GATE PH 2022 Question Paper & Solutions

Verbal Aptitude

You should _________ when to say _________.

Quantitative Aptitude

Two straight lines pass through the origin (x_0, y_0) = (0,0). One of them passes through the point (x1, y1) = (1,3) and the other passes through the point (x2, y2) = (1,2). What is the area enclosed between the straight lines in the interval [0, 1] on the x-axis?

Quantitative Aptitude

If p ∶ q = 1 ∶ 2 q ∶ r = 4 ∶ 3 r ∶ s = 4 ∶ 5 and u is 50% more than s, what is the ratio p ∶ u?

Analytical & Spatial Aptitude

Given the statements:  P is the sister of Q.  Q is the husband of R.  R is the mother of S.  T is the husband of P. Based on the above information, T is ______ of S.

Analytical & Spatial Aptitude

In the following diagram, the point R is the center of the circle. The lines PQ and ZV are tangential to the circle. The relation among the areas of the squares, PXWR, RUVZ and SPQT is

Verbal Aptitude

Healthy eating is a critical component of healthy aging. When should one start eating healthy? It turns out that it is never too early. For example, babies who start eating healthy in the first year are more likely to have better overall health as they get older. Which one of the following is the CORRECT logical inference based on the information in the above passage?

Quantitative Aptitude

P invested ₹ 5000 per month for 6 months of a year and Q invested ₹ x per month for 8 months of the year in a partnership business. The profit is shared in proportion to the total investment made in that year. If at the end of that investment year, Q receives 4 of the total profit, what is the value of x (in ₹)?

Quantitative Aptitude

The above frequency chart shows the frequency distribution of marks obtained by a set of students in an exam. From the data presented above, which one of the following is CORRECT?

Quantitative Aptitude

In the square grid shown on the left, a person standing at P2 position is required to move to P5 position. 3 9 11 7 14 2 4 0 2 4 6 8 10 12 14 16 3 4 5 6 7 8 9 Frequency Marks The only movement allowed for a step involves, "two moves along one direction followed by one move in a perpendicular direction". The permissible directions for movement are shown as dotted arrows in the right. For example, a person at a given position Y can move only to the positions marked X on the right. Without occupying any of the shaded squares at the end of each step, the minimum number of steps required to go from P2 to P5 is 1 2 3 4 5 P Q R S T Example: Allowed steps for a person at Y

Verbal Aptitude

Consider a cube made by folding a single sheet of paper of appropriate shape. The interior faces of the cube are all blank. However, the exterior faces that are not visible in the above view may not be blank. Which one of the following represents a possible unfolding of the cube? (A) (B) (C) (D) END OF THE QUESTION PAPER

Linear Algebra & Vector Calculus

For the Op-Amp circuit shown below, choose the correct output waveform corresponding to the input Vin0 = 1.5 sin 20𝜋t (in Volts). The saturation voltage for this circuit is V01 = ±10 V. (A) (B) (C) (D)

Nuclear Models & Decays

Match the order of \beta - decays given in the left column to appropriate clause in the right column. Here X(I\pi) and Y(I\pi) are nuclei with intrinsic spin I and parity 𝜋. 1. X ( 1 2 + ) → Y ( 1 2 + ) i) First forbidden \beta-decay 2. X ( 1 2 - ) → Y ( 2 + ) ii) Second forbidden \beta-decay 3. X(3+) → Y(0+) iii) Third forbidden \beta-decay 4. X(4-) → Y(0+) iv) Allowed \beta-decay

Linear Algebra & Vector Calculus

What is the maximum number of free independent real parameters specifying an n-dimensional orthogonal matrix?

Angular momentum & Hydrogen Atom

An excited state of Ca atom is [Mg]3p54s23d1. The spectroscopic terms corresponding to the total orbital angular momentum are

Electrostatics and Magnetostatics

On the surface of a spherical shell enclosing a charge free region, the electrostatic potential values are as follows: One quarter of the area has potential 𝜙_0, another quarter has potential 2𝜙_0 and the rest has potential 4𝜙_0. The potential at the centre of the shell is (You can use a property of the solution of Laplace's equation.)

Electrostatics and Magnetostatics

A point charge q is performing simple harmonic oscillations of amplitude A at angular frequency \omega. Using Larmor's formula, the power radiated by the charge is proportional to

Classical Thermodynamics

Which of the following relationship between the internal energy U and the Helmholtz's free energy F is true?

Nuclear Models & Decays

If nucleons in a nucleus are considered to be confined in a three-dimensional cubical box, then the first four magic numbers are

Differential equations

Consider the ordinary differential equation $y'' - 2x y' + 4y = 0$ and its solution $y(x) = a + bx + cx^2$. Then

Operational Amplifiers & Analog Circuits

For an Op-Amp based negative feedback, non-inverting amplifier, which of the following statements are true?

Angular momentum & Hydrogen Atom

From the pairs of operators given below, identify the ones which commute. Here l and j correspond to the orbital angular momentum and the total angular momentum, respectively.

Electromagnetic Waves in Media

For normal Zeeman lines observed ∥ and ⊥ to the magnetic field applied to an atom, which of the following statements are true?

Angular momentum & Hydrogen Atom

Pauli spin matrices satisfy (A) σα σβ - σβ σα = i εαβγ σγ (B) σα σβ - σβ σα = 2i εαβγ σγ (C) σα σβ + σβ σα = εαβγ σγ (D) σα σβ + σβ σα = 2 δαβ

Complex analysis

For the refractive index $n = n_r(\omega) + i n_{im}(\omega)$ of a material, which of the following statements are correct?

Complex analysis

Complex function $f(z) = z + |z - a|^2$ ($a$ is a real number) is

Fourier and Laplace transforms

If g(k) is the Fourier transform of f(x), then which of the following are true?

Differential equations

The ordinary differential equation $(1 - x^2) y'' - x y' + 9y = 0$ has a regular singularity at

Semiconductor Devices & Diodes

For a bipolar junction transistor, which of the following statements are true?

Band Theory of Solids & Semiconductors

Potassium metal has electron concentration of 1.4 × 1028m-3 and the corresponding density of states at Fermi level is 6.2 × 1048 Joule-1 m-3. If the Pauli paramagnetic susceptibility of Potassium is n × 10-L in standard scientific form, then the value of k (an integer) is __________ (Magnetic moment of electron is 9. 3 × 10-24 Joule T-1; permeability of free space is 4𝜋 × 10- T m A-1)

Linear Algebra & Vector Calculus

A power supply has internal resistance RS and open load voltage VS = 5 V. When a load resistance R is connected to the power supply, a voltage drop of V = 4 V is measured across the load. The value of is __________ (Round off to the nearest integer)

Electrostatics and Magnetostatics

Electric field is measured along the axis of a uniformly charged disc of radius 25 cm. At a distance d from the centre, the field differs by 10% from that of an infinite plane having the same charge density. The value of d is _______cm. (Round off to one decimal place)

Molecular Spectroscopy

In a solid, a Raman line observed at 300 cm-1 has intensity of Stokes line four times that of the anti-Stokes line. The temperature of the sample is _______K. (Round off to the nearest integer) (1 cm-1 ≡ 1.44 K)

Schrodinger equation & Wave Mechanics

An electromagnetic pulse has a pulse width of 10-3 s. The uncertainty in the momentum of the corresponding photon is of the order of 10- kg m s-1, where N is an integer. The value of N is ____________ (speed of light = 3 × 108 m s-1, ℎ = 6.6 × 10-^34 J s)

Small oscillations

The wavefunction of a particle in a one-dimensional infinite well of size 2 a at a certain time is 𝜓(x) = 1 √8 √2sin ( \pi ) +√3cos ( \pi 2) + cos ( ^3\pi 2 ). Probability of finding the particle in n = 2 state at that time is ______% (Round off to the nearest integer)

Electromagnetic Waves in Media

A spectrometer is used to detect plasma oscillations in a sample. The spectrometer can work in the range of 3 × 1012 rad s-1 to 30 × 1012 rad s-1. The minimum carrier concentration that can be detected by using this spectrometer is n × 1021 m-. The value of n is _____________ (Round off to two decimal places) (Charge of an electron = -1.6 × 10-1 C, mass of an electron = 9.1 × 10-^31 kg and \epsilon_0 = 8.85 × 10-12 C2 N-1 m-2)

Electrostatics and Magnetostatics

Consider a non-interacting gas of spin 1 particles, each with magnetic moment 𝜇, placed in a weak magnetic field B, such that L ≪ 1. The average magnetic moment of a particle is

Classical Thermodynamics

Water at 300 K can be brought to 320 K using one of the following processes. Process 1: Water is brought in equilibrium with a reservoir at 320 K directly. Process 2: Water is first brought in equilibrium with a reservoir at 310 K and then with the reservoir at 320 K. Process 3: Water is first brought in equilibrium with a reservoir at 350 K and then with the reservoir at 320 K. The corresponding changes in the entropy of the universe for these processes are ∆S_1, ∆S_2 and ∆S_3, respectively. Then

Lasers and Quantum Optics

A student sets up Young's double slit experiment with electrons of momentum p incident normally on the slits of width w separated by distance d. In order to observe interference fringes on a screen at a distance D from the slits, which of the following conditions should be satisfied?

Schrodinger equation & Wave Mechanics

Consider a particle in three different boxes of width L. The potential inside the boxes vary as shown in figures (i), (ii) and (iii) with V_0 ≪ ℏ^2\pi^2 _2^2 . The corresponding ground-state energies of the particle are E_1, E_2 and E_3, respectively. Then

Linear Algebra & Vector Calculus

In cylindrical coordinates (s, 𝜑, z), which of the following is a Hermitian operator?

Lagrangian & Hamiltonian Dynamics

A particle of mass 1 kg is released from a height of 1 m above the ground. When it reaches the ground, what is the value of Hamilton's action for this motion in J s? (g is the acceleration due to gravity; take gravitation potential to be zero on the ground)

Lagrangian & Hamiltonian Dynamics

If (x ̇ ẏ + 𝛼xy) is a constant of motion of a two-dimensional isotropic harmonic oscillator with Lagrangian L = m(ẋ_2 + ẏ_2) 2 - k(x_2 + y_2) 2 then 𝛼 is

Electromagnetic Waves in Media

In a two-dimensional square lattice, frequency \omega of phonons in the long wavelength limit changes linearly with the wave vector k. Then the density of states of phonons is proportional to

Dielectric, Magnetic Properties & Superconductivity

At T = 0 K, which of the following diagram represents the occupation probability P(E) of energy states of electrons in a BCS type superconductor? (A) (B) (C) (D) P(E) E EF 1 P(E) E EF 1 P(E) E EF 1 P(E) E EF 1

Harmonic oscillator

For a one-dimensional harmonic oscillator, the creation operator $a^\dagger$ acting on the $n$-th state $|\psi_n\rangle$ gives $a^\dagger |\psi_n\rangle = \sqrt{n+1} |\psi_{n+1}\rangle$. The position operator $x$ in the basis of the first three states is proportional to

Small oscillations

A piston of mass m is fitted to an airtight horizontal cylindrical jar. The cylinder and piston have identical unit area of cross-section. The gas inside the jar has volume V and is held at pressure P = P10. The piston is pushed inside the jar very slowly over a small distance. On releasing, the piston performs an undamped simple harmonic motion of low frequency. Assuming that the gas is ideal and no heat is exchanged with the atmosphere, the frequency of the small oscillations is proportional to

Classical Thermodynamics

A paramagnetic salt of mass m is held at temperature T in a magnetic field H. If S is the entropy of the salt and M is its magnetization, then dG = -SdT - MdH, where G is the Gibbs free energy. If the magnetic field is changed adiabatically by ΔH → 0 and the corresponding infinitesimal changes in entropy and temperature are ΔS and ΔT, then which of the following statements are correct

Schrodinger equation & Wave Mechanics

A particle of mass m is moving inside a hollow spherical shell of radius a so that the potential is V(r) = 0 for r < a ∞ for r ≥ a The ground state energy and wavefunction of the particle are E_0 and R(r), respectively. Then which of the following options are correct?

Small oscillations

A particle of unit mass moves in a potential V(r) = -V_0e-^2 . If the angular momentum of the particle is L = 0.5ඥV_0 , then which of the following statements are true?

Small oscillations

In a diatomic molecule of mass M, electronic, rotational and vibrational energy scales are of magnitude E, E and EV, respectively. The spring constant for the vibrational energy is determined by E. If the electron mass is m then

Bose-Einstein and Fermi-Dirac statistics

Electronic specific heat of a solid at temperature T is C = 𝛾T, where 𝛾 is a constant related to the thermal effective mass (m) of the electrons. Then which of the following statements are correct?

Band Theory of Solids & Semiconductors

In a Hall effect experiment on an intrinsic semiconductor, which of the following statements are correct?

Electrostatics and Magnetostatics

A parallel plate capacitor with spacing $d$ and area of cross-section $A$ is connected to a source of voltage $V$. If the plates are pulled apart quasistatically to a spacing of $2d$, then which of the following statements are correct?

Lagrangian & Hamiltonian Dynamics

A system with time independent Hamiltonian $H(q, p)$ has two constants of motion $f(q, p)$ and $g(q, p)$. Then which of the following Poisson brackets are always zero?

Lagrangian & Hamiltonian Dynamics

In the action-angle variables (I1, I2, 𝜃1, 𝜃2), consider the Hamiltonian H = 4I1I2 and 0 ≤ 𝜃1, 𝜃2 < 2𝜋. Let ^2 = 1 2 . Which of the following are possible plots of the trajectories with different initial conditions in 𝜃1-𝜃2 plane? (A) (B) (C) (D)

Linear Algebra & Vector Calculus

A particle of mass m in the x-y plane is confined in an infinite two-dimensional well with vertices at (0, 0), (0, L), (L, L), (L, 0). The eigenfunctions of this particle are 𝜓0,0 = _2 sin ( 0\pi ) sin ( 0\pi ) . If perturbation of the form V = Cxy, where C is a real constant, is applied, then which of the following statements are correct for the first excited state?

Electrostatics and Magnetostatics

A junction is formed between a metal on the left and an n-type semiconductor on the right. Before forming the junction, the Fermi level EF of the metal lies below that of the semiconductor. Then which of the following schematics are correct for the bands and the I-V characteristics of the junction? (A) (B) (C) (D) EF EF Conduction Band Valence Band Metal n-type semiconductor EF Conduction Band EF Valence Band Metal n-type semiconductor I V -V0 I V -V0

Electromagnetic Waves in Media

A plane polarized electromagnetic wave propagating in y-z plane is incident at the interface of two media at Brewster's angle. Taking z = 0 as the boundary between the two media, the electric field of the reflected wave is given by E\vec⃗ = A cos k_0 √3 2 y - 1 2 z - 𝜔t x ො then which among the following statements are correct?

Digital Electronics

The minimum number of two-input NAND gates required to implement the following Boolean expression is ________________ Y = [AB(C + BD) + A̅B]C

Semiconductor Devices & Diodes

In a nucleus, the interaction V0l⃗ ⋅ s⃗ is responsible for creating spin-orbit doublets. The energy difference between p1/2 and p^3/2 states in units of V0 \hbar^2 2 is ____________ (Round off to the nearest integer)

Special theory of relativity

Two identical particles of rest mass m_0 approach each other with equal and opposite velocity v = 0.5c, where c is the speed of light. The total energy of one particle as measured in the rest frame of the other is E = \alpham_0c2. The value of \alpha is _______________ (Round off to two decimal places)

Crystal Structure & Lattice Dynamics

In an X-Ray diffraction experiment on a solid with FCC structure, five diffraction peaks corresponding to (111), (200), (220), (311) and (222) planes are observed using 1.54 Å X-rays. On using 3 Å X-rays on the same solid, the number of observed peaks will be ________________

Classical Thermodynamics

For 1 mole of Nitrogen gas, the ratio $\left(\frac{\Delta S_I}{\Delta S_{II}}\right)$ of entropy change of the gas in processes (I) and (II) is (in integer) _

Lasers and Quantum Optics

Frequency bandwidth $\Delta \nu$ of a gas laser of frequency $\nu$ Hz is $\Delta \nu = \frac{2\nu}{c} \sqrt{\frac{\alpha}{A}}$, where $\alpha = 3.44 \times 10^6 \text{ m}^2 \text{s}^{-2}$. The value of $\Delta \nu$ in kHz is _

Electrostatics and Magnetostatics

A current of 1 A is flowing through a very long solenoid made of winding density 3000 turns/m. As shown in the figure, a parallel plate capacitor, with plates oriented parallel to the solenoid axis and carrying surface charge density 6\epsilon_0 C m-2, is placed at the middle of the solenoid. The momentum density of the electromagnetic field at the midpoint X of the capacitor is n × 10-1^3 N s m-3. The value of n is ______________ (Round off to the nearest integer) (speed of light c = 3 × 108 m s-1)