BEng (Hons) Telecommunications. Examinations for / Semester 2

Transcription

1 BEng (Hons) Telecommunications Cohort: BTEL/14B/FT Examinations for / Semester 2 MODULE: NUMBERS, LOGICS AND GRAPHS THEORIES 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 9 pages Page 1 of 9

2 ANSWER ALL QUESTIONS Question 1: (25 Marks) (a) An algorithm to compute the greatest common divisor (gcd) of two given positive integers is given by Algorithm: The Euclidean Algorithm To find the gcd of two positive integers a and b: A1: If a < b, swap the values of a and b A2: Find the remainder, r, of dividing a by b A3: a takes the value of b and b takes the value of r A4: Repeat A2-A3 until r = 0 A5: Then b is the gcd of the two original numbers Let a = and b = 8316 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 (iii) Assuming that you have the following Matlab R code: function value = GCD(a,b) if b==0, value=a; return else if a<b % Supply Matlab statements here to swap a and b end % Supply Matlab statement here to compute the remainder % Supply Matlab statements here to repeatedly find % the gcd until the remainder becomes zero end value = b Supply the additional Matlab R statements to implement the Euclidean algorithm Page 2 of 9

3 (iv) Trace the values of the variables a, b and r for finding the gcd(10920, 8316) for each iteration using the table given below and verify your answer to part (a)(i) a b r r > 0? (v) Using your answer to part (a)(i), find the least common multiple (lcm) of a and b and give the single Matlab R statement that will use the GCD function definition in part (a)(iii) to calculate the lcm of a and b ( marks) (b) (i) Let A = {1, 2,, 9, 10} Determine its truth value of the following statement ( x A)( y A)(x + y < 14), giving a reason for your answer (ii) Find a counterexample for the statement x U, x + 3 7, where U = {3, 5, 7, 9} is the universal set (iii) Give the truth table for p (p q) (1+1+2 marks) (c) Prove by mathematical induction that the statement n j(j!) = (n + 1)! 1, j=1 holds for n 1 (6 marks) Page 3 of 9

4 Question 2: (25 Marks) (a) (i) Write a function in Matlab R to calculate n! using the concept of recursion and the following function prototype: function factvalue = factorial(n) (ii) The permutation function P (n, k) = n (n 1) (n k + 1) is the product of the k largest integers that are n Implement the permutation function using for loops inside the function with prototype function permutevalue = permute(n, k) (iii) Hence, give the Matlab R statement that will use the above two function definitions to calculate the combination C(n, k) = n (n 1) (n k + 1), 1 2 k assuming that n and k have already been assigned values such that k n (iv) Show that ( n 1 ) ( n ) ( n ) = n 3 ( marks) (b) Solve the recurrence relation with a 1 = 4 and a 2 = 15 a n 5a n 1 + 6a n 2 = 2n 9, n 3, (9 marks) (c) Using the principle of pigeonhole, find the minimum number of students in a class to be sure that three of them are born in the same month (3 marks) Page 4 of 9

5 (d) (i) An algorithm for finding the square root of a positive real number b is given by : x 0 = 1, x n+1 = 1 [ x n + b ], 2 x n n = 0, 1, 2, Given that the following code excerpt has already been written, supply the necessary Matlab R statements that implement the above recursive function and output the result when x 2 n b < 10 7 function ans = sqroot(x, b) if % Supply the statements for the stopping criteria else % Supply the statement for the recursive step end (ii) Give the Matlab R command for calculating the square root of 5 (2+1 marks) Page 5 of 9

6 Question 3: (25 Marks) (a) Consider the following binary search tree: Figure 1: Binary Search Tree List down the sequence of elements encountered when the tree is traversed in (i) pre-order, (ii) in-order, (iii) post-order (3 marks) (b) (i) Draw the resulting top-down trees after inserting the following: 70, 40, 3, 30, 60, 50, 80, 90, 25, 100, 120, in the given order, into an initially empty tree showing each steps clearly Page 6 of 9

7 (ii) Figure 2 shows a tree Draw the resulting tree after the insertion of H and X F S B B D I J O T Figure 2: tree (9+4 marks) (c) Find a, b, c and d such that r 3 (r + 1) = ap (r, 1) + bp (r, 2) + cp (r, 3) + dp (r, 4), where P (r, k) denotes the permutation function Deduce that the sequence {a r } 0 given by a r = r 3 (r + 1) has the generating function g(x) = 8x3 + 14x 2 + 2x (1 x) 5 (9 marks) Page 7 of 9

8 Question 4: (25 Marks) (a) A cable television company wants to provide a service to each of five towns in Mauritius namely Curepipe, Port-Louis, Rose-Hill, Quatre-Bornes and Vacoas, and for this purpose the towns must be linked (directly or indirectly) by cables For reasons of economy, the company is anxious to find the layout that will minimise the length of cable needed Figure 3 shows in the form of a graph, the distances in miles between the towns 7 Port-Louis Curepipe Vacoas 5 4 Quatre- Bornes Rose-Hill Figure 3: Graph for Question 4(a) Choosing Port-Louis as the starting point, apply Prim s algorithm to determine the minimum spanning tree for the graph in Figure 3 and hence calculate the minimum length of cable needed (8 marks) Page 8 of 9

9 (b) Write the adjacency matrix for the graph in Figure 4 and determine whether it is symmetric or not Figure 4: Graph for Question 4(b) (4 marks) (c) A group of friends wants to set up a message system so that anyone of them can communicate with any of the others either directly or indirectly or via others in the group from their houses A, B, C, D, E and F The cost in dollar ($) of setting up the system between the pair of houses for which connections are possible are shown in Table 1 (i) House A B C D E F A B C D E F Table 1: Costs between pair of houses Give the pseudocode for Kruskal s algorithm (ii) Use Kruskal s algorithm to decide where they should make the links so that the total cost of setting the system will be as small as possible (iii) Hence find the minimal cost (5+7+1 marks) ***END OF QUESTION PAPER*** Page 9 of 9