# 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 1 Explanation:
 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 2 Explanation:
 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 3 Explanation:
 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 4 Explanation:
 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 5 Explanation:
 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 6 Explanation:
 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 7 Explanation:
 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 8 Explanation:
 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 $G_2$it 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 9 Explanation:
 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
Question 10 Explanation:
There are 10 questions to complete.