Q. Consider two languages L1 and L2 each on the alphabet ∑. Let f : ∑ → ∑ be a polynomial time computable bijection such that (∀ x) [x ∈ L1 iff f(x) ∈ L2]. Further, let f^-1 be also polynomial time computable. Which of the following CANNOT be true?

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Q. In a permutation a1.....an of n distinct integers, an inversion is a pair (ai, aj) such that i < j and ai > aj. If all permutations are equally likely, what is the expected number of inversions in a randomly chosen permutation of 1.....n ?

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Q. In a permutation a1.....an of n distinct integers, an inversion is a pair (ai, aj) such that i < j and ai > aj. What would be the worst case time complexity of the Insertion Sort algorithm, if the inputs are restricted to permutations of 1.....n with at most n inversions?

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Q. The cube root of a natural number n is defined as the largest natural number m such that m^3 ≤ n. The complexity of computing the cube root of n (n is represented in binary notation) is:

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Q. The following resolution rule is used in logic programming:

Derive clause (P v Q) from clauses (P v R), (Q v ¬ R)

Which of the following statements related to this rule is FALSE?

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Q. Identify the correct translation into logical notation of the following assertion.

"Some boys in the class are taller than all the girls"

Note : taller(x,y) is true if x is taller than y.

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Q. If a fair coin is tossed four times. What is the probability that two heads and two tails will result?

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Q. The number of different n × n symmetric matrices with each element being either 0 or 1 is:
(Note: power(2, x) is same as 2x)

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Q. Let A, B, C, D be n × n matrices, each with non-­zero determinant. If ABCD = 1, then B-1 is

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Q. What is the result of evaluating the following two expressions using three-digit floating point arithmetic with rounding?

(113. + -111.) + 7.51
113. + (-111. + 7.51)

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Q. The tightest lower bound on the number of comparisons, in the worst case, for comparison-based sorting is of the order of

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Q. The problems 3-SAT and 2-SAT are

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Q. The following propositional statement is

(P → (Q v R)) → ((P ^ Q) → R)

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Q. How many solutions does the following system of linear equations have ?

-x + 5y = -1
x - y = 2
x + 3y = 3

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Q. An examination paper has 150 multiple-choice questions of one mark each, with each question having four choices. Each incorrect answer fetches -0.25 mark. Suppose 1000 students choose all their answers randomly with uniform probability. The sum total of the expected marks obtained by all these students is:

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Q. Mala has a colouring book in which each English letter is drawn two times. She wants to paint each of these 52 prints with one of k colours, such that the colour-pairs used to colour any two letters are different. Both prints of a letter can also be coloured with the same colour. What is the minimum value of k that satisfies this requirement ?

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Q. In an M'N matrix such that all non-zero entries are covered in a rows and b columns. Then the maximum number of non-zero entries, such that no two are on the same row or column, is

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Q. Two n bit binary strings, S1 and S2, are chosen randomly with uniform probability. The probability that the Hamming distance between these strings (the number of bit positions where the two strings differ) is equal to d is

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Q. A point is randomly selected with uniform probability in the X-Y plane within the rectangle with corners at (0,0), (1,0), (1,2) and (0,2). If p is the length of the position vector of the point, the expected value of p^2 is

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Q. Let G1 = (V, E1) and G2 = (V, E2) be connected graphs on the same vertex set V with more than two vertices. If G1 ∩ G2 = (V, E1 ∩ E2) is not a connected graph, then the graph G1 U G2 = (V, E1 U E2)

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