PYQ 2002 - VedPrep

GATE PHYSICS 2002
Previous Year Question Paper with Solution.

1. This question consists of TWENTY FIVE sub-questions (1.1 to 1.25) of ONE mark each. For each of these sub-questions, four possible answers (a, b, c and d) are given, out of which only one is correct.

1.1. If two matrices A and B can be diagonalized simultaneously, which of the following is true?

(a) A²B = B²A

(b) A²B² = B²A

(d) AB²AB = BABA²

Ans. (c)

Sol. A and B matrices can be diagonalized simultaneously i.e. both A and B can be diagonalized using same matrix.

D_A P^–1AP, D_B = P^–1BP

The diagonal matrices D_A and D_B will commute with each other.

D_AD_B = D_BD_A

P^–1APP^–1BP = P^–1BPP^–1AP

P^–1ABP = P^–1BAP AB = BA

1.2. Which one of the following matrices is the inverse of the matrix ?

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

1.3. If a function f(z) = u(x, y) + iv(x, y) of the complex variable z = x + iy, where x, y, u and v are real, is analytic in a domain D of z, then which of the following is true?

(a)

(b)

(c)

(d)

Ans. (b)

Sol. A complex function f(z) = u(x, y) + iv(x, y) will be analytic in nature if its real part and imaginary part satisfies Cauchy-Riemann equation i.e.

1.4. The homogeneity of time leads to the law of conservation of

(a) linear momentum

(b) angular momentum

(d) parity

Ans. (c)

Sol. Homogeneity of time means time is not explicitly present in Lagrangian. Therefore energy is conserved.

1.5. Hamilton canonical equations of motion for a conservative system are

(a)

(b)

(c)

(d)

Ans. (d)

Sol. Hamilton's equations (canonical equations) are

1.6. If R₁ is the value of the Rydberg constant assuming mass of the nucleus to be infinitely large compared to that of an electron, and if R₂ is the Rydberg constant taking nuclear mass to be 7500 times the mass of the electron, then the ratio R₂/R₁ is

(a) a little less than unity

(b) a little more than unity

(d) infinitely large

Ans. (a)

Sol. Rydberg constant for an atom with infinitely large nucleus is given by

where m is mass of electron.

and Rydberg constant for an atom with nucleus of finite mass M is given by

According to given question and and

Thus, equation (iii), give

1.7. Consider an infinitely long straight cylindrical conductor of radius R with its axis along the z-direction, which carries a current of 1A uniformly distributed over its cross section. Which of the following statements is correct?

(a)

(b)

(c)

(d)

where r is the radial distance from the axis of the cylinder.

Ans. (c)

Sol. We can write from Ampere law,

1.8. Consider a set of two stationary point charges q₁ and q₂ as shown in the figure. Which of the following statements is correct?

(a) The electric field at P is independent of q₂

(b) The electric flux crossing the closed surface S is independent of q₂

(d)

Ans. (b)

Sol. (i) The electric field at P depends on both charges.

(ii) The electric flux through the closed surface S.

Therefore, the electric flux crossing the closed surface S is independent of q₂.

1.9. If the wave function of a particle trapped in space between x = 0 and x = L is given by where A is a constant, for which value(s) of x will the probability of finding the particle be the maximum?

(a)

(b)

(c)

(d)

Ans. (d)

Sol. Probability density corresponding to the given wave function of the particle is

Therefore, probability density will have a maximum value when i.e. at and

1.10. In a Stern-Gerlach experiment, the magnetic field is in +z direction. A particle comes out of this experiment in state. Which of the following statements is true?

(a) The particle has a definite value of the y-component of the spin angular momentum

(b) The particle has a definite value of the square of the spin angular momentum

(d) The particle has definite values of x-and y-components of spin angular momentum

Ans. (b)

Sol. In a Stern-Gerlach experiment, a particle comes out in state i.e. spin up sate and it can mathematically expressed as

We know that, is a simultaneous eigen state of S² and S_z corresponding to eigenvalues and respectively. Thus, the particle will have definite values of S² and S_z.

1.11. If is the total cross-section and being the angle of scattering, is the scattering amplitude for a quantum mechanical elastic scattering by a spherically symmetric potential, then which of the following is true? Note that k is the magnitude of the wave vector along the direction.

(a)

(b)

(c)

(d)

Ans. (c)

Sol. According to Partial wave analysis, for a spherically symmetric potential the scattering amplitude and total cross section can be written as

where is the phase shift of the individual partial waves due to scattering. Now for = 0, the scattering amplitude will be

1.12. In a classical micro-canonical ensemble for a system of N non-interacting particles, the fundamental volume in phase space which is regarded as "equivalent to one micro-state" is

(a) h^3N

(b) h^2N

(d) h

where h is the Planck's constant

Ans. (a)

Sol. For one particle, V = dx dp_x.dy dp_y.dz dp_z = h³

For N particles, V = (h³)^N = h^3N

1.13. Which of the following conditions should be satisfied by the temperature T of a system of N non-interacting particles occupying a volume V, for Bose-Einstein condensation to take place?

(a)

(b)

(c)

(d)

where m is the mass of each particle of the system, k_B is the Boltzmann constant, h is the Planck's constant and is the well known Zeta function.

Ans. (a)

Sol. For Bose-Einstein condensation to take place, the temperature of the Boson gas should be less than T_B, where T_B is given by

1.14. A large circular coil of N turns and radius R carries a time varying current . A small circular coil of n turns and radius r (r << R) is placed at the center of the large coil such that the coils are concentric and coplanar. The induced emf in the small coil

(a) leads the current in the large coil by

(b) lags the current in the large coil by

(d) lags the current in the large coil by

Ans. (d)

Sol.

The magnetic field at the centre of large circular surface is

Therefore, induce emf in the small coil lags behind the current in the large coil by

1.15. Four charges are placed at the four corners of a square of side a as shown in the figure. The electric dipole moment of this configuration is

(a)

(b)

(c)

(d)

Ans. (c)

Sol. The electric dipole moment of the given configuration with respect to origin,

1.16. Which of the following statements is true?

(a) In a micro-canonical ensemble the total number of particles N and the energy E are constants while in a canonical ensemble N and temperature T are constants

(b) In a micro-canonical ensemble the total number of particles N is a constant but the energy E is variable while in a canonical ensemble N and T are constants

(d) In a micro-canonical ensemble N and E are constants while in a canonical ensemble N is a constant but T varies

Ans. (a)

Sol. Micro-canonical ensemble: It is an ensemble in which all element have same macrostate represented by same number of particles, same volume and same energy.

Canonical ensemple: It is an ensemble in which all element have same macrostate represented by same number of particles, same volume and same temperature.

1.17. In a one-dimensional Kronig Penny model, the total number of possible wave functions is equal to

(a) twice the number of unit cells

(b) number of unit cells

(d) independent of the number of unit cells

Ans. (b)

Sol. For one dimensional Kronig Penney model, total number of possible states or wave functions is equal to the number of unit cells.

1.18. The potential in a divalent solid at a particular temperature is represented by a one-dimensional periodic model. The solid should behave electrically as

(a) a semiconductor

(b) a conductor

(d) a superconductor

Ans. (b)

Sol. Divalent solids are Alkaline earth metals. They behave like a conductor

1.19. In a cubic system with cell edge a, two phonons with wave vectors and collide and produce a third phonon with a wave vector such that

where is a lattice vector. Such a collision process will lead to

(a) finite thermal resistance

(b) zero thermal resistance

(d) a finite thermal resistance for certain only

Ans. (a)

Sol. In the phonon destroys momentum . So conservation of momentum does not hold. Therefore, a finite value of thermal resistance comes out.

1.20. The baryon number of proton, the lepton number of proton, the baryon number of electron and the lepton number of electron are respectively

(a) zero, zero, one and zero

(b) one, one, zero and one

(d) zero, one, one and zero

Ans. (c)

Sol. Since, proton belong to Baryon family so its Baryon number is one.

Since, proton does not belong to lepton family. Therefore, Lepton number of proton is zero.

Since, electron is a lepton, its lepton number is equal to one and its Baryon number is zero

Therefore, correct results are one, zero, zero and one.

1.21. Typical energies released in a nuclear fission and a nuclear fusion reaction are respectively

(a) 50 MeV and 1000 MeV

(b) 200 MeV and 1000 MeV

(d) 200 MeV and 10 MeV

Ans. (d)

Sol. 200 MeV and 10 MeV

1.22. Nuclear forces are

(a) spin dependent and have no non-central part

(b) spin dependent and have a non-central part

(d) spin independent and have a non-central part

Ans. (b)

Sol. Nuclear force have the following properties

(1) These forces are short range forces

(2) These forces are charge independent

(3) these forces are strongest known forces

(4) These forces are spin dependent

(5) These forces are velocity dependent

(6) These are non-central forces

1.23. The nuclear spins of ₆C¹⁴ and ₁₂Mg²⁵ nuclei are respectively

(a) zero and half-integer

(b) half-integer and zero

(d) both half-integers

Ans. (a)

Sol.

(₆C¹⁴ have even number of protons and neutrons)

Therefore, spin = half integer

1.24. The asymmetry terms in the Weizsacker semi-empirical mass formula is because of

(a) non-spherical shape of the nucleus

(b) non-zero spin of nucleus

(d) odd number of protons inside the nucleus

Ans. (c)

Sol. The E_asymmetry term originates from the asymmetry between the number of protons and the number of neutrons in the nucleus. Nuclear data for stable nuclei indicate that for lighter nuclei, the number of protons is almost equal to that of neutrons, i.e. N = Z. As mass number 'A' increases, the symmetry of proton and neutron number is lost and as the number of neutrons exceeds that of protons to maintain the nuclear the stability. This excess of neutrons over protons i.e. (N – Z) is the measure of the asymmetry and it decreases the stability or the binding energy of medium or heavy nuclei.

The asymmetry terms in the Weizsacker semi-empirical mass formula arises due to unequal number of protons and neutrons

1.25. Which of the following options is true for a two input XOR gate?

Input Output

A B

(a) 0 1 1

(b) 1 0 0

(d) 1 1 1

Ans.

Sol. XOR Exclusive OR gate

2. This question consists of TWENTY FIVE sub-questions (2.1 to 2.25) of ONE mark each. For each of these sub-questions, four possible answers (a, b, c and d) are given, out of which only one is correct.

2.1. Which of the following vectors is orthogonal to the vector where a and b are constants, and and are unit orthogonal vectors?

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

[ two vectors are said to be orthogonal if their dot product is zero]

2.2. Fourier transform of which of the following functions does not exist?

(a) e^–|x|

(b)

(c)

(d)

Ans. (c)

Sol. For the existance of fourier transform of a function f(x), f(x) should be either finite or zero at x = . But is infinite at x = , therefore, Fourier transform of will not exist.

2.3. The unit vector normal to the surface 3x² + 4y = z at the point (1, 1, 7) is

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Unit vector normal to the surface at the point (1, 1, 7) is

2.4. The solution of the differential equation

(a) Ax² + B

(b) Ax + Be^–x

(d) Ax + Bx²

where A and B are constants

Ans. (b)

Sol.

= (1 + x) (Be^–x) + x (A – Be^–x) – (Ax + Be^–x)

= Be^–x + xBe^–x + Ax – xBe^–x – Ax – Be^–x = 0

2.5. A particle of mass M moving in a straight line with speed v collides with a stationary particle of the same mass. In the center of mass coordinate system, the first particle is deflected by 90º.The speed of the second particle, after collision, in the laboratory system will be

(a)

(b)

(d) v/2

Ans. (a)

Sol.

angle of scattering in lab frame

Using conservation of momentum we can conclude that the two masses move with equal speed. Using conservation of kinetic energy

2.6. The scalar potential corresponding to the force field

(a) is y²/2

(b) is 1

(d) does not exist

Ans. (d)

Sol.

Force is not conservative so we cannot define potential.

2.7. Two particles of equal mass are connected by an inextensible string of length L. One of the masses is constrained to move on the surface of a horizontal table. The string passes through a small hole in the table and the other mass is hanging below the table. The only constraint is that the first mass moves on the surface of the table. The number of degrees of freedom of the masses-string system is

(a) five

(b) four

(d) one

Ans. (b)

Sol.

Particle A moves on a surface therefore it has 2 degree of freedom. Particle B moves in space so it has three degree of freedom. However, between the particles there is a constaint - length of string is constant. Therefore total degree of freedom for the system is 2 + 3 – 1 = 4.

2.8. An electron is accelerated from rest by 10.2 million volts. The percent increase in its mass is

(a) 20,000

(b) 2,000

(d) 20

Ans. (b)

Sol. Kinetic energy = qV

mc² – m₀c² = qV

2.9. An infinitely long closely wound solenoid carries a sinusoidally varying current. The induced electric field is

(a) zero everywhere

(b) non-zero inside and zero outside the solenoid

(d) zero inside and non-zero outside the solenoid

Ans. (c)

Sol. According to Maxwell equation,

And for r > a

2.10. A laser beam of wavelength 600 nm with a circular cross section having a radius of 10mm falls normally on a lens of radius 20 mm and focal length 10 cm. The radius of the focussed spot is approximately

(a) 0.3 µm

(b) 0.6 µm

(d) 6.0 µm

Ans. (b)

Sol. The radius of the focal spot

2.11. A left circularly polarized light beam of wavelength 600 nm is incident on a crystal of thickness d and propagates perpendicular to its optic axis. The ordinary and extraordinary refractive indices of the crystal are n₀ = 1.54 and n_e = 1.55 respectively. The emergent light will be right circularly polarized if d is

(a) 120 µm

(b) 60 µm

(d) 15 µm

Ans. (c)

Sol.

Let after travelling 'd' distance the left circular polarized light become right circular polarized.

Therefore, the phase introduced between two component

2.12. In a two beam interference pattern, the maximum and minimum intensity values are found to be 25I₀ and 9I₀ respectively, where I₀ is a constant. The intensities of the two interfering beams are

(a) 16I₀ and I₀

(b) 5I₀ and 3I₀

(d) 8I₀ and 2I₀

Ans. (a)

Sol.

Taking positive values only,

2.13. An electron propagating along the x-axis passes through a slit of width . The uncertainty in the y-component of its velocity after passing through the slit is

(a) 7.322 × 10⁵ m/s

(b) 1.166 × 10⁵ m/s

(d) 2.326 × 10⁴ m/s

Ans. (b)

Sol. uncertainty in the y component of position of the electron is = 1 nm. According to Heisenberg's uncertainty principle, the uncertainty in the y-component of velocity of the electron after passing through the slit is

2.14. and are two quantum mechanical operators. If stands for the commutator of and , then is equal to

(a)

(b)

(d)

Ans. (c)

Sol.

2.15. An electron is in a state with spin wave function in the S_z representation. What is the probability of finding the z-component of its spin along the direction?

(a) 0.75

(b) 0.50

(d) 0.25

Ans. (d)

Sol. Given state of the electron:

Probability of finding the z-component of spin along direction = 0.25

2.16. If the wavelength of the first line of the Balmer series in the hydrogen spectrum is , then the wavelength of the first line of the Lyman series is

(a)

(b)

(c)

(d)

Ans. (b)

Sol. For Balmer series, the first line is given by

Now, the first line of the Lyman series is given by

2.17. A vector is solenoidal if the constant a has a value

(a) 4

(b) –4

(d) –8

Ans. (c)

Sol.

2.18. The mean free path of the particles of a gas at a temperature T₀ and pressure p0 has a value . If the pressure is increased to 1.5p₀ and the temperature is reduced to 0.75 T₀, the mean free path

(a) remains unchanged

(b) is reduced to half

(d) is equal to 1.125

Ans. (b)

Sol. The mean free path is where di is the diameter of the molecule.

Now, if P = 1.5 p₀ and T = 0.75 T₀, we have

2.19. Which of the following relations between the particle number density n and temperature T must hold good for a gas consisting of non-interacting particles to be described by quantum statistics?

(a) n/T^1/2 << 1

(b) n/T^3/2 << 1

(d) n/T^1/2 and n\T^3/2 can have any value

Ans. (c)

Sol. The quantum statistics applies where the interparticle distance between the particles is comparable to their de-Broglie wavelengths.

If n is the number density of the particle then,

n = where N is the total number of particles

, where r is the interparticle distance.

Also, the de-Broglie wavelength is

2.20. For a prefect free-electron gas in a metal, the magnitudes of phase velocity (v_p) and group velocity (v_g) are such that

(a) v_p = v_g

(b)

(c)

(d) v_p = 2v_g

Ans. (b)

Sol. For a perfect free electron gas in a metal, the angular frequency and wavenumber are related by the following relation:

So, Phase velocity v_p = and Group velocity v_g = .

Therefore, relation between phase velocity and group velocity is v_p = .

2.21. A metal has free-electron density n = 10²⁹ m^–3. Which of the following wavelengths will excite plasma oscillations?

(a) 0.033 µm

(b) 0.330 µm

(d) 33.000 µm

Ans. (b)

Sol. The wavelength of plasma oscillation.

2.22. For an NaCl crystal, the cell-edge a = 0.563 nm. The smallest angle at which Bragg reflection can occur corresponds to a set of planes whose incides are

(a) 100

(b) 110

(d) 200

Ans. (a)

Sol. For cubic system, the relation between interatomic distance (d) and cell-edge (a) is given by

For minimum , n = 1, d must be maximum.

For maximum d, will be minimum. According to given options only "(a) (100)" shows minimum , which gives smallest .

2.23. Silicon has diamond structure with unit-cell-edge a = 0.563 nm. The interatomic separation is

(a) 0.122 nm

(b) 0.234 nm

(d) 0.542 nm

Ans. (c)

Sol. Unit-cell-edge (a) = 0.543 nm

The interatomic separation (d) = ?

Since diamond has fcc structure : 4r =

2.24. The spin and parity of ₄Be⁹ nucleus, as predicted by the shell model, are respectively.

(a) 3/2 and odd

(b) 1/2 and odd

(d) 1/2 and even

Ans. (a)

Sol. ₄Be⁹ has Z = 4, N = 9 – 4 = 5

5N (1s_1/2)² (1p_3/2)³

Therefore, J = , l = 1 (for p) and parity = (–1)' = (–1)⁺¹ = –1 (odd)

Therefore, spin and parity = and odd.

2.25. The feedback ratio of an amplifier, which on application of a negative feedback, changes the voltage gain from –250 to –100 is

(a) –0.250

(b) –0.025

(d) –0.006

Ans. (d)

Sol. Closed loop gain for negative feed-back A_CI =

where, A_OL (open loop again), A_CL (closed loop gain), feedback ratio

3. Given the differential equation find its solution that satisfies the initial conditions y = 0 and x = 0 and at x = 0

Sol.

Let, y = C . e^mx be the trial solution

So, y = A e^{(–1 + 2i)x} + B e^{(–1 – 2i)x}

4. Find the matrix that diagonalizes the matrix

Sol.

The eigen value equation,

Let, the eigen vector correcponding, l = 1 is X₁

Applying normalization condition i.e.

Similarly, the eigen vector corresponding to the eigen value = –1 is X₂ =

Diagonalizing matrix,

5. Using the residue theorem, compute the integral .

Sol.

Let t = (1 + x)

dt = dx

6. A particle of mass M is attached to two identical springs of unstretched length L₀ and spring constant k. The entire system is placed on a horizontal frictionless table as shown in the figure. The mass is slightly puled along the surface of the table and perpendicular to the lengths of the springs and then let go. Using the Lagrangian equation (s) of motion, show whether the mass will execute simple harmonic motion. If so, find the time period.

Sol.

Lagrange's equation

Motion is not simple harmonic.

7. A uniform thin circular disc of mass M and radius R lies in the X-Y plane with its centre at the origin. Find the moments of inertia tensor. What are the values of the principal moments of inertia? Find the principal axes.

Sol. x, y, z axes are symmetric therefore these are principal axes

Thus I_xy = I_yx = I_yz = I_zy = I_xz = I_zx = 0

Here I_xx, I_yy, I_zz are principal moments of inertia.

8. Two events, 10^–7 s apart in time, take place at two points 50 m apart on the X-axis. Find the speed of an observer moving along the X-axis who observes the two events simultaneously. What will be the spatial separation between these two events as seen by this observer?

Sol. = 10^–7 sec, = 50 meter

In observer's frame = 0

Spatial separation between these events in observer's frame

9. Consider a parallel plate air filled capacitor with a plate area of 10 cm² separated by a distance of 2 mm. The potential difference across the plates varies as

V = 360 sin (2 10⁶ t) volts,

where t is measured in seconds. Neglecting fringe effects, calculate the displacement current flowing through the capacitor.

Sol. We know that displacement current,

10. The potential of a spherically symmetric charge distribution is given by

a and R being constants. Find the corresponding charge distribution.

Sol.

According to Poisson's equation,

11. Consider a plane electromagnetic wave propagating in free space and having an electric field distribution given by

where and a are constants. Calculate the corresponding magnetic field .

Sol.

For plane progressive wave

12. A particle in the ground state of an infinitely deep one dimensional potential well of width a is subject to a perturbation of the form

where V₀ is a constant. Find the shift in energy of the particle in the lowest order perturbation theory.

Sol. The ground state wave function of infinitely deep potential well is given

13. A quantum particle is in a state which is the superposition of the eigenstates of the momentum operator . If the probability of finding the momentum of the particle is 90%, compute its wave function.

Sol.

The wave function is for a free particle and it can carry momentum or

Applying normalization condition,

14. The wave function of a free particle is given by , where C is a constant. Compute the momentum space probability density, normalize it to 1 and hence find the value of C.

Sol.

Therefore, the probability density in the momentum space

Therefore, according to normalization condition,

15. Carbon monoxide has a bond length of 0.1132 nm. What will be the frequency of rotation of the molecule for its lowest excited state?

Sol. The quantum-mechanical energy of a rotating diatomic molecule is given by

where, I is the moment of inertia of the molecule about the axis of rotation.

The lowest rotational excited energy level corresponds to J = 1, and for this level in CO

Substituting the value of h in M.K.S. units and the given value of 1, we have

The reduced mass of CO:

Moment of inertia

I = µr²; where r is bond length

= (1.14 × 10^–26 kg) (0.1132 × 10^–9 m)²

= 1.46 × 10^–46 kg-m²

The angular frequency of the CO molecule

16. 1 Kg of water at a temperature of 353 K is mixed adiabatically with an equal mass of water at 293 K. Find the change in entropy of the universe assuming the specific heat of water to be constant equal to 238 J.kg^–1. K^–1.

Sol.

Let the final temperature after mixing waters be T_f. Then,

By the principle of calorimetry, m₁s₁ (T_f – T₁) = m₂s₂ (T₂ – T_f), where s₁ = s₂ = specific heat of water.

Now change ln entropy is

17. A conductor having a free electron gas is maintained at a very low temperature (T 0K). Find the average energy per electron in terms of the electrons density and the electron mass.

Sol. Electrons are fermions. So they follow Fermi-Dirac statistics. Their distribution is

The density of state is

Since, electron spin degeneracy is 2 so we should multiply the density of state by the factor of two

Also, the total energy is

The average energy per electron

18. A small concentration of minority carries is injected into a homogeneous semiconductor held at 300K. An electric field of 30 V/cm is applied across the width of the crystal. As a result, the minority carriers move a distance of 1.5 cm in a time of 300 µs. What is the diffusion coefficient of the minority carriers in the semiconductor?

Sol. The Einstein relationship between mobility and diffusion constant,

19. A phonon with wave vector gets absorbed on collision with an electron of wave vector . The electron is considered free and its energy is much larger than that of the phonon. If the electron is scattered at an angle , show that .

Sol. According to momentum conservation,

where, is the wave vector of phonon, are the wave vector of electron before and after collision, respectively.

Since, energy of the electron is much larger than the phonon. So, we can take

20. In spherical coordinates, the wave function describing a state of a system is

where a0 is a constant. Find the parity of the system in this state.

Sol.

Operate parity operator on :

So, the parity of the function is odd.

21. Calculate the minimum kinetic energy that the neutron should have in order to induce the reaction

O¹⁶(n¹, He⁴)C¹³

in which C¹³ is left in an excited state of energy 1.79 MeV. Given:

Mass of O¹⁶ = 16.000000 amu

Mass of n¹ = 1.008986 amu

Mass of He⁴ = 4.003874 amu

Mass of C¹³ = 13.007490 amu

Sol. The Q value of the reaction is given by

Q = [M_o + M_n – (M_He + M_C)] C² – Eexcitation

= (–2.215 – 1.79) MeV = –4.005107 MeV

22. Calculate the dc collector voltage (V_c) with respect to ground in the amplifier circuit shown in the figure. The current gain for the transistor is 200.