81.  Let P(E) denote the probability of the event E. Given P(A) = 1, P(B) = 1/2, the values of P(A  B) and P(B  A) respectively are 
a.  1/4, 1/2 
b.  1/2, 1/14 
c.  1/2, 1 
d.  1, 1/2 
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Answer: (d).1, 1/2

82.  Let A be a sequence of 8 distinct integers sorted in ascending order. How many distinct pairs of sequences, B and C are there such that (i) each is sorted in ascending order, (ii) B has 5 and C has 3 elements, and (iii) the result of merging B and C gives A? 
a.  2 
b.  30 
c.  56 
d.  256 
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Answer: (c).56

83.  Let T(n) be the number of different binary search trees on n distinct elements. Then , where x is 
a.  nk+1 
b.  nk 
c.  nk1 
d.  nk2 
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Answer: (b).nk

84.  Let G be an arbitrary graph with n nodes and k components. If a vertex is removed from G, the number of components in the resultant graph must necessarily lie between 
a.  k and n 
b.  k  1 and k + 1 
c.  k  1 and n  1 
d.  k + 1 and n  k 
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Answer: (c).k  1 and n  1

85.  Which of the following is a valid first order formula ? (Here α and β are first order formulae with x as their only free variable) 
a.  A 
b.  B 
c.  C 
d.  D 
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Answer: (d).D

86.  Consider the following formula a and its two interpretations I1 and I2. Which of the following statements is true? 
a.  I1 satisfies α, I2 does not 
b.  I2 satisfies α, I1 does not 
c.  Neither I2 nor I2 satisfies α 
d.  Both I1 and I2 satisfy α 
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Answer: (d).Both I1 and I2 satisfy α

87.  m identical balls are to be placed in n distinct bags. You are given that m ≥ kn, where, k is a natural number ≥ 1. In how many ways can the balls be placed in the bags if each bag must contain at least k balls? 
a.  A 
b.  B 
c.  C 
d.  D 
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Answer: (b).B

88.  Consider the following recurrence relation. The value of T(m^2) for m ≥ 1 is 
a.  (m/6) (21m  39) + 4 
b.  (m/6) (4m^2  3m + 5) 
c.  (m/2) (m^2.5  11m + 20)  5 
d.  (m/6) (5m^3  34m^2 + 137m  104) + (5/6) 
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Answer: (b).(m/6) (4m^2  3m + 5)

89.  Let f : A → B be an injective (onetoone) function.
Define g : 2^A → 2^B as : g(C) = {f(x)  x ∈ C}, for all subsets C of A. Define h : 2^B → 2^A as : h(D) = {x  x ∈ A, f(x) ∈ D}, for all subsets D of B. Which of the following statements is always true ? 
a.  g(h(D)) ⊆ D 
b.  g(h(D)) ⊇ D 
c.  g(h(D)) ∩ D = ф 
d.  g(h(D)) ∩ (B  D) ≠ ф 
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Answer: (a).g(h(D)) ⊆ D

90.  Consider the set {a, b, c} with binary operators + and × defined as follows :
+ a b c × a b c a b a c a a b c b a b c b b c a c a c b c c c b For example, a + c = c, c + a = a, c × b = c and b × c = a. Given the following set of equations : (a × x) + (a × y) = c (b × x) + (c × y) = c The number of solution(s) (i.e., pair(s) (x, y)) that satisfy the equations is : 
a.  0 
b.  1 
c.  2 
d.  3 
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Answer: (c).2
