GATE-IT-2004

Question 1

What is the maximum number of edges in an acyclic undirected graph with n vertices?

• n-1
• n
• n + 1
• 2n-1

Question 2

Let a(x, y), b(x, y,) and c(x, y) be three statements with variables x and y chosen from some universe. Consider the following statement:

(∃x)(∀y)[(a(x, y) ∧ b(x, y)) ∧ ¬c(x, y)] Which one of the following is its equivalent?  

• (∀x)(∃y)[(a(x, y) ∨ b(x, y)) → c(x, y)]
• (∃x)(∀y)[(a(x, y) ∨ b(x, y)) ∧¬ c(x, y)]
• ¬ (∀x)(∃y)[(a(x, y) ∧ b(x, y)) → c(x, y)]
• ¬ (∀x)(∃y)[(a(x, y) ∨ b(x, y)) → c(x, y)]

Question 3

Let R₁ be a relation from A = {1, 3, 5, 7} to B = {2, 4, 6, 8} and R₂ be another relation from B to C = {1, 2, 3, 4} as defined below:

1. An element x in A is related to an element y in B (under R₁) if x + y is divisible by 3.
2. An element x in B is related to an element y in C (under R₂) if x + y is even but not divisible by 3.

Which is the composite relation R₁R₂ from A to C?  

• R1R2 = {(1, 2), (1, 4), (3, 3), (5, 4), (7, 3)}
• R1R2 = {(1, 2), (1, 3), (3, 2), (5, 2), (7, 3)}
• R1R2 = {(1, 2), (3, 2), (3, 4), (5, 4), (7, 2)}
• R1R2 = {(3, 2), (3, 4), (5, 1), (5, 3), (7, 1)}

Question 4

In a class of 200 students, 125 students have taken Programming Language course, 85 students have taken Data Structures course, 65 students have taken Computer Organization course; 50 students have taken both Programming Language and Data Structures, 35 students have taken both Data Structures and Computer Organization; 30 students have taken both Data Structures and Computer Organization, 15 students have taken all the three courses.How many students have not taken any of the three courses?

• 15
• 20
• 25
• 35

Question 5

In a population of N families, 50% of the families have three children, 30% of the families have two children and the remaining families have one child. What is the probability that a randomly picked child belongs to a family with two children?

• 3/23
• 6/23
• 3/10
• 3/5

Question 6

What values of x, y and z satisfy the following system of linear equations? 

[caption width="800"] [/caption]

• x=6, y=3, z=2

• x=12, y=3, z=-4

• x=6, y=6, z=-4

• x=12, y=-3, z=0

Question 7

Which one of the following regular expressions is NOT equivalent to the regular expression (a + b + c) *?

• (a* + b* + c*)*
• (a*b*c*)*
• ((ab)* + c*)*
• (a*b* + c*)*

Question 8

What is the minimum number of NAND gates required to implement a 2-input EXCLUSIVE-OR function without using any other logic gate?

• 3
• 4
• 5
• 6

Question 9

Which one of the following statements is FALSE?

• There exist context-free languages such that all the context-free grammars generating them are ambiguous

• An unambiguous context free grammar always has a unique parse tree for each string of the language generated by it.

• Both deterministic and non-deterministic pushdown automata always accept the same set of languages

• A finite set of string from one alphabet is always a regular language.

Question 10

What is the minimum size of ROM required to store the complete truth table of an 8-bit x 8-bit multiplier?

• 32 K x 16 bits
• 64 K x 16 bits
• 16 K x 32 bits
• 64 K x 32 bits