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

Set-8 Gate-2012 Discrete Mathematics

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

Answer : c)


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

Answer : c)


  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

Answer : a)


  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

Answer : b)


  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

Answer : a)


  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

Answer : b)


  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

Answer : d)


  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

Answer : a)
Discrete Mathematics |Gate-2020|


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