Part-A, B & C

ByKamlesh Sinha

CSIR-UGC NETJoint Test for Junior Research Fellowship and

Eligibility for Lectureship

Physical SciencesFor Part-A, B & C

(With Authentic & Detailed Solutions)

Solved Papers11June, 2017—June, 2012

CONTENTS

Important Information About Test ........................................................ (i)

Syllabus ............................................................................................... (ii)

Solved Paper, June, 2017.................................................................1-17

Solved Paper, December, 2016 ........................................................1-16










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Important Information About Test

Educational Qualification

M. Sc. or equivalent degree/ Integrated BS-MS/BS-4 years/BE/B. Tech/B. Pharma/MBBS with at least 55% marks

for General (UR) and OBC candidates and 50% for SC/ST, Persons with disability (PwD) candidates.

Candidates enrolled for M. Sc. or having completed 10+2+3 years of the above qualifying examination are also

eligible to apply in the above subject under the Result Awaited (RA) category on the condition that they complete the

qualifying degree with requisite percentage of marks within the validity period of two years to avail the fellowship from

the effective date of award.

Such candidates will have to submit the attestation format (Given at the reverse of the application form) duly

certified by the Head of the Department/Institute from where the candidate is appearing or has appeared.

B. Sc. (Hons) or equivalent degree holders or students enrolled in Integrated MS-PhD program with at least 55%

marks for General (UR) and OBC candidates; 50% marks for SC/ST, Persons with disability (PwD) candidates are also

eligible to apply. Candidates with bachelor’s degree will be eligible for CSIR fellowship only after getting registered/

enrolled for PhD/Integrated PhD program within the validity period of two years.

Candidates having Bachelor's degree only shall not be eligible for Lectureship.

The eligibility for lectureship of NET qualified candidates will be subject to fulfilling the cretereia laid down by UGC.

PhD degree holders who have passed Master's degree prior to 19th September 1991, with at least 50% marks are eligible

to apply for Lectureship only.

Age Limit & Relaxation

For JRF (NET) : Maximum 28 years {upper age limit may be relaxed up to 5 years in case of SC/ST/Persons with

disability (PwD)/ female applicants and 03 years in case of OBC (non creamy layer) applicants}

For LS (NET) : No upper age limit.

Scheme of the Test

The single paper MCQ based test will be held as under :

Subject Marks

(i) Life Sciences

(ii) Earth, Atmospheric, Ocean and Planetary Sciences

(iii) Mathematical Sciences 200

(iv) Chemical Sciences

(v) Physical Sciences

The Test Booklet shall be divided in three parts. (A, B & C) as per Syllabus & Scheme of Exam.

Part ‘A’ shall be common to all subjects. This part shall contain questions pertaining to General Aptitude with

emphasis on logical reasoning, graphical analysis, analytical and numerical ability, quantitative compari-

son, series formation, puzzles etc.

Part ‘B’ shall contain subject-related conventional Multiple Choice Questions (MCQs), generally covering the top-

ics given in the syllabus.

Part ‘C’ shall contain higher value questions that may test the candidate's knowledge of scientific concepts and/or

application of the scientific concepts. The questions shall be of analytical nature where a candidate is

expected to apply the scientific knowledge to arrive at the solution to the given scientific problem.

Syllabus & Scheme of Exam of single MCQ Paper may be seen at CSIR HRDG website :

www.csirhrdg.res.in.

Note : The Exam Scheme for Chemical Sciences has been revised from June, 2017 CSIR-UGC (NET) Exam

onwards. The revised exam scheme and model Question paper may be seen at CSIR HRDG website www.csirhrdg.res.in

Negative marking for wrong answers, wherever required, shall be applicable as per subject wise scheme of Exam.

If a question for any reason found wrong, the benefit of marks will be given to only those candidates who attempt the

question.

(ii)

SYLLABUS

PART ‘A’ (CORE)

I. Mathematical Methods of Physics :

Dimensional analysis, Vector algebra and vector 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 of complex analy-sis, analytic functions; Taylor & Laurent series; poles, residues and evaluation of integrals, Elementary probability

theory, random variables, binomial, Poisson and normal distributions, Central limit theorem.

II. Classical Mechanics :

Newton’s laws, Dynamical systems, Phase space dynamics, stability analysis, Central force motions, Two bodyCollisions - scattering in laboratory and Centre of mass frames. Rigid body dynamics moment of inertia tensor, Non-

inertial frames and pseudo forces, Variational principle. Generalized coordinates, Lagrangian and Hamiltonian formal-ism and equations of motion, Conservation laws and cyclic coordinates, Periodic motion: small oscillations, normal

modes, Special theory of relativity - Lorentz transformations, relativistic kinematics and mass - energy equivalence.

III. Electromagnetic Theory :

Electrostatics: Gauss’s law and its applications, Laplace and Poisson equations, boundary value problems,

Magnetostatics: Biot-Savart law, Ampere’s theorem, Electromagnetic induction, Maxwell’s equations in free space andlinear isotropic media; boundary conditions on the fields at interfaces, Scalar and vector potentials, gauge invariance,

Electromagnetic waves in free space, Dielectrics and conductors, Reflection and refraction, polarization, Fresnel’s law,interference, coherence, and diffraction, Dynamics of charged particles in static and uniform electromagnetic fields.

IV. Quantum Mechanics :

Wave-particle duality, Schrodinger equation (time-dependent and time-independent) Eigenvalue problems (particle

in a box, harmonic oscillator, etc.), Tunneling through a barrier, Wave-function in coordinate and momentumrepresentations. commutators and Heisenberg uncertainty principle, Diracnotation for state vectors, Motion in a

central potential: orbital angular momentum, angular momentum algebra, spin, addition of angular momenta;Hydrogen atom, Stern-Gerlach experiment, Time independent perturbation theory and applications, Variational method,

Time dependent perturbation theory and Fermi’s golden rule, selection rules, Identical particles, Pauli exclusionprinciple, spin-statistics connection.

V. Thermodynamic and Statistical Physics :

Laws of thermodynamics and their consequences, Thermodynamic potentials, Maxwell relations, chemical poten-

tial, phase equilibria, Phase space, micro and macro-states. Micro-canonical, canonical and grand-canonicalensembles and partition functions, Free energy and its connection with thermodynamic quantities, Classical and

quantum statistics. Ideal Bose and Fermi gases. Principle of detailed balance, Blackbody radiation and Planck’sdistribution law.

VI. Electronics and Experimental Methods :

Semiconductor devices (diodes, junctions, transistors, field effect devices, homo- and hetero-junction devices), de-vice structure, device characteristics, frequency dependence and applications, Opto-electronic devices (solar cells, photo-

detectors, LEDs), Operational amplifiers and their applications, Digital techniques and applications (registers, counters,comparators and similar circuits), A/D and D/A converters, Microprocessor and microcontroller basics.

Data interpretation and analysis, Precision and accuracy, Error analysis, propagation of errors, Least squares fitting.

PART ‘B’ (ADVANCED)

I. Mathematical Methods of Physics :

Green’s function. Partial differential equations (Laplace, wave and heat equations in two and three dimensions),

Elements of computational techniques: root of functions, interpolation, extrapolation, integration by trapezoid and

Simpson’s rule, Solution of first order differential equation using RungeKutta method, Finite difference methods. Ten-

sors, Introductory group theory : SU(2), O(3).

(iii)

II. Classical Mechanics :

Dynamical systems, Phase space dynamics, stability analysis, Poisson brackets and canonical transformations,

Symmetry, invariance and Noether’s theorem, Hamilton-Jacobi theory.

III. Electromagnetic Theory :

Dispersion relations in plasma, Lorentz invariance of Maxwell’s equation, Transmission lines and wave guides,

Radiation- from moving charges and dipoles and retarded potentials.

IV. Quantum Mechanics :

Spin-orbit coupling, fine structure, WKB approximation, Elementary theory of scattering: phase shifts, partial

waves, Born approximation, Relativistic quantum mechanics: Klein-Gordon and Dirac equations, Semi-classical theory

of radiation.

V. Thermodynamic and Statistical Physics :

First- and second-order phase transitions, Diamagnetism, paramagnetism, and ferromagnetism, Ising model,

Bose-Einstein condensation, Diffusion equation, Random walk and Brownian motion, Introduction to nonequilibrium

processes.

VI. Electronics and Experimental Methods :

Linear and nonlinear curve fitting, chi-square test. Transducers (temperature, pressure/ vacuum, magnetic fields,

vibration, optical, and particle detectors), Measurement and control, Signal conditioning and recovery. Impedance matching,

amplification (Op-amp based, instrumentation amp, feedback), filtering and noise reduction, shielding and grounding,

Fourier transforms, lock-in detector, box-car integrator, modulation techniques.

High frequency devices (including generators and detectors).

VII. Atomic & Molecular Physics :

Quantum states of an electron in an atom, Electron spin, Spectrum of helium and alkali atom, Relativistic correc-

tions for energy levels of hydrogen atom, hyperfine structure and isotopic shift, width of spectrum lines, LS & JJ

couplings, Zeeman, Paschen-Bach & Stark effects, Electron spin resonance, Nuclear magnetic, resonance, chemical

shift, Frank-Condon principle, Born-Oppenheimer approximation, Electronic, rotational, vibrational and Raman spectra

of diatomic molecules, selection rules. Lasers : spontaneous and stimulated emission, Einstein A & B coefficients,

Optical pumping, population inversion, rate equation, Modes of resonators and coherence length.

VIII. Condensed Matter Physics :

Bravais lattices, Reciprocal lattice, Diffraction and the structure factor, Bonding of solids, Elastic properties, phonons,

lattice specific heat. Free electron theory and electronic specific heat, Response and relaxation phenomena, Drude

model of electrical and thermal conductivity. Hall effect and thermoelectirc power, Electron motion in a periodic poten-

tial, band theory of solids : metals, insulators and semiconductors, Superconductivity: type-I and type-II superconduc-

tors, Josephson junctions, Superfluidity, Defects and dislocations, Ordered phases of matter: translational and orienta-

tional order, kinds of liquid crystalline order, Quasi crystals.

IX. Nuclear and Particle Physics :

Basic nuclear properties : size, shape and charge distribution, spin and parity, Binding energy, semi-empirical mass

formula, liquid drop model, Nature of the nuclear force, form of nucleon-nucleon potential, charge-independence and

charge-symmetry of nuclear forces, Deuteron problem. Evidence of shell structure, single-particle shell model, its

validity and limitations. Rotational spectra. Elementary ideas of alpha, beta and gamma decays and their selection

rules, Fission and fusion, Nuclear reactions, reaction mechanism, 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 particle reactions, Parity non-conservation in weak interaction, Relativistic

kinematics.

1. An ant starts at the origin and moves along the y-axisand covers a distance l. This is its first stage in itsjourney. Every subsequent stage requires the ant toturn right and move a distance which is half of itsprevious stage. What would be its coordinates at theend of its 5th stage ?

(A)

3 13,

8 16

l l(B)

13 3,

16 8

l l

(C)

13 3,

8 16

l l(D)

3 13,

16 8

l l

2. In a group of siblings there are seven sisters andeach sister has one brother. How many siblings arethere in total ?

(A) 15 (B) 14

(C) 8 (D) 7

3. What is the average value of y for the range of xshown in the following plot ?

(A) 0 (B) 1 (C) 1.5 (D) 2

4. A bread contains 40% (by volume) edible matter andthe remaining space is filled with air. If the densityof edible matter is 2 g/cc, what will be the bulk den-sity of the bread (in g/cc) ?

(A) 0.4 (B) 0.8 (C) 1.2 (D) 1.6

5. A board has 8 rows and 8 columns. A move is definedas two steps along a column followed by one stepalong a row or vice-versa. What is the minimumnumber of moves needed to go from one corner tothe diagonally opposite corner ?

(A) 5 (B) 6

(C) 7 (D) 9

6. A job interview is taking place with 21 male and 17female candidates. Candidates are called randomly.What is the minium number of candidates to be calledto ensure that at least two males or two females havebeen interviewed?

(A) 17 (B) 2

(C) 3 (D) 21

CSIR-UGC-JRF/NET Examination, June 2017

PHYSICAL SCIENCES

SOLVED PAPERPART - A

7.

The graph shows cumulative frequency % of researchscholars and the number of papers published by them.Which of the following statements is true?

(A) Majority of the scholars published more than 4papers.

(B) 60% of the scholars published at least 2 papers.

(C) 80% of the scholars published at least 6 papers.

(D) 30% of scholars have not published any paper.

8. A tells only lies on Monday, Tuesday and Wednesdayand speaks only the truth for the rest days ofthe week. B tells only lies on Thursday, Friday andSaturday and speaks only the truth for the rest daysof the week. If today both of them state that theyhave lied yesterday, what day is it today?

(A) Monday (B) Thursday

(C) Sunday (D) Tuesday

9. A fair dice was thrown three times and the outcomewas repeatedly six. If the dice is thrown again whatis the probability of getting six?

(A) 1/6 (B) 1/216

(C) 1/1296 (D) 1

10. Which is the odd one out based on a divisibility test?

154, 286, 363, 474, 572, 682

(A) 474 (B) 572

(C) 682 (D) 154

11. My birthday is in January. What would be a suffi-cient number of questions with ‘Yes/No’ answers thatwill enable one to find my birth date?

(A) 6 (B) 3

(C) 5 (D) 2

12. A square is drawn with one of its sides as the hypot-enuse of a right angled triangle as shown in the fig-ure. What is the area of the shaded circle?

2 PRATIYOGITA SAHITYA SERIES

(A)25

1cm2 (B)

25

2cm2

(C)25

3cm2 (D)

25

4 cm2

13. What should be added to the product of the two num-bers 983713 and 983719 to make it a perfect square?

(A) 9 (B) 13

(C) 19 (D) 27

14. InABC, AB = AC and BAC = 90º; EF AB and

DF AC.

The total area of the shaded region is :

(A) AF2/2 (B) AF2

(C) BC2/2 (D) BC2

15. Consider a circle of radius r. Fit the largest possiblesquare inside it and the largest possible circle insidethe square. What is the radius of the innermost circle?

(A)2

r(B)

2

r

(C)2 2

r(D)

2

r

16. In how many ways can you place N coins on a boardwith N rows and N columns such that every row andevery column contains exactly one coin?

(A) N

(B) N (N–1) (N–2) .... 2 × 1

(C) N2

(D) NN

17. Two identical wheels B and C move on the peripheryof circle A. Both start at the same point on A andreturn to it, B moving inside A and C outside it. Whichis the correct statement ?

(A) B wears out times C

(B) C wears out times B

(C) B and C wear out about equally

(D) C wears out two times B

18. Which of the following is the odd one out?

(A) Isosceles traingle (B) Square

(C) Regular hexagon (D) Rectangle

19. Find the missing word : A, AB, ............ ,

ABBABAAB

(A) AABB (B) ABAB

(C) ABBA (D) BAAB

20. A 100 m long train crosses a 200 m long and 20 mwide bridge in 20 seconds. What is the speed of thetrain in km/hr?

(A) 45 (B) 36

(C) 54 (D) 57.6

21. Which of the following cannot be the eigen valuesof a real 3 × 3 matrix.

(A) 2i, 0, –2i (B) 1, 1, 1

(C) ei, e–i, 1 (D) i, 1, 0

22. Let u (x, y) = eax cos (by) be the real part of a functionf(z) = u(x, y) + iv (x, y) of the complex variable

z = x + iy, where a, b are real constants and a 0.

The function f(z) is complex analytic everywhere in

the complex plane if and only if

(A) b = 0 (B) b = a

(C) b = ±2a (D) b = a ± 2

23. The integral

( / 2)

2 – 1

zzedz

zalong the closed contour

shown in the figure is :

(A) 0 (B) 2

(C) –2 (D) 4i

24. The function y(x) satisfies the differential equation

2dy

x ydx

= cos

.x

x If y(1) = 1, the value of y(2) is :

(A) (B) 1

(C) 1/2 (D) 1/4

PART - B

SOLVED PAPER JUNE 2017 3

25. The random variable x (– < x < ) is distributedaccording to the normal distribution P(x) =

2

2

2

–2

1.

2

x

e

The probability density of the random

variable y = x2 is :

(A)

2– / 2

2

1,

2

yey

0 y

(B)

2– / 2

2

1

2 2

yey

, 0 y

(C)

2– / 2

2

1,

2

ye 0 y

(D)

2– /

2

1,

2

yey

0 y

26. The Hamiltonian for a system described by thegeneralised coordinate x and generalised momentump is :

H =

22 2 21

2(1 2 ) 2

pax p x

x where , and are

constants. The corresponding Lagrangian is :

(A) 2 2 2 21 1( – ) (1 2 ) –

2 2x ax x x

(B)

2 2 2 21 1– –

2(1 2 ) 2x x ax x

x

(C) 2 2 2 2 21 1( – ) (1 2 ) –

2 2x a x x x

(D)

2 2 2 21 1–

2(1 2 ) 2x x ax x

x

27. An inertial observer sees two events E1 and E2 hap-pening at the same location but 6 s apart in time.Another observer moving with a constant velocity v(with respect to the first one) sees the same eventsto be 9 s apart. The spatial distance between theevents, as measured by the second observer, is ap-proximately :

(A) 300 m (B) 1000 m

(C) 2000 m (D) 2700 m

28. A ball weighing 100 gm, released from a height of5 m, bounces perfectly elastically off a plate. Thecollision time between the ball and the plate is 0.5 s.The average force on the plate is approximately :

(A) 3 N (B) 2 N

(C) 5 N (D) 4 N

29. A solid vertical rod, of length L and cross-sectionalarea A, is made of a material of Young’s modulus Y.The rod is loaded with a mass M, and, as a result,extends by a small amount L in the equilibriumcondition. The mass is then suddenly reduced toM/2. As a result the rod will undergo longitudinaloscillation with an angular frequency :

(A) 2 /YA ML (B) /YA ML

(C) 2 /YA M L (D) /YA M L

30. If the root-mean-squared momentum of a particle inthe ground state of a one-dimensional simple har-monic potential is p0, then its root-mean-squaredmomentum in the first excited state is :

(A) 0 2p (B) 0 3p

(C) 0 2 / 3p (D) 0 3 / 2p

31. Consider a potential barrier A of height V0 andwidth b, and another potential barrier B of height2V0 and the same width b. The ratio TA/ TB oftunnelling probabilities TA and TB, through barriersA and B respectively, for a particle of energy V0/100,is best approximated by :

(A) exp 2 20[( 1.99 – 0.99) 8 / ]mV b

(B)2 2

0exp [( 1.98 – 0.98) 8 / ]mV b

(C)2 2

0exp [( 2.99 – 0.99) 8 / ]mV b

(D) 2 20exp [( 2.98 – 0.98) 8 / ]mV b

32. A constant perturbation H' is applied to a system fortime t (where Ht << h) leading to a transitionfrom a state with energy Ei to another with energyEf. If the time of application is doubled, the probabil-ity of transition will be :

(A) unchanged

(B) doubled

(C) quadrupled

(D) halved

33. The two vectors 0

a

and bc

are orthonormal if :

(A) a = ±1, b = ±1/ 2, c = ±1/ 2

(B) a = ±1, b = ±1, c = 0

(C) a = ±1, b = 0, c = ±1

(D) a = ±1, b = ±1/2, c = 1/2

34. Two long hollow co-axial conducting cylinders of ra-dii R1 and R2 (R1 < R2) are placed in vacuum as shownin the figure below :

The inner cylinder carries a charge + per unit length

and the outer cylinder carries a charge – per unitlength. The electrostatic energy per unit length of

this sytem is :

(A)

2

0ln (R2/R1) (B)

22 22 1

0

( / )4

R R

(C)

2

04

ln (R2/R1) (D)

2

02ln (R2/R1)

35. A set of N concentric circular loops of wire, each car-rying a steady current I in the same direction, isarranged in a plane. The radius of the first loop is

. .

. .

.

.

.

.


r1 = a and the radius of then nth loop is given byrn = nrn – 1. The magnitude B of the magnetic field atthe centre of the circles in the limit N, is :

(A) 0l(e2 – 1) / 4a (B) 0l(e – 1)/a

(C) 0l(e2 – 1)/8a (D) 0l(e – 1)/2a

36. An electromagnetic wave (of wavelength 0 infree space) travels through an absorbing mediumwith dielectric permittivity given by = R + il where

l

R= 3. If the skin depth is

0 ,

4 the ratio of the ampli-

tude of electric field E to that of the magnetic field B, in themedium (in ohms) is :

(A) 120 (B) 377

(C) 30 2 (D) 30

37. The vector potential A = – ˆatke rr (where a and k are

constants) corresponding to an electromagnetic field

is changed to

'A = – ˆ– atke rr . This will be a gauge trans-formation if the corresponding change ' – in thescalar potential is :

(A) akr2e–at (B) 2akr2e–at

(C) –akr2e–at (D) –2akr2e–at

38. A thermodynamic function

G (T, P, N) = U – TS + PV

is given in terms of the internal energy U, tempera-ture T, entropy S, pressure P, volume V and thenumber of particles N. Which of the followingrelations is true? (In the following is the chemicalpotential)

(A) S = ,

–N P

G

T

(B) S =

,N P

G

T

(C) V = ,

–N T

G

P

(D) =

,

–P T

G

N

39. A box, separated by a movable wall, has two com-

partments filled by a monoatomic gas of .p

V

C

C

Initially the volumes of the two compartments areequal, but the pressures are 3P0 and P0, respectively.When the wall is allowed to move, the final pres-sures in the two compartments become equal. Thefinal pressure is :

(A)

0

2

3P (B)

0

23

3P

(C) 1 /

0

1(1 3 )

2P (D)

1 /

01 /

3

1 3P

40. A gas of photons inside a cavity of volume V is inequilibrium at temperature T. If the temperature ofthe cavity is changed to 2T, the radiation pressurewill change by a factor of :

(A) 2 (B) 16

(C) 8 (D) 4

41. In a thermodynamic system in equailibrium, eachmolecule can exist in three possible states with prob-abilities 1/2, 1/3 and 1/6 respectively. The entropyper molecule is :

(A) kB ln 3

(B)1 2

n 2 + n 32 3

B Bk l k l

(C)2 1

n 2 + n 33 2

B Bk l k l

(D)1 1

n 2 + n 32 6

B Bk l k l

42. In the n-channel JFET shown in figure below,Vi = – 2V, C = 10pF, VDD = + 16V and RD = 2 k.

If the drain D – source S saturation current IDSS is10 mA and the pinch-off voltage VP is – 8V, then thevoltage across points D and S is :

(A) 11.125 V (B) 10.375 V

(C) 5.75 V (D) 4.75 V

43. The gain of the circuit given below is

1– .

RC

The modification in the circuit required to introducea dc feedback is to add a resistor :

(A) between a and b

(B) between positive terminal of the op-amp andground

(C) in series with C

(D) parallel to C

44. A 2 × 4 decoder with an enable input can function asa :

(A) 4 × 1 multiplexer (B) 1 × 4 demultiplexer

(C) 4 × 2 encoder (D) 4 × 2 priority encoder

45. The experimentally measured values of thevariables x and y are 2.00 0.05 and 3.00 0.02,respectively. What is the error in the calculatedvalue of z = 3y – 2x from the measurements?

(A) 0.12 (B) 0.05

(C) 0.03 (D) 0.07


46. The Green’s function satisfying

2

02( , )

dg x x

dx= (x – x0)

with the boundary conditions g(–L, x0) = 0 = g (L, x0),is :

(A)

0 0

0 0

1( – ) ( ), –

2

1( ) ( – ),

2

x L x L L x xL

x L x L x x LL

(B)

0 0

0 0

1( ) ( ), –

2

1( – ) ( – ),

2

x L x L L x xL

x L x L x x LL

(C)

0 0

0 0

1( – ) ( ), –

2

1( ) ( – ),

2

L x x L L x xL

x L L x x x LL

(D)1

( – ) ( ), –2

x L x L L x LL

47. Let x, y, z be the Pauli matrices and

x'x + y'y + z'z

=

exp

2zi

[xx + yy + zz] exp

– .

2zi

Then the coordinates are related as follows :

(A)

'

'

'

x

y

z

=

cos –sin 0

sin cos 0

0 0 1

x

y

z

(B)

'

'

'

x

y

z

=

cos sin 0

–sin cos 0

0 0 1

x

y

z

(C)

'

'

'

x

y

z=

cos sin 02 2

–sin cos 02 2

0 0 1

x

y

z

(D)

'

'

'

x

y

z=

cos –sin 02 2

sin cos 02 2

0 0 1

x

y

z

48. The interval [0, 1] is divided into 2n parts of equal

length to calculate the integral

1 2

0

i xe dx using

Simpson’s 1

3rule. What is the minimum value of n

for the result to be exact?

(A) (B) 2

(C) 3 (D) 4

49. Which of the following sets of 3 × 3 matrices (inwhich a and b are real numbers) form a group undermatrix multiplication?

(A)

1 0

0 1 0 ; ,

0 1

a

a b

b(B)

1 0

0 1 ; ,

0 0 1

a

b a b

(C)

1 0

0 1 ; ,

0 0 1

a

b a b (D)

1 0

1 0 ; ,

0 0 1

a

b a b

50. The Lagrangian of a free relativistic particle (in one

dimension) of mass m is given by L = 2– 1 –m x

where x = dx/dt. If such a particle is acted upon by a con-stant force in the direction of its motion, the phase spacetrajectories obtained from the corresponding Hamiltonian are :

(A) ellipses (B) cycloids

(C) hyperbolas (D) parabolas

51. A Hamiltonian system is described by the canonicalcoordinate q and canonical momentum p. A new co-ordinate Q is defined as Q(t) = q(t + ) + p (t + ),where t is the time and is a constant, that is, the newcoordinate is a combination of the old coordinate and mo-mentum at a shifted time. The new canonical momentum P(t)can be expressed as :

(A) p(t + ) – q (t + ) (B) p(t + ) – q (t – )

(C)1

[ ( – ) – ( )]2

p t q t (D) 1

[ ( ) – ( )]2

p t q t

52. The energy of a one-dimensional system, governedby the Lagrangian

L = 2 21 1

–2 2

nmx kx

where k and n are two positive constants, is E0. Thetime period of oscillation satisfies.

(A)1

–nk (B)

12

1–– 2

0n

n

nk E

(C)1

2

–2– 2

0n

n

nk E (D)1

1– 2

0n

n

nk E

53. An electron is decelerated at a constant rate startingfrom an initial velocity u (where u << c) to u/2 duringwhich it travels a distance s. The amount of energylost to radiation is :

(A)2 2

023

e u

mc s

(B)

2 20

26

e u

mc s

(C)2

0

8

e u

mcs

(D)

20

16

e u

mcs

PART - C

.

.

.


54. The figure below describes the arrangement of slitsand screens in a Young’s double slit experiment. Thewidth of the slit in S1 is a and the slits in S2 are ofnegligible width.

If the wavelength of the light is , the value of d forwhich the screen would be dark is :

(A)

2

– 1a

b

(B)

2

– 12

b a

(C)

2

2

a b

(D)ab

55. A constant current I is flowing in a piece of wire thatis bent into a loop as shown in the figure.

The magnitude of the magnetic field at the point Ois :

(A)0I

n4 5

al

b

(B)

0I 1 1–

4 5 a b

(C)0I 1

4 5 a

(D)

0I 1

4 5 b

56. Consider the potential

( )V r

= 3 (3)

0 ( – )iiV a r r

where ir

are the position vectors of the vertices of a

cube of length a centered at the origin and V0 is a

constant. If V0a2 << 2

,h

m the total scattering cross-

section, in the low-energy limit, is

(A)2

2 02

16mV a

ah

(B)

2220

2 2

16 mV aa

h

(C)

22202

64 mV aa

h

(D)

220

2 2

64 mV aa

h

57. The Coulomb potential V(r) = –e2/r of a hydrogenatom is perturbed by adding H' = bx2 (where b is aconstant) to the Hamiltonian. The first order correc-tion to the ground state energy is (The ground state

wave function is 0 = 0– /

30

1 r aea

)

(A) 202ba (B) 2

0ba

(C) 20 / 2b a (D) 2

02 ba

58. Using the trial function

x = 2 2( – ), –

0 otherwise

A a x a x a

the ground state energy of a one-dimensional har-monic oscillator is :

(A) (B) 5

14 (C)

1

2 (D)

5

7

59. In the usual notation |n l m > for the states of ahydrogen like atom, consider the spontaneous tran-sitions |2 1 0 > |1 0 0 > and | 3 1 0 > | 1 0 0 >.If t1 and t2 are the lifetimes of the first and thesecond decaying states respectively, then the ratio

1

2

t

t is proportional to :

(A)

332

27

(B)

327

32

(C)

32

3

(D)

33

2

60. A random variable n obeys Poisson statistics. Theprobability of finding n = 0 is 10–6. The expectationvalue of n is nearest to :

(A) 14 (B) 106

(C) e (D) 102

61. The single particle energy levels of a noninteractingthree-dimensional isotropic system, labelled by

momentum k, are proportional to k3. The ratio /P

of the average pressure P

to the energy density at a fixed

temperature, is

(A) 1/3 (B) 2/3

(C) 1 (D) 3

62. The Hamiltonian for three Ising spins S0, S1 and S2,

taking values 1 , is

H = – JS0 (S1 + S2).

If the system is in equilibrium at temperature T,the average energy of the system, in terms of = (kBT)–1, is

(A)1 cos (2 )

–2 sin (2 )

j

j

(B) –2J [1 + cos (2J)]

(C) – 2/ (D) – 2Jsin (2 )

1 cos (2 )

J

J

63. Let I0 be the saturation current, the ideality factorand vF and vR the forward and reverse potentials,respectively, for a diode. The ratio RR/RF of its re-verse and forward resistances RR and RF respectively,varies as (In the following kB is the Boltzmann con-stant, T is the absolute temperature and q is thecharge.)

(A) expR F

F B

v qv

v k T

(B) expF F

R B

v qv

v k T

(C) exp –R F

F B

v qv

v k T

(D) exp –F F

R B

v qv

v k T


64. In the figures below, X and Y are one bit inputs. Thecircuit which corresponds to a one bit comparatoris :

(A)

(B)

(C)

(D)

65. Both the data points and a linear fit to the current vsvoltage of a resistor are shown in the graph below.

If the error in the slope is 1.255 × 10–3 –1, then the

value of resistance estimated from the graph is :

(A) (0.04 ± 0.8) (B) (25.0 ± 0.8)

(C) (25 ± 1.25) (D) (25 ± 0.0125)

66. In the following operational amplifier circuit Cin = 10

nF, Rin = 20 k, RF = 200 k and CF = 100 pF.

The magnitude of the gain at a input signal frequencyof 16 kHz is :

(A) 67 (B) 0.15 (C) 0.3 (D) 3.5

67. An atomic spectral line is observed to split into ninecomponents due to Zeeman shift. If the upper stateof the atom is 3D2 then the lower state will be :

(A) 3F2 (B) 3F1 (C) 3P1 (D) 3P2

68. If the coefficient of stimulated emission for a par-ticular transition is 2.1 × 1019 m3 W–1 s–3 and the

emitted photon is at wavelength 3000 Å, then thelifetime of the excited state is approximately.

(A) 20 ns (B) 40 ns(C) 80 ns (D) 100 ns

69. If the binding energies of the electron in the K andL shells of silver atom are 25.4 keV and 3.34 keV,respectively, then the kinetic energy of the Augerelectron will be approximately.(A) 22 keV (B) 9.3 keV(C) 10.5 keV (D) 18.7 keV

70. The energy gap and lattice constant of an indirectband gap semiconductor are 1.875 eV 0.52 nm,respectively. For simplicity take the dielectricconstant of the material to be unity. When it isexcited by broadband radiation, an electron initiallyin the valence band at k = 0 makes a transition tothe conduction band. The wavevector of the electronin the conduction band, in terms of the wavevectorkmax at the edge of the Brillouin zone, after the tran-sition is closest to :

(A) kmax/10 (B) kmax/100

(C) kmax/1000 (D) 0

71. The electrical conductivity of copper is approximately95% of the electrical conductivity of silver, while theelectron density in silver is approximately 70% ofthe electron density in copper. In Drude’s model, theapproximate ratio Cu/Ag of the mean collision timein copper (Cu) to the mean collision time in silver(Ag) is :

(A) 0.44 (B) 1.50 (C) 0.33 (D) 0.66

72. The charge distribution inside a material of conduc-tivity and permittivity at intitial time t = 0 is (r, 0) = 0, a constant. At subsequent times (r, t)is given by :

(A) exp (– )to

(B) 0

1[1 exp ]

2

t

(C)0

[1 – exp ]t

(D) 0 cosht

73. If in a spontaneous -decay of 23292U at rest, the total

energy released in the reaction is Q, then the en-

ergy carried by the -particle is

(A) 57 Q/58 (B) Q/57

(C) Q/58 (D) 23Q/58

74. The range of the nuclear force between two nucle-

ons due to the exchange of pions is 1.40 fm. If themass of the pion is 140 MeV/c2 and the mass of the

rho-meson is 770 MeV/c2, then the range of the forcedue to exchange of rhomesons is :

(A) 1.40 fm (B) 7.70 fm

(C) 0.25 fm (D) 0.18 fm

75. A baryon X decays by strong interaction asX + + – + 0, where + is a member of the isotriplet(+, 0, –). The third component I3 of the isospin of X is :

(A) 0 (B) 1/2

(C) 1 (D) 3/2


1. (A) Initially the coordinates are (0, 0). It moves l unitsupwards towards Y-axis and then move right half

of it. It is the second stage. Now the coordinatesare (l/2, l). Then it moves right and covers half of

distance. Now the coordinates are (l/2, 3l/4). Thenit again moves right and covers half the distance

the coordinates are now (3l/8, 3l/4) and again itmoves right and covers half distance. The final

coordinates are (3l/8, 13l/16) so option A is theright option

2. (C) There are seven sisters and each sister has 1brother. That means total no. of siblings are 8 so

option C is the right option.

3. (C) Here the range is lying between 1 and 2 so aver-

age of 1 and 2 is 1.5 so option C is the right option.

4. (B) Let say mass of the bread is 100 grams. So weightof edible matter is 40 grams. Now density =

mass/volume, so volume = mass/density = 100/2= 50 c.c. Now bulk density of bread = weight of

edible matter/volume = 40/50 = 0.8. So right op-tion is option B.

5. (A) If we draw the figure and see that there are 5moves in which it goes from one corner to other

corner diagonally. So right answer is option A.

6. (D) It is option D because at least 2 males or 2 fe-

males have called for interview therefore it couldbe 10 males also or 15 females also or even could

21 males and the only option that goes in accor-dance with this is option D.

7. (B) It has been shown in the graph that from

40%–100% of research scholars has publishedmore than 2 papers, so 60% of the scholars have

published at least 2 papers.

8. (B) Clearly it is Thursday because on Thursday A

speaks truth that he lies 1 day before that is onWednesday and it is given in the question. Also,

B lies on Thursday that it lies on 1 day before onWednesday because B speaks truth on Wednes-

day. So, the right option is B.

9. (A) In this question, we just need to find out the prob-ability of 6 in a dice and that is 1/6. Therefore,right option is A

10. (A) In this all the numbers except 474 are divisibleby 11 so 474 is the odd one out.

11. (C) "Consider all answers to be yes;

First question : Is your birthday in the first 15 days?Second question : Is your birthday in the first 7days?Third question : Is your birthday in the first 4 days?Fourth question : Is your birthday in the first 2 days?

Fifth question : Is your birthday on the first day?

In case any answer is no, the alternate half of thequestion will be considered and the next ques-tion will be framed accordingly.First question : Is your birthday in the first 15 days?– NoSecond question : Is your birthday in the first 8days? (16th to 24th)– NoThird question : Is your birthday in the first 4days? (25th to 28th)– NoFourth question : Is your birthday in the first 2days? (29th to 30th)– NoFifth question : Is your birthday on the 31st?– YesIt could be 4 questions as well, but none of theoptions match.

12. (D) Here diameter of the circle is calculated usingPythagoras theorem in the triangle containingthe sides 3 cm and 4 cm. From Pythagoras theo-rem the diameter of circle is coming as 5 cm.Therefore the radius of the circle is 5/2 cm. Nowapplying the area of circle as pi r square where ris 5/2 cm so the area comes out to be 25pi/4 cmsquare.

13. (A) Here if we multiply the numbers we get the unitsdigit as 7 and 7 + 9 (from the given option) = 16a perfect square therefore 9 is the correct answer.

ANSWERS

1. (A) 2. (C) 3. (C) 4. (B) 5. (A) 6. (D) 7. (B) 8. (B) 9. (A) 10. (A)

11. (C) 12. (D) 13. (A) 14. (C) 15. (A) 16. (B) 17. (C) 18. (A) 19. (C) 20. (C)

21. (D) 22. (B) 23. (C) 24. (D) 25. (A) 26. (A) 27. (C) 28. (D) 29. (A) 30. (B)

31. (A) 32. (B) 33. (C) 34. (C) 35. (D) 36. (D) 37. (C) 38. (A) 39. (C) 40. (B)

41. (C) 42. (D) 43. (C) 44. (C) 45. (B) 46. (0) 47. (0) 48. (0) 49. (0) 50. (0)

51. (0) 52. (B) 53. (D) 54. (B) 55. (B) 56. (C) 57. (B) 58. (B) 59. (B) 60. (A)

61. (A) 62. (D) 63. (B) 64. (C) 65. (A) 66. (D) 67. (C) 68. (C) 69. (D) 70. (*)

71. (D) 72. (A) 73. (A) 74. (C) 75. (B)

EXPLANATIONS

PART - A


14. (C) Here BDF is 90 degrees because DF is parallel

to AC and FEC = 90 because EF is parallel to

AB. Therefore BD = DF because B = DFB =

45 degrees and EF = EC because EFC = C =

45 degrees. Now area BDF = ½ *BD*DF = 1/2

BD square and area CEF = 1/2 FE *EC = ½ FE

square. Now using Pythagoras theorem, BD square

= BF square/2 and EF square = CF square/2. By

adding BD square + EF square = BF square /2 +

CF square/2 and from there we get the answer as

BC square /2, so option C is the right option.

15. (A) Here the diagonal of the square will be 2r. From

Pythagoras theorem, we get 4r square = 2 a square.

From this 2 multiplied by r. Now radius of the

inner circle is 2 *r divided by 2 that will be

r/ 2 . Now again applying Pythagoras theorem,

we get radius of inner circle = square root

(r square – r square/2) that is coming out to be

r/ 2 . That is the answer.

16. (B) In this question it is like arranging N alphabetsof a word so the answer is N factorial. Option B isthe correct option.

17. (C) It is clear from the figure both B and C wear out

equally.

18. (A) In this we can say isosceles triangle does not con-

tain any diagonals while others contain diagonals

so A is the right option.

19. (C) In this, if we look at first and second A and AB in

first only the first half part of AB is given so in

3rd and 4th if we compare, only the first half part

of 4th will be in the 3rd so right option is ABBA.

20. (C) Total length is length of the train plus length of

the bridge. So, total length is 100 + 200 = 300 m

and time taken is 20 sec. Therefore, speed is

300/20 = 15 m/sec = 54 km/hour.

21. (D) The real and complex eigen values are not pos-sible for a real matrix. As the given value of 3 × 3real matrix must have the form

The eigen values of the matrix are degneratevalues.

Hence, the eigen value (i, 1, 0) is not possible.

22. (B) Let f(z) = eaz

= ea(x ± iy)

= eax ei(± a)y

= eax (cos by + i sin by)

f(z) is analytic everywhere, since,

f '(z) = aeaz

This is only possible if b = ± a.

23. (C) Given, I =

/ 2

2 – 1

i zzedz

z

= ( )f z dz

The pole of the f(z) is located where the denomi-nator of f(z) diminishes.

z2 – 1 = 0

z = 1

The value of integraties I = 2i Residue (z = 1)Residue (z – 1) residue at (z = 1) is ...(i)

Residue (z = +1) =– / 2

21lt ( – 1)

( – 1)

i z

x

zez

z

= – / 2

1lt

1

i z

x

ze

z

= – / 2

2

ie...(ii)

The Residue at (z = – 1)

Residue (z = – 1) =

– / 2

–1

( 1)( )lt

( 1) ( – 1)

i z

z

z ze

z z

=

– / 2

z –1lt

( – 1)

i zze

z

= – / 2

2

ie...(iii)

Putting the values of equation (ii) and (iii) in equa-tion (i)

I = /2 – /2

22 2

i ie ei

= 2i cos /2 = 0

= /2 /2

22 2

i ie ei

= – 2

24. (D) From equation

2dy

x ydx

= cos x

xwe get,

2dy y

dx x=

cos x

x

( )dy

P x ydx

= Qx ...(i)

Now on integrating factor of equation (i)

we have

I.F. = 2x

dxe = elnx = x2

(I.F.) y = (I.F.) . Q( )x dx

x2y =

2

2

cos xx dx

x

=

sin xc

x2y =

sin xc ...(ii)

PART-B

CSIR-UGC NET/JRF Physical Sciences Solved Papers

Publisher : Sahitya Bhawan ISBN : 9789387362130 Author : Kamlesh Sanha

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