Let ({p,q{,*) be a semigroup where p * q = q. Show that:

 (a) p * q = q * p and
 (b) q * q = q 

Show that proposition C is a logical consequence of the formula

     A ∧ (A →(B ∨ C)) ∧ (B → ~A)  

Using truth tables.

The less-than relation, <, on reals is

Let A and B be sets with cardinalities m and n respectively. The number of one-one mappings (injections) from A to B, when m < n, is:

The proposition p ∧(~p ∨ q) is:

Let S be an infinite set and S₁ ..., S_n be sets such that S₁ ∪ S₂ ∪ ... ∪ S_n = S. Then,

Let A be a finite set of size n. The number of elements in the power set of A × A is:

The Laplace transform of the periodic function f(t) described by the curve below, i.e.,

is _________

Given and S the surface of a unit cube with one corner at the origin and edges parallel to the coordinate axes, the value of integral

The differential equation yⁿ + y = 0 is subjected to the boundary conditions.

    y (0) = 0        y(λ) = 0         

In order that the equation has non-trivial solution(s), the general value of λ is __________

For X = 4.0, the value of I in the FORTRAN 77 statement
1 = -2**2 + 5.0*X/X*3 + 3/4 is _______

The value of the double integral is

If the matrix A⁴, calculated by the use of Cayley-Hamilton theorem or otherwise, is _________

If the linear velocity is given by

The angular velocity at the point (1, 1, -1) is ________

Given the differential equation, y′ = x − y with the initial condition y(0) = 0. The value of y(0.1) calculated numerically upto the third place of decimal by the second order Runga-Kutta method with step size h = 0.1 is ________

The radius of convergence of the power series

The function f(x,y) = x²y - 3xy + 2y + x has

Which of the following improper integrals is (are) convergent?

The differential equation
d²y/dx² + dy/dx + siny = 0 is:

Simpson’s rule for integration gives exact result when f(x) is a polynomial of degree

Fourier series of the periodic function (period 2π) defined by

But putting x = π, we get the sum of the series.

The eigen vector(s) of the matrix

is (are)

Every element a of some ring (R,+,0) satisfies the equation aoa = a.
Decide whether or not the ring is commutative.

A 3-ary tree is a tree in which every internal node has exactly three children. Use induction to prove that the number of leaves in a 3-ary tree with n interval nodes is 2(n-1)+3.

Let p and q be propositions. Using only the truth table decide whether p ⇔ q does not imply p → q is true or false.

(a) Let * be a Boolean operation defined as
If C = A * B then evaluate and fill in the blanks:
(i) A * A = _______
(ii) C * A = _______
(b) Solve the following boolean equations for the values of A, B and C:

Find the inverse of the matrix

Match the following items

The Hasse diagrams of all the lattices with up to four elements are __________ (write all the relevant Hasse diagrams).

Let A, B and C be independent events which occur with probabilities 0.8, 0.5 and 0.3 respectively. The probability of occurrence of at least one of the event is __________

The number of subsets {1, 2, ... n} with odd cardinality is __________.

The number of edges in a regular graph of degree d and n vertices is _________.

The probability of an event B is P₁. The probability that events A and B occur together is P₂ while the probability that A and occur together is P₃. The probability of the event A in terms of P₁, P₂ and P₃ is __________.

On the set N of non-negative integers, the binary operation __________ is associative and non-commutative.

Amongst the properties {reflexivity, symmetry, anti-symmetry, transitivity} the relation R = {(x,y) ∈ N² | x ≠ y } satisfies __________.

The number of substrings (of all lengths inclusive) that can be formed from a character string of length n is

The tank of matrix is:

Some group (G,o) is known to be abelian. Then, which one of the following is true for G?

In a compact single dimensional array representation for lower triangular matrices (i.e all the elements above the diagonal are zero) of size n × n, non-zero elements (i.e elements of the lower triangle) of each row are stored one after another, starting from the first row, the index of the (i, j)^th element of the lower triangular matrix in this new representation is: