11-20-2009, 01:23 PM
I have posted one Sample Questions for Joint Entrance Screening Test

Subject: Theoretical Computer Science.

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This is JAY.

My ID is- putquery@gmail.com

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Sample Questions for Joint Entrance Screening Test

Subject: Theoretical Computer Science

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1. Select the correct alternative in each of the following:

(a) Let a and b be positive integers such that a > b and a2 − b2 is a prime number.

Then a2 − b2 is equal to

(A) a − b (B) a + b © a × b (D) none of the above

(b) When is the following statement true? (A [ B) \ C = A \ C

(A) If A¯ \ B \ C = (B) If A \ B \ C =

© always (D) never

© If a fair die (with 6 faces) is cast twice, what is the probability that the two

numbers obtained differ by 2?

(A) 1/12 (B) 1/6 © 2/9 (D) 1/2

(d) T(n) = T(n/2) + 2; T(1) = 1

When n is a power of 2, the correct expression for T(n) is:

(A) 2(log n + 1) (B) 2 log n © log n + 1 (D) 2 log n + 1

2. Consider the following function, defined by a recursive program:

function AP(x,y: integer) returns integer;

if x = 0 then return y+1

else if y = 0 then return AP(x-1,1)

else return AP(x-1, AP(x,y-1))

(a) Show that on all nonnegative arguments x and y, the function AP terminates.

(b) Show that for any x, AP(x, y) > y.

3. How many subsets of even cardinality does an n-element set have ? Justify answer.

4. A tournament is a directed graph in which there is exactly one directed edge between

every pair of vertices. Let Tn be a tournament on n vertices.

(a) Use induction to prove the following statement:

Tn has a directed hamiltonian path (a directed path that visits all vertices).

(b) Describe an algorithm that finds a directed hamiltonian path in a given tourna-

ment. Do not write whole programs; pseudocode, or a simple description of the

steps in the algorithm, will suffice. What is the worst case time complexity of

your algorithm?

5. Describe two different data structures to represent a graph. For each such representa-

tion, specify a simple property about the graph that can be more efficiently checked

in that representation than in the other representation. Indicate the worst case time

required for verifying both of your properties in either representation.

6. Two gamblers have an argument. The first one claims that if a fair coin is tossed

repeatedly, getting two consecutive heads is very unlikely. The second, naturally, is

denying this.

They decide to settle this by an actual trial; if, within n coin tosses, no two consecutive

heads turn up, the first gambler wins.

(a) What value of n should the second gambler insist on to have more than a 50%

chance of winning?

(b) In general, let P(n) denote the probability that two consecutive heads show up

within n trials. Write a recurrence relation for P(n).

© Implicit in the second gambler’s stand is the claim that for all sufficiently large n,

there is a good chance of getting two consecutive heads in n trials; i.e. P(n) > 1/2.

In the first part of this question, one such n has been demonstrated. What

happens for larger values of n? Is it true that P(n) only increases with n? Justify

your answer.

7. Consider the following program:

function mu(a,b:integer) returns integer;

var i,y: integer;

begin

---------P----------

i = 0; y = 0;

while (i < a) do

begin --------Q------------

y := y + b ;

i = i + 1

end

return y

end

Write a condition P such that the program terminates, and a condition Q which is

true whenever program execution reaches the place marked Q above.

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