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:

The number of distinct simple graphs with upto three nodes is

Let A and B be real symmetric matrices of size n × n. Then which one of the following is true?

Backward Euler method for solving the differential equation dy/dx = f(x,y) is specified by, (choose one of the following).

Let A and B be any two arbitrary events, then, which one of the following is true?

(a) Find the minimum value of 3 - 4x + 2x².
(b) Determine the number of positive integers (≤ 720) which are not divisible by any of numbers 2, 3, and 5.

Let G₁ and G₂ be subgroups of a group G.
(a) Show that G₁ ∩ G₂ is also a group of G.
(b) Is G₁ ∪ G₂ always a subgroup of G?

Prove using mathematical induction for n≥5, 2ⁿ > n²

Prove that in finite graph, the number of vertices of odd degree is always even.

Obtain the principal (canonical) conjunctive normal form of the propositional formula

  (p ∧ q) V (¬q ∧ r) 

Where ‘∧’ is logical and, ‘v’ is inclusive or and ¬ is negation.

(a) Determine the number of divisors of 600.
(b) Compute without using power series expansion

The solution of differential equation y'' + 3y' + 2y = 0 is of the form

If the proposition ¬p ⇒ ν is true, then the truth value of the proposition ¬p ∨ (p ⇒ q), where ¬ is negation, ‘∨’ is inclusive or and ⇒ is implication, is

Let A be the set of all non-singular matrices over real number and let* be the matrix multiplication operation. Then

The iteration formula to find the square root of a positive real number b using the Newton Raphson method is

A bag contains 10 white balls and 15 black balls. Two balls are drawn in succession. The probability that one of them is black and the other is white is:

A unit vector perpendicular to both the vectors a = 2i - 2j + k and b = 1 + j - 2k is:

A language with string manipulation facilities uses the following operations

 head(s): first character of a string
 tail(s): all but the first character of a string
 concat(s1,s2):s1 s2
 for the string acbc what will be the output of
 concat(head(s), head(tail(tail(s)))) 

If the cube roots of unity are 1, ω and ω², then the roots of the following equation are (x - 1)³ + 8 = 0

The probability that a number selected at random between 100 and 999 (both inclusive) will not contain the digit 7 is:

Let R be a symmetric and transitive relation on a set A. Then

The number of elements in he power set P (S) of the set S = {(φ), 1, (2, 3)} is:

In the interval [0, π] the equation x = cos x has

If at every point of a certain curve, the slope of the tangent equals −2x/y the curve is