# Engineering-Mathematics |Gate-2019| Previous Year Questions| Set-2

Engineering-Mathematics |Gate-2019|

1.

a. 1
b. Limit does not exist
c. 53/12
d. 108/7

2. Let G be an arbitrary group. Consider the following relations on G:

R1: ∀a,b ∈ G, aR1b if and only if ∃g ∈ G such that a = g-1bg
R2: ∀a,b ∈ G, aR2b if and only if a = b-1

Which of the above is/are equivalence relation/relations? [GATE – 2019]

a. R2 only
b. Neither R1 and R2
b. R1 only
c. R1 and R2

3. Let X be a square matrix. Consider the following two statements on X. [GATE – 2019]

I. X is invertible.
II. Determinant of X is non-zero.

Which one of the following is TRUE?

a. I implies II; II does not imply I.
b. I and II are equivalent statements.
c. I does not imply II; II does not imply I.
d. II implies I; I does not imply II.

4. Let G be an undirected complete graph on n vertices, where n > 2. Then, the number of different Hamiltonian cycles in G is equal to : [GATE – 2019]

a. (n-2)!/2
b. (n-1)!
c. 1
d. n!

5. Let U = {1,2,…,n}. Let A = {(x,X)|x ∈ X, X ⊆ U}. Consider the following two statements on |A|.

Which of the above statements is/are TRUE?   [GATE – 2019]

a. Only II
b. Only I
c. Neither I nor II
d. Both I and II

6. Suppose Y is distributed uniformly in the open interval (1,6). The probability that the polynomial 3x2 + 6xY + 3Y + 6 has only real roots is (rounded off to 1 decimal place) _____. [GATE – 2019]

a. 0.3
b. 0.9
c. 0.1
d. 0.8

7. Consider the first order predicate formula φ:

∀x[(∀z z|x ⇒ ((z = x) ∨ (z = 1))) ⇒ ∃w (w > x) ∧ (∀z z|w ⇒ ((w = z) ∨ (z = 1)))]

Here ‘a|b’ denotes that ‘a divides b’, where a and b are integers. Consider the following sets:

S1.  {1, 2, 3, …, 100}
S2.  Set of all positive integers
S3.  Set of all integers

Which of the above sets satisfy φ?   [GATE – 2019]

a. S1 and S3
b. S1, S2 and S3
c. S2 and S3
d. S1 and S2

Engineering-Mathematics |Gate-2019|

8. Consider the following matrix:

The absolute value of the product of Eigen values of R is ______. [GATE – 2019]

a. 12
b. 17
c. 10
d. 8