GATE PH 2013 Question Paper & Solutions

Quantitative Aptitude

f (x) is a symmetric periodic function of x i.e. f (x) = f (-x). Then, in general, the Fourier series of the function f(x) will be of the form

Quantitative Aptitude

In the most general case, which one of the following quantities is NOT a second order tensor? (A) Stress (B) Strain (C) Moment of inertia (D) Pressure

Quantitative Aptitude

An electron is moving with a velocity of 0.85c in the same direction as that of a moving photon. The relative velocity of the electron with respect to photon is (A) c (B) -c (C) 0.15c (D) -0.15c

Quantitative Aptitude

If Planck's constant were zero, then the total energy contained in a box filled with radiation of all frequencies at temperature T would be ( k is the Boltzmann constant and T is nonzero) (A) Zero (B) Infinite (C) ^3 2ܶkD) ܶ k

Quantitative Aptitude

Across a first order phase transition, the free energy is

Quantitative Aptitude

Two gases separated by an impermeable but movable partition are allowed to freely exchange energy. At equilibrium, the two sides will have the same

Quantitative Aptitude

The entropy function of a system is given by ) where aand _0 are positive constants. The temperature of the system is

Quantitative Aptitude

Consider a linear collection of N independent spin 1/2 particles, each at a fixed location. The entropy of this system is (k is the Boltzmann constant) (A) Zero (B) kܰC) 1 2kܰD) (2)

Quantitative Aptitude

The decay process \nu̅ violates (A) baryon number (B) lepton numb er (C) isospin (D) strangeness 2013 PHYSICS - PH PH 5/13

Quantitative Aptitude

The isospin (I) and baryon number (B) of the up quark is (A) I=1, B=1 (B) I=1, B=1/3 (C) I=1/2, B=1 (D) I=1/2, B=1/3

Electrostatics and Magnetostatics

Consider the scattering of neutrons by protons at very low energy due to a nuclear potential of range r0. Given that, ߛ k where δ is the phase shift, k the wave number and ) the logarithmic derivative of the deuteron ground state wave function, the phase shift is (A) L _0 (B) L_0 (C) \pi 2_0 (D) \pi 2_0

Nuclear Models & Decays

In the β decay process, the transition 2+ →3 +, is

Electrostatics and Magnetostatics

At a surface current, which one of the magnetostatic boundary condition is NOT CORRECT?

Complex analysis

Interference fringes are seen at an observation plane z = 0, by the superposition of two plane waves Ak\vecԦrԦ) 1 and Ak\vecԦrԦ) 1, where A1 and A2 are real amplitudes. The condition for interference maximum is (A)k\vecԦk\vecԦrԦ\pi B)k\vecԦk\vecԦrԦ\pi݉ C)k\vecԦk\vecԦrԦ\pi D)k\vecԦk\vecԦrԦ\pi݉

Linear Algebra & Vector Calculus

For a scalar function ߮satisfying the Laplace equation, ߮׏has

Electromagnetic Waves in Media

A circularly polarized monochromatic plane wave is incident on a dielectric interface at Brewster angle. Which one of the following statements is CORRECT ?

Schrodinger equation & Wave Mechanics

Which one of the following commutation relations is NOT CORRECT ? Here, symbols have their usual meanings.

Lagrangian & Hamiltonian Dynamics

The Lagrangian of a system with one degree of freedom q is given by L = 2, where \alphaand \betaare non-zero constants. If denotes the canonical momentum conjugate to q then which one of the following statements is CORRECT?

Complex analysis

What should be the clock frequency of a 6-bit A/D converter so that its maximum conversion time is 32 μs? (A) 1 MHz (B) 2 MHz (C) 0.5 MHz (D) 4 MHz

Crystal Structure & Lattice Dynamics

A phosphorous doped silicon semiconductor ( doping density: 1017/cm3) is heated from 100C to 200C. Which one of the following statements is CORRECT ?

Dielectric, Magnetic Properties & Superconductivity

Considering the BCS theory of superconductors, which one of the following statements is NOT CORRECT? (h is the Planck's constant and e is the electronic charge)

Crystal Structure & Lattice Dynamics

Group I contains elementary excitations in solids. Group II gives the associated fields with these excitations. MATCH the excitations with their associated field and select your answer as per codes given below. Group I Group II (P) phonon (i) photon + lattice vibration (Q) plasmon (ii) electron + elastic deformation (R) polaron (iii) collective electron oscillations (S) polariton (iv) elastic wave Codes

Complex analysis

The number of distinct ways of placing four indistinguishable balls into five distinguishable boxes is ____________. 2013 PHYSICS - PH PH 7/13

Semiconductor Devices & Diodes

A voltage regulator has ripple rejection of -50dB. If input ripple is 1 mV, what is the output ripple voltage in μV? The answer should be up to two decimal places. __________

Atomic Spectra & Multi-Electron Atoms

The number of spectral lines allowed in the spectrum for the 22 transition in sodium is ____________. Q. 26 to Q. 55 carry two marks each.

Fourier and Laplace transforms

Which of the following pairs of the given function F(t) and its Laplace transform f (s) is NOT CORRECT?

Complex analysis

If AԦ and B\vecԦ are constant vectors, then (AԦB\vecԦrԦ) is (A) AԦB\vecԦ (B) AԦB\vecԦ (C) rԦ (D) Zero

Complex analysis

+ 1 2) is equal to [Given  (n+1) = n (n) and  (1/2) = √\pi] (A) 0! 2 √\piB) 20! 0! 2 √\piC) 20! 0! 2^2 √\piD) 0! 2^2 √\pi

Complex analysis

The relativistic form of Newton's second law of motion is

Electrostatics and Magnetostatics

Consider a gas of atoms obeying Maxwell-Boltzmann statistics. The average value of in \vecԦ . Ԧ over all the momenta ݌Ԧ of each of the particles (where aԦ is a constant vector and ais its magnitude, m is the mass of each atom, T is temperature and k is Boltzmann's constant) is, (A) One (B) Zero (C) - ^2^2 L (D) - ^2 ^2 L

Nuclear Models & Decays

The electromagnetic form factor F(q2) of a nucleus is given by, 2 2 where ܳis a constant. Given that \pi ݍrݍ _0 ) =1 where ρ(r) is the charge density, the root mean square radius of the nucleus is given by, (A) ܳB) √2ܳC) √3ܳD) √6ܳ 2013 PHYSICS - PH PH 8/13

Lagrangian & Hamiltonian Dynamics

A uniform circular disk of radius R and mass M is rotating with angular speed  about an axis, passing through its center and inclined at an angle 60 degrees with respect to its symmetry axis. The magnitude of the angular momentum of the disk is, (A) √^3 4 2 (B) √^3 8 2 (C) √ 8 2 (D) √ 4 2

Small oscillations

Consider two small blocks, each of mass M, attached to two identical springs. One of the springs is attached to the wall, as shown in the figure. The spring constant of each spring is k. The masses slide along the surface and the friction is negligible. The frequency of one of the normal modes of the system is, (A) ට^3+√2 2 ටL (B) ට^3+√^3 2 ටL (C) ට^3+√ 2 ටL (D) ට^3+√8 2 ටL

Electrostatics and Magnetostatics

A charge distribution has the charge density given by _0)} . For this charge distribution the electric field at _0,0 , 0) (A) 2ො \pi\varepsilon_00^2 (B) ො 4\pi\varepsilon_00 (C) ො 4\pi\varepsilon_00^2 (D) ො 18\pi\varepsilon_00^2

Electrostatics and Magnetostatics

A monochromatic plane wave at oblique incidence undergoes reflection at a dielectric interface. If ෠in , ෠ and ො are the unit vectors in the directions of incident wave, reflected wave and the normal to the surface respectively, which one of the following expressions is correct? (A)ො0 (B)k෠k෠݊ො =0 (C)෠ =0 (D)k෠݊ොk෠ 0

Electrostatics and Magnetostatics

In a normal Zeeman effect experiment, spectral splitting of the line at the wavelength 643.8 nm corresponding to the transition 2 1 1 of cadmium atoms is to be observed. The spectrometer has a resolution of 0.01 nm. The minimum magnetic field needed to observe this is (݃k ,ܥ ,ݏ/݉ ) A) 0.26 T (B) 0.52 T (C) 2.6 T (D) 5.2 T

Central force problem

The spacing between vibrational energy levels in CO molecule is found to be 8.44x10-2 eV. Given that the reduced mass of CO is 1.14x10-28 kg , Planck's constant is 6.626 1 0-^34 Js and 1eV = 1.6x10-1 J. The force constant of the bond in CO molecule is (A) 1.87 N/m (B) 18.7 N/m (C) 187 N/m (D) 1870 N/m

Crystal Structure & Lattice Dynamics

A lattice has the following primitive vectors (in Å): aԦ =2 (̂k෠), ܾ\vecԦ෠ + ̂), ܿԦ =2 (̂ + ̂). The reciprocal lattice corresponding to the above lattice is

Complex analysis

The total energy of an ionic solid is given by an expression = ^2 4\pi0 + where  is Madelung constant, r is the distance between the nearest neighbours in the crystal and B is a constant. If r0 is the equilibrium separation between the nearest neighbours then the value of B is

Complex analysis

A proton is confined to a cubic box, whose sides have length 10-12 m. What is the minimum kinetic energy of the proton? The mass of proton is 1.67 10-2 kg and Planck's constant is 6.631 0-^34Js. (A) 1.1 10-1 J (B) 3.3 10-1 J (C) 9.9 10-1 J (D) 6.6 10-1 J

Linear Algebra & Vector Calculus

For the function f (z)= 18 (+^3)(-1)^2 , the residue at the pole z = 1 is (your answer should be an integer) ___________.

Linear Algebra & Vector Calculus

The degenerate eigenvalue of the matrix 4 1 1 14 1 1 1 4 3 is (your answer should be an integer) _________________

Nuclear Models & Decays

Consider the decay of a pion into a muon and an anti-neutrino \pi \nu̅ in the pion rest frame. 2 , 2 , 0 The energy (in MeV) of the emitted neutrino, to the nearest integer is ____________

Linear Algebra & Vector Calculus

In a constant magnetic field of 0.6 Tesla along the z direction, find the value of the path integral ∮AԦ݈݀ \vec\vec\vecԦ in the units of (Tesla m2) on a square loop of side length (1/ √2 ) meters. The normal to the loop makes an angle of 60° to the z-axis, as shown in the figure. The answer should be up to two decimal places. ____________

Schrodinger equation & Wave Mechanics

A spin-half particle is in a linear superposition 0.8|↑〉 + 0.6|↓〉 of its spin-up and spin-down states. If |↑〉 and |↓〉 are the eigenstates of ߪ then what is the expectation value, up to one decimal place, of the operator 10 + 5 ? Here, symbols have their usual meanings. ____________

Schrodinger equation & Wave Mechanics

Consider the wave function AinL(_0/r), where Ais the normalization constant. For r = 2_0, the magnitude of probability current density up to two decimal places, in units of (A݉/kis _____________. ẑ 60 2013 PHYSICS - PH PH 10/13

Statistical Ensembles & Partition Functions

An n-channel junction field effect transistor has 5mA source to drain current at shorted gate (IDSS) and 5V pinch off voltage (VP). Calculate the drain current in mA for a gate-source voltage (VGS) of -2.5V. The answer should be up to two decimal places. ________________ Common Data Questions Common Data for Questions 48 and 49: There are four energy levels E, 2E, 3E and 4E (where E >0). The canonical partition function of two particles is , if these particles are

Statistical Ensembles & Partition Functions

There are four energy levels E, 2E, 3E and 4E (where E > 0). The canonical partition function of two identical fermions is (A) e^(-5βE) (1 + e^(-βE) + e^(-2βE) + e^(-3βE) + e^(-4βE)) (B) e^(-5βE) (e^(2βE) + e^(βE) + 2 + e^(-βE) + e^(-2βE)) (C) (e^(-βE) + e^(-2βE) + e^(-3βE) + e^(-4βE))^2 (D) e^(-7βE) (1 + e^(-βE) + e^(-2βE))

Statistical Ensembles & Partition Functions

There are four energy levels E, 2E, 3E and 4E (where E > 0). The canonical partition function of two distinguishable particles is (A) e^(-5βE) (e^(2βE) + e^(βE) + 2 + e^(-βE) + e^(-2βE)) (B) (e^(-βE) + e^(-2βE) + e^(-3βE) + e^(-4βE))^2 (C) 1/2 (e^(-βE) + e^(-2βE) + e^(-3βE) + e^(-4βE))^2 (D) e^(-8βE) (1 + e^(-βE) + e^(-2βE))^2

Lagrangian & Hamiltonian Dynamics

The ground state eigenvector of the unperturbed Hamiltonian is (A) (1/ √2, 1/√2,0 ) (B) (1/ √2, -1/√2,0 ) (C) (0,0,1) (D) (1,0,0)

Perturbation theory & Approximation Methods

A pair of eigenvalues of the perturbed Hamiltonian, using first order perturbation theory, is (A) 3+2ε, 7+2ε (B) 3+2 ε, 2+ε (C) 3, 7+2 ε (D) 3, 2+2 ε 2013 PHYSICS - PH PH 11/13 Linked Answer Questions Statement for Linked Answer Questions 52 and 53: In the Schmidt model of nuclear magnetic moments, we have, Ԧ= ԰ 2Ԧ) where the symbols have their usual meaning

Lagrangian & Hamiltonian Dynamics

For the case ܬ+1 / 2 , where ܬis the total angular momentum, the expectation value of ܵԦ ∙ܬԦ in the nuclear ground state is equal to, (A) (ܬ1 ) / 2 (B) (ܬ+1 ) / 2 (C) ܬ2 (D) ܬ2

Nuclear Models & Decays

For the O17 nucleus (A=17, Z=8), the effective magnetic moment is given by, ߤԦ = ԰ 2݃ܬԦ , where ݃is equal to, (0 =5 . 5 9 for proton and -3.83 for neutron) (A) 1.12 (B) -0.77 (C) -1.28 (D) 1.28 Statement for Linked Answer Questions 54 and 55: Consider the following circuit

Complex analysis

For this circuit the frequency above which the gain will decrease by 20 dB per decade is (A) 15.9 kHz (B) 1.2 kHz (C) 5.6 kHz (D) 22.5 kHz

Operational Amplifiers & Analog Circuits

At 1.2kHz the closed loop gain is (A) 1 (B) 1.5 (C) 3 (D) 0.5 1kΩ 2kΩ V(out) 1000pF V(in) 10kΩ 2013 PHYSICS - PH PH 12/13 General Aptitude (GA) Questions Q. 56 - Q. 60 carry one mark each.

Complex analysis

A number is as much greater than 75 as it is smaller than 117. The number is: (A) 91 (B) 93 (C) 89 (D) 96

Complex analysis

The professor ordered to the students to go out of the class. I II III IV Which of the above underlined parts of the sentence is grammatically incorrect? (A) I (B) II (C) III (D) IV

Complex analysis

Which of the following options is the closest in meaning to the word given below: Primeval (A) Modern (B) Historic (C) Primitive (D) Antique

Complex analysis

Friendship, no matter how _________it is, has its limitations.

Complex analysis

Select the pair that best expresses a relationship similar to that expressed in the pair: Medicine: Health (A) Science: Experiment (B) Wealth: Peace (C) Education: Knowledge (D) Money: Happiness Q. 61 to Q. 65 carry two marks each.

Complex analysis

X and Y are two positive real numbers such that 6 and 8 . For which of the following values of ) the function ݂ܻwill give maximum value?

Complex analysis

If 7| =5 then the values of | is: (A) 2, 1/3 (B) 1/2, 3 (C) 3/2, 9 (D) 2/3, 9 2013 PHYSICS - PH PH 13/13

Complex analysis

Following table provides figures (in rupees) on annual expenditure of a firm for two years - 2010 and 2011. Category 2010 2011 Raw material 5200 6240 Power & fuel 7000 9450 Salary & wages 9000 12600 Plant & machinery 20000 25000 Advertising 15000 19500 Research & Development 22000 26400 In 2011, which of the following two categories have registered increase by same percentage?

Complex analysis

A firm is selling its product at Rs. 60 per unit. The total cost of production is Rs. 100 and firm is earning total profit of Rs. 500. Later, the total co st increased by 30%. By what percentage the price should be increased to maintained the same profit level. (A) 5 (B) 10 (C) 15 (D) 30

Complex analysis

Abhishek is elder to Savar. Savar is younger to Anshul. Which of the given conclusions is logically valid and is inferred from the above statements?