GATE PH 2014 Question Paper & Solutions

Quantitative Aptitude

A student is required to demonstrate a high level of comprehension of the subject, especially in the social sciences. The word closest in meaning to comprehension is (A) understanding (B) meaning (C) concentration (D) stability

Quantitative Aptitude

Choose the most appropriate word from the options given below to complete the following sentence. One of his biggest ______ was his ability to forgive. (A) vice (B) virtues (C) choices (D) strength

Quantitative Aptitude

Rajan was not happy that Sajan decided to do the project on his ow n. On observing his unhappiness, Sajan explained to Rajan that he preferred to work independently. Which one of the statements below is logically valid and can be inferred from the above sentences? (A) Rajan has decided to work only in a group. (B) Rajan and Sajan were formed into a group against their wishes. (C) Sajan had decided to give in to Rajan’s request to work with him. (D) Rajan had believed that Sajan and he would be working together.

Quantitative Aptitude

If y = 5x2 + 3, then the tangent at x = 0, y = 3 (A) passes through x = 0, y = 0 (B) has a slope of +1 (C) is parallel to the x-axis (D) has a slope of −1

Quantitative Aptitude

A foundry has a fixed daily cost of Rs 50,000 whenever it operates and a variable cost of Rs 800Q, where Q is the daily production in tonnes. What is the cost of production in Rs per tonne for a daily production of 100 tonnes?

Quantitative Aptitude

Find the odd one in the following group: ALRVX, EPVZB, ITZDF, OYEIK (A) ALRVX (B) EPVZB (C) ITZDF (D) OYEIK

Quantitative Aptitude

Anuj, Bhola, Chandan, Dilip, Eswar and Faisal live on different floors in a six -storeyed building (the ground floor is numbered 1, the floor above it 2, and so on). Anuj lives on an even -numbered floor. Bhola does not live on an odd numbered floor. Chandan does not live on any of the floors below Faisal’s floor. Dilip does not live on floor number 2. Eswar does not live on a floor immediately above or immediately below Bhola. Faisal lives three floors above Dilip. Which of the following floor-person combinations is correct? Anuj Bhola Chandan Dilip Eswar Faisal (A) 6 2 5 1 3 4 (B) 2 6 5 1 3 4 (C) 4 2 6 3 1 5 (D) 2 4 6 1 3 5

Quantitative Aptitude

The smallest angle of a triangle is equal to two thirds of the smallest angle of a quadrilateral. The ratio between the angles of the quadrilateral is 3:4:5:6. The largest angle of the triangle is twice its smallest angle. What is the sum, in degrees, of the second largest angle of the triangle and the largest angle of the quadrilateral?

Quantitative Aptitude

One percent of the people of country X are taller than 6 ft. Two percent of the people of country Y are taller than 6 ft. There are thrice as many people in country X as in country Y. Taking both countries together, what is the percentage of people taller than 6 ft? (A) 3.0 (B) 2.5 (C) 1.5 (D) 1.25

Quantitative Aptitude

The monthly rainfall chart based on 50 years of rainfall in Agra is shown in the following figure. Which of the following are true? (k percentile is the value such that k percent of the data fall below that value) (i) On average, it rains more in July than in December (ii) Every year, the amount of rainfall in August is more than that in January (iii) July rainfall can be estimated with better confidence than February rainfall (iv) In August, there is at least 500 mm of rainfall (A) (i) and (ii) (B) (i) and (iii) (C) (ii) and (iii) (D) (iii) and (iv)

Complex analysis

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+ +

Electrostatics and Magnetostatics

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

Atomic Spectra & Multi-Electron Atoms

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

Complex analysis

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)

Complex analysis

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

Complex analysis

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.

Electrostatics and Magnetostatics

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.

Linear Algebra & Vector Calculus

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

Complex analysis

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

Complex analysis

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

Complex analysis

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)

Complex analysis

The length element ds of an arc is given by, ( ) ( ) 2 122212 32 ) ( dx dxdx dx ds+ + = .The metric tensor ijg is (A)       1 3 3 2 (B)             12 3 2 32 (C)             2 3 2 3 1 2 (D)             22 3 2 31

Electrostatics and Magnetostatics

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−

Electrostatics and Magnetostatics

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

Schrodinger equation & Wave 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).

Statistical Ensembles & Partition Functions

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

Nuclear Models & Decays

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 - -

Lagrangian & Hamiltonian Dynamics

The Hamilton’s canonical equations of motion in terms of Poisson Brackets are (A) 𝑞𝑞̇ = {𝑞𝑞, 𝐻𝐻} ; 𝑝𝑝̇ = {𝑝𝑝, 𝐻𝐻} (B) 𝑞𝑞̇ = {𝐻𝐻, 𝑞𝑞} ; 𝑝𝑝̇ = {𝐻𝐻, 𝑝𝑝} (C) 𝑞𝑞̇ = {𝐻𝐻, 𝑝𝑝} ; 𝑝𝑝̇ = {𝐻𝐻, 𝑞𝑞} (D) 𝑞𝑞̇ = {𝑝𝑝, 𝐻𝐻} ; 𝑝𝑝̇ = {𝑞𝑞, 𝐻𝐻}

Crystal Structure & Lattice Dynamics

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)

Statistical Ensembles & Partition Functions

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

Lagrangian & Hamiltonian Dynamics

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)

Crystal Structure & Lattice Dynamics

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)

Fourier and Laplace transforms

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)

Complex analysis

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

Digital 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)

Complex analysis

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

Schrodinger equation & Wave 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−\alpha\alpha 𝑟𝑟 𝑟𝑟𝑛𝑛 𝑑𝑑𝑟𝑟 = 𝛤𝛤(𝑛𝑛) \alpha\alpha 𝑛𝑛 +1 = (𝑛𝑛−1)! \alpha\alpha 𝑛𝑛 +1 ∞ 0 ) (A) −𝑎𝑎0 2/2 (B) −𝑎𝑎0 2 (C) −3𝑎𝑎0 2/2 (D) −2𝑎𝑎0 2

Complex analysis

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)

Electrostatics and Magnetostatics

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

Statistical Ensembles & Partition Functions

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)

Electrostatics and Magnetostatics

A particle of mass 𝑚𝑚 is subjected to a potential, 𝑚𝑚(𝑥𝑥, 𝑦𝑦) = 1 2 𝑚𝑚\omega\omega2(𝑥𝑥2 + 𝑦𝑦2), −∞ ≤ 𝑥𝑥 ≤ ∞, −∞ ≤ 𝑦𝑦 ≤ ∞ The state with energy 4\hbar\omega\omega is 𝑔𝑔-fold degenerate. The value of 𝑔𝑔 is ___________ . r 450 ω 𝑧𝑧̂ 𝑥𝑥� m PH 6/10 PH (GATE 2014)

Lagrangian & Hamiltonian Dynamics

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.

Lagrangian & Hamiltonian Dynamics

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

Small oscillations

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

Linear Algebra & Vector Calculus

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

Electrostatics and Magnetostatics

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)

Linear Algebra & Vector Calculus

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

Complex analysis

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 ___________ .

Lagrangian & Hamiltonian Dynamics

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)

Statistical Ensembles & Partition Functions

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

Complex analysis

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

Complex analysis

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 ______________ .

Crystal Structure & Lattice Dynamics

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)

Lagrangian & Hamiltonian Dynamics

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 Spectra & Multi-Electron Atoms

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)

Statistical Ensembles & Partition Functions

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 Ensembles & Partition Functions

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

Electrostatics and Magnetostatics

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

Particle Physics & Fundamental Interactions

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)

Schrodinger equation & Wave 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

Lagrangian & Hamiltonian Dynamics

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? (A) 1 2 + (B) 1 2 − (C) 3 2 + (D) 3 2 −

Complex analysis

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

Complex analysis

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 _____________ .

Lagrangian & Hamiltonian Dynamics

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 .

Electrostatics and Magnetostatics

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)