# Discrete Mathematics |Gate-2020| previous year questions| Set-1

Discrete Mathematics |Gate-2020|

1. Which one of the following predicate formulae is NOT logically valid?

Note that W is a predicate formula without any free occurrence of x. [GATE -2020]

a. ∀x(p(x)∨W) ≡ ∀x p(x) ∨ W
b. ∃x(p(x) ∧ W) ≡ ∃x p(x) ∧W
c. ∀x(p(x)→ W) ≡ ∀x p(x) → W
d. ∃x (p(x) → W) ≡ ∃x p(x) → W

2. 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 n∑i=1aixi∑i=1naixi is an odd number is _______.[GATE -2020]

a. 0.3
b. 0.4
c. 0.5
d. 0.6

1. Let G be a group of 35 elements. Then the largest possible size of a subgroup of G other than G itself is ______.[GATE -2020]

a. 7
b. 8
c. 9
d. 10

1. 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 _____.[GATE -2020]

a. 0124
b. 0.125
c. 0.126
d. 0.127

1. 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 _______. [GATE -2020]

a. 12
b. 13
c. 14
d. 15

1. 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? [GATE -2020]

a. I and II only
b. II and III only
c. I and IV only
d. II and IV only

1. Graph G is obtained by adding vertex s to K3,4 and making s adjacent to every vertex of K3,4. The minimum number of colours required to edge-colour G is _____. [GATE -2020]

a. 4
b. 5
c. 6
d. 7

1. Consider the functions

I. e−xe−x
II. x2−sinxx2−sin⁡x
III. √x3+1×3+1

Which of the above functions is/are increasing everywhere in [0,1]? [GATE -2020]

a. III only
b. II and III only
c. II only
d. I and III only