# GATE IT 2006

 Question 1
In a certain town, the probability that it will rain in the afternoon is known to be 0.6. Moreover, meteorological data indicates that if the temperature at noon is less than or equal to $25^{\circ}C$, the probability that it will rain in the afternoon is 0.4. The temperature at noon is equally likely to be above $25^{\circ}C$, or at/below $25^{\circ}C$. What is the probability that it will rain in the afternoon on a day when the temperature at noon is above $25^{\circ}C$?
 A 0.4 B 0.6 C 0.8 D 0.9
Discrete Mathematics   Probability Theory
Question 1 Explanation:
 Question 2
For the set N of natural numbers and a binary operation $f : N \times N \to N$, an element $z \in N$ is called an identity for f, if f (a, z) = a = f(z, a), for all $a \in N$. Which of the following binary operations have an identity?
i. $f (x, y) = x + y - 3$
ii. $f (x, y) = \max(x, y)$
iii. $f (x, y) = x^y$
 A I and II only B II and III only C I and III only D None of these
Discrete Mathematics   Set Theory
Question 2 Explanation:
 Question 3
In the automaton below, s is the start state and t is the only final state.

Consider the strings $u = abbaba, v = bab, \text{ and } w = aabb$. Which of the following statements is true?
 A The automaton accepts u and v but not w B The automaton accepts each of u, v, and w C The automaton rejects each of u, v, and w D The automaton accepts u but rejects v and w
Theory of Computation   Finite Automata
Question 3 Explanation:
 Question 4
In the context-free grammar below, S is the start symbol, a and b are terminals, and$\epsilon$ denotes the empty string
$S \rightarrow aSa \mid bSb \mid a \mid b \mid \epsilon$
Which of the following strings is NOT generated by the grammar?
 A aaaa B baba C abba D babaaabab
Theory of Computation   Context Free Grammar
Question 4 Explanation:
 Question 5
Which regular expression best describes the language accepted by the non-deterministic automaton below?

 A $(a + b)^* \ a(a + b)b$ B $(abb)^*$ C $(a + b)^* \ a(a + b)^* \ b(a + b)^*$ D $(a + b)^*$
Theory of Computation   Regular Expression
Question 5 Explanation:
 Question 6
Given a boolean function f $(x_1, x_2, \ldots, x_n)$, which of the following equations is NOT true?
 A $f (x_1, x_2, \ldots, x_n) = x_1'f(x_1, x_2, \ldots, x_n) + x_1f(x_1, x_2, \ldots, x_n)$ B $f (x_1, x_2, \ldots, x_n) = x_2f(x_1, x_2, \ldots , x_n) + x_2'f(x_1, x_2, \ldots ,x_n)$ C $f (x_1, x_2, \ldots, x_n) = x_n'f(x_1, x_2, \ldots, 0) + x_nf(x_1, x_2, \ldots,1)$ D $f (x_1, x_2, \ldots , x_n) = f(0, x_2, \ldots , x_n) + f(1, x_2, \ldots, x_n)$
Digital Logic   Boolean Algebra
Question 6 Explanation:
 Question 7
The addition of 4-bit, two's complement, binary numbers 1101 and 0100 results in
 A 0001 and an overflow B 1001 and no overflow C 0001 and no overflow D 1001 and an overflow
Digital Logic   Number System
Question 7 Explanation:
 Question 8
Which of the following DMA transfer modes and interrupt handling mechanisms will enable the highest I/O band-width?
 A Transparent DMA and Polling interrupts B Cycle-stealing and Vectored interrupts C Block transfer and Vectored interrupts D Block transfer and Polling interrupts
Computer Organization   IO Interface
Question 8 Explanation:
 Question 9
In a binary tree, the number of internal nodes of degree 1 is 5, and the number of internal nodes of degree 2 is 10. The number of leaf nodes in the binary tree is
 A 10 B 11 C 12 D 15
Data Structure   Binary Tree
Question 9 Explanation:
 Question 10
A problem in NP is NP-complete if
 A it can be reduced to the 3-SAT problem in polynomial time B the 3-SAT problem can be reduced to it in polynomial time C it can be reduced to any other problem in NP in polynomial time D some problem in NP can be reduced to it in polynomial time
Algorithm   P-NP Theory
Question 10 Explanation:
There are 10 questions to complete.