ECE-103-1005-Final_exam

ECE-103-1005-Final_exam - (9103/2 CE 193 UNIVERSITY OF...

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Unformatted text preview: (9103/2 CE 193 UNIVERSITY OF WATERLOO FINAL EXAMINATION SPRING TERM 2000 E85013 103 NAME: Name: (Signature) Id.#: K-Section: Course Number: E&CE 103 Course Title: Discrete Mathematics Instructor: Professor J. Cheriyan Date of Exam: July 31, 2000 Time Period: 9:00 am — 12:00 noon Number of Exam Pages: 9 pages (including this cover sheet) Exam Type: Closed Book Additional Instructions 1. Please print your name, student identification number and K—section as indicated above. 2. This examination has 9 pages, including this cover page. Make sure you have a complete copy. 3. If you require more space to present your solution, please use the back of the previous page. Indicate clearly where your solution continues. 4. No calculators are allowed. 5. When solving a numerical example, show all of your work. Ma mad Ma ma fl-Ifl -|fl E&CE 103 2 1. (a) Use the extended Euclidean algorithm to find integers :r and y such that 11m + 429g _ gcd(1170,429). (b) Find one solution to the following linear integer equation, and then write down the complete set of solutions: 11m + 429g 2 100-gcd(1170,429). E&CE 103 3 [10] 2. Solve the following simultaneous linear congruences and find the smallest positive in- teger solution: x E 13 (mod 55) :v E 23 (mod 31). [5] 3. Use the second principle of mathematical induction to show the following: If an is defined recursively by an : 3an_1—2an_2 VneZ,n > 3 with the initial conditions a1 : 37 a2 = 5, then an 2 2" + 1 for all positive integers n. E&CE 103 4 4. (a) Solve the recurrence relation VneZ,n > 2 an : an—l + 6an—23’ with the initial conditions a0 = 1, a1 = 23. (b) There are n lines drawn in the plane such that every pair of lines has an inter— section and no three of the lines have a common point. Let Z)n be the number of regions in the plane determined by these n lines. Find a recurrence relation for the sequence be, b1, b2, . . . , bm . . .. Explain your solution in brief. What are the initial conditions for the recurrence relation? (Do not solve the recurrence relation.) E&CE 103 5 [10] 5. Let $1, 1172, .. ., xn be a sequence of n integers such that for each 3 : 1, 2, ..., n I ij i 0 (modn). #1 Prove: There exists a consecutive sequence of one or more integers act-424.1, . . . ,ka i+k (where k 2 0, and l g i g i + k S 72) whose sum, E xj, is an integer multiple of n. j=i E&CE 103 6 [2] 6. (a) Find the number of difierent permutations of a_ll of the letters in the word GREENERY. [2] (b) Find the number of different permutations using exactly seven of the letters in the word GREENERY. [3] (c) Are the answers to parts (a) and (b) the same? Explain in brief. (For both parts (a),(b) you may give your answer in terms of binomial coefficients or factorials.) [2] 7. (a) Define What is meant by a derangement. [2] (b) Write down the number of derangements of 5 objects. [2] (C) Find the number of permutations of 8 objects such that exactly three are in their original positions. E&CE 103 7 [3] 8. (a) State without proof the number of solutions to the equation $1+x2+x3+x4=30 where 11:1, $2, $3, 3:4 are nonnegative integers. [10] (b) Find the number of integers between 1 and 9999 whose digits sum to 30. (For both parts (a),(b) you may give your answer in terms of binomial coefficients or factorials.) E&CE 103 8 9. (a) Are the graphs 01 2 (V1, E1) and 02 2 (V2, E2) in Figure 1 isomorphic? If the graphs are isomorphic, give an isomorphism f : V1 —-> V2, otherwise give reasons why the graphs are not isomorphic. (b) Does there exist a simple graph with six vertices and the following degree se— quence? If yes, draw such a graph, and if not, briefly explain why there is no such graph. (i) 5, 5, 4, 3, 2, 2 (ii) 3, 3, 3, 2, 2, 1 (iii) 4, 4, 4, 2, 1, 1 [6] E&CE 103 9 10. (a) Define what is meant by (i) a simple graph, (ii) a cut edge (or of a connected graph, (iii) an Euler circuit of a graph. (b) Prove or disprove: Let G be a simple graph that is connected. If G has an Euler circuit, then G has w cut edge. (c) Is the converse of the statement in part (b) true? That is, if a connected simple graph has rm cut edge, then it has an Euler circuit. Give a proof (if true) or a counterexample (if false). ...
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ECE-103-1005-Final_exam - (9103/2 CE 193 UNIVERSITY OF...

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