Engineering-Mathematics |Gate-2014| Previous Year Questions| Set-7

Set-15 GATE-2006 Engineering-Mathematics

Engineering-Mathematics |Gate-2014|

1. Let G = (V,E) be a directed graph where V is the set of vertices and E the set of edges. Then which one of the following graphs has the same strongly connected components as G? [GATE – 2014]

a. G1=(V,E1) where E1={(u,v)|(u,v)∉E}
b. G2=(V,E2 )where E2={(u,v)│(u,v)∈E}
c. G3=(V,E3) where E3={(u,v)|there is a path of length≤2 from u to v in E}
d. G4=(V4,E) where V4 is the set of vertices in G which are not isolated

Answer : b)


2. The value of the dot product of the eigenvectors corresponding to any pair of different eigenvalues of a 4-by-4 symmetric positive definite matrix is _____________________. [GATE – 2014]

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

Answer : a)


3. Let the function : [GATE – 2014]

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

Answer : c)


4. The function f(x) = x sinx satisfies the following equation: f ”(x) + f(x)+ tcosx = 0. The value of t is __________. [GATE – 2014]

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

Answer : d)


5. A function f(x) is continuous in the interval [0,2]. It is known that f(0) = f(2) = -1 and f(1) = 1. Which one of the following statements must be true? [GATE – 2014]

a. There exists a y in the interval (0,1) such that f(y)=f(y+1)
b. For every y in the interval (0,1),f(y)=f(2-y)
c. The maximum value of the function in the interval (0,2) is 1
d. There exists a y in the interval (0, 1) such that f(y)=-f(2-y)

Answer : a)


6. Four fair six-sided dice are rolled. The probability that the sum of the results being 22 is X⁄1296. The value of X is ___________. [GATE – 2014]

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

Answer : c)


7. A pennant is a sequence of numbers, each number being 1 or 2. An n-pennant is a sequence of numbers with sum equal to n. For example, (1,1,2) is a 4-pennant. The set of all possible 1-pennants is {(1)}, the set of all possible 2-pennants is {(2), (1,1)}and the set of all 3-pennants is {(2,1), (1,1,1), (1,2)}. Note that the pennant (1,2) is not the same as the pennant (2,1).  The number of 10-pennants is ______________. [GATE – 2014]

a. 89
b. 90
c. 91
d. 92

Answer : a)


8. Let S denote the set of all functions  f:{0,1}→ {0,1}. Denote by N the number of functions from S to the set {0,1}. The value of loglog2N is ______. [GATE – 2014]

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

Answer : a)


9. Consider an undirected graph where self-loops are not allowed. The vertex set of G is {(i,j): 1 ≤ i ≤ 12, 1 ≤ j ≤ 12}. There is an edge between (a,b) and (c,d) if |a – c| ≤ 1 and |b – d| ≤ 1. The number of edges in this graph is __________. [GATE – 2014]

a. 507
b. 508
c. 506
d. 509

Answer : c)


10. An ordered -tuple (d1, d2, …, dn) with d1 ≥ d2 ≥ … dn is called graphic if there exists a simple undirected graph with n vertices having degrees d1, d2, …, dn respectively. Which of the following 6-tuples is NOT graphic? [GATE – 2014]  

a. (1, 1, 1, 1, 1, 1)
b. (2, 2, 2, 2, 2, 2)
c. (3, 2, 1, 1, 1, 0)
d. (3, 3, 3, 1, 0, 0)

Answer : d)


11. Which one of the following propositional logic formulas is TRUE when exactly two of p, q, and r are TRUE? [GATE – 2014]

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

Answer : b)


12. The security system at an IT office is composed of 10 computers of which exactly four are working. To check whether the system is functional, the officials inspect four of the computers picked at random (without replacement). The system is deemed functional if at least three of the four computers inspected are working.  Let the probability that the system is deemed functional be denoted by p. Then 100p =_____________. [GATE – 2014]

a. 11.90
b. 11.91
c. 11.92
d. 11.93

Answer : a)


13. Each of the nine words in the sentence “The quick brown fox jumps over the lazy dog” is written on a separate piece of paper. These nine pieces of paper are kept in a box. One of the pieces is drawn at random from the box. The expected length of the word drawn is _____________. (The answer should be rounded to one decimal place.) [GATE – 2014]

a. 4.3
b. 4.2
c. 4.1
d. 3.9

Answer : d)


14. The maximum number of edges in a bipartite graph on 12 vertices is ______. [GATE – 2014]

a. 36
b. 37
c. 38
d. 39

Answer : a)


15. If the matrix A is such that :

 then the determinant of A is equal to _____________. [GATE – 2014]

a. 1
b. 2
c. 0
d. 4

Answer : c)


16. A non-zero polynomial f(x) of degree 3 has roots at x = 1, x = 2 and x = 3.  Which one of the following must be TRUE? [GATE – 2014]

a. f(0)f(4) > 0
b. f(0)f(4) < 0
c. f(0) + f(4) > 0
d. f(0) + f(4) < 0

Answer : b)


17. In the Newton-Raphson method, an initial guess of x0 = 2 is made and the sequence x0, x1, x2 … is obtained for the function

0.75x3 – 2x2 – 2x + 4 = 0
 
Consider the statements
(I) x3 = 0.
(II) The method converges to a solution in a finite number of iterations.

Which of the following is TRUE? [GATE – 2014]    

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

Answer : a)


18. The product of the non-zero eigenvalues of the matrix

1 0 0 0 1
0 1 1 1 0
0 1 1 1 0
0 1 1 1 0
1 0 0 0 1

is ______ [GATE – 2014]

a. 6
b. 7
c. 8
d. 9

Answer : a)


19. The probability that a given positive integer lying between 1 and 100 (both inclusive) is NOT divisible by 2, 3 or 5 is ______ . [GATE – 2014]

a. 0.260 to 0.262
b. 0.261 to 0.263
c. 0.262 to 0.264
d. 0.259 to 0.261

Answer : d)


20. The number of distinct positive integral factors of 2014 is _________ [GATE – 2014]

a. 0.26
b. 0.27
c. 8
d. 0.29

Answer : c)


21. Consider the following relation on subsets of the set S of integers between 1 and 2014. For two distinct subsets U and V of S we say U < V if the minimum element in the symmetric difference of the two sets is in U. Consider the following two statements:

S1: There is a subset of S that is larger than every other subset.
S2: There is a subset of S that is smaller than every other subset.

Which one of the following is CORRECT? [GATE – 2014]

a. Both S1 and S2 are true
b. S1 is true and S2 is false
c. S2 is true and S1 is false
d. either S1 nor S2 is true

Answer : a)
Engineering-Mathematics |Gate-2014|


21. A cycle on n vertices is isomorphic to its complement. The value of n is _____. [GATE – 2014]

a. 5
b. 6
c. 7
d. 8

Answer : a)


22. Which one of the following Boolean expressions is NOT a tautology? [GATE – 2014]

a. ((a → b) ∧ (b → c)) ⟶ (a → c)
b. (a →c) → (∽ b → (a ∧ c))
c. (a ∧ b ∧ c) → (c ∨ a)
d. a → (b → a)

Answer : b)
Engineering-Mathematics |Gate-2014|


23. Consider the following statements: P:Good mobile phones are not cheap Q:Cheap mobile phones are not good L:P implies Q M:Q implies P N:P is equivalent to Q Which one of the following about L, M, and N is CORRECT? [GATE – 2014]

a. Only L is TRUE.
b. Only M is TRUE.
c. Only N is TRUE.
d. L, M and N are TRUE.

Answer : d)


24. Let X and Y be finite sets and f: X→Y be a function. Which one of the following statements is TRUE? [GATE – 2014]

a. For any subsets A and B of X, |f(A ∪ B)| = |f(A)|+|f(B)|
b. For any subsets A and B of X, f(A ∩ B) = f(A) ∩ f(B)
c. For any subsets A and B of X, |f(A ∩ B)| = min{ |f(A)|,f|(B)|}
d. For any subsets S and T of Y, f -1 (S ∩ T) = f -1 (S) ∩ f -1 (T)

Answer  : d)
Engineering-Mathematics |Gate-2014|


25. Let G be a group with 15 elements. Let L be a subgroup of G. It is known that L ≠ G and that the size of L is at least 4.  The size of L  is __________. [GATE – 2014]

a. 5
b. 6
c. 7
d. 8

Answer : a)


26. Which one of the following statements is TRUE about every n × n matrix with only real eigenvalues? [GATE – 2014]

a. If the trace of the matrix is positive, all its eigenvalues are positive.
b. If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigenvalues is negative.
c. If the determinant of the matrix is positive, all its eigenvalues are positive.
d. If the product of the trace and determinant of the matrix is positive, all its eigenvalues are positive.

Answer : a)


27. If V1 and V2 are 4-dimensional subspaces of a 6-dimensional vector space V, then the smallest possible dimension of V1∩V2   is ______. [GATE – 2014]

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

Answer : c) 
Engineering-Mathematics |Gate-2014|


28. Let S be a sample space and two mutually exclusive events A and B be such that A∪B = S. If P(∙) denotes the probability of the event, the maximum value of P(A)P(B) is __________.   [GATE – 2014]

a. 0.25
b. 0.26
c. 0.27
d. 0.28

Answer : a)


29. There are two elements , in a group ( ,∗) such that every element in the group can be written as a product of some number of x’s and ‘s in some order. It is known that x*x=y*y=x*y*x=y*x*y*x=e where    is the identity element. The maximum number of elements in such a group is  : [GATE – 2014]

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

Answer : a)
Engineering-Mathematics |Gate-2014|


30. Let δ denote the minimum degree of a vertex in a graph. For all planar graphs on n vertices with δ ≥ 3, which one of the following is TRUE? [GATE – 2014]

a. In any planar embedding, the number of faces is less than n/2+ 2
b. There is a planar embedding in which the number of faces is less than n/2+ 2
c. In any planar embedding, the number of faces is at least n/2+ 2
d. There is a planar embedding in which the number of faces is at most n/(δ+1)

Answer : c)


31. The CORRECT formula for the sentence, “not all rainy days are cold” is : [GATE – 2014]

a. ∀d (Rainy(d) ∧∼Cold(d))
b. ∀d (∼Rainy(d) → Cold(d))
c. ∃d (∼Rainy(d) → Cold(d))
d. ∃d (Rainy(d) ∧∼Cold(d))

Answer : d)
Engineering-Mathematics |Gate-2014|


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