**Engineering-Mathematics |Gate-2018|**

**1. **Consider a matrix A = uv^{T} 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

*Answer : a)*

**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

*Answer : c)*

Engineering-Mathematics |Gate-2018|

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

a. 1

b. 2

c. 3

d. 4

*Answer : c)*

**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

*Answer : b)*

Engineering-Mathematics |Gate-2018|

**5.** Which one of the following is a closed form expression for the generating function of the sequence {a_{n}}, where a_{n }= 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}

*Answer : d)*

**6. **Assume that multiplying a matrix G_{1} of dimension p×q with another matrix G_{2} of dimension q×r requires pqr scalar multiplications. Computing the product of n matrices G_{1}G_{2}G_{3}…G_{n} can be done by parenthesizing in different ways. Define G_{i}G_{i+1} as an explicitly computed pair for a given parenthesization if they are directly multiplied. For example, in the matrix multiplication chain G_{1}G_{2}G_{3}G_{4}G_{5}G_{6} using parenthesization(G_{1}(G_{2}G_{3}))(G_{4}(G_{5}G_{6})), G_{2}G_{3} and G_{5}G_{6} are the only explicitly computed pairs.

Consider a matrix multiplication chain F_{1}F_{2}F_{3}F_{4}F_{5}, where matrices F_{1}, F_{2}, F_{3}, F_{4} and F_{5} are of dimensions 2×25, 25×3, 3×16, 16×1 and 1×1000, respectively. In the parenthesization of F_{1}F_{2}F_{3}F_{4}F_{5} that minimizes the total number of scalar multiplications, the explicitly computed pairs is/ are [GATE – 2018]

a. F_{1}F_{2} and F_{3}F_{4} only

b. F_{2}F_{3} only

c. F_{3}F_{4} only

d. F_{1}F_{2} and F_{4}F_{5} only

*Answer : c)*

**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.

*Answer : b)*

**8.** Consider Guwahati (G) and Delhi (D) whose temperatures can be classified as high (H), medium (M) and low (L). Let P(H_{G}) denote the probability that Guwahati has high temperature. Similarly, P(M_{G}) and P(L_{G}) denotes the probability of Guwahati having medium and low temperatures respectively. Similarly, we use P(H_{D}), P(M_{D}) and P(L_{D}) 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 (H_{G}) then the probability of Delhi also having a high temperature (H_{D}) is 0.40; i.e., P(H_{D} ∣ H_{G}) = 0.40. Similarly, the next two entries are P(M_{D} ∣ H_{G}) = 0.48 and P(L_{D} ∣ H_{G}) = 0.12. Similarly for the other rows.

If it is known that P(H_{G}) = 0.2, P(M_{G}) = 0.5, and P(L_{G}) = 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

*Answer : a)*

**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

*Answer : d)*

**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

*Answer : d)*

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

*Answer : a)*

OS – GATE Previous Year Questions

DS – GATE Previous Year Questions