GATE-CS-2005

Question 41

What is the first order predicate calculus statement equivalent to the following? Every teacher is liked by some student

• ∀(x) [teacher (x) → ∃ (y) [student (y) → likes (y, x)]]
• ∀ (x) [teacher (x) → ∃ (y) [student (y) ^ likes (y, x)]]
• ∃ (y) ∀ (x) [teacher (x) → [student (y) ^ likes (y, x)]]
• ∀ (x) [teacher (x) ^ ∃ (y) [student (y) → likes (y, x)]]

Question 42

Let R and S be any two equivalence relations on a non-empty set A. Which one of the following statements is TRUE?

• R ∪ S, R ∩ S are both equivalence relations

• R ∪ S is an equivalence relation

• R ∩ S is an equivalence relation

• Neither R ∪ S nor R ∩ S is an equivalence relation

Question 43

Let f: B → C and g: A → B be two functions and let h = f o g. Given that h is an onto function. Which one of the following is TRUE?

• f and g should both be onto functions.
• f should be onto but g need not be onto
• g should be onto but f need not be onto
• both f and g need not be onto

Question 44

What is the minimum number of ordered pairs of non-negative numbers that should be chosen to ensure that there are two pairs (a, b) and (c, d) in the chosen set such that "a ≡ c mod 3" and "b ≡ d mod 5"

• 4
• 6
• 16
• 24

Question 45

Consider three decision problems P1, P2 and P3. It is known that P1 is decidable and P2 is undecidable. Which one of the following is TRUE?

• P3 is decidable if P1 is reducible to P3
• P3 is undecidable if P3 is reducible to P2
• P3 is undecidable if P2 is reducible to P3
• P3 is decidable if P3 is reducible to P2\'s complement

Question 46

Consider the set H of all 3 × 3 matrices of the type 

where a, b, c, d, e and f are real numbers and abc ≠ 0. Under the matrix multiplication operation, the set H is 

• a group
  

• a monoid but not a group
  

• a semigroup but not a monoid
  

• neither a group nor a semigroup
  

Question 47

Which one of the following graphs is NOT planar?

• G1
• G2
• G3
• G4

Question 48

Consider the following system of equations in three real variables x₁, x₂ and x₃

2x1 - x2 + 3x3 = 1 
3x1- 2x2 + 5x3 = 2
-x1 + 4x2 + x3 = 3 

This system of equations has

• no solution

• a unique solution

• more than one but a finite number of solutions

• an infinite number of solutions

Question 49

What are the eigenvalues of the following 2 × 2 matrix? 

• -1 and 1

• 1 and 6

• 2 and 5

• 4 and -1

Question 50

Let G(x) = 1/(1 - x)² = , where | x | < 1. What is g(i) ?

• i
• i+1
• 2i
• 2ⁱ