1. A curve is given by . The unit vector of the tangent to the curve at t = 1 is

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

2.

(a)

(b) x = 2y

(c)

(d)

Ans. (b)

Sol. f(x, y) = x³ – 2y³

Correct option is (b)

3. Two vehicles A and B are approaching an observer O at rest with equal speed as shown in the figure. Both vehicles have identical sirens blowing at a frequency f_s. The observer hears these sirens at frequency f_A and f_B, respectively from the two vehicles. Which one of the following is correct?

(a) f_A = f_B = f_s

(b) f_A = f_B > f_s

(c) f_A > f_B > f_s

(d) f_A < f_B < f_s

Ans. (b)

Sol. f_A = f_B > f_s

Correct option is (b)

4. Two boxes A and B contain an equal number of molecules of the same gas. If the volumes are V_A and V_B, and and denotes respective mean free paths, then

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

where N is number of molecules.

5. Three infinite plane sheets carrying uniform charge densities are placed parallel to the x-z plane at y = a, 3a, 4a, respectively. The electric field at the point (0, 2a, 0) is

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

Electric field at point P is due to all the surfaces

The electric field due to surface 1,2,3,4,5 and 6 are

6. There are three planets in circular orbits around a star at distances a, 4a and 9a, respectively. At time t = t₀, the star and the three planets are in a straight line. The period of revolution of the closest planet is T. How long after t₀ will they again be in the same straight line?

(a) 8T

(b) 27T

(c) 216T

(d) 512T

Ans. (c)

Sol. According to Kepler's Third Law

So, time period of revolution of three planets are

T₁ = T

T₂ = 8T

T₃ = 27T

The lcm of T₁, T₂ and T₃ is 216T

So, the configuration will repeat after 216T.

7. Which one of the following arrangements of optical components can be used to distinguish between an unpolarised light and a circularly polarised light?

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

8. Which one of the following graphs shows the correct variation of V_o with V_i? Here, V_d is the voltage drops across the diode and the Op-Amp is assumed to be ideal.

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

9. Let T_g and T_e be the kinetic energies of the electron in the ground and the third excited states of a hydrogen atom, respectively. According to the Bohr model, the ratio T_g/T_e is

(a) 3

(b) 4

(c) 9

(d) 16

Ans. (d)

Sol.

10. A current I is flowing through the sides of an equilateral triangle of side a. The magnitude of the magnetic field at the centroid of the triangle is

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Magnetic field due to finite conductor

Magnetic field due to each side

The magnetic field at the centroid

B = 3 × magnetic field due to each side

11. Consider a convex lens of focal length f. A point object moves towards the lens along its axis between f and f. If the speed of the object is V₀, then its image would move with speed V₁. Which of the following is correct?

(a) V₁ = V₀; the image moves away from the lens.

(b) V₁ = –V₀; the image moves towards the lens.

(c) V₁ > V₀; the image moves away from the lens

(d) V₁ < V₀; the image moves away from the lens

Ans. (c)

Sol.

For f < u < 2f, v > 2f

12. A disc of radius R₁ having uniform surface density has a concentric hole of radius R₂ < R₁. If its mass is M, the principal moments of inertia are

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

13.

(a)

(b)

(c) a₀ = 0, b_n = 0

(d)

Ans. (a)

Sol.

Since, the integrand is odd function.

Correct option is (a)

14. A raindrop falls under gravity and captures water molecules from atmosphere. Its mass changes at the rate m(t), where is a positive constant and m(t) is the instanteneous mass. Assume that acceleration due to gravity is constant and water molecules are at rest with respect to earth before capture. Which of the following statements is correct?

(a) The speed of the raindrop increases linearly with time

(b) The speed of the raindrop increases exponentially with time

(c) The speed of the raindrop approaches a constant value when .

(d) The speed of the raindrop approaches a constant value when .

Ans. (c)

Sol.

15.

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

16.

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

17. Which one of the figures correctly represents the T-S diagram of a Carnot engine?

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

18.

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

19. Which one of the following curves correctly represents (schematically) the solution for the equation ?

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

20. Consider an ensemble of thermodynamic systems, each of which is characterized by the same number of particles, pressure and temperature. The thermodynamic function describing the ensemble is

(a) Enthalpy

(b) Helmholtz free energy

(c) Gibb's free energy

(d) Entropy

Ans. (c)

Sol. Gibb's free energy

21. The equation of state for one mole of a non-ideal gas is given by , where the coefficients A and B are temperature dependent. If the volume changes from V₁ to V₂ in an isothermal process, the work done by the gas is

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Equation of state of non-ideal gas is given by

For isothermal process, A and B are constant

Work done by the gas

22.

(a)

(b)

(c)

(d)

Ans. (c)

Sol. For superposed wave

y = y₁ + y₂

23.

(a) A + B

(b)

(c)

(d) AB

Ans. (c)

Sol. Let us make a truth table

24. A particle P of mass m is constrained to move on the surface of a cylinder under a force as shown in figure (k is the positive constant). Which of the following statements is correct? (Neglect friction)

(a) Total energy of the particle is not conserved

(b) The motion along z-direction is simple harmonic

(c) Angular momentum of the particle about Q increases with time

(d) Linear momentum of the particle is conserved

Ans. (b)

Sol.

So, motion along z direction will be simple harmonic.

25.

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Consider a Gaussian surface of radius r < R

Now, let us find the value of k

26. The plane of polarization of a plane polarized light rotates by 60º after passing through a wave plate. The pass-axis of the wave plate is at an angle with respect to the plane of polarization of the incident light. The wave plate and are

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Only a half wave plate rotates the plane of polarization of light.

Correct option is (a)

27. A long solenoid is carrying a time dependent current such that the magnetic field inside has the form is along the axis of the solenoid. The displacement current at the point P on a circle of radius r in a plane perpendicular to the axis

(a) is inversely proportional to r and radially outward

(b) is inversely proportional to r and tangential

(c) increases linearly with time and is tangential

(d) is inversely proportional to r² and tangential

Ans. (b)

Sol. According to Faraday law

Displacement current density

28. An ideal gas consists of three dimensional polyatomic molecules. The temperature is such that only one vibrational mode is excited. If R denotes the gas constant, then the specific heat at constant volume of one mole of the gas at this temperature is

(a) 3R

(b)

(c) 4R

(d)

Ans. (c)

Sol. One vibrational mode consists of 2 degree of freedom

Degree of freedom of polyatomic molecules

= 3 + 3 + 2 = 8

29. The mean momentum of a nucleon in a nucleus of mass number A and atomic number Z depends on A, Z as

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

Correct option is (c)

30. A rectangular loop of dimensions l and w moves with a constant speed of v through a region containing a uniform magnetic field B directed into the paper and extending a distance of 4w. Which of the following figures correctly represents the variation of emf with the position (x) of the front end of the loop?

(a)

(b)

(c)

(d)

Ans. (c)

Sol. If we take anticlockwise orientation of loop as positive which usually is norm, then emf would be positive when loop enters the field region and negative loop leaves the field region.

31. Which of the combinations of crystal structure and their coordination number is (are) correct?

(a) body centered cubic-8

(b) face centred cubic–6

(c) diamond-4

(d) hexagonal closed packed–12

Ans. (a, c, d)

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

32. Which of the following relations is (are) true for thermodynamic variables?

(a)

(b)

(c) dF = –SdT + PdV

(d) dG = –SdT + V dP

Ans. (a, b, d)

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

33. Two beams of light in the visible range (400 nm - 700 nm) interfere with each other at a point. The optical path difference between them is 500 nm. which of the following wavelength will interfere constructively at the given point?

(a) 416.67 nm

(b) 555.55 nm

(c) 625 nm

(d) 666.66 nm

Ans. (a, b, c)

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

For constructive interference, optical path difference,

34. Consider a convex lens of focal length f. The lens is cut along a diameter in two parts. The two lens parts and an object are kept as shown in the figure. The images are formed at following distances from the object.

(a) 2f

(b) 3f

(c) 4f

(d)

Ans. (d)

Sol. Image formed due to upper part will be at infinity and image formed due to lower part will be at 4f. One more image will be formed at 3f which will be due to rays which are converged by first lens and further converged by second lens.

Correct option is (d)

35. In presence of a magnetic field and an electric field , a particle moves undeflected. Which of the following statements is (are) correct?

(a)

(b)

(c)

(d)

Ans. (b, d)

Sol.

36.

(a) M is symmetric

(b) M is invertible

(c) One eigenvalue is 13

(d) Its eigenvectors are orthogonal

Ans. (a, c, d)

Sol. det(M) = 0

The matrix is symmetric and hence its eigenvectors are orthogonal. The eigenvalues are 13 and 0.

Correct options are (a, c, d)

37. Let the electric field in some region R be given by . From this we may conclude that

(a) R has a non-uniform charge distribution

(b) R has no charge distribution

(c) R has a time dependent magnetic field

(d) The energy flux in R is zero everywhere

Ans. (b, c)

Sol.

So, R has no charge distribution

So, B is not space dependent.

38. Let f(x) = 3x⁶ – 2x² – 8. Which of the following statements is (are) true?

(a) The sum of all its roots is zero

(b) The product of its root is

(c) The sum of all its root is

(d) Complex roots are conjugates of each other

Ans. (a, b, d)

Sol.

Since, all the coefficients are real so, the complex roots are conjugates of each other.

Correct options are (a, b and d)

39. In a pn junction dopant concentration on the p-side is higher than on the n-side. Which of the following statements is (are) correct, when the junction is unbiased?

(a) The width of the depletion layers is larger on the n-side

(b) At thermal equilibrium the Fermi energy is higher on the p-side

(c) In the depletion region number of negative charge per unit area on the p-side is equal to number of positive charges per unit area on the n-side.

(d) The value of the built-in potential barrier depends on the dopant concentration.

Ans. (a, c, d)

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

40. Two projectiles of identical mass are projected from the ground with same initial angle with respect to earth surface and same initial velocity (u) in the same plane. They collide at the highest point of their trajectories and stick to each other. Which of the following statements is (are) correct?

(a) The momentum of the combined object immediately after the collision is zero

(b) Kinetic energy is conserved in the collision

(c) The combined object moves vertically downward

(d) The combined object moves in a parabolic path.

Ans. (a, c)

Sol. Correct option is (a, c)

41. For the given circuit value of the base current (I_b) of the npn transistor will be ______ mA. ( is the current gain and assume Op-Amp as ideal)

(Specify your answer in mA upto two digits after the decimal point).

Ans. (0.1)

Sol.

42. The coefficient of x³ in the Taylor expansion of sin(sin x) around x = 0 is ______. (Specify your answer upto two digits after the decimal point.)

Ans. (–0.33)

Sol.

f(x) = sin(sin x)

f'(x) = cos(sinx).cosx

f"(x) = –sin(sin x)cos²x – cos(sinx)sinx

f'''(x) = –cos(sin x)cos³x + sin(sinx) 2cos x sin x + sin(sin x)sin x cos x – cos(sin x)cos x

f"'(0) = –2

Correct answer is (–0.33)

43. Consider the first order phase transition of the sublimation of zinc. Assume the vapor to be an ideal gas and the molar volume of solid to be negligible. Experimentally, it is found that , where P is the vapor pressure Pascal, T is the vapor pressure in Pascal, T is in K, C₁ = 6790K and C₂ = 9. The latent heat of sublimation of zinc from the Clausius-Clapeyron equation is _______kJ/mole. (R = 8.314J/moleK)

(Specify your answer in kJ/mole upto one digit after the decimal point)

Ans. (130)

Sol. According to Clausius-Clapeyron equaiton

= C₁R log_e 10 = 6790 × 8.314 × 2.303

= 130 kJ

44. A system of 8 non-interacting electrons is confined by a three dimensional potential . The ground state energy of the system is units of is ________. (Specify your answer as an integer).

Ans. (18)

Sol. The energy eigenvalues are given by

Two electrons will occupy first energy level and six electrons will occupy second energy level

So, ground state energy, E = 2 × E₁ + 6 × E₂

45. In a grating with grating constant d = a + b, where a is the slit width and b is the separation between the slits, the diffraction pattern has the fourth order missing. The value of is ______. (Specify your answer as an integer.)

Ans. (3)

Sol. Position of diffraction minima

Position of interference maxima,

If 4^th order is missing then 4^th order is coinciding with first order

46. The lattice constant of unit cell of NaCl crystal is 0.563 nm. X-rays of wavelength 0.141 nm are diffracted by this crystal. The angle at which the first order maximum occurs is ____ degree. (Specify your answer in degrees upto two digits after the decimal point).

Ans. (7.19°)

Sol. According to Bragg's law

For NaCl crystal, d_hkl = a

47. Consider monoatomic ideal gas operating in a closed cycle as shown in the P-V diagram given below the ratio is _______. (Specify your answer upto digits after the decimal point).

Ans. (0.16)

Sol.

48. Consider an electromagnetic plane wave is the wavelength, c is the speed of light and b is a constant. The value of b is _____.

(Specify your answer upto two digits after the decimal point).

Ans. (0.577)

Sol.

49. A planet has average density same as that of the earth but it has only 1/8 of the mass of the earth. If the acceleration due to gravity at the surface is g_p and g_e for the planet and earth, respectively, then _______. (Specify your answer upto one digit after the decimal point).

Ans.

Sol.

50.

Ans.

Sol. The position of equilibrium are given by

51. A syringe is used to exert 1.5 atmospheric pressure to release water horizontally. The speed of water immediately after ejection is _____. (take 1 atmospheric = 10⁵ Pascal, density of water 10³kg/m³) (Specify your answer in ms^–1 as an integer.)

Ans. (10)

Sol. Applying Bernoulli principle at point A and B

52. Two events E₁ and E₂ take place in an inertial frame S with respective time-space coordinates (in SI units) : and E₂ (t₂ = 0, x₂ = 10⁸, y₂ = 0, z₂ = 0). Another intertial frame S' is moving with respect to S with a velocity . The time difference (t'₂ – t'₁) as observed in S' is _____s. (c = 3 × 10⁸ms^–1). (Specifiy your answer in seconds upto digits after the decimal point).

Ans. (–4)

Sol.

53. A body of mass 1 kg is moving under a central force in an elliptic orbit with semi major axis 1000 m and semi minor axis 100 m. The orbital angular momentum of the body is 100 kg m²s^–1. The time period of motion of the body is _______ hours.

(Specify your answer in hours upto two digits after the decimal point.)

Ans. (3.034)

Sol.

54. The moon moves around the earth in a circular orbit with a period of 27 days. The radius of the earth (R) is 6.4 × 10⁶m and the acceleration due to gravity on earth surface is 9.8 ms^–2. If D is the distance of the moon from the center of the earth, the value of D/R will be _____. (Specify your answer upto one digit after the decimal point).

Ans. (59.56)

Sol. For circular orbit

55.

Ans. (0.577)

Sol. Let us attach a frame S' to R₁.

Velocity of R₂ in frame attached to R₁

Length of R₂ is frame attached to R₁

56. Consider a slit of width 18µm which is being illuminated simultaneously with light of orange color (wavelength 600 nm) and of blue color (wavelength 450 nm). The diffraction pattern is observed on a screen kept at a distance in front of the slit. The smallest angle at which only the orange color is observed is and the smallest angle at which only the blue color is observed is . The angular difference (in degrees) is ______.

Ans. (0.477°)

Sol. Only orange color will be observed when central maxima of blue will end and only blue color would be obtained when central maxima of orange will end

57.

Ans. (5.28)

Sol. In S₁ frame,

Velocity of particle in S' frame

58. In the following circuit, the time constant RC is much greater than the period of the input. signal. Assume diode as ideal and resistance R to be large. The dc output voltage across resistance R will be ____V.

(Specify your answer in volts upto one digits after the decimal point).

Ans. (12)

Sol. Correct answer is 12.

59. For a metal, the electron density is 6.4 × 10²⁸ m^–3. The Fermi energy is ____ eV. (h = 6.626 × 10^–34 Js, m_e = 9.11 × 10^–31kg, 1eV = 1.6 × 10^–19J) (Specify your answer in electron volts (eV) upto one digit after the decimal point).

Ans. (5.85)

Sol.

60. For the following circuit, the collector voltage with respect to ground will be _______V. (Emitter diode voltage is 0.7V and of the transistor is large).

(Specify your answer in volts upto one digit after the decimal point).

Ans. (3.1)

Sol.