51.  The minimum number of cards to be dealt from an arbitrarily shuffled deck of 52 cards to guarantee that three cards are from some same suit is 
a.  3 
b.  8 
c.  9 
d.  12 
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Answer: (c).9

52.  An n x n array v is defined as follows: v[i, j] = ij for all i, j, 1 <= i <= n, 1 <= j <= n The sum of the elements of the array v is 
a.  0 
b.  n1 
c.  n^2  3n + 2 
d.  n^2 (n+1)/2 
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Answer: (a).0

53.  The determinant of the matrix is 
a.  5 
b.  0 
c.  4 
d.  20 
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Answer: (c).4

54.  X, Y and Z are closed intervals of unit length on the real line. The overlap of X and Y is half a unit. The overlap of Y and Z is also half a unit. Let the overlap of X and Z be k units. Which of the following is true? 
a.  k must be 1 
b.  k must be 0 
c.  k can take any value between 0 and 1 
d.  None of the above 
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Answer: (d).None of the above

55.  E1 and E2 are events in a probability space satisfying the following constraints: Pr(E1) = Pr(E2) Pr(EI U E2) = 1 E1 and E2 are independent The value of Pr(E1), the probability of the event E1 is 
a.  0 
b.  1/4 
c.  1/2 
d.  1 
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Answer: (d).1

56.  Which of the following statements is true ? 
a.  S > T 
b.  S = T 
c.  S < T and 2S > T 
d.  2S <= T 
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Answer: (a).S > T

57.  A polynomial p(x) satisfies the following: p(1) = p(3) = p(5) = 1 p(2) = p(4) = 1 The minimum degree of such a polynomial is 
a.  1 
b.  2 
c.  3 
d.  4 
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Answer: (d).4

58.  A relation R is defined on the set of integers as xRy if f(x + y) is even. Which of the following statements is true? 
a.  R is not an equivalence relation 
b.  R is an equivalence relation having 1 equivalence class 
c.  R is an equivalence relation having 2 equivalence classes 
d.  R is an equivalence relation having 3 equivalence classes 
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Answer: (c).R is an equivalence relation having 2 equivalence classes

59.  Let P(S) denotes the power set of set S. Which of the following is always true? 
a.  a 
b.  b 
c.  c 
d.  d 
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Answer: (b).b

60.  Let a, b, c, d be propositions. Assume that the equivalences a ↔ (b Vb) and b ↔ c hold. Then the truth value of the formula (a ∧ b) → (a ∧ c) ∨ d) is always 
a.  True 
b.  False 
c.  Same as the truth value of b 
d.  Same as the truth value of d 
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Answer: (a).True
