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Question 1 of 65
1. Question
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
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Question 2 of 65
2. Question
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). (ε_{0} = 8.854 × 10^{−12} F/m)
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Question 3 of 65
3. Question
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).
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NAT Type Question

Question 4 of 65
4. Question
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} Pa. (up to two decimal places) (ε_{0} = 8.854 × 10^{−12} F/m).
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NAT Type Question

Question 5 of 65
5. Question
The best resolution that a 7 bit A/D convertor with 5 V full scale can achieve is _____mV. (up to two decimal places).
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NAT Type Question

Question 6 of 65
6. Question
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?
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Question 7 of 65
7. Question
The atomic mass and mass density of Sodium are 23 and 0.968 g cm^{−}^{3}, respectively. The number density of valence electrons is ______×10^{22} cm^{−}^{3}. (Up to two decimal places.)
(Avogadro number, N_{A} = 6.022 × 10^{23})
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NAT Type Question

Question 8 of 65
8. Question
Consider a onedimensional lattice with a weak periodic potential The gap at the edge of the Brillouin zone is :
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Question 9 of 65
9. Question
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:
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Question 10 of 65
10. Question
If the Lagrangian is modified to which one of the following is TRUE?
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Question 11 of 65
11. Question
Two identical masses of 10 gm each are connected by a massless spring of spring constant 1 N/m. The nonzero angular eigenfrequency of the system is ______ rad/s. (up to two decimal places).
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Question 12 of 65
12. Question
The phase space trajectory of an otherwise free particle bouncing between two hard walls elastically in one dimension is a
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Question 13 of 65
13. Question
The Poisson bracket [x, xp_{y} + yp_{x}] is equal to
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Question 14 of 65
14. Question
The wavefunction of which orbital is spherically symmetric :
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Question 15 of 65
15. Question
The contour integral evaluated along a contour going from −∞ to +∞ along the real axis and closed in the lower halfplane by a half circle is equal to _____.(up to two decimal places).
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NAT Type Question

Question 16 of 65
16. Question
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 × 10^{8} ms^{−1})
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NAT Type Question

Question 17 of 65
17. Question
Which one of the following conservation laws is violated in the decay τ^{+} → μ^{+}μ^{+}μ^{−}
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Question 18 of 65
18. Question
Electromagnetic interactions are :
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Question 19 of 65
19. Question
A one dimensional simple oscillator with Hamiltonian is subjected to a small perturbation, H_{1} = αx + βx^{3} + γx^{4}. The first order correction to the ground state energy is dependent on
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Question 20 of 65
20. Question
For the Hamiltonian where a_{0} ∈ R, is a real vector, I is the 2 × 2 identity matrix, and are the Pauli matrices, the ground state energy is
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Question 21 of 65
21. Question
The coefficient of e^{ikx} in the Fourier expansion of u(x) = Asin^{2} (αx) for k = −2α is
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Question 22 of 65
22. Question
The degeneracy of the third energy level of a 3dimensional isotropic quantum harmonic oscillator is
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Question 23 of 65
23. Question
The electronic ground state energy of the Hydrogen atom is −13.6 eV. The highest possible electronic energy eigenstable has an energy equal to
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Question 24 of 65
24. Question
A reversible Carnot engine is operated between temperatures T_{1} and T_{2} (T_{2} > T_{1}) with a photon gas as the working substance. The efficiency of the engine is
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Question 25 of 65
25. Question
In the nuclear reaction ^{13}C_{6} + v_{e} → ^{13}N_{7} + X, the particle X is
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Question 26 of 65
26. Question
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.
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Question 27 of 65
27. Question
An infinite solenoid carries a time varying current I(t) = At^{2}, 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?
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Question 28 of 65
28. Question
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) (ε_{0} = 8.854 × 10^{−12} F/m)
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NAT Type Question

Question 29 of 65
29. Question
Water freezes at 0℃ at atmospheric pressure (1.01 × 10^{5} Pa). The densities of water and ice at this temperature and pressure are 1000 kg/m^{3} and 934 kg/m^{3} respectively. The latent heat of fusion is 3.34 × 10^{5} J/kg. The pressure required for depressing the melting temperature of ice by 10℃ is _____ GPa. (up to two decimal places)
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Question 30 of 65
30. Question
The minimum number of NAND gates required to construct an OR gate is
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Question 31 of 65
31. Question
Consider a 2dimensional electron gas with a density of 10^{19} 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^{−19}C)
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NAT Type Question

Question 32 of 65
32. Question
The total energy of an inertgas crystal is given by , were R is the interatomic spacing in Angstroms. The equilibrium separation between the atoms is ______ Angstroms. (up to two decimal places).
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NAT Type Question

Question 33 of 65
33. Question
Consider N noninteracting, distinguishable particles in a twolevel system at temperature T. The energies of the levels are 0 and ε, where ε > 0. In the high temperature limit (k_{B}T >> ε), what is the population of particles in the level with energy ε?
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Question 34 of 65
34. Question
A free electron of energy 1 eV is incident upon a onedimensional finite potential step of h eight 0.75 eV. The probability of its reflection from the barrier is ______(up to two decimal places).
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Question 35 of 65
35. Question
Consider a onedimensional 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):
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Question 36 of 65
36. Question
Consider a metal with free electron density of 6 × 10^{22} cm^{−3}. The lowest frequency electromagnetic radiation to which this metal is transparent is 1.38 × 10^{16} Hz. If this metal had a free electron density of 1.8 × 10^{23} cm^{−3} instead, the lowest frequency electromagnetic radiation to which it would be transparent is _________× 10^{16} Hz. (up to two decimal places).
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Question 37 of 65
37. Question
An object travels along the xdirection 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).
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NAT Type Question

Question 38 of 65
38. Question
The integral is equal to ______.(up to two decimal places).
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NAT Type Question

Question 39 of 65
39. Question
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).
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NAT Type Question

Question 40 of 65
40. Question
Let X be a column vector of dimension n > 1 with a t least one nonzero entry. The number of nonzero eigenvalues of the matrix M = XX^{T} is
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Question 41 of 65
41. Question
J^{P} for the ground state of the ^{13}C_{6} nucleus is
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Question 42 of 65
42. Question
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).
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NAT Type Question

Question 43 of 65
43. Question
The energy density and pressure of a photon gas are given by u = aT^{4} and P = u/3, where T is the temperature and a is the radiation constant. The entropy per u nit volume is given by αaT^{3}. The value of α is ______.(up to two decimal places).
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NAT Type Question

Question 44 of 65
44. Question
Which one of the following gases of diatomic molecules is Raman, infrared, and NMR active?
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Question 45 of 65
45. Question
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/c^{2}, m_{μ} = 105 MeV/c^{2}).
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NAT Type Question

Question 46 of 65
46. Question
Using Hund’s rule, the total angular momentum quantum number J for the electronic ground stat of the nitrogen atom is
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Question 47 of 65
47. Question
Which one of the following operators is Hermitian?
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Question 48 of 65
48. Question
The real space primitive lattice vectors are The reciprocal space unit vectors for this lattice are, respectively
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Question 49 of 65
49. Question
Consider two particles and two nondegenerate 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?
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Question 50 of 65
50. Question
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}).
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NAT Type Question

Question 51 of 65
51. Question
The geometric crosssection 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
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Question 52 of 65
52. Question
For the transistor amplifier circuit shown below with R_{1} = 10 kΩ, R_{2} = 10 kΩ, R_{3} = 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).
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NAT Type Question

Question 53 of 65
53. Question
Consider an ideal operational amplifier as shown in the figure below with R_{1} = 5 kΩ, R_{2} = 1 kΩ, R_{L} = 100 kΩ. For an applied input voltage V = 10 mV, current passing through R_{2} is ______μA. (up two decimal places).
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NAT Type Question

Question 54 of 65
54. Question
Consider the differential equation dy/dx + ytan x) = cos(x). If y(0) = 0, y(π/3) is ______. (up to two decimal places).
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Question 55 of 65
55. Question
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).
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Question 56 of 65
56. Question
The ninth and the tenth of this month are Monday and Tuesday _______.
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Question 57 of 65
57. Question
It is _______ to read this year’s textbook ______ the last year’s
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Question 58 of 65
58. Question
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?
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Question 59 of 65
59. Question
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?
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Question 60 of 65
60. Question
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?
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Question 61 of 65
61. Question
“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?
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Question 62 of 65
62. Question
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?
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Question 63 of 65
63. Question
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?
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Question 64 of 65
64. Question
The number of 3digit numbers such that the digit 1 is never to the immediate right of 2 is
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Question 65 of 65
65. Question
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?
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