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
3/23
B
6/23
C
3/10
D
3/5
General Aptitude    GATE-IT-2004    
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Question 1 Explanation: 
x- total families. Given:
  • 50% -3 Children -> (5*x/10) * 3=(1*x/2)*3
  • 30% -2 Children-> (3*x /10) * 2
  • Rear 20%- 1child -> (2*x/10)*1 =1*x/5
Probability = No of ways it can happen/Total Outcomes =( (3/10)*2 ) / ( (1/2)*3 + (3/10)*2 +1/5 ) = 6/23 So Answer is B
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 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?
 
A
15
B
20
C
25
D
35
General Aptitude    GATE-IT-2004    
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Question 2 Explanation: 
P(AUBUC)=P(A)+P(B)+P(C)- P(A∩B)-P(A∩C)- P(B∩C)+P(A∩B∩C)
125+85+65-503530+15 =175
No of students not taking any courses-> 200175=25 So C is the Answer.
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:
(∃x)(∀y)[(a(x, y) ∧ b(x, y)) ∧ ¬c(x, y)] Which one of the following is its equivalent?  
A
(∀x)(∃y)[(a(x, y) ∨ b(x, y)) → c(x, y)]
B
(∃x)(∀y)[(a(x, y) ∨ b(x, y)) ∧¬ c(x, y)]
C
¬ (∀x)(∃y)[(a(x, y) ∧ b(x, y)) → c(x, y)]
D
¬ (∀x)(∃y)[(a(x, y) ∨ b(x, y)) → c(x, y)]
Linear Algebra    GATE-IT-2004    
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Question 4
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?  
A
R1R2 = {(1, 2), (1, 4), (3, 3), (5, 4), (7, 3)}
B
R1R2 = {(1, 2), (1, 3), (3, 2), (5, 2), (7, 3)}
C
R1R2 = {(1, 2), (3, 2), (3, 4), (5, 4), (7, 2)}
D
R1R2 = {(3, 2), (3, 4), (5, 1), (5, 3), (7, 1)}
Set Theory & Algebra    GATE-IT-2004    
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Question 4 Explanation: 

R1 is a relation from A = {1, 3, 5, 7} to B = {2, 4, 6, 8} . Under R1, an element x in A is related to an element y in B if x + y is divisible by 3.
Thus, R1 = {(1, 2), (1, 8), (3, 6), (5, 4), (7, 2), (7, 8)}
R2 is a relation from B = {2, 4, 6, 8} to C = {1, 2, 3, 4} Under R2, an element y in B is related to an element z in C if y + z is even but not divisible by 3.
Thus, R2 = {(2, 2), (4, 4), (6, 2), (6, 4), (8, 2)}
it
Thus, R1R2 = {(1, 2), (3, 2), (3, 4), (5, 4), (7, 2)}
 
Thus, option (C) is correct.
 
Please comment below if you find anything wrong in the above post.
Question 5
What is the maximum number of edges in an acyclic undirected graph with n vertices?
A
n-i
B
n
C
n + 1
D
2n-1
Graph    GATE-IT-2004    
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Question 5 Explanation: 
n * (n - 1) / 2 when not cyclic. An acyclic graph with the maximum number of edges is actually a spanning tree and therefore, correct answer is n-1 edges. So Answer is A
Question 6
What values of x, y and z satisfy the following system of linear equations?
eq  
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=O
Linear Algebra    GATE-IT-2004    
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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*)*
Regular languages and finite automata    GATE-IT-2004    
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Question 7 Explanation: 
C- (ab)* + c*)* will always give strings with "ab" together.Whereas (a+b+c)* would generate language where a,b,c may not be always together. A,B,D may generate same language as (a+b+c)*   So, Answer is (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?
A
3
B
4
C
5
D
6
Digital Logic & Number representation    GATE-IT-2004    
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Question 8 Explanation: 
itq Pic Courtesy: http://www.electronics-tutorials.ws/logic/logic_7.html Other way around: x XOR y = x’y+xy’ = x’y+xy’+xx’+yy’ = (x+y) (x’+y’) Using NAND gates F= (x+y)(xy)’ = x. (xy)’ + y. (xy)’ Taking compliment F’= ( x. (xy)’ + y. (xy)’ )’ = (x. (xy)’)’. (y. (xy)’) Compliment again F=( (x. (xy)’)’. (y. (xy)’) )’ So Answer is B
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 one alphabet is always a regular language.
Context free languages and Push-down automata    GATE-IT-2004    
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Question 9 Explanation: 

A) For real-world programming languages, the reference CFG is often ambiguous, due to issues such as the dangling else problem. //Wikipedia

B) A string is ambiguous if it has two distinct parse trees;The grammar is unambiguous,if a string has distinct parse trees.

C) Deterministic pushdown automata can recognize all deterministic context-free languages while        nondeterministic ones can recognize all context-free languages

Therefore it's FALSE

D)Properties of Regular Language:

  • The set of regular languages over an alphabet is closed under operations union, concatenation and Kleene star.
  • Finite languages are regular
So Answer is C
Question 10
What is the minimum size of ROM required to store the complete truth table of an 8-bit x 8-bit multiplier?
A
32 K x 16 bits
B
64 K x 16 bits
C
16 K x 32 bits
D
64 K x 32 bits
Computer Organization and Architecture    GATE-IT-2004    
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Question 10 Explanation: 
Input to ROM - 2 lines ,8 bit each. Possible combinations in ROM - (2^8)x(2^8) Size of truth table = (2^8)*(2^8)=2^16=64 KB Maximum output size = 16 bit So, Answer is B
There are 90 questions to complete.

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