PYQ 2008 - VedPrep

GATE PHYSICS 2008
Previous Year Question Paper with Solution.

1. For arbitrary matrices E, F, G and H, if EF – FE = 0 then Trace(EFGH) is equal to

(a) Trace(HGFE)

(b) Trace(E).Trace(F).Trace(G).Trace(H)

(d) Trace(EGHF)

Ans. (d)

Sol. We have, EF–FE–0 EF = FE

Trace (EFGH) = Trace(FEGH) = Trace(EGHF)

2. An unitary matrix is given, where and are real. The inverse of the matrix is

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

3. The curl of a vector field is . Identify the appropriate vector field from the choices given below

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

4. A rigid body is rotating about its centre of mass, fixed at the origin, with an angular velocity and angular acceleration . If the torque acting on it is and its angular momentum is , the rate of change of its kinetic energy is

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Rotational Kinetic energy is given as

5. A cylinder of mass M and radius R is rolling down without slipping on an inclined plane of angle of inclination . The number of generalized coordinates required to describe the motion of this system is

(a) 1

(b) 2

(d) 6

Ans. (b)

Sol. We need two generalised coordino the condition for rolling, one coordinate gets eliminated.

6. A parallel plate capacitor is being discharged. What is the direction of the energy flow in terms of the Poynting vector in the space between the plates?

(a) Along the wire in the positive z-axis

(b) Radially inward

(d) Circumferential

Ans. (c)

Sol. The electric field and magnetic field have following directions

Direction of Poynting vector

7. Unpolarized light falls from air to a planar air-glass interface (refractive index of glass is 1.5) and the reflected light is observed to be plane polarized. The polarization vector and the angle of incidence are

(a) perpendicular to the plane of incidence and = 42°

(b) parallel to the plane of incidence and = 56°

(d) parallel to the plane of incidence and = 42°

Ans. (c)

Sol.

8. A finite wave train, of an unspecified nature, propagates along the positive x-axis with a constant speed v and without any change of shape. The differential equation among the four listed below.

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Since, the wave train is propagating along the positive x-axis with constant speed v.

So, wave equation will be

9. Let denote the ground state of the hydrogen atom. Choose the correct statement from those given below:

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Since, the ground state of hydrogen atom does not depends on the

10. Thermodynamic variables of a system can be volume V, pressure P, temperature T, number of particles N, internal energy E and chemical potential µ, etc. For a system to be specified by Microcanonical (MC), Canonical (CE) and Grand Canonical (GC) ensembles, the parameters required for the respective ensembles are:

(a) MC : (N, V, T); CE : (E, V, N); GC : (V, T, µ)

(b) MC : (E, V, N); CE : (N, V, T); GC : (V, T, µ)

(d) MC : (E, V, N); CE : (V, T, µ); GC : (N, V, T)

Ans. (b)

Sol. The parameters required are

MC : (E, V, N)

CE : (N, V, T)

CE : (N, V, µ)

11. The pressure versus temperature diagram of a given system at certain low temperature range is found to be parallel to the temperature axis in the liquid-to-solid transition region. The change in the specific volume remains constant in the region. The conclusion one can get from the above is

(a) the entropy of solid is zero in this temperature region

(b) the entropy increases when the system goes from liquid to solid phase in this temperature region

(d) the change in entropy is zero in the liquid-to-solid transition region

Ans. (d)

Sol. P-T curve is parallel to the temperature axis

From Clausius-Clapeyron equation,

12. The radial wave function of the electrons in the state of n = 1 and l = 0 in a hydrogen atom is is the Bohr radius. The most probable value of r for an electron is

(a) a0

(b) 2a0

(d) 8a0

Ans. (a)

Sol. Radial probability density of finding the electron

13. The last two terms of the electronic configuration of manganese (Mn) atom is 3d54s2. The term factor of Mn4+ ion is

(a) 4D1/2

(b) 4F3/2

(d) 3D7/2

Ans. (b)

Sol. For electronic configuration is 3d³. To fill 3 electrons

For less than half filled ground state is for

Therefore, term is ⁴F_3/2

14. The coherence length of laser light is

(a) directly proportional to the length of the active lasing medium

(b) directly proportional to the width of the spectral line

(d) inversely proportional to the length of the active lasing medium

Ans. (c)

Sol. Length 'L' of the active medium is related with the wavelength of laser light by

i.e., coherent length is inversely proportional to the width of the spectral line.

15. Metallic monovalent sodium crystallizes in body centered cubic structure. If the length of the unit cell is 4 × 10–8 cm, the concentration of conduction electrons in metallic sodium is

(a) 6.022 × 1023 cm–3

(b) 3.125 × 1022 cm–3

(d) 1.250 × 1020 cm–3

Ans. (b)

Sol. The number of atom in the unit cell of B.C.C. = 2. Each atom will give one free electrons. So, the concentration of conduction electrons.

16. The plot of inverse magnetic susceptibility versus temperature T of an antiferromagnetic sample corresponds to

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

17. According to the quark model, the K+ meson is composed of the following quarks:

(a) u u d

(b)

(c)

(d)

Ans. (c)

Sol. K⁺ has charge (Q) = + 1, Strangeness (S) = 1, Baryon number (B) = 0

Since, the Baryon number is zero. Therefore, we need one quark and one anti-quark. One with positive sign and other with negative sign. Also strangeness is 1, we can take one strange anti-quark i.e. . Now in order to give it charge (Q) = +1, we should add one up quark u su that its charge become

Therefore, quark structure of K⁺ meson is .

18. An O16 nucleus is spherical and has a charge radius R and a volume . According to the empirical observations of the charge radii, the volume of the 54Xe128 nucleus, assumed to be spherical, is

(a) 8V

(b) 2V

(d) 1.89V

Ans. (a)

Sol.

19. A common emitter transistor amplifier circuit is operated under a fixed bias. In this circuit, the operating point.

(a) remains fixed with an increase in temperature

(b) moves towards cut-off region with an increase in temperature

(d) moves towards the saturation region with an increase in temperature

Ans. (d)

Sol. As temperature increase the β of the transistor increase and as a result I_C also start to increase. So, Q point will be shift towards saturated region

20. Under normal operating conditions, the gate terminal of an n-channel junction field effect transistor (JFET) and an n-channel metal oxide semiconductor field effect transistor (MOSFET) are

(a) both biased with positive potentials

(b) both biased with negative potentials

(d) biased with negative and positive potentials, respectively

Ans. (b)

Sol.

Hence, both biased with negative potential.

21. The eigenvalues of the matrix are

(a)

(b)

(d)

Ans. (b)

Sol. Let the is the eigenvalue of the matrix. Therefore, the eigenvalue equation,

22. If the Fourier transform then will correspond to

(a)

(b) a constant

(c)

(d)

Ans. (d)

Sol.

23. If where C is the unit circle taken anticlockwise and Ln(z) is the principal branch of the Logarithm function, which one of the following is correct?

(a) I = 0 by residue theorem

(b) I is not defined since Ln(z) has a branch cut

(c)

(d)

Ans. (a)

Sol. For the principal branch of f(z), z = 0 will behave like a simple pole

24. The value of is

(a) 0

(b)

(c)

(d)

Ans. (b)

Sol.

25. Consider the Bessel equation . Which one of the following statements is correct?

(a) Equation has regular singular points at z = 0 and

(b) Equation has 2 linearly independent solutions that are entire

(d) Limit taken along x-axis, exists for both the linearly independent solutions

Ans. (c)

Sol. Consider the equation,

This is a Bessel differential equation with v = 0

Since, is not defined at z = 0, therefore z = 0 is a regular singular point ofthe differential equation.

The given Bessel differential equation has two linearly independent solutions i.e.

(i) J₀(z) which isdefined everywhere and it is entire solution

(ii) Y₀(z) which is not defined at z = 0 and it issingular(notanalytic) at z = 0

26. Under a certain rotation of coordinate axes, a rank-1 tensor va(a = 1, 2, 3) transforms according to the orthogonal transformation defined by the relations . Under the same rotation a rank-2 tensor Ta, b would transform such that

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

Rotation in rank-2 tensor matrix is given as

27. The Lagrangian of a system is given by . It describes the motion of

(a) a harmonic oscillator

(b) a damped harmonic oscillator

(d) a system with unbounded motion

Ans. (a)

Sol.

Equation of motion

This is equation of motion of a harmonic oscillator

28. The moment of inertia tensor of a rigid body is given by . The magnitude of the moment of inertia about an axis is

(a) 6

(b) 5

(d) 8/3

Ans. (b)

Sol. Moment of inertia about given axis is

29. A hoop of radius R is pivoted at a point on the circumference. The period of small oscillations in the plane of the hoop is

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Time period of oscillation is given as

I = moment of inertia about axis of rotation

d = distance between point of suspension and center of mass.

Here I = 2MR², d = R.

30. A mass m is constrained to move on a horizontal frictionless surface. It is set in circular motion with radius r0 and angular speed by an applied force communicated through an inextensible thread that passes through a hole on the surface as shown in the figure. This force is then suddenly doubled. The magnitude of the radial velocity of the mass

(a) increases till the mass falls into the hole

(b) decreases till the mass falls into the hole

(d) becomes zero at a radius r1 where 0 < r1 < r0

Ans. (d)

Sol. Since the force is radial, angular momentum is conserved. If mass fall into the hole, its angular momentum will become zero. Which will violate conservation of angular momentum. Therefore it will not fall into the hole.

31. For a simple harmonic oscillator the Lagrangian is given by . If and H(p, q) is the Hamiltonian of the system, the Poisson bracket {A(p, q), H(p, q)} is given by

(a) iA(p, q)

(b) A*(p, q)

(d) –iA(p, q)

Ans. (a)

Sol.

32. A plane electromagnetic wave is given by . At a given location, the number of times vanishes in one second is

(a) An integer near when and zero when is integer

(b) An integer near and is independent of

(d) An integer near and is independent of

Ans. (a)

Sol.

It will not be linear polarized light. E will not be zero.

33. A dielectric sphere is placed in a uniform electric field directed along the positive y-axis. Which one of the following represents the correct equipotential surfaces?

(a)

(b)

(c)

(d)

Ans. (d)

Sol. Since electronic permeability of dielectric material ε is greater than the . So, the field intensity inside the material will be increase.

34. A rod of length L with uniform charge density per unit length is in the xy-plane and rotating about z-axis passing through one of its edge with an angular velocity as shown in the figure below. refer to the unit vectors at Q, is the vector potential at a distance d from the origin O along z-axis for d >> L and is the current density due to the motion of the road. Which one of the following statements is correct?

(a)

(b)

(c)

(d)

Ans. (b)

Sol. Since rod is rotated in the xy–plane the direction of J and A will be along .

Since, d >> L, so we can consider rod as magnetic dipole, the vector potential due to magnetic dipole as

35. A circular disc of radius a on the xy plane has a surface charge density . The electric dipole moment of this charge distribution is

(a)

(b)

(c)

(d)

Ans. (b)

Sol. The elementary area on the disc.

Total charge on the disc is given by

So, dipole moment independent of choice of origin.

So, the position of any point on the disc is given by

The dipole moment,

36. At time t = 0, a charge distribution exists within an ideal homogeneous conductor of permittivity and conductivity . At a later time is given by

(a)

(b)

(c)

(d)

Ans. (a)

Sol. We know that, from equation of continuity

37. A nonrelativistic charged particle moves along the positive x-axis with a constant positive acceleration . The particle is at the origin at t = 0. Radiation is observed at t = 0 at a distant point (0, d, 0) on the y-axis. Which one of the following statements is correct?

(a) The radiation is unpolarized

(b) The radiation is plane polarized with polarization parallel to the x-axis

(d) The radiation is elliptically polarized

Ans. (b)

Sol. Correct option is (b)

38. For a physical system, two observables Q1 and Q2 are known to be compatible. Choose the correct implication from amongst those given below:

(a) Every eigenstate of O1 must necessarily be an eigenstate of O2

(b) Every non-degenerate eigenstate of O1 must necessarily be an eigenstate of O2

(c) When an observation of O1 is carried out on an arbitrary state of the physical system, a subsequent observation of O2 leads to an unambiguous result

(d) Observation of O1 and O2, carried out on an arbitrary state of the physical system, lead to the identical results irrespective of the order in which the observations are made.

Ans. (d)

Sol. Since, Q₁ and Q₂ are compatible variable, therefore, they will commute with each other.

39. An exact measurement of the position of a simple harmonic oscillator (SHO) is made with the result x = x0. [The SHO has energy levels En (n = 0, 1, 2...) and associated normalized wave-functions ]. Subsequently, an exact measurement of energy E is made. Using the general notation Pr(E = E') denoting the probability that a result E' is obtained for this measurement, the following statements are written. Which one of the following statements is correct?

(a) Pr(E = E0) = 0

(b) Pr(E = En) = 1 for some value of n

(c)

(d) Pr(E > E") > 0 for any E"

Ans.

Sol. Wrong Questions

40. Consider the combined system of proton and electron in the hydrogen atom in its (electronic) ground state. Let I denote the quantum number associated with the total angular momentum and let denote the magnitude of the expectation value of the net magnetic moment in the state. Which of the following pairs represents a possible state of the system (µB is Bohr magneton)?

(a)

(b)

(c)

(d)

Ans. (b)

Sol. For ground state of hydrogenatonv

41. A particle is placed in a one dimensional box of size L along the x-axis (0 < x < L). Which of the following is true?

(a) In the ground state, the probability of finding the particle in the interval (L/4, 3L/4) is half.

(b) In the first excited state, the probability of finding the particle in the interval (L/4, 3L/4) is half. This also holds for states with n = 4, 6, 8...

(d) In the ground state, the particle has a definite momentum.

Ans. (c)

Sol.

The probability of finding the particle between 0 to is

42. An inelastic ball of mass m has been thrown vertically upwards from the ground at z = 0. The initial kinetic energy of the ball is E. The phase trajectory of the ball after successive bouncing on the ground is

(a)

(b)

(c)

(d)

Ans. (c)

Sol. At z = 0, momentum will be maximum and when z will be maximum then momentum will be zero. Again when z will start decrease momentum will be start to increase but direction will be opposite. And in the inelastic collision. The momentum of the ball after collision will be less than value of before collision.

43. A system containing N non-interacting localized particles of spin and magnetic moment µ each is kept in constant external magnetic field B and in thermal equilibrium at temperature T. The magnetization of the system is,

(a)

(b)

(c)

(d)

Ans. (b)

Sol. The energy of a particle with magnetic moment u in an external magnetic field B is

The partition function of a particle is

The partition function for N non-interacting such particles is

44. Two identical particles have to be distributed among three energy levels. Let rB, rF and rC represent the ratios of probability of finding two particles to that of finding one particle in a given energy state. The subscripts B, F and C correspond to whether the particles are bosons, fermions and classical particles, respectively. Then, rB : rF : rC is equal to

(a)

(b)

(c)

(d)

Ans. (d)

Sol. The distributions are as follows:

Here we have choosen 1-state to be the given energy state.

45. A photon gas is at thermal equilibrium at temperature T. The mean number of photons in an energy state is

(a)

(b)

(c)

(d)

Ans. (d)

Sol. The photons are bosons and hence follow Bose-Einstoin statistics. The mean number of photons are

46. Consider a system of N atoms of an ideal gas of type A at temperature T and volume V. It is kept in diffusive contact with another system of N atoms of another ideal gas of type B at the same temperature T and volume V. Once the combined system reaches equilibrium.

(a) the total entropy of the final system is the same as the sum of the entropy of the individual system always

(b) the entropy of mixing is 2NkB ln 2

(d) the entropy of mixing is non-zero when the atoms A and B are of the same type.

Ans. (b)

Sol.

Since the gases are ideal in nature, then internal energy depends only on the temperature. Furthermore, temperature of the mixture remains T as T_A = T_B = T.

From first law of thermodynamics,

dQ = dU + pdV

The entropy change of the system is

47. Consider a system of two non-interacting classical particles which can occupy any of the three energy levels with energy values E = 0, and and having degeneracies g(E) = 1, 2 and 4 respectively. The mean energy of the system is

(a)

(b)

(c)

(d)

Ans. (b)

Sol. The mean energy is

48. Three consecutive absorption lines at 64.275 cm–1, 77.130 cm–1 and 89.985 cm–1 have been observed in a microwave spectrum for a linear rigid diatomic molecule. The moments of inertia IA and IB are (IA is with respect to the bond axis passing through the centre of mass and IB is with respect to an axis passing through the centre of mass and perpendicular to bond axis).

(a)

(b)

(c)

(d)

Ans. (b)

Sol. The wave number of three consecutive lines are 64.275 cm^–1 ,77.130 cm^–1 and 89.985 cm^–1.

Therefore, the difference in wave number

49. A pure rotational Raman spectrum of a linear diatomic molecule is recorded using electromagnetic radiation of frequency vc. The frequency of two consecutive Stokes lines are

(a) vc – 10B, vc – 14B

(b) vc – 2B, vc – 4B

(d) vc + 2B, vc + 4B

Ans. (a)

Sol. The wave number of the rotational Raman lines are given by

50. Which one of the following statement is incorrect in vibrational spectroscopy with anharmonicity?

(a) The selection rule for vibrational spectroscopy is

(b) Anharmonicity leads to multiple absorption lines

(d) The frequencies of hot band lines are smaller than the fundamental absorption

Ans. (c)

Sol. For vibrational spectroscopy with anharmonicity the selection rule is = ± 1 ,± 2 ,± 3,.... The anharmonicity leads to multiple absorption lines. The intensity of hot band lines are stronger than the fundamental absorption is incorrect statement as fundamental lines are most intense.

51. The molecular spectra of two linear molecules O-C-O and O-C-S are recorded in the microwave region. Which one of the following statement is correct?

(a) Both the molecules would show absorption lines

(b) Both the molecules would not show absorption lines

(d) O-C-S would show absorption lines, but not O-C-O

Ans. (d)

Sol. The CO₂ molecule do not have permanent dipole moment while COS have permanent dipole moment. Thus CO₂ molecule will not show absorption lines.

52. When the refractive index of the active medium changes by in a laser resonator of length L, the change in the spectral spacing between the longitudinal modes of the laser is (c is the speed of light in free space)

(a)

(b)

(c)

(d) zero

Ans. (c)

Sol. Length of the resonator must satisfy relation L = , where is wavelength of laser light, n is number of mode. Velocity of light in a medium of refractive index µ is where v is frequency of light.

Spectral spacing between n^th and (n + 1)^th mode is

If length L is kept constant but refractive index u is changed from spectral spacing becomes

Change in spectral spacing,

53. The primitive translation vectors of the body centered cubic lattice are and . The primitive translation vectors of the reciprocal lattice are

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

Primitive translation vectors of reciprocal lattice are

54. The structure factor of a single cell of identical atoms of form factor f is given by

where (xj, yj, zj) is the coordinate of an atom, and hkl are the Miller indices. Which one of the following statement is correct for the diffraction peaks of body centered cubic (BCC) and face centered cubic (FCC) lattices?

(a)

(b)

(c)

(d)

Ans. (a)

Sol. For bcc structure: (0, 0, 0) and are atomic positions

For odd value of (h + k + l) above term will be zero. For, even value of (h + k + l), S_skt = 2f.

Therefore, for bcc structure, reflections like (100), (111). (210) etc. are missing whereas the diffraction lines corresponding to (II0), (200), (222) etc. are present.

For fee structure: An fee unit cell has four identical atoms. One of the atomsis contributed by corners and may arbitrarily be assigned coordinates (0, 0, 0), whereas other three are contributed by face-centers and have the coordinates .

From above equation (ii), it is clear structure factor is non-zero only if h, k and l are all even or all odd and has a value equal to 4. Hence reflections of the type (III), (200), (222) etc. are present, whereas those of the type (100), (110), (211) etc. are absent for an fee crystal.

55. The lattice specific heat C of a crystalline solid can be obtained using the Dulong Petit model, Einstein model and Debye model. At low temperature , which one of the following statements is true (a and A are constants)

(a)

(b)

(c)

(d)

Ans. (c)

Sol. According to Dulong-Petit law: C_V = 3R = constant ...(i)

56. A linear diatomic lattice of lattice constant a with masses M and m(M > m) are coupled by a force constant C. The dispersion relation is given by

Which one of the following statements is incorrect?

(a) The atoms vibrating in transverse mode correspond to the optical branch

(b) The maximum frequency of the acoustic branch depends on the mass of the lighter atom m

(d) No normal modes exist in the acoustic branch for any frequency greater than the maximum frequency at

Ans. (b)

Sol. Maximum frequency of aquostic branch

It depends on mass of heavier atoms, not on the mass of lighter atoms

Dispersion curves for linear diatomic lattice showing acoustical and optical modes.

From above figure, it is clear that is larger than .

57. The kinetic energy of a free electron at a corner of the first Brillouin zone of a two dimensional square lattice is larger than that of an electron at the mid-point of a side of the zone by a factor b. The value of b is

(a)

(b) b = 2

(d) b = 8

Ans. (b)

Sol. The kinetic energy of a free electron at a corner and center of the first Brillouin zone in two-dimensional square lattice is given by

The ratio of kinetic energies = b

58. An intrinsic semiconductor with mass of a hole mh and mass of an electron me is at a finite temperature T. If the top of the valence band energy is Ev and the bottom of the conduction band energy is Ec, the Fermi energy of the semiconductor is

(a)

(b)

(c)

(d)

Ans. (c)

Sol. We know that the electron density in the conduction band

And the density of hole at the valence band

For an intrinsic semiconductor the numbers of electrons in the conduction band is equal to the number of holes in the valence band only when an electron make a transition to the condition band.

Taking log both side we get

59. Choose the correct statement from the following

(a) The reaction can proceed irrespective of the kinetic energies of K+ and K–.

(b) The reaction is forbidden by the baryon number conservation

(d) The decay proceeds via weak interactions

Ans. (d)

Sol. For (a) the reaction K⁺K^– pp^– can proceed irrespective of the kinetic energies of K⁺ and K^– this statement is not correct. There should be threshold kinetic energy of the incidence particle for reaction to proceed.

For (b) K⁺ K^– pp^–

Baryon number (B): 0 + 0 → 1 – 1 = 0 (allowed). This statement is not correct.

For (c) K⁺ K^–

Strangeness (S) : 1 – 1 0 r 0 – 0 (allowed). This statement is not correct.

(d) The decay K⁰ proceeds via weak interact ion. This statement is correct as strangeness is not conserved and also = 1

Strangeness (S): + 1, 0 + 0 = 1

60. The following gives a list of pairs containing (i) a nucleus (ii) one of its properties. Find the pair which is inappropriate.

(a) (i) 10Ne20 nucleus; (ii) stable nucleus

(b) (i) A spheroidal nucleus; (ii) an electric quadrupole moment

(d) (i) U238 nucleus; (ii) Binding energy = 1785 MeV (approximately)

Ans. (c)

Sol. For (a), 10_Ne²⁰ nucleus is stable (A~ 20 is a magic number

For (b), a spheroidal nucleus shows an electric quadrupole moment.

For (c), ⁸O₁₆

For Z = N = even i.e. even-even nuclei, nucler spin J = 0 and not ½

61. The four possible configurations of neutrons in the ground state of 4Be9 nucleus, according to the shell model, and the associated nuclear spin are listed below. Choose the correct one:

(a) (1s1/2)2(1p3/2)3; J = 3/2

(b) (1s1/2)2(1p1/2)2(1p3/2)1; J = 3/2

(d) (1s1/2)2(1p3/2)2(1p1/2)1; J = 1/2

Ans. (a)

Sol. Since, ₄Be⁹ has Z = 4, N = 9 – 4 = 5

Therefore, for N = 5, (^ls_1/2)²(^lp_3/2)³

and nuclear spin

62. The mass difference between the pair of mirror nuclei 6C11 and 5B11 is given to be . According to the semi-empirical mass formula, the mass difference between the pair of mirror nuclei 9F17 and 8O17 will approximately be (rest mass of proton mp = 938.27 MeV/c2 and rest mass of neutrons mn = 939.57 MeV/c2)

(a)

(b)

(c)

(d)

Ans. (b)

Sol. For mirror nucleus ₆C¹¹ and ₅B¹¹

Mass difference between mirror nuclei is

M (A, Z) – M(A, Z – 1) = M_p – Mn + a₃A^2/3

For ₆C¹¹ and ₅B¹¹ mass difference is

Now for ₉F¹⁷ and _gO¹⁷

Mass difference = M_p – Mn + a₃ (17)^2/3

Putting value of a₃.

63. A heavy nucleus is found to contain more neutrons than protons. This fact is related to which one of the following statements

(a) The nuclear force between neutrons is stronger than that between protons

(b) The nuclear force between protons is of a shorter range than those between neutrons, so that a smaller number of protons are held together by the nuclear force

(d) It costs more energy to add a person to a (heavy) nucleus than a neutron because of the Coulomb repulsion between protons

Ans. (d)

Sol. Since, proton is charged particle. Therefore, it costs more energy to add extra proton in nucleus in order to minimize the Coulomb repulsion.

64. A neutral pi meson has a rest-mass of approximately 140 MeV/c2 and a lifetime of sec. A produced in the laboratory is found to decay after sec into two photons. Which of the following sets represents a possible set of energies of the two photons as seen in the laboratory?

(a) 70 MeV and 70 MeV

(b) 35 MeV and 100 MeV

(d) 25 MeV and 150 MeV

Ans. (c)

Sol. The life time of -meson in its rest frame . The life time with respect to Lab frame,

According to energy conservation,

mc² = E₁ + E₂ (as the rest mass of photon is zero)

65. An a.c. voltage of 220 Vrms is applied to the primary of a 10:1 step-down transformer. The secondary of the transformer is centre tapped and connected to a full wave rectifier with a load resistance. The d.c. voltage appearing across the load is

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

For full wave rectifier output will be

66. Let I1 and I2 represent mesh currents in the loop abcda and befcb respectively. The correct expression describing Kirchoff's voltage loop law in one of the following loops is

(a) 30I1 – 15I2 = 10

(b) –15I1 + 20I2 = –20

(d) –15I1 + 20I2 = 20

Ans. (a)

Sol.

Applying KVL at Loop (2)

20I₂ + 20 + 15(I₂ – I₁) = 0

–15I₁ +35I₂ = 20

Applying KVL at Loop (1)

10I₁ + 15(I₁– I₂) + 5(I₁, –2) = 0

30I₁ – 15I₂ = 10

67. The simplest logic gate circuit corresponding to the Boolean expression, is

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

Y = P + Q

{A + BG = (A + B) (A + C), by distribution theorem}

Check the output of given option i.e. y = p + q

Y = 1

68. An analog voltage V is converted into 2-bit binary number. The minimum number of comparators required and their reference voltage are

(a)

(b)

(c)

(d)

Ans. (a)

Sol. 2-bit flash "ADC"

Number of comparator

Note: (i) Number of resistor = 2ⁿ

(ii) Reference voltage are mentioned at point (a), (b) and (c) (using voltage division rule)

69. The following circuit (where RL >> R) performs the operation of

(a) OR gate for a negative logic system

(b) NAND gate for a negative logic system

(d) AND gate for a negative logic system

Ans. (d)

Sol. [Analysing CKT for Positive Logic]

Case-(1)

V₁ High V₁ = low

V₂ = low V₂ = low

Case - (3):

V₁ and V₂ High

Truth Table

Note: CKT is (+)ve logic–OR operation or, (–)ve logic AND operation.

Duality convert (+)ve logic to (–)ve logic and vice versa.

70. In the T type master-slave JK flip flop is shown along with the clock and input waveforms. The Qn output of flip flop was zero initially. Identify the correct output waveform.

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

Common Data Questions

Common Data for Questions 71, 72 and 73: A beam of identical particles of mass m and energy E is incident from left on a potential barrier of width L (between 0 < x < L) and height V0 as shown in the figure (E < V0)

For x > L there is tunneling with a transmission coefficient T > 0. Let A0, AB and AT denote the amplitudes for the incident, reflected and the transmitted waves, respectively.

71. Throughout 0 < x < L, the wave-function

(a) can be chosen to be real

(b) is exponentially decaying

(d) is zero

Ans. (a)

Sol. The schrodinger equation between 0 < x < L

72. Let the probability current associated with the incident wave be S0. Let R be the reflection coefficient. Then

(a) the probability current vanishes in the classically forbidden region

(b) the probability current is TS0 for x > L

(d) for x > L, the probability current is complex

Ans. (b)

Sol.

transmitted current = TS₀

73. The ratio of the reflected to the incident amplitude AB/A0 is

(a) 1 – AT/A0

(b)

(d)

Ans. (b)

Sol. According to energy conservation

R + T = 1

Common Data for Questions 74 and 75: Consider two concentric conducting spherical shells with inner and outer radii a, b and c, d as shown in the figure. Both the shells are given Q amount of positive charges.

74. The electric field in different regions are

(a)

(b)

(c)

(d)

Ans. (d)

Sol. Since, charge remain outer surface of a conductor.

So, E = 0 for 0 < r < a (since there are no charge for r < a)

Also, E = 0 for a < r < b (since there are no charge for a < r < b)

for b < r < c (due to charge on the outer surface of the inner sphere)

And for r > d(Q charge for outer sphere and another Q induce charge will be appear at the outer surface)

75. In order to have equal surface charge densities on the outer surfaces of both the shells, the following conditions should be satisfied

(a) d = 4b and c = 2a

(b)

(c)

(d)

Ans. (c)

Sol. Since, both surface have equal surface charge density

Statement for Linked Answer Questions 76 and 77:

Consider the -decay of a free neutron at rest in the laboratory.

76. Which of the following configurations of the decay products corresponds to the largest energy of the anti-neutrino ? (rest mass of electron me = 0.51 MeV/c2, rest mass of proton mp = 938.27 MeV/c2 and rest mass of neutron mn = 939.57 MeV/c2)

(a) In the laboratory, proton is produced at rest

(b) In the laboratory, momenta of proton, electron and the anti-neutrino all have the same magnitude

(d) In the laboratory, electron is produced at rest

Ans. (d)

Sol. P-decay of a free neutron

which is written in terms of n and p as

For anti-neutrino to have maximum energy, electron should be produced at rest, as P or nucleus Y will have almost negligible energy as it is heavier. Energy released Q is always distributed among electron and as their kinetic energies. For maximum anti–neutrino energy electron should be produced at rest.

77. Using the result of the above problem, answer the following. Which of the following represents approximately the maximum allowed energy of the anti-neutrino ?

(a) 1.3 MeV

(b) 0.8 MeV

(d) 2.0 MeV

Ans. (b)

Sol.

Q value = [m_n – (m_p + m_e + )] c² = [939.57 – (938.27 + 0.51 + 0)]c²

= 0.79. MeV = 0.8 MeV

This whole energy goes to for maximum energy.

Statement for Linked Answer Questions 78 and 79:

Consider a two dimensional electron gas of N electrons of mass m each in a system of size L × L.

78. The density of states between energy is

(a)

(b)

(c)

(d)

Ans. (a)

Sol. The density of state for 2-D electron gas is

Since electron spin degeneracy is 2 so we should multiply density of states by a factor of 2.

79. The ground state energy E0 of the system in terms of the Fermi energy EF and the number of electrons N is given by

(a)

(b)

(c)

(d)

Ans. (b)

Sol. Total number of particles is

Also the ground state energy is

Statement for Linked Answer Questions 80 and 81:

The rate of a clock in a spaceship "Suryashakti" is observed from each to be 3/5 of the rate of the clocks on earth.

80. The speed of the spaceship "Suryashakti" relative to earth is

(a)

(b)

(c)

(d)

Ans. (a)

Sol. According to question

using time dilation formula we get

81. The rate of a clock in a spaceship "Aakashganga" is observed from earth to be 5/13 of the rate of the clocks on earth. If both Aakashganga and Suryashakti are moving in the same direction relative to someone on earth, then the speed of Aakashganga relative to Suryashakti is

(a)