TRIBHUVAN UNIVERSITY
INSTITUTE OF SCIENCE AND TECHNOLOGY
2081 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.
- 1List any four rules of inferences. Using direct and indirect proof show that, for any real number x, If $x^{3}-7x^{2}+x-7=0$ then $x=7$
- 2What are the necessary and sufficient conditions for graphs to have Euler path only and Euler circuit? Let R be a relation defined on set of natural numbers N, such that $a, b \in N$, $aRb \leftrightarrow \frac{a}{b} = 2^{k}$ where $k \in \{0, 1, 2, \dots\}$ Show that R is a partial ordering relation on N.
- 3Determine whether the function $f(x)=x-1$ from set of integers to the set of integers is injective, surjective or bijective. Write the types of fuzzy set operations with their definitions. Give the floor and ceiling value of 1.2 and -1.2.
Group B
Short Answer QuestionsAttempt any Eight questions.
- 4Find the GCD of 12 and 16 using Extended Euclidean Algorithm.
- 5Using mathematical induction show that $2+5+8+\dots+(3n-1)=\frac{n(3n+1)}{2}$
- 6State sum rule and product rule. If 26 integers are chosen from the set of consecutive integers {1, 2, 3,...... 50}, prove that there are sure to be two numbers so that one is multiple of the other.
- 7Discuss about meet and join operation between Boolean matrixes. What are the values of 5 MOD 7 and -5 MOD 7.
- 8Define chromatic number. How does Kruskal's algorithm find Minimum Spanning Tree?
- 9Solve the recurrence relation $a_{n}=a_{n-1}+2a_{n-2}$ with initial conditions with $a_{0}=2$ and $a_{1}=7.$
- 10What is network flow? Give an example of saturated edge, unsaturated edge and slack.
- 11When do we use permutation rather than combination? How many 5 digit numbers can be generated using the digits each number starts with 98 and no digit appears more than once?
- 12Define subset and power set. How do you prove correctness of recursive algorithm using Induction? Illustrate with an example.