GATE PH 2016 Question Paper & Solutions

Quantitative Aptitude

The volume of a sphere of diameter 1 unit is _ (A) least (B) less General Aptitude - GA Set-2 (C) lesser than the volume of a cube of side 1 unit. (D) low 0.2

Quantitative Aptitude

Which of the following magnetic vector potentials gives rise to a uniform magnetic field k Bˆ 0 ? (A) k z Bˆ 0 (B) j x Bˆ0- (C)  j x i yB ˆ ˆ 2 0  - (D)  j x i yB ˆ ˆ 2 0 

Quantitative Aptitude

The unruly crowd demanded that the accused be (A) hanged (B) hanging (C) hankering without trial. (D) hung Choose the statement(s) where the underlined word is used correctly: (i) (ii) (iii) A prone is a dried plum. He was lying prone on the floor. People who eat a lot of fat are prone to heart disease. (A) (i) and (iii) only (B) (iii) only (C) (i) and (ii) only (D) (ii) and (iii) only

Quantitative Aptitude

Fact: If it rains, then the field is wet. Read the following statements: It rains (ii) (iii) (iv) The field is not wet The field is wet It did not rain Which one of the options given below is NOT logically possible, based on the given fact? (A) If (iii), then (iv). (B) If (i), then (iii). (C) If (i), then (ii). (D) If (ii), then (iv).

Quantitative Aptitude

A window is made up of a square portion and an equilateral triangle portion above it. The base of the triangular portion coincides with the upper side of the square. If the perimeter of the window is 6 m, the area of the window in m' is (A) 1.43 (B) 2.06 (C) 2.68 (D) 2.88 1/3 GATE 2016 General Aptitude - GA Set-2

Quantitative Aptitude

Students taking an exam are divided into two groups, P and Q such that each group has the same number of students. The performance of each of the students in a test was evaluated out of 200 marks. It was observed that the mean of group P was 105, while that of group Q was 85. The standard deviation of group P was 25, while that of group Q was 5. Assuming that the marks were distributed on a normal distribution, which of the following statements will have the highest probability of being TRUE? (A) No student in group Q scored less marks than any student in group P. (B) No student in group P scored less marks than any student in group Q.

Quantitative Aptitude

(C) Most students of group Q scored marks in a narrower range than students in group P. (D) The median of the marks of group P is 100. A smart city integrates all modes of transport, uses clean energy and promotes sustainable use of resources. It also uses technology to ensure safety and security of the city, something which critics argue, will lead to a surveillance state. Which of the following can be logically inferred from the above paragraph? (i) (ii) (iii) (iv) All smart cities encourage the formation of surveillance states. Surveillance is an integral part of a smart city. Sustainability and surveillance go hand in hand in a smart city. There is a perception that smart cities promote surveillance. (A) (i) and (iv) only (B) (ii) and (iii) only (C) (iv) only (D) (1) only

Quantitative Aptitude

The total power emitted by a spherical black body of radius R at a temperature T is P1. Let P2 be the total power emitted by another spherical black body of radius R/2 kept at temperature 2T. The ratio, P1/P2 is _______. (Give your answer upto two decimal places) 3/10

Quantitative Aptitude

Find the missing sequence in the letter series. B, FH, LNP, _ (A) SUWY (B) TUVW (C) TVXZ (D) TWXZ The binary operation o is defined as a O b = abt(a+b), where a and b are any two real numbers. The value of the identity element of this operation, defined as the number x such that a O x = a, for any a, is_ (A) 0 (B) 1 (C) 2 (D) 10

Quantitative Aptitude

Which of the following curves represents the function [| (||)|]|) for \pi ? Here, xrepresents the abscissa and yrepresents the ordinate. (A) (B) (C) (D) END OF THE QUESTION PAPER 1/10 Q. 1 - Q. 25 carry one mark each.

Differential equations

Consider the linear differential equation $\frac{dy}{dx} = xy$. If $y=2$ at $x=0$, then the value of $y$ at $x=2$ is given by

Electrostatics and Magnetostatics

Which of the following magnetic vector potentials gives rise to a uniform magnetic field $B_0\hat{k}$?

Lasers and Quantum Optics

The molecule $^{17}\text{O}_2$ is

Complex analysis

There are four electrons in the $3d$ shell of an isolated atom. The total magnetic moment of the atom in units of Bohr magneton is _______.

Waveguides and radiation

Which of the following transitions is NOT allowed in the case of an atom, according to the electric dipole radiation selection rule?

Atomic Spectra & Multi-Electron Atoms

In the $\text{SU}(3)$ quark model, the triplet of mesons $(\pi^+, \pi^0, \pi^-)$ has

Special theory of relativity

The kinetic energy of a particle of rest mass 0m is equal to its rest mass energy. Its momentum in units of c m0 , where c is the speed of light in vacuum, is _______. (Give your answer upto two decimal places)

Statistical Ensembles & Partition Functions

The total power emitted by a spherical black body of radius R at a temperature T is P,. Let P, be the total power emitted by another spherical black body of radius R/2 kept at temperature 2T. The ratio, P,/P, is (Give your answer upto two decimal places) PH 1/10 GATE 2016 Physics

Statistical Ensembles & Partition Functions

The entropy S of a system of N spins, which may align either in the upward or in the downward direction, is given by S = -k,N[pln p +(I- p)In(1- p)]. Here ke is the Boltzmann constant. The probability of alignment in the upward direction is p. The value of p, at which the entropy is maximum. is _ (Give your answer upto one decimal place)

Classical Thermodynamics

For a system at constant temperature and volume, which of the following statements is correct at equilibrium? (A) The Helmholtz free energy attains a local minimum. (B) The Helmholtz free energy attains a local maximum. (C) The Gibbs free energy attains a local minimum. (D) The Gibbs free energy attains a local maximum.

Statistical Ensembles & Partition Functions

N atoms of an ideal gas are enclosed in a container of volume V. The volume of the container is changed to 4V, while keeping the total energy constant. The change in the entropy of the gas, in units of Nk In 2, is_ -, where kg is the Boltzmann constant.

Complex analysis

Which of the following is an analytic function of z everywhere in the complex plane? (A) z (B) (E*) (C) |7|" (D) VE

Complex analysis

In a Young's double slit experiment using light, the apparatus has two slits of unequal widths. When only slit-l is open, the maximum observed intensity on the screen is 4/. When only slit-2 is open, the maximum observed intensity is lo. When both the slits are open, an interference pattern appears on the screen. The ratio of the intensity of the principal maximum to that of the nearest

Statistical Ensembles & Partition Functions

C'onsider a metal which obeys the Sommerfeld model exactly. If Er is the Fermi energy of the metal at T= 0 K and Rg is its Hall coefficient, which of the following statements is correct? (A) RgcE,'3 (B) RgsE,"3 (C) RgxE,*2 (D) R is independent of Eg

Crystal Structure & Lattice Dynamics

A one-dimensional linear chain of atoms contains two types of atoms of masses mj and me (where m? > m.), arranged alternately. The distance between successive atoms is the same. Assume that the harmonic approximation is valid. At the first Brillouin zone boundary, which of the following statements is correct? (A) The atoms of mass mz are at rest in the optical mode, while they vibrate in the acoustical mode. (B The atoms of mass m, are at rest in the optical mode, while they vibrate in the acoustical mode. (C) Both types of atoms vibrate with equal amplitudes in the optical as well as in the acoustical modes. (D) Both types of atoms vibrate, but with unequal, non-zero amplitudes in the optical as well as in the acoustical modes.

Schrodinger equation & Wave Mechanics

Which of the following operators is Hermitian? (A) d dx d (B) dx (c) i22 dx? d' (D) PH 2/10 GATE 2016 Physics

Special theory of relativity

The kinetic energy of a particle of rest mass m, is equal to its rest mass energy. Its momentum in units of m,c, where c is the speed of light in vacuum, is _ (Give your answer upto two decimal places)

Statistical Ensembles & Partition Functions

The number density of electrons in the conduction band of a semiconductor at a given temperature is2x10' m 3 . Upon lightly doping this semiconductor with donor impurities, the number density of conduction electrons at the same temperature becomes 4x 102° m-3 The ratio of majority to minority charge carrier concentration is

Complex analysis

Two blocks are connected by a spring of spring constantk. One block has mass m and the other block has mass 2m. If the ratio k/m = 4 s-3 , the angular frequency of vibration w of the two block spring system in s' is \cdot (Give your answer upto two decimal places)

Lagrangian & Hamiltonian Dynamics

A particle moving under the influence of a central force F(F) = -kr (where f is the position vector of the particle and k is a positive constant) has non-zero angular momentum. Which of the following curves is a possible orbit for this particle? (A) A straight line segment passing through the origin. (B) An ellipse with its center at the origin. (C) An ellipse with one of the foci at the origin. (D) A parabola with its vertex at the origin.

Complex analysis

Consider the reaction 3Mn + è → LACr + X. The particle X is (A) Y (B) Ve (C) n (D) п

Electrostatics and Magnetostatics

The scattering of particles by a potential can be analyzed by Born approximation. In particular, if the scattered wave is replaced by an appropriate plane wave, the corresponding Born approximation is known as the first Born approximation. Such an approximation is valid for (A) large incident energies and weak scattering potentials. (B) large incident energies and strong scattering potentials. (C) small incident energies and weak scattering potentials. (D) small incident energies and strong scattering potentials.

Nuclear Models & Decays

Consider an elastic scattering of particles in / = 0 states. If the corresponding phase shift б, is 90° and the magnitude of the incident wave vector is equal to V2 fm' then the total scattering cross section in units of fm' is

Electrostatics and Magnetostatics

A hydrogen atom is in its ground state. In the presence of a uniform electric field E = E,Ê, the leading order change in its energy is proportional to (E,)". The value of the exponent n is PH 3/10 GATE 2016 Physics

Statistical Ensembles & Partition Functions

A solid material is found to have a temperature independent magnetic susceptibility, x = C. Which of the following statements is correct? (A) If C is positive, the material is a diamagnet. (B) If C is positive, the material is a ferromagnet. (C) If C is negative, the material could be a type I superconductor. (D) If C is positive, the material could be a type I superconductor.

Electrostatics and Magnetostatics

An infinite, conducting slab kept in a horizontal plane carries a uniform charge density o. Another infinite slab of thickness t, made of a linear dielectric material of dielectric constant k, is kept above the conductine slab. The bound charse density on the upper surface of the dielectric slak is (A) ° 2k (B) (C) ° (k-2) 2k o (k -1) (D) k

Atomic Spectra & Multi-Electron Atoms

The number of spectroscopic terms resulting from the L.S coupling of a 3p electron and a 3d electron is

Nuclear Properties & Nuclear Forces

Which of the following statements is NOT correct? (A) A deuteron can be disintegrated by irradiating it with gamma rays of energy 4 MeV. (B) A deuteron has no excited states. (C) A deuteron has no electric quadrupole moment. (D) The 'S, state of deuteron cannot be formed.

Schrodinger equation & Wave Mechanics

If $, and 5, are the spin operators of the two electrons of a He atom, the value of (5, \cdot5,) for the ground state is (1) -33 (B) - 3 (C) 0 (D)

Complex analysis

A two-dimensional square rigid box of side L contains six non-interacting electrons at T = 0 K. The mass of the electron is m. The ground state energy of the system of electrons, in units of "'h? 2ml is

Electrostatics and Magnetostatics

An alpha particle is accelerated in a cyclotron. It leaves the cyclotron with a kinetic energy of 16 MeV. The potential difference between the D electrodes is 50 kilovolts. The number of revolutions the alpha particle makes in its spiral path before it leaves the cyclotron is_

Linear Algebra & Vector Calculus

Let Vi be the ith component of a vector field V, which has zero divergence. If dj = a/axj, the expression for Eijk Elmk dja,Vm is equal to (A) - đ ,0KV; (B) д,0 V; (C) 037, (D)-0}V 4/10 GATE 2016 Physics

Complex analysis

The direction of Üf for a scalar field f(x, y,z) = =x2-xy +=2? at the point P (1, 1,2) is (1) (-]-27) (B) (1+2k) (C) (-2k) (D) (+2Ê)

Angular momentum & Hydrogen Atom

0,, 0, and o, are the Pauli matrices. The expression 2o,o, + o,0, is equal to (A) - 3iо, (B) - io: (C) io, (D) 3io,

Complex analysis

A particle of mass m = 0.1 kg is initially at rest at origin. It starts moving with a uniform acceleration à =10î ms7 at t = O. The action S of the particle, in units of J-s, at t = 2 s is (Give your answer upto two decimal places)

Fourier and Laplace transforms

A periodic function f (x) of period 2m is defined in the interval (-n < x < 1t) as: F(x) =51:-0≤*500 0 <X<П The appropriate Fourier series expansion for f (x) is

Crystal Structure & Lattice Dynamics

(A) f (x) = (4/ п)[sin x + (sin 3x) / 3 + (sin 5x) /5+...] (B) f(x) = (4/ п)[sin x - (sin 3x) /3 + (sin 5x) /5- ...] (C) f(x) = (4/п)[cOs x+(cos 3x) /3+(cos5x) /5+...] (D) f (x) = (4/пt)[COs x - (cos3x)/3+(cos5x) /5-...] Atoms, which can be assumed to be hard spheres of radius R, are arranged in an fcc lattice with lattice constant a, such that each atom touches its nearest neighbours. Take the center of one of the atoms as the origin. Another atom of radius r (assumed to be hard sphere) is to be accommodated at a position (0, a/2, 0) without distorting the lattice. The maximum value of r/R is _ (Give your answer upto two decimal places)

Complex analysis

In an inertial frame of reference S, an observer finds two events occurring at the same time at co- ordinates x, = 0 and Xz = d. A different inertial frame S' moves with velocity v with respect to S along the positive x-axis. An observer in S' also notices these two events and finds them to occur at times t, and t' and at positions x' and x', respectively. IfAt' =t2-t, Ax'=x2-x, and y= 1 , which of the following statements is true ? (A) At' =0,Ax' =yd (C) Дt' =-yvd/c ,Ar' =yd (B) Дt' = 0, Ax' = d/r (D) At' =-yvd/c*,Ax'=d/v PH 5/10 GATE 2016 Physics

Crystal Structure & Lattice Dynamics

The energy vs. wave vector (E-k) relationship near the bottom of a band for a solid can be approximated as E = A(ka)* + B(ka)* *, where the lattice constant a = 2.1 À. The values of A and B are 6.3x10-19 J and 3.2x10-20 J, respectively. At the bottom of the conduction band, the ratio of the effective mass of the electron to the mass of free electron is _ - (Give your answer upto two decimal places) (Take h=1.05x10-34 J-s, mass of free electron= 9.1x10-3' kg)

Electrostatics and Magnetostatics

The electric field component of a plane electromagnetic wave travelling in vacuum is given by E(z,t) = E, cos(kz -@t)f. The Poynting vector for the wave is (A) (c8,/2) E} cos'(kz-wt)Î (B) (c80/2)E?cos'(kz-wt)k (C) CE, Ez cos'(kz-at)] (D) CE, E, Cos' (kZ-00t)K

Complex analysis

Consider a system having three energy levels with energies O, 2e and 3&, with respective degeneracies of 2, 2 and 3. Four bosons of spin zero have to be accommodated in these levels such that the total energy of the system is10g. The number of ways in which it can be done is

Lagrangian & Hamiltonian Dynamics

The Lagrangian of a system is given by zm1ö2+sin-04' -mglcos A, where m,l and g are constants. Which of the following is conserved? (A) ésin' 0 (B) @ sin 0 (C) sin 0 (D) - sin' 0

Nuclear Models & Decays

Protons and a-particles of equal initial momenta are scattered off a gold foil in a Rutherford scattering experiment. The scattering cross sections for proton on gold and a-particle on gold are o, and Oa respectively. The ratio oalo, is_ PH 6/10 GATE 2016 Physics

Complex analysis

For the digital circuit given below, the output X is A. (A)A+B\cdotC (C) А\cdot(B+C) (B) A.(B+C) (D) A+(B\cdotC)

Statistical Ensembles & Partition Functions

The Fermi energies of two metals X and Y are 5 eV and 7 eV and their Debye temperatures are 170 K and 340 K, respectively. The molar specific heats of these metals at constant volume at low temperatures can be written as (C,)x=YxT+AxT3 and (C,)y=YyT + AyT', where y and A are constants. Assuming that the thermal effective mass of the electrons in the two metals are same, which of the following is correct? (A) Yx = 7 Ax =8 Vy 5' Ay (C) Yx = S Ax -= 7' Ay 1 8 (B) Yx = 7 1 Ax =- 5'À, 8 (D) YX_SAX=8 7'A

Statistical Ensembles & Partition Functions

A two-level system has energies zero and E . The level with zero energy is non-degenerate, while the level with energy E is triply degenerate. The mean energy of a classical particle in this system at a temperature T is Ee-ElkgT (A) 1+3e-E/KgT (C) 3EeBIKT 1+e-E/kgT Ee-EIKgT (B) IteBlkIT 3EeElkgT (D) 1+32-B1K4T PH 7/10 GATE 2016 Physics

Nuclear Models & Decays

A particle of rest mass M is moving along the positive x-direction. It decays into two photons ri and y, as shown in the figure. The energy of y, is 1 GeV and the energy of Y is 0.82 GeV. The value of M (in units of GeV/c') is _ (Give your answer upto two decimal places) AY1 M 45° 60° *Vz

Electrostatics and Magnetostatics

If x and p are the x components of the position and the momentum operators of a particle respectively, the commutator [x2 ',p'] is (A) ih(xp - px) (C) ih(xp + px) (B) 2ih(xp - px) (D) 2ih(xp + px) The x-y plane is the boundary between free space and a magnetic material with relative permeability M, . The magnetic field in the free space is B,î + B,k. The magnetic field in the magnetic material is (A) B,i+ B,k 1 (C) B,Î+ B,K (B) B,Î+ M, B.K (D) M, B,i + B,K M,

Electrostatics and Magnetostatics

The x-y plane is the boundary between free space and a magnetic material with relative permeability \mu_r. The magnetic field in the free space is B_x \hat{i} + B_z \hat{k}. The magnetic field in the magnetic material is

Harmonic oscillator

Let |l, m> be the simultaneous eigenstates of L^2 and L_z. Here L is the angular momentum operator with Cartesian components (L_x, L_y, L_z), l is the angular momentum quantum number and m is the azimuthal quantum number. The value of <1, 0 | (L_x + i L_y) | 1, -1> is

Lagrangian & Hamiltonian Dynamics

Let |l,m) be the simultaneous eigenstates of I' and L. Here I is the angular momentum operator with Cartesian components (L,,L,,L.), l is the angular momentum quantum number and m is the azimuthal quantum number. The value of (1,0(I, +iL,)1,-1) is (A) 0 (B) h (C) V2h (D) 13h For the parity operator P, which of the following statements is NOT true? (A) pt = P (B) p2 = -P (C) p2 = 1 (D) pt = P-1 GATE 2016 Physics

Complex analysis

For the transistor shown in the figure, assume VBE = 0.7 V and Pdc = 100. If Vin = 5V, Vout (in Volts) is _. (Give your answer upto one decimal place) 10 V > 3kг Vin owv 200 k S O Vour =

Complex analysis

The state of a system is given by (4)=(41)+2182)+3183) where Ф,), (Ф2) and (Фs) form an orthonormal set. The probability of finding the system in the state |Ф2) is (Give your answer upto two decimal places) NICNI- + Nin Ni- (D) wis, Nie 'O nuclei are

Nuclear Models & Decays

According to the nuclear shell model, the respective ground state spin-parity values of '§0 and GATE 2016 Physics

Electrostatics and Magnetostatics

A particle of mass m and energy E, moving in the positive x direction, is incident on a step potential at x = 0, as indicated in the figure. The height of the potential is Vo, where Vo > E. At x =xo, wherex, > 0, the probability of finding the electron is l/e times the probability of finding it at x = 0. If a = 2m(V. - E) , the value of xo is Vo E x=0 -- - ------ x = x0 (1) 2 (C) 20 (D) 1 40 END OF THE QUESTION PAPER PH 10/10