Graduate Aptitude Test in Engineering 2017

Question Paper Name: Physics 5th Feb 2017

Subject Name: Physics

Duration : 180

Total Marks: 100

1. Identical charges q are placed at five vertices of a regular hexagon of side a. The magnitude of the electric field and the electrostatic potential at the centre of the hexagon are respectively

(A)

(B)

(C)

(D)

Ans: (C)

2. A parallel plate capacitor with square plates of side 1 m separated by 1 micro meter is filled with a medium of dielectric constant of 10. If the charges on the two plates are 1C and −1C, the voltage across the capacitor is _____kV. (upto two decimal places). (ε₀ = 8.854 × 10⁻¹² F/m)

Ans: (11.25 to 11.34)

3. Light is incident from a medium of refractive index n = 1.5 onto vacuum. The smallest angle of incidence for which the light is not transmitted into vacuum is ____ degrees. (up to two decimal places).

Ans: (41.60 to 42.00)

4. A monochromatic plane wave in the free space with electric field amplitude of 1 V/m is normally incident on a fully reflecting mirror. The pressure exerted on the mirror is _______× 10⁻¹² (up to two decimal places) (ε₀ = 8.854 × 10⁻¹² F/m).

Ans: (8.80 to 8.90)

5. The best resolution that a 7 bit A/D convertor with 5 V full scale can achieve is _____mV. (up to two decimal places).

Ans: (39.30 to 39.50)

6. In the figure given below, the input to the primary of the transformer is a voltage varying sinusoidally with time. The resistor R is connected to the centre tap of the secondary. Which one of the following plots represents the voltage across the resistor R as a function of time?

(A)

(B)

(C)

(D)

Ans: (A)

7. The atomic mass and mass density of Sodium are 23 and 0.968 g cm⁻³, respectively. The number density of valence electrons is ______×10²² cm⁻³. (Up to two decimal places.)

(Avogadro number, N_A = 6.022 × 10²³)

Ans: (2.50 to 2.55)

8. Consider a one-dimensional lattice with a weak periodic potential The gap at the edge of the Brillouin zone is :

(A) U₀

(B) U₀/2

(D) U₀/4

Ans: (A)

9. Consider a triatomic molecule of the shape shown in the figure below in three dimensions. The heat capacity of this molecule at high temperature (temperature much higher than the vibrational energy scales of the molecule but lower than its bond dissociation energies) is:

(A)

(B) 3k_B

(C)

(D) 6k_B

Ans: (D)

10. If the Lagrangian is modified to which one of the following is TRUE?

(A) Both the canonical momentum and equation of motion do not change

(B) Canonical momentum changes, equation of motion does not change

(D) Both the canonical momentum and equation of motion change

Ans: (B)

11. Two identical masses of 10 gm each are connected by a massless spring of spring constant 1 N/m. The non-zero angular eigenfrequency of the system is ______ rad/s. (up to two decimal places).

Ans: (14.10 to 14.20)

12. The phase space trajectory of an otherwise free particle bouncing between two hard walls elastically in one dimension is a

(A) straight line

(B) parabola

(D) circle

Ans: (C)

13. The Poisson bracket [x, xp_y + yp_x] is equal to

(A) −x

(B) y

(D) p_y

Ans: (B)

14. The wavefunction of which orbital is spherically symmetric :

(A) p_x

(B) p_y

(D) d_xy

Ans: (C)

15. The contour integral evaluated along a contour going from −∞ to +∞ along the real axis and closed in the lower half-plane by a half circle is equal to _____.(up to two decimal places).

Ans: (3.13 to 3.15)

16. The Compton wavelength of a proton is ____ fm. (up to two decimal places).

(m_p = 1.67 × 10⁻²⁷ kg, h = 6.626 × 10⁻³⁴ Js, ,e = 1.602 × 10⁻¹⁹ C, c = 3 × 10⁸ ms⁻¹)

Ans: (1.30 to 1.34)

17. Which one of the following conservation laws is violated in the decay τ⁺ → μ⁺μ⁺μ⁻

(A) Angular momentum

(B) Total Lepton number

(D) Tau number

Ans: (D)

18. Electromagnetic interactions are :

(A) C conserving

(B) C non-conserving but CP conserving

(D) CPT non-conserving

Ans: (A)

19. A one dimensional simple oscillator with Hamiltonian is subjected to a small perturbation, H₁ = αx + βx³ + γx⁴. The first order correction to the ground state energy is dependent on

(A) only β

(B) α and γ

(D) only γ

Ans: (D)

20. For the Hamiltonian where a₀ ∈ R, is a real vector, I is the 2 × 2 identity matrix, and are the Pauli matrices, the ground state energy is

(A) |b|

(B) 2a₀ − |b|

(D) a₀

Ans: (C)

21. The coefficient of e^ikx in the Fourier expansion of u(x) = Asin² (αx) for k = −2α is

(A) A/4

(B) −A/4

(D) −A/2

Ans: (B)

22. The degeneracy of the third energy level of a 3-dimensional isotropic quantum harmonic oscillator is

(A) 6

(B) 12

(D) 10

Ans: (A)

23. The electronic ground state energy of the Hydrogen atom is −6 eV. The highest possible electronic energy eigenstable has an energy equal to

(A) 0

(B) 1 eV

(D) ∞

Ans: (D)

24. A reversible Carnot engine is operated between temperatures T₁ and T₂ (T₂ > T₁) with a photon gas as the working substance. The efficiency of the engine is

(A)

(B)

(C)

(D)

Ans: (B)

25. In the nuclear reaction ¹³C₆ + v_e → ¹³N₇ + X, the particle X is

(A) an electron

(B) an anti-electron

(D) a pion

Ans: (A)

26. Three charges (2C, −1C, −1C) are placed at the vertices of an equilateral triangle of side 1m as shown in the figure. The component of the electric dipole moment about the marked origin along the direction is ____ Cm.

27. An infinite solenoid carries a time varying current I(t) = At², with A ≠ 0. The axis of the solenoid is along the are the usual radial and polar directions in cylindrical polar coordinates. is the magnetic field at a point outside the solenoid. Which one of the following statements is true?

(A) B_r = 0, B_θ = 0, B_z = 0

(B) B_r ≠ 0, B_θ ≠ 0, B_z = 0

(D) B_r = 0, B_θ = 0, B_z ≠ 0

Ans: (D)

28. A uniform volume charge density is placed inside a conductor (with resistivity 10⁻² Ωm). The charge density becomes 1/(2.718) of its original value after time _____ femto seconds. (up to two decimal places) (ε₀ = 8.854 × 10⁻¹² F/m)

Ans: (87.50 to 89.50)

29. Water freezes at 0℃ at atmospheric pressure (1.01 × 10⁵ Pa). The densities of water and ice at this temperature and pressure are 1000 kg/m³ and 934 kg/m³ The latent heat of fusion is 3.34 × 10⁵ J/kg. The pressure required for depressing the melting temperature of ice by 10℃ is _____ GPa. (up to two decimal places)

Ans: (0.15 to 0.19)

30. The minimum number of NAND gates required to construct an OR gate is

(A) 2

(B) 4

(D) 3

Ans: (D)

31. Consider a 2-dimensional electron gas with a density of 10¹⁹ m⁻². The Fermi energy of the system is _____ eV(up to two decimal places).

(m_e = 9.31 × 10⁻³¹ kg, h = 6.626 × 10⁻³⁴ Js, e = 1.602 × 10⁻¹⁹C)

Ans: (2.32 to 2.38)

32. The total energy of an inert-gas crystal is given by , were R is the inter-atomic spacing in Angstroms. The equilibrium separation between the atoms is ______ Angstroms. (up to two decimal places).

Ans: (0.90 to 1.10)

33. Consider N non-interacting, distinguishable particles in a two-level system at temperature T. The energies of the levels are 0 and ε, where ε > 0. In the high temperature limit (k_BT >> ε), what is the population of particles in the level with energy ε?

(A) N/2

(B) N

(D) 3N/4

Ans: (A)

34. A free electron of energy 1 eV is incident upon a one-dimensional finite potential step of h eight 0.75 eV. The probability of its reflection from the barrier is ______(up to two decimal places).

Ans: (0.10 to 0.12)

35. Consider a one-dimensional potential well of width 3 mm. Using the uncertainty principle (∆x∙∆p ≥ h/2), an estimate of the minimum depth of the well such that it has at least one bound state for an electron is (m_e = 9.31 × 10⁻³¹ kg, h = 6.626 × 10⁻³⁴ J s, e = 1.602 × 10⁻¹⁹ C):

(A) 1 μeV

(B) 1 meV

(D) 1 MeV

Ans: (B)

36. Consider a metal with free electron density of 6 × 10²² cm⁻³. The lowest frequency electromagnetic radiation to which this metal is transparent is 1.38 × 10¹⁶ If this metal had a free electron density of 1.8 × 10²³ cm⁻³ instead, the lowest frequency electromagnetic radiation to which it would be transparent is _________× 10¹⁶ Hz. (up to two decimal places).

Ans: (2.35 to 2.45)

37. An object travels along the x-direction with velocity c/2 in a frame O. An observer in a frame O’ sees the same object travelling with velocity c/4. The relative velocity of Oʹ with respect to O in units of c is ______.(up to two decimal places).

Ans: (0.27 to 0.31)

38. The integral is equal to ______.(up to two decimal places).

Ans: (0.43 to 0.45)

39. The imaginary part of an analytic complex function is v(x, y) = 2xy + 3y. The real part of the function is zero at the origin. The value of the real part of the function at 1 + i is _____.(up to two decimal places).

Ans: (2.90 to 3.10)

40. Let X be a column vector of dimension n > 1 with a t least one non-zero entry. The number of non-zero eigenvalues of the matrix M = XX^T is

(A) 0

(B) n

(D) n – 1

Ans: (C)

41. J^P for the ground state of the ¹³C₆ nucleus is

(A)

(B)

(C)

(D)

Ans: (D)

42. A uniform solid cylinder is released on a horizontal surface with speed 5 m/s without any rotation (slipping without rolling). The cylinder eventually starts rolling without slipping. If the mass and radius of the cylinder are 10 gm and 1 cm respectively, the final linear velocity of the cylinder is ______ m/s. (up to two decimal places).

Ans: (3.30 to 3.35)

43. The energy density and pressure of a photon gas are given by u = aT⁴ and P = u/3, where T is the temperature and a is the radiation constant. The entropy per u nit volume is given by αaT³. The value of α is ______.(up to two decimal places).

Ans: (1.30 to 1.36)

44. Which one of the following gases of diatomic molecules is Raman, infrared, and NMR active?

(A) ¹H – ¹H

(B) ¹²C – ¹⁶O

(D) ¹⁶O – ¹⁶O

Ans: (C)

45. The π⁺ decays at rest to μ⁺ and v_μ. Assuming the neutrino to be massless, the momentum of the neutrino is ____ MeV/c. (up to decimal places)

(m_π = 139 MeV/c², m_μ = 105 MeV/c²).

Ans: (29.50 to 30.10)

46. Using Hund’s rule, the total angular momentum quantum number J for the electronic ground stat of the nitrogen atom is

(A) 1/2

(B) 3/2

(D) 1

Ans: (B)

47. Which one of the following operators is Hermitian?

(A)

(B)

(C)

(D)

Ans: (A)

48. The real space primitive lattice vectors are The reciprocal space unit vectors for this lattice are, respectively

(A)

(B)

(C)

(D)

Ans: (A)

49. Consider two particles and two non-degenerate quantum levels 1 and 2. Level 1 always contains a particle. Hence, what is the probability that level 2 also contains a particle for each of the two cases:

(i) when the two particles are distinguishable and (ii) when the two particles are bosons?

(A) (i) 1/2 and (ii) 1/3

(B) (i) 1/2 and (ii) 1/2

(D) (i) 1 and (ii) 0

Ans: (C)

50. A person weighs w_p at Earth’s north pole and w_e at the equator. Treating the Earth as a perfect sphere of radius 6400 km, the value 100 × (w_p – w_e)/w_p is _____. (up to two decimal places).

(Take g = 10 ms⁻²).

Ans: (0.32 to 0.36)

51. The geometric cross-section of two colliding protons at large energies is very well estimated by the product of the effective sizes of each particle. This is closest to

(A) 10 b

(B) 10 mb

(D) 10 pb

Ans: (B)

52. For the transistor amplifier circuit shown below with R₁ = 10 kΩ, R₂ = 10 kΩ, R₃ = 1 kΩ, and β = 99. Neglecting the emitter diode resistance ,the input impedance of the amplifier looking into the base for small ac signal is _____kΩ. (up to two decimal places).

Ans: (4.75 to 5.01)

53. Consider an ideal operational amplifier as shown in the figure below with R₁ = 5 kΩ, R₂ = 1 kΩ, R_L = 100 kΩ. For an applied input voltage V = 10 mV, current passing through R₂ is ______μA. (up two decimal places).

Ans: (9.80 to 10.20)

54. Consider the differential equation dy/dx + ytan x) = cos(x). If y(0) = 0, y(π/3) is ______. (up to two decimal places).

Ans: (0.51 to 0.53)

55. Positronium is an atom made of an electron and a positron. Given the Bohr radius for the ground state of the Hydrogen atom to be 0.53 Angstroms, the Bohr radius for the ground state of posistronium is _____ Angstroms. (up to two decimal places).

Ans: (0.99 to 1.10)

56. The ninth and the tenth of this month are Monday and Tuesday _______.

(A) figuratively

(B) retrospectively

(D) rightfully

Ans: (C)

57. It is _______ to read this year’s textbook ______ the last year’s

(A) easier, than

(B) most easy, than

(D) easiest, from

Ans: (A)

58. A rule states that in order to drink beer, one must be over 18 years old. In a bar, there are 4 people. P is 16 years old, Q is 25 years old, R is drinking milkshake and S is drinking a beer. What must be checked to ensure that the rule is being followed?

(A) Only P’s drink

(B) Only P’s drink and S’s age

(D) Only P’s drink, Q’s drink and S’s age

Ans: (B)

59. Fatima starts from point P, goes North for 3 km, and then East for 4 km to reach point Q. She then turns to face point P and goes 15 km in that direction. She then goes North for 6 km. How far is she from point P, and in which direction should she go to reach point P?

(A) 8 km, East

(B) 12 km, North

(D) 10 km, North

Ans: (A)

60. 500 students are taking one or more courses out of Chemistry, Physics and Mathematics. Registration records indicate course enrolment as follows: Chemistry (329), Physics (186), Mathematics (295), Chemistry and Physics (83), Chemistry and Mathematics (217), and Physics and Mathematics (63). How many students are taking all 3 subjects?

(A) 37

(B) 43

(D) 53

Ans: (D)

61. “If you are looking for a history of India, or for an account of the rise and fall of the British Raj, or for the reason of the cleaving of the subcontinent into two mutually antagonistic parts and the effects this mutilation will have in the respective sections, and ultimately on Asia, you will not find it in these pages; for though I have spent a lifetime in the country. I lived too near the seat of events, and was too intimately associated with the actors, to get the perspective needed for the impartial recording of these matters.”

Which of the following statements best reflects the author’s opinion?

(A) An intimate association does not allow for the necessary perspective.

(B) Matters are recorded with an impartial perspective.

(D) Actors are typically associated with the impartial recording of matters.

Ans: (A)

62. Each of P, Q, R, S, W, X, Y and Z has been married at most once. X and Y are married and have two children P and Q. Z is the grandfather of the daughter S of P. Further, Z and W are married and are parents of R. Which one of the following must necessarily be FALSE?

(A) X is the mother-in-law of R

(B) P and R are not married to each other

(D) Q cannot be married to R

Ans: (D)

63. 1200 men and 500 women can build a bridge in 2 weeks. 900 men and 250 women will take 3 weeks to build the same bridge. How many men will be needed to build the bridge in one week?

(A) 3000

(B) 3300

(D) 3900

Ans: (C)

64. The number of 3-digit numbers such that the digit 1 is never to the immediate right of 2 is

(A) 781

(B) 791

(D) 891

Ans: (C)

65. A contour line joins locations having the same height above the mean sea level. The following is a contour plot of geographical region. Contour lines are shown at 25 m intervals in this plot.

Which of the following is the steepest path leaving from P?

(A) P to Q

(B) P to R

(D) P to T

Ans: (B)

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