GATE-CS-2014-(Set-3)

Question 51

• I Only
• II Only
• Both I and II
• Neither I or II

Question 52

The value of integral given below is: [Tex]\\int_{0}^{\\pi} x^2 cosx dx[/Tex] (A)[Tex]-2 \\pi[/Tex] (B)[Tex]\\pi[/Tex] (C)[Tex]-\\pi[/Tex] (D)[Tex]2\\pi[/Tex]

• A
• B
• C
• D

Question 53

Let S be the sample space and two mutually exclusive events A and B be such that A U B = S. If P(.) denotes the probability of the event. The maximum value of P(A)P(B) is ______

• 0.5
• 0.25
• 0.225
• 0.125

Question 54

There are two elements x, y in a group (G,∗) such that every element in the group can be written as a product of some number of x\'s and y\'s in some order. It is known that

  x ∗ x = y ∗ y = x ∗ y ∗ x ∗ y = y ∗ x ∗ y ∗ x = e

where e is the identity element. The maximum number of elements in such a group is __________.

• 2
• 3
• 4
• 5

Question 55

If G is a forest with n vertices and k connected components, how many edges does G have?

• floor(n/k)
• ceil(n/k)
• n-k
• n-k+1

Question 56

Let d denote the minimum degree of a vertex in a graph. For all planar graphs on n vertices with d ≥ 3, which one of the following is TRUE?

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

Question 57

Consider the set of all functions f: {0,1, … ,2014} → {0,1, … ,2014} such that f(f(i)) = i, for all 0 ≤ i ≤ 2014. Consider the following statements:

P. For each such function it must be the case that 
   for every i, f(i) = i.
Q. For each such function it must be the case that 
   for some i, f(i) = i.
R. Each such function must be onto. 

Which one of the following is CORRECT?

• P, Q and R are true
• Only Q and R are true
• Only P and Q are true
• Only R is true

Question 58

The CORRECT formula for the sentence, “not all rainy days are cold” is

• A
• B
• C
• D

Question 59

Let X denote the Exclusive OR (XOR) operation. Let ‘1’ and ‘0’ denote the binary constants. Consider the following Boolean expression for F over two variables P and Q:

F(P, Q) = ( ( 1 X P) X (P X Q) ) X ( (P X Q) X (Q X 0) ) 

The equivalent expression for F is

• P + Q
• (P + Q)\'
• P X Q
• (P X Q)\'

Question 60

Let A be a square matrix of size n x n. Consider the following program. What is the expected output? 

C = 100
for i = 1 to n do
    for j = 1 to n do
    {
        Temp = A[i][j] + C
        A[i][j] = A[j][i]
        A[j][i] = Temp - C
    } 
for i = 1 to n do
    for j = 1 to n do
        Output(A[i][j]);

• The matrix A itself

• Transpose of matrix A

• Adding 100 to the upper diagonal elements and subtracting 100 from diagonal elements of A

• None of the above