GATE CSE 1996

Question 1
Let A and B be sets and let A^c and B^c denote the complements of the sets A and B. The set (A-B) \cup (B-A) \cup (A \cap B) is equal to
A
A \cup B
B
A^c \cup B^c
C
A \cap B
D
A^c \cap B^c
Discrete Mathematics   Set Theory
Question 2
Let X = \{2, 3, 6, 12, 24\}, Let \leq be the partial order defined by X \leq Y if x divides y. Number of edges in the Hasse diagram of (X, \leq) is
A
3
B
4
C
9
D
None of the above
Discrete Mathematics   Set Theory
Question 3
Suppose X and Y are sets and |X| and |Y| are their respective cardinality. 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
Discrete Mathematics   Functions
Question 4
Which of the following statements 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
Discrete Mathematics   Group Theory
Question 5
Two dice are thrown simultaneously. The probability that at least one of them will have 6 facing up is
A
\frac{1}{36}
B
\frac{1}{3}
C
\frac{25}{36}
D
\frac{11}{36}
Discrete Mathematics   Probability Theory
Question 6
The formula used to compute an approximation for the second derivative of a function f at a point x_0 is
A
\dfrac{f(x_0 +h) + f(x_0 - h)}{2}
B
\dfrac{f(x_0 +h) - f(x_0 - h)}{2h}
C
\dfrac{f(x_0 +h) + 2f(x_0) + f(x_0 - h)}{h^2}
D
\dfrac{f(x_0 +h) - 2f(x_0) + f(x_0 - h)}{h^2}
Engineering Mathematics   Calculus
Question 7
Let AX = b be a system of linear equations where A is an m \times n matrix and b is a m \times 1 column vector and X is an n \times 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 < n and b is the zero vector, then the system has infinitely many solutions.
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.
Engineering Mathematics   Linear Algebra
Question 8
Which two of the following four regular expressions are equivalent? (\varepsilon is the empty string).
(i). (00)^ * (\varepsilon +0)
(ii). (00)^*
(iii). 0^*
(iv). 0(00)^*
A
(i) and (ii)
B
(ii) and (iii)
C
(i) and (iii)
D
(iii) and (iv)
Theory of Computation   Regular Expression
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 G_1 and G_2it is undecidable whether L(G_1) = L(G_2)
D
Given two regular grammars G_1 and G_2 it is undecidable whether L(G_1) = L(G_2)
Theory of Computation   Undecidability
Question 10
Let L \subseteq \Sigma^* where \Sigma = \left\{a,b \right\}. Which of the following is true?
A
L = \left\{x \mid x \text{ has an equal number of } a\text{'s and }b\text{'s}\right \} is regular
B
L = \left\{a^nb^n \mid n \geq 1\right \} is regular
C
L = \left\{x \mid x \text{ has more number of }a\text{'s than }b\text{'s}\right \} is regular
D
L = \left\{a^mb^n \mid m \geq 1, n \geq 1 \right \} is regular
Theory of Computation   Regular Language
There are 10 questions to complete.

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