# Engineering-Mathematics |Gate-2016| Previous Year Questions| Set-5

Engineering-Mathematics |Gate-2016|

1. Let p,q,r,s represent the following propositions.

p: x ∈ {8,9,10,11,12}
q: x is a composite number
r: x is a perfect square
s: x is a prime number

The integer x≥2 which satisﬁes ¬((p ⇒ q) ∧ (¬r ∨ ¬s))  is _________. [GATE – 2016]

a. 11
b. 12
c. 13
d. 14

2. Let an be the number of n-bit strings that do NOT contain two consecutive 1s. Which one of the following is the recurrence relation for an? [GATE – 2016]

a. an = a(n-1) + 2a(n-2)
b. an = a(n-1) + a(n-2)
c. n = 2a(n-1) + a(n-2)
d. an = 2a(n-1) + 2a(n-2)

Engineering-Mathematics |Gate-2016|

3. lim (x→4) sin(x-4)/ x-4 = ____________  [GATE – 2016]

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

4. A probability density function on the interval [a,1] is given by 1/x2 and outside this interval the value of the function is zero. The value of a is _________. [GATE – 2016]

a. 0.7
b. 0.8
c. 0.5
d. 0.6

5. Two eigenvalues of a 3 × 3 real matrix P are (2 + √-1) and 3. The determinant of P is __________. [GATE – 2016]

a. 15
b. 16
c. 17
d. 18

Engineering-Mathematics |Gate-2016|

6. The coefﬁcient of x12 in (x3 + x4 + x5 + x6 + …)3 is _________. [GATE – 2016]

a. 10
b. 11
c. 12
d. 13

7. Consider the recurrence relation a1 = 8, an = 6n2 + 2n + an-1. Let a99 = K × 104. The value of K is ___________. [GATE – 2016]

a. 198
b. 199
c. 200
d. 201

8. A function f:N+ → N+, deﬁned on the set of positive integers N+, satisﬁes the following properties:

f(n) = f(n/2)           if n is even
f(n) = f(n+5)           if n is odd

Let R = {i|∃j: f(j)=i} be the set of distinct values that f takes. The maximum possible size of R is __________. [GATE – 2016]

a. 2
b. 3
c. 4
d. 5

Engineering-Mathematics |Gate-2016|

9. Consider the following experiment.

Step1. Flip a fair coin twice.
Step2. If the outcomes are (TAILS, HEADS) then output Y and stop.
Step3. If the outcomes are either (HEADS, HEADS) or (HEADS, TAILS), then output N and stop.
Step4. If the outcomes are (TAILS, TAILS), then go to Step 1.

The probability that the output of the experiment is Y is (up to two decimal places) ________. [GATE – 2016]

a. 0.33
b. 0.34
c. 0.35
d. 0.36

10. Consider the following expressions:

(i) false
(ii) Q
(iii) true
(iv) P ∨ Q
(v) ¬Q ∨ P

The number of expressions given above that are logically implied by P ∧ (P ⇒ Q) is _________. [GATE – 2016]

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

Engineering-Mathematics |Gate-2016|

11. Let f(x) be a polynomial and g(x) = f'(x) be its derivative. If the degree of (f(x) + f(-x)) is 10, then the degree of (g(x) – g(-x)) is __________.[GATE – 2016]

a. 11
b. 10
c. 9
d. 12

12. The minimum number of colours that is sufﬁcient to vertex-colour any planar graph is ________. [GATE – 2016]

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

13. Consider the systems, each consisting of m linear equations in n variables.

I. If m < n, then all such systems have a solution
II. If m > n, then none of these systems has a solution
III. If m = n, then there exists a system which has a solution

Which one of the following is CORRECT? [GATE – 2016]

a. I, II and III are true
b. Only II and III are true
c. Only III is true
d. None of them is true

14. Suppose that a shop has an equal number of LED bulbs of two different types. The probability of an LED bulb lasting more than 100 hours given that it is of Type 1 is 0.7, and given that it is of Type 2 is 0.4. The probability that an LED bulb chosen uniformly at random lasts more than 100 hours is _________. [GATE – 2016]

a. 0.55
b. 0.56
c. 0.57
d. 0.58

15. Suppose that the eigenvalues of matrix A are 1, 2, 4. The determinant of (A-1)T is _________. [GATE – 2016]

a. 0.125
b. 0.126
c. 0.127
d. 0.128

16. A binary relation R on ℕ × ℕ is deﬁned as follows: (a,b)R(c,d) if a≤c or b≤d. Consider the following propositions:

P: R is reﬂexive
Q: Ris transitive

Which one of the following statements is TRUE? [GATE – 2016]

a. Both P and Q are true.
b. P is true and Q is false.
c. P is false and Q is true.
d. Both P and Q are false.

17. Which one of the following well-formed formulae in predicate calculus is NOT valid? [GATE – 2016]

a. (∀x p(x) ⇒ ∀x q(x)) ⇒ (∃x ¬p(x) ∨ ∀x q(x))
b. (∃x p(x) ∨ ∃x q(x)) ⇒ ∃x (p(x) ∨ q(x))
c. ∃x (p(x) ∧ q(x)) ⇒ (∃x p(x) ∧ ∃x q(x)
d. ∀x (p(x) ∨ q(x)) ⇒ (∀x p(x) ∨ ∀x q(x))

18. Consider a set U of 23 different compounds in a Chemistry lab. There is a subset S of U of 9 compounds, each of which reacts with exactly 3 compounds of U. Consider the following statements:

I. Each compound in U\S reacts with an odd number of compounds.
II. At least one compound in U\S reacts with an odd number of compounds.
III. Each compound in U\S reacts with an even number of compounds.

Which one of the above statements is ALWAYS TRUE? [GATE – 2016]

a. Only I
b. Only II
c. Only III
d. None

19. The value of the expression 1399(mod 17), in the range 0 to 16, is ________. [GATE – 2016]

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