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?

The number of distinct simple graphs with upto three nodes 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 substrings (of all lengths inclusive) that can be formed from a character string of length n 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 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 __________.

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

Match the following items

Find the inverse of the matrix

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:

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.

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

Let G be a group of 35 elements. Then the largest possible size of a subgroup of G other than G itself is ______.

Let R be the set of all binary relations on the set {1,2,3}. Suppose a relation is chosen from R at random. The probability that the chosen relation is reflexive (round off to 3 decimal places) is _____.

Consider the functions

I. e^-x
II. x²-sin x
III. √(x³+1)

Which of the above functions is/are increasing everywhere in [0,1]?

For n>2, let a ∈ {0,1}ⁿ be a non-zero vector. Suppose that x is chosen uniformly at random from {0,1}ⁿ. Then, the probability that  is an odd number is _____.

Graph G is obtained by adding vertex s to K_3,4 and making s adjacent to every vertex of K_3,4. The minimum number of colours required to edge-colour G is _____.

Which one of the following predicate formulae is NOT logically valid?
Note that W is a predicate formula without any free occurrence of x.

The number of permutations of the characters in LILAC so that no character appears in its original position, if the two L’s are indistinguishable, is _______.

Let A and B be two n×n matrices over real numbers. Let rank(M) and det(M) denote the rank and determinant of a matrix M, respectively. Consider the following statements,

I. rank(AB) = rank(A) rank(B)
II. det(AB) = det(A) det(B)
III. rank(A + B) ≤ rank(A) + rank(B)
IV. det(A + B) ≤ det(A) + det(B)

Which of the above statements are TRUE?

Let A and B be sets and let A^c and B^c denote the complements of the sets A and B. The set (A – B) ∪ (B - A) ∪ (A∩B) is equal to

Let X = {2,3,6,12,24}, Let ≤ be the partial order defined by X ≤ Y if x divides y. Number of edge as in the Hasse diagram of (X,≤) is

Suppose X and Y are sets and X Y and are their respective cardinalities. It is given that there are exactly 97 functions from X to Y. from this one can conclude that

Which of the following statements is false?

Two dice are thrown simultaneously. The probability that at least one of them will have 6 facing up is

The formula used to compute an approximation for the second derivative of a function f at a point X₀ is

Let Ax = b be a system of linear equations where A is an m × n matrix and b is a m × 1 column vector and X is a n × 1 column vector of unknowns. Which of the following is false?

Let R denotes the set of real numbers. Let f: R×R → R×R be a bijective function defined by f(x,y) = (x+y,x-y), The inverse function of f is given by

Let R be a non-empty relation on a collection of sets defined by A R B if and only if A ∩ B = ф. Then, (pick the true statement)

Which of the following is false? Read ∧ as AND, ∨ as OR, ~ as NOT, → as one way implication and ↔ as two way implication.