# GATE CS 1996

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Question 1 |

Let

*A*and*B*be sets and let*A*and^{c}*B*denote the complements of the sets^{c}*A*and*B*. The set*(A−B)*∪*(B−A)*∪*(A*∩*B*) is equal to a). A ∪ B b). A^{c }∪ B^{c}c). A ∩ B d). A^{c }∩ B^{c}a | |

b | |

c | |

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

3 | |

4 | |

9 | |

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

|X|=1,|Y|=97 | |

|X|=97,|Y|=1 | |

|X|=97,|Y|=97 | |

None of the above |

**Set Theory & Algebra**

**GATE CS 1996**

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Question 4 |

Which of the following statement is false?

The set of rational numbers is an abelian group under addition | |

The set of integers in an abelian group under addition | |

The set of rational numbers form an abelian group under multiplication | |

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

1/36 | |

1/3 | |

25/36 | |

11/36 |

**Probability**

**GATE CS 1996**

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Question 5 Explanation:

There can be two cases:

This explanation is contributed by

**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.

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 X_{0}isa | |

b | |

c | |

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?The system has a solution if and only if, both A and the augmented matrix [Ab] have the same rank | |

If m | |

If m=n and b is a non-zero vector, then the system has a unique solution | |

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)*(i) and (ii) | |

(ii) and (iii) | |

(i) and (iii) | |

(iii) and (iv) |

**Regular languages and finite automata**

**GATE CS 1996**

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Question 9 |

Which of the following statements is false?

The Halting Problem of Turing machines is undecidable | |

Determining whether a context-free grammar is ambiguous is undecidable | |

Given two arbitrary context-free grammars G1 and G2 it is undecidable whether L(G1)=L(G2) | |

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= {*a*| n ≥ 1 } is regular c). L= { x | x has more^{n}b^{n}*a*'s than*b*'s } is regular d). L= {*a*| m ≥ 1, n ≥ 1 } is regular^{m}b^{n}a | |

b | |

c | |

d |

**Regular languages and finite automata**

**GATE CS 1996**

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There are 75 questions to complete.