jrfCS_sampleq_final2008

8/11/2019 jrfCS_sampleq_final2008

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Test Code : RC (Short Answer Type) 2008

JRF in Computer and Communication Sciences

The Candidates for Junior Research Fellowship in Computer Science andCommunication Sciences will have to take two tests - Test MIII (objectivetype) in the forenoon session and Test RC (short answer type) in the after-noon session. The RC test booklet will have two groups as follows:

GROUP AA test for all candidates in logical reasoning and basics of programming,carrying 20 marks.

GROUP BA test, divided into five sections, carrying equal marks of 80 in the followingareas at M.Sc./M.E./M.Tech. level:

(i) Mathematics, (ii) Statistics, (iii) Physics, (iv)

Radiophysics/ Telecommunication/ Electronics/ Electrical Engg., and

(v) Computer Science.

A candidate has to answer questions from only one of these sec-tions, according to his/her choice.

The syllabi and sample questions of the RC test are given overleaf.

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Syllabus

Elements of Computing:Logical reasoning, basics of programming (using pseudo-codes), Elementarydata types and arrays.

Mathematics:Graph theory and combinatorics: Graphs and digraphs, paths and cycles,trees, Eulerian graphs, Hamiltonian graphs, chromatic numbers, planargraphs, tournaments, inclusion-exclusion principle, pigeon-hole principle.

Linear programming: Linear programming, simplex method, duality.

Linear algebra: Vector spaces, basis and dimension, linear transformations,

matrices, rank, inverse, determinant, systems of linear equations, character-istic roots (eigen values) and characteristic vectors (eigen vectors), orthog-onality and quadratic forms.

Abstract algebra: Groups, subgroups, cosets, Lagranges theorem, normalsubgroups and quotient groups, permutation groups, rings, subrings, ideals,integral domains, fields, characteristic of a field, polynomial rings, uniquefactorization domains, field extensions, finite fields.

Elementary number theory: Elementary number theory, divisibility, congru-ences, primality.

Calculus and real analysis: Real numbers, basic properties, convergenceof sequences and series, limits, continuity, uniform continuity of functions,differentiability of functions of one or more variables and applications, in-definite integral, fundamental theorem of calculus, Riemann integration, im-proper integrals, double and multiple integrals and applications, sequencesand series of functions, uniform convergence.

Differential equations: Solutions of ordinary and partial differential equa-tions and applications.

Statistics:Probability Theory and Distributions: Basic probability theory, discrete and

continuous distributions, moments, characteristic functions, Markov chains.

Estimation and Inference: Sufficient statistics, unbiased estimation, max-imum likelihood estimation, consistency of estimates, most powerful anduniformly most powerful tests, unbiased tests and uniformly most powerfulunbiased tests, confidence sets.

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Linear Models: Gauss-Markov set up and least squares theory, multiple

linear regression, one and two way analysis of variance.

Multivariate Analysis: Multivariate normal distribution, Wishart distribu-tion, HotellingsT2 test, principal component analysis, multiple and canon-ical correlations, discriminant analysis, cluster analysis, factor analysis.

Physics:Classical Mechanics: Variational principle and Lagranges equation, centralforce problem, rigid body motion, Hamilton equation of motion, canonicaltransformations, Hamilton Jacobi theory and action angle variables, La-grangian and Hamiltonian formulation for continuous systems and fields,relativistic mechanics.

Electrodynamics: Electromagnetic fields and potentials, electromagnetic ra-diation, scattering, dispersion, relativistic electrodynamics.

Thermodynamics and Statistical Mechanics: Reviews of thermodynamics,statistical basis of thermodynamics, density matrix formulation, ensem-bles, partition function and its uses, Maxwell-Boltzmann, Bose-Einstein andFermi Dirac statistics, simple gases, Ising model.

Non-Relativistic Quantum Mechanics: Basics of quantum mechanics, thetwo body problem and central potential, quantum particles in electromag-netic fields, matrix mechanics and spin, approximate methods: stationary

states, approximative methods: time dependent problems.

Solid State Physics: Crystal structures, interacting forces, lattice vibrations,electronic band structures, density of states, elementary excitations, trans-port properties.

Electronics: Basics of semiconductor physics, amplifiers, communicationprinciples.

Vibrations and Waves: Forced vibrations, coupled vibrations, stretchedstrings, small oscillations.

Radiophysics/Telecommunication/Electronics/Electrical Engg.:Boolean algebra, digital circuits and systems, circuit theory, amplifiers, oscil-lators, digital communication, digital signal processing, linear control theory,electrical machines.

Computer Science:Discrete Mathematics: Elementary counting techniques, Principles of inclusion-

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exclusion, recurrence relations, generating functions, propositional logic.

Data Structures: Stack, queue, linked list, binary tree, heap, AVL tree, B-tree.

Design and Analysis of Algorithms: Order notation, sorting, selection, search-ing, hashing, string handling algorithms, graph algorithms, NP-completeness.

Programming Languages: Fundamental concepts - abstract data types, pro-cedure call and parameter passing, C language.

Computer Organization and Architecture: Number representation, computerarithmetic, memory organization, I/O organization, microprogramming,pipelining, instruction level parallelism.

Operating Systems: Memory management, processor management, criticalsection, deadlocks, device management, concurrency control.

Formal Languages and Automata Theory: Finite automata and regular ex-pression, context-free grammars, Turing machines, elements of undecidabil-ity.

Principles of Compiler Construction: Lexical analyzer, symbol table, parser,code optimization.

Database Systems: Relational model, relational algebra, relational calculus,functional dependency, normalization (upto 3rd normal form).

Computer Networks: Layered network structures, network security, LANtechnology - Bus/tree, Ring, Star; data communications - data encoding,flow control, error detection/correction.

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Sample Questions

Note that all questions in the sample set are not of equal difficulty.They may not carry equal marks in the test.

GROUP A

ELEMENTS OF COMPUTING

A1. The strength of a mug (made of glass) is defined as follows.

There is a building having an infinite number of floors. A mug is saidto possess strengthh units if it does not break when it is dropped fromthe h-th floor, but it breaks when it is dropped from the -th floor,whereh + 1. The strength of a mug is known to be finite.If you are given only one mug, you can determine its strength bydropping it successively from 1st, 2nd, . . ., floors until it breaks. Thus,if the strength of the mug is h, then the number of times you needto perform the experiment is h+ 1, where an experiment consists ofdropping the mug from a floor and observing whether it breaks afterreaching the ground. Note that we may use the same mug for many

experimentsuntil it breaks.

Now consider that instead of one, you are given two mugs of the samestrength. Design a scheme to determine the strength of these mugswith minimum number of experiments. Also report the exact numberof experiments you have performed in your scheme.

A2. How many isomers are there for the organic compound C6H14? Inother words, how many distinct non-isomorphic unlabelled trees arethere with 6 vertices of degree 4, and 14 vertices of degree 1?

A3. In the following table, find the entry in the square marked with *.

Justify your answer.

BD1 CE5 DF21EG2 F H8 GI34HJ3 IK13 *

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A4. Consider the pseudo-code given below.

Input: Integers b andc.

1. a0max(b, c), a1min(b, c).2. i1.3. Divideai1 byai . Letqi be the quotient andri the remainder.

4. Ifri= 0 then go to Step 8.

5. ai+1ai1 qi ai.6. ii + 1.7. Go to Step 3.

8. Printai .

What is the output of the above algorithm? What is the mathematicalrelation between the outputai and the two inputs b andc?

A5. Write the output of the following pseudo-code:

1. for (n= 15, downto 2, step -2)

2. if (n > 10)

3. thennn + 1 and print n;4. else nn 1 and print n;5. endfor

A6. Given an array of n integers, write a pseudo-code for reversing thecontents of the array without using another array. For example, forthe array10 15 3 30 3the output should be3 30 3 15 10.You may use one temporary variable.

A7. Consider the sequencean= an1 an2+ n for n2, with a0 = 1 anda1 = 1. Is a2006 odd? Justify your answer.

A8. Derive an expression for the maximum number of regions that can beformed within a circle by drawing n chords.

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A9. A functionPrintRec is defined as follows:

PrintRec(A, B, C, n)begin

ifn >0begin

print B;PrintRec(B, A, C, n 1);print A;PrintRec(C, A, B, n 2);

endend

Find the output for the function call PrintRec(x, y, z, 3). Show theintermediate steps of your derivation.

GROUP B

(i) MATHEMATICS

M1. (a) Letf : R R be a differentiable function for which there doesnot exist any x [0, 1] such that both f(x) = 0 and f(x) = 0.Show that fhas only a finite number of zeros in [0,1].[f(x) denotes the derivative off at x].

(b) Let f : R

R

be such that f(x) exists and is continuous in[0,1). Show that

limx0

1

x2

x0

(x 3y)f(y)dy=f(0)2

.

[x0 denotes: x decreases to zero.]

M2. (a) Letf : [0, 1][0, 1] be such that

f(x) =nx [nx]; 1n

< x 1n 1 ; n= 2, 3, 4, . . . ; x= 0

andf(0) = 0. Show that 10 f(x)dx exists and find its value.

Note: [y] = Largest integery; y R.(b) Let f : [0, 1](0, ) be continuous. Let

an=

10

(f(x))ndx

1n

; n= 1, 2, 3, . . . .

Find limn an.

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M3. (a) Suppose f is a continuous real valued function on [0, 1]. Show

that 10

f(x)xdx= 1

2f()

for some [0, 1].(b) For every x0, prove that

x + 1 x= 1

2

x + (x),

for a unique (x), 0< (x)< 1. Prove that, in fact

(i) 14 (x) 12 ,

(ii) limx0 (x) = 1

4 and limx (x) = 1

2 .

M4. (a) Show that f(x) = e|x| x5 x 2 has at least two real roots,wheree is the base of natural logarithms.

(b) Let

an be a convergent series such thatan0 for alln. Showthat

an/n

p converges for every p > 12 .

M5. (a) Let G be the set of all non-singular 2 2 matrices

a bc d

where the elements a,b, c, d belong to the field of order 3. Usingmatrix multiplication as the operation in G what is the order of

groupG? Is G abelian? Justify your answers.(b) Prove thatAut(Q, +)Z2, where Z2 is the group consisting of

only two elements; andAut(Q, +) is the automorphism group of(Q, +).

M6. LetR= (S, +, , 0, 1) be a commutative ring andnbe a positive integersuch thatn = 2k for some positive integer k .

(a) Show that for every aSn1

i=0

ai =k1

i=0

1 + a2i .(b) Let m = w

n2 + 1 where wS, w= 0. Show that for 1p < n,

n1i=0

wip 0 modm.

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M7. (a) Show that there is a basis consisting of only symmetric and skew-

symmetric matrices for the vector space of all n nreal matricesover R.

(b) Does there exist a linear transformation from R3 to R3 such that(0,1,1), (1,1,0), (1,2,1) are mapped to (1,0,0), (0,1,0), (0,1,1) re-spectively? Justify your answer.

M8. (a) LetA be a square matrix such that A2 =A. Show that all eigenvalues ofA are 0 or 1.

(b) LetA be a symmetric matrix whose eigenvalues are 0 or 1. Showthat A2 =A.

(c) Suppose A is an nn matrix such that A2

= A. Show thatr+ s= n, where rank(A) =r and rank(I A) =s.

M9. Letkbe a positive integer. LetG= (V, E) be the graph whereV is theset of all strings of 0s and 1s of length k , andE={(x, y) :x, yV,x and y differ in exactly one place}.

(a) Determine the number of edges in G.

(b) Prove thatG has no odd cycle.

(c) Prove thatG has a perfect matching.

(d) Determine the maximum size of an independent set in G.

M10. LetT be a tree with n vertices. For vertices u, v ofT, define d(u, v)to be the number of edges in the path from u to v . Let W(T) be thesum ofd(u, v) over all

n2

pairs of vertices{u, v}.

(a) Suppose the treeTis the path on n vertices. Show that

W(T) =1

2

n1k=1

(k2 + k).

(b) Now, suppose T is the star graph on n vertices. Show thatW(T) = (n

1)2. (N.B. The edge set of the star graph is equal

to{(u1, ui) : 2in}.)(c) Hence, for any treeT, show that

(n 1)2 W(T)12

n1k=1

(k2 + k).

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M11. (a) Show that, given 2n + 1 points with integer coordinates in Rn,

there exists a pair of points among them such that all the co-ordinates of the midpoint of the line segment joining them areintegers.

(b) Find the number of functions from the set{1, 2, 3, 4, 5} onto theset{1, 2, 3}.

M12. Consider the following LP:

P: minimize x1+ x3subject to

x1+ 2x25,x2+ 2x3 = 6,x1, x2, x30.

(a) Write down the dual D ofP and find the optimal solution ofDgraphically.

(b) Using the optimal solution ofD , find the optimal solution ofP.

M13. (a) A setScontains integers 1 and 2. S also contains all integers ofthe form 3x+y where x and y are distinct elements ofS , andevery element ofSother than 1 and 2 can be obtained as above.What is S? Justify your answer.

(b) Let (n) denote the number of positive integers m relatively

prime ton; m < n. Letn = pqwherep and qare prime numbers.Then show that

(n) = (p 1)(q 1) =pq(1 1q

)(1 1p

)

M14. Consider the nn matrix A = ((aij)) with aij = 1 for i < j andaij = 0 for ij . Let

V ={f(A) :fis a polynomial with real coefficients}.Note that V is a vector space with usual operations. Find the dimen-sion ofV , when

(a) n= 3,(b) n= 4.

Justify your answer.

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(iv) STATISTICS

S1. (a) Let {Xn}n1be a sequence of random variables satisfying Xn+1 =Xn+ Zn (addition is modulo 5), where{Zn}n1 is a sequence ofindependent and identically distributed random variables withcommon distributionP(Zn= 0) = 1/2, P(Zn=1) =P(Zn= +1) = 1/4.Assume that X1 is a constant belonging to{0, 1, 2, 3, 4}. Whathappens to the distribution ofXn as n ?

(b) Let{Yn}n1 be a sequence of independent and identically dis-tributed random variables with a common uniform distribution

on{1, 2, . . . , m}. Define a sequence of random variables{Xn}n1as Xn+1 = M AX{Xn, Yn} where X1 is a constant belonging to{1, 2, . . . , m}. Show that{Xn}n1 is a Markov chain and classifyits states.

S2. Let there ber red balls andb black balls in a box. One ball is removedat random from the box. In the next stage (a + 1) balls of the coloursame as that of the removed ball were put into the box (a1). Thisprocess was repeatedn times. Let Xn denote the total number of redballs at the n-th instant.

(a) Compute E(Xn).

(b) Show that if (r+ b) is much larger than a and n,

1

rE(Xn) =

1 +

na

r+ b

+ O

1

r+ b

.

S3. Let x1, x2, . . . , xn be a random sample of size n from the gamma dis-tribution with density function

f(x, ) = k

(k)exxk1, 0< x 0 is unknown andk >0 is known. Find a minimum varianceunbiased estimator for 1 .

S4. Let 0 < p < 1 and b > 0. Toss a coin once where the probabilityof occurrence of head is p. If head appears, thenn independent and

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identically distributed observations are generated from Uniform (0, b)

distribution. If the outcome is tail, thenn independent and identicallydistributed observations are generated from Uniform (2b, 3b) distribu-tion. Suppose you are given these n observationsX1, . . . , X n, but notthe outcome of the toss. Find the maximum likelihood estimator ofbbased onX1, . . . , X n. What happens to the estimator as n goes to?

S5. Let X1, X2, . . . be independent and identically distributed randomvariables with common density functionf. Define the random variableN as

N=n, ifX1

X2

Xn

1< Xn; forn = 2, 3, 4, . . . .

FindP rob(N=n). Find the mean and variance ofN.

S6. (a) LetX and Ybe two random variables such thatlog X

log Y

N(, ).

Find a formula for (t, r) = E(XtYr), where t and r are realnumbers, andEdenotes the expectation.

(b) Consider the linear modelyn1= Anpp1 + n1 and the usual

Gauss-Markov set up where E() = 0 and D() =

2

Inn, Edenotes the ExpectationandD denotes the dispersion.Assume that A has full rank. Show that V ar

LS1

= (

TB1)12 where

ATA=

11 T1p11p1 Bp1p1

andLS1 = the least square estimate of1, the first component ofthe vector, V ar denotes the variance and T denotes transpose.

S7. Letp1(x) andp2(x) denote the probability density functions for classes

1 and 2 respectively. LetP and (1 P) be the prior probabilities ofthe classes 1 and 2, respectively. Consider

p1(x) =

x, x[0, 1);2 x, x[1, 2];0, otherwise;

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and

p2(x) =

x 1, x[1, 2);3 x, x[2, 3];0, otherwise.

(a) Find the optimal Bayes risk for this classification problem.

(b) For which values ofP, is the above risk

(I) minimized?

(II) maximized ?

S8. Let X= (X1, . . . , X n) and Y = (Y1, . . . , Y n) be two independent and

identically distributed multivariate random vectors with mean 0 andcovariance matrix 2In, where

2 > 0 and In is the nn identitymatrix.

(a) Show thatXTY/(X Y) andV =(X2i + Y2i ) are indepen-dent.(Here,(a1, . . . , an)=

a21+ + a2n ).

(b) Obtain the probability density ofn

i=1 X2i/n

i=1 Y2i

.

S9. LetX1, X2, . . . , X nbe independent random variables. LetE(Xj) =jand V(Xj) = j

32, j = 1, 2, . . . , n, < 0. HereE(X) denotes the expectation and V(X) denotes the variance of therandom variable X. It is assumed that and 2 are unknown.

(a) Find the best linear unbiased estimator for .

(b) Find the uniformly minimum variance unbiased estimate for under the assumption that Xis are normally distributed; 1in.

S10. A hardware store wishes to order Christmas tree lights for sale duringChristmas season. On the basis of past experience, they feel that thedemandv for lights can be approximately described by the probabilitydensity function f(v). On each light ordered and sold they make a

profit ofa cents, and on each light ordered but not sold they sustaina loss ofb cents. Show that the number of lights they should order tomaximize the expected profit is given by x, which is the solution ofthe equation: x

0f(v)dv=

a

a + b.

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S11. Let (X, Y) follow the bivariate normal distribution. Let mean of X

= mean ofY = 0. Let variance of X = variance of Y = 1, and thecorrelation coefficient between X and Y be . Find the correlationcoefficient betweenX3 and Y3.

S12. Let X have probability density function f(x)( < x


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(a) What is the rank of the error-space? Justify your answer.

(b) Write down any linear function of observations that belongs to(i) estimation space, (ii) error space.

(c) Write down a parametric function that is not estimable. Justifyyour answer.

S17. LetA={H H H , H H T , H T H , H T T , T H H , T H T , T T H , T T T } be thespace obtained by tossing a coin three times. Let f :A(0, ) andx1A. For any xiA, xi+1 is found in the following way.Toss a fair coin three times and let the outcome be z .

Iff(z)f(xi) thenxi+1= z, otherwisexi+1= xi.What can you say about limx0 f(xi)? Justify your answer.

S18. Let there be two classesC1 andC2. Let the density function for classCi be pi for i = 1, 2 where pi(x) =ie

ix; x >0, i = 1, 2. Let the priorprobability forC1 be 0.4 and the prior probability for C2 be 0.6. Findthe decision rule for classification of an observation, which providesthe minimum probability of misclassification and find its value forthat decision rule.

(iii) PHYSICS

P1. (a) Two particles A and B of equal mass m are attached with twoidentical massless springs of stiffness constant k in a mannershown in the figure below. When set in longitudinal vibration,find the frequencies and the ratio of the amplitudes, in the normalmodes, of the two particles.

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A

B

(b) A uniform string fixed at both ends is struck at its centre so asto obtain an initial velocity which varies from zero at the ends

to v0 at the centre. Find the equation of motion of the resultingvibration.

(c) Show that the zero temperature spin susceptibility of a non-interacting electron gas is

= Bg(EF)

whereB = Bohr magneton, g(EF) = density of states per unitenergy at the Fermi energy EF.

P2. (a) In the electron spin-orbit interaction the two possible values ofj

arel +12 andl

12 . Show that the expectation values ofSZin the

states j =l+ 12 and j = l 12 are +mj2l+1 and

mj2l+1 respectively.

[All the symbols have their usual meanings.]

(b) A particle in the infinite square well has the initial wave function

(x, 0) =A sin3x

a , (0xa)

(i) DetermineA.(ii) Find (x, t).(iii) Calculate< x > as a function of time.

(c) Consider the operator function

(a, a+) = (1 e)ea+a,

where a, a+ are the annihilation and creation operators of thefield respectively. Prove that is a density operator.

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P3. A horse-shoe magnet is formed out of a bar of wrought iron of 50 cm

length having cross section 6.28 cm2. An exciting coil of 500 turns isplaced on each limb and connected in series. Find the exciting currentnecessary for the magnet to lift a load of 19.6 kg (see the figure givenbelow) assuming that the load has negligible reluctance and makesclose contact with the magnet. Relative permeability of iron is 700.

P4. (a) Consider a conducting electron gas at the absolute zero temper-ature in a weak magnetic field B . The concentrations of spin upand spin down electrons may be parameterised respectively as

N+= (1/2)N(1 + x), N= (1/2)N(1 x)

whereNis the total number of electrons.Evaluate the factorx and calculate the total energy of gas.

(b) Suppose that there are N spinless particles satisfying Bose-Einsteinstatistics. The density of available states between EandE+ dEis g(E), where

g(E) = 0, E 0,

andN0 is the number of particles at energy E0.

(i) Find the chemical potential as a function of N, E0, N0and .

(ii) Can this system have Bose Einstein condensation? Justifyyour answer.

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P5. (a) Three particles A, B and C of equal mass m are placed on a

smooth horizontal plane. A is joined to B and C by light threadsAB and AC respectively and BAC = 600. An impulse I isapplied to A in the direction BA. Find the initial velocities (im-mediately after the application of I) of the particles and show

that A begins to move in a direction making an angle tan137

with BA.

(b) A particle on a frictionless horizontal plane at a latitude isgiven an initial speed u in the northern direction. Prove that itdescribes a circle of radius u2sin with a period T =

sin where

is the angular velocity of the earth.

P6. Two electrons are confined in a one dimensional box of lengtha. Aclever experimentalist has made arrangements such that both the elec-trons are in the same spin state. Ignore the Coulomb interaction be-tween the electrons.

(a) Write down the ground state wave function of the two-electronsystem.

(b) What is the probability that both the electrons are found on thesame half of the box?

(c) Will the nature of construction of the wave function in (a) hold ifCoulomb interaction is included? Give reasons for your answer.

(d) In the above problem, consider two charged -mesons insteadof two electrons. Write down the ground state wave functionignoring the Coulomb interactions.

P7. (a) Consider an eigenstate of a two-particle system, in one dimension,represented by the wave function

(x1, x2) =eiP(x1+x2)/(2)e(Mk/2)

12(x1x2)2/(2).

Herex1and x2are the positions of the two particles of equal mass(M) moving in one dimension and interacting with a harmonic

oscillator force F =k(x1 x2).(i) Calculate the total energy associated with the relative mo-

tion.

(ii) Also calculate the mean absolute value of the relative mo-mentum.

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(b) A particle of massm1MeV/c2 and kinetic energyk1MeV collides

with a stationary particle of massm2MeV/c2. After the collision,the two particles stick together. Calculate

(i) the initial momentum of the two-particle system, and

(ii) the final velocity of the two-particle system.

P8. (a) A monatomic classical gas (made up of N atoms, each havingmassm) is contained in a cylinder with cross sectional area Aandheight h. The system is subject to a linear potentialU(z) =bz ,wherez is the vertical coordinate. Assume that the temperatureTis uniform in the cylinder. Find the free energy of this system.

(b) Consider a sequence of 2N ions of alternating charges qarrangedon a line with a repulsive potential

A

Rn between nearest neigh-

bours in addition to the usual Coulomb potential. Find the equi-librium separation R0 and the equilibrium energy. Also evalu-ate the nearest neighbour distance when the potential energy iszero.(Neglect the surface effect).

P9. (a) Consider an electromagnetic wave in free space of the form,

E(x,y ,z ,t) = (E0x(x, y)i + E0y(x, y)j)e

i(kzt),

B(x,y ,z ,t) = (B0

x

(x, y)i + B0

y

(x, y)j)ei(kzt).

Here E0 and B0 are in the xy plane.Show that E0 and B0 satisfy the time independent Maxwellsequations.

(b) Two point charges of magnitudee are located at the end pointsof a line of length 2l in the xy plane with its midpoint passingthrough the origin. The line is rotating about thez-axis in theanti-clockwise direction with a constant angular velocity .

Calculate the following:

(i) electric dipole moment of the system.

(ii) magnetic dipole moment of the system.(c) A parallel plate capacitor (having perfectly conducting plates)

with plate separationd is filled with layers of two different mate-rials. The first layer has dielectric constant 1 and conductivity1; the second layer has dielectric constant 2 and conductivity2. Their thicknesses are d1 and d2, respectively. A potential

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difference of V is applied across the capacitor. Neglecting the

edge effect,

(i) calculate the electric field in each of the two dielectric media.

(ii) what is the current flowing through the capacitor?

(iii) what is the total surface charge density on the interface be-tween the two layers?

P10. (a) Suppose a planet is moving in a circular orbit of radiusR. It isstopped suddenly in its orbit. Show that it would fall onto the

sun in a time which is

2

8 times of its period.

(b) A block of mass M is rigidly connected to a massless circulartrack of radius a fixed in a vertical plane on a horizontal tableas shown in the figure. A particle of mass m is confined to movewithout friction on the circular track (in the vertical plane).

a

(i) Set up the Lagrangian using x and as the coordinates.

(ii) Find the equations of motion.

(iii) Solve the equation of motion for ( for small ).

P11. For an intrinsic semiconductor with a gap width of 1 eV, calculate theposition of Fermi level atT = 00Kand T= 3000K, ifmh= me, whereme andmh are effective masses of an electron and a hole respectively.Also calculate the density of free electrons and holes at T = 3000K

and T= 6000K, given that log10e= 0.40,

2

memhkT

h2

32

= 0.5

10

19

/cc. If the above semiconductor is now doped with a group Velement with a doping concentration of 1014/cc, then compute theelectron and hole densities of the doped semiconductor specimen.

P12. (a) A negative feedback amplifier has a voltage gain of 100. Varia-tions of the voltage gain up to2% can be tolerated for somespecific application. If the open-loop gain variations of10% are

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expected owing to production spreads in device characteristics,

determine the minimum value of the feedback ratio and alsothe open loop gain to satisfy the above specification.

(b) Calculate the output voltageV0 for the following network:

5V 10K 20K

10K20K

5V

20K5V

+

-

V

V

V 2

1

o

+

-

-

-

-

+

+

+

P13. (a) Consider the following circuit for deriving a +5 volt power supplyto a resistive load RL from an input d-c voltage source whosevoltage may vary from 8V to 11V. The load RL may draw a

maximum power of 250 mW. The Zener diode has a breakdownvoltage of 5 volts.

DC

voltage

source

-

+R

Zener

diodeR

L

5 V

+

-

8 - 11 V

Compute the maximum value of the resistance R and also thepower dissipation requirements for R and the Zener diode. As-sume that the minimum breakdown current of the Zener diode isnegligible compared to the load current.

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(b) Consider the following circuit. Calculate the potential difference

between the points F and C, as shown by the ideal voltmeter.

10

2

2

4

2

2

4

5 V

+ -

A

EF

D

CB

1

(iv) RADIOPHYSICS/TELECOMMUNICATIONS/ELECTRONICS/ELECTRICAL ENGINEERING

R1. Design a sequential machine that produces an output 1 whenever asubstring of 5 consecutive symbols in the input starts with two 1sand contains exactly three 1s. If a substring of 5 symbols starts withtwo 1s, the analysis of the next substring does not begin until theprocessing of the current substring is complete. Realize this circuit

with the minimum number of NAND gates and flip flops.

R2. Consider the following circuit, where all the resistances are in ohms.Calculate the potential difference betweenAandB . Also compute thecurrent i drawn from the battery.

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12V

8

10

8

88

4 4

i

6 6

4

-

A B

4 4

R3. Draw the state table for the synchronous sequential circuit shown inthe figure below.

R4. Consider the following circuit of an ideal OP-AMP and an RC two-port network. Assume that the RC two-port network is representedin terms of its y parameters, i.e., y11 =

I1V1

|V2=0, y12 = I1V2 |V1=0,y21 =

I2V1

|V2=0, and y22 = I2V2 |V1=0. Show that the voltage gain of theabove circuit is given by

VoVs=y21(1 + k) + ky22y22 ,

wherek = R

R.

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VS

RC

R

R

V1

V2

V0

I1

I2

+ +

+

- - -

R5. Consider a voltage amplifier circuit shown in the figure below, whereRi and R0 represent the input and output impedances respectively,C0 is the total parasitic capacitance across the output port, is theamplifier gain and the output is terminated by a load resistance RL.

(a) Calculate the current, voltage and power gain in decibels (dB) ofthe amplifier, whenRi= 1M, RL= 600, Ro= 100M, Co= 10pf, = 10.

(b) Calculate the 3-dB cutoff frequency of the amplifier whenRi= 5K, RL= 1K, Ro= 100, Co = 10pf, = 2.

R6. A 50 HP (37.3 KW), 460 V DC shunt motor running freely takes acurrent of 4 A and runs at a speed of 660 rpm. The resistance of thearmature circuit (including brushes) is 0.3 ohm and that of the shuntfield circuit is 270 ohm.

(a) Determine (i) the total current, and (ii) the speed of the motorwhen it is running at full load.

(b) Determine the armature current at which the efficiency is maxi-mum (ignore the effect of armature reaction).

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R7. (a) Three 100 ohm, non-inductive resistances are connected in (i)

Star and (ii) Delta configurations across 400 V, 50 Hz, 3-phasemain. Calculate the power taken from the supply system in eachcase.

(b) In the event of one of the three resistances getting open circuited,what variation would be in the value of the total power taken ineach of the two configurations?

R8. Consider the following circuit with an OP-AMP.

R = 10 K2

R = 100 K1

Vin V

out

The plot of output voltage Vout vs. input voltage Vin for the givencircuit is as follows.

Vin

Vout

VA

VB

Vc Vd

Let VA = 10V and VB =10V. Assume that Vin < Vc, and isgradually increasing. The output voltageVout = VAuntilVin= Vdandthen falls to VB. The output remains at VB for Vin> Vd. Similarly, ifVin is initially > Vd and gradually reduced, Vout remains at VB untilVin= Vc, and then rises to VA for all values Vin< Vc.

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(a) Explain why the circuit behaves in this fashion, and

(b) calculate the values ofVc and Vd.

R9. Assume that an analog voice signal which occupies a band from 300Hz to 3400 Hz, is to be transmitted over a Pulse Code Modulation(PCM) system. The signal is sampled at a rate of 8000 samples/sec.Each sample value is represented by 7 information bits plus 1 paritybit. Finally, the digital signal is passed through a raised cosine roll-offfilter with the roll-off factor of 0.25. Determine

(a) whether the analog signal can be exactly recovered from the dig-ital signal;

(b) the bit duration and the bit rate of the PCM signal before filter-ing;

(c) the bandwidth of the digital signal before and after filtering;

(d) the signal to noise ratio at the receiver end (assume that theprobability of bit error in the recovered PCM signal is zero).

R10. A logic circuit is to be designed having four inputsx1, x0, y1 and y0and the three outputs z1, z2 and z3. The pair of bits x1x0 and y1y0represent two binary numbers X and Y with x1 and y1 as the mostsignificant bits. z1 is 1 if X is larger than Y and z2z3 represent the

difference between the two numbersXandY. Find the minimum sumof product expressions for z1, z2 andz3.

R11. A messagebbccfe\needs to be encoded using arithmetic coding. Theprobabilities of message symbols are shown in the following table.

symbol a b c d e f \Probability 0.05 0.2 0.1 0.05 0.3 0.2 0.1

Using the symbol probabilities shown in the above table, find

(a) a fractional value that is to be transmitted after encoding themessage bbccfe

\,

(b) the exact decoding scheme of the message from the fractionalvalue estimated at the encoding stage, and

(c) the number of bits required to represent the encoded messageafter arithmetic coding.

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R12. Consider two identical parallel plate air capacitors in series. The com-

bination is maintained at the constant potential difference of 35 volts.Now a dielectric sheet of dielectric constant 4 and thickness equal to0.8 of the air gap is inserted into one of the capacitors, so that it spansthe whole area of the plates of the capacitors. Calculate the voltageacross this capacitor and the ratio of electrostatic energies stored inthe two capacitors.

R13. Two linear, time-invariant (LTI) discrete-time systems with frequencyresponses as indicated below, are cascaded to form another LTI system.

H1(ej

) = 1 if||< /2,0 otherwise.

H2(ej) =

j if0< < ,j if <


28/34

(v) COMPUTER SCIENCE

C1. (a) Write the smallest real number greater than 6.25 that can berepresented in the IEEE-754 single precision format (32-bit wordwith 1 sign bit and 8-bit exponent).

(b) Convert the sign-magnitude number10011011 into a 16-bit 2scomplement binary number.

(c) The CPU of a machine is requesting the following sequence ofmemory accesses given as word addresses: 1, 4, 8, 5, 20, 17,

19, 56. Assuming a direct-mapped cache with 8 one-word blocks,that is initially empty, trace the content of the cache for the abovesequence.

C2. Consider a collection of n binary strings S1, . . . , S n. Each Si is oflengthli bits where 1lik.

(a) Write a functionprefix(S,T) in C programming language thattakes two binary strings S, T and returns 1 ifS is a prefix of T,else it returns 0. For example, prefix (001,00101)returns 1 butprefix(010,00101)returns 0.

(b) Suppose we want to report all the pairs (i, j) for which Si is a

prefix ofSj, (1i=jn). How many times do we need to callthe prefix function described above?

(c) Present anO(nk) time algorithm to report all the (i, j)s as men-tioned in (b). (Hint: Use a binary tree with each edge markedas 0 or 1; a path from the root to a node in the tree represents abinary string.)

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C3. (a) Write a computer program in the C language that takes an array

Aof 2ndistinct floating point numbers, and prints the maximumand the minimum values of the array A, along with their indices.(Full credit will be given only if your program does not make morethan(3n 2) floating point comparisons.)

(b) Briefly argue that your program indeed computes the maximumand the minimum values correctly.

C4. Let a1= 1, a2 = 2, and an= an1+ an2+ 1 for n >2.

(a) Express 63 as a sum of distinctais.

(b) Write an algorithm to express any positive integerk as a sum ofat mostlog2 k many distinct ais.

(c) Prove the correctness of your algorithm.

C5. LetS={x1, x2, . . . xn}be a set ofn integers. A pair (xi, xj) is said tobe the closest pair if|xi xj | |xi xj|, for all possible pairs (xi, xj),i, j= 1, 2, . . . , n , i=j .

(a) Describe a method for finding the closest pair among the set ofintegers inSusing O(n log2 n) comparisons.

(b) Now suggest an appropriate data structure for storing the ele-

ments in Ssuch that if a new element is inserted to the setS oran already existing element is deleted from the set S, the currentclosest pair can be reported inO(log2 n) time.

(c) Briefly explain the method of computing the current closest pair,and necessary modification of the data structure after each up-date. Justify the time complexity.

C6. LetA be ann nmatrix such that for every 2 2 sub-matrix

a bc d

of A, if a < b then c d. Note that for every pair of rows i and

j, ifaik and ajl are the largest elements in i-th and j -th rows ofA,

respectively, thenkl (as illustrated in the 5 5 matrix below).

3 4 2 1 17 8 5 6 42 3 6 6 55 6 9 10 74 5 5 6 8

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(a) Write an algorithm for finding the maximum element in each rowof the matrix with time complexity O(n log n).

(b) Establish its correctness, and justify the time complexity of theproposed algorithm.

C7. Consider a file consisting of 100 blocks. Assume that each disk I/Ooperation accesses a complete block of the disk at a time. How manydisk I/O operations are involved with contiguous and linked allocationstrategies, if one block is

(a) added at the beginning?

(b) added at the middle?

(c) removed from the beginning?

(d) removed from the middle?

C8. (a) Consider the context-free grammarG = ({S, A}, {a, b}, S , P ),where P = {SAS,

Sb,ASA,Aa}

Show that G is left-recursive. Write an equivalent grammar Gfree of left-recursion.(b) Consider the grammarG = ({S, T}, {a,, (, ), +}, S , P ),

whereP = {Sa||(T),

T T+ S|S}Find the parse tree for the sentence:

(((a + a) + + (a)) + a)

C9. (a) Five batch jobs P1, . . . , P 5 arrive almost at the same time. Theyhave estimated run times of 10, 6, 2, 4 and 8 ms. Their prioritiesare 3, 5, 2, 1 and 4 respectively, where 1 indicates the highestpriority and 5 indicates the lowest. Determine the average turn-around and waiting time for the following scheduling algorithms:

(i) Round robin with time quantum of 5 ms,

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(ii) Priority scheduling.

(b) The access time of a cache memory is 100 ns and that of mainmemory is 1000 ns. It is estimated that 80% of the memoryrequests are for read and the remaining 20% are for write. Thehit ratio for read access is 0:9. A write through procedure is used.

(i) What is the average access time of the system consideringonly memory read cycles?

(ii) What is the average access time of the system consideringboth read and write requests?

C10. (a) A programPconsisting of 1000 instructions is run on a machine

at 1 GHz clock frequency. The fraction of floating point (FP)instructions is 25%. The average number of clock-cycles per in-struction (CPI) for FP operations is 4.0, and that for all otherinstructions is 1.0.

(i) Calculate the average CPI for the overall programP.

(ii) Compute the execution time needed byP in seconds.

(b) Consider a 100mbps token ring network with 10 stations havinga ring latency of 50 s (the time taken by a token to make onecomplete rotation around the network when none of the stations isactive). A station is allowed to transmit data when it receives thetoken, and it releases the token immediately after transmission.The maximum allowed holding time for a token (THT) is 200 s.

(i) Express the maximum efficiency of this network when only asingle station is active in the network.

(ii) Find an upper bound on the token rotation time when allstations are active.

(iii) Calculate the maximum throughput rate that one host canachieve in the network.

C11. Consider a graphG (called an interval graph) whose nodes correspondto a set of intervals on the real line. Thei-th interval is denoted by

[i, i], where 0i < i. An edge between two nodes (i, j) impliesthat the corresponding intervals [i, i] and [j, j]] overlap.

(a) Consider the set of intervals [3, 7], [2, 4], [2, 3], [1, 5], [1, 2], [6, 7],[10, 16], [11, 12]. Draw the corresponding interval graph and iden-tify the largest subgraph where all the nodes are connected toeach other.

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(b) Write an algorithm which takes the interval graph G as input

and finds the largest subgraph of G in which all the nodes areconnected to each other. What is the time complexity of youralgorithm?

(c) Given a list of intervals, write an algorithm to list all the con-nected components in the corresponding interval graph. What isthe time complexity of your algorithm?

C12. (a) A functional dependency is called a partialdependency ifthere is a proper subsetof such that. Show that everypartial dependency is a transitive dependency.

(b) Let R = (A,B,C,D ,E) be a schema with the set F ={

ABC, CD E, B D, E A} of functional dependencies.

Suppose R is decomposed into two schema R1 = (A,B,C) andR2 = (A,D,E)

(i) Is this decomposition loss-less? Justify.

(ii) Is this decomposition dependency preserving? Justify.

(c) Consider the relations r1(A,B,C),r2(C,D ,E) and r3(E, F). As-sume that the set of all attributes constitutes the primary keysof these relations, rather than the individual ones. LetV(C, r1)be 500, V(C, r2) be 1000, V(E, r2) be 50, and V(E, r3) be 150,whereV(X, r) denotes the number of distinct values that appear

in relation r for attribute X . Ifr1 has 1000 tuples, r2 has 1500tuples, and r3 has 750 tuples, then give the ordering of the nat-ural join r1 r2 r3 for its efficient computation. Justify youranswer.

C13. (a) (i) Write a Context Free Grammar (CFG) for structure defini-tions in C. Assume that the only allowable types are char,int, and float (you need not handle pointers, arrays, struc-ture, fields, etc.).

(ii) Assume that chars are stored using 1 byte each; ints andfloats are stored using 4 bytes each and are aligned at 4 byteboundaries. Add semantic rules to your grammar to calculatethe number of bytes required to store the structure definedby your grammar.

(b) (i) Compute the canonical collection of sets ofLR(1) items (i.e.canonical LR items) for the following grammar: SaXcd,

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S

aY ce, X

b, Y

b. Is the grammarLR(1)? Briefly

justify.(ii) Give an example of a grammar that is unambiguous but not

LR(2). Briefly justify/explain your example.

C14. An operating system allocates memory in units of 1 KB pages. Theaddress space of a process can be up to 64 MB in size; however, at anypoint of time, a process can be allocated at most 16 MB of physicalmemory. In addition the kernel uses 65 KB of physical memory tostore page table entries of the current process. The OS also uses atranslation-lookaside buffer (TLB) to cache page table entries. Youare also given the following information:

size of a page table entry is 4 bytes, TLB hit ratio is 90%, time for a TLB lookup is negligible, time for a memory read is 100 nanoseconds, time to a read a page from the swapping device into physical

memory is 10 milliseconds.

Calculate the effective memory access time for a process whose ad-dress space is 20 MB? Assume that memory accesses are random and

distributed uniformly over the entire address space.

C15. (a) What are the conditions which must be satisfied by a solution tothe critical section problem?

(b) Consider the following solution to the critical section problem fortwo processes. The two processes, P0 andP1, share the followingvariables:

var flag : array [0..1] of Boolean;

(* initially false *)

turn : 0..1;

The program below is for processPi (i= 0 or 1) with process Pj(j= 1 or 0) being the other one.

repeat

flag[i]


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then begin

flag[i]

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