GATE 2008
Question 2 
If P, Q, R are subsets of the universal set U, then (P∩Q∩R) ∪ (P^{c}∩Q∩R) ∪ Q^{c }∪ R^{c} is
Q^{c} ∪ R^{c}
 
P ∪ Q^{c} ∪ R^{c}  
P^{c} ∪ Q^{c} ∪ R^{c}  
U 
(P∩Q∩R)∪(P^{c}∩Q∩R)∪Q^{c}∪R^{c}
It can be written as the p.q.r + p'.q.r +q' + r'
=> (p+p').q.r + q' + r'
=> q.r + (q'+r')
=> q.r + q' + r' = 1 i.e., U
Question 3 
The following system of equations

x_{1} + x_{2} + 2x_{3} = 1
x_{1} + 2x_{2} + 3x_{3} = 2
x_{1} + 4x_{2} + ax_{3} = 4
has a unique solution. The only possible value(s) for a is/are
0  
either 0 or 1  
one of 0, 1 or 1  
any real number 
When a5 = 0, then rank(A) = rank[AB]<3,
So infinite number of solutions.
But, it is given that the given system has unique solution i.e., rank(A) = rank[AB] = 3 will be retain only if a5 ≠ 0.
Question 4 
In the IEEE floating point representation, the hexadecimal value 0×00000000 corresponds to
the normalized value 2^{  127}  
the normalized value 2^{  126}  
the normalized value + 0  
the special value + 0 
Question 5 
In the Karnaugh map shown below, X denotes a don’t care term. What is the minimal form of the function represented by the Karnaugh map?
Question 6 
Let r denote number system radix. The only value(s) of r that satisfy the equation is/are
decimal 10  
decimal 11  
decimal 10 and 11  
any value > 2 
(r^{2} + 2r + 1)^{1/2} = r + 1
(r + 1)^{2} * 1/2 = r + 1
r + 1 = r + 1 Any value of r will satisfy the above equation. But the radix should be greater than 2 because the 121 has 2. So r > 2 is correct.
Question 7 
The most efficient algorithm for finding the number of connected components in an undirected graph on n vertices and m edges has time complexity
θ(n)  
θ(m)  
θ(m+n)  
θ(mn) 
Suppose if we are using Adjacency matrix means it takes θ(n^{2}).
Question 8 
Given f_{1}, f_{3} and f in canonical sum of products form (in decimal) for the circuit
 f_{1} = Σm(4,5,6,7,8)
f_{3} = Σm(1,6,15)
f = Σm(1,6,8,15)
then f_{2} is
Σm(4,6)  
Σm(4,8)  
Σm(6,8)  
Σm(4,6,8) 
f_{1}*f_{2} is intersection of minterms of f_{1} and f_{2}
f = (f_{1}*f_{2}) + f_{3} is union of minterms of (f_{1}*f_{2}) and f_{3}
Σm(1,6,8,15) = Σm(4,5,6,7,8) * f_{2} + Σm(1,6,15)
Options A, B and D have minterm m_{4} which result in Σm(1,4,6,15), Σm(1,4,6,8, 15) and Σm(1,4,6,8, 15)respectively and they are not equal to f.
Option C : If f_{2} = Σm(6,8)
RHS: Σm(4,5,6,7,8) * Σm(6,8) + Σm(1,6,15)
= Σm(6,8) + Σm(1,6,15)
= Σm(1,6,8,15)
= f = LHS
Question 9 
Which of the following is true for the language {a^{p}p is a prime} ?
It is not accepted by a Turing Machine  
It is regular but not contextfree
 
It is contextfree but not regular  
It is neither regular nor contextfree, but accepted by a Turing machine 
Question 10 
Which of the following are decidable?

I. Whether the intersection of two regular languages is infinite
II. Whether a given contextfree language is regular
III. Whether two pushdown automata accept the same language
IV. Whether a given grammar is contextfree
I and II  
I and IV  
II and III  
II and IV 
Statement IV is also decidable, we need to check that whether the given grammar satisfies the CFG rule (TYPE 2 grammar productions).
But statements II and III are undecidable, as there doesn’t exist any algorithm to check whether a given contextfree language is regular and whether two pushdown automata accept the same language.
Question 11 
Which of the following describes a handle (as applicable to LRparsing) appropriately?
It is the position in a sentential form where the next shift or reduce operation will occur.
 
It is nonterminal whose production will be used for reduction in the next step.  
It is a production that may be used for reduction in a future step along with a position in the sentential form where the next shift or reduce operation will occur.
 
It is the production p that will be used for reduction in the next step along with a position in the sentential form where the right hand side of the production may be found.

Question 12 
Some code optimizations are carried out on the intermediate code because
They enhance the portability of the compiler to other target processors  
Program analysis is more accurate on intermediate code than on machine code
 
The information from dataflow analysis cannot otherwise be used for optimization  
The information from the front end cannot otherwise be used for optimization 
Question 13 
If L and are recursively enumerable, then L is
regular  
contextfree  
contextsensitive  
recursive 
If are recursively enumerable, then it implies that there exist a Turing Machine (lets say M2) which always HALT for the strings which is NOT in L(as L is complement of Since we can combine both Turing machines (M1 and M2) and obtain a new Turing Machine (say M3) which always HALT for the strings if it is in L and also if it is not in L. This implies that L must be recursive.
Question 14 
What is the maximum size of data that the application layer can pass on to the TCP layer below?
Any size  
2^{16} bytessize of TCP header  
2^{16} bytes  
1500 bytes 
Transport Layer  65515 byte
Network layer  65535 byte
Data link layer  1500 byte
Question 15 
Which of the following tuple relational calculus expression(s) is/are equivalent to ∀t ∈ r(P(t))?
 I. ¬∃t ∈ r(P(t))
II. ∃t ∉ r(P(t))
III. ¬∃t ∈ r(¬P(t))
IV. ∃t ∉ r(¬P(t))
I only  
II only  
III only  
III and IV only 
∀xP(x) ≡ ∼∃x(∼P(x))
∼∀x(∼P(x)) ≡ ∃x(P(x))
Given: ∀t ∈ r(P(t)) (1)
As per Demorgan law
(1) ⇒ ∼∃t ∈ r(∼P(t))
which is option (III).
Question 16 
A clustering index is defined on the fields which are of type
nonkey and ordering  
nonkey and nonordering  
key and ordering
 
key and nonordering

Question 17 
Which of the following system calls results in the sending of SYN packets?
socket  
bind  
listen  
connect 
1) The client requests a connection by sending a SYN (synchronize) message to the server.
2) The server acknowledges this request by sending SYNACK back to the client.
3) The client responds with an ACK, and the connection is established.
Question 18 
Which combination of the integer variables x, y and z makes the variable a get the value 4 in the following expression?
a = (x > y) ? ((x > z) ? x : z) : ((y > z) ? y : z)
x = 3, y = 4, z = 2  
x = 6, y = 5, z = 3  
x = 6, y = 3, z = 5  
x = 5, y = 4, z = 5 
→ We can directly eliminate the options B & C, because none of the variable can assign a value 4.
→ Given explanation is
a = (x>y)?((x>z)?x:z):((y>z)?y:z)
Option A:
x=3; y=4; z=2
a=(3>4)? ⇒ No
Then evaluate second expression ⇒ (4>2)?Yes
⇒a=y
a=4 (True)
Option D:
x=5; y=4; z=5
a=(5>4) ⇒ Yes
Then evaluate first expression ⇒ (5>5)? No
⇒ a=z ⇒ a=5 (Not true)
⇒ Answer is Option A.
Question 19 
The Breadth First Search algorithm has been implemented using the queue data structure. One possible order of visiting the nodes of the following graph is
MNOPQR  
NQMPOR  
QMNPRO  
QMNPOR 
Option C: QMNPRO
→ Queue starts with Q then neighbours of Q is MNP and it is matching with the given string .
→ Now , Next node to be considered is M . Neighbours of M are N, Q and R , but N and Q are already in Queue. So, R is matching with one given string
→ Now, next node to be considered is N. Neighbours of N are M, Q and O, but M and Q are already in Queue. So, O is matching with a given string.
Hence , Option C is Correct.
Similarly, check for option (D).
Question 20 
The data blocks of a very large file in the Unix file system are allocated using
contiguous allocation  
linked allocation  
indexed allocation  
an extension of indexed allocation 
Question 21 
The minimum number of equal length subintervals needed to approximate to an accuracy of atleast using the trapezoidal rule is
1000e  
1000  
100e  
100 