Let a(x, y), b(x, y,) and c(x, y) be three statements with variables x and y chosen from some universe. Consider the following statement:

 (∃x)(∀y)[(a(x, y) ∧ b(x, y)) ∧ ¬c(x, y)] 

Let R₁ be a relation from A = {1, 3, 5, 7} to B = {2, 4, 6, 8} and R₂ be another relation from B to C = {1, 2, 3, 4} as defined below:
1. An element x in A is related to an element y in B (under R₁) if x + y is divisible by 3.
2. An element x in B is related to an element y in C (under R₂) if x + y is even but not divisible by 3.
Which is the composite relation R₁R₂ from A to C?

What is the maximum number of edges in an acyclic undirected graph with n vertices?

What values of x, y and z satisfy the following system of linear equations?

Let p, q and s be four primitive statements. Consider the following arguments:
P: [(¬p ∨ q) ∧ (r → s) ∧ (p ∨ r)] → (¬s → q)
Q: [(¬p ∧ q) ∧ [q → (p → r)] → ¬r
R: [[(q ∧ r) → p] ∧ (¬q ∨ p)] → r
S: [p ∧ (p → r) ∧ (q ∨ ¬r)] → q
Which of the above arguments are valid?

Let A be an Let A be an n × n matrix of the following form.

What is the value of the determinant of A?

Let X and Y be two exponentially distributed and independent random variables with mean α and β, respectively. If Z = min(X,Y), then the mean of Z is given by

Let H₁, H₂, H₃, ... be harmonic numbers. Then, for n ∈ Z⁺, can be expressed as

In how many ways can we distribute 5 distinct balls, B₁, B₂, …, B₅ in 5 distinct cells, C₁, C₂, …, C₅ such that Ball B, is not in cell C_i, ∀i = 1, 2, …, 5 and each cell contains exactly one ball?

If matrix and X₂ - X + I = 0 (I is the Identity matrix and 0 is the zero matrix), then the inverse of X is:

What is the number of vertices in an undirected connected graph with 27 edges, 6 vertices of degree 2, 3 vertices of degree 4 and remaining of degree 3?

If f(1) = 2, f(2) = 4 and f(4) = 16, what is the value of f(3) using Lagrange’s interpolation formula?

Consider the following iterative root finding methods and convergence properties:

Iterative root finding          Convergence properties methods 
(Q) False Position                (I) Order of convergence = 1.62 
(R) Newton Raphson               (II) Order of convergence = 2 
(S) Secant                      (III) Order of convergence = 1 
                                      with guarantee of convergence 
(T) Successive Approximation     (IV) Order of convergence = 1 
                                      with no guarantee of convergence 

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.

Which one of the following is false?

Newton-Raphson iteration formula for finding 3√c, where c > 0 is,

The matrices and commute under multiplication

The probability that top and bottom cards of a randomly shuffled deck are both aces in

Let f be a function defined by

Find the values for the constants a, b, c and d so that f is continuous and differentiable every where on the real line.

Let F be the collection of all functions f: {1,2,3} → {1,2,3}. If f and g ∈ F, define an equivalence relation ~ by f ~ g if and only if f(3) = g(3).
a) Find the number of equivalence classes defined by ~.
b) Find the number of elements in each equivalence class.

The Fibonacci sequence {f₁,f₂,f₃,...,f_n} is defined by the following recurrence:

   fn+2 = fn+1 + fn, n ≥ 1; f2=1 : f1=1 

Prove by induction that every third element of the sequence is even.

Let and be two matrices such that AB = I. Let and CD = 1. Express the elements of D in terms of the elements of B.