4. Fourier transform of unit step 2. The volume of the parallelepiped function : is : (A) (B) (D)

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1 1. If a is a constant vector, then a r (A) a is equal to : (B) (B) 2a 2 /3 (C) /2 (C) 0 (D) 3 /3 (D) a/2 4. Fourier transform of unit step 2. he volume of the parallelepiped function : with sides : a iˆ 2ˆj 3 kˆ, b 4iˆ 5ˆj 6kˆ and c 7iˆ 8ˆj 10kˆ is : (A) he constant term (a 0 ) in the Fourier series expansion of the function f(x) = x 2 for < x < is : (A) 2 is : (A) (B) 1 x 0 x 0 x 0 1 2k 1 2 k (B) 2 (C) 3 (D) (C) 2k2 (D) 1 e 2k k

2 5. he residue of 2 z z a 2 2 at z = ia 7. y(x) satisfies the differential is : equation : (A) ia/2 d y y cosec x dx2 (B) ia he complementary function part (C) ia 2 y c of y(x) is given by : (D) ia 2 /2 (A) y c (x) = c 1 sinx + c 2 cosx 6. A deck of 5 cards (each carrying a (B) y c (x) = xe ix distinct number from 1 to 5) is (C) y c (x) = (c 1 sinx + c 2 cosx) e x shuffled thoroughly. wo cards are (D) y then removed one at a time from the c (x) = c 1 e x + c 2 cosx deck. What is the probability that 8. A, B, C, D are (n n) matrices, each the two cards are selected with the with non-zero determinant. If : number on the first card being one ABCD = I, higher than the number on the second card? (A) 1/5 2 then B 1 is : (A) D 1 C 1 A 1 (B) 4/25 (C) 1/4 (D) 2/5 (B) CDA (C) ADC (D) does not exist

3 9. he degrees of freedom for a particle falling under gravity (without air resistance) are : (A) Zero (B) One (C) wo he Hamilton s equations of motion (D) hree are : psinq 10. A particle of mass m is released from (A) q 1, 1 p2 height h. Its velocity at the position when it has covered a distance equal to (h/2) is : (A) v = 2g h 11. Which of the following properties of the Lagrangian is due to space is isotropic characteristic of the inertial frame of reference? (A) Invariant to translation (B) Invariant to time (C) Invariant to rotation (D) Explicit function of time 12. For a given system, Hamiltonian H(q, p) is defined as : H(q, p) = p + 1 p2 sinq p 1. (B) (C) p pcosq q 1, 1 p2 2pcosq q 1, 1 p2 2 1 p cosq p 1 p2cosq (B) v = 2g h (C) v = gh. (D) q p psinq 1, 1 p2 2 1 p cosq (D) v = g.h p 1 p2cosq

4 13. For a particle under the action of central force, which of the following quantities is invariant? (A) Linear velocity 15. he plane of oscillation of Focoult pendulum rotates with infinite period, when it is located at ( is the lattitude of location) : (B) Angular velocity (C) Linear momentum (D) Angular momentum 14. Which of the following statements holds true for a rigid rotator (or 16. An observer is moving with rotor)? relativistic speed u. His length (A) All the three principal moments measurements along the direction of inertia are different of velocity are half of the (B) All the three principal moments measurements made when he of inertia are equal (C) wo of the three principal moments of inertia are equal and the third one is non-zero and different (D) wo of the three principal moments of inertia are equal (A) = 90 N (B) = 60 N (C) = 30 N (D) = 0 was at rest. His time measurement of 1 hr. when he was at rest, now becomes : (A) 30 minutes (B) 01 hour (C) 02 hours and the third one is zero (D) Infinite hours

5 17. he magnetic induction B at point O in the figure is given as : 18. A cubical box of sides 5 cm encloses 100 electric dipoles of unit moments. What would be the flux of electric (C) Zero cm 200 (D) cm A uniform current 20 A flows I through a copper pipe of inner (A) 4 R1 R2 radius 10 cm. Which of the following statements is valid? (B) (C) (D) 0I R R I R R 1 2 0I R +R field emanating from the cube? (A) (B) 20 cm cm 0 (A) he magnetic field is non-zero except along the axial points of the pipe. (B) he magnetic field everywhere inside the pipe is zero. (C) he magnetic field everywhere inside the pipe is constant and non-zero. (D) he magnetic field is nonuniform inside the pipe.

6 20. An uncharged sphere of homogeneous linear dielectric material is placed in an uniform electric field E. he resultant In electrodynamics, the curling magnetic field is more appropriately expressed as : (A) a bound magnetization current electric field E inside the dielectric sphere is : (A) Zero (B) Uniform non-zero and parallel to 0 E density J M (B) a free current density J (C) stationary electric charges (D) a free current density J and a time-varying electric field D 22. he continuity equation for charges and currents is not explicitly written in the Maxwell s equations. his is because : (A) It is not a universal law and holds good in certain cases only (B) It is difficult to prove the continuity equation rigorously (C) Uniform non-zero and antiparallel to E 0 (D) Non-uniform but along E 0 (C) It is not required for timevarying electromagnetic fields (D) It is already included implicitly in the Maxwell s equations

7 23. If A and are the vector and 24. A plane polarized EM wave is propagating in certain medium such that the ratio of the electric scalar potentials, respectively, energy density to the magnetic then the Lorentz gauge in timevarying electromagnetic case is 100 times larger than the given by : displacement current (B) Conduction and displacement (A) A 0 0 t (B) A 0 0 t energy density is Which of the following statements is valid for this medium? (A) Conduction current is about currents are of the same magnitude (C) Displacement current is about 100 times larger than the (C) A t 0 0 conduction current (D) he medium through which the (D) A 0 0 t EM wave is propagating is a dielectric

8 25. A proton of mass m and charge Q is moving in counter-clockwise 26. If two operators Â and ˆB have a common eigenfunction. What is the value of the commutator direction in a crossed uniform magnetic field of induction B pointed into the plane of the paper. If v and R are the velocity and (D) Zero radius of the orbit, then angular 27. Standing wave with wave function momentum p acquired by the particle is given by : (A) p = QBR (B) p = QB R [Â, ˆB]? (A) 1 (B) i (C) Â ˆB (x) = Asinkx, (corresponding to particles of mass m) gives rise to current density equal to : (A) A 2k/ m (B) A 2k (C) p = QB mr (D) p = QBR m (C) A 2 sinkxcoskx m (D) Zero

9 28. A particle moves in one-dimensional 29. he wave function of a particle potential (x). At x = a, if has a moving in one-dimensional time finite discontinuity (jump), then independent potential (x) is which of the following is true for its represented by (x) = e iax + b wave function! and its first where a and b are real constants. derivative!" at x = a? his means that the potential (x) (A) Both! and!" are discontinuous is of the form : (B) Both! and!" are continuous (A) (x) # x (C)! is continuous and!" must be (B) (x) # x 2 discontinuous (D)! is discontinuous and!" must (C) (x) = constant be continuous (D) (x) = e ax + b

10 30. In a quantum mechanical one-dimensional system, the Hamiltonian is : 31. Which of the following is an eigenfunction of linear momentum H = p2 x 2m + ½m$2 x 2 + %x = H 0 + %x where % is very small. he eigenvalues of H 0 are $ n ½ with (0) n x as eigenfunctions. Which of the following is correct? direction of positive x-axis, with (A) he energy levels are not changed since all eigen- zero uncertainty in the linear functions (0) n x have definite parity (B) All energy levels are shifted by a constant amount (C) Only n = odd energy operator, such that the i x eigenfunction describes a particle moving in free space in the momentum? (A) e kx (B) e ikx eigenvalues are shifted (C) coskx (D) Only n = even energy eigenvalues are shifted (D) sinkx

11 32. A system in one-dimension is in a 33. he expectation value of momentum for a bound state of finite square state described by : well potential (width a and depth (x, t) = Aexp{i(px Et)/} + 0 ) is : (A) Zero Bexp{ i(px + Et)/} (B) Depends on 0 A and B are complex numbers, p (C) Depends on a and E are real. he probability (D) Depends on 0 a L is quantum mechanical operator current density is given by : of orbital angular momentum. (A) p/m L L is equal to : (B) ( A B ) p/m (A) 0 (C) ( A 2 + B 2 ) p/m (B) il (C) l(l + 1) 2 (D) ( A 2 B 2 ) p/m (D) L 2

12 35. A constant volume ideal gas 36. Consider a process in which a system thermometer in thermal equilibrium is in thermal equilibrium with a with system A at temperature A measures a pressure p A. When the same thermometer is brought in thermal equilibrium with another system B at temperature A /3, the reservoir p. Such a process is pressure in the thermometer will called : now be : (A) p A /3 (B) p2 A 3 heat reservoir at temperature. he pressure of the system remains constant at the pressure of the (A) Isothermal process (B) Adiabatic and isochoric process (C) 3p A (C) Isochoric process (D) pa 3 (D) Isobaric and isothermal process

13 37. A Carnot engine operates between 38. A gas is contained in a box of volume at pressure P and temperature. two temperatures 1 and 2 as One wall of the box is moved at a shown in the diagram. very high speed so that the volume is doubled. Which of the following is true? (A) P, both increase (B) P, both decrease (C) P increases, decreases (D) P decreases, increases 39. A thermodynamic system is characterized by independent variables entropy S and volume he maximum efficiency of the and dependent variables pressure P and temperature. hen which of engine & max is given by : the following is true? 1 (A) & max = 1 2 (B) & max = 1 2 (A) (B) P S S P S S 2 (C) & max = (D) & max = (C) (D) P P S S

14 40. A system consists of 3 independent particles. Each particle can occupy any of the two energy states with energy 0 and. he system is in 42. In the Debye model of specific heat of solids, the specific heat at low temperature has the following thermal equilibrium with a heat reservoir at temperature. Its partition function will be : (A) Z = 1 + 3e ' + 3e 2' + e 3' (B) Z = 1 + e ' (C) Z = 1 + e 3' (D) Z = 1 + e ' + e 2' + e 3' (D) 1/2 41. Considering the Maxwell-Boltzmann 43. In Signal generators : distribution of speeds of molecules, the average speed of the molecule of mass m at temperature is : (A) (B) 3k m 8k m temperature dependence : (A) 3/2 (B) 2 (C) 3 (A) Energy is created (B) Energy is generated (C) Energy is converted from a simple d.c. source into a.c. energy at some specific (C) 0 (D) 2k m frequency (D) Energy is maintained

15 44. In an experiment, the voltage 45. When two rotary pumps are connected in series to a vacuum across a 10 k( resistor is applied to chamber, the ultimate pressure can CRO. he screen shows a sinusoidal be achieved is closest to : signal of total vertical occupancy (A) orr 3 cm and total horizontal occupancy (B) orr of 2 cm. he front-panel controls of (C) orr /div and time/div are on 2 /div (D) orr and 2 ms/div respectively. Find out 46. Which of the following noise can be the frequency of the waveform : reduced by integration of the (A) 250 khz signal? (A) White noise (B) 250 Hz (B) Shot noise (C) 2.5 khz (C) Flicker noise (D) 25 khz (D) Environmental noise

16 47. In which of the following detector P-N junction diode is used? (A) G.M. counter 49. A McLead gauge has a volume of 100 ml and a capillary of 0.5 mm diameter. he pressure indicated by a reading of 25 cm of mercury (B) Scintillation counter (C) Lithium drifted detector (D) Ionisation chamber 48. An oil drop has a weight of electron charge units is : (A) 2 electrons (B) 4 electrons (C) 3 electrons corresponds to : (A) (B) (C) (D) A transparent film of glass of N. With an electric field refractive index 1.5 is introduced of /m between the plates normally in the path of one of the of Millikan s oil-drop apparatus, this interfering beams of Michelson s drop is observed to be essentially interferometer which is illuminated with light of wavelength 4800 Å. balanced. he charge on the drop in his causes 500 dark fringes to sweep across the field of view. he thickness of the film is : (A) cm (B) cm (C) cm (D) 6 electrons (D) cm

M04M.1 Particles on a Line

Part I Mechanics M04M.1 Particles on a Line M04M.1 Particles on a Line Two elastic spherical particles with masses m and M (m M) are constrained to move along a straight line with an elastically reflecting