GATE-CS-2006

Question 21

For each element in a set of size 2n, an unbiased coin is tossed. The 2n coin tosses are independent. An element is chosen if the corresponding coin toss were head. The probability that exactly n elements are chosen is:

• (2nCn) / (4^n)
• (2nCn) / (2^n)
• 1 / (2nCn)
• 1/2

Question 22

Let E, F and G be finite sets. Let X = (E ∩ F) - (F ∩ G) and Y = (E - (E ∩ G)) - (E - F). Which one of the following is true?

• X ⊂ Y

• X ⊃ Y

• X = Y

• X - Y ≠ φ and Y - X ≠ φ

Question 23

F is an n*n real matrix. b is an n*1 real vector. Suppose there are two n*1 vectors, u and v such that, u ≠ v and Fu = b, Fv = b. Which one of the following statements is false?

• Determinant of F is zero.
• There are an infinite number of solutions to Fx = b
• There is an x≠0 such that Fx = 0
• F must have two identical rows

Question 24

Given a set of elements N = {1, 2, ..., n} and two arbitrary subsets A⊆N and B⊆N, how many of the n! permutations π from N to N satisfy min(π(A)) = min(π(B)), where min(S) is the smallest integer in the set of integers S, and π(S) is the set of integers obtained by applying permutation π to each element of S?

• (n - |A ∪ B|) |A| |B|
• (|A|²+|B|²)n²
• n! |A∩B| / |A∪B|
• |A∩B|²nC|A∪B|

Question 25

Let S = {1, 2, 3, ...., m}, m>3. Let x1, x2,....xn be the subsets of S each of size 3. Define a function f from S to the set of natural numbers as, f(i) is the number of sets [Tex]X_j[/Tex] that contain the element i. That is, f(i) = |{j|i [Tex]\\epsilon[/Tex] [Tex]X_j[/Tex]}|. Then, [Tex]\\sum_{i=1}f(i)[/Tex] is :

• 3m
• 3n
• 2m + 1
• 2n + 1

Question 26

Which one of the first order predicate calculus statements given below correctly express the following 
English statement? 

Tigers and lions attack if they are hungry or threatened. 

• B
  

• A
  

• C
  

• D
  

Question 27

Consider the following propositional statements: P1 : ((A ∧ B) → C)) ≡ ((A → C) ∧ (B → C)) P2 : ((A ∨ B) → C)) ≡ ((A → C) ∨ (B → C)) Which one of the following is true?

• P1 is a tautology, but not P2

• P2 is a tautology, but not P1

• P1 and P2 are both tautologies

• Both P1 and P2 are not tautologies

Question 28

A logical binary relation □ ,is defined as follows: 

Let ~ be the unary negation (NOT) operator, with higher precedence than □. 

Which one of the following is equivalent to A∧B ? 

(A) (~A □ B)  
(B) ~(A □ ~B) 
(C) ~(~A □ ~B)   
(D) ~(~A □ B) 

• C
  

• D
  

• B
  

• A
  

Question 29

If s is a string over (0 + 1)* then let n₀(s) denote the number of 0’s in s and n₁(s) the number of 1’s in s. Which one of the following languages is not regular? 7" />

• A
• B
• C
• D

Question 30

For S ∈ (0 + 1) * let d(s) denote the decimal value of s (e.g. d(101) = 5). Let L = {s ∈ (0 + 1)* d(s)mod5 = 2 and d(s)mod7 != 4}. Which one of the following statements is true?

• L is recursively enumerable, but not recursive
• L is recursive, but not context-free
• L is context-free, but not regular
• L is regular