# Engineering-Mathematics |Gate-2017| Previous Year Questions| Set-4

Engineering-Mathematics |Gate-2017|

1. Let T be a binary search tree with 15 nodes. The minimum and maximum possible heights of T are:

Note: The height of a tree with a single node is 0. [GATE – 2017]

a. 4 and 15 respectively
b. 3 and 14 respectively
c. 4 and 14 respectively
d .3 and 15 respectively

2. The statement (¬p) ⇒ (¬q) is logically equivalent to which of the statements below? [GATE – 2017]

I. p ⇒ q
II. q ⇒ p
III. (¬q) ∨ p
IV. (¬p) ∨ q

a. I only
b. I and IV only
c. II only
d. II and III only

Engineering-Mathematics |Gate-2017|

3. Consider the first-order logic sentence F: ∀x(∃yR(x,y)). Assuming non-empty logical domains, which of the sentences below are implied by F? [GATE – 2017]

I. ∃y(∃xR(x,y))
II. ∃y(∀xR(x,y))
III. ∀y(∃xR(x,y))
IV. ¬∃x(∀y¬R(x,y))

a. IV only
b. I and IV only
c. II only
d. II and III only

4. Let c1, cn be scalars not all zero. Such that the following expression holds:

where ai is column vectors in Rn. Consider the set of linear equations. Ax = B. where A = [a1…….an]

and Then, Set of equations has   : [GATE – 2017]

a. a unique solution at x = Jn where Jn denotes a n-dimensional vector of all 1
b. no solution
c. infinitely many solutions
d. finitely many solutions

5. Let X be a Gaussian random variable with mean 0 and variance σ2. Let Y = max(X, 0) where max(a,b) is the maximum of a and b. The median of Y is __________. [GATE – 2017]

a. 0
b. 1
c. 2
d. 3

Engineering-Mathematics |Gate-2017|

6. The value of  lim(x→1)  x7 -2 x8 +1 / x8  -3 x2 + 2 : [GATE – 2017]

a. is 0
b. is -1
c. is 1
d. does not exist

7. Let p, q and r be prepositions and the expression (p → q) → r be a contradiction. Then, the expression (r → p) → q is : [GATE – 2017]

a. a tautology
c. always TRUE when p is FALSE
d. always TRUE when q is TRUE

8. Let u and v be two vectors in R2 whose Euclidean norms satisfy ||u||=2||v||. What is the value of α such that w = u + αv bisects the angle between u and v? [GATE – 2017]

a. 2
b. 1/2
c. 1
d. -1/2

Engineering-Mathematics |Gate-2017|

9. Let A be m×n real valued square symmetric matrix of rank 2 with expression given below.

Consider the following statements

(i)  One eigenvalue must be in [-5, 5].
(ii) The eigenvalue with the largest magnitude
must be strictly greater than 5.

Which of the above statements about engenvalues of A is/are necessarily CORRECT?  [GATE – 2017]

a. Both (I) and (II)
b. (I) only
c. (II) only
d. Neither (I) nor (II)

10. The number of integers between 1 and 500 (both inclusive) that are divisible by 3 or 5 or 7 is _________. [GATE – 2017]

a. 271
b. 272
c. 273
d. 274

11. If f(x) = Rsin(πx/2) + S, f(1/2) = √2 and ∫1 f(x) dx = 2R/ π , then the constants R and S are respectively.  [GATE – 2017]

a. 2/ π and 16/ π
b. 2/ π and 0
c. 4/ π and 0
d. 4/ π and 16/ π

Engineering-Mathematics |Gate-2017|

12. Let p, q, r denote the statements “It is raining”, “It is cold”, and “It is pleasant”, respectively. Then the statement “It is not raining and it is pleasant, and it is not pleasant only if it is raining and it is cold” is represented by : [GATE – 2017]

a. (¬p ∧ r) ∧ (¬r → (p ∧ q))
b. (¬p ∧ r) ∧ ((p ∧ q) → ¬r)
c. (¬p ∧ r) ∨ ((p ∧ q) → ¬r)
d. (¬p ∧ r) ∨ (r → (p ∧ q))

13. Consider the set X = {a,b,c,d e} under the partial ordering

R = {(a,a),(a,b),(a,c),(a,d),(a,e),(b,b),(b,c),(b,e),(c,c),(c,e),(d,d),(d,e),(e,e)}.

The Hasse diagram of the partial order (X,R) is shown below.

The minimum number of ordered pairs that need to be added to R to make (X,R) a lattice is _________. [GATE – 2017]

a. 0
b. 1
c. 2
d. 3

14. G is an undirected graph with n vertices and 25 edges such that each vertex of G has degree at least 3. Then the maximum possible value of n is ___________. [GATE – 2017]

a. 16
b. 17
c. 18
d. 19

15. P and Q are considering to apply for job. The probability that p applies for job is 1/4. The probability that P applies for job given that Q applies for the job 1/2 and The probability that Q applies for job given that P applies for the job 1/3.The probability that P does not apply for job given that Q does not apply for the job: [GATE – 2017]

a. 4/5
b. 5/6
c. 7/8
d. 11/12

16. If a random variable X has a Poisson distribution with mean 5, then the expectation E[(X + 2)2] equals _________. [GATE – 2017]

a. 54
b. 55
c. 56
d. 57

17. If the characteristic polynomial of a 3 × 3 matrix M over R (the set of real numbers) is λ3 – 4λ2 + aλ + 30, a ∈ ℝ, and one eigenvalue of M is 2, then the largest among the absolute values of the eigenvalues of M is ________. [GATE – 2017]

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