# Engineering-Mathematics |Gate-2018| Previous Year Questions| Set-3

Engineering-Mathematics |Gate-2018|

1. Consider a matrix  A = uvT where u= (1 / 2) , v = (1 / 1) . Note that v T denotes the transpose of v. The largest eigenvalue of A is __________________ [GATE – 2018]

a. 3
b. 4
c. 5
d. 6

2. Two people, P and Q, decide to independently roll two identical dice, each with 6 faces, numbered 1 to 6. The person with the lower number wins. In case of a tie, they roll the dice repeatedly until there is no tie. Define a trial as a throw of the dice by P and Q. Assume that all 6 numbers on each dice are equi-probable and that all trials are independent. The probability (rounded to 3 decimal places) that one of them wins on the third trial is __________. [GATE – 2018]

a. 0.021
b. 0.022
c. 0.023
d. 0.024

Engineering-Mathematics |Gate-2018|

3. The chromatic number of the following graph is _______.

a. 1
b. 2
c. 3
d. 4

4. Let G be a finite group on 84 elements. The size of a largest possible proper subgroup of G is _________. [GATE – 2018]

a. 41
b. 42
c. 43
d. 44

Engineering-Mathematics |Gate-2018|

5. Which one of the following is a closed form expression for the generating function of the sequence {an}, where a= 2n+3 for all n = 0, 1, 2, …? [GATE – 2018]

a. 3/(1-x)2
b. 3x/(1-x)2
c. 2-x/(1-x)2
d. 3-x/(1-x)2

6. Assume that multiplying a matrix G1 of dimension p×q with another matrix G2 of dimension q×r requires pqr scalar multiplications. Computing the product of n matrices G1G2G3…Gn can be done by parenthesizing in different ways. Define GiGi+1 as an explicitly computed pair for a given parenthesization if they are directly multiplied. For example, in the matrix multiplication chain G1G2G3G4G5G6 using parenthesization(G1(G2G3))(G4(G5G6)), G2G3 and G5G6 are the only explicitly computed pairs.

Consider a matrix multiplication chain F1F2F3F4F5, where matrices F1, F2, F3, F4 and F5 are of dimensions 2×25, 25×3, 3×16, 16×1 and 1×1000, respectively. In the parenthesization of F1F2F3F4F5 that minimizes the total number of scalar multiplications, the explicitly computed pairs is/ are [GATE – 2018]

a. F1F2 and F3F4 only
b. F2F3 only
c. F3F4 only
d. F1F2 and F4F5 only

7. Consider the first-order logic sentence

φ ≡ ∃s∃t∃u∀v∀w∀x∀y ψ(s,t,u,v,w,x,y)

where ψ(s,t,u,v,w,x,y) is a quantifier-free first-order logic formula using only predicate symbols, and possibly equality, but no function symbols. Suppose φ has a model with a universe containing 7 elements.

Which one of the following statements is necessarily true? [GATE – 2018]

a. There exists at least one model of φ with universe of size less than or equal to 3.
b. There exists no model of φ with universe of size less than or equal to 3.
c. There exists no model of φ with universe of size greater than 7.
d. Every model of φ has a universe of size equal to 7.

8. Consider Guwahati (G) and Delhi (D) whose temperatures can be classified as high (H), medium (M) and low (L). Let P(HG) denote the probability that Guwahati has high temperature. Similarly, P(MG) and P(LG) denotes the probability of Guwahati having medium and low temperatures respectively. Similarly, we use P(HD), P(MD) and P(LD) for Delhi.

The following table gives the conditional probabilities for Delhi’s temperature given Guwahati’s temperature. [GATE – 2018]

Consider the first row in the table above. The first entry denotes that if Guwahati has high temperature (HG) then the probability of Delhi also having a high temperature (HD) is 0.40; i.e., P(HD ∣ HG) = 0.40. Similarly, the next two entries are P(MD ∣ HG) = 0.48 and P(LD ∣ HG) = 0.12. Similarly for the other rows.

If it is known that P(HG) = 0.2, P(MG) = 0.5, and P(LG) = 0.3, then the probability (correct to two decimal places) that Guwahati has high temperature given that Delhi has high temperature is _______ .

a. 0.60
b. 0.61
c. 0.62
d. 0.63

9. Let N be the set of natural numbers. Consider the following sets, [GATE – 2018]

P: Set of Rational numbers (positive and negative)
Q: Set of functions from {0, 1} to N
R: Set of functions from N to {0, 1}
S: Set of finite subsets of N

Which of the above sets are countable?

a. Q and S only
b. P and S only
c. P and R only
d. P, Q and S only

10.  Consider  a matrix P whose only eigenvectors are the multiples of [1]
[4] [GATE – 2018]

Consider the following statements.

(I) P does not have an inverse
(II) P has a repeated eigenvalue
(III) P cannot be diagonalized

Which one of the following options is correct?

a. Only I and III are necessarily true
b. Only II is necessarily true
c. Only I and II are necessarily true
d. Only II and III are necessarily true

Engineering-Mathematics |Gate-2018|

11. Let G be a graph with 100! vertices, with each vertex labeled by a distinct permutation of the numbers 1, 2, …, 100. There is an edge between vertices u and v if and only if the label of u can be obtained by swapping two adjacent numbers in the label of v. Let y denote the degree of a vertex in G, and z denote the number of connected components in G.

Then, y + 10z = ___________. [GATE – 2018]

a. 109
b. 110
c. 111
d. 112