No. of Printed Pages : 8 I MTE-13 n -1:CF;7 BACHELOR'S DEGREE PROGRAMME (BDP) Term-End Examination June, 2016 ELECTIVE COURSE : MATHEMATICS MTE-1 3 : DISCRETE MATHEMATICS Time : 2 hours Maximum Marks : 50 (VVeightage : 70%) Note : Question no. 1 is compulsory. Answer any four questions from questions no. 2 to 7. Use of calculators is not allowed. 1. Which of the following statements are true and which are false? Justify your answer. 5x2=10 (a) (b) (c) (d) (e) The statement 'If x2 + y2 = 0 for any two positive integers x and y, then xy is prime,' is true. For any graph G, A(G) = x(g). The recurrence relation an = 4 an_i + n3 is of order 1 and degree 3. The number of possible outcomes obtained by casting a die 9 times is C(9, 7). There is a graph G with degree sequence {1, 2, 3, 3, 4, 5). MTE-13 1 P.T.O.
2. (a) Prove that the following statements are equivalent for a graph G : 4 (i) G is a tree. (ii) Any two vertices in G are connected by a unique path. (b) Verify whether [p - (p v q)] p is a tautology using a truth table. (c) Using the principle of inclusion-exclusion find the number of integer solutions of the equation x + y + z = 18, with 0 < x 6, 0 < y 7, 0 < z 8. 3 3. (a) Show, by Mathematical induction that, (n + 1)2 < 2n2, V n > 3. 4 (b) (c) Solve the recurrence relation an - 10 an_i + 31 an_2-30 an_3 = 0, n 3 with the initial conditions ao = 0, al = 1, a2 = -1. 4 There are 38 different time periods during which classes at a university can be scheduled. If there are 677 different classes, how many different rooms will be needed? 2 4. (a) If a five digit number is chosen at random, what is the probability that the product of digits is 28? 3 (b) (i) Define the hypercube Q n. (ii) Draw Q3. (iii) Is Q3 Eulerian? Justify your answer. (iv) Is Q3 Hamiltonian? Justify your answer. 4 MTE-13 2
(c) Solve the recurrence relation, an = an_i +,3 (n - 1), ao = 1. 3 5. (a) Check the validity of the following argument If I do not get 96%, I will not get a scholarship. If I work hard, I will get 96%. I worked hard. Therefore I got a scholarship. (b) Consider the following graphs : MTE-13 (i) (ii) Is G connected? If not, show that G is a disjoint union of its components. Find the length of the longest path in G. (iii) Give the longest cycle in G. (iv) Is G bipartite? Justify. 5 3 P.T.O.
6. (a) Using the generating functions approach, find the number of integer solutions to the linear equation, x 1 + x2 + x3 = n, where 1 xi 3, 1 x2 1 and x3 3. 6 (b) Show that the number of partitions of 10 into distinct parts (integers) is equal to the number of partitions of 10 into odd parts. 4 7. (a) An English word is called a palindrome if it reads the same whether read from left to right or from right to left, ROTOR, for example. Let an be the number of English words of length n, not necessarily meaningful, which are palindromes. We consider a single letter a palindrome. (i) What are al and a2? (ii) Set up a recurrence for a n. (iii) Check that an = V26 {(1 + 2126) ( 1)n (1-1 2 is a solution to the recurrence. (iv) Find a3 using the above expression for an- 5 (b) Express the polynomial x4 + x2 + x in terms of the factorial polynomials [x] 4, [x] 3, etc. 3 (c) Show that Kim, 101 is not Hamiltonian. 2 MTE-13 4
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