GATE-CS-2004

Question 71

An examination paper has 150 multiple-choice questions of one mark each, with each question having four choices. Each incorrect answer fetches -0.25 mark. Suppose 1000 students choose all their answers randomly with uniform probability. The sum total of the expected marks obtained by all these students is:

• 0
• 2550
• 7525
• 9375

Question 72

Mala has a colouring book in which each English letter is drawn two times. She wants to paint each of these 52 prints with one of k colours, such that the colour-pairs used to colour any two letters are different. Both prints of a letter can also be coloured with the same colour. What is the minimum value of k that satisfies this requirement ?

• 9
  

• 8
  

• 7
  

• 6
  

Question 73

In an M\'N matrix such that all non-zero entries are covered in a rows and b columns. Then the maximum number of non-zero entries, such that no two are on the same row or column, is

• ≤ a + b
  

• ≤ max {a, b}
  

• ≤ min {M-a, N-b}
  

• ≤ min {a, b}
  

Question 74

The minimum number of colours required to colour the following graph, such that no two adjacent vertices are assigned the same colour, is

• 2
• 3
• 4
• 5

Question 75

Two n bit binary strings, S1 and S2, are chosen randomly with uniform probability. The probability that the Hamming distance between these strings (the number of bit positions where the two strings differ) is equal to d is

• nCd /2ⁿ
• nCd / ^d
• d/2ⁿ
• 1/2^d

Question 76

How many graphs on n labeled vertices exist which have at least (n² - 3n)/2 edges ?

• A
• B
• C
• D

Question 77

A point is randomly selected with uniform probability in the X-Y plane within the rectangle with corners at (0,0), (1,0), (1,2) and (0,2). If p is the length of the position vector of the point, the expected value of p² is

• 2/3
• 1
• 4/3
• 5/3

Question 78

Let G1 = (V, E1) and G2 = (V, E2) be connected graphs on the same vertex set V with more than two vertices. If G1 ∩ G2 = (V, E1 ∩ E2) is not a connected graph, then the graph G1 U G2 = (V, E1 U E2)

• cannot have a cut vertex
• must have a cycle
• must have a cut-edge (bridge)
• has chromatic number strictly greater than those of G1 and G2

Question 79

Let A[1, ..., n] be an array storing a bit (1 or 0) at each location, and f(m) is a function whose time complexity is θ(m). Consider the following program fragment written in a C like language:

C

counter = 0;
for (i = 1; i < = n; i++)
{ 
      if (A[i] == 1) 
         counter++;
      else {
         f(counter); 
         counter = 0;
      }
}

The complexity of this program fragment is

• Ω(n²)
• Ω(nlog n) and O(n²)
• θ(n)
• O(n)

Question 80

The time complexity of the following C function is (assume n > 0)

C

int recursive (int n) {
   if (n == 1)
      return (1);
   else
      return (recursive (n - 1) + recursive (n - 1));
}

• O(n)
• O(n log n)
• O(n²)
• O(2ⁿ)