TRIBHUVAN UNIVERSITY
INSTITUTE OF SCIENCE AND TECHNOLOGY
2080 B.S.
Candidates are required to give their answers in their own words as far as practicable.
Group A
Long Answer QuestionsAttempt any Two questions.
- 1How can you use mathematical induction to prove statements? Use mathematical induction to show that the sum of first n positive integers is n(n+1)/2. (4+6)
- 2Explain linear homogeneous recurrence relation with constant coefficients. What is the solution of the recurrence relation a_n = 6a_(n-1) - 9a_(n-2) with initial conditions a_0 = 1 and a_1 = 6? (2+8)
- 3What is shortest path problem? Use Dijkstra's shortest path algorithm to find the shortest path between the vertices a and z in the weighted graph given below. (2+8)
Group B
Short Answer QuestionsAttempt any Eight questions.
- 4Let us assume that R be a relation on the set of ordered pairs of positive integers such that ((a,b),(c,d)) ∈ R if and only if ad=bc. Is R an equivalence relation? (5)
- 5Define function. Let f1 and f2 be functions from R to R such that f1(x) = x^2 and f2(x) = x - x^2. What are the functions f1+f2 and f1*f2? (2+3)
- 6Explain fuzzy set with example. How do you find complement of a fuzzy set? (2.5+2.5)
- 7What is congruent modulo? Determine whether 37 is congruent to 3 modulo 7 and whether -29 is congruent to 5 modulo 17. (2+1.5+1.5)
- 8Define network flow with example. What are saturated edge, unsaturated edge and slack value? (2+3)
- 9Give an example of tautology and contradiction. Show that implication and contrapositive are equivalence. (2+3)
- 10What is direct proof? Give a direct proof that if m and n are both perfect squares, then mn is also a perfect square. (1.5+3.5)
- 11What is product rule? How many strings are there of four lowercase letters that have the letter x in them? (1.5+3.5)
- 12Explain matrix representation of relations with example. (5)