NET-DEC-2012 Past Paper PDF

fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES NET-DEC-2012 Part A Q1. A granite block of 2m × 5m × 3m size is cut into 5 cm thick slabs of 2m × 5m size. These...

fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES NET-DEC-2012 Part A Q1. A granite block of 2m × 5m × 3m size is cut into 5 cm thick slabs of 2m × 5m size. These slabs are laid over a 2m wide pavement. What is the length of the pavement that can be covered with these slabs? (a) 100 m (b) 200 m (c) 300 m (d) 500 m Q2. Which is the least among the following? 0.33 0..33, 0.44 0.44, π-1/π, e-1/e (a) 0.330.33 (b) 0.44 0.44 (c) π-1/π (d) e-1/e Q3. What is the next number in this “see and tell” sequence? 1 11 21 1211 111221 _______ (a) 312211 (b) 1112221 (c) 1112222 (d) 1112131 Q4. A vertical pole of length a stands at the centre of a horizontal regular hexagonal ground of side a. A rope that is fixed taut in between a vertex on the ground and the tip of the pole has a length (a) a (b) 2a (c) 3a (d) 6a Q5. A peacock perched on the top of a 12 m high tree spots a snake moving towards its hole at the base of the tree from a distance equal to thrice the height of the tree. The peacock flies towards the snake in a straight line and they both move at the same speed. At what distance from the base of the tree will the peacock catch the snake? (a) 16 m (b) 18 m (c) 14 m (d) 12 m fiziks c/o Anand Institute of mathematics, 28-B/6 Jia Sarai Near IIT, Hauz Khas, New Delhi, PIN- 110016 (INDIA) Phone: 011-32718565, +91-9871145498 Website: http://www.physicsbyfiziks.com 1 Email: [email protected]  fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES Q6. The cities of a country are connected by intercity roads. If a city is directly connected to an odd number of other cities, it is called an odd city. If a city is directly connected to an even number of other cities, it is called an even city. Then which of the following is impossible? (a) There are an even number of odd cities (b) There are an odd number of odd cities (c) There are an even number of even cities (d) There are an odd number of even cities Q7. In the figure ABC   / 2 , AD = DE = EB C What is the ratio of the area of triangle ADC to that of triangle CDB? (a) 1:1 (b) 1:2 (c) 1:3 (d) 1:4 A D E B Q8. A rectangular sheet ABCD is folded in such a way that vertex A D C meets vertex C, thereby forming a line PQ. Assuming AB = 3 and P BC = 4, find PQ. Note that AP = PC and AQ = QC. (a) 13/4 (b) 15/4 Q (c) 17/4 (d) 19/4 A B Q9. A string of diameter 1mm is kept on a table in the shape of a close flat spiral i.e. a spiral with no gap between the turns. The area of the table occupied by the spiral is 1m2. Then the length of the string is (a) 10 m (b) 10 2 m (c) 103 m (d) 10 4 m Q10. 25% of 25% of a quantity is x% of the quantity where x is (a) 6.25 % (b) 12.5 % (c) 25 % (d) 50 % fiziks c/o Anand Institute of mathematics, 28-B/6 Jia Sarai Near IIT, Hauz Khas, New Delhi, PIN- 110016 (INDIA) Phone: 011-32718565, +91-9871145498 Website: http://www.physicsbyfiziks.com 2 Email: [email protected]  fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES Q11. In sequence {an}every term is equal to the sum of all previous terms. If a0 = 3, then a n 1 lim is n a n (a) 3 (b) 2 (c) 1 (d) e Q12. In the figure given, angle ABC = π/2. I, II, III are the areas of A semicircles on the sides opposite angles B, A and C, respectively. I III Which of the following is always true? (a) II2 + III2 = I2 (b) II + III = I B II C (c) II2 + III2 > I2 (d) II + III < I Q13. What is the minimum number of days between one Friday the 13th and the next Friday the 13 th? (Assume that the year is a leap year). (a) 28 (b) 56 (c) 91 (d) 84 Q14. Suppose a person A is at the North-East corner of a square (see A the figure below). From that point he moves along the diagonal N and after covering 1/3rd portion of the diagonal, he goes to his left W E and after sometime he stops, rotates 90° clockwise and moves straight. After a few minutes he stops, rotates 180° anticlockwise. S Towards which direction he is facing now? (a) North-East (b) North-West (c) South-East (d) South-West Q15. Cucumber contains 99% water. Ramesh buys 100 kg of cucumbers. After 30 days of storing the cucumbers lose some water. They now contain 98% water. What is the total weight of cucumbers now? (a) 99 kg (b) 50 kg (c) 75 kg (d) 2 kg fiziks c/o Anand Institute of mathematics, 28-B/6 Jia Sarai Near IIT, Hauz Khas, New Delhi, PIN- 110016 (INDIA) Phone: 011-32718565, +91-9871145498 Website: http://www.physicsbyfiziks.com 3 Email: [email protected]  fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES Q16. In a museum there were old coins with their respective years engraved on them, as follows: (i) 1837 AD (ii) 1907 AD (iii) 1947 AD (iv) 200 BC identify the fake coin(s) (a) coin (i) (b) coin (iv) (c) coins (i) and (ii) (d) coin (iii) Q17. A student observes the movement of four snails and plots the graphs of distance moved as a function of time as given in figures (A), (B), (C) and (D). Distance Distance Distance Distance Time Time Time Time A  B C D Which of the following is not correct? (a) Graph (A) (b) Graph (B) (c) Graph (C) (d) Graph (D) Q18. Find the missing letter: A EGK C ? P U R Q V B OJF D (a) H (b) L (c) Z (d) Y fiziks c/o Anand Institute of mathematics, 28-B/6 Jia Sarai Near IIT, Hauz Khas, New Delhi, PIN- 110016 (INDIA) Phone: 011-32718565, +91-9871145498 Website: http://www.physicsbyfiziks.com 4 Email: [email protected]  fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES Q19. Consider the following equation x 2  4 y 2  9 z 2  14 x  28 y  42 z  147 where x, y and z are real numbers. Then the value of x + 2y + 3z is (a) 7 (b) 14 (c) 21 (d) not unique Q20. The map given below shows a meandering river following a semi-circular path, along which two villages are located at A and B. The distance between A and B along the east- west direction in the map is 7 cm. What is the length of the river between A and B in the ground? N A B SCALE  1 : 50000 (a) 1.1 km (b) 3.5 km (c) 5.5 km (d) 11.0 km fiziks c/o Anand Institute of mathematics, 28-B/6 Jia Sarai Near IIT, Hauz Khas, New Delhi, PIN- 110016 (INDIA) Phone: 011-32718565, +91-9871145498 Website: http://www.physicsbyfiziks.com 5 Email: [email protected]  fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES Part B Q21. A 2×2 matrix A has eigenvalues e i / 5 and ei / 6. The smallest value of n such that An = I is (a) 20 (b) 30 (c) 60 (d) 120 Q22. The graph of the function f(x) shown below is best described by 1.00 0.25 0.50 0.25 f (x) 0.00  0.25  0.50  0.75  1.00 0 1 2 3 4 5 6 7 8 9 10 x (a) The Bessel function J0 (x) (b) cos x 1 (c) e  x cos x (d) cos x x Q23. In a series of five Cricket matches, one of the captains calls “Heads” every time when the toss is taken. The probability that he will win 3 times and lose 2 times is (a) 1/8 (b) 5/8 (c) 3/16 (d) 5/16  a b c  Q24. The unit normal vector at the point  , ,  on the surface of the  3 3 3 x2 y2 z2 ellipsoid    1 , is a2 b2 c2 bciˆ  caˆj  abkˆ aiˆ  bˆj  ckˆ (a) (b) a 2c 2  b 2c 2  a 2b 2 a 2  b2  c2 biˆ  cˆj  akˆ iˆ  ˆj  kˆ (c) (d) a 2  b2  c2 3 fiziks c/o Anand Institute of mathematics, 28-B/6 Jia Sarai Near IIT, Hauz Khas, New Delhi, PIN- 110016 (INDIA) Phone: 011-32718565, +91-9871145498 Website: http://www.physicsbyfiziks.com 6 Email: [email protected]  fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES Q25. A solid cylinder of height H, radius R and density ρ, floats vertically on the surface of a liquid of density  0. The cylinder will be set into oscillatory motion when a small instantaneous downward force is applied. The frequency of oscillation is g  g g 0 g (a) (b) (c) (d) 0 H 0 H 0H H Q26. Three particles of equal mass m are connected by two identical massless springs of stiffness constant k as shown in the figure: If x1, x2 and x3 denote the horizontal displacements of the masses from their respective equilibrium positions, the potential energy of the system is k k m m m 1 1 (a) 2  k x12  x 22  x32  (b) 2  k x12  x 22  x32  x2  x1  x3   1 1 (c) 2  k x12  2 x22  x32  2 x2 x1  x3   (d) 2  k x12  2 x22  x32  2 x 2 x1  x3   Q27. Let v, p and E denote the speed, the magnitude of the momentum, and the energy of a free particle of rest mass m. Then (a) dE  constant (b) p = mv dp (c) v  cp 2 2 2 (d) E = mc2 p m c Q28. A binary star system consists of two stars S1 and S2, with masses m and 2m respectively separated by a distance r. If both S1 and S2 individually follow circular orbits around the centre of mass with instantaneous speeds v1 and v2 respectively, the speeds ratio v1/v2 is (a) 2 (b) 1 (c) 1/2 (d) 2 fiziks c/o Anand Institute of mathematics, 28-B/6 Jia Sarai Near IIT, Hauz Khas, New Delhi, PIN- 110016 (INDIA) Phone: 011-32718565, +91-9871145498 Website: http://www.physicsbyfiziks.com 7 Email: [email protected]  fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES Q29. Three charges are located on the circumference of a circle of radius R as shown in the figure below. The two charges Q subtend an angle 90° at the centre Q Q of the circle. The charge q is symmetrically placed with respect to the charges Q. If the electric field at the centre of the circle is zero, what is the magnitude of Q? (a) q / 2 (b) 2q (c) 2q (d) 4q q Q30. Consider a hollow charged shell of inner radius a and outer radius b. The volume charge k density is  r   (k is constant) in the region a < r < b. The magnitude of the electric r2 field produced at distance r > a is k b  a  (a) for all r > a 0r2 k b  a  kb (b) 2 for a < r < b and for r > b  0r  0r 2 k r  a  k b  a  (c) 2 for a < r < b and for r > b  0r 0r2 k r  a  k b  a  (d) 2 for a < r < b and for r > b  0a 0r2 Q31. Consider the interference of two coherent electromagnetic waves whose electric field   vectors are given by E1  iˆE 0 cos  t and E 2  ˆjE0 cos t    where  is the phase 0 2 difference. The intensity of the resulting wave is given by E , where E 2 is the 2 time average of E 2. The total intensity is (a) 0 (b)  0 E 02 (c)  0 E 02 sin 2  (d)  0 E02 cos 2  fiziks c/o Anand Institute of mathematics, 28-B/6 Jia Sarai Near IIT, Hauz Khas, New Delhi, PIN- 110016 (INDIA) Phone: 011-32718565, +91-9871145498 Website: http://www.physicsbyfiziks.com 8 Email: [email protected]  fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES Q32. Four charges (two + q and two –q) are kept fixed at the four vertices of a square of side a as shown q q a R P q q At the point P which is at a distance R from the centre (R >> a), the potential is proportional to (a) 1/R (b) 1/R2 (c) 1/R3 (d) 1/R4 Q33. A point charges q of mass m is kept at a distance d below a grounded infinite conducting sheet which lies in the xy - plane. For what value of d will the charge remains stationary? (a) q / 4 mg 0 (b) q / mg 0 (c) There is no finite value of d (d) mg 0 / q Q34. The wave function of a state of the hydrogen atom is given by    200  2 211  3 210  2 211 where  nlm is the normalized eigen function of the state with quantum numbers n, l and m in the usual notation. The expectation value of Lz in the state  is (a) 15 / 16 (b) 11 / 16 (c) 3 / 8 (d)  / 8 1 Q35. The energy eigenvalues of a particle in the potential V x   m 2 x 2  ax are 2  1 a2  1 a2 (a) E n   n    (b) E n   n     2 2m 2  2 2m 2  1 a2  1 (c) E n   n    (d) E n   n    2 m 2  2 fiziks c/o Anand Institute of mathematics, 28-B/6 Jia Sarai Near IIT, Hauz Khas, New Delhi, PIN- 110016 (INDIA) Phone: 011-32718565, +91-9871145498 Website: http://www.physicsbyfiziks.com 9 Email: [email protected]  fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES Q36. If a particle is represented by the normalized wave function   15 a 2  x 2   for  a  x  a  x   4a 5 / 2 0 otherwise  the uncertainty p in its momentum is (a) 2 / 5a (b) 5 / 2a (c) 10 / a (d) 5 / 2 a Q37. Given the usual canonical commutation relations, the commutator  A, B of A  i xp y  yp x  and B   yp z  zp y  is (a)  xp z  p x z  (b)   xp z  p x z  (c)  xp z  p x z  (d)   xp z  p x z  Q38. The entropy of a system, S, is related to the accessible phase space volume  by S  k B ln E , N , V  where E, N and V are the energy, number of particles and volume respectively. From this one can conclude that  (a) does not change during evolution to equilibrium (b) oscillates during evolution to equilibrium (c) is a maximum at equilibrium (d) is a minimum at equilibrium Q39. Let W be the work done in a quasistatic reversible thermodynamic process. Which of the following statements about W is correct? (a) W is a perfect differential if the process is isothermal (b) W is a perfect differential if the process is adiabatic (c) W is always a perfect differential (d) W cannot be a perfect differential fiziks c/o Anand Institute of mathematics, 28-B/6 Jia Sarai Near IIT, Hauz Khas, New Delhi, PIN- 110016 (INDIA) Phone: 011-32718565, +91-9871145498 Website: http://www.physicsbyfiziks.com 10 Email: [email protected]  fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES Q40. Consider a system of three spins S1, S2 and S3 each of which can take values +1 and -1. The energy of the system is given by E   J S1 S 2  S 2 S 3  S 3 S1  where J is a positive constant. The minimum energy and the corresponding number of spin configuration are, respectively, (a) J and 1 (b) -3J and 1 (c)-3J and 2 (d) -6J and 2 1 Q41. The minimum energy of a collection of 6 non-interacting electrons of spin - and mass 2 m placed in a one dimensional infinite square well potential of width L is (a) 14 2  2 / mL2 (b) 91 2  2 / mL2 (c) 7 2  2 / mL2 (d) 3 2  2 / mL2 Q42. A live music broadcast consists of a radio-wave of frequency 7 MHz, amplitude- modulated by a microphone output consisting of signals with a maximum frequency of 10 kHz. The spectrum of modulated output twill be zero outside the frequency band (a) 7.00 MHz to 7.01 MHz (b) 6.99 MHz to 7.01 MHz (c) 6.99 MHz to 7.00 MHz (d) 6.995 MHz to 7.005 MHz Q43. In the op-amp circuit shown in the figure, Vi is a sinusoidal input signal of frequency 10 Hz and V0 is the output signal. The magnitude of the 0.01F gain and the phase shift, respectively, close to the values (a) 5 2 and  / 2 10K 1K (b) 5 2 and   / 2 Vi Vo (c) 10 and zero (d) 10 and π fiziks c/o Anand Institute of mathematics, 28-B/6 Jia Sarai Near IIT, Hauz Khas, New Delhi, PIN- 110016 (INDIA) Phone: 011-32718565, +91-9871145498 Website: http://www.physicsbyfiziks.com 11 Email: [email protected]  fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES Q44. The logic circuit shown in the figure below Implements the Boolean expression A HIGH y B (a) y  A  B (b) y  A  B (c) y  A  B (d) y  A  B Q45. A diode D as shown in the circuit has an i-v relation that can be approximated by v 2  2v D , for v D  0 iD   D 0, for v D  0 1 iD  10 V D vD  The value of v D in the circuit is  (a)  1 11 V  (b) 8 V (c) 5 V (d) 2 V Q46. The Taylor expansion of the function ln(cosh x), where x is real, about the point x = 0 starts with the following terms: 1 2 1 4 1 2 1 4 (a)  x  x .... (b) x  x .... 2 12 2 12 1 2 1 4 1 2 1 4 (c)  x  x .... (d) x  x .... 2 6 2 6 fiziks c/o Anand Institute of mathematics, 28-B/6 Jia Sarai Near IIT, Hauz Khas, New Delhi, PIN- 110016 (INDIA) Phone: 011-32718565, +91-9871145498 Website: http://www.physicsbyfiziks.com 12 Email: [email protected]  fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES Q47. Given a 2×2 unitary matrix U satisfying U†U = UU† = 1 with det U  e i , one can   construct a unitary matrix V V †V  VV †  1 with det V = 1 from it by (a) multiplying U by e i / 2 (b) multiplying any single element of U by e  i (c) multiplying any row or column of U by e i / 2 (d) multiplying U by e i z 3 dz Q48. The value of the integral  , where C is a closed contour defined by the C z 2  5z  6 equation 2 z  5  0, traversed in the anti-clockwise direction, is (a)  16 i (b) 16 i (c) 8 i (d) 2 i d2 f Q49. The function f(x) obeys the differential equation  3  2i  f  0 and satisfies the dx 2 conditions f(0) = 1 and f(x) → 0 as x → ∞. The value of f(π) is (a) e 2 (b) e 2 (c)  e 2 (d)  e 2 i Q50. A planet of mass m moves in the gravitational field of the Sun (mass M). If the semi- major and semi-minor axes of the orbit are a and b respectively, the angular momentum of the planet is (a) 2GMm 2 a  b (b) 2GMm 2 a  b  2GMm 2 ab 2GMm 2 ab (c) (d) ab ab fiziks c/o Anand Institute of mathematics, 28-B/6 Jia Sarai Near IIT, Hauz Khas, New Delhi, PIN- 110016 (INDIA) Phone: 011-32718565, +91-9871145498 Website: http://www.physicsbyfiziks.com 13 Email: [email protected]  fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES Q51. The Hamiltonian of a simple pendulum consisting of a mass m attached to a massless p2 string of length l is H   mgl 1  cos . If L denotes the Lagrangian, the value of 2 ml 2 dL is: dt 2g g (a)  p sin  (b)  p sin 2 l l g (c) p cos (d) lp2 cos l Q52. Which of the following set of phase-space trajectories is not possible for a particle obeying Hamilton’s equations of motion? (a) (b) P P x x (c) (d) P P x x Q53. Two bodies of equal mass m are connected by a massless rigid rod of length l lying in the xy-plane with the centre of the rod at the origin. If this system is rotating about the z-axis with a frequency ω, its angular momentum is (a) ml 2 / 4 (b) ml 2 / 2 (c) ml 2 (d) 2ml 2 fiziks c/o Anand Institute of mathematics, 28-B/6 Jia Sarai Near IIT, Hauz Khas, New Delhi, PIN- 110016 (INDIA) Phone: 011-32718565, +91-9871145498 Website: http://www.physicsbyfiziks.com 14 Email: [email protected]  fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES Q54. An infinite solenoid with its axis of symmetry along the z-direction carries a steady current I.  ẑ The vector potential A at a distance R from the axis (a) is constant inside and varies as R outside the solenoid R (b) varies as R inside and is constant outside the solenoid 1 (c) varies as inside and as R outside the solenoid R 1 (d) varies as R inside and as outside the solenoid R Q55. Consider an infinite conducting sheet in the xy-plane with a time dependent current density Kt iˆ , where K is a constant. The vector potential at (x, y, z) is given  K  by A  0 ct  z  iˆ. The magnetic field B is 2 4c  0 Kt ˆ  0 Kz ˆ (a) j (b)  j 2 2c 0 K 0 K (c)  ct  z iˆ (d)  ct  z  ˆj 2c 2c  Q56. When a charged particle emits electromagnetic radiation, the electric field E and the  1   1 Poynting vector S  E  B at a larger distance r from emitter vary as n and 0 r 1 respectively. Which of the following choices for n and m are correct? rm (a) n = 1 and m = 1 (b) n = 2 and m = 2 (c) n = 1 and m = 2 (d) n = 2 and m = 4 fiziks c/o Anand Institute of mathematics, 28-B/6 Jia Sarai Near IIT, Hauz Khas, New Delhi, PIN- 110016 (INDIA) Phone: 011-32718565, +91-9871145498 Website: http://www.physicsbyfiziks.com 15 Email: [email protected]  fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES 1 Q57. The energies in the ground state and first excited state of a particle of mass m  in a 2 potential V x  are -4 and -1, respectively, (in units in which ħ = 1). If the corresponding wavefunctions are related by  1 x    0  x sinh x, then the ground state eigenfunction is (a)  0 x   sec hx (b)  0 x   sec hx (c)  0 x   sec h 2 x (d)  0 x   sec h 3 x Q58. The perturbation ba  x , a  x  a H '  0, otherwise acts on a particle of mass m confined in an infinite square well potential 0, a  x  a V x    , otherwise The first order correction to the ground state energy of the particle is ba ba (a) (b) (c) 2ba (d) ba 2 2 Q59. Let 0 and 1 denote the normalized eigenstates corresponding to the ground and the first excited states of a one-dimensional harmonic oscillator. The uncertainty x in the 1 state 0  1  is 2 (a) x   / 2m (b) x   / m (c) x  2 / m (d) x  4 / m fiziks c/o Anand Institute of mathematics, 28-B/6 Jia Sarai Near IIT, Hauz Khas, New Delhi, PIN- 110016 (INDIA) Phone: 011-32718565, +91-9871145498 Website: http://www.physicsbyfiziks.com 16 Email: [email protected]  fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES Q60. What would be the ground state energy of the Hamiltonian 2 d 2 H     x  2m dx 2 2 if variational principle is used to estimate it with the trial wavefunction   x   Ae bx with b as the variational parameter?  1 2 n  1  x e dx  2b  2 n  2bx [Hint: 2  n   ]   2 (a)  m 2 / 2 2 (b)  2m 2 /   2 (c)  m 2 /   2 (d) m 2 /   2 Q61. The free energy difference between the superconducting and the normal states of a 2  4 material is given by F  FS  FN      , where  is an order parameter and α 2 and β are constants such that   0 in the normal and   0 in the superconducting state, while   0 always. The minimum value of F is (a)   2 /  (b)   2 / 2  (c)  3 2 / 2  (d)  5 2 / 2  Q62. A given quantity of gas is taken from the state A → C reversibly, by two paths, A → C directly and A → B → C as shown in the figure. P During the process A → C the work done by the gas is 100 J and the A heat absorbed is 150 J. If during the process A → B → C the work done by the gas is 30 J, the heat absorbed is B C (a) 20 J (b) 80 J V (c) 220 J (d) 280 J fiziks c/o Anand Institute of mathematics, 28-B/6 Jia Sarai Near IIT, Hauz Khas, New Delhi, PIN- 110016 (INDIA) Phone: 011-32718565, +91-9871145498 Website: http://www.physicsbyfiziks.com 17 Email: [email protected]  fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES Q63. Consider a one-dimensional Ising model with N spins, at very low temperatures when almost all spins are aligned parallel to each other. There will be a few spin flips with each flip costing an energy 2J. In a configuration with r spin flips, the energy of the system is N E = - NJ + 2rJ and the number of configuration is C r ; r varies from 0 to N. The partition function is N  J  (a)   (b) e  NJ / kBT  k BT  N N  J   J  (c)  sinh  (d)  cosh   k B T   k B T  Q64. A magnetic field sensor based on the Hall Effect is to be fabricated by implanting As into a Si film of thickness 1 µm. The specifications require a magnetic field sensitivity of 500 mV/Tesla at an excitation current of 1 mA. The implantation dose is to be adjusted such that the average carrier density, after activation, is (a) 1.25 × 1026 m-3 (b) 1.25 × 1022 m-3 (c) 4.1 × 10 21 m-3 (d) 4.1 × 10 20 m-3 Q65. Band-pass and band-reject filters can be implemented by combining a low pass and a high pass filter in series and in parallel, respectively. If the cut-off frequencies of the low pass and high pass filters are  0LP and  0HP , respectively, the condition required to implement the band-pass and band-reject filters are, respectively, (a)  0HP   0LP and  0HP   0LP (b) 0HP  0LP and  0HP   0LP (c)  0HP   0LP and 0HP  0LP (d)  0HP   0LP and  0HP  0LP fiziks c/o Anand Institute of mathematics, 28-B/6 Jia Sarai Near IIT, Hauz Khas, New Delhi, PIN- 110016 (INDIA) Phone: 011-32718565, +91-9871145498 Website: http://www.physicsbyfiziks.com 18 Email: [email protected]  fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES Q66. The output characteristics of a solar panel at a certain level of irradiance is shown in the figure below. 3 I 2 1 0 0 5 10 15 20 V If the solar cell is to power a load of 5 Ω, the power drawn by the load is (a) 97 W (b) 73 W (c) 50 W (d) 45 W Q67. Consider the energy level diagram shown below, which corresponds to the molecular nitrogen laser. 2 R  21 1 1 0 20 -3 -1 If the pump rate R is 10 atoms cm s and the decay routes are as shown with  21  20 ns and  1  1s , the equilibrium populations of states 2 and 1 are, respectively, (a) 1014 cm-3 and 2  1012 cm-3 (b) 21012 cm-3 and 1014 cm-3. (c) 21012 cm-3 and 2  10 6 cm-3 (d) zero and 1020 cm-3 Q68. Consider a hydrogen atom undergoing a 2 P 1S transition. The lifetime tsp of the 2P state for spontaneous emission is 1.6 ns and the energy difference between the levels is 10.2 eV. Assuming that the refractive index of the medium n0 = 1, the ratio of Einstein coefficients for stimulated and spontaneous emission B21   / A21   is given by (a) 0.683 × 1012 m3J-1s-1 (b) 0.146 × 10-12 Jsm-3. (c) 6.83 × 1012 m3J -1s -1 (d) 1.463 × 10-12 Jsm-3. fiziks c/o Anand Institute of mathematics, 28-B/6 Jia Sarai Near IIT, Hauz Khas, New Delhi, PIN- 110016 (INDIA) Phone: 011-32718565, +91-9871145498 Website: http://www.physicsbyfiziks.com 19 Email: [email protected]  fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES Q69. Consider a He-Ne laser cavity consisting of two mirrors of reflectivities R1 = 1 and R2 = 0.98. The mirrors are separated by a distance d = 20 cm and the medium in between has a refractive index n0 = 1 and absorption coefficient α = 0. The values of the separation between the modes δv and the width v p of each mode of the laser cavity are: (a) v  75kHz, v p  24kHz (b) v  100kHz, v p  100kHz (c) v  750MHz, v p  2.4 MHz (d) v  2.4 MHz, v p  750MHz Q70. Non-interacting bosons undergo Bose-Einstein Condensation (BEC) when trapped in a three dimensional isotropic simple harmonic potential. For BEC to occur, the chemical potential must be equal to (a)  / 2 (b)  (c) 3 / 2 (d) 0 Q71. In a band structure calculation, the dispersion relation for electrons is found to be  k   cos k x a  cos k y a  cos k z a , where β is a constant and a is the lattice constant. The effective mass at the boundary of the first Brillouin zone is 2 2 4 2 2 2 (a) (b) (c) (d) 5 a 2 5 a 2 2 a 2 3 a 2 Q72. The radius of the Fermi sphere of free electrons in a monovalent metal with an fcc structure, in which the volume of the unit cell is a3, is 1/ 3 1/ 3 1/ 3  12 2   3 2   2  1 (a)  3  (b)  3  (c)  3  (d)  a   a  a  a Q73. The muon has mass 105 MeV/c2 and mean lifetime 2.2 µs in its rest frame. The mean distance traversed by a muon of energy 315 MeV/c2 before decaying is approximately (a) 3 × 10 5 km (b) 2.2 cm (c) 6.6 µm (d) 1.98 km fiziks c/o Anand Institute of mathematics, 28-B/6 Jia Sarai Near IIT, Hauz Khas, New Delhi, PIN- 110016 (INDIA) Phone: 011-32718565, +91-9871145498 Website: http://www.physicsbyfiziks.com 20 Email: [email protected]  fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES Q74. Consider the following particles: the proton p, the neutron n, the neutral pion π0 and the delta resonance Δ+. When ordered in terms of decreasing lifetime, the correct arrangement is as follows: (a) π0, n, p, Δ+. (b) p, n, Δ+, π0. (c) p, n, π0, Δ+. (d) Δ+, n, π0, p Q75. The single particle energy difference between the p-orbitals (i.e. P3/2 and P1/2) of the 114 nucleus 50 Sn is 3 MeV. The energy difference between the states in its 1f orbitals is (a) -7 MeV (b) 7 MeV (c) 5 MeV (d) -5 MeV fiziks c/o Anand Institute of mathematics, 28-B/6 Jia Sarai Near IIT, Hauz Khas, New Delhi, PIN- 110016 (INDIA) Phone: 011-32718565, +91-9871145498 Website: http://www.physicsbyfiziks.com 21 Email: [email protected]