PYQ 2020 - VedPrep

GATE PHYSICS 2020
Previous Year Question Paper with Solution.

1. Which one of the following is a solution of , for k real?

(a) e^–kx

(b) sin kx

(d) sinh x

Ans. (a)

Sol.

Correct option is (a)

2. A real, invertible 3 × 3 matrix M has eigenvalues , (i = 1,2,3) and the corresponding eigenvectors are , (i = 1,2,3) respectively. Which one of the following is correct?

(a)

(b)

(c)

(d) the eigenvalues of M and M^–1 are not related.

Ans. (b)

Sol. Correct option is (b).

3. A quantum particle is subjected to the potential

The ground state wave function of the particle is proportional to

(a)

(b)

(c)

(d)

Ans. (d)

Sol. Ground state has even parity so

Correct option is (d)

4. Let and , respectively denote the lowering and raising operators of a one-dimensional simple harmonic oscillator. Let be the energy eigenstate of the simple harmonic oscillator. Given that is also an eigenstate of , the corresponding eigenvalue is

(a) n(n – 1)

(b) n(n + 1)

(d) n²

Ans. (a)

Sol.

Correct option is (a)

5. Which one of the following is a universal logic gate?

(a) AND

(b) NOT

(d) NAND

Ans. (d)

Sol. Correct option is (d)

6. Which one of the following is the correct binary equivalent of the hexadecimal F6C?

(a) 0110 1111 1100

(b) 1111 0110 1100

(d) 0110 1100 0111

Ans. (b)

Sol.

Correct option is (b)

7. The total angular momentum j of the ground state of the nucleus is

(a)

(b) 1

(c)

(d)

Ans. (d)

Sol.

For N = 9 : (1s_1/2)²(1p_3/2)⁴(1p_1/2)²(1d_5/2)¹

The angular momentum is

Correct option is (d)

8. A particle X is produced in the process via the strong interaction. If the quark content of the K⁺ is , the quark content of X is

(a)

(b) uud

(d)

Ans. (c)

Sol. Lets first identify the particle X

Thus the particle X is . The quark content of is uus.

Correct option is (c)

9. A medium ( > 1, µ_r = 1, σ > 0) is semi-transparent to an electromagnetic wave when

(a) Conduction current >> Displacement current

(b) Conduction current << Displacement current

(d) Both Conduction current and Displacement current are zero

Ans. (b)

Sol. Conduction current

Displacement current

For semi-transparent mekdium i.e for poor conductor

Let

Correct option is (b)

10. A particle is moving in a central force field given by , where 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)

Ans. (d)

Sol.

Correct option is (d)

11. Choose the correct statement related to the Fermi energy (E_F) and the chemical potential (µ) of a metal.

(a) µ = E_F only at 0 K

(b) µ = E_F at finite temperature

(d) µ > E_F at finite temperature

Ans. (a)

Sol. In metal the Fermi energy (E_F) is also known as chemical potential (µ) at absolute zero.

Correct option is (a)

12. Consider a diatomic molecule formed by identical atoms. If E_V and E_e represent the energy of the vibrational nuclear motion and electronic motion respectively, then in terms of the electronic mass m and nuclear mass M, is proportional to

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Correct option is (a)

13. 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 and (see figure)?

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

Correct option is (d)

14. If and are the electric and magnetic fields respectively, then is

(a) odd under parity and even under time reversal

(b) even under parity and odd under time reversal

(d) even under parity and even under time reversal

Ans. (c)

Sol. The electric field changes sign under P-operation as r changes its sign and does not under T-operation as r does not

The magnetic field B = i × r does not changes its sign under p-operation as i –i and r –r under P-operation but under T-operation changes its sign because i –i but r r.

Thus changes its sign under P-operation as well as under T-operation.

Correct option is (c)

15. 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 I is the moment of inertia and is the restoring torque per unit deflection. The disc is subjected to angular deflections due to thermal collisions from the surrounding gas at temperature T and p_θ is the momentum conjugate to . The average and the root-mean-square angular deflection, _avg and _rms, respectively are

(a)

(b)

(c)

(d)

Ans. (b)

Sol. Correct option is (b)

16. 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?

(a) The temperature of the gas remains unchanged

(b) Internal energy of the gas decreases

(d) Internal energy of the gas increases as its atoms have more space to move around

Ans. (a)

Sol. Free expansion case, temperature remains unchanged.

Correct option is (a)

17. Particle A with angular momentum decays into two particles B and C with angular momenta j₁ and j₂, respectively. If , the value of is ______.

Ans. 1

Sol. Correct answer is (1)

18. Far from the Earth, the Earth's magnetic field can be approximated as due to a bar magnet of magnetic pole strength 4×10¹⁴ Am. Assume this magnetic field is generated by a current carrying loop encircling the magnetic equator. the current required to do so is about 4×10ⁿ A, where n is an integer. The value of n is _______.

(Earth's circumference: 4 × 10⁷ m)

Ans. 7

Sol.

19. The number of distinct ways the primitive unit cell can be constructed for the two dimensional lattice as shown in the figure is _________.

Ans. 5

Sol.

Correct answer is (5)

20. 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 ______.

Ans. 3

Sol. Number of Zeeman levels in strong field can be found from

E = (m_L + 2m_s)µ_BB

For d-level: L = 2 and S = ½

The number of doubly degenerate states are +µ_BB, 0, µ_BB only three.

Correct answer is (3)

21. A particle Y undergoes strong decay . The isospin of Y is ______.

Ans. 2

Sol.

In strong interaction, isospin is conserved, thus the isospin of Yis 2.

Correct answer is (2)

22. For a complex variable z and the contour c : |z| = 1 taken in the counter clockwise direction, _______.

Ans. –2

Sol. b₁ = –2

Correct answer is (–2)

23. Let p be the momentum conjugate to the generalized coordinate q. If the transformation

is canonical, then m = _______.

Ans. 0.5

Sol.

Correct answer is (0.5)

24. 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 7×10ⁿ, where n is an integer. The value of n is _______.

Assume:

Breakdown electric field strength in air is

Permittivity of free space = 8.85×10^–12 F/m

Electron charge e = 1.60×10^–19 C

Ans. 14

Sol.

25. If a particle is moving along a sinusoidal curve, the number of degrees of freedom of the particle is _____.

Ans. 1

Sol. Equation of constrain is y = A sin x and z = 0

DOF = 3.N – k N = 1, k = 2

3 – 1 – 2 = 1 So one degree of freedom

Correct answer is (1)

26. The product of eigenvalues of is

(a) –1

(b) 1

(d) 2

Ans. (a)

Sol.

Correct option is (a)

27. Let . Let . Let denote the three-dimensional real vector space. Which one of the following is correct?

(a) S is a linearly dependent set

(b) S is a linearly dependent set

(d)

Ans. (c)

Sol.

c₁ + c₂ + c₃ = 0

c₃ = 0, c₂ = 0, c₁ = 0

c₂ + c₃ = 0

c₃ = 0 so they are linear independent so they form a basis

Correct option is (c)

28. denotes the spin operator defined as . Which one of the following is correct?

(a) The eigenstates of spin operator are and

(b) The eigenstates of spin operator are and

(d) in the spin state , upon the measurement of , the probability for obtaining is .

Ans. (d)

Sol.

Eigen value of and corresponds eigen state

Correct option is (d)

29. The input voltage (V_in) to the circuit shown in the figure is 2cos(100t) V. The output voltage (V_out) is . If R = 1 , the value of C (in µF) is

(a) 0.1

(b) 1

(d) 100

Ans. (c)

Sol.

Correct option is (c)

30. 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 = t₀ the outputs are in logic low state (Q₀ = Q₁ = Q₂ = Q₃ = 0). Then at t = t₁, the logic state of the outputs is

(a) Q₀ = 1, Q₁ = 0, Q₂ = 0 and Q₃ = 0

(b) Q₀ = 0, Q₁ = 0, Q₂ = 0 and Q₃ = 1

(d) Q₀ = 0, Q₁ = 1, Q₂ = 1 and Q₃ = 1

Ans. (b)

Sol.

Correct option is (b)

31. Consider the Lagrangian , where a, b and c are constants. If p_x and p_y are the momenta conjugate to the coordinates x and y respectively, then the Hamiltonian is

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

Correct option is (a)

32. Which one of the following matrices does not represent a proper rotation in a plane?

(a)

(b)

(c)

(d)

Ans. (d)

Sol. Rotational matrix is orthogonal matrix

Correct option is (d)

33. A uniform magnetic field exists in an inertial frame K. A perfect conducting sphere moves with a constant velocity 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

The induced surface charge density on the sphere (to the lowest order in v/c) in the frame K' is

(a) maximum along z'

(b) maximum along y'

(d) uniform over the sphere

Ans. (a)

Sol.

It is indicated that, the electric and magnetic fields in K and K' are related as maximum along z'

Correct option is (a)

34. 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 , through the entire surface of the cylinder?

(a)

(b)

(c)

(d)

Ans. (d)

Sol. Flux through the closed surface constant when charge is inside otherwise zero.

Correct option is (d)

35. 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) the crystal is metallic for any value of n

(b) the crystal is non-metallic for any value of n

(d) the crystal is metallic for odd values of n

Ans. (d)

Sol. The conduction band is partially filled for odd value of n and hence behaves as a metal.

The band is totally filled for even value of n and known as non-metallic.

Correct option is (d)

36. According to the Fermi gas model of the nucleus, the nucleons move in a spherical volume of radius R (= R₀A^1/3, where A is the mass number and R₀ is an empirical constant with the dimensions of length). The Fermi energy of the nucleus E_F is proportional to

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

Correct option is (c)

37. 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) (n, m) = (2, 4)

(b) (n, m) = (3, 3)

(d) (n, m) = (1, 5)

Ans. (c)

Sol. For p-atoms per basis

Total degree of freedom = 2p

Number of acoustical branches (n) = 2

Number of optical branches (n) = 2p – 2

For = 2 and n = 2 × 3 – 2 = 4

(n, m) = (4, 2)

Correct option is (c)

38. The internal energy U of a system is given by U(S,V) = V^–2/3S², where 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?

(a)

(b)

(c)

(d)

Ans. (a)

Sol. dU = TdS – PdV

Correct option is (a)

39. The potential energy of a particle of mass m is given by

U(x) = a a > 0, k² > 0.

The angular frequency of small oscillations of the particle about x = 0 is

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

Correct option is (b)

40. The radial wave function of a particle in a central potential is given by , where A is the normalization constant and a is positive constant of suitable dimensions. If a is the most probable distance of the particle from the force center, the value of is ______.

Ans. 4

Sol.

41. 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 ________.

Ans. 6

Sol. Energy eigen value for particle in cubical

Ground state E_1,1,1 = 3E₀

First E_2,1,1 = E_1,2,1 = E_1,1,2 = 6E₀

Second Excited state E_2,2,1 = E_2,1,2 = E_1,2,2 = 9E₀

Third Excited state E_3,1,1 = E_1,3,1 = E_1,1,3 = 11E₀

Fourth Excited state E_2,2,2 = 12E₀ non-degenerate

Fifth Excited state E_{1, 2, 3} = E_{1, 3, 2} = E_{2, 1, 3} = E_{2, 3, 1} = E_{3, 1, 2} = E_{3, 2, 1} = 14E₀

So fifth excited state has 6 fold degeneracy.

Correct answer is (6)

42. 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 _______.

Ans. 8

Sol. Current

Correct answer is (8)

43. Let u^µ denote the 4-velocity of a relativistic particle whose square u^µu_µ = 1. If is the Levi-Civita tensor then the value of is _______.

Ans. 0

Sol. Correct answer is (0)

44. Consider a simple cubic monoatomic Bravais lattice which has a basis with vectors , 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 , where , and are primitive reciprocal lattice vectors. For n₁ = 3, n₂ = 3 and n₃ = 2, the geometrical structure factor is __________.

Ans. 2

Sol. Geometric structure factor

Correct answer is (2)

45. A plane electromagnetic wave of wavelength is incident on a circular loop of conducting wire. The loop radius is a (a << ). 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 _______.

Ans. (–270 or –90 or 90 or 270)

Sol.

Correct answer is (–270 or –90 or 90 or 270)

46. An electron in a hydrogen atom is in the state n = 3, l = 2, m = –2. Let denote the y-component of the orbital angular momentum operator. If , the value of is _____.

Ans. 1

Sol.

47. A sinusoidal voltage of the form V(t) = V₀cos(t) 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 . The lowest non-vanishing exponent in is __________.

Ans. 2

Sol.

Consider an amperian loop of radius r(r < R), then I_enc = 0 and since

The lowest non-vanishing exponent in is n = 2.

Correct answer is (2)

48. If , for , the value of a₂ is _______.

Ans. (–1)

Sol. x sin kx = a_k sin²kx

Correct answer is (–1)

49.

The value of is ________.

Ans. 0

Sol.

Correct answer is (0)

50. Consider the Hamiltonian where

and is the time independent perturbation given by

, where k > 0. If, the maximum energy eigenvalue of is 3 eV corresponding to E = 2eV, the value of k (rounded off to three decimal places) in eV is ______.

Ans. 1

Sol.

Eigen value of H'

E₁ = E – k, E₂ = E + 0, E₃ = E + k where E = 2eV and E₃ = 3eV

E₃ = E + k ⇒ k = 3eV – 2eV = 1eV

Correct answer is (1)

51. A hydrogen atom is in an orbital angular momentum state . If lies on a cone which makes a half angle 30º with respect to the z-axis, the value of l is _______.

Ans. 3

Sol.

52. 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/c²)

Ans. 6999

Sol. Assuming that that protons and anti-protons are produced at rest with mass 1GeV/c²

p + p p + p + n-number of protons +n-number of anti protons

E : 7000 + 7000 = 1 + 1 + 6999 + 6999

Correct answere is (6999)

53. 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 8×10ⁿ, where n is an integer. The value of n is _______.

(Boltzmann constant: 8.617×10^–5 eV/K)

Ans. 14

Sol.

Correct answer is (14)

54. 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

If E is the energy of a particle, then the maximum in the corresponding energy distribution in units of E/k_BT occurs at ________ (rounded off to one decimal place).

Ans. 0.5

Sol.