GATE PH 2025 Question Paper & Solutions

Verbal Aptitude

Is there any good show _______ television tonight? Select the most appropriate option to complete the above sentence.

Verbal Aptitude

As the police officer was found guilty of embezzlement, he was _______ dismissed from the service in accordance with the Service Rules. Select the most appropriate option to complete the above sentence.

Quantitative Aptitude

The sum of the following infinite series is: 1 1! + 1 2! + 1 3! + 1 4! + 1 5! + ⋯

Analytical & Spatial Aptitude

A thin wire is used to construct all the edges of a cube of 1 m side by bending, cutting and soldering the wire. If the wire is 12 m long, what is the minimum number of cuts required to construct the wire frame to form the cube?

Quantitative Aptitude

The figures I, II and III are parts of a sequence. Which one of the following options comes next in the sequence at IV? (A) (B) (C) (D) I II III IV ?

Quantitative Aptitude

"Why do they pull down and do away with crooked streets, I wonder, which are my delight, and hurt no man living? Every day the wealthier nations are pulling down one or another in their capitals and their great towns: they do not know why they do it; neither do I. It ought to be enough, surely, to drive the great broad ways which commerce needs and which are the life -channels of a modern city, without destroying all history and all the humanity in between: the islands of the past." (From Hilaire Belloc's "The Crooked Streets") Based only on the information provided in the above passage, which one of the following statements is true?

Quantitative Aptitude

Rohit goes to a restaurant for lunch at about 1 PM. When he enters the restaurant, he notices that the hour and minute hands on the wall clock are exactly coinciding. After about an hour, when he leaves the restaurant, he notices that the clock hands are again exactly coinciding. How much time ( in minutes) did Rohit spend at the restaurant?

Quantitative Aptitude

A color model is shown in the figure with color codes: Yellow (Y), Magenta (M), Cyan (Cy), Red (R), Blue (Bl), Green (G), and Black (K). Which one of the following options displays the color codes that are consistent with the color model? (A) (B) (C) (D)

Quantitative Aptitude

A circle with center at (x, y) = (0.5, 0) and radius = 0.5 intersects with another circle with center at (x, y) = (1, 1) and radius = 1 at two points. One of the points of intersection (x, y) is:

Quantitative Aptitude

An object is said to have an n-fold rotational symmetry if the object, rotated by an angle of 2𝜋 n , is identical to the original. Which one of the following objects exhibits 4 -fold rotational symmetry about an axis perpendicular to the plane of the screen? Note: The figures shown are representative. (A) (B) (C) (D)

Crystal Structure & Lattice Dynamics

For a two-dimensional hexagonal lattice with lattice constant a, the atomic density is

Crystal Structure & Lattice Dynamics

Consider a crystal that has a basis of one atom. Its primitive vectors are a1⃗⃗⃗⃗ = aî , a2⃗⃗⃗⃗ = aĵ , a3⃗⃗⃗⃗ = a 2 (î + ĵ + k̂), where î, ĵ, k̂ are the unit vectors in the x, y and z directions of the Cartesian coordinate system and a is a positive constant. Which one of the following is the correct option regarding the type of the Bravais lattice?

Harmonic oscillator

A particle of mass m is in a potential V(x) = 1 2 m\omega2x2 for x > 0 and V(x) = ∞ for x ≤ 0, where \omega is the angular frequency. The ratio of the energies corresponding to the lowest energy level to the next higher level is

Central force problem

A particle is scattered from a potential V(r ) = g𝛿3(r ), where g is a positive constant. Using the first Born approximation, the angular (𝜃, \phi) dependence of differential scattering cross section d𝜎 dΩ is

Classical Thermodynamics

The Joule-Thomson expansion of a gas is (A) Isentropic (B) Isenthalpic (C) Isobaric (D) Isochoric

Electromagnetic Waves in Media

Which one of the following is correct for the phase velocity vp and group velocity vg? (c is the speed of light in vacuum)

Electrostatics and Magnetostatics

As per the Drude model of metals, the electrical resistance of a metallic wire of length L and cross-section area A is (Consider 𝜏 as the relaxation time , m as electron mass, n as carrier concentration and e as electronic charge)

Nuclear Models & Decays

Which one of the following baryons has strangeness quantum number S = -1?

Digital Electronics

A logic gate circuit is shown in the figure below. The correct combination for the input (P, Q) for which the output T = 1 is (A) (0, 0) (B) (0, 1) (C) (1, 1) (D) (1, 0)

Nuclear Models & Decays

The nuclear energy levels of mirror nuclei are similar. Using this empirical fact alone, the nuclear force can be said to be independent of which one of the following properties of the nucleons?

Complex analysis

Consider the function f(z) = 1 z2(z-2)3 of a complex variable z. The residues of the function at z = 0 and z = 2, respectively, are

Statistical Ensembles & Partition Functions

Consider one mole of a monovalent metal at absolute zero temperature, obeying the free electron model. Its Fermi energy is EF. The energy corresponding to the filling of NA 2 electrons, where NA is the Avogadro number, is 2nEF. The value of n is

Electrostatics and Magnetostatics

A paramagnetic material containing paramagnetic ions with total angular momentum J = 1 2 is kept at absolute temperature T. The ratio of the magnetic field required for 80% of the ions to be in the lowest energy state to that required for having 60% of the ions to be in the lowest energy state at the same temperature is

Linear Algebra & Vector Calculus

Which of the following option(s) is/are correct for the ground state of a hydrogen atom?

Special theory of relativity

Which of the following option(s) is/are correct for photons?

Blackbody radiation & Phase transitions

A schematic Pressure-Temperature diagram of water is shown in the figure. Which of the following option(s) is/are correct?

Special theory of relativity

Which of the following consideration (s) is/are showing t hat nuclear beta decay , n → p + e- + 𝜈̅e, has to be a three-body decay?

Electrostatics and Magnetostatics

Potential energy of two diatomic molecules P and Q of the same reduced mass is shown in the figure. According to this diagram, which of the following option (s) is/are correct?

Special theory of relativity

Nuclear radiation emitted from a Co60 radioactive source is detected by a photomultiplier tube (PMT) coupled to a scintillator crystal. Which of the following option(s) is/are correct?

Classical Thermodynamics

One mole of an ideal monatomic gas at absolute temperature T undergoes free expansion to double its original volume , so that the entropy change is ΔS1. An identical amount of the same gas at absolute temperature 2 T undergoes isothermal expansion to double its original volume , so that the entropy change is ΔS2. The value of ΔS1 ΔS2 (in integer) is ____

Electrostatics and Magnetostatics

A linear dielectric sphere of radius R has a uniform frozen-in polarization along the z-axis. The center of the sphere initially coincides with the origin, about which the electric dipole moment is p 1. When the sphere is shifted to the point (2R, 0, 0), the corresponding dipole moment with respect to the origin is p 2. The value of |p 1| |p 2| (in integer) is _____

Angular momentum & Hydrogen Atom

The effective magnetic moment (in units of Bohr magneton) for the ground state of an isolated 4f ion with 6 unpaired electrons in the 4f shell according to Hund 's rules is (in integer) _____

Crystal Structure & Lattice Dynamics

Powder X-ray diffraction pattern of a cubic solid with lattice constant a has the (111) diffraction peak at 𝜃 = 30°. If the lattice expands such that the lattice constant becomes 1.25a, the angle (in degrees) corresponding to the (111) peak changes to sin-1 ( 1 n). The value of n (rounded off to one decimal place) is _____

Crystal Structure & Lattice Dynamics

Consider a monatomic chain of length 30 cm. The phonon density of states is 1.2 × 10-4 s. Assuming the Debye model, the velocity of sound in m/s (rounded off to one decimal place) is _____

Angular momentum & Hydrogen Atom

The Δ+ baryon with spin 3 2, at rest, decays to a proton and a pion (Δ+ → p + 𝜋0). The Δ+ has positive intrinsic parity and 𝜋0 has negative intrinsic parity. The orbital angular momentum of the proton-pion system (in integer) is _____

Electrostatics and Magnetostatics

The screened nuclear charge of neutral Helium atom is given as 1.7e, where e is the magnitude of the electronic charge. Assuming the Bohr model of the atom for which the energy levels are En = - Z2 2 1 n2 atomic units (Z is the atomic number), the first ionization potential of Helium in atomic units is

Complex analysis

The wire loop shown in the figure carries a steady current I. Each straight section of the loop has length d. A part of the loop lies in xy plane and the other part is tilted at 30° with respect to the xz plane. The magnitude of the magnetic dipole moment of the loop (in appropriate units) is

Linear Algebra & Vector Calculus

The figure shows a system of two equal masses m and three massless horizontal springs with spring constants k1 , k2 , k1. Ignore gravity. The masses can move only in the horizontal direction and there is no dissipation. If m = 1, k1 = 2 and k2 = 3 (all in appropriate units), the frequencies of the normal modes of the system in the same system of units are

Lagrangian & Hamiltonian Dynamics

Two projectile protons P1 and P2 both with spin up (along the +z direction) are scattered from another fixed target proton T with spin up at rest in the xy plane, as shown in the figure. They scatter one at a time. The nuclear interaction potential between both the projectiles and the target proton is 𝜆L⃗ ⋅ S , where L⃗ is the orbital angular momentum of the system with respect to the target, S is the spin angular momentum of the system and 𝜆 is a negative constant in appropriate units. Which one of the following is correct?

Electrostatics and Magnetostatics

A thin circular ring of radius R lies in the xy plane with its centre coinciding with the origin. The ring carries a uniform line charge density 𝜆. The quadrupole contribution to the electrostatic potential at the point (0, 0, d), where d ≫ R, is

Digital Electronics

An \alpha particle is scattered from an Au target at rest as shown in the figure. D1 and D2 are the detectors to detect the scattered \alpha particle at an angle 𝜃 and along the beam direction, respectively, as shown. The signals from D1 and D2 are converted to logic signals and fed to logic gates . When a particle is detected, the signal is 1 and is 0 otherwise. Which one of the following circuits detects the particle scattered at the angle 𝜃 only? (A) (B) (C) (D)

Statistical Ensembles & Partition Functions

Consider a two -level system with energy states +𝜀 and - 𝜀. The number of particles at +𝜀 level is N+ and the number of particles at - 𝜀 level is N-. The total energy of the system is E and the total number of particles is N = N+ + N-. In the thermodynamic limit, the inverse of the absolute temperature of the system is (Given: ln N! ≃ N ln N - N)

Schrodinger equation & Wave Mechanics

Let |m〉 and |n〉 denote the energy ei genstates of a one -dimensional simple harmonic oscillator . The position and momentum operators are X̂ and P̂, respectively. The matrix element 〈m|P̂X̂|n〉 is non-zero when

Linear Algebra & Vector Calculus

A two -level quantum system has energy eigenvalues E1 and E2. A perturbing potential H' = 𝜆Δ𝜎x is introduced, where Δ is a constant having dimensions of energy, 𝜆 is a small dimensionless parameter, and 𝜎x = (0 1 1 0). The magnitudes of the first and the second order corrections to E1 due to H', respectively, are

Lagrangian & Hamiltonian Dynamics

An electron with mass m and charge q is in the spin up state (1 0) at time t = 0. A constant magnetic field is applied along the y-axis, B = B0 ĵ, where B0 is a constant. The Hamiltonian of the system is H = - \hbar\omega𝜎y, where \omega = qB0 2m > 0 and 𝜎y = (0 -i i 0 ). The minimum time after which the electron will be in the spin down state along the x-axis, i.e., 1 √2 ( 1 -1) , is

Linear Algebra & Vector Calculus

A system of three non -identical spin 1 2 particles has the Hamiltonian H = A \hbar2 ( S 1 + S 2) ⋅ S 3, where S 1, S 2 and S 3 are the spin operators of particles labelled 1, 2 and 3 respectively and A is a constant with appropriate dimensions. The set of possible energy eigenvalues of the system is

Dielectric, Magnetic Properties & Superconductivity

Which of the following option(s) is/are correct for a Type I superconductor?

Lagrangian & Hamiltonian Dynamics

Consider two hypothetical nuclei X1 and X2 undergoing \beta decay, resulting in nuclei Y1 and Y2, respectively. The decay scheme and the corresponding JP values of the nuclei are given in the figure. Which of the following option(s) is/are correct? (J is the total angular momentum and P is parity)

Electrostatics and Magnetostatics

A point charge q is placed at a distance d above an infinite, grou nded conducting plate placed on the xy plane at z = 0. The electrostatic potential in z > 0 region is given by \phi = \phi1 + \phi2, where \phi1 = 1 4𝜋\epsilon0 q √x2+y2+(z-d)2 and \phi2 = - 1 4𝜋\epsilon0 q √x2+y2+(z+d)2. Which of the following option(s) is/are correct?

Linear Algebra & Vector Calculus

In coordinates (t , x), a contravariant second rank tensor A has non-zero diagonal components Att = P and Axx = Q, with all other components vanishing, and P, Q being real constants. Here, t is time and x is space coordinate. Consider a Lorentz transformation (t, x) → (t', x') to another frame that moves with relative speed v in the +x direction, so that A → A'. If A'tt and A'xx are the diagonal components of A', then setting the speed of light c = 1, and with \gamma = 1 √1-v2 , which of the following option(s) is/are correct?

Lagrangian & Hamiltonian Dynamics

The Lagrangian of a particle of mass m and charge q moving in a uniform magnetic field of magnitude 2B that points in the z direction, is given by L = m 2 v2 + qB(xvy - yvx) where vx , vy , vz are the components of its velocity v . If px , py , pz denote the conjugate momenta in the x, y, z directions and H is the Hamiltonian, which of the following option(s) is/are correct ?

Lagrangian & Hamiltonian Dynamics

A bead is constrained to move along a long, massless, frictio nless horizontal rod parallel to the x axis. The rod itself is moving vertically upward along the z direction against gravity with a constant speed, starting from z = 0 at t = 0, and remains horizontal. The conjugate momenta are denoted by px , py , pz and the Hamiltonian by H. Which of the following option(s) is/are correct?

Lagrangian & Hamiltonian Dynamics

In a one -dimensional Hamiltonian system with position q and momentum p, consider the canonical transformation (q , p) → (Q = 1 p , P = q p2), where Q and P are the new position and momentum , respectively. Which of the following option(s) regarding the generating function F is/are correct?

Lagrangian & Hamiltonian Dynamics

The energy of a free, relativistic particle of rest mass m moving along the x axis in one dimension, is denoted by T. When moving in a given potential V(x), its Hamiltonian is H = T + V(x). In the presence of this potential, i ts speed is v, conjugate momentum p, and the Lagrangian L. Then, which of the following option(s) is/are correct?

Linear Algebra & Vector Calculus

Consider the integral I = 1 2𝜋i ∮ z4 - 1 (z - a b) (z - b a) dz, where z is a complex variable and a, b are positive real numbers. The i ntegral is taken over a unit circle with center at the origin. Which of the following option(s) is/are correct?

Electrostatics and Magnetostatics

A neutral conducting sphere of radius R is placed in a uniform electric field of magnitude E0, that points along the z axis. The electrostatic potential at any point r outside the sphere is given by V(r, 𝜃) = V0 - E0r (1 - R3 r3) cos 𝜃, where V0 is the constant potential of the sphere. Which of the following option(s) is/are correct?

Electrostatics and Magnetostatics

A point charge q is placed at the origin, inside a linear dielectric medium of infinite extent, having relative permittivity \epsilonr. Which o f the following option (s) is/are correct?

Linear Algebra & Vector Calculus

A linear magnetic material in the form of a cylinder of radius R and length L is placed with its axis parallel to the z axis. The cylinder has uniform magnetization Mk̂. Which of the following option(s) is/are correct?

Electrostatics and Magnetostatics

Cyclotrons are used to accelerate ions like deuterons (d) and \alpha particles. Keeping the magnetic field same for both, d and \alpha are extracted with energies 10 MeV and 20 MeV with extraction radii rd and r\alpha, respectively. Taking the masses Md = 2000 MeV/c2 and M\alpha = 4000 MeV/c2, t he value of r\alpha rd (in integer) is ______

Complex analysis

In the transistor circuit shown in the figure, VBE = 0.7 V and \betaDC = 400. The value of the base current in \muA (rounded off to one decimal place) is ___________

Complex analysis

Consider the set {1, x, x2}. An orthonormal basis in x ∈ [-1,1] is formed from these three terms , where the normalization of a function f(x) is defined via ∫ x2[f(x)]2dx = 1 1 -1 . If the orthonormal basis set is (√ 3 2 , √ 5 2 x , 1 2 √ 21 N (5x2 - 3)) , then the value of N (in integer) is ______

Lagrangian & Hamiltonian Dynamics

The Hamiltonian for a one dimensional system with mass m, position q and momentum p is H(p, q) = p2 2m + q2A(q), where A(q) is a real function of q. If m d2q dt2 = -5qA(q), then dA(q) dq = n A(q) q . The value of n (in integer) is ______

Small oscillations

A system of f ive identical, non -interacting particles with mass m and spin 3 2 is confined to a one-dimensional potential well of length L. If the lowest energy of the system is N 𝜋2\hbar2 2mL2, the value of N (in integer) is _____

Complex analysis

A wheel of mass 4M and radius R is made of a thin uniform distribution of mass 3M at the rim and a point mass M at the center . The spokes of the wheel are massless. The center of mass of the wheel is connected to a horizontal massless rod of length 2R, with one end fixed at O, as show n in the figure . The wheel rolls without slipping on horizontal ground with angular speed Ω. If L⃗ is the total angular momentum of the wheel about O, then the magnitude | dL⃗ dt | = N(MR2Ω2). The value of N (in integer) is ____

Operational Amplifiers & Analog Circuits

The figure shows an opamp circuit with a 5.1 V Zener diode in the feedback loop. The opamp runs from ±15 V supplies. If a +1V signal is applied at the input, the output voltage (rounded off to one decimal place) is ____