GATE PH 2019 Question Paper & Solutions

Quantitative Aptitude

The fishermen, _______ the flood victims owed their lives, were rewarded by the government.

Quantitative Aptitude

Some students were not involved in the strike. If the above statement is true, which of the following conclusions is/are logically necessary? 1. Some who were involved in the strike were students. 2. No student was involved in the strike. 3. At least one student was involved in the strike. 4. Some who were not involved in the strike were students.

Quantitative Aptitude

The radius as well as the height of a circular cone increases by 10%. The percentage increase in its volume is _______.

Quantitative Aptitude

Five numbers 10, 7, 5, 4 and 2 are to be arranged in a sequence from left to right following the directions given below: 1. No two odd or even numbers are next to each other. 2. The second number from the left is exactly half of the left-most number. 3. The middle number is exactly twice the right-most number. Which is the second number from the right?

Quantitative Aptitude

Until Iran came along, India had never been _______ in kabaddi.

Quantitative Aptitude

Since the last one year, after a 125 basis point reduction in repo rate by the Reserve Bank of India, banking institutions have been making a demand to reduce interest rates on small saving schemes. Finally, the government announced yesterday a reduction in interest rates on small saving schemes to bring them on par with fixed deposit interest rates. Which one of the following statements can be inferred from the given passage?

Quantitative Aptitude

In a country of 1400 million population 70% own mobile phones. Among the mobile phone owners, only 294 million access the Internet. Among these Internet users, only half buy goods from e-commerce portals. What is the percentage of these buyers in the country? (a) 10.50 (b)14.70 (c)15.00

Quantitative Aptitude

The nomenclature of Hindustani music has changed over the centuries. Since the medieval period dhrupad styles were identified as baanis. Terms like gayaki and baaj were used to refer to vocal and instrumental styles, respectively. With the institutionalization of music education the term gharana

Quantitative Aptitude

Two trains started at 7 AM from the same point. The first train travelled north at a speed of 80 / km h and the second train travelled south at a speed of 100 / km h . The time at which they were 540km apart is __________A.M (a) 9 (b) 10 (c) 11 (d) 11.30

Quantitative Aptitude

“I read somewhere that in ancient times the prestige of a kingdom depended upon the number of taxes that it was able to levy on its people. It was very much like the prestige of a head-hunter in his won community.” Based on the paragraph above, the prestige of a head-hunter depended upon ___

Complex analysis

The relative magnetic permeability of a type-I super conductor is (a) 0 (b) 1 − (c) 2π (d) 1 4π GATE 2019

Nuclear Models & Decays

Considering baryon number and lepton number conservation laws, which of the following process is/are allowed? (i) 0 e p e π ν + → + + (ii) e e vµ ν µ + + + → + (a) both (i) and (ii) (b) only (i) (c) only (ii) (d) neither (i) nor (ii) GATE 2019

Complex analysis

For the following circuit, what is the magnitude of out V if in 1.5 V V = ? (a) 0.015V (b) 0.15V (c) 15V (d) 150V GATE 2019

Linear Algebra & Vector Calculus

For the differential equation ( ) 2 2 2 1 0 d y y n n dx x − + = , where n is a constant, the product of its two independent solutions is (a) 1 x (b) x (c) nx (d) 1 1 nx + GATE 2019

Lagrangian & Hamiltonian Dynamics

Consider a one-dimensional gas of N non-interacting particles of mass m with the Hamiltonian for a single particle given by ( ) 2 2 2 1 2 2 2 p H m x x m ω = + + The high temperature specific heat in units of B R Nk = ( B k is the Boltzmann constant) is (a) 1 (b) 1.5 (c) 2 (d) 2.5 GATE 2019

Electrostatics and Magnetostatics

An electric field $\vec{E} = E_0\hat{z}$ is applied to a Hydrogen atom in $n = 2$ excited state. Ignoring spin, the $n = 2$ state is fourfold degenerate, which in the $(l, m)$ basis are given by $|0,0\rangle$, $|1,1\rangle$, $|1,0\rangle$ and $|1,-1\rangle$. If $H'$ is the interaction Hamiltonian corresponding to the applied electric field, which of the following matrix elements is nonzero?

Bose-Einstein and Fermi-Dirac statistics

A large number N of ideal bosons, each of mass m , are trapped in a three-dimensional potential ( ) 2 2 2 m r V r ω = . The bosonic system is kept at temperature T which is much lower than the Bose-Einstein condensation temperature C T . The chemical potential ( ) µ satisfies (a) 3 2 µ ω ≤  (b) 3 2 2 ω µ ω > >   (c) 3 2 ω µ ω > >   (d) 3 µ ω =  GATE 2019

Linear Algebra & Vector Calculus

During a rotation, vectors along the axis of rotation remain unchanged. For the rotation matrix 0 1 0 0 0 1 1 0 0     −     −   , the unit vector along the axis of rotation is (a) ( ) 1 ˆ ˆ ˆ 2 2 3 i j k − + (b) ( ) 1 ˆ ˆ ˆ 3 i j k + − (c) ( ) 1 ˆ ˆ ˆ 3 i j k − − (d) ( ) 1 ˆ ˆ ˆ 2 2 3 i j k + − GATE 2019

Complex analysis

For a spin-1/2 particle, let $|\uparrow\rangle$ and $|\downarrow\rangle$ denote its spin up and spin down states, respectively. If $|a\rangle = \frac{1}{\sqrt{2}}(|\uparrow\downarrow\rangle + |\downarrow\uparrow\rangle)$ and $|b\rangle = \frac{1}{\sqrt{2}}(|\uparrow\downarrow\rangle - |\downarrow\uparrow\rangle)$ are composite states of two such particles, which of the following statements is true for their total spin $S$?

Complex analysis

Consider a transformation from one set of generalized coordinate and momentum ( ,q p ) to another set ( , Q P ) denoted by, ; s r Q pq P q = = where s and r are constants. The transformation is canonical if (a) 0 s = and 1 r = (b) 2 s = and 1 r = − (c) 0 s = and 1 r = − (d) 2 s = and 1 r = GATE 2019

Statistical Ensembles & Partition Functions

In order to estimate the specific heat of phonons, the appropriate method to apply would be (a) Einstein model for acoustic phonons and Debye model for optical phonons (b) Einstein model for optical phonons and Debye model for acoustic phonons (c) Einstein model for both optical and acoustic phonons (d) Debye model for both optical and acoustic phonons GATE 2019 GATE - 2019 [SOLUTION] Physics by fiziks 8 Revised Edition-2022 H.No. 40-D, G.F., Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498, Website: www.physicsbyfiziks.com, Email: [email protected]

Complex analysis

The pole of the function ( ) cot f z z = at 0 z = is (a) a removablesingularity (b) an essential singularity (c) a simple pole (d) a second order pole GATE 2019

Nuclear Models & Decays

A massive particle X in free space decays spontaneously into two photons. Which of the following statements is true for X ? (a) X is charged (b) Spin of X must be greater than or equal to 2 (c) X is a boson (d) X must be a baryon GATE 2019

Electrostatics and Magnetostatics

The electric field of an electromagnetic wave is given by ( ) ˆ 3sin E kz t x ω = − +  ( ) ˆ 4cos kz t y ω − . The wave is (a) Linearly polarized at an angle 1 4 tan 3 −      from the x - axis (b) Linearly polarized at an angle 1 3 tan 4 −      from the x - axis (c) Elliptically polarized in clockwise direction when seen travelling towards the observer (d) Elliptically polarized in counter-clockwise direction when seen travelling towards the observer GATE 2019

Nuclear Properties & Nuclear Forces

The nuclear spin and parity of 40 20Ca in its ground state is (a) 0+ (b) 0− (c) 1+ (d) 1− GATE 2019

Electrostatics and Magnetostatics

An infinitely long thin cylindrical shell has its axis coinciding with the z -axis. It carries a surface charge density 0 cos σ φ , where φ is the polar angle and 0 σ is a constant. The magnitude of the electric field inside the cylinder is (a) 0 (b) 0 0 2 σ ∈ (c) 0 0 3 σ ∈ (d) 0 0 4 σ ∈ GATE 2019 y x GATE - 2019 [SOLUTION] Physics by fiziks 10 Revised Edition-2022 H.No. 40-D, G.F., Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498, Website: www.physicsbyfiziks.com, Email: [email protected]

Complex analysis

Consider a three-dimensional crystal of N inert gas atoms. The total energy is given by ( ) 12 6 2 U R N p q R R σ σ       = ∈ −               , where 12.13, 14.45 p q = = and R is the nearest neighbour distance between two atoms. The two constants, ∈ and R , have the dimensions of energy and length, respectively. The equilibrium separation between two nearest neighbour atoms in units of σ (rounded off to two decimal places) is____________ GATE 2019

Complex analysis

The energy-wavevector ( ) E k − dispersion relation for a particle in two dimensions is E Ck = , where C is a constant. If its density of states ( ) D E is proportional to p E then the value of p is____________ GATE 2019

Electrostatics and Magnetostatics

A circular loop made of a thin wire has radius 2cm and resistance2Ω. It is placed perpendicular to a uniform magnetic field of magnitude 0 0.01 B =  Tesla. At time 0 t = the field starts decaying as 0 / 0 t t B B e− =   , where 0 1 t s = . The total charge that passes through a cross section of the wire during the decay isQ . The value of Q in C µ (rounded off to two decimal places) is____________ GATE 2019

Electrostatics and Magnetostatics

The electric field of an electromagnetic wave in vacuum is given by ( ) 9 0 ˆ cos 3 4 1.5 10 E E y z t x = + − ×  The wave is reflected from the 0 z = surface. If the pressure exerted on the surface is 2 0 E α ∈ , the value of α (rounded off to one decimal place) is___________ GATE 2019

Schrodinger equation & Wave Mechanics

The Hamiltonian for a quantum harmonic oscillator of mass $m$ in three dimensions is $H = \frac{p^2}{2m} + \frac{1}{2}m\omega^2 r^2$, where $\omega$ is the angular frequency. The expectation value of $r^2$ in the first excited state of the oscillator in units of $\frac{\hbar}{m\omega}$ (rounded off to one decimal place) is _______.

Lagrangian & Hamiltonian Dynamics

The Hamiltonian for a particle of mass m is 2 2 p H kqt m = + where q and p are the generalized coordinate and momentum, respectively, t is time and k is a constant. For the initial condition, 0 q = and 0 p = at ( ) 0, t q t tα = ∝ . The value of α is ________ GATE 2019

Statistical Ensembles & Partition Functions

At temperature T Kelvin ( ) K , the value of the Fermi function at an energy 0.5eV above the Fermi energy is 0.01. Then T , to the nearest integer, is __________ ( 5 8.62 10 / B k eV K − = × ) GATE 2019

Complex analysis

Let $|\psi_1\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and $|\psi_2\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$ represent two possible states of a two-level quantum system. The state obtained by the incoherent superposition of $|\psi_1\rangle$ and $|\psi_2\rangle$ is given by a density matrix that is defined as $\rho = c_1|\psi_1\rangle\langle\psi_1| + c_2|\psi_2\rangle\langle\psi_2|$. If $c_1 = 0.4$ and $c_2 = 0.6$, the matrix element $\rho_{22}$ (rounded off to one decimal place) is _______.

Electrostatics and Magnetostatics

A conventional type-I superconductor has a critical temperature of 4.7 K at zero magnetic field and a critical magnetic field of 0.3 Tesla at 0 K . The critical field in Tesla at 2K (rounded off to three decimal places) is__________ GATE 2019

Digital Electronics

Consider the following Boolean expression: ( ) ( ) ( ) A B A B C A B C   + + + +   It can be represented by a single three-input logic gate. Identify the gate (a) AND (b) OR (c) XOR (d) NAND GATE 2019

Linear Algebra & Vector Calculus

The value of the integral ( ) 2 2 cos kx dx x a ∞ −∞ + ∫ , where 0 k > and 0 a > , is (a) ka e a π − (b) 2 ka e a π − (c) 2 ka e a π − (d) 3 2 ka e a π − GATE 2019 GATE - 2019 [SOLUTION] Physics by fiziks 14 Revised Edition-2022 H.No. 40-D, G.F., Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498, Website: www.physicsbyfiziks.com, Email: [email protected]

Complex analysis

Official Question 38 from GATE PH 2019 Examination

Complex analysis

A solid cylinder of radius R has total charge Q distributed uniformly over its volume. It is rotating about its axis with angular speedω . The magnitude of the total magnetic moment of the cylinder is (a) 2 QR ω (b) 2 1 2 QR ω (c) 2 1 4 QR ω (d) 2 1 8 QR ω GATE 2019

Complex analysis

Official Question 40 from GATE PH 2019 Examination

Complex analysis

A 3-bit analog-to-digital converter is designed to digitize analog signals ranging from 0 V to 10 V. For this converter, the binary output corresponding to an input of 6 V is

Complex analysis

Official Question 42 from GATE PH 2019 Examination

Electrostatics and Magnetostatics

A particle of mass m moves in a lattice along the x - axis in a periodic potential ( ) ( ) V x V x d = + with periodicity d . The corresponding Brillouin zone extends from 0k − to 0k with these two k - points being equivalent. If a weak force F in the x - direction is applied to the particle, it starts a periodic motion with the time period T . Using the equation of motion crystal dp F dt = for a particle moving in a band, where crystal p is the crystal momentum of the particle, the period T is found to be ( h is Planck constant) (a) 2md F (b) 2 2 md F (c) 2h Fd (d) h Fd GATE 2019

Schrodinger equation & Wave Mechanics

Consider a rectangular potential barrier $V(x) = V_0$ for $a < x < b$ and 0 elsewhere, with $V_0 > 0$. For particles of energy $E < V_0$ incident on this barrier from the left ($x < a$), which schematic diagram best represents the probability density $|\psi(x)|^2$ as a function of $x$?

Lagrangian & Hamiltonian Dynamics

The spin-orbit interaction term of an electron moving in a central field is written as ( ) f r l s⋅ , where r is the radial distance of the electron from the origin. If an electron moves inside a uniformly charged sphere, then (a) ( ) f r = constant (b) ( ) 1 f r r− ∝ (c) ( ) 2 f r r− ∝ (d) ( ) 3 f r r − ∝ GATE 2019

Complex analysis

For the following circuit, the correct logic values for the entries 2 X and 2Y in the truth table are (a) 1 and 0 (b) 0 and 0 (c) 0 and 1 (d) 1 and 1 GATE 2019

Electrostatics and Magnetostatics

In a set of N successive polarizers, the th m polarizer makes an angle 2 m N π       with the vertical. A vertically polarized light beam of intensity 0I is incident on two such sets with 1 N N = and 2 N N = , where 2 1 N N > . Let the intensity of light beams coming out be ( ) 1 I N and ( ) 2 I N , respectively. Which of the following statements is correct about the two outgoing beams? (a) ( ) ( ) 2 1 I N I N > ; the polarization in each case is vertical (b) ( ) ( ) 2 1 I N I N < ; the polarization in each case is vertical (c) ( ) ( ) 2 1 I N I N > ; the polarization in each case is horizontal B G A C P Y X G A B P C X Y 1 0 1 0 0 0 1 0 0 0 1 0 2 X 2Y 1 0 0 0 1 0 1 C B G A P Y X 0 1 0 0 0 0 1 1 0 0 0 GATE - 2019 [SOLUTION] Physics by fiziks Revised Edition-2022 H.No. 40-D, G.F., Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498, Website: www.physicsbyfiziks.com, Email: [email protected] 17 (d) ( ) ( ) 2 1 I N I N < ; the polarization in each case is horizontal GATE 2019

Electrostatics and Magnetostatics

A ball bouncing of a rigid floor is described by the potential energy function ( ) for 0 for 0 mgx x V x x >  = ∞ ≤  Which of the following schematic diagrams best represents the phase space plot of the ball? (a) (b) (c) (d) GATE 2019

Electrostatics and Magnetostatics

An infinitely long wire parallel to the x -axis is kept at z d = and carries a current I in the positive x direction above a superconductor filling the region 0 z ≤ (see figure). The magnetic field B  inside the superconductor is zero so that the field just outside the superconductor is parallel to its surface. The magnetic field due to this configuration at a point ( ) , , 0 x y z > is (a) ( ) ( ) 0 2 2 ˆ ˆ 2 z d j yk I y z d µ π − − +        + −   (b) ( ) ( ) ( ) ( ) 0 2 2 2 2 ˆ ˆ ˆ ˆ 2 z d j yk z d j yk I y z d y z d µ π   − − + + −   +       + − + +     (c) ( ) ( ) ( ) ( ) 0 2 2 2 2 ˆ ˆ ˆ ˆ 2 z d j yk z d j yk I y z d y z d µ π   − − + + −   −       + − + +     (d) ( ) ( ) ( ) ( ) 0 2 2 2 2 ˆ ˆ ˆ ˆ 2 yj z d k yj z d k I y z d y z d µ π   + − − +   +       + − + +     GATE 2019

Electrostatics and Magnetostatics

The vector potential inside a long solenoid with n turns per unit length and carrying current I , written in cylindrical coordinates is ( ) 0 ˆ , , 2 nI A s z s µ φ φ =  . If the term ( ) 0 ˆ ˆ cos sin 2 nI s s µ α φφ β φ + , where 0, 0 α β ≠ ≠ is added to ( ) , , A s z φ  , the magnetic field remains the same if (a) α β = (b) α β = − (c) 2 α β = (d) 2 β α = ( ) 1 ˆ ˆ ˆ Useful formulae: ; 1 1 ˆ ˆ ˆ s s z z t t t t s z s s z sv v v v v v v s z s z z s s s φ φ φ φ φ φ φ ∂ ∂ ∂   ∆= + +   ∂ ∂ ∂       ∂ ∂   ∂ ∂ ∂ ∂       ∇× = − + − + −       ∂ ∂ ∂ ∂ ∂ ∂              GATE 2019 I ˆz d ˆx superconductor GATE - 2019 [SOLUTION] Physics by fiziks Revised Edition-2022 H.No. 40-D, G.F., Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498, Website: www.physicsbyfiziks.com, Email: [email protected] 19

Lagrangian & Hamiltonian Dynamics

Low energy collision ( s - wave scattering) of pion (π + ) with deuteron (d ) results in the production of two proton ( d p p π + + → + ). The relative orbital angular momentum (in units of ) of the resulting two-proton system for this reaction is (a) 0 (b) 1 (c) 2 (d) 3 GATE 2019

Lagrangian & Hamiltonian Dynamics

Consider the Hamiltonian ( ) 2 4 2 , 2 p q H q p q α β = + , where α and β are parameters with appropriate dimensions, and q and p are the generalized coordinate and momentum, respectively. The corresponding Lagrangian ( ) , L q q is (a) 2 4 2 1 2 q q q β α −  (b) 2 4 2 2 q q q β α +  (c) 2 4 2 1 q q q β α +  (d) 2 4 2 1 2 q q q β α − +  GATE 2019

Complex analysis

For a given load resistance 4.7 L R = ohm, the power transfer efficiencies load total P P η   =     of a dc voltage source and a dc current source with internal resistances 1 R and 2 R , respectively, are equal. The product 1 2 R R in units of 2 ohm (rounded off to one decimal place) is_________ GATE 2019

Complex analysis

The ground state electronic configuration of the rare-earth ion ( 3 Nd + ) is [ ] 3 2 6 4 5 5 Pd f s p . Assuming LS coupling, the Lande g - factor of this ion is 8 11. The effective magnetic moment in units of Bohr magneton B µ (rounded off to two decimal places) is ____________. GATE 2019

Linear Algebra & Vector Calculus

A projectile of mass 1kg is launched at an angle of 0 30 from the horizontal direction at 0 t = and takes time T before hitting the ground. If its initial speed is 1 10 ms−, the value of the action integral for the entire flight in the units of 2 1 kgm s− (round off to one decimal place) is___________. [Take 2 10 g ms− = ] GATE 2019

Complex analysis

Let θ be a variable in the range π θ π − ≤ < . Now consider a function ( ) 1 for 2 2 0 otherwise π π θ ψ θ −  ≤ <  =   if its Fourier-series is written as ( ) im m m C e θ ψ θ ∞ − =−∞ = ∑ , then the value of 2 3 C (rounded off to three decimal places) is__________. GATE 2019

Complex analysis

Two spaceships A and B , each of the same rest length L , are moving in the same direction with speeds 4 5 c and 3 5 c , respectively, where c is the speed of light. As measured by B , the time taken by A to completely overtake B [see figure below] in units of / L c (to the nearest integer) is ___________ (i) (ii) GATE 2019

Nuclear Models & Decays

A radioactive element X has a half-life of 30 hours. It decays via alpha, beta and gamma emissions with the branching ratio for beta decay being 0.75. The partial half-life for beta decay in unit of hours is ____________ GATE 2019

Statistical Ensembles & Partition Functions

In a thermally insulated container, 0.01 kg of ice at 273 K is mixed with 0.1 kg of water at 300 K . Neglecting the specific heat of the container, the change in the entropy of the system in / J K on attaining thermal equilibrium (rounded off to two decimal places) is____________ GATE 2019

Electrostatics and Magnetostatics

Consider a system of three charges as shown in the figure below: For 10 ; 60 r m θ = = degrees; 6 10 q − = Coulomb, and 3 10 d m − = , the electric dipole potential in volts (rounded off to three decimal places) at a point ( ) ,r θ is _________ [Use: 2 9 2 0 1 9 10 4 Nm C π = × ∈ ] GATE 2019

Statistical Ensembles & Partition Functions

Consider two system A and B each having two distinguishable particles. In both the systems, each particle can exist in states with energies 0,1,2 and 3 units with equal probability. The total energy of the combined system is 5 units. Assuming that the system A has energy 3 units and the system B has energy 2 units, the entropy of the system is ln B k λ . The value of λ is__________ GATE 2019

Complex analysis

Official Question 62 from GATE PH 2019 Examination

Complex analysis

Official Question 63 from GATE PH 2019 Examination

Complex analysis

Two events, one on the earth and the other one on the Sun, occur simultaneously in the earth’s frame. The time difference between the two events as seen by an observer in a spaceship moving with velocity 0.5c in the earth’s frame along the line joining the earth to the Sun is t ∆, where c is the speed of light. Given that light travels from the Sun to the earth in 8.3 minutes in the earth’s frame, the value of t ∆ in minutes (rounded off to two decimal places) is____________ (Take the earth’s frame to be inertial and neglect the relative motion between the earth and the sun) GATE 2019

Crystal Structure & Lattice Dynamics

In a certain two-dimensional lattice, the energy dispersion of the electrons is ( ) 1 3 2 cos 2cos cos 2 2 x x y k t k a k a k a ε   = − +      where ( ) , x y k k k =  denotes the wave vector, a is the lattice constant and t is a constant in units of eV . In this lattice the effective mass tensor ij m of electrons calculated at the center of the Brillouin zone has the form 2 2 0 0 ij m ta α α   =      . The value of α (rounded off to two decimal places) is ____________ GATE 2019