GATE PH 2018 Question Paper & Solutions

Quantitative Aptitude

"When she fell down the , she received many _ The words that best fill the blanks in the above sentence are General Aptitude (GA) Set-4 but little help." (A) stairs, stares (C) stares, stairs (B) stairs, stairs (D) stares, stares

Quantitative Aptitude

"In spite of being warned repeatedly, he failed to correct his The word that best fills the blank in the above sentence is behaviour." (A) rational (B) reasonable (C) errant (D) good

Quantitative Aptitude

For 0 ≤ x ≤ 2nt, sin x and cos x are both decreasing functions in the interval (A) (0,7) (в) (f,т) (C) (T, 3) (D) (7,27)

Quantitative Aptitude

The area of an equilateral triangle is V3. What is the perimeter of the triangle? (A) 2 (B) 4 (C) 6 (D) 8

Quantitative Aptitude

Arrange the following three-dimensional objects in the descending order of their volumes: (i) (ii) A cuboid with dimensions 10 cm, 8 cm and 6 cm (iii) A cube of side 8 cm A cylinder with base radius 7 cm and height 7 cm (iv) A sphere of radius 7 cm (A) (i), (ii), (ii), (iv) (B) (ii), (i), (iv), (iii) (C) (iii), (ii), (i), (iv) (D) (iv), (iii), (ii), (i)

Quantitative Aptitude

An automobile travels from city A to city B and returns to city A by the same route. The speed of the vehicle during the onward and return journeys were constant at 60 km/h and 90 km/h, respectively. What is the average speed in km/h for the entire journey? (A) 72 (B) 73 (C) 74 (D) 75 GA 112 GATE 2018 General Aptitude (GA) Set-4

Quantitative Aptitude

A set of 4 parallel lines intersect with another set of 5 parallel lines. How many parallelograms are formed? (A) 20 (B) 48 (C) 60 (D) 72

Quantitative Aptitude

To pass a test, a candidate needs to answer at least 2 out of 3 questions correctly. A total of 6,30,000 candidates appeared for the test. Question A was correctly answered by 3,30,000 candidates. Question B was answered correctly by 2,50,000 candidates. Question C was answered correctly by 2,60,000 candidates. Both questions A and B were answered correctly by 1,00,000 candidates. Both questions B and C were answered correctly by 90,000 candidates. Both questions A and C were answered correctly by 80,000 candidates. If the number of students answering all questions correctly is the same as the number answering none, how many candidates failed to clear the test? (A) 30,000 (B) 2,70,000 (C) 3,90,000 (D) 4,20,000

Quantitative Aptitude

I522 +x-1= 0 what is the value of ** + 2? (A) 1 (B) 5 (C) 7 (D) 9

Quantitative Aptitude

In a detailed study of annual crow births in India, it was found that there was relatively no growth during the period 2002 to 2004 and a sudden spike from 2004 to 2005. In another unrelated study, it was found that the revenue from cracker sales in India which remained fairly flat from 2002 to 2004, saw a sudden spike in 2005 before declining again in 2006. The solid line in the graph below refers to annual sale of crackers and the dashed line refers to the annual crow births in India. Choose the most appropriate inference from the above data. Annual sale of crackers in India Annual crow births in India 2001 2003 2005 2007 (A) There is a strong correlation between crow birth and cracker sales. (B) Cracker usage increases crow birth rate. (C) If cracker sale declines, crow birth will decline. (D) Increased birth rate of crows will cause an increase in the sale of crackers. END OF THE QUESTION PAPER GA 212

Linear Algebra & Vector Calculus

The eigenvalues of a Hermitian matrix are all

Angular momentum & Hydrogen Atom

(B) imaginary (C) of modulus one (D) real and positive Which one of the following represents the 3p radial wave function of hydrogen atom? (ao is the Bohr radius) (A) (R(r) (B) 1R(r) r/ao r/ao (C) (D) 1R(r) L r/ao 0 r/ao

Lagrangian & Hamiltonian Dynamics

Given the following table, Group I P: Stern-Gerlach experiment Q: Zeeman effect R: Frank-Hertz experiment S: Davisson-Germer experiment Group II 1: Wave nature of particles 2: Quantization of energy of electrons in the atoms 3: Existence of electron spin 4: Space quantization of angular momentum which one of the following correctly matches the experiments from Group I to their inferences in Group II? (A) P-2, Q-3, R-4, S-1 (C) P-3, Q-4, R-2, S-1 (B) P-1, Q-3, R-2, S-4 (D) P-2, Q-1, R-4, S-3

Complex analysis

In spherical polar coordinates (r, 0, Ф), the unit vector Ô at (10, 7/4, n/2) is (A) K ((+) (C(-+) PH 1/12 GATE 2018 Physics

Linear Algebra & Vector Calculus

The scale factors corresponding to the covariant metric tensor gij in spherical polar coordinates are (A) 1,r2 ,r' sin? 0 (B) 1,r2, sin? 0 (C) 1,1, 1 (D) 1,r,r sin e

Electrostatics and Magnetostatics

In the context of small oscillations, which one of the following does NOT apply to the normal coordinates? (A) Each normal coordinate has an eigen-frequency associated with it (B) The normal coordinates are orthogonal to one another (C) The normal coordinates are all independent (D) The potential energy of the system is a sum of squares of the normal coordinates with constant coefficients

Crystal Structure & Lattice Dynamics

For the given unit cells of a two dimensional square lattice, which option lists all the primitive cells? (A) \cdot and @ (C) 0,0,3 and ® (B) O,® and B (D) D,2,®,Đ and ©

Lagrangian & Hamiltonian Dynamics

Among electric field (E), magnetic field (B), angular momentum ({), and vector potential (Ä), which is/are odd under parity (space inversion) operation? (A) E only (B) E & A only (C) E & B only (D) B & L only

Molecular Spectroscopy

The expression for the second overtone frequency in the vibrational absorption spectra of a diatomic molecule in terms of the harmonic frequency we and anharmonicity constant Xe is (A) 2we(1 - Xe) (B) 2we(1 - 3xe) (C) 3We (1 -2xe) (D) 3We(1- 4xe)

Atomic Spectra & Multi-Electron Atoms

Match the physical effects and order of magnitude of their energy scales given below, where a = Arteohc is fine structure constant; Me and M, are electron and proton mass, respectively. Group I P: Lamb shift Q: Fine structure R: Bohr energy S: Hyperfine structure (A) P-3, Q-1, R-2, S-4 (C) P-4, Q-2, R-1, S-3 Group II 1: ~0(a2=mec2) 2: ~0(a"mec') 3: ~0(a*mec2/mp) 4:~0(a5mec3) (B) P-2, 2-3, R-1, S-4 (D) P-2, Q-4, R-1, S-3 PH 2/12 GATE 2018 Physics

Complex analysis

The logic expression ABC + ABC + ABC + ABC can be simplified to (A) A XORC (B) A AND & (C) 0 (D) 1

Statistical Ensembles & Partition Functions

At low temperatures (T), the specific heat of common metals is described by (with a and B as constants) (A) aT + BT3 (B) BT3 (С) еxр(-а/T) (D) aT +BTS

Complex analysis

In a 2-to-1 multiplexer as shown below, the output X = Ao if C = 0, and X = A, if C = 1. A1 - Which one of the following is the correct implementation of this multiplexer? (A) (B) Ao- A, 17D.X (C) A,. 57D.X (D) Ao- IB7DEX CED.X

Nuclear Models & Decays

The elementary particle E° is placed in the baryon decuplet, shown below, at Д++ Strangeness P PR (A) P (B) Q S 3rd component of isospin (C)R (D) S PH 3/12 GATE 2018 Physics

Angular momentum & Hydrogen Atom

The intrinsic/permanent electric dipole moment in the ground state of hydrogen atom is (ao is the Bohr radius) (A) - Зеао (B) zero (C) eao (D) 3eao

Statistical Ensembles & Partition Functions

The high temperature magnetic susceptibility of solids having ions with magnetic moments can be described by x oc * with T as absolute temperature and as constant. The three behaviors i.e. paramagnetic, ferromagnetic and anti-ferromagnetic are described, respectively, by (A) 0 < 0, 0 > 0, 0 = 0 (C) 0 = 0, 0 < 0, 0 > 0 (B) 0 > 0, 0 < 0, 0 = 0 (D) 0 = 0, 0 > 0, 0 < 0

Atomic Spectra & Multi-Electron Atoms

Which one of the following is an allowed electric dipole transition? (A) 'So → 3S1 (B) 2P3/2 →2D5/2 (C) =D5/2 →2P1/2 (D) 3Po → SDo

Nuclear Models & Decays

In the decay, ut → et + Ve + X, what is X? (A) Y (B) Ve (C) Vu (D) Vu

Complex analysis

A spaceship is travelling with a velocity of 0.7c away from a space station. The spaceship ejects a probe with a velocity 0.59c opposite to its own velocity. À person in the space station would see the probe moving at a speed Xc, where the value of X'is. - (up to three decimal places).

Operational Amplifiers & Analog Circuits

For an operational amplifier (ideal) circuit shown below, 4 kg +10 V 0 2 kQ Viam ½am 5 kg if V1 = 1 V and V2 = 2 V, the value of Vo is -oVo RIS 010 V _V (up to one decimal place).

Complex analysis

An infinitely long straight wire is carrying a steady current I. The ratio of magnetic energy density at distance rj to that at rz(= 2 rj) from the wire is PH 4/12 GATE 2018 Physics

Maxwell's Equations & Conservation Laws

A light beam of intensity lo is falling normally on a surface. The surface absorbs 20% of the intensity and the rest is reflected. The radiation pressure on the surface is given by X 1o/c, where Xis (up to one decimal place). Here c is the speed of light.

Electrostatics and Magnetostatics

The number of independent components of a general electromagnetic field tensor is

Complex analysis

If X is the dimensionality of a free electron gas, the energy (E) dependence of density of states is given by EX-Y, , where Y is

Nuclear Models & Decays

For nucleus 164Er, a J™ = 2† state is at 90 keV. Assuming 164Er to be a rigid rotor, the energy of its 4+ state is keV (up to one decimal place). PH 5/12 GATE 2018

Complex analysis

- Q. 55 carry two marks each. Physics Given V1 = î - j and V2 = - 2i + 3j + 2k, which one of the following ½z makes (V,V2,Vz) a complete set for a three dimensional real linear vector space ? (A) V3 =î+j+4k (B) Vz = 2i-^+ 2k (C) V3 = î+ 2j+6k (D) V3 = 2i+j+4K

Complex analysis

An interstellar object has speed v at the point of its shortest distance R from a star of much larger mass M. Given v2 = 2 GM/R, the trajectory of the object is (A) circle (B) ellipse (C) parabola (D) hyperbola

Electrostatics and Magnetostatics

A particle moves in one dimension under a potential V (x) = a|x| with some non-zero total energy. Which one of the following best describes the particle trajectory in the phase space? (A) (B) (C) (D) x

Electrostatics and Magnetostatics

Consider an infinitely long solenoid with N turns per unit length, radius R and carrying a current I(t) = a cos wt, where a is a constant and w is the angular frequency. The magnitude of electric field at the surface of the solenoid is (A) EMoNRwa sin wt (C) MoNRwa sin wt (B) EMOWNR cos wt (D) MowNR cos wt

Electrostatics and Magnetostatics

A constant and uniform magnetic field B = Bok pervades all space. Which one of the following is the correct choice for the vector potential in Coulomb gauge? (A) -Bo(x+y)î (B) Bo(x + y)Î (C) Boxj (D) --B.(хі-УÎ)

Lagrangian & Hamiltonian Dynamics

If H is the Hamiltonian for a free particle with mass m, the commutator [x, [x, H]] is (A) h2/m (B) -h/m (C) -A2/(2m) (D) R/(2m) PH 6/12 GATE 2018 Physics

Electrostatics and Magnetostatics

A long straight wire, having radius a and resistance per unit length r, carries a current 1. The magnitude and direction of the Poynting vector on the surface of the wire is (A) 12%/2ma , perpendicular to axis of the wire and pointing inwards (B) [r/2ma , perpendicular to axis of the wire and pointing outwards (C) 17r/a, perpendicular to axis of the wire and pointing inwards (D) 12%/a , perpendicular to axis of the wire and pointing outwards

Complex analysis

Three particles are to be distributed in four non-degenerate energy levels. The possible number of ways of distribution: (i) for distinguishable particles, and (ii) for identical Bosons, respectively, is (A) (i) 24, (ii) 4 (B) (i) 24, (ii) 20 (C) (i) 64, (ii) 20 (D) (i) 64, (ii) 16

Atomic Spectra & Multi-Electron Atoms

The term symbol for the electronic ground state of oxygen atom is (A) 1So (B) 'Dz (C) 3рo (D) 3P2

Crystal Structure & Lattice Dynamics

The energy dispersion for electrons in one dimensional lattice with lattice parameter a is given by E(k) = Eo - {W coska, where W and E, are constants. The effective mass of the electron near the bottom of the band is (1) W2 (B) waz (C) 2waz (D) Na

Statistical Ensembles & Partition Functions

Amongst electrical resistivity (p), thermal conductivity (k), specific heat (C), Young's modulus (Y), and magnetic susceptibility (X), which quantities show a sharp change at the superconducting transition temperature? (A) p, к, C, Y (B) р,С, X (С) р, к, С, Х (D) к, Y,x PH 7/12 GATE 2018 Physics

Electromagnetic Waves in Media

A quarter wave plate introduces a path difference of 2/4 between the two components of olarization parallel and perpendicular to the optic axis. An electromagnetic wave with = (@ +f) Eo ei(kz-wt) is incident normally on a quarter wave plate which has its optis axis making an angle 135° with the x-axis as shown. The emergent electromagnetic wave would be (A) elliptically polarized (B) circularly polarized (C) linearly polarized with polarization as that of incident wave (D) linearly polarized but with polarization at 90° to that of the incident wave

Electrostatics and Magnetostatics

A p-doped semiconductor slab carries a current / = 100 mA in a magnetic field B = 0.2 T as shown. One measures Vy = 0.25 mV and V, = 2 mV. The mobility of holes in the semiconductor is m'V-'s' (up to two decimal places). 1→ J t = 1 mm w = 4 mm / = 10 mm pH 8/12 GATE 2018 Physics

Complex analysis

An n-channel FET having Gate-Source switch-off voltage VGs(OFF) = - 2 V is used to invert a 0 - 5 V square-wave signal as shown. The maximum allowed value of R would be kZ (up to two decimal places). 8V^ Vin R → Vout Iam ov 1KDg 7 10002 012 V -8 V

Statistical Ensembles & Partition Functions

Inside a large nucleus, a nucleon with mass 939 MeVcª has Fermi momentum 1.40 fm 'at absolute zero temperature. Its velocity is Xc, where the value of X is (up to two decimal places). (hc = 197 MeV-fm)

Lagrangian & Hamiltonian Dynamics

4 MeV y-rays emitted by the de-excitation of '9F are attributed, assuming spherical symmetry, to the transition of protons from 1d3/2 state to 1ds/2 state. If the contribution of spin-orbit term to the total energy is written as C{l \cdot 5), the magnitude of C is MeV (up to one decimal place).

Electrostatics and Magnetostatics

An a particle is emitted by a 290Th nucleus. Assuming the potential to be purely Coulombic beyond the point of separation, the height of the Coulomb barrier is MeV (up to two decimal places). = 1.44 MeV-fm, ro = 1.30 fm)

Complex analysis

For the transformation Q =/2q e-1+2a cosp, P = JZqe-a-1 sinp (where a is a constant) to be canonical, the value of a is PH 9/12 GATE 2018 Physics

Complex analysis

Given d'f (x) _ , df(x) dx? - + f(x) = 0, and boundary conditions f (0) = 1 and f (1) = 0, the value of f (0.5) is two decimal places). _ (up to

Linear Algebra & Vector Calculus

The absolute value of the integral 57 +-322 22-4 -dz, over the circle |z - 1.5| = 1 in complex plane, is (up to two decimal places).

Complex analysis

A uniform circular disc of mass m and radius R is rotating with angular speed w about an axis passing through its center and making an angle 0 = 30° with the axis of the disc. If the kinetic energy of the disc is amw? R?, the value of a is

Electrostatics and Magnetostatics

The ground state energy of a particle of mass m in an infinite potential well is Eo. It changes to Eo(1 + a X 10-3), when there is a small potential bump of height Vo = T2 n2 50mLz and width a = L/100, as shown in the figure. The value of a is to two decimal places). (up ^ V(x) L- PH 10/12 GATE 2018 Physics

Electrostatics and Magnetostatics

An electromagnetic plane wave is propagating with an intensity I = 1.0 x 105 Wm 2 in a medium with € = 3€o and M = Mo. The amplitude of the electric field inside the medium is x 103 Vm' (up to one decimal place). (Eo = 8.85 × 10-12 C'N'm", Mo = 4T X 10-7NA-3,c = 3 x 108 ms')

Statistical Ensembles & Partition Functions

A microcanonical ensemble consists of 12 atoms with each taking either energy 0 state, or energy e state. Both states are non-degenerate. If the total energy of this ensemble is 4e, its entropy will be Kg (up to one decimal place), where kg is the Boltzmann constant.

Linear Algebra & Vector Calculus

A two-state quantum system has energy eigenvalues ‡e corresponding to the normalized states (Wt). Attime t = O, the system is in quantum state ⅖ll4+) + I4-)1. The probability that the system will be in the same state at t = h/(6€) is (up to two decimal places).

Statistical Ensembles & Partition Functions

An air-conditioner maintains the room temperature at 27 °C while the outside temperature is 47 °C. The heat conducted through the walls of the room from outside to inside due to temperature difference is 7000 W. The minimum work done by the compressor of the air- conditioner per unit time is

Statistical Ensembles & Partition Functions

Two solid spheres A and B have same emissivity. The radius of A is four times the radius of B, and temperature of A is twice the temperature of B. The ratio of the rate of heat radiated from A to that from B is

Statistical Ensembles & Partition Functions

The partition function of an ensemble at a temperature T is 2=(2cosh 7)" . where kg is the Boltzmann constant. The heat capacity of this ensemble at T = = is X Nks, where the value of X is (up to two decimal places).

Electrostatics and Magnetostatics

An atom in its singlet state is subjected to a magnetic field. The Zeeman splitting of its 650 nm spectral line is 0.03 nm. The magnitude of the field is Tesla (up to two decimal places). (e =1.60 x 10-19 C, Me =9.11 X 10-31 kg, c = 3.0 x 108 ms') PH 11/12 GATE 2018 Physics

Statistical Ensembles & Partition Functions

The quantum effects in an ideal gas become important below a certain temperature Ta hen de Broglie wavelength corresponding to the root mean square thermal speer ecomes equal to the inter-atomic separation. For such a gas of atoms of mass 2 x 10- kg and number density 6.4 x 1025 m 3, Ta = _× 10-3 K (up to one decimal place). (kg = 1.38 x 10-23 J/K, h = 6.6 × 10-34 J-s) END OF THE QUESTION PAPER PH 12/12 Q.No. Type 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 MCQ MCQ MCQ MCQ MCQ MCQ MCQ MCQ MCQ MCQ MCQ MCQ MCQ MCQ MCQ MCQ MCQ MCQ MCQ MCQ MCQ MCQ MCQ MCQ Section GA GA GA GA GA GA GA GA GA GA PH PH PH PH PH PH PH PH PH PH PH PH PH PH Key/Range A B D Marks 1 1 1 1 1 2 2 2 D C A A B C D B D C A A 2 1 1 1 1 1 1 1 1 1 1 C 1 1 1