### Computer Science (GATE Exam) 2008 - Previous Question Paper Solution

Description: GATE Exam Previous Year Question Paper Solution Computer Science(CS) - 2008 | |

Number of Questions: 85 | |

Created by: Aliensbrain Bot | |

Tags: Computer Science GATE CS Previous Year Paper |

$\lim_{x \to \infty}\frac{x-\sin x}{x+\cos x}$ equals

If P, Q, R are subsets of the universal set U, then $(P\cap Q\cap R) \cup (P^c \cap Q \cap R) \cup Q^c \cup R^c$ is

The most efficient algorithm for finding the number of connected components in an undirected graph on n vertices and m edges has time complexity

Given f_{1}, f_{3} and f in canonical sum of products form (in decimal) for the circuit.

$f_1 = \Sigma m(4, 5, 6, 7, 8)$ $f_3 = \Sigma m(1, 6, 15)$ $f = \Sigma m(1, 6, 8, 15)$

Then f_{2} is

In the IEEE floating point representation, the hexadecimal value 0x00000000 corresponds to

Some code optimizations are carried out on the intermediate code because

What is the maximum size of data that the application layer can pass on to the TCP layer below?

Which of the following system calls results in the sending of SYN packets?

Let $r$ denote number system radix. The only value(s) of $r$ that satisfy the equation $\sqrt{121_r}={11}_r$, is/are

Which of the following describes a handle (as applicable to LR-parsing) appropriately?

The Newton-Raphson iteration $x_{n+1} = \frac{1}{2}\left(x_n+\frac{R}{x_n}\right)$ can be used to compute the

The minimum number of equal length subintervals needed to approximate $\int_1^2 xe^x\,dx$ to an accuracy of at least $\frac{1}{3}\times10^{-6}$ using the trapezoidal rule is

Which of the following statements is true for every planar graph on n vertices?

Let $P =\sum_{\substack{1≤i≤2k \\ i\;odd}} i$ and $Q = \sum_{\substack{1≤i≤2k \\ i\;even}} i$, where $k$ is a positive integer. Then

If $P, Q, R$ are Boolean variables, then

$(P + \bar{Q}) (P.\bar{Q} + P.R) (\bar{P}.\bar{R} + \bar{Q})$ simplifies to

Aishwarya studies either computer science or mathematics everyday. If she studies computer science on a day, then the probability that the studies mathematics the next day is 0.6. If she studies mathematics on a day, then the probability that the studies computer science the next day is 0.4. Given that Aishwarya studies computer science on Monday, what is the probability that she studies computer science on Wednesday?

A point on a curve is said to be an extremum if it is a local minimum or a local maximum. What is the number of distinct extrema for the curve 3x^{4} − 16x^{3} + 24x^{2} + 37?

How many of the following matrices have an eigenvalue 1? $\left[\begin{array}{cc}1 & 0 \\ 0 & 0 \end{array} \right]\left[\begin{array}{cc}0 & 1 \\ 0 & 0 \end{array} \right] \left[\begin{array}{cc}1 & -1 \\ 1 & 1 \end{array} \right]$ and $\left[\begin{array}{cc}-1 & 0 \\ 1 & -1 \end{array} \right]$

$P$ and $Q$ are two propositions. Which of the following logical expressions are equivalent?

I. $P ∨ \neg Q$ ||. $\neg(\neg P ∧ Q)$ |||. $(P ∧ Q) ∨ (P ∧ \neg Q) ∨ (\neg P ∧ \neg Q)$ IV. $(P ∧ Q) ∨ (P ∧ \neg Q) ∨ (\neg P ∧ Q)$

Let fsa and pda be two predicates such that fsa(x) means x is a finite state automaton, and pda(y) means that y is a pushdown automaton. Let equivalent be another predicate such that equivalent (a, b) means a and b are equivalent. Which of the following first order logic statements represents the following: Each finite state automaton has an equivalent pushdown automaton

The minimum number of comparisons required to determine if an integer appears more than $\frac{n}{2}$ times in a sorted array of $n$ integers is

A B-tree of order 4 is built from scratch by 10 successive insertions. What is the maximum number of node splitting operations that may take place?

G is a graph on n vertices and 2n-2 edges. The edges of G can be partitioned into two edge-disjoint spanning trees. Which of the following is NOT true for G?

Dijkstra's single source shortest path algorithm when run from vertex a in the above graph, computes the correct shortest path distance to

Consider the following functions:
f(n) = 2^{n}
g(n) = n!
h(n) = n^{logn}
Which of the following statements about the asymptotic behaviour of f(n), g(n), and h(n) is true?

The subset-sum problem is defined as follows: Given a set S of n positive integers and a positive integer W, determine whether there is a subset of S Whose elements sum to W. An algorithm Q solves this problem in O(nW) time. Which of the following statements is false?

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

Which of the following tuple relational calculus expression(s) is/are equivalent to $\forall t \in r \left(P\left(t\right)\right)$?

I. $\neg \exists t \in r \left(P\left(t\right)\right)$ II. $\exists t \notin r \left(P\left(t\right)\right)$ III. $\neg \exists t \in r \left(\neg P\left(t\right)\right)$ IV. $\exists t \notin r \left(\neg P\left(t\right)\right)$

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?

Consider the Quicksort algorithm. Suppose there is a procedure for finding a pivot element which splits the list into two sub-lists each of which contains at least one-fifth of the elements. Let T(n) be the number of comparisons required to sort n elements. Then

You are given the postorder traversal, P, of a binary search tree on the n elements 1, 2,….,n. You have to determine the unique binary search tree that has P as its postorder traversal. What is the time complexity of the most efficient algorithm for doing this?

We have a binary heap on n elements and wish to insert n more elements (not necessarily one after another) into this heap. The total time required for this is

A client process P needs to make a TCP connection to a server process S. Consider the following situation: the server process S executes a socket (), a bind () and a listen () system call in that order, following which it is preempted. Subsequently, the client process P executes a socket () system call followed by connect () system call to connect to the server process S. The server process has not executed any accept () system call. Which one of the following events could take place?

If a class B network on the Internet has a subnet mask of 255.255.248.0, what is the maximum number of hosts per subnet?

An LALR(1) parser for a grammar G can have shift-reduce (S-R) conflicts if and only if

In the slow start phase of the TCP congestion control algorithm, the size of the congestion window

A computer on a 10Mbps network is regulated by a token bucket. The token bucket is filled at a rate of 2Mbps. It is initially filled to capacity with 16Megabits. What is the maximum duration for which the computer can transmit at the full 10Mbps?

Which of the following are true? I. A programming language which does not permit global variables of any kind and has no nesting of procedures/functions, but permits recursion can be implemented with static storage allocation II. Multi-level access link (or display) arrangement is needed to arrange activation records only if the programming language being implemented has nesting of procedures/functions III. Recursion in programming languages cannot be implemented with dynamic storage allocation IV. Nesting procedures/functions and recursion require a dynamic heap allocation scheme and cannot be implemented with a stack-based allocation scheme for activation records V. Programming languages which permit a function to return a function as its result cannot be implemented with a stack-based storage allocation scheme for activation records

Which of the following statements about synchronous and asynchronous I/O is NOT true?

Which of the following is NOT true for deadlock prevention and deadlock avoidance schemes?

A process executes the following code for (i = 0; i < n; i++) for (); The total number of child processes created is

The P and V operations on counting semaphores, where s is a counting semaphore, are defined as follows: P(s) : s = s - 1; ifs < 0 then wait; V(s) : s = s + 1; ifs <= 0 then wakeup a process waiting on s;

Assume that P_{b} and V_{b} the wait and signal operations on binary semaphores are provided. Two binary
semaphores x_{b} and y_{b} are used to implement the semaphore operations P(s) and V(s) as follows:
P(s) : Pb(Xb);
s = s - 1;
if (s < 0) {
Vb(Xb) ;
Pb(Yb) ;
}
else Vb(Xb);V(s) : Pb(Xb) ;
s = s + 1;
if (s <= 0) Vb(Yb) ;
Vb(Xb) ;

The initial values of x_{b} and y_{b} are respectively

Consider the following C functions: int f1 (int n) { If(n == 0 | |n == 1) return n; else return(2*f1(n-1) + 3* f1(n-2)); } int f2 (int n) { int i; int X[N], Y[N], Z[N]; X[0] = Y[0] = Z[0] = 0; X[1] = 1; Y[1] = 2; Z[1] = 3; for(i = 2; i<=n; i++) { X[i] = Y[i-1] + Z [i-2]; Y[i] = 2*X [i]; Z[i] = 3*X[i]; } Return X[n]; }

f_{1} (8) and f_{2} (8) return the values

A processor uses 36 bit physical addresses and 32 bit virtual addresses, with a page frame size of 4 Kbytes. Each page table entry is of size 4 bytes. A three level page table is used for virtual to physical address translation, where the virtual address is used as follows:

Bits 30-31 are used to index into the first level page table Bits 21-29 are used to index into the second level page table Bits 12-20 are used to index into the third level page table, and Bits 0-11 are used as offset within the page

The number of bits required for addressing the next level page table (or page frame) in the page table entry of the first, second and third level page tables are respectively

Let x_{n} denote the number of binary strings of length n that contain no consecutive 0s.

Which of the following recurrences does x_{n} satisfy?

Consider the following C functions: int f1 (int n) { If(n == 0 | |n == 1) return n; else return(2*f1(n-1) + 3* f1(n-2)); } int f2 (int n) { int i; int X[N], Y[N], Z[N]; X[0] = Y[0] = Z[0] = 0; X[1] = 1; Y[1] = 2; Z[1] = 3; for(i = 2; i<=n; i++) { X[i] = Y[i-1] + Z [i-2]; Y[i] = 2*X [i]; Z[i] = 3*X[i]; } Return X[n]; }

The running time of f1 (n) and f2 (n) are

The subset-sum problem is defined as follows. Given a set of n positive integers, S = {a_{1}, a_{2}, a_{3}, …,
a_{n}}, and positive integer W, is there a subset of S whose elements sum to W? A dynamic program for
solving this problem uses a 2-dimensional Boolean array, X, with n rows and W+1 columns.
X [i, j], 1$\le$i $\le$ n,0 $\le$j $\le$ W, is TRUE if and only if there is a subset of {a_{1}, a_{2}, …, a}
whose elements sum to j.

Which entry of the array X, if TRUE, implies that there is a subset whose elements sum to W?

Let x_{n} denote the number of binary strings of length n that contain no consecutive 0s.

The value of x_{5} is

Consider the following C program that attempts to locate an element x in an array Y[ ] using binary search. The program is erroneous.

- f(int Y[10], int x) {
- int u, j, k;
- i = 0; j = 9;
- do {
- k = (i + j)/2;
- if (Y[k] ! = x) && (i < j);
- } while ((Y[k] ! = x) && (i < j));
- if (Y[k] == x) print f(“x is in the array”);
- else print f(“x is not in the array”);
- }

On which of the following contents of Y and x does the program fail?

Consider the following C program that attempts to locate an element x in an array Y[ ] using binary search. The program is erroneous.

- f(int Y[10], int x) {
- int u, j, k;
- i = 0; j = 9;
- do {
- k = (i + j)/2;
- if (Y[k] ! = x) && (i < j);
- } while ((Y[k] ! = x) && (i < j));
- if (Y[k] == x) print f(“x is in the array”);
- else print f(“x is not in the array”);
- }

The correction needed in the program to make it work properly is

The subset-sum problem is defined as follows. Given a set of n positive integers, S = {a_{1}, a_{2}, a_{3}, …,
a_{n}}, and positive integer W, is there a subset of S whose elements sum to W? A dynamic program for
solving this problem uses a 2-dimensional Boolean array, X, with n rows and W+1 columns.
X [i, j], 1$\le$i $\ge$ n,0 $\le$j $\le$ W, is TRUE if and only if there is a subset of {a_{1}, a_{2}, …, a}
whose elements sum to j.

Which of the following is valid for 2 $\le$ i $\le$ n and ai $\le$ j $\le$ W?

For a magnetic disk with concentric circular tracks, the seek latency is not linearly proportional to the seek distance due to

The use of multiple register windows with overlap causes a reduction in the number of memory accesses for I. Function locals and parameters II. Register saves and restores III. Instruction fetches

Which of the following are NOT true in a pipelined processor? I. Bypassing can handle all RAW hazards II. Register renaming can eliminate all register carried WAR hazards III. Control hazard penalties can be eliminated by dynamic branch prediction

Which of the following must be true for the RFE (Return From Exception) instruction on a general purpose processor? I. It must be a trap instruction II. It must be a privileged instruction III. An exception cannot be allowed to occur during execution of an RFE instruction

For inclusion to hold between two cache levels L1 and L2 in a multi-level cache hierarchy, which of the following are necessary? I. L1 must be a write-through cache II. L2 must be a write-through cache III. The associativity of L2 must be greater than that of L1 IV. The L2 cache must be at least as large as the L1 cache

In an instruction execution pipeline, the earliest that the data TLB (Translation Look a side Buffer) can be accessed is

Which of the following is/are true of the auto-increment addressing mode? I. It is useful in creating self-relocating code II. If it is included in an Instruction Set Architecture, then an additional ALU is required for effective address calculation III. The amount of increment depends on the size of the data item accessed

Consider a machine with a 2-way set associative data cache of size 64Kbytes and block size 16bytes. The cache is managed using 32 bit virtual addresses and the page size is 4Kbyts. A program to be run on this machine begins as follows: double ARR[1024] [1024]; int i, j; /* Initialize array ARR to 0.0 */ for (i = 0; i < 1024; i++) for (j = 0; j < 1024; j++) ARR[i] [j] = 0.0; The size of double is 8Bytes. Array ARR is located in memory starting at the beginning of virtual page 0xFF000 and stored in row major order. The cache is initially empty and no pre-fetching is done. The only data memory references made by the program are those to array ARR

The total size of the tags in the cache directory is

Delayed branching can help in the handling of control hazards

For all delayed conditional branch instructions, irrespective of whether the condition evaluates to true or false

Consider a machine with a 2-way set associative data cache of size 64Kbytes and block size 16bytes. The cache is managed using 32 bit virtual addresses and the page size is 4Kbyts. A program to be run on this machine begins as follows: double ARR[1024] [1024]; int i, j; /* Initialize array ARR to 0.0 */ for (i = 0; i < 1024; i++) for (j = 0; j < 1024; j++) ARR[i] [j] = 0.0; The size of double is 8Bytes. Array ARR is located in memory starting at the beginning of virtual page 0xFF000 and stored in row major order. The cache is initially empty and no pre-fetching is done. The only data memory references made by the program are those to array ARR

The cache hit ratio for this initialization loop is

Consider a machine with a 2-way set associative data cache of size 64Kbytes and block size 16bytes. The cache is managed using 32 bit virtual addresses and the page size is 4Kbyts. A program to be run on this machine begins as follows: double ARR[1024] [1024]; int i, j; /* Initialize array ARR to 0.0 */ for (i = 0; i < 1024; i++) for (j = 0; j < 1024; j++) ARR[i] [j] = 0.0; The size of double is 8Bytes. Array ARR is located in memory starting at the beginning of virtual page 0xFF000 and stored in row major order. The cache is initially empty and no pre-fetching is done. The only data memory references made by the program are those to array ARR

Which of the following array elements has the same cache index as ARR [0] [0]?

Delayed branching can help in the handling of control hazards

The following code is to run on a pipelined processor with one branch delay slot:
11 : ADD R2 ^{$\leftarrow$} R7 + R8
12 : SUB R4 ^{$\leftarrow$} R5 – R6
13 : ADD R1 ^{$\leftarrow$} R2 + R3
14 : STORE Memory [R4] ^{$\leftarrow$} R1
BRANCH to Label if R1 == 0
Which of the instructions 11, 12, 13 or 14 can legitimately occupy the delay slot without any other program modification?

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)

Choose the correct option to fill ?1 and ?2 so that the program below prints an input string in reverse order. Assume that the input string is terminated by a new line character.

```
void reverse(void) {
int c;
if(?1) reverse();
?2
}
main() {
printf("Enter text"); ptintf("\n");
reverse(); printf("\n");
}
```

The following C function takes a single-linked list of integers as a parameter and rearranges the elements of the list. The function is called with the list containing the integers 1,2,3,4,5,6,7 in the given order. What will be the contents of the list after the function completes execution?

```
struct node {
int value;
struct node * next;
};
Void rearrange struct node * list {
struct node * p, * q;
int temp;
if (!list || !!list - > next) return;
p = list; | q = list - > next;
while q {
temp = p - > value;
p - > value = q - > value;
q - > value = temp;
p = q - > next;
q = p ? p - > next : 0;
}
}
```

What is printed by the following C program?

```
int f(int x, int py, int **ppz) void main() {
{
Int y, z;
int c, * b, * * a; * * ppz + = 1;
z = * ppz c = 4;
b = &c;
a = &b;
*py + = 2;
y = *py;
printf(“ % d”, f(c, b, a));
x + = 3;
}
return x + y + z;
}
```

A clustering index is defined on the fields which are of the type

Consider the following E-R diagram

The minimum number of tables needed to represent M, N, P, R1, R2 is

Consider a file of 16384 records. Each record is 32 bytes long and its key field is of size 6 bytes. The file is ordered on a non-key field and the file organisation is unspanned. The file is stored in a file system with block size 1024 bytes and the size of a block pointer is 10 bytes. If the secondary index is built on the key field of the file and a multi-level index scheme is used to store the secondary index, then the numbers of first-level and second-level blocks in the multi-level index are respectively

Let R and S be two relations with the following schema
R (__P__, __Q__, R1, R2, R3)
S(__P__, __Q__, S1, S2)
Where {P, Q} is the key for both schemas. Which of the following queries are equivalent?

I. $\Pi_P \left(R \bowtie S\right)$ II. $\Pi_P \left(R\right) \bowtie \Pi_P\left(S\right)$ III. $\Pi_P \left(\Pi_{P, Q} \left(R\right) \cap \Pi_{P,Q} \left(S\right) \right)$ IV. $\Pi_P \left(\Pi_{P, Q} \left(R\right) - \left(\Pi_{P,Q} \left(R\right) - \Pi_{P,Q} \left(S\right)\right)\right)$

Consider the following relational schemes for a library database:
Book Title, Author, Catalog_ no, Publisher, Year, Pr ice
Collection Title, Author, Catalog_ no
within the functional dependencies:
I. Title Author ^{$\rightarrow$} Catalog_no
II. Catalog_no ^{$\rightarrow$} Title Author Publisher Year
III. Publisher Title Year ^{$\rightarrow$} Pr ice

Assume {Author, Title} is the key for both schemes. Which of the following statements is true?

Consider the following E-R diagram

Which of the following is a correct attribute set for one of the tables for the correct answer to the above question?

Which of the following statements is false?

Which of the following is true for the language $\left\{ a^p \text{ | p is a prime} \right\} $?

Which of the following are decidable? I. Whether the intersection of two regular languages is infinite II. Whether a given context-free language is regular III. Whether two push-down automata accept the same language IV. Whether a given grammar is context-free

If L and $\bar L$ are recursively enumerable then L is

Which of the following are regular sets?

I. $\left\{a^nb^{2m} \mid n \geq 0, m \geq 0 \right\}$ II. $\left\{a^nb^m \mid n =2m \right\}$ III. $\left\{a^nb^m \mid n \neq m \right\}$ IV. $ \left\{xcy \mid x, y, \in \left\{a, b\right\} ^* \right\} $

Which of the following statements are true?

I. Every left-recursive grammar can be converted to a right-recursive grammar and vice-versa II. All $\epsilon$ productions can be removed from any context-free grammar by suitable transformations III. The language generated by a context-free grammar all of whose productions are of the form X --> w or X --> wY (where, w is a string of terminals and Y is a non-terminal), is always regular IV. The derivation trees of strings generated by a context-free grammar in Chomsky Normal Form are always binary trees

Match the following NFAs with the regular expressions they correspond to

- $\epsilon + 0\left(01^
*1+00\right)^*01^*$ - $\epsilon + 0\left(10^*1+00\right)^*0$
- $\epsilon + 0\left(10^*1+10\right)^*1$
- $\epsilon + 0\left(10^
*1+10\right)^*10^*$

Match the following:

E. | Checking that identifiers are declared before their use | P. | $L : = : \left\{a^nb^mc^nd^m \mid n: \geq1, m \geq 1\right\}$ | |

F. | Number of formal parameters in the declaration of a function agrees with the number of actual parameters in a use of that function | Q. | $X: \rightarrow XbX \mid XcX \mid dXf \mid g$ | |

G. | Arithmetic expressions with matched pairs of parentheses | R. | $L: = \left\{wcw\mid w : \in \left(a\mid b\right)^* \right\}$ | |

H. | Palindromes | S. | $X : \rightarrow : bXb \mid :cXc : \mid \epsilon $ |

Given below are two finite state automata (→ indicates the start state and F indicates a final state)

Y: | | a | b | |-----------------|---|---| | $\rightarrow$ 1 | 1 | 2 | | 2(F) | 2 | 1 |

Z: | | a | b | |-----------------|---|---| | $\rightarrow$ 1 | 2 | 2 | | 2(F) | 1 | 1 |

Which of the following represents the product automaton Z × Y?

The following system of equations $$x_1 + x_2 + 2x_3 = 1$$ $$x_1 + 2x_2 + 3x_3 = 2$$ $$x_1 + 4x_2 + αx_3 = 4$$ has a unique solution. The only possible value(s) for α is/are

Let $X$ be a random variable following normal distribution with mean $+1$ and variance $4$. Let $Y$ be another normal variable with mean $-1$ and variance unknown. If $P (X ≤ -1) = P (Y ≥ 2)$ , the standard deviation of $Y$ is

The data blocks of a very large file in the Unix file system are allocated using