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 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 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 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 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 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 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 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 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 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
There are 10 questions to complete.

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