SUBJECT CODE BOOKLET CODE

2012 (11) PHYSICAL SCIENCES

TEST BOOKLET Time : 3:00 Hours ' Mcrxzmurn Marks: 200

INSTRUCTIONS

1. You have opted for English as medium of Question Paper. This Test Booklet contains seventy five (20 Part6A'+25 Part 'B' + 30 Part 'C') Multiple Choice Questions (MCQs). You are required to answer a maximum of 15,20 and 20 questions from-part 'A' 'By and 'C' respectively. If more than required number of questions are answered, only first 15, 20 and 20 questions in Parts 'A' 'B' and 'C' respectively, will be taken up for evaluation.

2. Answer sheet has been provided separately. Before you start filling up your particulars, please ensure that the booklet contains requisite number of pages and that these are not tom or mutilated. If it is so, you may request the Invigilator to change the booklet. Likewise, check the answer sheet also. Sheets for rough work have been appended to tht: test booklet.

3. Write your Roll No., Name, Your address and Serial Number of this Test Booklet on the Answer sheet in the space provided on the side 1 of Answer sheet. Also put your signatures in the space identified.

4. You must darken the appropriate circles with a pencil related to Roll Number, Subiect Code, Booklet Code and Centre Code on the OMR answer sheet. It is the sole responsibilitv of the candidate to meticulously follow the instructions ~ i v e n on the Answer Sheet, failinp which, the computer shall not be able to deci~her the correct details which may ultimately result in loss, including rejection of the OMR answer sheet.

5. Each question in Part 'A' carries 2 marks, Part 'B' 3.5 marks and Part 'C' 5 marks respectively. There will be negative marking @ 25% for each wrong answer.

6. Below each question in Part 'A', 'B' and 'C' four alternatives or responses are given. Only one of these alternatives is the "correct" optipn to the question. You have to find, for each question, the correct or the best answer.

7. Candidates found copvina or resorting to any unfair means are liable to be disqualified from t h s and future examinations.

8. Candidate should not write anything anywhere except on answer sheet or sheets for rough work.

9. After the test is over, you MUST hand over the Test Booklet and the answer sheet (OMR) to the invigilator.

\ 10. Use of calculator is not permitted.

Roll No. .......................

Name ...........................

I have verified all the information filled in by the candidate.

.............................. Signature of the Inv~giiariir-

C

lev amu

G

"

A

Po,

R

9.1 I x 10 Kg

6.63 10 '' J sec

1.6 x 10-'nC

1.38 x JIK

3.0 x 1 O%i/Sec

1.097 x 1 0"m-I

6.023 x 1 02' mole-'

USEFUL FUNDAMENTAL CONSTANTS

Mass of electron 9. 1 l x lo-" ~g

Planck's constant 6.63 x 10 "' J sec

Charge of elect1.011 1.G y 10 "'C

Boltzmann constant I.-38y lOUJ/K

Velocity of Light 3.0 r 1 0' m/Scc

Rydberg conslant

Avogadro number

8.854 x lo-'' Fm-'

471 lo-' Hm-I

Molar Gas constants

1.097. 10' m '

6.023 x 10'' mole-'

8.3 14 J K ' mole-'

& f&%? i..lf ww 3% 8 I .m #cv yi?az7 $ m (20 1 ~ m 'A'+ + 25 ?rm 'B' + 30 V T T 'C' ;) j QEYT f l i ~ i ? ~ ~ 1797 (h'IC'Q)& f? 1 W 'A' r)l d 1 5 3 3 ~ ~ '13' 1$' 20 ri?n 37i;'i 'i" 4 & 20 P$ d ): # 8 I / &$f&i ?? 3Lf$+F

~ y . $ &y ~ , y (Tcr a C ~ i . ; 1js-d v 7 ~ ' t i ' d 1 5 , ~ 'B' 3 f 20 M w 'C' i+

20 =& F~$ 3)-q $7 /

C

1,v amu

G

R,

A

E"

P o

R

lev amu

USEFUL, FUNDAMENTAL CONSTANTS

9.1 I 10 ' I Kg

6.63 x I0 " J scc

1.6 x 10 IqC

1.38 x lo-" .I/K

3.0 x 1 OK m/Sec

8.3 14 J K-' mole '

Mass of electron 9.1 1 x 10:' Kg

Planck's constant 6.63 x I0 I4Jsec

Charge of electron l . 6 ~ 10-'"C

Boltzmann constant

Velocity of Light

1.6 x 10-19~

Rydberg conslant

Avogadro number

8.854 x 10-I? Fm-I

4n x lo-' Hm-I

Molar Gas constants

1.097 x 10' m ' 6.023 x lo2' mole-'

LIST OF THE ATOMIC WEIGHTS OF THE ELEMENTS

Element Symbol Atomic Atomic Element Symbol Atomic Atomic Number Weight Number Weight

Actinium Ac 89 (227) Mercury Hg 80 200.59 Aluminium A l 13 26.98 Molybdenum M o 42 95.94 Americium A m 95 (243) Neodymium Nd 60 144.24 Antimony Sb 5 1 121.75 Neon Ne 10 20.183

Argon A r 18 39.948 Neptunium NP 93 (237) Arsenic A s 33 74.92 Nickel Ni 28 58.71 Astatine A t 85 (210) Niobium Nb 41 92.91 Barium Ba 56 137.34 Nitrogen N 7 14.007 Berkel ium Bk 97 (249) Nobelium No 102 Beryll ium

(253) Be 4 9.012 Osmium 0 s 76 190.2

Bismuth Bi 83 208.98 Oxygen 0 8 15.9994 Boron B 5 10.81 Palladium Pd 46 106.4 Bromine Br 35 79.909 Phosphorus P 15 30.974 Cadmium Cd 48 112.40 Platinum Pt 78 195.09 Calcium C a 20 40.08 Plutonium P u 94 Californium C f 98

(242) (251) ~ o l o n i u m Po 84

Carbon C 6 (210)

I2.Oq1 potassium K 19 39.102 Cerium C e 58 I 4 O . l 2 Praseodymium Pr 59 140.91 Cesium C s 55 132.91 Promethium Pm 61 Chlorine C I 17

(147) 35.453 Protactinium Pa 91

. Chromium C r 24 (231)

52.00 Radium Ra 88 Cobalt C o 27

(226) 58.93 Radon Rn 86

Copper C u 29 (222)

63'54 Rhenium Re 75 186.23 Curium C m 96

(247' Rhodium Rh 45 '102.91 Dysprosium DY 66 Einsteinium

162'50 Rubidium Rb 37 85.47 Es 99

(254) Ruthenium Ru 44 101.1 Erbium Er 68

167'26 samarium Sm [i 2 Europium 6 3

'1 50.35 Eu

Fermium 100 151'96 Scandium Sc 2 1 44.96

Fm Fluorine F 9

(253) selenium Se 34 78.96

Francium F r 87 19'00 sil icon Si 14 28.09

Gadolinium G d 64 (223) silver A g 47 107.870

157.25 Gallium Ga 3 1 69.72 Sodium Na 11 22.9898

Germanium Ge 32 72.59 Strontium Sr 38 87.62

Gold A u 79 196.97 Sulfur S 16 32.064

Hafnium Hf 72 178.49 Tantalum Ta 73 180.95

Helium He 2 4.003 Technetium Tc 43 (99)

Holmium Ho 67 164,93 Tellurium Te 52 127.60

Hydrogen H 1 1,0080 Terbium Tb 65 158.92

Indium In 4 9 114.82 Thallium TI 8 1 204.37

Iodine I 53 126,90 Thorium T h 90 232.04 Iridium I r 77 192.2 Thulium T m 59 168.93 Iron F e 26 55.85 Tin Sn 50 118.69 Krypton Kr 3 6 83,e0 Titanium Ti 22 47.90 Lar~thanum La 57 138.91 W 74 183.85 Lawrencium L r 103 (257) Uranium U 92 238.03 Lead Pb 8 2 207.19 Vanadium V 23 50,94 Lithium L i 3 6.939 Xe 54 131.30 Lutet ium L u 71 174.97 Ytterbium Yb 7 0 173.04 Magnesium M g 12 24,312 Yttrium Y 3 9 88.91 Manganese lLl n 2 5 54.94 Zinc Zr: 3 0 65.37

Mendelevium M d 201 (256) Zirconium Z T 4 0 91.22 - -- Rased on mass of Cf2 at 12.000 ... . The ratio of these weights of those on !!ic order chemical scale (in which oxygen of

natural isotopic composition was assigned a mass of 16.0000 ...) is 1.000055. (Values in parentheses represent the most stable known isotopes.)

A granite block of 2 m x 5 n~ x 3 n~ size is cut into 5 cm thick slabs of 2 m x 5 m size. 'These slabs are laid over a 2 m wide pavement. What is the length of the pavement that chn be covered with these slabs?

3 -. *+~If-w-8?

Which is the least among the following?

0 .33 ' .~~, 0.44O.~~, e-i /e 2

3 ,+'~&m3d'm?23mT;f)?icwmT2?

What is the next number in this "see and tell" sequence?

1 11 21 1211 111221

(1) 312211 (2) 1112221 (3) 1 1 12222 (4) 1112131

A vertical pole of Isngth u stands at the centre of a horizontal regular hexagonal ground of side u. A 1upc that is fixed taut in between a vertex on the ground and the tip oi'the pole has a length

A peacock perched on the top of' a 12 m liigli 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?

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?

1 & m 8 / There are an even number of odd cities 2. RW ;nf & M f3cm 81 / There are an odd number of odd cities

3. 3m ;nf & ?$zw rn 81 / There are an even number of even cities 4. rn ;m3) & ?$zw @W 81 / There are an odd number of even cities

In the figure LABC1=n / 2 A D = D E = E B

What is the ratio of the area of triangle ADC to that of triangle CDB?

A rectangular sheet ABCD is folded in such a way that vertex A meets vertex C, thereby forming a line PQ. Assuming AB= 3 and BC=4, find PQ. Note that AP = PC and AQ = QC.

A string of diameter lmm 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 1 m2. Then the length of the string is

lo. W C T % $ ~ ~ % J ; ~ T ~ ~ % & V ~ % ~ T X % ~ W X ~

25% of 25% of a quantity is x% of the quantity where x is

11. qz5J7 {a,) F)'m Ln d iV$ @?8 a) G% &r%?7 $ m 81 Q'f?a0=3, d dhTT7

In sequence {a,) every term is equal to the sum of all its previous terms.

an+, is If a. = 3, then limn,, - an

In the iigure below, angle ABC = 7112. I, 11, I11 are the areas of semicircles on the sides opposite angles B, A, and C, respectively. Which of the following is always true?

What is the minimum number of days between one Friday the 1 3 ' ~ and the next Friday the 13'~? (Assume that the year is a leap year).

Suppose a person A is at the North-East comer of a square (see the figure below). From that point he moves along the diagonal and after covering 113'~ portion of the diagonal, he goes to his left and after sometime he stops, rotates 90" clockwise and moves straight. After a few minutes he stops, rotates 180" anticlockwise. Towards which direction he is facing now?

( I u , f l - -~d (2) m3--r&si (:) L-":;;::: 9,~; / ; \ \ .+ :<I$> ( I?-- ~/f$37~ ,. .

( 1 ) North-East (2) North- W esr i.; i :tc:!~q.;,- !:.A:;: (~4) (,;rjuih-\;G'es[

C ' L I C U ~ I ~ ~ ~ contains 99% water. Ramesh buys 1 ii0 kg of cucumbers. After 30 days of storing, the cucumbers lose some water. They 11ow contain 98'31 walcr. What is the total weight of cucumbers now?

(A) 1837 .$id? ~T;T (B) 1907 $?8 ?q' (C) 1947 u'2%8 (D) 200 $m @

~ ~ / ~ ? + ~

In a museum there were old coins with their respective years engraved on them, as follows.

(A) 1837 AD (B) 1907 AD (C) 1947 AD (D) 200 BC

Identify the fake coin(s)

(1) %m lcoin A (2) i?%m/coin D (3) 4 A a S (4) Ahm /coin C /coins A and D

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).

( y k 8 !'I / C - - -

n - 0 n n

Time Time Time Time

Which of the following is not correct?

(1) *(A) (2) *(B) (3) *(C) (4) *(Dl ( 1 ) Graph (A) (2) Graph (B) (3) Graph (C) (4) Graph (Dl

Find the missing letter :

Consider the following equation x2 + 4y2 + 9z2 = 14x + 28y+42z + 147 where x, y and z are real numbers. Then the value of x+2y+3z is

(4) 3T;m 769 /not unique

C

P R V D

A ? U Q B

EGK

OJF

71'lic niap gi\,cn bclo~v shows a meandering river following a scn;i-circular path, along \vl?icIi [ \LO villagcs arc loca(ctl at A and 13. l ' hc distance between A and B along the cast-west Ji~.cction i l l thc niap 1s 7 cm. What is the Icngth o f t h e river between A and f3 in the g ~ ~ u n d ' !

SCALE = 1:50000 '

l ~ / 5 and e 1 ~ / 6 2 1. A 2 x 2 matrix A has eigenvalues e . The smallest value of n such that A" = I

is

1. 20 2. 30 3. 60 4. 120

1. 8m vm JO ( x ) 2. cosx 1

3. e-"cos x 4. --cosx X

22. The graph of the ftlinction J'(x) shown below is best described by

1. The Bessel fui~ctioi~ J , (x) 2. C0.Y X

1 3. e--'cosx 4. -cosx

X

23. In a serles of five Cricket matches, one i)f'the capta~lls calls "lleads" every time when the toss is taken. ' f ie probability that he will win 3 times and lose 2 times is

24. The unit nonnal vector at the point [ --- 2, --- $; . -- 1 on the surface of the ellipsoid

25. A solid cylinder of height H , radius R and density p , floats vertically on the surface of a

liquid of density Pi,. The cylinder will be set inlo oscillatory motion when a small

i~lstalltalleous downward force is applied. The frequency of oscillatio~l is

26. Three particles of equal mass ni are connected by two den tical massless springs of stiffness

constant k as shown in the figure:

Ic k k m m m

If X, ,x, and x3 denote the horizontal displacements of the masses from their respective

equilibrium positions, the potential energy of the systenl is

27. 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

1 . dE = constant A P

28. A binary star system consists of two stars S, and S, , with masses rn and 2m respectively separated

by a distance r . If both S, and S, individually follow circular orbits around the centre of mass

with instantaneous speeds v , and v2 respectively, the speeds ratio v, / V , i -

29. 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 of the circle. The charge g is

symmetrically placed with respect to the chargese . If the electric field at the centre of the

circle is zero, what is the magnitude of '1

30. Consider a hollow charged shell of inner radius a and outer radius b . The volume charge

k density is p(r) = ( k is a constant) in the region n < r < b . The magnitude of the electric

r - field produced at distance r > n is

k(b - a ) 1. for all r > a

I" r2 k(b - a )

2. kb

for n <r < b and -- for r > b I,, r I, r2

Ii(r - ( 1 ) 3 , - - --- k(b - (1 )

3 for a < r- < h and for I- > h

Ell /-- lo r2

I < ( / . - ( 1 ) ..I, -- ~

k (b - c r ) i f I I 1 for 1, > I?

L', , (1 E l , (1

3 1 . Consider the interference of two coherent electromagnetic waves whose electric field vectors - - are given by E, = E,, C O S C O ~ and E, = E,, C O S ( W ~ + (o) where (O is the phase

difference. ?Ille inte~~eity of tlie resulting wave is given by U ( ~ ' ) , where ( E ' ) is the time 2

average of E' . The total i~lte~lsi ty is

32. Four charges (two +q and two --q ) are kept fixed at the four vertices of a square of side u as

shown

At the point P which is at a distance R from the centre ( R >> a ), the potential is proportional to ,

33. A point charge 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?

- 3. There is no finite value of d 4. ,/mgr~,, / q

34. 6 * v ~ & ~ r n w a ? j T - m

34. The wave function of a state of the hydrogen atom is given by

h e r / IS the norn~alii.ed elgen lunctloil ofthe state with quantum numbers n.1 and rzlm

m in the usual notation. The expectation value of L- in the state Y is

1 35. . The energy eigenvalues of a particle in the potential V ( x ) =-rnu2x2 - a x are

2

4. E,, = ( n + : ) h m

36. If a particle is represented by the normalized wave function

the uncertainty Ap in its momentum I S

37. Given the usual canonical commutatioo relations, the commutator [ A , B] of A =i(xPP -ypX)'

and B = (-ypZ + z p Y ) is

38. The entropy of a system, S , is related to the accessible phase space volume r by

S = k,ln T ( E , N, V ) where E, Nand Vare the energy, number of particles and volume

respectively. From this one can conclude that r

1. does not change during evolution to equilibrium 2. oscillates during evolution to equilibrium 3. is a maximum at equilibrium 4. is a minimum at equilibrium

30. Let A W be the work done in a quasistatic reversible thermody~~an~ic process. Which of the following statenients about A W is correct'?

1 . A W is a perfect differential if the process is isotheniial 2. A W is a perfect differential if the process is adiabatic 3. A w is always a perfect differential 4. A W cannot be a perfect differential

40. Consider a system of three spins S, , S, and S3 each of which can take values +1 and -1. The

energy of the system is given by E = -J[S, S2 + S2 S3 + S3 S, ] , where J is a positive

constant. The minimum energy and the corresponding number of spin configurations are, respectively,

1. J and 1 2. - 3 J a n d l 3. - 3 J a n d 2 4. - 6 J a n d 2

4 1. The minimum energy of a collection of 6 non-interacting electrons of spin- f and mass m

placed in a one dimensional infinite square well potential of width L is

42. 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 will be zero outside the frequency band

1. 7.00 MHz to 7.01 MHz 3. 6.99 MHz to 7.00 MHz

2. 6.99 MHz to 7.01 MHz 4. 6.995 MHz to 7.005 MHz

43. In the op-amp circuit shown in the figure, V, is a sinusoidal input signal of frequency 10

Hz and Vo is the output signal.

The magnitude of the gain and the phase shift, respectively, close to the values

1. 5& and a/2 2 . 5& and - a/2

3. 10 andzero 4. 10 and r

44. The logic circuit shown in the figure below

implements the Boolean expression

u,,' + 2v,, , for v,, > 0 i =( 0,

for v, I 0

45. A diode D as shown in the circuit has an i - v relation that can be approximated by

. {;r2 + 2vD, for vLj > 0 1 , = for v,, 5 0

The value of V,, in the circuit is

I . ( - 1 + & ) ~ 2. 8 V 3. 5 V

46. The Taylor expansion of the function in (cosh x) , where x is real, about the point x =O starts with the following tenns:

47. Given a :! < 2 unitary matrix U satisfying Cj 'l-' -: (i'U = i witil det 0' - c ~ " " , one can

constluct a unitary ~uatrix V ( V f V = V V f - 1 ) wlth det J ' --- 1 from it by

1 . rn~~lt~plylng U by e -1p/ 2

2. ml~lt~plvlng any single elcment of U by c "' -1u 1.2

3. tnult~plylng s11) row or co1u.t~: of U by e 4. multlpiylng U by e-lp

7"z 48. The value of the integral , where C is a closed contour defined by the

i z ' -5z+6

equation 2 1.1 - 5 = 0 , traversed in the anti-clockwise direction, is

d ' f 49. e-ii f ( x ) 3 i a a i r * m ( 3 - 2 i ) f = 0 r n i q j i ¶ % = i m d ~ i r ~ f (0)=1

dx' Y ~ ( x ) - + o m x - t m m@minmi/ f ( r ) m m ? :

1. e2?r 2. e-2n - 7n 2 zi 3 . -e 4. -e

49. The function f ( x ) obeys the differential equation - d 2 f - (3 - 2i) f = 0 and satisfies the dx2

conditions f ( 0 ) = 1 and f (x) -+ o as x + m . The value of f ( i r ) is

I. a2?r 2. e-2n -2n 2 ni 3 . -e 4. -e

5 0 . A planet of mass 171 moves in the gravitational field of the Sun (mass M ). If the semi-major

and semi-minor axes of the orb~t are a and b respectively, the angular mornenturn of the planet 1s:

5 1 . The Hamiltonian of a simple pendulum consisting of a mass rn attached to a nlassless string

Po - of length C is H = - + mgl(l - case) . If L denotes the Lagrangian, the value 2me'

52. Which of the fo l lowi~~g set of pllase-space trajectories is possible for a particle obeying Ikunilton's equations of motion ?

53. 7'wo bodles of equal mass ~i are connected by a massless ngld rod of len@h t lying 111 tile A?? -

plane with the centre of the rod at the origin. If this system is rotating about the z-axis with a

frequency W , its an~wlar momentum is

54. An infinite solenoid with its axis of synlnletry along the z -direction carries a steady current I .

- The vector potential A at a distance R from the axis

I . is constant inside and varies as R outside the solenoid 2. varies as R inside and is constant outside the solenoid

I 3. vanes as - ins~de and as X o ~ ~ t s ~ d e the solelloid

R

1 4. vanes as R inside and as - outside the solenoid

R

55. ('o~isider an inlinite conduct~ng sheet in the XJ' -plane with a time dependent current density

K t i , wllerc k' is a constant 1.11~ rector potaitinl at (I, -y. 2) is given by

* 'LL K 1 ^ - A = ! L ( I - ) i . Fl.be magnetic field B is 4 c

- 56. When a charged particle elnits electromagnetic radiation, the electric field E and the

.- - - 1 1 Poynting vector S = ' - E X B at a large distance Y from the emitter vary as - and

llo /"I'

respectively. Which of the fhllowing clioices for n and rtr are correct'!

1 . /1=1 311d / 1 1 = 1

3 . t l = 1 and n1=2 2 . /1=2 and 171'2 4 . rz = 2 and tn = 4

2. yo ( x ) = sech x

4 . ( x ) = sech3x

1 . 57. The energies in the ground state and first excited state of a particle of mass rn =- in a potential

2

v (x) are -4 and - 1 , respectively, (in units in which h =I ). If the corresponding

wavefunctions are related by Y , ( x ) = yo ( x ) sinh X , then the ground state eigenfunction is

2. yo ( x ) = sech x

4 . yo ( x ) = sech3x

0, - a < x < a v (x) = a, 3m.W

5 8. The perturbation

j"(0-x), - a < x < a [ I t =

\(I , otherwise

j o . - o < x < n v(x)=

l m , otherwise.

, . I he first order cot-rection to the ground-state energy of the particle is

59. Let 1 0 ) and 11) denote the normalized eigenstates corresponding to the ground and the first

excited states of a one-dimensional harmonic oscillator. The uncertainty h in the state

60. What would be thc ground state energy ot'~!i~.: : ;. ..: .,,lian

A e-bx2 if variational principle is used to estimate it wit11 tlie trial wnvef~inction ( x ) = with b as tlie variational parameter?

m I 2" -2"' & = (2b)-"-

[Hint: x e -m

61. The free energy difference between the superconducting and tlie ~iormal states of a material is

4 given by A F = Fs - F, = a ly/(- +-It+/( , where I// IS an order parameter and a and P are 2

constants such that a > O in the normal and a < 0 in the superconducting state, while P > 0 always. The min~mulli value of A F 1s

62. 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 below.

During the process A + C the work done by the gas is 100 J and the 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

1. 205 2. 80 J 3. 220J 4. 280 J

63. Consider a one-dimensional Ising model with N spins, at very low temperatures when almost all the spins are aligned parallel to each other. There will be a few spin flips with each flip costing an energy 2 J . In a configuration with r spin flips, the energy of the system is E=- NJ + 2rJ

.V ,,--; and the number of configuratioii is c,,. ; i- varies i :m 0 to I\'. The partition hnction is

64. A magnetic field sensor based on the Hall effect is to be fabricated by implanting As into a Si film of thickness 1 pm. The specificatiosls require a magnetic field sensitivity of 500 mV/Tesla at an excitation current of 1 nlA. The implantation dose is to be adjusted such that the average carrier density, after activation, is

65 . Band-pass and band-reject filters can be implemented by combining a low pass and a high pass j filter in series and in parallel, respectively. If the cut-off frequencies of the low pass and high

pass filters are and O f P , respectively, the condition required to implement the band-

pass and band-reject filters are, respectively,

HP LP 1. ooHP < W k P and Wo <oO 2. ofp <okP and ofP >&loLP

3 lo:p > OkP and W t P < c i k P 4. u:P > 0 k P and uOHP > a);P

66. The output characteristics of a solar panel at a certain level of irradiance is shown in the figure below.

III

v

If the solar cell is to power a load of 5 R, the power drawn by the load is

1. 97 W 2. 73 W 3. 50 W 4. 45 W

67. ~ ~ & w * . $ f l f 3 9 ~ ~ & m d w r n f 3 J r 3 /

1. l0l4 cm-' and 2x10" cni-' 2. 2 x 1 012 ~111.~ and 1 OI4 cm"

3. 2x10I2 cm-' and 2x10" cm-' 4. zero and 10" ~ r n ' ~

67. Consider the energy level diagranl shown below, which corresponds to the molecular nitrogen laser.

If the pump rate R is 10"' atoms c111-' S" and the decay routes are as shown with Zzl =20 ns

and s, =1 pS , the equilibrium populations of states 2 and 1 are, respectively, [

1 . 10'4cm-' and 2 ~ 1 0 " c n 1 - ' 2. 2~ 10" cn~-%nd 1 014 cm-'

3. 2x10" cm-' and 2x10' cm-' 4. zero and 10"' cm"

68. Consider a hydrogen aton1 undergoing a 2P-+lS transition. The lifetime ts,, of the 2P state for spontaneous emission is 1.6 ns and the enerby difference between the levels is 10.2

eV. Assuming that the refractive index of the medium n, = 1, the ratio of Einstein coefficients

for stimulated and spontaneous emission B,, ( w ) / A , (0) is given by

69. rn- R, = 1 w R, = 0.98 J: d * m? w He-Ne 7;)il?r n %v?l v h rJi @ + d = 2 0 M ~ w ~ . $ p ~ d w w ~ i ~ ~ n ~ = l ~ 3 1 J m ) B o T ~

a=O r ? ~ m h r ) d d t J J : m S v p&w3ffmr&nmiwiftdz$Avp%:

I . 6 v = 7 5 k H z , A v p = 2 4 k H z 2. Sv =I00 kHz, Avp=lOO kHz

3. 6v=750MHz,Avi ,=2 .4MHz 4. 6v=2.4MHz,Avp=75OMHz

69. Consider a He-Ne laser cavity consisting of two mirrors of reflectivities R, = 1 and

R2 = 0.98. The mirrors are separated by a distance d = 20 cm and the medium in between

has a refractive index no = 1 and absorption coefficient = 0 . The values of the separation

between the modes SV and the width A Vp of each mode of the laser cavity are :

I . 6v=75kHz, ,4vp=24kHz 2. 6v =I00 kHz, 4vp=100 kHz

3 . 6 ~ = 7 5 0 M H z , A v , ~ = 2 . 4 M H z 4. ~ v = ~ . ~ M H ~ , A v , ~ = ~ ~ o M H z

70. F&TI%* M GT ;?ia F *-M w&%+z ?mi M M $ WV d 2 F/V

ah-a~$-t~~q-r m (BEC) m 8! BEC J: sRi.r 8.13 &J 7- @W i ? h #

70. Non-interacting bosons undergo Bose-Einstcin Condensation (BEC) when trapped in a three- dimensional isotropic simple harmonic potential. kor BEC to occur, the chemical potential must be equal to

71. ~ ~ ~ F 1 : 3 ~ ~ J : ~ ~ ~ 4 m v ~ m 8 ~

.E~ = p(cos kxa + cos kva + cos kza) m p p m $ v a ~ m m J i y r b i ~ e ~ 7 & p & & i f t w m a r r j t 8 :

71. In a band structure calculation, the dispersion relation for electrons is found to be

E~ = ~ ( C O S kxa + cos kJa + cos k : a ) ,

where p is a constant and a is the lattice constant. The effective mass at the boundary of the first Brillouin zone is

ti" 4. -

3pf12

72. The radius of the Fermi sphere of free electrons in a monovalent metal with an fcc

structure, in which the volun~e of the unit cell is a3 , is

73. 'The l~luon has mass 105 MeV /c' and mean lifetime 2.2 ps in its rest frame. The mean

distance traversed by a muon of energy 3 15 MeV / C' before decaying is approximately

74. Consider the following particles: the proton p , the neutron n , the neutral pion no and the

delta resonance A'. When ordered in terms of decreasing lifetime, the correct arrangement is as follows:

75. * ' : t ~ n f p - ~ & ( m m ~ p,,, W p, , , ) d d h in W-yioftq crzd 3m? 3 MeV 81 & I f m & J : & J : ~ ~ d - ~ $ :

1. -7 MeV 2. 7MeV 3. 5 MeV 4. -5 MeV

75. The single particle energy difference between the p-orbitals (i.e., p,,, and PI, , ) of the nucleus 114

I 50 Sn is 3 MeV. The energy difference between the states in its If orbital is

I I 1. -7 MeV 2. 7MeV 3. 5 MeV 4. -5 MeV

38

TVi ~ I R o U G H WORK

Tq; ~IROUGH WORK

~ q ; +/ROUGH WORK

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