Give me question paper for Joint Entrance Screening Test Theoretical Science examination in PDF file format ??

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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 (C) 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 =
(C) always (D) never
(c) 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 (C) 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 (C) 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?