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2003 Course Design & Analysis of Algorithms

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Total No. of Questions : 12] [Total No. of Pages : 3 [3764]-411 P1541 B.E. (Computer) DESIGN AND ANALYSIS OF ALGORITHMS (2003 Course) Time : 3 Hours] [Max. Marks : 100 Instructions to candidates : 1) Answer THREE questions from each section. 2) Answers to the TWO sections should be written in SEPARATE answer books. 3) Figures to the right indicate full marks. 4) Assume suitable data, if necessary. SECTION - I Q1) a) b) Prove by generalized mathematical induction that every positive integer can be expressed as a product of prime numbers . [8] Consider the recurrence: T(n) = O(n) T(l) = (l) Show that above recurrence is asymptotically bound by (n). c) [8] State whether the following function is e CORRECT or INCORRECT and justify your answer. [2] 3n+ 2 = O(n) OR Q2) a) Prove by contradiction : There exist two irrational numbers X and Y such that Xy is rational. [8] b) Prove by mathematical induction that the sum of the cubes of the first n positive integers is equal to the square of the sum of these integers. [8] c) State whether the following function is e CORRECT or INCORRECT and justify your answer. [2] 10n2+ 4n + 2 = O(n2) Q3) a) Write an algorithm for sorting n numbers using Quick sort method. Determine its time complexity. [8] P.T.O.

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