# GATE-IT-2004

• Last Updated : 22 Jun, 2021

 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
GATE-IT-2004    General Aptitude
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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
GATE-IT-2004    General Aptitude
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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)]
GATE-IT-2004    Linear Algebra
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Question 3 Explanation:
Choice (C) is = ¬ (∀x)(∃y)[(a(x, y) ∧ b(x, y)) → c(x, y)] = ¬ (∀x)(∃y)[ a∧ b → c ] = ¬ (∀x)(∃y)[ (ab)' + c] = ∃ x ∀ y[ (ab)' + c]' = ∃ x ∀ y[ abc' ] = ∃ x ∀ y[ a ∧ b ∧ c ¬c] Which is same as the given expression. (∃x)(∀y)[(a(x, y) ∧ b(x, y)) ∧ ¬c(x, y)]
 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)}
GATE-IT-2004    Set Theory & Algebra
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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)}
Then the composition of R1 with R2, denoted R2R1, is the relation from A to C defined by the following property: (x, z) R2R1 If and only if there is a y E B such that (x, y) R1 and (y, z) R2.
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-1 B n C n + 1 D 2n-1
GATE-IT-2004    Top MCQs on Graph Data Strcuture with Answers
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Question 5 Explanation:
n * (n - 1) / 2 when cyclic. But acyclic graph with the maximum number of edges is actually a spanning tree and therefore, correct answer is n-1 edges.
 Question 6

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

 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
GATE-IT-2004    Linear Algebra
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Question 6 Explanation:

1 * x + 2 * y + 3 * z = 6

1 * x + 3 * y + 4 * z = 8

2 * x + 2 * y + 3 * z = 12

Put x =6 , y = 6 and z = -4

The above three equation are satisfied.

 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*)*
GATE-IT-2004    Regular languages and finite automata
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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
GATE-IT-2004    Digital Logic & Number representation    Logic functions and Minimization
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Question 8 Explanation:
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.
GATE-IT-2004    Context free languages and Push-down automata
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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