GATE PH 2024 Question Paper & Solutions

Verbal Aptitude

If '→' denotes increasing order of intensity, then the meaning of the words [smile → giggle → laugh] is analogous to [disapprove → ________ → chide]. Which one of the given options is appropriate to fill the blank?

Analytical & Spatial Aptitude

Find the odd one out in the set: {19, 37, 21, 17, 23, 29, 31, 11}

Analytical & Spatial Aptitude

In the following series, identify the number that needs to be changed to form the Fibonacci series. 1, 1, 2, 3, 6, 8, 13, 21,…

Analytical & Spatial Aptitude

The real variables x, y, z, and the real constants p, q, r satisfy x pq - r2 = y qr - p2 = z rp - q2 Given that the denominators are non-zero, the value of px + qy + rz is

Quantitative Aptitude

Take two long dice (rectangular parallelepiped), each having four rectangular faces labelled as 2, 3, 5, and 7. If thrown, the long dice cannot land on the square faces and has 1/4 probability of landing on any of the four rectangular faces. The label on the top face of the dice is the score of the throw. If thrown together, what is the probability of getting the sum of the two long dice scores greater than 11?

Verbal Aptitude

In the given text, the blanks are numbered (i)-(iv). Select the best match for all the blanks. Prof. P (i) merely a man who narrated funny stories. (ii) in his blackest moments he was capable of self-deprecating humor. Prof. Q (iii) a man who hardly narrated funny stories. (iv) in his blackest moments was he able to find humor.

Quantitative Aptitude

How many combinations of non-null sets A, B, C are possible from the subsets of {2, 3, 5} satisfying the conditions: (i) A is a subset of B, and (ii) B is a subset of C?

Quantitative Aptitude

The bar chart gives the batting averages of VK and RS for 11 calendar years from 2012 to 2022. Considering that 2015 and 2019 are world cup years, which one of the following options is true?

Analytical & Spatial Aptitude

A planar rectangular paper has two V-shaped pieces attached as shown below. This piece of paper is folded to make the following closed three-dimensional object. The number of folds required to form the above object is

Analytical & Spatial Aptitude

Four equilateral triangles are used to form a regular closed three-dimensional object by joining along the edges. The angle between any two faces is

Lagrangian & Hamiltonian Dynamics

If F1(Q, q) = Qq is the generating functio n of a canonical transformation from (p, q) to (P, Q), then which one of the following relations is correct?

Electromagnetic Waves in Media

An unpolarized plane electromagnetic wave in a dielectric medium 1 is incident on a plane interface that separates medium 1 from another dielectric medium 2. Medium 1 and medium 2 have refractive indices n1 and n2, respectively, with n2 > n1. If the angle of incidence is tan-1 ( n2 n1 ), which one of the following statements is true?

Differential equations

The wavefunction of a particle in an infinite one-dimensional potential well at time t is Ψ(x, t) = √2 3 e-iE1t \hbar⁄ 𝜓1(x) + 1 √6 ei𝜋 6⁄ e-iE2t \hbar⁄ 𝜓2(x) + 1 √6 ei𝜋 4⁄ e-iE3t \hbar⁄ 𝜓3(x) where 𝜓1, 𝜓2 and 𝜓3 are the normalized ground state, the normalized first excited state and the normalized second excited state, respectively. E1, E2 and E3 are the eigen-energies corresponding to 𝜓1, 𝜓2 and 𝜓3, respectively. The expectation value of energy of the particle in state Ψ(x, t) is

Classical Thermodynamics

If a thermodynamical system is adiabatically isolated and experiences a change in volume under an externally applied constant pressure, then the thermodynamical potential minimized at equilibrium is the

Angular momentum & Hydrogen Atom

The mean distance between the two atoms of HD molecule is r, where H and D denote hydrogen and deuterium, respectively . The mass of the hydrogen atom is mH. The energy difference between two low est lying rotational states of HD in multiples of \hbar2/(mHr2) is

Crystal Structure & Lattice Dynamics

Crystal structures of two metals A and B are two-dimensional square lattices with same lattice constant a. Electrons in metals behave as free electrons. The Fermi surfaces corresponding to A and B are shown by solid circles in figures. The electron concentrations in A and B are nA and nB, respectively. The value of ( nB nA ) is

Nuclear Models & Decays

Consider the induced nuclear fission reaction U92 235 + n → Rb37 93 + Cs55 141 + 2n where neutron momenta in both initial and final state s are negligible. The ratio of the kinetic energies (KE) of the daughter nuclei, KE( Rb37 93 ) KE( Cs55 141 ) is ___________

Differential equations

The symbols C, D, Vin and V0 shown in the figure denote capacitor, ideal diode, input voltage and output voltage, respectively. Which one of the following output waveforms (V0) is correct for the given input waveform (Vin)? (A) (B) (C) (D)

Nuclear Models & Decays

Let Ne and Te, respectively, denote number and kinetic energy of electrons produced in a nuclear beta decay. Which one of the following distributions is correct? (A) (B) (C) (D)

Electrostatics and Magnetostatics

An infinitely long cylinder of radius R carries a frozen -in magnetization M⃗⃗ = ke-sẑ, where k is a constant and s is the distance from the axis of cylinder. The magnetic permeability of free space is \mu0. There is no free current present anywhere. The magnetic flux density (B⃗ ) inside the cylinder is

Lasers and Quantum Optics

Atomic numbers of V, Cr, Fe and Zn are 23, 24, 26 and 30, respectively. Which one of the following materials does NOT show an electron spin resonance (ESR) spectra?

Differential equations

A particle is subjected to a potential V(x) = { ∞, x ≤ 0 V0, a ≤ x ≤ b 0, elsewhere Here, a > 0 and b > a. If the energy of the particle E < V0, which one of the following schematics is a valid quantum mechanical wavefunction ( Ψ) for the system? (A) (B) (C) (D)

Lagrangian & Hamiltonian Dynamics

Let 𝜌(p , q , t) be the phase space density of an ensemble of a system. The Hamiltonian of the system is H(p , q ). If {A, B} denotes the Poisson bracket of A and B, then d𝜌 dt = 0 implies

Digital Electronics

Consider the following circuit: Suppose the input signal P is and the input signal Q is Which one of the following output signals is correct? (A) (B) (C) (D)

Special theory of relativity

An inertial observer sees two spacecrafts S and T flying away from each other along x-axis with individual speed 0.5c, where c is the speed of light. The speed of T with respect to S is

Electrostatics and Magnetostatics

Let P, Q and R be three different nuclei. Which one of the following nuclear processes is possible?

Differential equations

The temperature dependence of the electrical conductivity (𝜎) of three intrinsic semiconductors A, B and C is shown in figure. Let EA, EB and EC be the bandgaps of A, B and C, respectively. Which one of the following relations is correct?

Perturbation theory & Approximation Methods

Following trial wavefunctions \phi1 = e-Z'(r1+r2) and \phi2 = e-Z'(r1+r2)(1 + g|r 1 - r 2|) are used to get a variational estimate of the ground state energy of the helium atom. Z' and g are the variational parameters, r 1and r 2 are the position vectors of the electrons. Let E0 be the exact ground state energy of the helium atom. E1 and E2 are the variational estimates of the ground state energy of the helium atom corresponding to \phi1 and \phi2, respectively. Which one of the fo llowing options is true?

Crystal Structure & Lattice Dynamics

The wavefunction for a particle is given by the form e-(i\alphax+\beta), where \alpha and \beta are real constants. In which one of the following potentials V(x), the particle is moving?

Differential equations

Consider a volume integral I = ∫ ∇2 (1 r) dV V over a volume V, where r = √x2 + y2 + z2. Which of the following statement is/are correct?

Complex analysis

The complex function e-( 2 z-1) has __________

Digital Electronics

The minimum number of basic logic gates required to realize the Boolean expression B ∙ (A + B) + A ∙ (B̅ + A) is _____ (in integer).

Classical Thermodynamics

The vapor pressure (P) of solid ammonia is given by ln (P) = 23.03 - 3754 T , while that of liquid a mmonia is given by ln (P) = 19.49 - 3063 T , where T is the temperature in K. The temperature of the triple point of ammonia is _____ K ( rounded off to two decimal places).

Crystal Structure & Lattice Dynamics

The electric field in a region depends only on x and y coordinates as E⃗ = k (x x̂ + y ŷ ) x2 + y2 where k is a constant. The flux of E⃗ through the surface of a sphere of radius R with its center at the origin is n𝜋kR, where the value of n is _____ (in integer).

Differential equations

The Hamiltonian of a system of N particles in volume V at temperature T is H = ∑ aiqi 2 2N i=1 + ∑ bipi 2 2N i=1 where ai and bi are positive constants. The ensemble average of the Hamiltonian is \alphaNkBT, where kB is the Boltzmann constant. The value of \alpha is ____ (in integer).

Nuclear Properties & Nuclear Forces

Binding energy and rest mass energy of a two-nucleon bound state are denoted by B and mc2, respectively, where c is the speed of light . The minimum energy of a photon required to dissociate the bound state is

Lagrangian & Hamiltonian Dynamics

The spin-orbit interaction in a hydrogen-like atom is given by the Hamiltonian H' = -kL⃗ ∙ S where k is a real constant. The splitting between levels p3/2 2 and p1/2 2 due to this interaction is

Lagrangian & Hamiltonian Dynamics

Consider the Lagrangian L = mẋẏ - m\omega0 2xy. If px and py denote the generalized momenta conjugate to x and y, respectively, then the canonical equations of motion are

Crystal Structure & Lattice Dynamics

The X -ray diffraction pattern of a mon atomic cubic crystal with rigid spherical atoms of radius 1.56 Å shows several Bragg reflections of which the reflection appearing at the lowest 2𝜃 value is from (111) plane. If the wavelength of X -ray used is 0.78 Å, the Bragg angle (in 2𝜃, rounded off to one decimal place ) corresponding to this reflection and the crystal structure, respectively, are

Electrostatics and Magnetostatics

In a parallel plate capacitor, the plate at x = 0 is grounded and the plate at x = d is maintained at a potential V0. The space between the two plates is filled with a linear dielectric of permittivity \epsilon = \epsilon0 (1 + x d), where \epsilon0 is the permittivity of free space. Neglecting the edge effects, the electric field (E⃗ ) inside the capacitor is

Differential equations

The equation of motion for the forced simple harmonic oscillator is ẍ(t) + \omega2x(t) = F cos (\omegat) where x(t = 0) = 0 and ẋ(t = 0) = 0. Which one of the following options is correct?

Differential equations

An atom is subjected to a weak uniform magnetic field B⃗ . The number of lines in its Zeeman spectrum for transition from n = 2, l = 1 to n = 1, l = 0 is

Electrostatics and Magnetostatics

Consider two matrices: P = [1 2 0 1] and Q = [1 0 0 1]. Which of the following statement is/are true?

Differential equations

An infinite one dimensional lattice extends along x-axis. At each lattice site there exists an ion with spin 1 2 . The spin can point either in +z or - z direction only. Let SP, S F, and S A denote the entropies of paramagnetic, ferromagnetic and antiferromagnetic configurations, respectively. Which of the f ollowing relation is/are true?

Differential equations

Consider a vector field F = (2xz + 3y2)ŷ + 4yz2ẑ. The closed path (Γ: A → B → C → D → A) in z = 0 plane is shown in figure. ∮ F . Γ dl⃗⃗⃗ denotes the line integral of F along the closed path Γ. Which of the following option is/are true? MSQ

Electrostatics and Magnetostatics

Two point charges of charge +q each are placed a distance 2d apart. A grounded solid conducting sphere of radius a is placed midway between them. Assume a2 ≪ d2. Which of the following statement is/are true?

Differential equations

A particle of mass m is moving in the potential V(x) = {V0 + 1 2 m\omega0 2x2, x > 0, ∞ , x ≤ 0, Figures P, Q, R and S show different combinations of the values of \omega0 and V0. Ej (P), Ej (Q), Ej (R) and Ej (S) with j = 0, 1, 2, …, are the eigen-energies of the j-th level for the potentials shown in figures P, Q, R and S, respectively. W hich of the statement is/are true?

Lagrangian & Hamiltonian Dynamics

The non-relativistic Hamiltonian for a single electron atom is H0 = p2 2m - V(r) where V(r) is the Coulomb potential and m is the mass of the electron. Considering the spin-orbit interaction term H' = 1 2m2c2 1 r dV dr L⃗ ∙ S added to H0, which of the following statement is/are true?

Particle Physics & Fundamental Interactions

Decays of mesons and baryons can be categorized as weak, strong and electromagnetic decays depending upon the interactions involved in the processes. Which of the following option is/are true?

Band Theory of Solids & Semiconductors

An extrinsic semiconductor shown in figure carries a current of 2 mA along its length parallel to +x axis. When the majority charge carrier concentration is 12.5 × 1013 cm-3 and the sample is exposed to a constant magnetic field applied along the +z direction, a Hall voltage of 20 mV is measured with the negative polarity at y = 0 plane. Take the electric charge as 1.6 × 10-19 C. The concentration of minority charge carrier is negligible. Which of the following statement is/are true?

Linear Algebra & Vector Calculus

A\alpha and B\beta (\alpha, \beta = 1,2,3, ⋯ , n) are contravariant and covariant vectors, respectively. By convention, any repeated indices are summed over. Which of the following expression is/are tensors?

Dielectric, Magnetic Properties & Superconductivity

The temperature T dependence of magnetic susceptibility 𝜒 (Column I) of certain magnetic materials (Column II) are given below. Which of the following option is/are correct? Column I Column II (1) (P) Diamagnetic (2) (Q) Paramagnetic (3) (R) Ferromagnetic (4) (S) Antiferromagnetic

Differential equations

The curves P and Q schematically show the variation of X -ray intensity with wavelength at two different accelerating voltages for a given target material. In the figure 𝜆1 = 0.25 Å, 𝜆2 = 0.5 Å, 𝜆3 = 1.0 Å, and 𝜆4 = 2.25 Å. Take Planck's constant as 6.6 × 10-34 Js, speed of light as 3 × 108 ms-1 and elementary charge as 1.6 × 10-19 C. Which of the following statement is/are true?

Differential equations

Apart from the acoustic modes, 9 optical modes are identified from the measurements of phonon dispersions of a solid with chemical formula AnBm, where A and B denote the atomic species, and n and m are integers. Which of the following combination of n and m is/are possible?

Differential equations

An oscillating electric dipole of moment d (t) = d0 cos(\omegat) ẑ is placed at origin as shown in figure. Consider a point P(r, 𝜃, \phi) at a very large distance from the dipole. Here r, 𝜃 and \phi are spherical polar coordinates. Which of the following statement is/are true for intensity of radiation?

Fourier and Laplace transforms

The Fourier transform and its inverse transform are respectively defined as f ̃ (\omega) = 1 √2𝜋 ∫ f(x)ei\omegaxdx +∞ -∞ and f(x) = 1 √2𝜋 ∫ f ̃ (\omega)e-i\omegaxd\omega +∞ -∞ . Consider two functions f and g. Another function f ∗ g is defined as (f ∗ g)(x) = 1 √2𝜋 ∫ f(y)g(x - y)dy +∞ -∞ Which of the following relation is/are true? Note: Tilde (~) denotes the Fourier transform.

Dielectric, Magnetic Properties & Superconductivity

A material behaves as a superconductor below a critical temperature Tc and as a normal conductor above Tc. A magnetic field B⃗ = Bẑ is applied when T > Tc. The material is then cooled below Tc in the presence of B⃗ . Which of the following figure represent the correct configuration of magnetic field lines? (A) (B) (C) (D)

Semiconductor Devices & Diodes

A typical biasing of a silicon transistor is shown in figure. The value of common-emitter current gain β for the transistor is 100. Ignore reverse saturation current. The output voltage V0 (in V) is ____ (in integer).

Statistical Ensembles & Partition Functions

The canonical partition function of an ideal gas is Q(T, V, N) = 1 N! [ V (𝜆(T))3] N where T, V, N and 𝜆(T) denote temperature, volume, number of particles, and thermal de Broglie wavelength, respectively. Let kB be the Boltzmann constant and \mu be the chemical potential. Take ln(N!) = N ln(N) - N. If the number density ( N V) is 2.5 × 1025 m-3 at a temperature T, then e\mu/(kBT) (𝜆(T))3 × 10-25 is _______ m-3 (rounded off to one decimal place).

Operational Amplifiers & Analog Circuits

Lagrangian of a particle of mass m is L = 1 2 mẋ 2 - 𝜆x4, where 𝜆 is a positive constant. If the particle oscillates with total energy E, the n the time period of oscillations is a ∫ dx √( 2 m) (E - 𝜆x4) (E 𝜆) 1 4 0 The value of a is _____ (in integer).

Schrodinger equation & Wave Mechanics

A particle of mass m in an infinite potential well of width a is subjected to a perturbation, V' = ℎ2 40ma2 as shown in figure, where ℎ is Planck's constant. The first order energy shift of the fourth energy eigenstate due to this perturbation is ( ℎ2 Nma2) The value of N is _____ (in integer).

Differential equations

Consider a three-dimensional system of non-interacting bosons with zero chemical potential. The energy of the system \epsilon ∝ k2, where k is the wavevector. The low temperature specific heat of the system at constant volume depends on the temperature as CV ∝ T n 2. The value of n is _____ (in integer).

Operational Amplifiers & Analog Circuits

Consider the operational amplifier circuit shown in figure. The output voltage V0 is _____ V (in integer).

Nuclear Models & Decays

An electron in the Coulomb field of a proton is in the following state of coherent superposition of orthonormal states 𝜓nlm Ψ = 1 3 𝜓100 + 1 √3 𝜓210 - √5 3 𝜓320 Let E1, E2, and E3 represent the first three energy levels of the system. A sequence of measurements is done on the same system at different times. Energy is measured first at time t1 and the outcome is E2. Then total angular momentum is measured at time t2 > t1 and finally energy is measured again at t3 > t2. The probability of finding the system in a state with energy E2 after the final measurement is P/9. The value of P is _____ (in integer).

Nuclear Models & Decays

According to the nuclear shell model, the absolute value of the difference in magnetic moments of O8 15 and N7 15 , in the units of nuclear magneton (\muN) is a/3. The magnitude of a is _____ (in integer).