value of the magnetic field required to maintain non-relativistic protons of energy 1 MeV in a circular orbit of radius 100 mm is ______________ Tesla. (Given: 2710 67 . 1-× =pm kg, ee = 1.6 × 10-19 CC)

Nuclear & Particle Physics

For a system of two bosons, each of which can occupy any of the two energy levels 0 and ε, the mean energy of the system at a temperature T with T kB 1=β is given by (A) βεβε βεβε ε ε 2 2 2 1 2 - - -- + + + e e e e (B) βεβε βεε 22 1 - - - + + e e e (C) βεβε βεβε ε ε 2 2 2 2 - - -- + + + e e e e (D) βεβε βεβε ε ε 2 2 2 2 - - -- + + + e e e e

Nuclear & Particle Physics

In an interference pattern formed by two coherent sources, the maximum and the minimum of the intensities are 9I0 and I0, respectively. The intensities of the individual waves are (A) 3I0 and I0 (B) 4I0 and I0 (C) 5I0 and 4I0 (D) 9I0 and I0

Quantum Mechanics

ψ1 and ψ2 are two orthogonal states of a spin 2 1 system. It is given that      +     = 1 0 3 2 0 1 3 1 1ψ , where       0 1 and       1 0 represent the spin-up and spin-down states, respectively. When the system is in the state ψ2, its probability to be in the spin-up state is ______________ .

Solid State Physics

Neutrons moving with speed 103 m/s are used for the determination of crystal structure. If the Bragg angle for the first order diffraction is 300, the interplanar spacing of the crystal is _________ Å. (Given: mn = 1.675 × 10-27 kg, ℎ = 6.626 × 10-34 JJ. ss) PH 8/10 PH (GATE 2014) 1 PHYSICS-PH

Classical Mechanics

The Hamiltonian of a particle of mass m is given by 22 22 H pq m α= - . Which one of the following figures describes the motion of the particle in phase space? (A) (B) (C) (D)

Atomic & Molecular Physics

The intensity of a laser in free space is 150 mW/m2. The corresponding amplitude of t he electric field of the laser is ____________ V/m. (𝜀𝜀0 = 8.854 × 10-12 CC2/NN. mm2)

Nuclear & Particle Physics

The emission wavelength for the transition 1D2 → 1F3 is 3122 Å. The ratio of populations of the final to the initial states at a temperature 5000 K is (ℎ = 6.626 × 10-34 JJ. ss, cc = 3 × 108 mm/ss, kkBB = 1.380 × 10-23 JJ/KK) (A) 2.03 × 10-5 (B) 4.02 × 10-5 (C) 7.02 × 10-5 (D) 9.83 × 10-5

Statistical Physics & Thermodynamics

Consider a system of 3 fermions, which can occupy any of the 4 available energy states with equal probability. The entropy of the system is (A) kkBBln2 (B) 2kkBBln2 (C) 2kkBBln4 (D) 3kkBBln4

Quantum Mechanics

A particle is confined to a one dimensional potential box with the potential ( ) otherwise, 0 , 0 ∞ = < < =a xx V If the particle is subjected to a perturbation, within the box, x Wβ= , where β is a small constant, the first order correction to the ground state energy is (A) 0 (B) 4 /βa (C) 2 /βa (D) βa

Nuclear & Particle Physics

Consider the process - + - ++ → +π π µµ . The minimum kinetic energy of the muons (µ) in the centre of mass frame required to produce the pion ( π) pairs at rest is __________ MeV. (Given: mµ = 105 MeV/c2, mπ = 140 MeV/c2). PH 9/10 PH (GATE 2014) 1 PHYSICS-PH

Quantum Mechanics

A one dimensional harmonic oscillator is in the superposition of number states, n , given by 32 322 1 + =ψ . The average energy of the oscillator in the given state is ______________ .ω

Quantum Mechanics

A nucleus X undergoes a first forbidden β-decay to a nucleus Y. If the angular momentum (I) and parity (P), denoted by I P as 7 2 - for X, which of the following is a possible IIPPvalue for Y?

Electronics

The current gain of the transistor in the following circuit is 𝛽𝛽ddcc = 100. The value of collector current Ic is _______________ mA.

Nuclear & Particle Physics

In order to measure a maximum of 1 V with a resolution of 1 mV using a n-bit A/D converter, working under the principle of ladder network, the minimum value of n is _____________ .

Quantum Mechanics

If L+ and L- are the angular momentum ladder operators, then, the expectation value of (LL+LL- + LL-LL+), in the state |ll = 1, mm = 1〉 of an atom is ________ 2 .

Nuclear & Particle Physics

A low pass filter is formed by a resistance R and a capacitance C. At the cut-off angular frequency 1 c RCω = , t he voltage gain and the phase of the output voltage relative to the input voltage respectively, are (A) 0.71 and 450 (B) 0.71 and - 450 (C) 0.5 and - 900 (D) 0.5 and 900 END OF THE QUESTION PAPER Vo Vi 20 µF 20 µF 12 V 3 kΩ 3 kΩ 150 kΩ PH 10/10 PH (GATE 2014) Section Q. No. Key / Range Marks Section Q. No. Key / Range Marks GA 1 A 1 PH 24 A 1 GA 2 B 1 PH 25 6.99 to 7.01 1 GA 3 D 1 PH 26 A 2 GA 4 C 1 PH 27 D 2 GA 5 1300 to 1300 1 PH 28 40 to 43 2 GA 6 D 2 PH 29 A 2 GA 7 B 2 PH 30 114 to 117 2 GA 8 180 to 180 2 PH 31 3.99 to 4.01 2 GA 9 D 2 PH 32 1.99 to 2.01 2 GA 10 B 2 PH 33 D 2 PH 1 D 1 PH 34 B 2 PH 2 B 1 PH 35 C 2 PH 3 A 1 PH 36 2.3 to 2.4 2 PH 4 2.1 to 2.3 1 PH 37 D 2 PH 5 C 1 PH 38 0.24 to 0.26 2 PH 6 1.99 to 2.01 1 PH 39 1.41 to 1.47 2 PH 7 1.49 to 1.51 1 PH 40 A 2 PH 8 D 1 PH 41 B 2 PH 9 C 1 PH 42 0.66 to 0.68 2 PH 10 D 1 PH 43 3.91 to 4.15 2 PH 11 B 1 PH 44 D 2 PH 12 B 1 PH 45 10.58 to 10.70 2 PH 13 D 1 PH 46 C 2 PH 14 0.99 to 1.01 1 PH 47 B 2 PH 15 1.99 to 2.01 1 PH 48 C 2 PH 16 B 1 PH 49 34.9 to 35.1 2 PH 17 D 1 PH 50 3.2 to 3.3 2 PH 18 A 1 PH 51 C 2 PH 19 C 1 PH 52 1.4 to 1.7 2 PH 20 A 1 PH 53 9.99 to 10.01 2 PH 21 A 1 PH 54 1.99 to 2.01 2 PH 22 B 1 PH 55 B 2 PH 23 C 1 ‐ Physics

Nuclear & Particle Physics

Question 56 from official GATE PH 2014 Examination.

Nuclear & Particle Physics

Question 57 from official GATE PH 2014 Examination.

Nuclear & Particle Physics

Question 58 from official GATE PH 2014 Examination.

Nuclear & Particle Physics

Question 59 from official GATE PH 2014 Examination.

Nuclear & Particle Physics

Question 60 from official GATE PH 2014 Examination.

Nuclear & Particle Physics

Question 61 from official GATE PH 2014 Examination.

Nuclear & Particle Physics

Question 62 from official GATE PH 2014 Examination.

Nuclear & Particle Physics

Question 63 from official GATE PH 2014 Examination.

Nuclear & Particle Physics

Question 64 from official GATE PH 2014 Examination.

Nuclear & Particle Physics

Question 65 from official GATE PH 2014 Examination.