111.  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 colourpairs 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 ? 
a.  9 
b.  8 
c.  7 
d.  6 
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Answer: (c).7

112.  In an M'N matrix such that all nonzero entries are covered in a rows and b columns. Then the maximum number of nonzero entries, such that no two are on the same row or column, is 
a.  ≤ a + b 
b.  ≤ max {a, b} 
c.  ≤ min {Ma, Nb} 
d.  ≤ min {a, b} 
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Answer: (d).≤ min {a, b}

113.  The minimum number of colours required to colour the following graph, such that no two adjacent vertices are assigned the same colour, is 
a.  2 
b.  3 
c.  4 
d.  5 
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Answer: (c).4

114.  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 
a.  nCd /2^n 
b.  nCd / d 
c.  d/2^n 
d.  1/2^d 
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Answer: (a).nCd /2^n

115.  How many graphs on n labeled vertices exist which have at least (n^2  3n)/2 edges ? 
a.  A 
b.  B 
c.  C 
d.  D 
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Answer: (d).D

116.  A point is randomly selected with uniform probability in the XY 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 p^2 is 
a.  2/3 
b.  1 
c.  4/3 
d.  5/3 
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Answer: (d).5/3

117.  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) 
a.  cannot have a cut vertex 
b.  must have a cycle 
c.  must have a cutedge (bridge) 
d.  has chromatic number strictly greater than those of G1 and G2 
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Answer: (b).must have a cycle

118.  For the composition table of a cyclic group shown below: * a b c d a a b c d b b a d c c c d b a d d c a b Which one of the following choices is correct? 
a.  a, b are generators 
b.  b, c are generators 
c.  c, d are generators 
d.  d, a are generators 
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Answer: (c).c, d are generators

119.  If P, Q, R are subsets of the universal set U, then 
a.  Q^c U R^c 
b.  P U Q^c U R^c 
c.  P^c U Q^c U R^c 
d.  U 
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Answer: (d).U

120.  
a.  A 
b.  B 
c.  C 
d.  D 
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Answer: (c).C
