# GATE CS 1996

 Question 1
Let A and B be sets and let Ac and Bc denote the complements of the sets A and B. The set (A−B)(B−A)(AB) is equal to a). A ∪ B b). A∪ Bc c). A ∩ B d). A∩ Bc
 A a B b C c D d
Set Theory & Algebra    GATE CS 1996
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 Question 2
Let X= {2, 3, 6, 12, 24}, Let ≤ be the partial order defined by X ≤ Y if x divides y. Number of edges in the Hasse diagram of (X,≤) is
 A 3 B 4 C 9 D None of the above
Set Theory & Algebra    GATE CS 1996
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 Question 3
Suppose X and Y are sets and |X| and |Y| are their respective cardinalities. It is given that there are exactly 97 functions from X to Y. From this one can conclude that
 A |X|=1,|Y|=97 B |X|=97,|Y|=1 C |X|=97,|Y|=97 D None of the above
Set Theory & Algebra    GATE CS 1996
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 Question 4
Which of the following statement is false?
 A The set of rational numbers is an abelian group under addition B The set of integers in an abelian group under addition C The set of rational numbers form an abelian group under multiplication D The set of real numbers excluding zero is an abelian group under multiplication
Set Theory & Algebra    GATE CS 1996
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 Question 5
Two dice are thrown simultaneously. The probability  that at least one of them will have 6 facing up is
 A 1/36 B 1/3 C 25/36 D 11/36
Probability    GATE CS 1996
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Question 5 Explanation:
There can be two cases:
• Case 1: Exactly one dice has 6 facing up and other dice can have any number from 1,2,3,4,5 facing up. There will be 5*2=10 such occurrences.
• Case 2: Both of the dices having 6 coming up. Only one possible case exists for the same.
Thus total favorable cases = 10+1 = 11 While total cases = 6*6=36 Which gives the desired probability as: 11/36.
This explanation is contributed by Pradeep Pandey.
 Question 6
The formula used to compute an approximation for the second derivative of a function f at a point X0 is A a B b C c D d
Numerical Methods and Calculus    GATE CS 1996
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 Question 7
Let Ax=b be a system of linear equations where A is an m×n matrix and b is a m×1 column vector and X is an n×1 column vector of unknowns. Which of the following is false?
 A The system has a solution if and only if, both A and the augmented matrix [Ab] have the same rank B If m C If m=n and b is a non-zero vector, then the system has a unique solution D The system will have only a trivial solution when m=n, b is the zero vector and rank(A) = n
Linear Algebra    GATE CS 1996
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 Question 8
Which two of the following four regular expressions are equivalent? (ε is the empty string). (i). (00)*(ε+0) (ii). (00)* (iii). 0* (iv). 0(00)*
 A (i) and (ii) B (ii) and (iii) C (i) and (iii) D (iii) and (iv)
Regular languages and finite automata    GATE CS 1996
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 Question 9
Which of the following statements is false?
 A The Halting Problem of Turing machines is undecidable B Determining whether a context-free grammar is ambiguous is undecidable C Given two arbitrary context-free grammars G1 and G2 it is undecidable whether L(G1)=L(G2) D Given two regular grammars G1 and G2 it is undecidable whether L(G1)=L(G2)
Undecidability    GATE CS 1996
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 Question 10
Let L ⊆ ∑* where ∑ = {a, b}. Which of the following is true? a). L= { x | x has an equal number of a's and b's } is regular b). L= { anbn | n ≥ 1 } is regular c). L= { x | x has more a's than b's } is regular d). L= {ambn | m ≥ 1, n ≥ 1 } is regular
 A a B b C c D d
Regular languages and finite automata    GATE CS 1996
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There are 75 questions to complete.

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