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Test Code : RC (Short Answer Type) 2012

Junior Research Fellowship (JRF) in Computer Science

The candidates for Junior Research Fellowship in Computer and Commu-nication Sciences will have to take two tests - Test MIII (objective type)in the forenoon session and Test RC (short answer type) in the afternoonsession. The RC test booklet will have two groups as follows:

GROUP AA test for all candidates in the basics of computer programming and discretemathematics.

GROUP BA test, divided into five sections in the following areas at M.Sc./M.E./M.Tech.level:

Mathematics, Statistics, Physics, Electrical and Electronics Engineering, and Computer Science.

A candidate has to answer questions from only one of these sections inGROUP B, according to his/her choice.

Group A carries 40 marks and Group B carries 60 marks.

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

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Syllabi

GROUP A:

Logical reasoning; elementary combinatorics including inclusion-exclusionprinciple and pigeonhole principle; basics of programming (using pseudo-code); elementary data structures including array, stack and queue; Booleanalgebra.

GROUP B:

Mathematics:

Graph theory and combinatorics: Graphs, paths and cycles, trees, Euleriangraphs, Hamiltonian graphs, chromatic numbers, planar graphs, digraphsand tournaments.

Linear algebra: Vector spaces, basis and dimension, linear transformations,matrices, rank, inverse, determinant, systems of linear equations, eigenvaluesand eigenvectors, orthogonality.

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, convergence of sequences andseries, limits, continuity, uniform continuity of functions, differentiability offunctions of one or more variables, indefinite integral, fundamental theoremof integral calculus, Riemann integration, improper integrals, sequences andseries of functions, uniform convergence.

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Statistics:

Probability theory and distributions: Basic probability theory, discrete andcontinuous 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, Bayesian methods.

Linear models: Gauss-Markov set up and least squares theory, multiplelinear regression, one and two way analysis of variance.

Multivariate analysis: Multivariate normal distribution, principal compo-

nent analysis, multiple and canonical correlations, discriminant analysis.

Physics:

Classical mechanics: Lagrangian and Hamiltonian formulation of Newtonianmechanics, symmetries and conservation laws, motion in central field offorce, small oscillations and normal modes, wave motion, special theory ofrelativity.

Electrodynamics: Electric and magnetic phenomena in dielectrics, Maxwellsequations, conservation laws, gauge invariance, electromagnetic waves.

Thermodynamics and statistical physics: Laws of thermodynamics, statisti-cal basis of thermodynamics, thermodynamic potentials and Maxwells re-lations, density matrix formulation, ensembles, partition function, classicaland quantum statistics, blackbody radiation and Plancks distribution law.

Quantum physics: Basics of quantum mechanics, two-body problem, centralpotential, angular momentum algebra, quantum particles in electromagneticfields, matrix mechanics.

Atomic and nuclear physics: Energy spectrum of an electron in hydrogenatom, electron spin, relativistic correction, selection rules, Zeeman effect,Stark effect, basic nuclear properties, nuclear force, nuclear models, radioac-tive decays.

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Electronics: Basics of semiconductor physics, transistors, amplifiers includ-

ing feedback, oscillators, operational amplifiers, RLC circuits, digital inte-grated circuits, A/D and D/A converters.

Electrical and Electronics Engineering:

Digital circuits and systems: Gates and logic circuits, combinational andsequential circuits, A/D and D/A converters.

Circuit theory: Kirchoffs laws, theorem of superposition, Thevenins theo-rem, Nortons theorem, A.C. circuits.

Linear electronic devices and circuits: Transistors, amplifiers including feed-

back amplifiers, oscillators, operational amplifiers.Digital communication: Information and coding theory, channel capacity,digital transmission.

Digital signal processing: Sampling, linear time invariant systems,Z-transform, discrete-time Fourier and discrete Fourier transforms.

Electrical machines: DC motors and generators, transformers, inductionmotors.

Computer Science:

Discrete mathematics: Functions and relations, recurrence relations, gen-erating functions, graph theory - paths and cycles, trees, digraphs, planargraphs, Eulerian graphs, Hamiltonian paths.

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

Data structures and algorithms: Linked list, queue, binary tree, heap, AVLtree, Order notation, sorting, selection, searching, hashing, graph algo-rithms.

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Computer organization and architecture: Number representation, computer

arithmetic, memory organization, I/O organization, pipelining.

Operating systems: Process concept and management, scheduling, processsynchronization, concurrency control, critical section problems, deadlocks,memory management, file systems.

Formal languages and automata theory: Finite automata and regular expres-sions, context-free grammars, Turing machines, elements of undecidability.

Database systems: Relational model, relational algebra, relational calcu-lus, functional dependency, normalization (including multi-valued depen-dencies), query processing.

Computer networks: Layered network structures, network security, LANtechnology - bus/tree, ring, star; data communications - data encoding,flow control, error detection/correction, TCP/IP networking.

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Sample Questions

Note that all questions in the sample set are not of same marks and same level of difficulty.

GROUP A

A1. Show how two complex numbers (a+ib) and (c+id) may be multipliedusing only three multiplications of real numbers, where i =

1. Youmay use any number of additions and subtractions.

A2. Consider the pseudo-code given below.Input: Integers b and c.

1. a0 max(b, c), a1 min(b, c).2. i 1.3. Divide ai1 by ai . Let qi be the quotient and ri be the remainder.

4. Ifri = 0 then go to Step 8.

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

8. Print ai .

What is the output of the above algorithm when b = 28 and c = 20?What is the mathematical relation between the output ai and the twoinputs b and c?

A3. Consider the sequence an = an1 an2 + n for n 2, with a0 = 1 anda1 = 1. Is a2011 odd? Justify your answer.

A4. Given an array ofn integers, write pseudo-code for reversing the con-tents of the array without using another array. For example, for thearray

10 15 3 30 3the output should be3 30 3 15 10.You may use one temporary variable.

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A5. The integers 1, 2, 3, 4 and 5 are to be inserted into an empty stack

using the following sequence of push() operations:push(1) push(2) push(3) push(4) push(5)

Assume that pop() removes an element from the stack and outputs thesame. Which of the following output sequences can be generated byinserting suitable pop() operations into the above sequence of push()operations? Justify your answer.

(a) 5 4 3 2 1

(b) 1 2 3 4 5

(c) 3 2 1 4 5

(d) 5 4 1 2 3.

A6. Derive an expression for the maximum number of regions that can beformed within a circle by drawing n chords.

A7. Given A = {1, 2, 3, . . . , 70}, show that for any six elements a1, a2, a3,a4, a5 and a6 belonging to A, there exists one pair ai and aj for which|ai aj | 14 (i = j).

A8. Consider the multiplication of two 2-bit integers a1a0 and b1b0 to get

a 4-bit output c3c2c1c0. Assuming that the rightmost bit is the leastsignificant bit, derive Boolean functions for the output bits c0 and c3.

A9. Calculate how many integers in the set {1, 2, 3, . . . , 1000} are not di-visible by 2, 5, or 11.

A10. Let M be a 4-digit positive integer. Let N be the 4-digit integerobtained by writing the digits ofM in reverse order. IfN = 4M, thenfind M. Justify your answer.

A11. Consider all the permutations of the digits 1, 2, . . . , 9. Find the numberof permutations each of which satisfies all of the following:

the sum of the digits lying between 1 and 2 (including 1 and 2)is 12,

the sum of the digits lying between 2 and 3 (including 2 and 3)is 23,

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the sum of the digits lying between 3 and 4 (including 3 and 4)is 34, and

the sum of the digits lying between 4 and 5 (including 4 and 5)is 45.

A12. A Boolean operation denoted by = , is defined as follows:

A B A = BT T T

F T T

T F F

F F T

Show that A B can be expressed in terms of = alone, withoutusing any 0 or 1 input.

A13. A Boolean function fD1 is said to be the dual of another Booleanfunction f1 if f

D1 is obtained from f1 by interchanging the operations

+ and ., and the constants 0 and 1. For example, if f1(a,b,c) =(a + b).(b + c) then fD1 (a,b,c) = a.b + b.c.

A Boolean function f is self-dual if f1 = fD1 . Given f1(a,b,c) =

ab + bc + x, find the Boolean expression x such that f1 is self-dual.

A14. (a) There are n students in a class. The students have formed kcommittees. Each committee consists of more than half of thestudents. Show that there is at least one student who is a memberof more than half of the committees.

(b) Let D = {d1, d2, . . . , dk} be the set of distinct divisors of a positiveinteger n (D includes 1 and n). Show that

ki=1

sin1

logn di =

4 k.

hint: sin1 x + sin1

1

x2 = 2

A15. (a) Give a strategy to sort four distinct integers a,b,c,d in increasingorder that minimizes the number of pairwise comparisons neededto sort any permutation of a,b,c,d.

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(b) An nn matrix is said to be tridiagonal if its entries aij are zeroexcept when |i j| 1 for 1 i, j n. Note that only 3n 2entries of a tridiagonal matrix are non-zero. Thus, an array L ofsize 3n 2 is sufficient to store a tridiagonal matrix. Given i, j,write pseudo-code to

i. store aij in L, and

ii. get the value of aij stored in L.

A16. (a) Consider an m n integer grid. A pathfrom the lower left cornerat (0, 0) to the grid point (m, n) can use three kinds of steps,namely (i) (p, q) (p + 1, q) (horizontal), (ii) (p, q) (p, q+ 1)(vertical), or (iii) (p, q) (p + 1, q + 1) (diagonal). Derive anexpression for Dm,n, the number of such distinct paths.

(b) The numbers 1, 2, . . . , 10 are circularly arranged. Show that thereare always three adjacent numbers whose sum is at least 17, ir-respective of the arrangement.

A17. (a) Consider six distinct points in a plane. Let m and M denoterespectively the minimum and the maximum distance betweenany pair of points. Show that M/m 3.

(b) Consider the following intervals on the real line:

A1 = (13.3, 18.3) A3 = (8.3, 23.3) A1 A2A2 = (10.8, 20.8)

A1 A4 = (5.8, 25.8)

A1

A2

A3

where (a, b) = {x : a < x < b}.Write pseudo-code that finds the interval to which a given inputx (5.8, 25.8) belongs, i.e., your pseudo-code should calculatei {1, 2, 3, 4} such that x Ai. Your method should not use anycomparison operation.

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GROUP B

(i) MATHEMATICS

M1. (a) Let f be a real-valued continuous function on [0, 1] such that10

f4(x)dx = 0. Show that f(x) = 0 for all x [0, 1].(b) Let f be a real-valued continuously differentiable function with

f(0) = 1. Find limx0

1

x2

x2x2

f(t)dt.

(c) Let D [0, 1] [0, 1] be defined as D = {(x, y) : xy 3/4}.Compute

D

2(xy)1dxdy, where is a constant.

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

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

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

and f(0) = 0. Show that10 f(x)dx exists and find its value.

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

an = 1

0

(f(x))ndx1

n

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

Find limn an.

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

0xf(x) dx =

1

2f()

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

x + 1 x = 1

2

x + (x)

,

for a unique (x), 0 < (x) < 1. Also prove that,(i) 14 (x) 12 , and(ii) limx0 (x) =

14 and limx (x) =

12 .

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M4. Let G be a finite group and f be an automorphism of G such that

f(f(x)) = x for all x G. Also, the only fixed point of f is e, i.e., ifg G and f(g) = g, then g = e.

(a) Show that the map g g1f(g) is a bijection.(b) Hence or otherwise, show that f(g) = g1 for all g G.(c) Hence or otherwise, show that G is abelian.

M5. Let R = (S, +, , 0, 1) be a commutative ring and n be a positive integersuch that n = 2k for some positive integer k.

(a) Show that for every a

S

n1i=0

ai =

k1i=0

1 + a2

i

.

(b) Let m = wn

2 + 1 where w S, w = 0. Show that for 1 p < n,n1i=0

wip 0 mod m.

M6. (a) Show that there is a basis consisting of only symmetric and skew-

symmetric matrices for the vector space of all n n real 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.

M7. (a) Let A be a square matrix such that A2 = A. Show that all eigenvalues of A are 0 or 1.

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

(c) Suppose A is an n

n matrix such that A2 = A. If rank(A) = rand rank(I A) = s, show that r + s = n.

M8. (a) Show that in a connected graph, any two longest paths have atleast one vertex in common.

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(b) Construct a cubic graph with 2n vertices having no triangles. (A

graph is cubic if every vertex has degree three.)(c) Let G be a graph on 9 vertices. Show that either G has a triangle

or G contains K4. (Here G is the complement of the graph G,and K4 is the complete graph with 4 vertices). [Hint: any vertexhas degree at least 4 in either G or G.]

(d) Let S = {0, 1, 2, 3} and let ak be the number of strings of length kover Shaving an even number of zeros. Find a recurrence relationfor ak and then solve for ak.

M9. Let T be a tree with n vertices. For vertices u, v ofT, define d(u, v) tobe the number of edges in a path from u to v. Let W(T) be the sum

of d(u, v) over all(n2)

pairs of vertices {u, v}.(a) Suppose the tree T is a path on n vertices. Show that

W(T) =1

2

n1k=1

(k2 + k).

(b) Now, suppose T is a star graph on n vertices. Show that W(T) =(n1)2. (N.B. The edge set of the star graph is equal to {(u1, ui) :2 i n}.)

(c) Hence or otherwise show that for any tree T,

(n 1)2 W(T) 12

n1k=1

(k2 + k).

M10. (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}.

M11. (a) A set S contains integers 1 and 2, and all integers of the form3x + y where x and y are distinct elements of S. What is S ?Justify your answer.

(b) Let (n) denote the number of positive integers m relativelyprime to n; m < n. Let n = pq where p and q are prime numbers.Then show that (n) = (p 1)(q 1) = pq(1 1q )(1 1p).

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M12. Consider the n n matrix A = ((aij)) with aij = 1 for i < j andaij = 0 for i j. Let

V = {f(A) : f is a polynomial with real coefficients}.

Note that V is a vector space with usual operations. Find the dimen-sion of V, when (a) n = 3, and (b) n = 4. Justify your answer.

M13. Consider the following system of equations over a field F.

a1x + b1y = c1a2x + b2y = c2,

where a1, a2, b1, b2, c1, c2

F. State the conditions for which the abovesystem of equations has (a) no solution, (b) a unique solution and (c)more than one solution.

Next consider the following equations over the field Zp, the field ofintegers modulo p, where p is a prime.

7x y = 111x + 7y = 3

For what value ofp does this system of equations have (a) no solution,(b) a unique solution and (c) more than one solution? Find the solutionin the case of (b).

M14. In each of the following four problems, two sets S and T are given.You need to define a continuous and onto function f : S T in eachof the cases. You need to provide mathematical reasons if you cannotdefine such an f.

(a) S = (0, 1), T = (0, 1].

(b) S = (0, 1], T = (0, 1).

(c) S = (0, 1) (2, 3), T = (0, 1).(d) S = (0, 1), T = (0, 1) (2, 3).

M15. You are given 49 balls of colour red, black and white. It is knownthat, for any 5 balls of the same colour, there exist at least two amongthem possessing the same weight. The 49 balls are distributed in twoboxes. Prove that there are at least 3 balls which lie in the same boxpossessing the same colour and having the same weight.

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M16. Consider a sequence {an} such that 0 < a1 < a2 and ai = ai1ai2for i 3.

(a) Show that {a2n1 : n = 1, 2, . . .} is an increasing sequence and{a2n : n = 1, 2, . . .} is a decreasing sequence.

(b) Show that limn

an exists.

(c) Find limn

an.

M17. If every vertex of a graph G = (V, E) has degree at least|V|2 , then

prove that there is a simple cycle containing all the vertices. Notethat, a simple cycle with a vertex set U V is a cycle where eachvertex of U appears only once if one traverses along the cycle.

M18. (a) Consider three non-null matrices (not necessarily square) A, Band C that satisfy ACCt = BCCt, where Ct denotes the trans-pose of C. Is it necessarily true that

i. A = B?

ii. AC = BC?

Justify your answers.

(b) Consider a real matrix Cnp, whose columns are linearly in-dependent and all elements in the first column are 1. DefineD = I C(CtC)1Ct, where I is the identity matrix of order nand Ct denotes the transpose of C.

i. Show that D1 = 0, where 1 = (1, 1, . . . , 1)t.

ii. Find the rank of D.

M19. (a) Consider a graph with 8 vertices. If the degrees of seven of thevertices are 1, 2, 3, 4, 5, 6 and 7, find the degree of the eighth ver-tex. Argue whether the graph is planar. Also find its chromaticnumber. [6+3+3=12]

(b) Let S be a subset of{10, 11, 12 . . . , 98, 99} containing 10 elements.Show that there will always exist two disjoint subsets A and Bof S such that the sum of the elements of A is the same as thatof B.

M20. (a) Determine the product of all distinct positive integer divisors of6304.

(b) Let p1 < p2 < . . . < p31 be prime numbers such that 30 dividesp41 +p

42 + +p431. Prove that p1 = 2, p2 = 3 and p3 = 5.

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(c) Find all primes p and q such that p + q = (p q)3. Justify youranswer.

M21. (a) Show that each of the equations sin(cos x) = x andcos(sin y) = y has exactly one root in [0, /2]. If x1 and x2are the roots of these two equations respectively, then show thatx1 < x2.

(b) Let f be a real-valued continuous function on R satisfying theinequality

f(x) 12h

x+hxh

f(y) dy, x R, h > 0.

Prove that for any bounded closed interval, the maximum of fon that interval is attained at one of its end points.

(c) Define f(x) = ex2/2

x e

t2/2 dt for x > 0. Show that 0 0.

M22. (a) For n N, let the sequences {an}n=1 and {bn}n=1 be given by

0 < b1 < a1, an+1 =a2n + b

2n

an + bn, and bn+1 =

an + bn2

.

Show that both the sequences are monotone and they have thesame limit.

(b) Consider a polynomial fn(x) = xn + a1x

n1 + + an1x + anwith integer coefficients. If fn(b1) = fn(b2) = fn(b3) = fn(b4) =fn(b5) = 19 for five distinct integers b1, b2, b3, b4 and b5, then howmany different integer solutions exist for fn(x) = 23? Justifyyour answer.

M23. (a) Show that f(x) = e|x| x5 x 2 has at least two real roots,where e is the base of natural logarithms.

(b) Let

an be a convergent series such that an 0 for all n. Showthat

an/np converges for every p > 12 .

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(ii) STATISTICS

S1. (a) Let {Xn}n1 be 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 of Xn as n ?

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

{1, 2, . . . , m

}. Define a sequence of random variables

{Xn

}n1

as 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 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 < ,

where > 0 is unknown and k > 0 is known. Find a minimum varianceunbiased estimator for 1 .

S3. Let 0 < p < 1 and b > 0. Toss a coin once where the probabilityof occurrence of head is p. If head appears, then n independent andidentically distributed observations are generated from Uniform (0, b)distribution. If the outcome is tail, then n independent and identicallydistributed observations are generated from Uniform (2b, 3b) distribu-tion. Suppose you are given these n observations X1, . . . , X n, but notthe outcome of the toss. Find the maximum likelihood estimator of bbased on X1, . . . , X n. What happens to the estimator as n goes to ?

S4. Let X1, X2, . . . be independent and identically distributed random

variables with common density function f. Define the random variableN as

N = n, if X1 X2 Xn1 < Xn; for n = 2, 3, 4, . . . .Find Prob(N = n). Find the mean and variance of N.

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S5. (a) Let X and Y be two random variables such thatlog Xlog Y

N(, ).

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

(b) Consider the linear model yn1 = Anpp1 + n1 and the usualGauss-Markov set up where E() = 0 and D() = 2Inn, Edenotes the expectation and D denotes the covariance matrix.Assume that A has full rank. Show that V ar

(LS1

)= (

TB1)12 where

AT

A = 11 T

Bp1p1

is a p 1 1 matrix and LS1 = the least square estimate of1, the first component of the vector , V ar denotes the variance,and AT denotes the transpose of the matrix A.

S6. Let X = (X1, . . . , X n) and Y = (Y1, . . . , Y n) be two independent andidentically distributed multivariate random vectors with mean 0 andcovariance matrix 2In, where

2 > 0 and In is the n n identitymatrix.

(a) Show that XTY/(X Y) and V =

(X2i + Y2i ) are indepen-

dent.(Here, (a1, . . . , an) =

a21 + + a2n ).

(b) Obtain the probability density function of(n

i=1 X2i /n

i=1 Y2i

).

S7. Let X1, X2, . . . , X n be independent random variables. Let E(Xj) = jand V(Xj) = j

32, j = 1, 2, . . . , n, < < and 2 > 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 estimate for .

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

n.

S8. Let (X, Y) follow a bivariate normal distribution. Let mean of X =mean ofY = 0, variance ofX = variance ofY = 1, and the correlationcoefficient between X and Y be . Find the correlation coefficientbetween X3 and Y3.

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S9. Let the probability density function of a random variable X be f(x),

< x < , and we have two hypothesesH0 : f(x) = (2)

1/2 exp(x2/2) andH1 : f(x) = (1/2)exp(|x|).In connection with testing H0 against H1, derive the most powerfultest at level = 0.05.

S10. Let X1, X2, . . . , X n be independent and identically distributed obser-vations with a common exponential distribution with mean . Showthat there is no uniformly most powerful test for testing H0 : = 1against HA : = 1 at a given level 0 < < 1, but there exists auniformly most powerful unbiased test. Derive that test.

S11. (a) An unbiased die is rolled once. Let the score be N {1, 2, . . . , 6}.The die is then rolled n times. Let X be the maximum of thesen scores. Find the probability of the event (X = 6).

(b) The unit interval (0,1) is divided into two sub-intervals by pickinga point at random from the interval. Let Y and Z be the lengthsof the longer and shorter sub-intervals, respectively. Find thedistribution of Z and show that YZ does not have a finite expec-tation.

S12. Let X1, X2, X3 be independent and identically distributed observationswith a common double exponential distribution with density

f(x, ) =1

2exp(|x |), < x < , < < .

Suppose that the observations are all distinct.

(a) Find the maximum likelihood estimator of . Give a completeargument for your answer.

(b) Suppose it is known that 10 10. Find the maximumlikelihood estimator of . Justify your answer.

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S13. Consider the following linear model

yij = i + j + eij , i = 1, 2; j = 1, 2, 3;

(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.

S14. Let A = {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,

) and

x1 A. For any xi A, xi+1 is found in the following way.Toss a fair coin three times and let the outcome be z.If f(z) f(xi) then xi+1 = z, otherwise xi+1 = xi.What can you say about limi f(xi)? Justify your answer.

S15. Let there be two classes C1 and C2. Let the density function forclass Ci be pi(x) for i = 1, 2 where pi(x) = ie

ix; x > 0, i = 1, 2.Let the prior probability for C1 be 0.4 and the prior probability forC2 be 0.6. Find the decision rule for classification of an observation,which provides the minimum probability of misclassification and find

its value for that decision rule.

S16. (a) Let x1

, x2

, x3

, . . . , x10

be a random sample of 10 observations froma 100 dimensional multivariate normal distribution with disper-sion matrix

= ((ij))100100. Let ij be maximum likelihood

estimate ofij based on x1

, x2

, x3

, . . . , x10

. Let

= ((ij))100100.

Show that rank (

) 10.(b) Show that X1 + 2X2 is not sufficient for where X1 and X2 are

two independent and identically distributed observations fromN(, 1) distribution.

S17. (a) Suppose X N(, 1), where 0. For any fixed x > 0, defineg() = P(X

2 > x), 0.Show that g() is a strictly increasing function of on [0, ).

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(b) Suppose X Np(, Ip), where IRp, and Ip is the identitymatrix of order p. For any fixed x > 0, define

h() = P(XTX> x), IRp.

Assuming h() depends on only through T, show that h()is a strictly increasing function ofT.

S18. (a) Suppose independent observations are generated from the lin-ear regression model yi = 0 + 1xi + i, i = 1, 2, . . . , n, wherex1, . . . , xn are known positive numbers, and E(i) = 0 i, Var(i) =2 i. However, we fit the model yi = 1xi + i. Show that thebias in the least squares estimate of 1 is at most 0/x.

(b) Consider the linear model (Y

,X

,

2In), where

Xis an n pmatrix with entries 0, 1 and 1. Assume that X is of full column

rank. Show that if j denotes the best linear unbiased estimate(BLUE) of j, then

Var(j) 2/n.

S19. Suppose that X has uniform distribution over the square defined bythe four vertices (0, ), (, 0), (0, ) and (, 0) in the plane, where > 0. Let X1,X2, . . . ,Xn be independent and identically distributed(i.i.d.) observations from this distribution. Find the maximum likeli-hood estimate of based on these observations.

S20. Let X1, . . . , X n be a random sample from a Bernoulli(p) distribution.We want to test H0 : p = 0.4 against H1 : p = 0.6.

(a) Find the most powerful test for testing H0 against H1.

(b) Use the Central Limit Theorem to determine the sample sizeneeded so that the two probabilities of error of the test obtainedin (a) are both approximately equal to . Find the result in termsof z, the (1 )-th quantile of N(0, 1).

S21. (a) Let {Xn}n=1 be independent and identically distributed as U(0, ),where > 0. Let X

(n)(k) denote the k-th order statistic based on

X1, . . . , X n. Show that for any fixed k, nX(n)(k) converges in dis-

tribution to a non-degenerate random variable as n .

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(b) Let X1 and X2 be independent and identically distributed as

U(0, 1). Find the distribution of Z = X1X2.S22. Let X = {1, 2, 3, . . .} be the state space of a Markov chain with

the transition probability matrix P = ((pi,j)) where pi,1 = 1/i2 and

pi,i+1 = 1 1/i2 for all i 1.(a) Derive the stationary distribution of the Markov chain.

(b) Is the Markov chain irreducible? Justify your answer.

(c) Does the Markov chain converge to the stationary distributionfor all initial states? Justify your answer.

S23. (a) Let X1, X2, . . . , X n be independent and identically distributed

as U(, + ||), where = 0. Find the MLE of based onX1, X2, . . . , X n.

(b) Let X be a random variable with probability density function f.If is the test function of a most powerful test at level fortesting H0 : = 0 versus H1 : = 1, then prove that either(1) = 1 or (0) = , where () = E(X).

S24. Consider a linear regression model containing an intercept. If c is thenumber of times the i-th row of the design matrix is replicated, thenshow that the i-th leverage value hii satisfies n

1 hii c1, wheren denotes the number of observations.

S25. (a) Let N(1, 2, 21, 22, ) denote a bivariate normal distributionwith marginal means 1 and 2, marginal variances

21 and

22 and

the correlation coefficient . Consider two distributions N(0, 0, 1, 1, )and N(0, 0, 1, 1, ), where > 0. Assume that the populationshave equal prior probabilities.

i. Find the classifier that has the lowest misclassification prob-ability.

ii. Calculate the misclassification probability of this classifierand show that it decreases with .

iii. Let X have a p-variate normal distribution with mean andcovariance matrix such that HX has the same distributionas that of X for all p p orthogonal matrices H. Show that = 0 and = kI, where k is a positive constant.

S26. (a) Let X1, . . . , X 5 be independent and identically distributed as N(0, 1).Calculate the probability P(X1 > X2X3X4X5).

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(b) Let X be a random variable taking values 1, 2, . . . , k, with prob-

abilities p1, p2, . . . , pk, respectively. Prove that

ki=1

p2i max{p1, p2, . . . , pk}.

(c) Let (i1, i2, . . . , i2n) and (j1, j2, . . . , j2n) be two random permuta-tions of (1, 2, . . . , 2n).

i. Let S denote the number of indices k such that ik = 2jk.For example, if n = 3, then S = 2 for the permutations(1, 4, 5, 3, 2, 6) and (3, 2, 6, 4, 1, 5). Determine E(S), the ex-pected value of S.

ii. Let Z be a discrete uniform random variable taking valuesin {1, 2, . . . , 2n}. Show that E(iZ/jZ) 1.

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(iii) PHYSICS

P1. A particle of mass m is constrained to move on a smooth sphere ofradius a and is acted on by gravity. Choosing and (polar and az-imuthal angles, with polar axis vertically up) as the generalized coordi-nates, write down the Hamiltonian for the particle. Obtain Hamiltonsequations of motion and show that

a =k2 cos

m2a3 sin3 + g sin ,

where k is the constant value ofp, the generalized momentum corre-

sponding to .P2. (a) The molar internal energy of a monatomic gas obeying van der

Waals equation is given by 32RT aV , where a is a constant, andR, T and V carry their usual meanings.

(i) If the gas is allowed to expand adiabatically into vacuum(i.e., free expansion) from volume V1 to V2, and T1 is itsinitial temperature, what is the final temperature of the gas?

(ii) What would be the final temperature for an ideal gas?

(b) A single classical particle of mass m with energy E, whereE = P2/2m, is enclosed in a volume V; P being the momentumcorresponding to the energy E.

(i) Determine the (asymptotic) number of accessible microstatesin the energy range to + d.

(ii) Using this result, obtain the partition function of the afore-said system if the energy varies between 0 and .

(iii) What would be the free energy of the particle?

P3. A nuclear fission explosion produces a fire ball that can be approxi-

mated at some instant to be a black body of 10 cm radius having atemperature 108K. Calculate

(i) the total electromagnetic power radiated,

(ii) the radiation flux at a distance of 1 km, and

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(iii) the wavelength corresponding to maximum energy radiated.

[Stefans constant ()= 0.57 107 Wattm2K4 andWiens (displacement law) constant = 2.9 103 meter Kelvin.]

P4. Assume that three spins S1, S2, S3 are arranged in the form of an equi-lateral triangle with each spin interacting with its two neighbors (seefigure below). Each spin can take values +1 or 1. The energy of thissystem in a magnetic field perpendicular to the plane of the triangle,is

H = J(S1S2 + S2S3 + S3S1) F(S1 + S2 + S3).Here J and F are constant parameters.

SS

S

3

1

2

(i) Find the partition function of the system.

(ii) Find the average spin.

P5. Consider a simple harmonic oscillator in one dimension with the Hamil-

tonian H = aa +

1

2, where a and a are annihilation and cre-

ation operators respectively. The other symbols have their usual mean-ings.

(a) Show that a |n> = n |n 1>, |n> being the eigen function ofH corresponding to the eigen value En = (n +

1

2).

(b) Consider the wave function of a harmonic oscillator at t = 0

(0) = N(|0> +2|1> +3|2>) ,

where N is the normalization constant.

(i) Find the value ofN.

(ii) Calculate the probability of observing the energy of the os-cillator to be 32 upon measurement.

(iii) Find the wave function (t) at time t and calculate the ex-pectation value of the energy for this wave function.

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P6. Consider a system of three non interacting particles that are confined

to move in a one dimensional infinite potential well of length a:V(x) = 0, 0 < x < a

= , otherwiseDetermine the energy and wave function of the ground state and thefirst excited state, when the particles are

(a) spinless and distinguishable with masses m1 < m2 < m3,

(b) identical bosons,

(c) distinguishable spin 12 particles.

P7. (a) A substance shows a Raman line at 4567

A when the exciting line

4358

A is used. Deduce the positions of Stokes and anti-Stokes

lines for the same substance when the exciting line 4047

A is used.

(b) The ground state of chlorine is 2P32

. Find its magnetic moment.

How many substates will the ground state split into in a weakmagnetic field?

P8. The -particle is an unstable sub-atomic particle that can decay spon-taneously into a proton and a negatively charged pion

p + .In a certain experiment, the outgoing proton and pion were observed,

both travelling in the same direction along the positive x-axis with mo-menta Pp and P respectively. Find the rest mass of -particle giventhe rest masses of proton and pion to be mp and m respectively.

P9. (a) Consider the multiplication of two 2-bit integers a1a0 and b1b0 toget a 4-bit output c3c2c1c0. Design a circuit forderiving the bit c2 using only 2-input NAND gates.

(b) Suppose in a voltage sensitive Wheatstone bridge, each arm hasa resistance R. Now the resistance of one arm is changed to R+r,where r

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P10. (a) A negative feedback amplifier has a voltage gain of 100. Varia-

tions of the voltage gain up to 2% can be tolerated for somespecific application. If the open-loop gain variations of10% areexpected owing to spread in device characteristics because of vari-ation in manufacturing conditions, determine the minimum valueof the feedback ratio and also the open loop gain to satisfy theabove requirements.

(b) Calculate the output voltage V0 for the following network:

5V 10K 20K

10K20K

5V

20K5V

+

-

V

V

V2

1

o

+

-

-

-

-

+

+

+

P11. (a) Consider the following circuit for deriving a +5 volt power supply

to a resistive load RL from an input d-c voltage source whosevoltage may vary from 8V to 11V. The load RL may draw amaximum power of 250 mW. The Zener diode has a breakdownvoltage of 5 volts. Compute the maximum value of the resistance

DC

voltage

source

-

+R

Zener

diodeR

L

5 V

+

-

8 - 11 V

R and also the power dissipation requirements for R and the

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Zener diode. Assume that the minimum breakdown current of

the Zener diode is negligible compared to the load current.(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

P12. (a) The point of support O of a simple pendulum of length l andpoint mass m moves with a horizontal displacement x = A sin t

in the same plane as that of the motion of the pendulum (seefigure below). Deduce the Lagrangian of the system. Using Euler-Lagrange equations of motion, show that

+g

lsin =

Aw2

lcos sin t.

Is this a constrained system? If so, what is the nature ofthe constraint?

(b) A projectile of mass m is thrown at an angle with the verticalaxis y with an initial velocity v0. Assuming that the xy-planecontains its whole trajectory, deduce the Hamiltonian function.Using Hamiltons equations, find the trajectory of the projectilein the xy-plane.

P13. (a) Consider an iron sphere of radius R that carries a charge Q and auniform magnetization M = Mz. Compute the angular momen-tum stored in the electromagnetic fields.

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O

x = A sin t

g

m

l

(b) Find the charge and current distributions that would give rise tothe potentials

= 0, A =

0k

4c(ct |x|)2z for |x| < ct

0 for |x| ct

where k is a constant and c =1

00. What would be the surface

current?

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P14. (a) Write the properly normalized Maxwell-Boltzmann distribution

f(u) for finding particles of mass m with magnitude of velocityin the interval [u, u + du] at a temperature T.

i. What is the most probable speed at temperature T?

ii. What is the average speed?

iii. What is the average squared speed?

(b) A container is divided into two compartments I and II by apartition having a small hole of diameter d. The two compart-ments are filled with Helium gas at temperatures T1 = 150K andT2 = 300K, respectively.

i. How does the diameter d determine the physical process by

which the gas comes to steady state?ii. If1 and 2 are the mean free paths in the compartments I

and II, respectively, find 1 : 2 when d 1 and d 2.iii. Find 1 : 2 when d 1 and d 2.

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(iv) ELECTRICAL AND ELECTRONICS ENGINEERING

E1. (a) Consider the multiplication of two 2-bit integers a1a0 and b1b0 toget a 4-bit output c3c2c1c0. Design a circuit forderiving the bit c2 using only 2-input NAND gates.

(b) Consider a bit sequence ai, i 0 which has the property ai+4 =ai+1 ai.

i. To generate this sequence, which bits of the sequence needto be initialized?

ii. Design a logic circuit using flip-flops and NAND gates to

generate the above bit sequence.iii. For any given value of i, identify the points in the circuit at

which the values of ai, ai+1, . . . , ai+4 may be obtained.

E2. A long shunt D.C. generator supplies power to 55 lamps of rating200 W, 220 V each at 88% overall efficiency. The shunt and seriesfield resistances are 110 and 0.06 respectively, while the armatureresistance is 0.04 . Calculate

(a) the copper loss,

(b) the iron and friction loss,

(c) the torque of the rotor if the generator runs at 1000 r.p.m., and

(d) the efficiency of the generator if the shunt field resistance ischanged to 220 .

E3. 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, R0 = 100M, C0 = 10pf, = 10.

(b) Calculate the 3-dB cutoff frequency of the amplifier whenRi = 5K, RL = 1K, R0 = 100, C0 = 10pf, = 2.

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E4. Open-circuit and short-circuit tests are conducted on a 220/440 V,4.4 KVA, single phase transformer. The following readings are ob-tained.

Open Circuit test with voltage applied on low-voltage side:Voltage = 110 V,Current = 1.1A, andPower = 150 W.

Short Circuit test with voltage applied on high-voltage side:

Voltage = 24 V,Current = 8A, andPower = 64 W.

At 0.8 p.f. lagging, calculate

(i) the efficiency of the transformer at full-load, and

(ii) the output voltage at the secondary terminal when supplying full-load secondary current.

E5. 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;

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(d) the signal to noise ratio at the receiver end (assume that the

probability of bit error in the recovered PCM signal is zero).

E6. Consider a hypothetical three-input H gate defined as

H(A ,B,C ) = AC + BC.

Using minimum number ofH gates, realize the conventional BooleanNOT, two-input AND, two-input OR and two-input XOR. The twoinput variables may be denoted by A and B.

E7. A prism is made of wire mesh with each side having equal resistance

R. (See the figure given below.) A battery of 6 V and zero internalresistance is connected across E and F. If R is 0.5 , find the currentin the branch AD.

AB

CD

E

F

E8. (a) Consider a Discrete Memory-less System (DMS) with eight sym-bols x0, x1, . . . , x7 occurring with probabilities 0.005,0.005, 0.04, 0.10, 0.10, 0.15, 0.25 and 0.35 respectively.

i. Devise an optimum variable length code for this system.

ii. What is the entropy of the DMS in bits?

iii. What is the average number of bits required by this codingsystem?

[log2 3 = 1.58496, log2 5 = 2.32193, and log2 7 = 2.80736.]

(b) Define Mutual Information (MI) between two randomvariables X and Y when

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i. both X and Y are discrete,

ii. X is discrete and Y is continuous.iii. In case (ii) above, consider that X has two outcomes x1 = A

and x2 = A. If the conditional probability density functionp(y/xi) is Gaussian with mean xi and variance

2 for i = 1, 2,then find the MI of the system.

E9. A 5 KVA distribution transformer has a full load efficiency at unitypower factor of 95%. It is given that Copper and Iron losses of thetransformer are of equal proportion (1:1) at full load. Calculate theaverage efficiency per day and explain the steps involved in the com-putation. Assume that the transformer is loaded throughout 24 hourswith the following load pattern:

(a) 1/4 th of full load for 7 hours

(b) No load for 10 hours

(c) 1/2 of full load for 5 hours

(d) Full load for 2 hours

Also, assume that the power factor of the load for the whole day to beunity.

E10. (a) A sequence x[n] of length 8 has a Discrete Fourier Transform(DFT) X[k] as follows:

X[k] =

1; k = 0, 1, 2, 3k 2; k = 4, 5, 6, 70; otherwise.

x[n] is now up-sampled by a factor of 2 by inserting a zero betweenevery pair of samples of x[n], and appending a zero at the end toproduce a sequence y[n] of length 16. Sketch Y[k], the 16-pointDFT of y[n].

(b) A zero-mean, wide-sense stationary sequence with variance 2xpasses through a stable and causal all-pass filter with z-transform

H(z) = z1

1 z1 .

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Determine the range of values that can take. Also calculate

the power spectrum of the output, i.e., the Discrete Time FourierTransform (DTFT) of the auto-correlation sequence of the out-put.

E11. (a) Consider the following circuit where the amplifier has a voltagegain of A with zero phase shift. Let a ramp voltage Vi = t, (> 0) being a constant, be applied to the input. Derive anexpression for the output voltage Vo. Also, find the initial slopeof Vo as time t 0.

C

R2

2

VoVi

C

R

1

1

+

+Amplifier

(b) Calculate the gain and cutoff frequency of the following circuitwhen R1 = R2 = 2 k, C1 = C2 = 0.05 F, Roi = 10 k andRof = 50 k.

+

V

C C

R R

R R

1

1

2

21 Vo

oi of

opamp

E12. (a) Design a digital circuit to compare two three bit numbers A =

a2a1a0 and B = b2b1b0; the circuit should have three outputsindicating A = B, A < B and A > B. Give a gate-level diagramof this comparator circuit.

(b) Using two such comparator circuits and as few basic gates aspossible, design a digital circuit to compare two six bit numbers

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A = a5a4a3a2a1a0 and B = b5b4b3b2b1b0. This circuit should also

have only three outputs indicating A = B, A < B and A > B.

E13. A 22 KVA, 2200/220V two-winding transformer is converted to anauto-transformer with additive polarity.

(a) Calculate the percent increase in KVA of the auto-transformerwith respect to the original two-winding transformer.

(b) The auto-transformer has a full-load efficiency of 90% at unitypower factor. Calculate the efficiency of the auto-transformerwhen the load is reduced to half at the same power factor. Assumethat iron loss is 100 W.

E14. (a) Consider the following circuit, where a sinusoidal source V1 sin tand a DC source V2 are connected as shown. Assume V1 > V2.Let VAB be the voltage between A and B. The value of eachresistance is R.

A B

D

R

R

R

R

R

V2V1 sint

The voltage sources and the diode D are assumed as ideal. Drawthe waveform of VAB and justify your answer.

(b) Consider the following circuit. An a.c. source V1 sin t is con-nected to the non-inverting input of the OP-AMP. Draw the volt-age waveform VAB and justify your answer.

E15. A DC shunt motor of rated power 20 KW is connected to a supplyvoltage S = 220 V. The armature and field copper losses are 0.8 KW

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CD A

B

R1 R2

+

V1 sint

and 0.22 KW, respectively. The rotor speed is 1360 r.p.m. Assume

that all other losses are negligible.

(a) Calculate the efficiency, field and armature current at the ratedoperation.

(b) Suppose the above machine has to perform as a DC generatorproviding an output voltage of 120 V and a power of 12 KW.Calculate the necessary rotor speed and the required torque atnormal operation.

E16. In the following circuit, determine the current through the 4 resistor.It is assumed that the two batteries are ideal and L is an ideal currentsource.

2

2

1 3

4

10

6 A L

10 V

24 V

I

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E17. (a) Consider the following circuit. The emf E = 15V, R1 = 1.5K,

and R2 = 3.0K. The switch is closed for t < 0, and steady-state conditions are established. The switch is now thrown openat t = 0.

I2 I1

R2

I1I2

R1

S

L = 0.45H

E

A

B

i. Find the initial voltage emf0 across L just after t = 0. Whichend of the coil is at higher potential: A or B?

ii. Draw freehand graphs of the currents in R1 and R2 as afunction of time, indicating the values before and after t = 0.

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(b) For the circuit shown below, assume = hFE = 100,

VCE, sat = 0.2V, VBE, sat = 0.8V, VBE, active = 0.7V, andVBE, cutoff = 0.0. [All symbols follow the standard notations.]

5V

10V

49K

3K

V0

T

Re = 1K

i. Determine whether the transistor T is in the cutoff, satura-

tion or active region.ii. Find the minimum value of Re for which the transistor is inthe active region. [Assume ICO IB]

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(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 number 10011011 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 oflength li bits where 1 li k.

(a) Write a function prefix(S,T) in C programming language thattakes two binary strings S and T and returns 1 if S is a prefix ofT, else it returns 0. For example, prefix (001,00101) returns 1but prefix(010,00101) returns 0.

(b) Suppose we want to report all the pairs (i, j) for which Si is aprefix ofSj, (1 i = j n). How many times do we need to callthe prefix function described above?

(c) Write an O(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.)

C3. Let S = {x1, x2, . . . xn} be a set of n integers. A pair (xi, xj) (wherei = j) is said to be the closest pair if |xi xj | |xi xj|, for allpossible pairs (xi, xj), i

, j = 1, 2, . . . , n , i = j.(a) Describe a method for finding the closest pair among the set of

integers in S using O(n log2 n) comparisons.

(b) Now suggest an appropriate data structure for storing the ele-ments in S such that if a new element is inserted to the set S or

an already existing element is deleted from the set S, the currentclosest pair can be reported in O(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.

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C4. Consider the C program given below.

#include

#define n 5

#define m n*n

int **a;

main(){

int i, j, k;

void printarray();

a = (int **)calloc(n, sizeof(int *));

for(i = 0; i < n; i++)

*(a+i) = (int *)calloc(n, sizeof(int));

for(k=1, i=0, j=(n-1)/2; k

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C5. Let A be an n

n matrix such that for every 2

2 sub-matrix a b

c d

of A, if a < b then c d. Moreover, for any pair of rows i and j,if aik and ajl are the largest elements in i-th and j -th rows of A,respectively, then k l (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

(a) Write an algorithm for finding the maximum element in each row

of the matrix with time complexity O(n log n).(b) Establish its correctness, and justify the time complexity of the

proposed algorithm.

C6. 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?

C7. Consider the context-free grammar G = ({S, A}, {a, b}, S , P ),where P = {S AS,

S b,A SA,A a}.

Show that G is left-recursive. Write an equivalent grammar G free ofleft-recursion.

C8. (a) Five batch jobs P1, . . . , P 5 arrive almost at the same time. They

have estimated run times of 10, 6, 2, 4 and 8 ms, respectively.Their priorities are 3, 5, 2, 1 and 4 respectively, where 1 indicatesthe highest priority and 5 indicates the lowest. Determine theaverage turnaround and waiting time for the following schedulingalgorithms:

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

(ii) Priority scheduling.(b) The access time of a cache memory is 100 ns and that of main

memory 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?

C9. (a) A program P consisting 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 program P.

(ii) Compute the execution time needed by P 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 a

single 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.

C10. Consider a graph G (called an interval graph) whose nodes correspondto a set of intervals on the real line. The i-th interval is denoted by[i, i], where 0 i < 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?

C11. (a) Let R = (A ,B,C ,D ,E ,F) be a schema with the set F = {A BC, CD E, B D, E A} of functional dependencies.Suppose R is decomposed into two schemata R1 = (A ,B,C ) andR2 = (A ,D ,E ,F)

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

(iii) Identify all the candidate keys for R.

(iv) Decompose R into normalized sets of relations using 3NF.

(v) If a new dependency A F (multi-valued dependency) isintroduced, what would be the new set of normalized rela-tions?

(b) 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. Let V(C, r1)be 500, V(C, r2) be 1000, V(E, r2) be 50, and V(E, r3) be 150,

where V(X, r) denotes the number of distinct values that appearin relation r for attribute X . If r1 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.

C12. 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 a

translation-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%,

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time for a TLB lookup is negligible, time for a memory read is 100 nanoseconds, time to read a page from the swapping device into physical mem-

ory 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 anddistributed uniformly over the entire address space.

C13. (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 and P1, share the following

variables:char flag[2] = {0,0};

char turn = 0;

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

do {

flag[i] = 1;

while (flag[j])

if (turn == j) {

flag[i] = 0;

while (turn == j) {};

}...

CRITICAL SECTION

...

turn = j;

flag[i] = 0;

...

REMAINDER SECTION

...

} while (1);

Does this solution satisfy the required conditions?

C14. Construct an AVL tree of height 5 with minimum number of nodes.

C15. (a) Construct a finite state machine that accepts all the binary stringsin which the number of 1s and number of 0s are divisible by 3and 2, respectively.

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(b) Describe the language recognized by the following machine.

a

b

b

b

a,b

ba

a

a

(c) Consider the grammar E E + n|E n|n. For a sentencen + n n, find the handles in the right sentential form of thereductions.

C16. (a) Station A is sending data to station B over a full duplex errorfree channel. A sliding window protocol is being used for flowcontrol. The send and receive window sizes are 6 frames each.Each frame is 1200 bytes long and the transmission time for sucha frame is 70 S. Acknowledgment frames sent by B to A are verysmall and require negligible transmission time. The propagationdelay over the link is 300 S. What is the maximum achievable

throughput in this communication?(b) Consider a large number of hosts connected through a shared

communication channel. The hosts transmit whenever they haveany data. However, if two data packets try to occupy the channelat the same time, there will be a collision and both will be gar-bled. The hosts retransmit these packets that suffered collisions.Assume that the generation of new packets by a host is a Poissonprocess and is independent of other hosts. The total number oftransmissions (old and new packets combined) per packet timefollows Poisson distribution with mean 2.0 packets per packettime. Compute the throughput of the channel. (Packet time is

the amount of time needed to transmit a packet.)

C17. For a graph G = (V, E), a vertex coloring with x colors is defined as afunction C : V {1, 2, . . . , x}, such that C(u) = C(v) for every edge(u, v) E. The chromatic polynomial P(G, x) denotes the number of

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distinct vertex colorings for G with at most x colors. For example,

P(K2, x) = x(x 1).(a) Derive the chromatic polynomial for the graphs

i. K1,5

ii. C4

where Km,n denotes the complete bipartite graph of m and nvertices in the two partitions, and Cn is the simple cycle with nvertices.

(b) Consider Kn to be a complete graph with n vertices. ExpressP(Kn, x) as a recurrence relation and solve it.

C18. The diameter of a tree T = (V, E) is given by maxu,vV{

(u, v)

}, where

(u, v) is the shortest path distance (i.e., the length of the shortestpath) between vertices u and v. So, the diameter is the largest of allshortest path distances in the tree.

(a) Write pseudo-code for an efficient algorithm to compute the di-ameter of a given tree T.

(b) Analyze the time complexity of your algorithm.

(c) What is its space complexity?

(d) Clearly mention the data structure(s) used by your algorithm.

(e) A vertex c is called a center of a tree T if the distance from c toits most distant vertex is the minimum among all vertices in V.Write an algorithm to determine and report a center of the giventree T.

C19. (a) Write a context-free grammar for the language consisting of allstrings over {a, b} in which the number of as is not the same asthat of bs.

(b) Let the set D = {p | p is a polynomial over the single variable x,and p has a root which is an integer (positive or negative)}.

i. Design a Turing machine (TM) to accept D.ii. Can D be decided by a TM? Justify your answer.

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C20. Consider three parallel lines L1, L2 and L3. On each line Li, a set of

n points {pi1, pi2, . . . , pin} is placed.The objective is to identify a triplet of indices (k,,m) (if exists) suchthat a straight line can pass through p1k, p2 and p3m on L1, L2 andL3 respectively. [See the following two figures for a demonstration.]

L1

L2

L3

L1

L2

L3

(a) (b)

In Figure (a), there does not exist any triplet (k,,m) such that astraight line can pass through p1k, p2 and p3m. In Figure (b), thetriplet (3, 3, 4) is a solution since a straight line passes through p13, p23and p34.

Present an efficient algorithm for solving this problem. Justify itscorrectness and worst case time complexity.

[Full credit will be given if your algorithm is correct, and of worst casetime complexity O(n2).]

C21. Consider a bit sequence ai, i 0 which has the property ai+4 =ai+1 ai.(a) To generate this sequence, which bits of the sequence need to be

initialized?

(b) Design a logic circuit using flip-flops and NAND gates to generatethe above bit sequence.

(c) For any given value ofi, identify the points in the circuit at whichthe values of ai, ai+1, . . . , ai+4 may be obtained.

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C22. (a) How many distinct labeled spanning trees does a complete graph

of n vertices have? Give a formal argument toestablish your answer.

(b) A least expensive connection route among n houses needs to bedesigned for a cable-TV network. Consider the following algo-rithm A for finding a spanning tree.Algorithm AInput: G = (V, E)Output: Set of edges M ESort E in decreasing order of cost of edges;i 0;while i 0 such that (q, zi) = (q, zj) = q. Prove that(q, zgcd(i,j)) = q.

(c) Recall that a typical URL has the following form. It starts witha protocol specifier, followed by a colon (:), two forward slashes(/), a hostname, a dot (.), and then a domain name. This isfollowed by an optional path specifier. Some example URLs are

given below.

http://www.isical.ac.in/

http://www.isical.ac.in/deanoffice/results.html

http://www.isical.ac.in/deanoffice/results/jrf/

Assuming that A-Z, a-z, and 0-9 are the only characters that canbe used in a host/domain/file/directory name, write a regularexpression for URLs.

C27. (a) A Boolean function f is said to be positive unate if f can beexpressed in a form where all the variables appear in uncompli-mented form, and only the AND and OR Boolean operators are

used. For example, the function g1 = X1X2 + X2X3 is positiveunate but g2 = X1X2 + X2X3 is not. Consider the followingcircuit and determine which of the two functions f1 and f2 arepositive unate.

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X

Y

Y

f1

f2

(b) Consider a machine with four registers (one of which is the accu-mulator A) and the following instruction set.

LOAD R and STORE R are indirect memory operations thatload and store using the address stored in the register R.Thus, LOAD R loads the contents of memory[R] into A andSTORE R stores the contents of A in memory[R].

MOV R1 R2 copies the contents of register R1 into register R2. ADD R and SUB R operate on the accumulator and one other

register R, such that A = A op R.

LDC n stores the 7-bit constant n in the accumulator. BRA, BZ, and BNE are branch instructions, each taking a 5-bit

offset.

Design an instruction encoding scheme that allows each of theabove instructions (along with operands) to be encoded in 8 bits.

C28. Let a1 = 1, a2 = 2, and an = an1 + an2 + 1 for n > 2.

(a) Express 63 as a sum of distinct ais.

(b) Write an algorithm to express any positive integer k as a sum ofat most log2 k many distinct ais.

(c) Prove the correctness of your algorithm.