1. A classical particle has total energy E. The plot of potential energy (U) as a function of distance (r) from the centre of force located at r = 0 is shown in the figure. Which of the regions are forbidden for the particle?

(a) I and II

(b) II and IV

(c) I and IV

(d) I and III

Ans. (d)

Sol. In the region I and III potential energy is more than total energy.

Correct option is (d)

2. For using a transistor as an amplifier, choose the correct option regarding the resistances of base-emitter (R_BE) and base-collector (R_BC) junctions

(a) Both R_BE and R_BC are very low

(b) Very low R_BE and very high R_BC

(c) Very high R_BE and very low R_BC

(d) Both R_BE and R_BC are very high

Ans. (b)

Sol. Correct option is (b)

3. Isothermal compressibility is given by

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Correct option is (c)

4. In a heat engine based on the Carnot cycle, heat is added to the working substance at constant

(a) Entropy

(b) Pressure

(c) Temperature

(d) Volume

Ans. (c)

Sol. Correct option is (c)

5.

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Correct option is (c)

6. If (x,y,z) is a scalar function which satisfies the Laplace equation, then the gradient of is

(a) Solenoidal and irrotational

(b) Solenoidal but not irrotational

(c) Irrotational but not solenoidal

(d) Neither solenoidal nor irrotational

Ans. (a)

Sol.

Correct option is (a)

7. A unit vector perpendicular to the plane containing

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

Correct option is (d)

8. Which one of the following crystallographic planes represent (101) Miller indices of a

(a)

(b)

(c)

(d)

Ans. (b)

Sol. Plane intercepts

Plane is parallel to y – axis and intersecting x and z – axis at a and c. Thus, option (b) is correct.

9. In the thermal neutron induced fission of ²³⁵U, the distribution of relative number of the observed fission fragments (Yield) versus mass number (A) is given by

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Correct option is (a)

10. The function is continuous everywhere except at

(a) x = 0

(b) x = ±9

(c) x = ± 9i

(d) x = ±3i

Ans. (d)

Sol. We know that a rational function is discontinuous at a point where the denominator is 0.

Therefore,

Correct option is (d)

11. Which one of the following is an impossible magnetic field ?

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

Correct option is (d)

12. The mass per unit length of a rod (length 2m) varies as = 3x kg/m. The moment of inertia (in kg m²) of the rod about a perpendicular-axis passing through the tip of the rod (at x = 0) is

(a) 10

(b) 12

(c) 14

(d) 16

Ans. (b)

Sol.

Correct option is (b)

13. The maximum number of intensity minima that can be observed in the Fraunhofer diffraction pattern of a single slit (width 10 µm) illuminated by a laser beam (wavelength 0.630 µm) will be

(a) 4

(b) 7

(c) 12

(d) 15

Ans. (d)

Sol.

Correct option is (d)

14. A small spherical ball having charge q and mass m, is tied to a thin massless non-conducting string of length l. The other end of the string is fixed to an infinitely extended thin non-conducting sheet with uniform surface charge density . Under equilibrium, the string makes an angle 45º with the sheet as shown in the figure. Then is given by (g is the acceleration due to gravity and is the permittivity of free space)

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

Correct option is (c)

15.

(a) circular

(b) Elliptical

(c) Helical

(d) Spiral

Ans. (c)

Sol.

Correct option is (c)

16.

(a) 45º

(b) 60º

(c) 30º

(d) 90º

Ans. (c)

Sol.

Correct option is (c)

17. A thin lens of refractive index 3/2 is kept inside a liquid of refractive index 4/3. If the focal length of the lens in air is 10 cm, then its focal length inside the liquid is

(a) 10 cm

(b) 30 cm

(c) 40 cm

(d) 50 cm

Ans. (c)

Sol.

f₁ = 4f_a = 4 × 10 = 40cm

Correct option is (c)

18. During the charging of a capacitor C in a series RC circuit, the typical variations in the magnitude of the charge q(t) deposited on one of the capacitor plates, and the current i(t) in the circuit, respectively are best represented by

(a) Fig. I and Fig. II

(b) Fig. I and Fig. IV

(c) Fig. III and Fig. II

(d) Fig. III and Fig. IV

Ans. (a)

Sol. Correct option is (a)

19. The output of following logic circuit can be simplified to

(a) X + YZ

(b) Y + XZ

(c) XYZ

(d) X + Y + Z

Ans. (b)

Sol. Output = XY + X (Y + Z) + Y (Y + Z) = XY + XY + XZ + Y + YZ

= Y[1 + Z + X + X] + XZ = Y + XZ

Correct option is (b)

20. A red star having radius r_R at a temperature T_R and a white star having radius r_W at a temperature T_W, radiate the same total power. If these stars radiate as perfect black bodies, then

(a) r_R > r_W and R_R > T_W

(b) r_R < r_W and T_R > T_W

(c) r_R > r_W and T_R < T_W

(d) r_R < r_W and T_R < T_W

Ans. (c)

Sol.

Correct option is (c)

21. Consider the normal incidence of a plane electromagnetic wave with electric field given by over an interface at z = 0 separating two media [wave velocities v₁ and v₂(v₂ > v₁) and wave vectors k₁ and k₂, respectively] as shown in figure. The magnetic field vector of the reflected wave is

( is the angular frequency)

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

Correct option is (c)

22.

(a) 2, 4 and 6

(b) 2i, 4i and 6

(c) 2i, 4 and 8

(d) 0, 4 and 8

Ans. (a)

Sol. For calculation of eigenvalues

Correct option is (a)

23. For a quantum particle confined inside a cubic box of side L, the ground state energy is given by E₀. The energy of the first excited state is

(a) 2E₀

(b)

(c) 3E₀

(d) 6E₀

Ans. (d)

Sol.

Correct option is (d)

24. The amount of work done to increase the speed of an electron from c/3 to 2c/3 is

(c = 3 × 10⁸ m/s and rest mass of electron is 0.511 MeV)

(a) 56.50 keV

(b) 143.58 keV

(c) 168.20 keV

(d) 511.00 keV

Ans. (b)

Sol. Change in kinetic energy is equal to work done

put v₁ = c/3, v₂ = 2c/3 = 0.511

W = 143.58 keV

Correct option is (b)

25. During free expansion of an ideal gas under adiabatic condition, the internal energy of the gas

(a) Decreases

(b) Initially decreases and then increases

(c) Increases

(d) Remains constant

Ans. (d)

Sol.

Work is done at the expense of internal energy.

Correct option is (d)

26. A ball of mass m is falling freely under gravity through a viscous medium in which the drag force is proportional to the instantaneous velocity v of the ball. Neglecting the buoyancy force of the medium, which one of the following best describes the variation of v as a function of time t?

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

Correct option is (d)

27. For a forward biased p-n junction diode, which one of the following energy-band diagrams is correct?

( is the Fermi energy)

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Correct option is (a)

28.

(a) 1.5

(b) 2.0

(c) 3.0

(d) 4.0

Ans. (b)

Sol.

Correct option is (b)

29. In the given phase diagram for a pure substance, regions I, II, III, IV, respectively represent

(a) Vapor, Gas, Solid, Liquid

(b) Gas, Vapor, Liquid, Solid

(c) Gas, Liquid, Vapor, Solid

(d) Vapor, Gas, Liquid, Solid

Ans. (b)

Sol. IV – Solid

III – Liquid

II – Vapour

I – Gas (superheated dry vapour)

Correct option is (b)

30. The location of Cs⁺ and Cl^– ions inside the unit cell of CsCl crystal is shown in the figure. The Bravais lattice of CsCl is

(a) simple cubic

(b) body centered orthorhombic

(c) face centered cubic

(d) base centered orthorhombic

Ans. (a)

Sol. Cesium – Chloride is made of two interpenetrating simple cubic lattices are displaced diagonally by half of the diagonal length. Thus, Bravais lattice of CsCl is simple cubic. The correct option is (a)

31. Which of the following statement(s) is/are true?

(a) Newton's laws of motion and Maxwell's equations are both invariant under Lorentz transformations.

(b) Newton's laws of motion and Maxwell's equations are both invariant under Galilean transformations.

(c) Newton's laws of motion are invariant under Galilean transformations and Maxwell's equations are invariant under Lorentz transformations.

(d) Newton's laws of motion are invariant under Lorentz transformations and Maxwell's equations are invariant under Galilean transformations.

Ans. (c)

Sol. Correct option is (c)

32. In the X-ray diffraction pattern recorded for a simple cubic (lattice parameter a = 1 Å) using X rays of wavelength 1 Å, the first order diffraction peak(s) would appear for the

(a) (100) planes

(b) (112) planes

(c) (210) planes

(d) (220) planes

Ans. (a)

Sol. In simple cubic cell, planes are present. The first order diffraction peak would appear for the first plane (100).

Correct option is (a)

33. For an underdamped harmonic oscillator with velocity v(t).

(a) Rate of energy dissipation varies linearly with v(t)

(b) Rate of energy dissipation varies as square of v(t)

(c) The reduction in the oscillator frequency, compared to the undamped case, is independent of v(t)

(d) For weak damping, the amplitude decays exponentially to zero

Ans. (b, c, d)

Sol.

Power dissipation is proportional to v², thus option (a) is wrong and option (b) is correct.

Also displacement decays exponentially to zero, thus option (d) is also correct.

The damped oscillation frequency is

It is independent of v(t). Thus option (c) is also correct

Correct option is (b), (c), (d)

34. The relation between the nuclear radius (R) and the mass number (A), given by R = 1.2 A^1/3 fm, implies that

(a) The central density of nuclei is independent of A

(b) The volume energy per nucleon is a constant

(c) The attractive part of the nuclear force has a long range

(d) The nuclear force is charge dependent

Ans. (a, b, d)

Sol. Correct option is (a), (b), (d)

35. A thermodynamic system is described by the P, V, T coordinates. Choose the valid expression(s) for the system.

(a)

(b)

(c)

(d)

Ans. (a, c)

Sol. Correct option is (a), (c)

36. Out of the following statements, choose the correct option(s) about a perfect conductor.

(a) The conductor has an equipotential surface

(b) Net charge, if any, resides only on the surface of conductor

(c) Electric field cannot exist inside the conductor

(d) Just outside the conductor, the electric field is always perpendicular to its surface

Ans. (a, b, c, d)

Sol. Correct option is (a), (b), (c), (d)

37. An atomic nucleus X with half-life Tx decays to a nucleus Y, which has half-life T_Y. The condition(s) for secular equilibrium is(are)

(a)

(b) T_X < T_Y

(c) T_X << T_Y

(d) T_Y >> T_Y

Ans. (d)

Sol. Correct option is (d)

38. The gradient of a scalar field S(x, y, z) has the following characteristic(s).

(a) Line integral of a gradient is path-independent

(b) Closed line integral of a gradient is zero

(c) Gradient of S is a measure of the maximum rate of change in the field S

(d) Gradient of S is a scalar quantity

Ans. (a, b, c)

Sol. Correct option is (a), (b), (c)

39. Consider an object moving with a velocity in a frame which rotates with a constant angular velocity . The Coriolis force experienced by the object is

(a)

(b)

(c)

(d) always directed towards the axis of rotation

Ans. (c)

Sol.

Correct option is (c)

40. Consider a classical particle subjected to an attractive inverse-square force field. The total energy of the particle is E and the eccentricity is . The particle will follow a parabolic orbit if

(a) E > 0 and = 1

(b) E < 0 and < 1

(c) E = 0 and = 1

(d) E < 0 and = 1

Ans. (c)

Sol.

Correct option is (c)

41. For the input voltage V_i = (200 mV) sin (400t), the amplitude of the output voltage (V₀) of the given Op-Amp circuit is ________V.

(Round off to 2 decimal places)

Ans. (11.03)

Sol.

Correct answer is (11.03)

42. Light of wavelength 680 nm is incident normally on a diffraction grating having 4000 lines/cm. The diffraction angle (in degrees) corresponding to the third-order maximum is ______.

(Round off to 2 decimal places)

Ans. (55⁰)

Sol.

Correct answer is (55⁰)

43. Two gases having molecular diameters D₁ and D₂, and mean free paths respectively, are trapped separately in identical containers. If D₂ = 2D₁, then = ________.

(Assume there is no change in other thermodynamic parameters)

Ans. (4)

Sol.

Correct answer is (4)

44. Sodium (Na) exhibits body-centered-cubic (BCC) crystal structure with atomic radius 0.186 nm. The lattice parameter of Na unit cell is ________ nm.

(Round off to 2 decimal places)

Ans. (0.43)

Sol.

Correct answer is (0.43)

45. In a typical human body, the amount of radioactive ⁴⁰K is 3.24 × 10^–5 percent of its mass. The activity due to ⁴⁰K in a human body of mass 70 kg is ______ kBq.

(Round off to 2 decimal places)

(Half-life of ⁴⁰K = 3.942 × 10¹⁶ s. Avogadro's number N_A = 6.022 × 10²³ mol^–1)

Ans. (6.0)

Sol.

= 6.0 × 10¹³ disintegration / s

= 6.0 × 10¹³ Bq

= 6.0 × 10¹⁰ Bq

Correct answer is (6.0)

46. The flux of the function passing through the surface ABCD along is _________.

(Round off to 2 decimal places)

Ans. (1.17)

Sol. y = 1 plane

Correct answer is (1.17)

47. The 7^th bright fringe in the Young's double slit experiment using a light of wavelength 550 nm shifts to the central maxima after covering the two slits with two sheets of different refractive indices n₁ and n₂ but having same thickness 6 µm. The value of |n₁ – n₂| is _________.

(Round off to 2 decimal places)

Ans. (0.64)

Sol.

Correct answer is (0.64)

48. An object of 2 cm height is placed at a distance of 30 cm in front of a concave mirror with radius of curvature 40 cm. The height of the image is ________ cm.

Ans. (4)

Sol. u = – 30 cm

f = – 20 cm

Correct answer is (4)

49. An infinitely long very thin straight wire carries uniform line charge density × 10^–2 C/m. The magnitude of electric displacement vector at a point located 20 mm away from the axis of the wire is _______ C/m².

Ans. (2)

Sol.

Correct answer is (2)

50. The electrostatic energy (in units of J) of a uniformly charged spherical shell of total charge 5C and radius 4 m is _______.

(Round off to 3 decimal places)

Ans. (3.125)

Sol.

Correct answer is (3.125)

51. Given the wave function of a particle for 0 < x < L and 0 elsewhere, the probability of finding the particle between x = 0 and x = L/2 is ________.

(Round off to 1 decimal place)

Ans. (0.125)

Sol.

Correct answer is (0.5)

52. If the wavelength of X-ray line of an element is 1.544Å, then the atomic number (Z) of the element is _________.

(Rydberg constant R = 1.097 × 10⁷m^–1 and velocity of light c = 3×10⁸ m/s)

Ans. (29)

Sol. According to Mosely's formula, the frequency of K α X – ray line is related to atomic number by the formula

Therefore, z – 1 = 28.06

or z = 29.06

Since atomic number must be an integer

z = 29

Correct answer is (29)

53. The value of , along the line 3y = x, where z = x + iy is _______.

(Round off to 1 decimal place)

Ans. (111.1)

Sol.

z = x + iy

z = 3y + iy

dz = 3dy + idy = (3 + i) dy

Correct answer is (111.1)

54. The decimal equivalent of the binary number 110,101 is ________.

Ans. (6.625)

Sol.

Correct answer is (6.625)

55. A di-atomic gas undergoes adiabatic expansion against the piston of a cylinder. As a result, the temperature of the gas drops from 1150 K to 400 K. The number of moles of the gas required to obtain 2300 J of work from the expansion is _______.

(The gas constant R = 8.314 J mol^–1K^–1.)

(Round off to 2 decimal places)

Ans. (0.1475)

Sol.

Correct answer is (0.1475)

56.

Ans. (2)

Sol.

Correct answer is (2)

57. The Zener current I_z for the given circuit is __________ mA.

Ans. (1)

Sol.

Correct answer is (1)

58. The value of emitter current in the given circuit is ________ µA.

(Round off to 1 decimal place)

Ans. (444.9)

Sol.

Correct answer is (444.9)

59. A proton is confined within a nucleus of size 10^–13 cm. The uncertainty in its velocity is ___________ × 10⁸ m/s.

(Round off to 2 decimal places)

(Planck's constant h = 6.626 × 10^–34 J s and proton mass m_p = 1.672 × 10^–27 kg)

Ans. (0.31)

Sol.

Correct answer is (0.31)

60. If the diameter of the Earth is increased by 4% without changing the mass, then the length of the day is _______ hours.

(Take the length of the day before the increment as 24 hours. Assume the Earth to be a sphere with uniform density.)

(Round off to 2 decimal places)

Ans. (25.95)

Sol.

Correct answer is (25.95)