A 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 $1^3 + 2^3 + 3^3 + ... + n^3 = (n^2(n+1)^2/4)$ for $n \geq 1$. [10]

State Generalized Pigeonhole principle. Find the MST from following graph using Kruskal Algorithm. [5]

How 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 $a_0 = 1, a_1 = 6$. [10]

Use Chinese remainder Theorem to find the value of x such that x = 0 (MOD 2), x = 2 (MOD 3), x = 3 (MOD 5). [5]

Find the value of -2 MOD 3 and $3^{15}$ MOD 5. Illustrate an example to show the join operation between any two Boolean matrices. [5]

Find the GCD of 24 and 32 using Extended Euclidean algorithm. [5]

List any one example of tautology. Represent the following sentences into predicate logic. a. Not all employees are loyal. b. All students having good attitude are lovable. [5]

Prove that 'If the product of two integers a and b is even then either a is even or b is even' using condition method. [5]

Give 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'. [5]

Show that the relation R = {(a, b): |a-b| 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 minimum capacity? [10]

Define bipartite graph with an example. State the necessary conditions for the graphs to be isomorphic. [5]

Give an example of fallacy. State the necessary and sufficient conditions for a graph to have Euler path and Euler circuit. [5]