UPKAR PRAKASHAN, AGRA–2

By

Dr. Surekha Tomar

Revised & Enlarged Edition

© Publishers

Publishers

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ISBN : 978-81-7482-527-8

Code No. 500

Printed at : UPKAR PRAKASHAN (Printing Unit) Bye-pass, AGRA

Preface to the Fourth Edition

This revised version of the book consists of material covering the latest syllabus

prescribed by CSIR-UGC/GATE.

Section A is fully revised and changed according to the given syllabus. It is hoped that this

change will help students to grasp the Section A and respond with a commendable score in the

examination.

While every effort has been made to weedout printing errors, it is possible that some

might have managed to dodge the vigilant eyes. We would be grateful to the readers for

bringing these errors to our notice and appreciate any suggestion for further improvement of

this book.

—Dr. Surekha Tomar

Preface to the Third Edition

The third edition of the book will make the readers to fathom the concepts of physics byapplying it to the practical problems.

The first and second editions of the book that appeared earlier received a vehementsupport from the physicist’s fraternity.

The book in its present form will provide the conceptual clarity on the subject matterthrough question and answer. The step by step solution of even the typical physics problemwas given. It will be about the issues and techniques. Where appropriate, some discussionswill be technical.

The success rate of the students who are reading the book has been on the rise. Theprevious editions of the book have become an important tool in the preparation of variouscompetitive examinations like NET, GATE etc.

The current edition is the enhanced version of its previous edition. The views andsuggestions of scholars and seasoned professors from various universities have beenincorporated. Besides, it will contain the solved questions from various prestigious recentlyconducted competition exams like NET, GATE…

In this edition the book has been thoroughly revised and following changes have beenmade :

1. A New chapter titled Experimental Techniques and Data Analysis has been added aschapter 7.

2. Chapter 6 titled Experimental Design is replaced by chapter titled as Electronics.

3. In all the chapters, the three sections namely Information at glance, objectivequestions and the descriptive part are updated according to revised syllabus.

4. The comprehensive solutions of all the unsolved problems taken from the latestquestion papers of various prestigious competition exams at national and internationallevel has been added.

5. The misprints and omissions have been removed as far as possible.

The book is designed keeping in view the need of the students who are preparing for thecompetition exams. The book is for the physics students pursuing graduation or postgraduation wishes to clear competitions to become eligible for job or fellowships.

There is perhaps no more eloquent testimony to the recognition gained by the book thanthe fact that it has become a ‘must read’ book for preparing the competition exams.Suggestions for the further improvement for the book are cordially invited.

—Dr. Surekha Tomar

CONTENTS

● Previous Years’ Solved Papers

PART-A

General Aptitude 1–96

PART-B & C(Core & Advanced Course)

1. Basic Mathematical Methods 3–138● Objective Type Questions 38● Descriptive Questions 101

2. Classical Dynamics 139–209● Objective Type Questions 148● Descriptive Questions 185

3. Electromagnetics 210–313● Objective Type Questions 224● Descriptive Questions 285

4. Quantum Physics andApplication 314–424● Objective Type Questions 330● Descriptive Questions 388

5. Thermodynamics andStatistical Physics 425–534● Objective Type Questions 440● Descriptive Questions 493

6. Electronics 535–700

● Objective Type Questions 570● Objective Type Questions

(Digital Electronics) 615● Descriptive Questions 637

7. Experimental Techniquesand Data Analysis 701–759● Objective Type Questions 717● Descriptive Questions 741

8. Atomic and MolecularPhysics 760–853● Objective Type Questions 771● Descriptive Questions 797

9. Condensed Matter Physics 854–945● Objective Type Questions 867● Descriptive Questions 907

10. Nuclear and ParticlePhysics 946–1033● Objective Type Questions 964● Descriptive Questions 986● Descriptive Questions

(Particle Physics) 1024

➠ Objective Type Questionsfor Advanced Part 1034–1109

➠ Model Paper-I 1110–1119

➠ Model Paper-II 1120–1127

GENERAL INFORMATION

EXAM SCHEMETime : 3 Hours Max. Marks : 200

Single Paper Test having Multiple ChoiceQuestions (MCQs) is divided in three parts.

Part 'A' This part shall carry 20 questions pertaining

to General aptitude with emphasis on logicalreasoning graphical analysis, analytical andnumerical ability, quantitative comparisons, seriesformation, puzzles etc. The candidates shall berequired to answer any 15 questions. Eachquestion shall be of two marks. The total marksallocated to this section shall be 30 out of 200.

Part 'B'This part shall contain 25 Multiple Choice

Questions (MCQs) generally covering the topicsgiven in the Part ‘B’ of syllabus. Candidates arerequired to answer any 20 questions. Eachquestion shall be of 3·5 marks. The total marksallocated to this section shall be 70 out of 200.

Part 'C'This part shall contain 30 questions from Part

‘C’ & ‘B’ of the syllabus that are designed to testa candidate's knowledge of scientific conceptsand/or application of the scientific concepts. Thequestions shall be of analytical nature where acandidate is expected to apply the scientificknowledge to arrive at the solution to the givenscientific problem. A candidate shall be requiredto answer any 20 questions. Each question shall beof 5 marks. The total marks allocated to thissection shall be 100 out of 200.● There will be negative marking @25% for

each wrong answer.

● To enable the candidates to go through thequestions, the question paper booklet shall bedistributed 15 minute before the scheduled

time of the Exam. The answer sheet (OMRsheet) shall be distributed at the scheduledtime of the Exam.

SYLLABUSPart 'A'

This part shall carry 20 questions pertainingto General aptitude with emphasis on logicalreasoning graphical analysis, analytical andnumerical ability, quantitative comparisons, seriesformation, puzzles etc. The candidates shall berequired to answer any 15 questions. Eachquestion shall be of two marks. The total marksallocated to this section shall be 30 out of 200.

Part 'B'I. Mathematical Methods of Physics

Dimensional analysis. Vector algebra andvector calculus. Linear algebra, matrices, Cayley-Hamilton Theorem. Eigenvalues and eigenvectors.Linear ordinary differential equations of first &second order, Special functions (Hermite, Bessel,Laguerre and Legendre functions). Fourier series,Fourier and Laplace transforms. Elements ofcomplex analysis, analytic functions; Taylor &Laurent series; poles, residues and evaluation ofintegrals. Elementary probability theory, randomvariables, binomial, Poisson and normal distri-butions. Central limit theorem.

II. Classical MechanicsNewton’s laws. Dynamical systems, Phase

space dynamics, stability analysis. Central forcemotions. Two body Collisions—scattering inlaboratory and Centre of mass frames. Rigid bodydynamics—moment of inertia tensor. Non-inertialframes and pseudoforces. Variational principle.Generalized co-ordinates. Lagrangian andHamiltonian formalism and equations of motion.

( vii )

Conservation laws and cyclic co-ordinates.Periodic motion: small oscillations, normalmodes. Special theory of relativity—Lorentztransformations, relativistic kinematics and mass–energy equivalence.

III. Electromagnetic Theory

Electrostatics : Gauss’s law and itsapplications, Laplace and Poisson equations,boundary value problems. Magnetostatics: Biot-Savart law, Ampere's theorem. Electromagneticinduction. Maxwell's equations in free space andlinear isotropic media; boundary conditions on thefields at interfaces. Scalar and vector potentials,gauge invariance. Electromagnetic waves in freespace. Dielectrics and conductors. Reflection andrefraction, polarization, Fresnel’s law, inter-ference, coherence and diffraction. Dynamics ofcharged particles in static and uniformelectromagnetic fields.

IV. Quantum Mechanics

Wave-particle duality. Schrödinger equation(time-dependent and time-independent).Eigenvalue problems (particle in a box, harmonicoscillator, etc.). Tunneling through a barrier.Wave-function in co-ordinate and momentumrepresentations. Commutators and Heisenberguncertainty principle. Dirac notation for statevectors. Motion in a central potential: orbitalangular momentum, angular momentum algebra,spin, addition of angular momenta; Hydrogenatom. Stern-Gerlach experiment. Time-inde-pendent perturbation theory and applications.Variational method. Time dependent perturbationtheory and Fermi's golden rule, selection rules.Identical particles, Pauli exclusion principle, spin-statistics connection.

V. Thermodynamic and Statistical Physics

Laws of thermodynamics and their con-sequences. Thermodynamic potentials, Maxwellrelations, chemical potential, phase equilibria.Phase space, micro and macro-states. Micro-canonical, canonical and grand-canonicalensembles and partition functions. Free energyand its connection with thermodynamic quantities.Classical and quantum statistics. Ideal Bose andFermi gases. Principle of detailed balance.Blackbody radiation and Planck's distribution law.

VI. Electronics and Experimental Methods

Semi-conductor devices (diodes, junctions,transistors, field effect devices, homo and hetero-junction devices), device structure, devicecharacteristics, frequency dependence andapplications. Opto-electronic devices (solar cells,photo-detectors, LEDs). Operational amplifiersand their applications. Digital techniques andapplications (registers, counters, comparators andsimilar circuits). A/D and D/A converters.Microprocessor and microcontroller basics. Datainterpretation and analysis. Precision andaccuracy. Error analysis, propagation of errors.Least squares fitting.

PART ‘C’

I. Mathematical Methods of PhysicsGreen’s function. Partial differential

equations (Laplace, wave and heat equations intwo and three dimensions). Elements ofcomputational techniques: root of functions,interpolation, extrapolation, integration bytrapezoid and Simpson’s rule, Solution of firstorder differential equation using Runge-Kuttamethod. Finite difference methods. Tensors.Introductory group theory : SU(2), O(3).

II. Classical MechanicsDynamical systems, Phase space dynamics,

stability analysis. Poisson brackets and canonicaltransformations. Symmetry, invariance andNoether’s theorem. Hamilton-Jacobi theory.

III. Electromagnetic TheoryDispersion relations in plasma. Lorentz

invariance of Maxwell’s equation. Transmissionlines and wave guides. Radiation from movingcharges and dipoles and retarded potentials.

IV. Quantum MechanicsSpin-orbit coupling, fine structure. WKB

approximation. Elementary theory of scattering :phase shifts, partial waves, Born approximation.Relativistic quantum mechanics : Klein-Gordonand Dirac equations. Semi-classical theory ofradiation.

V. Thermodynamic and Statistical PhysicsFirst and second-order phase transitions.

Diamagnetism, paramagnetism and ferromag-netism. Ising model. Bose-Einstein condensation.

( viii )

Diffusion equation. Random walk and Brownianmotion. Introduction to non-equilibrium processes.

VI. Electronics and Experimental MethodsLinear and non-linear curve fitting, chi-square

test. Transducers (temperature, pressure/ vacuum,magnetic fields, vibration, optical and particledetectors). Measurement and control. Signalconditioning and recovery. Impedance matching,amplification (Op-amp based, instrumentationamp, feedback), filtering and noise reduction,shielding and grounding. Fourier transforms, lock-in detector, box-car integrator, modulationtechniques.

High frequency devices (including generatorsand detectors).

VII. Atomic & Molecular PhysicsQuantum states of an electron in an atom.

Electron spin. Spectrum of helium and alkaliatom. Relativistic corrections for energy levels ofhydrogen atom, hyperfine structure and isotopicshift, width of spectrum lines, LS & JJ couplings.Zeeman, Paschen-Bach & Stark effects. Electronspin resonance. Nuclear magnetic resonance,chemical shift. Frank-Condon principle. Born-Oppenheimer approximation. Electronic,rotational, vibrational and Raman spectra ofdiatomic molecules, selection rules. Lasers :spontaneous and stimulated emission, Einstein A& B coefficients. Optical pumping, populationinversion, rate equation. Modes of resonators andcoherence length.

VIII. Condensed Matter PhysicsBravais lattices. Reciprocal lattice. Diffraction

and the structure factor. Bonding of solids. Elastic

properties, phonons, lattice specific heat. Freeelectron theory and electronic specific heat.Response and relaxation phenomena. Drudemodel of electrical and thermal conductivity. Halleffect and thermo-electric power. Electron motionin a periodic potential, band theory of solids :metals, insulators and semi-conductors.Superconductivity : type-I and type-II super-conductors. Josephson junctions. Superfluidity.Defects and dislocations. Ordered phases ofmatter : translational and orientational order,kinds of liquid crystalline order. Quasi crystals.

IX. Nuclear and Particle Physics

Basic nuclear properties : size, shape andcharge distribution, spin and parity. Bindingenergy, semi-empirical mass formula, liquid dropmodel. Nature of the nuclear force, form ofnucleon-nucleon potential, charge-independenceand charge-symmetry of nuclear forces. Deuteronproblem. Evidence of shell structure, single-particle shell model, its validity and limitations.Rotational spectra. Elementary ideas of alpha, betaand gamma decays and their selection rules.Fission and fusion. Nuclear reactions, reactionmechanism, compound nuclei and direct reactions.

Classification of fundamental forces.Elementary particles and their quantum numbers(charge, spin, parity, isospin, strangeness, etc.).Gellmann-Nishijima formula. Quark model,baryons and mesons. C, P, and T invariance.Application of symmetry arguments to particlereactions. Parity non-conservation in weakinteraction. Relativistic kinematics.

CSIR-UGC-NET/JRF Exam. June 2016 Solved PaperPhysical Sciences(Held on 19 June, 2016)

PART A

1. “My friend Raju has more than 1000 books”,said Ram. “Oh no, he has less than 1000books”, said Shyam. “Well, Raju certainlyhas at least one book”, said Geeta. If only oneof these statements is true, how many booksdoes Raju have ?(A) 1 (B) 1000(C) 999 (D) 1001

2. Of the following, which is the odd one out ?(A) Cone (B) Torus(C) Sphere (D) Ellipsoid

3. An infinite number of identical circular discs

each of radius 12 are tightly packed such that

the centres of the discs are at integer values ofcoordinates x and y. The ratio of the area ofthe uncovered patches to the total area is—

(A) 1 – π4

(B)π4

(C) 1 – π (D) π

4. It takes 5 days for a steamboat to travel fromA to B along a river. It takes 7 days to returnfrom B to A. How many days will it take for araft to drift from A to B (all speeds stayconstant) ?(A) 13 (B) 35(C) 6 (D) 12

5. N is a four digit number. If the leftmost digitis removed, the resulting three digit number is1/9th of N. How many such N are possible ?(A) 10 (B) 9(C) 8 (D) 7

6. What is the minimum number of movesrequired to transform figure 1 to figure 2 ? Amove is defined as removing a coin andplacing it such that it touches two other coinsin its new position.

(A) 1 (B) 2(C) 3 (D) 4

7.

(A) (B)

(C) (D)

8.

2 | CSIR Physical Sci. (J-16)

Which of the following inferences can bedrawn from the above graph ?(A) The total number of students qualifying

in Physics in 2015 and 2014 is the same(B) The number of students qualifying in

Biology in 2015 is less than that in 2013(C) The number of Chemistry students

qualifying in 2015 must be more than thenumber of students who qualified inBiology in 2014

(D) The number of students qualifying inPhysics in 2015 is equal to the number ofstudents in Biology that qualified in 2014

9. A student appearing for an exam is declaredto have failed the exam if his/her score is lessthan half the median score. This implies—(A) 1/4 of the students appearing for the

exam always fail(B) if a student scores less than 1/4 of the

maximum score, he/she always fails(C) if a students scores more than 1/2 of the

maximum score, he/she always passes(D) it is possible that no one fails

10. The relationship among the numbers in eachcorner square is the same as that in the othercorner squares. Find the missing number—

(A) 10 (B) 8(C) 6 (D) 12

11. Which one of the following statements islogically incorrect ?(A) I always speak the truth(B) I occasionally lie(C) I occasionally speak the truth(D) I always lie

12. What comes next in the sequence ?

(A) (B)

(C) (D)

13. AB and CD are two chords of a circlesubtending 60° and 120° respectively at thesame point on the circumference of the circle.Then AB : CD is—

(A) √⎯ 3 : 1 (B) √⎯ 2 : 1

(C) 1 : 1 (D) √⎯ 3 × √⎯ 214. Which of the following best approximates sin

(0·5°) ?

(A) 0·5 (B) 0·5 × π90

(C) 0·5 × π

180(D) 0·5 ×

π360

15. The set of numbers (5, 6, 7, m, 6, 7, 8, n) hasan arithmetic mean of 6 and mode (mostfrequently occurring number) of 7. Then m ×n =(A) 18 (B) 35(C) 28 (D) 14

16. Fill in the blank : F2, …, D8, C16, B32, A64.(A) C4 (B) E4(C) C2 (D) G16

17. How many times starting at 1:00 pm wouldthe minute and hour hands of a clock make anangle of 40° with each other in the next 6hours ?(A) 6 (B) 7(C) 11 (D) 12

18. A solid contains a spherical cavity. The cavityis filled with a liquid and includes a sphericalbubble of gas. The radii of cavity and gasbubble are 2 mm and 1 mm, respectively.What proportion of the cavity is filled withliquid ?

(A)18

(B)38

(C)58

(D)78

19. The diagram shows a block of marble havingthe shape of a triangular prism. What is themaximum number of slabs of 10 × 10 × 5 cm3size that can be cut parallel to the face onwhich the block is resting ?

CSIR Physical Sci. (J-16) | 3

(A) 50 (B) 100

(C) 125 (D) 250

20. Brothers Santa and Chris walk to school fromtheir house. The former takes 40 minuteswhile the latter, 30 minutes. One day Santastarted 5 minutes earlier than Chris. In howmany minutes would Chris overtake Santa ?

(A) 5 (B) 15

(C) 20 (D) 25

PART B

21. Let (x, t) and (x′, t′) be the coordinate systemsused by the observers O and O′, respectively.Observer O′ moves with a velocity v = βcalong their common positive x-axis. If x+ = x+ ct and x– = x – ct are the linear combinationsof the coordinates, the Lorentz transformationrelating O and O′ takes the form—

(A) x′+ = x_ – βx+

1 – β2 and x′– =

x+ – βx–

1 – β2

(B) x′+ = 1 + β1 – β

x+ and x′– = 1 – β1 + β

x–

(C) x′+ = x+ – βx–

1 – β2 and x′– =

x– – βx+

1 – β2

(D) x′+ = 1 – β1 + β

x+ and x′– = 1 + β1 – β

x–

22. Four equal charges of +Q each are kept at thevertices of a square of side R. A particle ofmass m and charge +Q is placed in the planeof the square at a short distance a (

4 | CSIR Physical Sci. (J-16)

∞

Σn = 0

a(a + 1) … (a + n – 1)b(b + 1) … (b + n – 1)

c(c + 1) … (c + n – 1)n! zn,

satisfies the recursion relation—

(A)ddz

F(a, b, c; z) = cab

F(a – 1, b – 1, c – 1; z)

(B)ddz

F(a, b, c; z) = cab

F(a + 1, b + 1, c + 1; z)

(C)ddz

F(a, b, c; z) = abc

F(a – 1, b – 1, c – 1; z)

(D)ddz

F(a, b, c; z) = abc

F(a + 1, b + 1, c + 1; z)

28. The Hamiltonian of a system with generalizedcoordinate and momentum (q, p) is H = p2q2.A solution of the Hamiltonian equation ofmotion is (in the following A and B areconstants)—

(A) p = Be– 2At, q = AB

e2At

(B) p = Ae– 2At, q = AB

e– 2At

(C) p = AeAt, q = AB

e– At

(D) p = 2Ae– A2t, q =

AB

eA2t

29. The value of the contour integral

12πi

O∫C e4z – 1

cosh (z) – 2sinh (z) dz

around the unit circle C traversed in the anti-clockwise direction, is—

(A) 0 (B) 2

(C) – 8

√⎯ 3(D) – tanh ( )12

30. The state of a particle of mass m in a one-dimensional rigid box in the interval 0 to L isgiven by the normalised wavefunction ψ(x)

= 2L ( )35 sin ( )2πxL + 45 sin ( )4πxL . If

its energy is measured, the possibleoutcomes and the average value of energy are,respectively—

(A)h2

2mL2,

2h2

mL2 and

7350

h2

mL2

(B)h2

8mL2,

h2

2mL2 and

1940

h2

mL2

(C)h2

2mL2,

2h2

mL2 and

1910

h2

mL2

(D)h2

8mL2,

2h2

mL2 and

73200

h2

mL2

31. A magnetic field B is Bẑ in the region x > 0and zero elsewhere. A rectangular loop, in thexy-plane, of sides l (along the x-direction) andh (along the y-direction) is inserted into thex > 0 region from the x < 0 region at a

constant velocity v = vx̂. Which of thefollowing values of l and h will generate thelargest EMF ?(A) l = 8, h = 3 (B) l = 4, h = 6(C) l = 6, h = 4 (D) l = 12, h = 2

32. The x – and z-components of a static magneticfield in a region are Bx = B0(x2 – y2) and Bz =0, respectively. Which of the followingsolutions for its y-component is consistentwith the Maxwell equations ?(A) By = B0xy(B) By = – 2B0xy(C) By = – B0 (x2 – y2)

(D) By = B0 ( )13 x3 – xy233. Suppose that the Coulomb potential of the

hydrogen atom is changed by adding aninverse-square term such that the total

potential is V(→r ) = –

ze2

r +

gr2

, where g is a

constant. The energy eigenvalues Enlm in themodified potential—(A) depend on n and l, but not on m(B) depend on n but not on l and m(C) depend on n and m, but not on l(D) depend explicitly on all three quantum

numbers n, l and m

34. If L̂x, L̂y and L̂z are the components of theangular momentum operator in three dimen-

sions, the commutator [̂L x, L̂xL̂yL̂z] may besimplified to—

(A) i hLx ( )L̂2z – L̂2y (B) ihL̂zL̂yL̂x(C) ihLx (2L̂

2z – L̂

2y) (D) 0

35. Two parallel plate capacitors, separated bydistances x and 1·1x respectively, have adielectric material of dielectric constant 3·0inserted between the plates, and are connected

CSIR Physical Sci. (J-16) | 5

to a battery of voltage V. The difference incharge on the second capacitor compared tothe first is—(A) + 66% (B) + 20%(C) – 3·3% (D) – 10%

36. The eigenstates corresponding to eigenvaluesE1 and E2 of a time-independent Hamiltonianare |1〉 and |2〉 respectively. If at t = 0, thesystem is in a state |ψ(t = 0)〉 = sin θ|1〉 + cosθ|2〉 the value of 〈ψ(t) | ( ψ(t)〉 at time t willbe—(A) 1

(B)(E1 sin2θ + E2 cos2θ)

√⎯⎯⎯⎯E21 + E22(C) eiE1t/h sin θ + eiE2t/h cos θ

(D) e– iE1t/h sin2 θ + e– iE2t/h cos2 θ

37. The half space regions x > 0 and x < 0 arefilled with dielectric media of dielectricconstants ∈1 and ∈2 respectively. There is auniform electric field in each part. In the righthalf, the electric field makes an angle θ1 tothe interface. The corresponding angle θ2 inthe left half satisfies—

(A) ∈1 sin θ2 = ∈2 sin θ1(B) ∈1 tan θ2 = ∈2 tan θ1(C) ∈1 tan θ1 = ∈2 tan θ2(D) ∈1 sin θ1 = ∈2 sin θ2

38. A box of volume V containing N moleculesof an ideal gas, is divided by a wall with ahole into two compartments. If the volume ofthe smaller compartment is V/3, the varianceof the number of particles in it, is—

(A)N3

(B)2N9

(C) √⎯ N (D) √⎯ N3

39. Given the input voltage Vi, which of thefollowing waveforms correctly represents theoutput voltage V0 in the circuit shown below ?

(A)

(B)

(C)

(D)

6 | CSIR Physical Sci. (J-16)

40. The intensity distribution of a red LED on anabsorbing layer of material is a Gaussiancentred at the wavelength λ0 = 660 nm andwidth 20 nm. If the absorption coefficientvaries with wavelength as α0 – K(λ – λ0),where α0 and K are positive constants, thelight emerging from the absorber will be—(A) blue shifted retaining the Gaussian

intensity distribution(B) blue shifted with an asymmetric intensity

distribution(C) red shifted retaining the Gaussian

intensity distribution(D) red shifted with an asymmetric intensity

distribution

41. The dependence of current I on the voltage Vof a certain device is given by

I = I0 ( )1 – VV02

where I0 and V0 are constants. In anexperiment the current I is measured as thevoltage V applied across the device is

increased. The parameters V0 and √⎯ I0 can begraphically determined as—(A) the slope and the y-intercept of the I-V2

graph(B) the negative of the ratio of the y-intercept

and the slope, and the y-intercept of theI-V2 graph

(C) the slope and the y-intercept of the √⎯ I – Vgraph

(D) the negative of the ratio of the y-interceptand the slope, and the y-intercept of the

√⎯ I – V graph42. A gas of non-relativistic classical particles in

one dimension is subjected to a potential V(x)= α |x| (where α is a constant). The partition

function is ( )β = 1kBT(A)

4mπβ3α2h2

(B)2mπ

β3α2h2

(C)8mπ

β3a2h2 (D)

3mπβ3α2h2

43. In the schematic figure given below, assumethat the propagation delay of each logic gateis tgate.

The propagation delay of the circuit will bemaximum when the logic inputs A and Bmake the transition—(A) (0, 1) → (1, 1) (B) (1, 1) → (0, 1)(C) (0, 0) → (1, 1) (D) (0, 0) → (0, 1)

44. When an ideal monoatomic gas is expandedadiabatically from an initial volume V0 to3V0, its temperature changes from T0 to T.Then the ratio T/T0 is—

(A)13

(B) ( )132/3

(C) ( )131/3

(D) 3

45. The specific heat per molecule of a gas ofdiatomic molecules at high temperatures is—(A) 8kB (B) 3·5 kB(C) 4·5 kB (D) 3kB

PART C

46. In finding the roots of the polynomial f(x) =3x3 – 4x – 5 using the iterative Newton-Raphson method, the initial guess is taken tobe x = 2. In the next iteration its value isnearest to—(A) 1·671 (B) 1·656(C) 1·559 (D) 1·551

47. For a particle of energy E and momentum p(in a frame F), the rapidity y is defined as y =

12 ln

⎝⎜⎛

⎠⎟⎞E + p3c

E – p3c . In a frame F′ moving with

velocity v = (0, 0, βc) with respect to F, therapidity y′ will be—

(A) y′ = y + 12 ln (1 – β2)

(B) y′ = y – 12 ln ( )1 + β1 – β

(C) y′ = y + ln ( )1 + β1 – β(D) y′ = y + 2 ln ( )1 + β1 – β

continue

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