BSc (Hons) Mathematics. Examinations for / Semester 2

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BSc (Hons) Mathematics Cohort: BM/13/FT Examinations for 2014-2015 / Semester 2 MODULE: FINITE MATHEMATICS MODULE CODE: Duration: 3 Hours Instructions to Candidates: 1. Answer ALL questions. 2. Questions may be answered in any order but your answers must show the question number clearly.

1 BSc (Hons) Mathematics Cohort: BM/13/FT Examinations for / Semester 2 MODULE: FINITE MATHEMATICS MODULE CODE: Duration: 3 Hours Instructions to Candidates: 1. Answer ALL questions. 2. Questions may be answered in any order but your answers must show the question number clearly. 3. All workings should be clearly shown. 4. Always start a new question on a fresh page. 5. All questions carry equal marks. 6. Total marks 100. This question paper contains 4 questions and 8 pages. Page 1 of 8

2 ANSWER ALL QUESTIONS Question 1: (25 Marks) (a) (i) Using arithmetic modulo two, show that the system of equations has no integer solutions. 6a 5b = 4, 2a + 3b = 3, (ii) Determine the truth value of the statement x y U, x 2 < y + 1 where U = {1, 2, 3} is the universal set. (2+1 marks) (b) Show that n i p Θ ( n p+1), p 1. i=1 (4 marks) (c) Consider the running time of an algorithm defined by the recurrence relation T (1) = 1, T (n) = 4T ( n/2 ) + n 2, n 2. (i) By first unfolding the recurrence by iterated substitution, show that T (n) = n 2 + n 2 log 2 n, and prove it by mathematical induction. (ii) Using the Master theorem, prove that T (n) Θ ( n 2 log 2 n ). (7+3 marks) Page 2 of 8

3 (d) Let a = and b = Using the Euclidean algorithm, (i) find d = gcd(a, b), the greatest common divisor of a and b, (ii) find the integers x and y such that d = xa + yb. (2+2 marks) (e) Let A k denote the set of integers in {1, 2,..., 1000} that are divisible by k, for a positive integer k and 1000 A k =, k where y denotes the greatest integer less than or equal to y and let A k A l = A lcm(k, l), where lcm(k, l) denotes the least common multiple of k and l. Using the principle of inclusion and exclusion, find the number of positive integers less than or equal to 1000 that are divisible by 7, 10 and 15. (4 marks) Page 3 of 8

4 Question 2: (25 Marks) (a) Solve the recurrence relation with a 0 = 6 and a 1 = 10. a n + 4a n 1 + 3a n 2 = 16n + 20, n 2, (10 marks) (b) Consider the following recurrence J(1) = 1, J(2n) = 2J(n) 1, for n 1, J(2n + 1) = 2J(n) + 1, for n 1. Evaluate J(300). (4 marks) (c) Consider the Fibonacci sequence given by F 0 = 1, F 1 = 1 and F n+1 = F n + F n 1, for n 1. Let G(z) = F n z n, n=0 denote the generating function for the Fibonacci sequence. (i) Show that G(z) = 1 (1 φz)(1 ˆφz), where φ + ˆφ = 1 and φ ˆφ = 1. (ii) Hence, deduce that F n = 1 (φ φ ˆφ n+1 ˆφ ) n+1. (5+6 marks) Page 4 of 8

5 Question 3: (25 Marks) (a) (i) Draw the resulting trees after inserting the following: showing each steps clearly. 50, 20, 1, 10, 40, 30, 60, 70, 5, 80, 100, (ii) Figure 1 shows a tree. Draw the resulting tree after the insertion of H and X. F S B B D I J O T Figure 1: tree. (6+4 marks) (b) If X = {α, β, γ, δ} and G = {g 1, g 2, g 3, g 4 } is a group of permutations of X, where find the cyclic index of G. g 1 = (α)(β)(γ)(δ), g 2 = (α β)(γ)(δ), g 3 = (α)(β)(γ δ), g 4 = (α β)(γ δ), (4 marks) Page 5 of 8

6 (c) The food trays in a university cafeteria are rectangular and divided into 4 equal rectangular compartments. Find the number of distinguishable ways of filling a tray with 4 foods if the long dimension of the tray must be parallel to the longer sides of the rectangular table edge. (5 marks) (d) Using Polya s second enumeration theorem, find the number of distinguishable bracelets that can be made consisting of 7 beads, of which 2 beads are red, 3 beads are blue and 2 beads are yellow when only rotational symmetries are considered. (6 marks) Page 6 of 8

7 Question 4: (25 Marks) (a) Consider a connected undirected graph G = (v, e) and a weight function f : e R. (i) Describe what is meant by a spanning tree of G. (ii) Describe Prim s algorithm for finding a minimum spanning tree. (iii) The Central Water Authority wants to lay on a water supply to caravans positioned in different regions of the country specifically at Albion, Bagatelle, Curepipe, Dagotière, Ebene, Moka and Plaine Magnien, as shown in Figure 2, with distances in kilometres between them. Choosing Albion as the starting point, use Prim s algorithm to determine how the caravans should be connected so that the total length of pipe required is a minimum. Albion 50 Ebene 40 Plaine Magnien Curepipe Moka Bagatelle 60 Dagotiere Figure 2: Locations of caravans. ( marks) Page 7 of 8

8 (b) A complicated business document, currently written in English is to be translated into each of the other community languages. Because it is harder to find translators for some languages than for others, some translations are more expensive than others. The costs in $ are shown below: From/ To Chinese Danish English French Greek Hindi Italian Portuguese Spanish Chinese Danish English French Greek Hindi Italian Portuguese Spanish Use Kruskal s algorithm to decide which translations should be made so as to obtain a version of the business document in each language at minimum total cost and hence find the minimal total cost. ***END OF QUESTION PAPER*** (9 marks) Page 8 of 8

Mathematics for Computer Science MATH 1206C [P.T.O] Page 1 of 5

Mathematics for Computer Science MATH 1206C [P.T.O] Page 1 of 5 x Page 1 of 5 Question 1: (25 Marks) ANSWER ALL QUESTIONS (a) Prove by mathematical induction that for n 1, n r=1 (b) State and prove the Well-Ordering Principle. r 3 = n2 4 (n + 1)2. (c) Using proof by