GATE PH 2020 Question Paper & Solutions

Verbal Aptitude

He is known for his unscrupulous ways. He always sheds _______ tears to deceive people.

Verbal Aptitude

Jofra Archer, the England fast bowler, is _______ than accurate.

Verbal Aptitude

Select the word that fits the analogy: Build : Building :: Grow : _______

Verbal Aptitude

I do not think you know the case well enough to have opinions. Having said that, I agree with your other point. What does the phrase "having said that" mean in the given text?

Quantitative Aptitude

Define [x] as the greatest integer less than or equal to x, for each x ∈ (-∞, ∞). If y = [x], then area under y for x ∈ [1, 4] is (A) 1 (B) 3 (C) 4 (D) 6

Verbal Aptitude

Crowd funding deals with mobilisation of funds for a project from a large number of people who would be willing to invest smaller amounts through web-based platforms for the project. Based on the above paragraph, which of the following is correct about crowd funding?

Quantitative Aptitude

P, Q, R and S are to be uniquely coded using α and β. If P is coded as αα and Q as αβ, then R and S, respectively, can be coded as (A) βα and αβ (B) ββ and αα (C) αβ and ββ (D) βα and ββ

Analytical & Spatial Aptitude

The sum of the first n terms in the sequence 8, 88, 888, 8888, ... is (A) 80/81 (10^n - 1) + 8/9 n (B) 80/81 (10^n - 1) - n (C) 80/81 (10^n - 1) + 8n (D) 80/81 (10^n - 1) - 8/9 n

Quantitative Aptitude

Select the graph that schematically represents BOTH y = x^m and y = x^(1/m) properly in the interval 0 ≤ x ≤ 1, for integer values of m, where m > 1.

Quantitative Aptitude

The bar graph shows the data of the students who appeared and passed in an examination for four schools P, Q, R and S. The average of success rates (in percentage) of these four schools is: - School P: Appeared = 500, Passed = 280 - School Q: Appeared = 600, Passed = 330 - School R: Appeared = 700, Passed = 455 - School S: Appeared = 400, Passed = 240 (A) 58.5 % (B) 58.8 % (C) 59.0 % (D) 59.3 %

Complex analysis

Which one of the following is a solution of du(x) 2= ku(x), for k real? (A) (B) (0) (D) e kx sinkx coskx sinh x

Linear Algebra & Vector Calculus

A real, invertible 3x3 matrix M has eigenvalues 2,, (i =1,2,3) and the corresponding eigenvectors are |e;), (i=1,2,3) respectively. Which one of the following is correct? M/e) le), for i =1,2,3 (A) (B) (C) (D) M-'le,) = 1, e;), for i = 1,2,3 The eigenvalues of M and M-' are not related.

Electrostatics and Magnetostatics

A quantum particle is subjected to the potential 00, x5-2 00, x2% The ground state wave function of the particle is proportional to (A) (B) (C) (D) sin( *) cos za) cos

Linear Algebra & Vector Calculus

Let â and ât, respectively denote the lowering and raising operators of a one- dimensional simple harmonic oscillator. Let n) be the energy eigenstate of the simple harmonic oscillator. Given that n) is also an eigenstate of a aãâ, the corresponding eigenvalue is n (n -1) n (n+1) (n+1)} (A) (B) (C) (D)

Complex analysis

He is known for his unscrupulous ways. He always sheds_ tears to deceive people. fox's crocodile's crocodile fox Jofra Archer, the England fast bowler, is_ more fast faster less fast more faster than accurate. Select the word that fits the analogy: Build : Building :: Grow : _ Grown Grew Growth Growed I do not think you know the case well enough to have opinions. Having said that, I agree with your other point. What does the phrase "having said that" mean in the given text? as opposed to what I have said despite what I have said in addition to what I have said contrary to what I have said Define [x] as the greatest integer less than or equal to x, for each x € (-00, 00). If y = [x], then area under y for x € [1,4] is 1 3 4 6 Q6 - Q10 carry two marks each.

Complex analysis

owd funding deals with mobilisation of funds for a project from a large number onle who would be willing to invest smaller amounts through web-based olatforms the project. (A) (B) Based on the above paragraph, which of the following is correct about crowd funding? Funds raised through unwilling contributions on web-based platforms. Funds raised through large contributions on web-based platforms. (C) (D) (A) (B) (C) (D)

Lagrangian & Hamiltonian Dynamics

(A) (B) (0) (D) Which one of the following is a universal logic gate? AND NOT OR NAND Which one of the following is the correct binary equivalent of the hexadecimal F6C? 0110 1111 1100 1111 0110 1100 1100 0110 1111 0110 1100 0111 The total angular momentum j of the ground state of the 1?O nucleus is 1 1

Complex analysis

(A) (B) (C) (D) Funds raised through coerced contributions on web-based platforms. Funds raised through voluntary contributions on web-based platforms. P, Q. R and S are to be uniquely coded using a and B. If P is coded as aa and Q as aß, then R and S, respectively, can be coded as Ba and aß ßß and aa aß and BB Ba and BB The sum of the first n terms in the sequence 8, 88, 888, 8888,.. is so (10" - 1) + 3n 81 (10" - 1) - n 80 81 (10 - 1) + 9n 80 - (10" - 1) - 81 Select the graph that schematically represents BOTH y = x™ and y = x/m properly in the interval 0 ≤ x ≤ 1, for integer values of m, where m > 1. y 1 xI/m xm xm x x1/m 1 xI/m xm 1

Complex analysis

Question 9 of Section Physics, GATE PH 2020.

Electrostatics and Magnetostatics

A particle X is produced in the process it + p→K*+ X via the strong interaction. If the quark content of the K* is us, the quark content of X is cs und uuS ud A medium (Er > 1, M, = 1,0 > 0) is semi-transparent to an electromagnetic wave when Conduction current » Displacement current Conduction current « Displacement current Conduction current = Displacement current Both Conduction current and Displacement current are zero A particle is moving in a central force field given by F =-. "si, where i is the unit vector pointing away from the center of the field. The potential energy of the particle is given by (A) (B) (C) (D) 202 242

Electrostatics and Magnetostatics

Choose the correct statement related to the Fermi energy (Ep) and the chemical potential (M) of a metal. (A) (B) (C) (D) # = Ep only at 0 K M = Ep at finite temperature M< EpatoK M> Ep at finite temperature

Complex analysis

Consider a diatomic molecule formed by identical atoms. If Ey and Ee represent the energy of the vibrational nuclear motion and electronic motion respectively, then in terms of the electronic mass m and nuclear mass M, Ee Ey is proportional to (A) (B) (C) (D) m M (*"

Electrostatics and Magnetostatics

Which one of the following relations determines the manner in which the electric field lines are refracted across the interface between two dielectric media having dielectric constants &1 and E2 (see figure)? 01 82 102 (A) (B) (C) (D) & sin Q, = &z sin 02 &, tan0, = E, tan 0, 8; cotO, = 8, cotO, O.No. 14 If E and B are the electric and magnetic fields respectively, then E. B is (A) (B) (C) (D) odd under parity and even under time reversal even under parity and odd under time reversal odd under parity and odd under time reversal even under parity and even under time reversal

Complex analysis

Question 14 of Section Physics, GATE PH 2020.

Lagrangian & Hamiltonian Dynamics

A small disc is suspended by a fiber such that it is free to rotate about the fiber axis (see figure). For small angular deflections, the Hamiltonian for the disc is given by where / is the moment of inertia and a is the restoring torque per unit deflection. The disc is subjected to angular deflections (0) due to thermal collisions from the surrounding gas at temperature T and Po is the momentum conjugate to 0. The average and the root-mean-square angular deflection, avg and Ormss respectively are (A) Puny 0 and em = kat) (B) (C) (0) Pavg # 0 and Orms =

Statistical Ensembles & Partition Functions

As shown in the figure, an ideal gas is confined to chamber A of an insulated container, with vacuum in chamber B. When the plug in the wall separating the chambers A and B is removed, the gas fills both the chambers. Which one of the following statements is true? Plug (A) (B) (C) (D)

Complex analysis

Question 17 of Section Physics, GATE PH 2020.

Lagrangian & Hamiltonian Dynamics

B The temperature of the gas remains unchanged Internal energy of the gas decreases Temperature of the gas decreases as it expands to fill the space in chamber B Internal energy of the gas increases as its atoms have more space to move around Particle A with angular momentum J = decays into two particles B and C with angular momenta j, and / respectively. If 1f ® ®(22)c) the value of a is Far from the Earth, the Earth's magnetic field can be approximated as due to a bar magnet of magnetic pole strength 4x10'4 Am. Assume this magnetic field is generated by a current carrying loop encircling the magnetic equator. The current required to do so is about 4x10" A, where n is an integer. The value of n is

Crystal Structure & Lattice Dynamics

(Earth's circumference: 4x10' m) The number of distinct ways the primitive unit cell can be constructed for the two dimensional lattice as shown in the figure is _ -

Lagrangian & Hamiltonian Dynamics

A hydrogenic atom is subjected to a strong magnetic field. In the absence of spin- orbit coupling, the number of doubly degenerate states created out of the d-level is

Nuclear Models & Decays

A particle Y undergoes strong decay Y → IT + T . The isospin of Y is

Linear Algebra & Vector Calculus

For a complex variable z and the contour c: z = 1 taken in the counter clockwise 2 direction, + 3 dz =

Complex analysis

Let p be the momentum conjugate to the generalized coordinate q. If the transformation Q=V2q''' cos p P=Sa''' sin p is canonical, then m =

Electrostatics and Magnetostatics

A conducting sphere of radius 1 m is placed in air. The maximum number of electrons that can be put on the sphere to avoid electrical breakdown is about 7x10", where n is an integer. The value of n is Assume: Breakdown electric field strength in air is E = 3x10° V/m Permittivity of free space 8, =8.85x10-12 F/m Electron charge e = 1.60x10-19 C If a particle is moving along a sinusoidal curve, the number of degrees of freedom of the particle is Q26 - Q55 carry two marks each.

Complex analysis

Question 25 of Section Physics, GATE PH 2020.

Linear Algebra & Vector Calculus

The product of eigenvalues of 0 0 0 - 0 1) 0 0, (A) (B) THON (A) (8) (D) the three-dimensional real vector space. Which one of the following is correct? S is an orthonormal set S is a linearly dependent set S is a basis for R' 0 Elevel- 0 0

Complex analysis

Question 27 of Section Physics, GATE PH 2020.

Complex analysis

Question 28 of Section Physics, GATE PH 2020.

Complex analysis

R +12V Vin \cdotVout R -12V C (A) (B) (0) (D)

Schrodinger equation & Wave Mechanics

S, denotes the spin operator defined as S = (i d Which one of the following is correct? The eigenstates of spin operator 5, are /1% 6) and (4),= (;) The eigensates of spin operator 5, are /7% = (-1) and 7% - (;) 1(1 In the spin state upon the measurement of Sx, the probability for obtaining 77x 1 is 4 In the spin state 6(3) V3)' upon the measurement of Sr, the probability for obtaining 7), is = 2+V3 The input voltage (Vin) to the circuit shown in the figure is 2 cos(100t) V. The output voltage (Vou) is 2 cos(1001-2) V. If R = 1 k2, the value of C (in uF) is 0.1 1 10 100 (A) (B) (C) (D)

Complex analysis

(A) (8) (C) (D)

Lagrangian & Hamiltonian Dynamics

Consider a 4-bit counter constructed out of four flip-flops. It is formed by connecting the J and K inputs to logic high and feeding the Q output to the clock input of the following flip-flop (see the figure). The input signal to the counter is a series of square pulses and the change of state is triggered by the falling edge. At time t=t, the outputs are in logic low state (Qo = Q, = Q2 = Q3 = 0). Then at t=t,, the logic state of the outputs is Input 1 \cdot (logic high) 4-bit ripple counter Input signal Qo = 1, QI = 0, Q2 = 0 and Q3 = 0 Qo = 0, Q1 = 0, Q2 = 0 and Q3 = 1 Qo = 1, QI = 0, Q2 = 1 and Q3 = 0 Q0 = 0, QI = 1, Q2 = 1 and Q3 = 1 Consider the agranian = t + (g +exy, where a, b and c are constants. If P, and P, are the momenta conjugate to the coordinates x and y respectively, then the Hamiltonian is 2 -- cxy 4a 2 2 Px + 2a Py 26 - - cxy Pỉ + Pỉ + cxy 26 Pỉ + P, 2 - + cxy Which one of the following matrices does NOT represent a proper rotation in a plane? -sin o cos o - cos d -sin 0) (coso sin 0 -sin A cosA sin o - cos A -sin ® - cos 0

Electrostatics and Magnetostatics

A uniform magnetic field B = Boy exists in an inertial frame K. A perfect conducting sphere moves with a constant velocity V = Yo with respect to this inertial frame. The rest frame of the sphere is K' (see figure). The electric and magnetic fields in K and K' are related as El=rE,+#xB) Bf = B The induced surface charge density on the sphere (to the lowest order in v/c) in the frame K'is (A) (B) (C) (D)

Complex analysis

maximum along Z' maximum along y' maximum along x' uniform over the sphere A charge q moving with uniform speed enters a cylindrical region in free space at t = 0 and exits the region at † = T (see figure). Which one of the following options best describes the time dependence of the total electric flux @(t), through the entire surface of the cylinder? (A) (B) ф(t) t = 0 t=- (C) t = 0 t = T (D) t= 0 t= T

Complex analysis

Consider a one-dimensional non-magnetic crystal with one atom per unit cell. Assume that the valence electrons i) do not interact with each other and ii) interact weakly with the ions. If n is the number of valence electrons per unit cell, then at 0 K, (A) (B) (C) (D) the crystal is metallic for any value of n the crystal is non-metallic for any value of n the crystal is metallic for even values of n the crystal is metallic for odd values of n

Statistical Ensembles & Partition Functions

According to the Fermi gas model of the nucleus, the nucleons move in a spherical volume of radius R (= R, 4?, where A is the mass number and R, is an empirical constant with the dimensions of length). The Fermi energy of the nucleus E, is proportional to (A) (B) (C) 1 Ro 1 (D)

Crystal Structure & Lattice Dynamics

Consider a two dimensional crystal with 3 atoms in the basis. The number of allowed optical branches (n) and acoustic branches (m) due to the lattice vibrations are (A) (B) (C) (D) (n, m) = (2,4) (n, m) = (3, 3) (n, m) = (4, 2) (n, m) = (1, 5)

Statistical Ensembles & Partition Functions

(A) (B) The internal energy U of a system is given by U(S,V) = 2V-2/3g2 , where 2 is a constant of appropriate dimensions; V and S denote the volume and entropy, respectively. Which one of the following gives the correct equation of state of the system? PV1/3 T2 = constant PV 71/3 = constant (0) (D) VIST - = constant PV2/3 - = constant T

Electrostatics and Magnetostatics

The potential energy of a particle of mass m is given by U(x) = asin(k}x - 7/2), a>0, k° >0. The angular frequency of small oscillations of the particle about x = 0 is (A) k2. 2a m (B) k2. m (C) (D) V2m 21: I

Electrostatics and Magnetostatics

The radial wave function of a particle in a central potential is given by R(r) = 1" exp Za , where A is the normalization constant and a is positive constant of suitable dimensions. If ya is the most probable distance of the particle from the force center, the value of y is

Electrostatics and Magnetostatics

A free particle of mass M is located in a three-dimensional cubic potential well with impenetrable walls. The degeneracy of the fifth excited state of the particle is

Complex analysis

Consider the circuit given in the figure. Let the forward voltage drop across each diode be 0.7 V. The current I (in mA) through the resistor is -+10.1 V 1 k5 > LAHA

Linear Algebra & Vector Calculus

Let u" denote the 4-velocity of a relativistic particle whose square U"U, = 1. If E upo is the Levi-Civita tensor then the value of Eupol"u' uPu" is _

Complex analysis

Question 44 of Section Physics, GATE PH 2020.

Complex analysis

Question 45 of Section Physics, GATE PH 2020.

Crystal Structure & Lattice Dynamics

Consider a simple cubic monoatomic Bravais lattice which has a basis with vectors r; = 0, 12 = (* + + 2), a is the lattice parameter. The Bragg reflection is observed due to the change in the wave vector between the incident and the scattered beam as given by K = n,G, + nGz + nGz, where Gi, G, and G, are primitive reciprocal lattice vectors. For n, = 3, n, = 3 and n, = 2, the geometrical structure factor is A plane electromagnetic wave of wavelength 2 is incident on a circular loop of conducting wire. The loop radius is a (a < 2). The angle (in degrees), made by the Poynting vector with the normal to the plane of the loop to generate a maximum induced electrical signal, is An electron in a hydrogen atom is in the state n=3,1 = 2, m =-2. Let L, denote the y-component of the orbital angular momentum operator. If (AL,)} = ah?, the value of a is

Maxwell's Equations & Conservation Laws

A sinusoidal voltage of the form V(t)=V, cos(ct) is applied across a parallel plate capacitor placed in vacuum. Ignoring the edge effects, the induced emf within the region between the capacitor plates can be expressed as a power series in c. The lowest non-vanishing exponent in co is

Complex analysis

If x=$ a, sinkx, for -n ≤x≤t, the value of a, is _

Complex analysis

0, Let f(x) =<n, 0, x<- 1 2n <X< 2n 2n 2n <x.

Linear Algebra & Vector Calculus

The value of lit, I (x) sin. dr is._ Consider the Hamiltonian A = , + Al' where 0 0 i, = and ' is the time independent perturbation given by E) Ê' = k , where k > 0. If, the maximum energy eigenvalue of  is 3 eV 0 k 0) corresponding to E = 2 eV, the value of k (rounded off to three decimal places) in eV

Complex analysis

Question 51 of Section Physics, GATE PH 2020.

Complex analysis

Question 52 of Section Physics, GATE PH 2020.

Complex analysis

Question 53 of Section Physics, GATE PH 2020.

Lagrangian & Hamiltonian Dynamics

A hydrogen atom is in an orbital angular momentum state L,m =l). If L lies on a cone which makes a half angle 30° with respect to the z-axis, the value of / is In the center of mass frame, two protons each having energy 7000 GeV, collide to produce protons and anti-protons. The maximum number of anti-protons produced is (Assume the proton mass to be 1 GeV/c?) Consider a gas of hydrogen atoms in the atmosphere of the Sun where the temperature is 5800 K. If a sample from this atmosphere contains 6.023×1023 of hydrogen atoms in the ground state, the number of hydrogen atoms in the first excited state is approximately 8x10", where n is an integer. The value of n is (Boltzmann constant: 8.617×10-5 eV/K) For a gas of non-interacting particles, the probability that a particle has a speed v in the interval v to v+ dv is given by f(v)dv= 4rv}dv If E is the energy of a particle, then the maximum in the corresponding energy distribution in units of E/kgT occurs at (rounded off to one decimal place). The Planck's onergy density disibion is iven by 4(a) eaT At long wavelengths, the energy density of photons in thermal equilibrium with a cavity at temperature T varies as Ta, where o is Copyright: GATE 2020, IT Delhi

Complex analysis

Question 55 of Section Physics, GATE PH 2020.