GATE PH 2026 Question Paper & Solutions

Verbal Aptitude

"He often _____ the numbers. False claims are not going to help. Honesty _____ trust", said the manager. Choose the option with the correct order of words to fill the blanks.

Analytical & Spatial Aptitude

In the sequence of tiles shown below, the missing tile indicated by the question mark should be (A) (B) (C) (D)

Analytical & Spatial Aptitude

A school has 100 students distributed among 1st to 10th standards. Based on this, which one of the following statements is always correct?

Analytical & Spatial Aptitude

How many 3-digit numbers can be formed using three distinct single digit prime numbers?

Analytical & Spatial Aptitude

In a group of students, 10 students like Mathematics, 12 students like English, 4 students like both Mathematics and English, and 6 students like neither Mathematics nor English. The number of students in the group is ____

Verbal Aptitude

Charity : P :: Retaliation : Q Choose the appropriate pair of words P and Q that fit the analogy.

Analytical & Spatial Aptitude

A paper shown in Panel I is folded along the dashed lines (- - -) to construct a cube. The shaded regions shown in Panel I appear on the outer surface of the cube. Referring to cubes shown in Panel II, which one of the options is correct?

Analytical & Spatial Aptitude

Consider the cube shown below with its 8 corners labelled a, b, c, d, e, f, g, and h. The figure is representative. All corners are to be colored such that any two corners that are connected by an edge must be of different colors. The minimum number of colors required to achieve this is ________

Analytical & Spatial Aptitude

Four hills H1, H2, H3, and H4 are present in an area. The following observations are made about them: i. Neither H2 nor H3 is the easternmost hill. ii. Neither H2 nor H3 is the westernmost hill. iii. Neither the easternmost hill nor the westernmost hill is the southernmost hill. iv. Two hills are located to the west of H2. v. The southernmost hill has at least two hills to its east. The southernmost hill is ________.

Analytical & Spatial Aptitude

As shown in the figure, circle C1 with center O1 and radius r1 touches the square VWXY at points P and Q while circle C2 with center O2 and radius r2 touches the square VWXY at points R and S. The two circles touch each other at T. Given r1 = 1 cm and VY̅̅̅̅ = VW̅̅̅̅̅ = 4 cm, r2 = _____ cm.

Lagrangian & Hamiltonian Dynamics

In free space, an electromagnetic wave is travelling whose wavevector is k⃗ = 10(x̂ + √3ŷ) m-1 . The electric field component of this electromagnetic wave is given by E⃗ (r , t) = ẑ 600 cos(k⃗ ⋅ r - \omegat) V.m-1 . The speed of light in free space is c = 3.0 × 108 m.s-1. The corresponding magnetic field B⃗ (r , t) is

Lagrangian & Hamiltonian Dynamics

An infinitely large non-conducting thin sheet in the xy plane (z=0) carries a uniform surface charge density 𝜎 = 17.70 × 10-12 C.m-2. The electric field in the region z < 0 is E⃗ 2 = x̂ + 2ŷ + 3ẑ. Then, the electric field E⃗ 1 in the region z > 0 will be (\epsilon0 = 8.85 × 10-12 C2. N-1. m-2)

Lagrangian & Hamiltonian Dynamics

Consider an operator  which is not Hermitian. Find the possible values of c and d such that the operator (c  - d †) is Hermitian.

Differential equations

For a scalar field 𝜓(r ) and a vector field A (r ), ∇⃗⃗ × (A 𝜓) is equivalent to the expression

Electrostatics and Magnetostatics

Which of the following options is correct for transformation of electric field E⃗ and magnetic field B⃗ under time reversal, i.e., t → -t ?

Electrostatics and Magnetostatics

On a horizontal plane, a projectile of mass m is launched from the ground with speed v0 at an angle 𝜃0 with the horizontal. In addition to the gravitational force (mg), it also experiences a drag force F drag = -\gammav , where v is its velocity and \gamma is a constant. It hits the ground at a distance R from the point of launch with its velocity making an angle 𝜃 with the horizontal, as shown schematically in the figure. Then which of the following options is correct?

Classical Thermodynamics

Consider the Otto cycle for an ideal gas engine consisting of two quasistatic adiabatic and two quasistatic isochoric processes. The correct temperature-entropy (T-S) phase diagram for the cycle is (A) (B) (C) (D)

Differential equations

The formula for energy E of a photon gas at temperature T in a two-dimensional box at equilibrium with g2d(𝜈) denoting the density of states of photons is given below where symbols 𝜈, ℎ and kB have their standard meaning. The specific heat (CV) of this photon gas obeys E = ∫ d𝜈 g2d(𝜈) ∞ 0 ℎ𝜈 exp ( ℎ𝜈 kBT) - 1

Differential equations

For the electric field of an electromagnetic wave given below, which of the following statements is correct? E⃗ = x̂ E0cos(\omegat) + ŷ 2E0cos (\omegat + 𝜋 2)

Differential equations

A gas of non-interacting 4He atoms (of mass m) is in a three-dimensional trap whose energy levels can be approximated by those of a harmonic oscillator potential V(x, y, z) = 1 2 m\omega2(x2 + y2 + z2). The chemical potential of the gas at T=0 K is

Differential equations

Given |v1⟩ = 1 √2 (1 i ) and |v2⟩ = 1 √2 ( 1 -i), the tensor product |v1⟩ ⊗ |v2⟩ is

Differential equations

Sketch of a two-dimensional vector field V⃗ is shown below. Here, length and arrow head of the arrows denote magnitude and direction of the vector field, respectively. Which of the following statements is correct for ∇⃗⃗ × V⃗ ?

Crystal Structure & Lattice Dynamics

Which one of the following is an allowed process?

Particle Physics & Fundamental Interactions

Given Q is the electromagnetic charge and S is the strangeness quantum number, identify the particle(s) for which (Q - S) = 0 is satisfied.

Classical Thermodynamics

Schematic variation of the specific heat Cp of an ideal gas of diatomic molecules with temperature T is shown in the figure below. For rotational energy ER and vibrational energy Ev of the molecule, which of the following options is/are correct? Here kB is the Boltzmann constant.

Differential equations

If the perturbation V = 𝜆x3 is added to the Hamiltonian of a one-dimensional harmonic oscillator, the matrix element ⟨m|V|0⟩ is/are non-zero for which of the following states? Here, the eigenstates of the harmonic oscillator are denoted by |n⟩.

Operational Amplifiers & Analog Circuits

For which of the following functions does the Laplacian vanish?

Operational Amplifiers & Analog Circuits

A projectile of mass m is launched from the ground with the initial speed v0 at an angle 30° from the horizontal. Take the ground to be horizontal. Ignoring the drag, the magnitude of Hamilton's action ∫ Ldt for the particle from the beginning till it hits the ground is f × ( mv0 3 g ). The value of f (rounded off to two decimal places) is _____

Angular momentum & Hydrogen Atom

Consider an electron in the energy eigenstate 𝜓211(r ) of the hydrogen atom. Given that the radial probability distribution of the electron in such a state takes its maximum value at r = n0a, where a is the Bohr radius, and n0 is an integer. The value of n0 (in integer) is _____ The radial part of the wavefunction 𝜓211(r ) is given by R21(r) = 1 √24a5 re-r/2a .

Electrostatics and Magnetostatics

A dielectric sphere carries a uniform polarization P = 26 μC. cm-2. The magnitude of the electric field at the center of the sphere is E × 109 N. C-1 . The value of E (rounded off to one decimal place) is _____ (\epsilon0 = 8.85 × 10-12C2. N-1. m-2)

Dielectric, Magnetic Properties & Superconductivity

Consider a metal-superconductor junction connected to a dc voltage V. At T < Tc, where Tc is the superconductor's transition temperature, the current I versus V behavior of this junction is shown schematically in the figure below. If the superconducting energy gap is D meV. The value of D (rounded off to one decimal place) is _____

Band Theory of Solids & Semiconductors

For the energy dispersion of an electron in a one-dimensional solid E(k) = E0 - 2\gamma cos(ka), the ratio of the effective mass of the electron in the solid to the free electron mass (me) at k = 0 is R0. Taking \gamma = 0.5 eV and a = 0.5 nm, the value of R0 (rounded off to two decimal place) is _____ (\hbar = 1.054 × 10-34 J. s, me = 9.1 × 10-31 kg, electron charge = 1.6 × 10-19 C)

Classical Thermodynamics

The specific heat Cp(T) of one mole of a material as a function of temperature T is given as Cp(T) = AT + BT3, where A = 0.695 mJ. mol-1. K-2 and B = 0.045 mJ. mol-1. K-4. When T is changed from 1 K to 10 K at constant pressure, then the change in entropy ΔS in mJ. mol-1. K-1 (rounded off to one decimal place) is _____

Molecular Spectroscopy

Raman spectrum of a molecule was recorded using a source of wavelength 5000 Å. The first Stokes line is observed at 5100 Å. The first anti-Stokes line will appear at a wavelength L (in Å). The value of L (rounded off to nearest integer) is _____

Nuclear Models & Decays

A rocket of length 18.0 m is moving at speed 0.9c (where c is the speed of light) parallel to its own length, relative to the earth. The length of the rocket measured in meters by an observer on earth (rounded off to two decimal places) is _____

Complex analysis

The function f(z) of complex variable z given below, f(z) = z2 - 5z + 4 z3 + 4z - z2 - 4 has singular points at z =

Differential equations

Consider the Pauli matrices 𝜎x = (0 1 1 0) , 𝜎y = (0 -i i 0 ) , 𝜎z = (1 0 0 -1). The value of Tr(𝜎z [ 𝜎x, 𝜎y]) is

Differential equations

Which one of the following statements is true?

Differential equations

The Hamiltonian for a quantum particle of mass m is given below, where \omega < Ω. The Schrödinger equation for this system can be solved exactly using the orthogonal transformations: x = x1- x2 √2 and y = x1+ x2 √2 . H = - \hbar2 2m [ 𝜕2 𝜕x2 + 𝜕2 𝜕y2 ] + 1 2 mΩ2(x2 + y2) + m\omega2xy The ground state energy of this system is

Angular momentum & Hydrogen Atom

Consider two particles with angular momenta j1 = 2\hbar and j2 = \hbar/2. If the expression |j = 5/2,m = 3/2⟩ = {c1|j1 = 2, m1 = 1⟩|j2 = 1/2, m2 = 1/2⟩ + c2|j1 = 2, m1 = 2⟩|j2 = 1/2, m2 = -1/2⟩ gives an eigenstate of the total angular momentum of the two particles, using standard notation. Which of the following is true? (Hint: Ĵ±|j, m⟩ = √j(j + 1) - m (m ± 1) |j, m ± 1⟩)

Differential equations

The energy E and degeneracy d of the second excited state of a three-dimensional, isotropic quantum harmonic oscillator with angular frequency \omega are

Differential equations

Two identical particles with a fixed total energy E = 2\hbar\omega are in thermal equilibrium in a one-dimensional harmonic oscillator potential 1 2 m\omega2x2. Let the entropy of the particles be denoted by SF if they are fermions with spin 1 2 (sz = ± \hbar 2 ) and SB if they are bosons with spin 0. Then, which of the following options is correct? (kB is the Boltzmann constant)

Electrostatics and Magnetostatics

In the circuit shown, V(t) = 2 sin(2000𝜋t) Volts, where t is in seconds, which of the following options is correct? (take the opamp to be ideal)

Electrostatics and Magnetostatics

Considering the circuit and the associated signals measured at different pins (numbered as 1, 2, 3, 4, 5) shown in the figure, the correct option is

Classical Thermodynamics

A gas of N classical particles that can occupy energy levels, \epsilon1 and \epsilon2 = \epsilon1 + Δ is in equilibrium with a reservoir at temperature T. From the schematics shown below, choose the correct dependence of the internal energy U on T . (A) (B) (C) (D)

Lagrangian & Hamiltonian Dynamics

The Lagrangian L0 = 1 2 mq̇ 2 - 1 2 m\omega2q2 with the generalized coordinate q is transformed to L = L0 + \alpha df(q) dt . Consider the following statements: (i) Expression for the canonical momentum does not change. (ii) The equation of the motion does not change. Which of the following options is correct for the above statements?

Differential equations

An infinitely large thin sheet in the xy-plane carries uniform positive charge density and is moving with constant velocity v in the +x direction (see figure below). The direction of the corresponding Poynting vector is

Classical Thermodynamics

A positive point charge is fixed at the origin. At some distance from it on the x axis, a point dipole is kept pointing in the +y direction. The force on the dipole is

Small oscillations

Two frames S (solid lines) and S' (dashed lines) with common origin are shown in the figure below. Frame S is inertial while S' is rotating about the common z-axis. There is a point mass fixed at P on the x-axis of the S frame. The magnitude of the centrifugal force and the Coriolis force experienced by the mass in the S' frame is Fcen and Fcor, respectively. Which of the following options is correct for these forces?

Crystal Structure & Lattice Dynamics

Which of the following operators is/are self-adjoint?

Crystal Structure & Lattice Dynamics

Consider two operators  and B̂ which are related as  = exp(i𝜃B̂). If 𝜃 is a non-zero real number, which of the following statements is/are true?

Crystal Structure & Lattice Dynamics

Consider operators Â, B̂, and Ĉ for three observables of a quantum system satisfying [Â, B̂] = 0, [B̂, Ĉ] = 0, and [Â, Ĉ] ≠ 0, with uncertainties ΔA, ΔB, ΔC, respectively. From the options given below, which is/are implied by the commutation relations among Â, B̂, and Ĉ ?

Operational Amplifiers & Analog Circuits

Consider the distribution of outcomes generated by N (N ≫ 1) independent throws of (i) a coin or (ii) a six-sided dice. A coin (dice) is unbiased if both (all) its sides have equal probability to show up in a throw; it is biased otherwise. For the cases (i) and (ii) above, which of the following statements is/are true?

Operational Amplifiers & Analog Circuits

The dispersion (E(k)) of the conduction band (CB) and valence band (VB) for a semiconductor are shown schematically in the figure. Considering the possibility of an electron making a transition from the bottom of the CB to the top of the VB, which of the following options is/are correct?

Small oscillations

A symmetric rigid body has moment of inertia I1, I2, I3 about its principal axes 1, 2, and 3, respectively, with I1 = I3 = I⊥ and I2 ≠ I⊥. It is rotating in space with no torque on it so that its angular momentum L⃗ is constant. Let \omega1, \omega2, \omega3 be the components of its angular velocity along the principal axes 1, 2, and 3, respectively. Which of the following quantities is/are constant during the motion of this rigid body?

Central force problem

An α particle moves towards a fixed nucleus carrying charge Ze, with initial speed v0 and impact parameter b. Starting from a large distance from the nucleus, its distance of closest approach is rm and its speed there is vm. Then which of the following options is/are correct? (k = 1 / (4πε0) and r0 = kZe^2 / (m v0^2)) (A) v0 b = vm rm (B) v0 b = 2 vm r0 (C) For b/r0 ≪ 1, rm = 4 r0 + b^2 / (2 r0) ignoring higher order corrections in b/r0 (D) For b/r0 ≪ 1, rm = 4 r0 + b^2 / (8 r0) ignoring higher order corrections in b/r0

Nuclear Models & Decays

Consider a particle of mass m = 9.0 × 10-5 kg and charge q = 3.0 × 10-4 C in a uniform electromagnetic field E⃗ = 2 x̂ V. m-1, B⃗ = 3 ẑ V. m-2. s. The particle is released from the coordinates (0, 5 m , 0) at time t = 0. Starting from initial speed zero, it comes back to the y-axis for the first time at time t. The value of t in seconds (rounded off to two decimal places) is _____

Lasers and Quantum Optics

An atom has two energy levels with energy difference 2.2 eV between them. A gas of these atoms is with 8 × 1020 atoms in the upper state and 5 × 1020 atoms in the lower state. Ignoring spontaneous emission, the maximum possible energy released by this gas of atoms by stimulated emission is E Joules. The value of E (rounded off to one decimal place) is _____ (e = 1.6 × 10-19 C)

Nuclear Models & Decays

The Hamiltonian of two interacting spin-1/2 particles is H = A \hbar2 S 1 ⋅ S 2 , where S 1 and S 2 are the spin angular momenta of particles 1 and 2, respectively. Here, A = 10.56 eV. The energy in eV required to induce an excitation from the ground state to the excited state (rounded off to two decimal places) is _____

Semiconductor Devices & Diodes

The output signal (current Id) of a reversed biased (with a voltage Vb) photodiode, on which light is incident, is fed to an amplifier (see figure). The output voltage is digitized by a 10 bit Analogue to Digital convertor (ADC) which has a reference voltage of 5 V. The smallest current Idwhich can be measured by the circuit in nano-Amperes (rounded off to one decimal place) is _____

Differential equations

A capacitor is made of two circular metal plates of radius 1 m separated by a distance of d = 1 mm. The space between them is filled by a dielectric with permittivity \epsilonr = 5. The capacitor is connected to a voltage V = 10 sin(2𝜋 × 106 × t) volts, where t is in seconds. The maximum value of the magnetic field in between the plates at a radial distance r = 0.5 m from the centre of the capacitor is B × 10-6 T. The value of B (rounded off to two decimal places) is _____ (Speed of light in vacuum c = 3 × 108 m. s-1)

Small oscillations

Rotational spectrum of a diatomic molecule consists of lines of equal spacing with an interval of 20.0 cm-1. Its moment of inertia is found to be I0 × 10-47kg. m2, the value of I0 (rounded off to one decimal place) is _____ (ℎ = 6.6 × 10-34 J. s, speed of light in vacuum c = 3 × 108 m. s-1)

Differential equations

Copper has an electron number density of 8.3 × 1028 m-3. Its Fermi energy in eV (rounded off to one decimal place) is _____ (\hbar = 1.06 × 10-34 J. s, mass of electron me = 9.10 × 10-31 kg, charge of electron = 1.60 × 10-19 C)

Nuclear Models & Decays

Two 1 kg blocks are connected to two massless springs of spring constants 8 N. m-1 and 4 N. m-1. The system is kept on a frictionless horizontal floor with one end of a spring attached to a wall (see figure below). They are performing oscillatory motion along the x-axis with the normal mode frequencies \omegaH and \omegaL (\omegaH > \omegaL). The ratio \omegaH \omegaL (rounded off to two decimal places) is _____

Nuclear Models & Decays

A 15 cm long scale is held horizontally with one of its ends on the edge of a 1 m high table and the other end resting on one's index finger. As the finger is removed (see figure below), the scale starts rotating about its end on the table. After 0.1 s, during which it has rotated by a negligibly small angle but has gained a rotational speed as it leaves the table and falls vertically towards the ground. When its centre of mass has fallen by 0.5 m, it has rotated by an angle 𝜃. The value of 𝜃 in degrees (rounded off to one decimal place) is _____ (g = 9.8 m.s-2)