# GATE IT 2004

 Question 1
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?
 A $\left(\dfrac{3}{23}\right)$ B $\left(\dfrac{6}{23}\right)$ C $\left(\dfrac{3}{10}\right)$ D $\left(\dfrac{3}{5}\right)$
Discrete Mathematics   Probability Theory
Question 1 Explanation:
 Question 2
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 Programming Language 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?
 A 15 B 20 C 25 D 30
Discrete Mathematics   Set Theory
Question 2 Explanation:
 Question 3
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:
$\qquad(\exists x)(\forall y)[(a(x, y) \wedge b(x, y)) \wedge \neg c(x, y)]$
Which one of the following is its equivalent?
 A $(\forall x)(\exists y)[(a(x, y) \vee b(x, y)) \to c(x, y)]$ B $(\exists x)(\forall y)[(a(x, y) \vee b(x, y)) \wedge\neg c(x, y)]$ C $\neg (\forall x)(\exists y)[(a(x, y) \wedge b(x, y)) \to c(x, y)]$ D $\neg (\forall x)(\exists y)[(a(x, y) \vee b(x, y)) \to c(x, y)]$
Discrete Mathematics   Propositional Logic
Question 3 Explanation:
 Question 4
Let $R_{1}$ be a relation from $A =\left \{ 1,3,5,7 \right \}$ to $B = \left \{ 2,4,6,8 \right \}. R_{2}$ be another relation from B to C = {1, 2, 3, 4} as defined below:

i. An element x in A is related to an element y in B (under $R_{1}$) if x + y is divisible by 3.
ii. An element x in B is related to an element y in C (under $R_{2}$) if x + y is even but not divisible by 3.

Which is the composite relation $R_{1}R_{2}$ from A to C?
 A $R_{1}R_{2} = {(1, 2), (1, 4), (3, 3), (5, 4), (7, 3)}$ B $R_{1}R_{2} = {(1, 2), (1, 3), (3, 2), (5, 2), (7, 3)}$ C $R_{1}R_{2} = {(1, 2), (3, 2), (3, 4), (5, 4), (7, 2)}$ D $R_{1}R_{2} = {(3, 2), (3, 4), (5, 1), (5, 3), (7, 1)}$
Discrete Mathematics   Relation
Question 4 Explanation:
 Question 5
What is the maximum number of edges in an acyclic undirected graph with n vertices?
 A n-1 B n C n+1 D 2n-1
Discrete Mathematics   Graph Theory
Question 5 Explanation:
 Question 6
What values of x, y and z satisfy the following system of linear equations?
$\begin{bmatrix} 1 &2 &3 \\ 1& 3 &4 \\ 2& 2 &3 \end{bmatrix} \begin{bmatrix} x\\y \\ z \end{bmatrix} = \begin{bmatrix} 6\\8 \\ 12 \end{bmatrix}$
 A x = 6, y = 3, z = 2 B x = 12, y = 3, z = - 4 C x = 6, y = 6, z = - 4 D x = 12, y = - 3, z = 0
Engineering Mathematics   Linear Algebra
Question 6 Explanation:
 Question 7
Which one of the following regular expressions is NOT equivalent to the regular expression $(a + b + c)^*?$
 A $(a^* + b^* + c^*)^*$ B $(a^*b^*c^*)^*$ C $((ab)^* + c^*)^*$ D $(a^*b^* + c^*)^*$
Theory of Computation   Regular Expression
Question 7 Explanation:
 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?
 A 2 B 4 C 5 D 6
Digital Logic   Boolean Algebra
Question 8 Explanation:
 Question 9
Which one of the following statements is FALSE?
 A There exist context-free languages such that all the context-free grammars generating them are ambiguous B An unambiguous context-free grammar always has a unique parse tree for each string of the language generated by it C Both deterministic and non-deterministic pushdown automata always accept the same set of languages D A finite set of string from some alphabet is always a regular language
Theory of Computation   Context Free Grammar
Question 9 Explanation:
 Question 10
What is the minimum size of ROM required to store the complete truth table of an 8-bit $\times$ 8-bit multiplier?
 A 32 K $\times$ 16 bits B 64 K $\times$ 16 bits C 16 K $\times$ 32 bits D 64 K $\times$ 32 bits
Computer Organization   Memory Chip Design
Question 10 Explanation:
There are 10 questions to complete. 