TRIBHUVAN UNIVERSITY
INSTITUTE OF SCIENCE AND TECHNOLOGY
2079 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 do you plot the function on graph? Determine whether the function $f(x)=x^2$ is injective, surjective or bijective with reasons. Solve the recurrence relation $a_{n}=6a_{n-1}+9a_{n-2}$ with initial conditions $a_{0}=1$ and $a_{1}=6$.
- 2A group of 8 scientist is composed of 5 chemist and 3 biologist. In how many ways can a committee of 5 be formed that has 3 chemist and 2 biologist? Using mathematical induction prove that $1^3+2^3+3^3+\dots+n^3=\frac{n^2(n+1)^2}{4}$ for $n\ge1$
- 3Show that the relation $R=\{(a,b): a-b \text{ is even}\}$ is an equivalence relation in the set of integers. Given the following transport network with the edges called with their capacities find all S-D cuts and their capacities and what the mimimum capacity? [diagram/image to be added manually]
Group B
Short Answer QuestionsAttempt any Eight questions.
- 4List any one example of tautology. Represent the following sentences into predicate logic. Not all employees are loyal. All students having good attitude are lovable.
- 5Prove that " If the product of two integers a and b is even then either a is even of b is even" using condition method.
- 6Use Chinese remainder Theorem to find the value of x such that $x=0\text{ MOD }2$, $x=2\text{ MOD }3$, $x=3\text{ MOD }5$
- 7Define bipartite graph with an example. State the necessary conditions for the graphs to be isomorphic.
- 8State Generalized Pigeonhole principle. Find the Ms
- 9Give the premise "If it rains or strike holds then the exam will be cancelled. If it doesn't rain then it will be sunny day. The exam was not cancelled. Show that it was sunny day".
- 10Find the value of -2 MOD 3 and 315 MOD 5. Illustrate an example to show the join operation between any two Boolean matrixes.
- 11Give an example of fallacy. State the necessary and sufficient conditions for a graph to have Euler path and Euler circuit.
- 12Find the GCD of 24 and 32 using Extended Euclidean algorithm.