GATE PH 2023 Question Paper & Solutions

Analytical & Spatial Aptitude

"You are delaying the completion of the task. Send _______ contributions at the earliest."

Verbal Aptitude

References : ______ : : Guidelines : Implement (By word meaning)

Analytical & Spatial Aptitude

In the given figure, PQRS is a parallelogram with PS = 7 cm, PT = 4 cm and PV = 5 cm. What is the length of RS in cm? (The diagram is representative.)

Analytical & Spatial Aptitude

In 2022, June Huh was awarded the Fields medal, which is the highest prize in Mathematics. When he was younger, he was also a poet. He did not win any medals in the International Mathematics Olympiads. He dropped out of college. Based only on the above information, which one of the following statements can be logically inferred with certainty?

Analytical & Spatial Aptitude

A line of symmetry is defined as a line that divides a figure into two parts in a way such that each part is a mirror image of the other part about that line. The given figure consists of 16 unit squares arranged as shown. In addition to the three black squares, what is the minimum number of squares that must be coloured black, such that both PQ and MN form lines of symmetry? (The figure is representative)

Analytical & Spatial Aptitude

Human beings are one among many creatures that inhabit an imagined world. In this imagined world, some creatures are cruel. If in this imagined world, it is given that the statement " Some human beings are not cruel creatures" is FALSE, then which of the following set of statement(s) can be logically inferred with certainty? (i) All human beings are cruel creatures. (ii) Some human beings are cruel creatures. (iii) Some creatures that are cruel are human beings. (iv) No human beings are cruel creatures.

Quantitative Aptitude

To construct a wall, sand and cement are mixed in the ratio of 3:1. The cost of sand and that of cement are in the ratio of 1:2. If the total cost of sand and cement to construct the wall is 1000 rupees, then what is the cost (in rupees) of cement used? (A) 400 (B) 600 (C) 800 (D) 200

Verbal Aptitude

The World Bank has declared that it does not plan to offer new financing to Sri Lanka, which is battling its worst economic crisis in decades, until the country has an adequate macroeconomic policy framework in place. In a statement, the World Bank said Sr i Lanka needed to adopt structural reforms that focus on economic stabilisation and tackle the root causes of its crisis. The latter has starved it of foreign exchange and led to shortages of food, fuel, and medicines. The bank is repurposing resources under existing loans to help alleviate shortages of essential items such as medicine, cooking gas, fertiliser, meals for children, and cash for vulnerable households. Based only on the above passage, which one of the following statements can be inferred with certainty?

Analytical & Spatial Aptitude

The coefficient of x4 in the polynomial (x - 1)3(x - 2)3 is equal to _______.

Analytical & Spatial Aptitude

Which one of the following shapes can be used to tile (completely cover by repeating) a flat plane, extending to infinity in all directions, without leaving any empty spaces in between them? The copies of the shape used to tile are identical and are not allowed to overlap.

Classical Thermodynamics

Which one of the following entropy (S) - temperature (T) diagrams CORRECTLY represents the Carnot cycle abcda shown in the 𝑷-𝑽 diagram? (A) (B) (C) (D)

Differential equations

Which one of the following is a dimensionless constant?

Semiconductor Devices & Diodes

Choose the most appropriate matching of the items in Column 1 with those in Column 2. Column 1 Column 2 (i) PIN diode P. Voltage regulation (ii) Tunnel diode Q. Radio frequency and microwave devices (iii) Zener diode R. Optoelectronic detection (iv) Photo diode S. Oscillator

Atomic Spectra & Multi-Electron Atoms

The atomic number of an atom is 6. What is the spectroscopic notation of its ground state, according to Hund's rules?

Angular momentum & Hydrogen Atom

H is the Hamiltonian, L⃗ the orbital angular momentum and Lz is the z-component of L⃗ . The 1s state of the hydrogen atom in the non-relativistic formalism is an eigen function of which one of the following sets of operators?

Lagrangian & Hamiltonian Dynamics

The Hall experiment is carried out with a non-magnetic semiconductor. The current I is along the x-axis and the magnetic field B is along the z-axis. Which one of the following is t he CORRECT representation of the variation of the magnitude of the Hall resistivity 𝜌xy as a function of the magnetic field? (A) (B) (C) (D)

Linear Algebra & Vector Calculus

Consider a two dimensional Cartesian coordinate system in which a rank 2 contravariant tensor is represented by the matrix (0 1 1 0). The coordinate system is rotated anticlockwise by an acute angle 𝜃 with the origin fixed. Which one of the following matrices represents the tensor in the new coordinate system?

Lagrangian & Hamiltonian Dynamics

A compound consists of three ions X, Y and Z. The Z ions are arranged in an FCC arrangement. The X ions occupy 1 6 of the tetrahedral voids and the Y ions occupy 1 3 of the octahedral voids. Which one of the following is the CORRECT chemical formula of the compound?

Lagrangian & Hamiltonian Dynamics

For a non-magnetic metal, which one of the following graphs best represents the behaviour of C T vs. T2, where C is the heat capacity and T is the temperature? (A) (B) (C) (D)

Differential equations

For nonrelativistic electrons in a solid, different energy dispersion relations (with effective masses ma∗ , mb ∗ , and mc∗) are schematically shown in the plots. Which one of the following options is CORRECT?

Electrostatics and Magnetostatics

The figure schematically shows the M (magnetization) - H (magnetic field) plots for certain types of materials. Here M and H are plotted in the same scale and units. Which one of the following is the most appropriate combination?

Crystal Structure & Lattice Dynamics

Graphene is a two dimensional material, in which carbon atoms are arranged in a honeycomb lattice with lattice constant a. As shown in the figure , a 1 and a 2 are two lattice vectors. Which one of the following is the area of the first Brillouin zone for this lattice?

Differential equations

A 60Co nucleus emits a \beta-particle and is converted to 60Ni* with JP = 4+, which in turn decays to the 60Ni ground state with JP = 0+ by emitting two photons in succession, as shown in the fi gure. Which one of the following statements is CORRECT?

Classical Thermodynamics

Which one of the following options is CORRECT for the given logic circuit?

Angular momentum & Hydrogen Atom

An atom with non-zero magnetic moment has an angular momentum of magnitude √12 \hbar. When a beam of such atoms is passed through a Stern - Gerlach apparatus, how many beams does it split into?

Linear Algebra & Vector Calculus

A 4 × 4 matrix M has the property M† = -M and M4 = 𝟏, where 𝟏 is the 4 × 4 identity matrix. Which one of the following is the CORRECT set of eigenvalues of the matrix M?

Particle Physics & Fundamental Interactions

The Ξ0∗ particle is a member of the Baryon decuplet with isospin state | I, I3〉 = | 1 2 , 1 2〉 and strangeness quantum number -2. In the quark model, which one of the following is the flavour part of the Ξ0∗ wavefunction?

Classical Thermodynamics

Which of the following is(are) the CORRECT option(s) for the Joule -Thomson effect?

Nuclear Properties & Nuclear Forces

The deuteron is a bound state of a neutron and a proton. Which of the following statements is(are) CORRECT?

Semiconductor Devices & Diodes

The Geiger-Muller counter is a device to detect \alpha, \beta and \gamma radiations. It is a cylindrical tube fi lled with monatomic gases like a rgon, and polyatomic gases such as ethyl alcohol. The inner electrode is along the axis of the cylindrical tube and the outer electrode is the tube . Which of the following statements is (are) CORRECT?

Electrostatics and Magnetostatics

Consider an isolated magnetized sphere of radius R with a uniform magnetization M⃗⃗ along the positive z direction, with the north and south poles of the sphere lying on the z axis. It is given that the magnetic field inside the sphere is B⃗ = 2\mu0 3 M⃗⃗ , where \mu0 is the permeability of vacuum. Which of the following statements is(are) CORRECT?

Differential equations

Which of the following options represent(s) linearly independent pair(s) of functions of a real variable x ?

Angular momentum & Hydrogen Atom

In the vector model of angular momentum applied to atoms, what is the minimum angle in degrees (in integer) made by the orbital angular moment um vector and the positive z axis for a 2p electron?

Crystal Structure & Lattice Dynamics

For a transistor amplifier, the frequency response is such that the mid band voltage gain is 200. The cutoff frequencies are 20 Hz and 20 kHz. What is t he ratio (rounded off to two decimal places) of the voltage gain at 10 Hz to that at 100 kHz?

Crystal Structure & Lattice Dynamics

An electric field as a function of radial coordinate r has the form E⃗ = \alpha e-r2 r r̂, where \alpha is a constant. Assume that dimensions are appropriately taken care of. The electric flux through a sphere of radius √2, centered at the origin, is Φ. What is the value of Φ 2𝜋\alpha (rounded off to two decimal places)?

Differential equations

It is given that the electronic ground state of a diatomic molecule X2 has even parity and the nuclear spin of X is 0. Which one of the following is the CORRECT statement with regard to the rotational Raman spectrum ( J is the rotational quantum number) of this molecule?

Lagrangian & Hamiltonian Dynamics

An input voltage in the form of a square wave of frequency 1 kHz is given to a circuit, which results in the output shown schematically below. Which one of the following options is the CORRECT representation of the circuit? (A) (B) (C) (D)

Differential equations

A simple harmonic oscillator with an angular frequency \omega is in thermal equilibrium with a reservoir at absolute temperature T, with \omega = 2kBT \hbar . Which one of the following is the partition function of the system?

Crystal Structure & Lattice Dynamics

Which one of the following options is the most appropriate match between the items given in Column 1 and Column 2? Column 1 Column 2 (i) Visible light P. Transition between core energy levels of atoms (ii) X-rays Q. Transition between nuclear energy levels (iii) Gamma rays R. Pair production (iv) Thermal neutrons S. Crystal structure determination T. Photoelectric effect

Special theory of relativity

A rod PQ of proper length L lies along the x-axis and moves towards the positive x direction with speed v = 3c 5 with respect to the ground (see figure) , where c is the speed of light in vacuum. An observer on the ground measures the positions of P and Q at different times tP and tQ respectively in the ground frame, and finds the difference betwen them to be 9L 10 . What is the value of tQ - tP?

Small oscillations

A symmetric top has principal moments of inertia I1 = I2 = 2\alpha 3 , I3 = 2\alpha about a set of principal axes 1, 2, 3 respectively, passing through its center of mass, where \alpha is a positive constant. There is no force acting on the body and the angular speed of the body about the 3 -axis is \omega3 = 1 8 rad/s. With what angular frequency in rad/s does the angular velocity vector \omega⃗⃗ 1 precess about the 3-axis?

Electrostatics and Magnetostatics

A particle of mass m is free to move on a frictionless horizontal two dimensional (r, 𝜃) plane, and is acted upon by a force F = - k 2r3 r̂ with k being a positive constant. If pr and p𝜃 are the generalised momenta corresponding to r and 𝜃 respectively, then what is the value of dpr dt ?

Electrostatics and Magnetostatics

Consider two real functions U(x, y) = xy(x2 - y2) , V(x, y) = ax4 + by4 + cx2y2 + k , where k is a real constant and a, b, c are real coefficients. If U(x, y) + i V(x, y) is analytic, then what is the value of a × b × c ?

Electrostatics and Magnetostatics

Young's double slit experiment is performed using a beam of C60 (fullerene) molecules, each molecule being made up of 60 carbon atoms. When the slit separation is 50 nm, fringes are formed on a screen kept at a distance of 1 m from the slit s. Now, t he experiment is repeated with C70 molecules with a slit separation of 92.5 nm. The kinetic energies of both the beams are the same. The position of the 4th bright fringe for C60 will correspond to the nth bright fringe for C70. What is the value of n (rounded off to the nearest integer) ?

Crystal Structure & Lattice Dynamics

A neutron beam with a wave vector k⃗ and an energy 20.4 meV diffracts from a crystal with an outgoing wave vector k⃗ '. One of the diffraction peaks is observed for the reciprocal lattice vector G of magnitude 3.14 Å -1 . What is the diffraction angle in degrees (rounded off to the nearest integer) that k⃗ makes with the plane? (Use mass of neutron = 1.67 × 10-27 Kg)

Crystal Structure & Lattice Dynamics

In the first Brillouin zone of a rectangular lattice (lattice constants a = 6 Å and b = 4 Å), three incoming ph onons with the same wave vector 〈1.2 Å-1, 0.6 Å-1〉 interact to give one phonon. Which one of the following is the CORRECT wave vector of the resulting phonon?

Classical Thermodynamics

For a covalently bonded solid consisting of ions of mass m, the binding potential can be assumed to be given by U(r) = - \epsilon ( r r0 ) e - r r0 , where \epsilon and r0 are positive constants. What is the Einstein frequency of the solid in Hz ?

Particle Physics & Fundamental Interactions

In a hadronic interaction, 𝜋0's are produced with different momenta, and they immediately decay into two photons with an opening angle 𝜃 between them. Assuming that all these decays occur in one plane, which one of the following figures depicts the behaviour of 𝜃 as a function of the 𝜋0 momentum p? (A) (B) (C) (D)

Differential equations

A particle has wavefunction 𝜓(x, y, z) = N z e-\alpha(x2+y2+z2) , where N is a normalization constant and \alpha is a positi ve constant. In this state, which one of the following options represent s the eigenvalues of L2 and Lz respectively? Some values of Yℓ m are : Y0 0 = √ 1 4𝜋 , Y1 0 = √ 3 4𝜋 cos 𝜃 , Y1 ±1 = ∓√ 3 8𝜋 sin 𝜃 e±i\phi

Differential equations

The wavefunction of a particle in one dimension is given by 𝜓(x) = {M, -a < x < a 0, otherwise. Here M and a are positive constants. If 𝜑(p) is the corresponding momentum space wavefunction, which one of the following plots best represents |𝜑(p)|2 ? (A) (B) (C) (D)

Schrodinger equation & Wave Mechanics

Consider a particle in a two dimensional infinite square well potential of side L, with 0 ≤ x ≤ L and 0 ≤ y ≤ L. The wavefunction of the particle is zero only along the line y = L 2 , apart from the boundaries of the well. If the energy of the particle in this state is E, what is the energy of the ground state?

Differential equations

Consider two non-identical spin 1 2 particles labelled 1 and 2 in the spin product state | 1 2 , 1 2〉 | 1 2 , - 1 2〉. The Hamiltonian of the system is H = 4λ \hbar2 S 1 ⋅ S 2 , where S 1 and S 2 are the spin operators of particles 1 and 2, respectively, and 𝜆 is a constant with appropriate dimensions. What is the expectation value of H in the above state?

Classical Thermodynamics

A spin 1 2 particle is in a spin up state along the x-axis (with unit vector x̂) and is denoted as | 1 2 , 1 2〉x. What is the probability of finding the parti cle to be in a spin up state along the direction x̂', which lies in the xy-plane and makes an angle 𝜃 with respect to the positive x-axis, if such a measurement is made?

Differential equations

Different spectral lines of the Balmer series (transitions n → 2, with n being the principal quantum number) fall one at a time on a Young's double slit apparatus. The separation between the slits is d and the screen is placed at a constant distance from the slits. What factor should d be multiplied by to maintain a constant fringe width for various lines, as n takes different allowed values?

Electrostatics and Magnetostatics

Under parity and time reversal transformations, which of the following statements is(are) TRUE about the electric dipole moment p and the magnetic dipole moment 𝝁 ?

Complex analysis

Consider the complex function f(z) = z2 sin z (z - 𝜋)4 . At z = 𝜋 , which of the following options is(are) CORRECT?

Linear Algebra & Vector Calculus

Consider the vector field V⃗ consisting of the velocities of points on a thin horizontal disc of radius R = 2 m, moving anticlockwise with uniform angular speed \omega = 2 rad/sec about an axis passing through its center. If V = |V⃗ | , then which of the following options is(are) CORRECT ? (In the options, r̂ and 𝜃̂ are unit vectors corresponding to the plane polar coordinates r and 𝜃). You may use the fact that in cylindrical coordinates (s, \phi, z) (s is the distance from the z-axis), the gradient, divergence, curl and Laplacian operators are: ∇⃗⃗ f = 𝜕f 𝜕s ŝ + 1 s 𝜕f 𝜕\phi \phî + 𝜕f 𝜕z ẑ ; ∇⃗⃗ . A = 1 s 𝜕 𝜕s (sAs) + 1 s 𝜕A\phi 𝜕\phi + 𝜕Az 𝜕z ; ∇⃗⃗ × A = ( 1 s 𝜕Az 𝜕\phi - 𝜕A\phi 𝜕z ) ŝ + ( 𝜕As 𝜕z - 𝜕Az 𝜕s ) \phî + 1 s ( 𝜕 𝜕s (sA\phi) - 𝜕As 𝜕\phi ) ẑ ; ∇⃗⃗ 2f = 1 s 𝜕 𝜕s (s 𝜕f 𝜕s) + 1 s2 𝜕2f 𝜕\phi2 + 𝜕2f 𝜕z2 .

Nuclear Properties & Nuclear Forces

A slow moving 𝜋- particle is captured by a deuteron (d) and this reaction produces two neutrons (n) in the final state, i.e., 𝜋- + d → n + n. Neutron and deuteron have even intrinsic parities, whereas 𝜋- has odd intrinsic parity. L and S are the orbital and spin angular momenta, respectively of the system of two neutrons. Which of the following statements regarding the final two-neutron state is(are) CORRECT?

Electrostatics and Magnetostatics

Two independent electrostatic configurations are shown in the figure. Configuration (I) consists of an isolated point charge q = 1 C, and configuration (II) consists of another identical charge surrounded by a thick conducting shell of inner radius R1 = 1 m and outer radius R2 = 2 m, with the charge being at the center of the shell. WI = \epsilon0 2 ∫ EI 2 dV and WII = \epsilon0 2 ∫ EII 2 dV , where EI and EII are the magnitudes of the electric fields for configurations (I) and (II) respectively, \epsilon0 is the permittivity of vacuum, and the volume integrations are carried out over all space. If 8𝜋 \epsilon0 |WI - WII| = 1 n , what is the value of the integer n?

Semiconductor Devices & Diodes

In pion nucleon scattering, the pion and nucleon can combine to form a short lived bound state called the Δ particle (𝜋 + N → Δ). The masses of the pion, nucleon and the Δ particle are 140 MeV/c2, 938 MeV/c2 and 1230 MeV/c2, respectively. In the lab frame, where the nucleon is at rest, what is the minimum energy (in MeV/c2, rounded off to one decimal place) of the pion to produce the Δ particle?

Operational Amplifiers & Analog Circuits

Consider an electromagnetic wave propagating in the z-direction in vacuum, with the magnetic field given by B⃗ = B⃗ 0ei(kz-\omegat). If B0 = 10-8 T, the average power passing through a circle of radius 1.0 m placed in the xy plane is P (in Watts). Using \epsilon0 = 10-11 C2 N m2 , what is the value of 103P 𝜋 (rounded off to one decimal place)?

Operational Amplifiers & Analog Circuits

An \alpha-particle is emitted from the decay of Americium ( Am) at rest, i.e., Am94 241 → U92 237 + \alpha. The rest mass es of Am94 241 , U92 237 and \alpha are 224.544 GeV/c2, 220.811 GeV/ c2 and 3.728 GeV/ c2 respectively. What is the kinetic energy (in MeV/c2, rounded off to two decimal places) of the \alpha-particle?

Operational Amplifiers & Analog Circuits

Consider 6 identical, non-interacting, spin 1 2 atoms arranged on a crystal lattice at absolute temperature T. The z-component of the magnetic moment of each of these atoms can be ± \muB. If P and Q are the probabilities of the net magnetic moment of the solid being 2\muB and 6\muB respectively, what is the value of P Q (in integer)?

Nuclear Models & Decays

Two identical, non-interacting 4He2 atoms are distributed among 4 different non- degenerate energy levels. The probability that they occupy different energy levels is p. Similarly, two 3He2 atoms are distributed among 4 different non-degenerate energy levels, and the probability that they occupy different levels is q. What is the value of p q (rounded off to one decimal place)?

Nuclear Models & Decays

Two identical bodies kept at temperatures 800 K and 200 K act as the hot and the cold reservoirs of an ideal heat engine , respectively . Assume that their heat capacity ( C) in Joules/K is independent of temperature and that they do not undergo any phase change. Then, the maximum work that can be obtained from the heat engine is n × C Joules. What is the value of n (in integer)? END OF QUESTION PAPER