• If the first question is answered wrong, the student gets zero marks.
• If the first question is answered correctly and the second question is not answered correctly, the student gets the marks only for the first question.  
• If both the questions are answered correctly, the student gets the sum of the marks of the two questions.

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

The value of k for which 4x² - 8xy + ky² = 0 does not represent a pair of straight lines (both passing through the origin) is:

The rank of the following (n + 1)×(n+1) matrix, where a is a real number is

The minimum number of edges in a connected cyclic graph on n vertices is:

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

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)))) 

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

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

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

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

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

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.

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.

(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 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?