### Computer Science (GATE Exam) 2007 - Previous Question Paper Solution

Description: GATE Exam Previous Year Question Paper Solution Computer Science(CS) - 2007 | |

Number of Questions: 85 | |

Created by: Aliensbrain Bot | |

Tags: Computer Science GATE CS Previous Year Paper |

Let S be a set of n elements. The number of ordered pairs in the largest and the smallest equivalence relations on S are:

What is the maximum number of different Boolean functions involving n Boolean variables?

Let G be the non-planar graph with the minimum possible number of edges. Then G has

How many 3-to-8 line decoders with an enable input are needed to construct a 6-to-64 line decoder without using any other logic gates?

Consider the DAG with V = {1, 2, 3, 4, 5, 6}, shown below.

Which of the following is NOT a topological ordering?

Consider the following Boolean function of four variables: f (w, x, y, z) = $\sum$(1,3,4,6,9,11,12,14) The function is:

Consider the following statements about user level threads and kernel level threads. Which one of the following statements is FALSE?

The maximum number of binary trees that can be formed with three unlabeled nodes is:

Which one of the following is a top-down parser?

Consider the following two statements about the function f (x) = x: P. f (x) is continuous for all real values of x Q. f (x) is differentiable for all real values of x Which of the following is TRUE?

Let A be a $4 \times 4$ matrix with eigen values -5,-2,1,4. Which of the following is an eigen value of the matrix$\begin{bmatrix} A & I \\ I & A \end{bmatrix}$, where $I$ is the $4 \times 4$ identity matrix?

The height of a binary tree is the maximum number of edges in any root to leaf path. The maximum number of nodes in a binary tree of height h is:

Which of the following sorting algorithms has the lowest worst-case complexity?

In Ethernet when Manchester encoding is used, the bit rate is:

How many different non-isomorphic Abelian groups of order 4 are there?

Consider the series $x_{n+1} = \frac{x_n}{2}+\frac{9}{8x_n},x_0 = 0.5$ obtained from the Newton-Raphson method. The series converges to

Suppose we uniformly and randomly select a permutation from the 20! Permutations of 1, 2, 3,….., 20. What is the probability that 2 appears at an earlier position than any other even number in the selected permutation?

Consider the set of (column) vectors defined by $$X = \left \{x \in R^3 \mid x_1 + x_2 + x_3 = 0, \text{ where } x^T = \left[x_1,x_2,x_3\right]^T\right \}$$.

Which of the following is TRUE?

Let Graph(x) be a predicate which denotes that x is a graph. Let Connected(x) be a predicate which denotes that x is connected. Which of the following first order logic sentences DOES NOT represent the statement: “Not every graph is connected”?

The control signal functions of a 4-bit binary counter are given below (where X is “don't care”): Clear Clock Load Count Function

The counter is connected as follows:

Assume that the counter and gate delays are negligible. If the counter starts at 0, then it cycles through the following sequence:

Which of the following graphs has an Eulerian circuit?

Group 1 contains some CPU scheduling algorithms and Group 2 contains some applications. Match the entries in Group 1 to the entries in Group 2.

**Group I
** **Group II
** (P) Gang Scheduling (1) Guaranteed Scheduling (Q) Rate Monotonic Scheduling (2) Real-time Scheduling (R) Fair Share Scheduling (3) Thread Scheduling

Define the connective * for the Boolean variables X and Y as: X * Y = XY + X'Y'. Let Z = X *Y. Consider the following expressions P, Q and R.

$\pi P$ : X = Y * Z Q : Y = X * Z R : X *Y * Z = 1

Which of the following is TRUE?

Suppose only one multiplexer and one inverter are allowed to be used to implement any Boolean function of n variables. What is the minimum size of the multiplexer needed?

Which one of the following uses UDP as the transport protocol?

Let f (w, x, y, z) = ^{$\sum$}(0,4,5,7,8,9,13,15). Which of the following expressions is/are NOT equivalent to f?

P. x’y’z’ + w’xy’ + wy’z + xz Q. w’y’z’ + wx’y’ + xz R. w’y’z’ + wx’y’ + xyz + xy’z S. x’y’z’ + wx’y’ + w’y

The in order and preorder traversal of a binary tree are d b e a f c g and a b d e c f g, respectively The post order traversal of the binary tree is:

Consider a hash table of size seven, with starting index zero, and a hash function (3x + 4) mod 7. Assuming the hash table is initially empty, which of the following is the contents of the table when the sequence 1, 3, 8, 10 is inserted into the table using closed hashing? Note that − denotes an empty location in the table.

In an unweighted, undirected connected graph, the shortest path from a node S to every other node is computed most efficiently, in terms of time complexity, by

A complete n-ary tree is a tree in which each node has n children or no children. Let I be the number of internal nodes and L be the number of leaves in a complete n-ary tree. If L = 41, and I = 10, what is the value of n?

In a look-ahead carry generator, the carry generate function Gi and the carry propagate function P_{i} for
inputs A_{i} and B_{i} are given by:

P_{i} = A_{i} $\oplus$ B_{i} and G_{i} = A_{i} B_{i}
The expressions for the sum bit S_{i} and the carry bit C_{i+1} of the look-ahead carry adder are given by:
S_{i} = P_{i} C_{i} and C_{i+1} = G_{i} + P_{i }C_{i}, where C_{o} is the input carry.
Consider a two-level logic implementation of the look-ahead carry generator.
Assume that all P_{i} and G_{i} are available for the carry generator circuit and that the AND and OR gates
can have any number of inputs. The number of AND gates and OR gates needed to implement the
look-ahead carry generator for a 4-bit adder with 3 2 1 0 4 S ,S ,S ,S and C as its outputs are
respectively:

Which of the following is TRUE about formulae in Conjunctive Normal Form?

What is the time complexity of the following recursive function? Int Do Something (int n) { return 1; else return (Do Something (floor sqrt (n))) + n);

Let w be the minimum weight among all edge weights in an undirected connected graph. Let e be a specific edge of weight w . Which of the following is FALSE?

Consider the process of inserting an element into a Max Heap, where the Max Heap is represented by an array. Suppose we perform a binary search on the path from the new leaf to the root to find the position for the newly inserted element, the number of comparisons performed is:

In the following C function, let n ^{$\ge$ }m.

Int gcd (n,m) { if (n% m ==0) return m; n = n %m; return gcd (m, n); }

How many recursive calls are made by this function?

An array of n numbers is given, where n is an even number. The maximum as well as the minimum of these n numbers needs to be determined. Which of the following is TRUE about the number of comparisons needed?

Consider the following two statements: P: Every regular grammar is LL(1) Q: Every regular set has a LR(1) grammar Which of the following is TRUE?

A single processor system has three resource types X, Y and Z, which are shared by three processes. There are 5 units of each resource type. Consider the following scenario, where the column alloc denotes the number of units of each resource type allocated to each process, and the column request denotes the number of units of each resource type requested by a process in order to complete execution. Which of these processes will finish LAST?

Consider the grammar with non-terminals N = { S, C, S_{1}), terminals T = {a, b, i, t, e}, with S as the start symbol, and the following set of rules:

S $\rightarrow$iCtSS_{1} | a
S_{1} $\rightarrow$ eS |$\in$
C $\rightarrow$ b

The grammar is NOT LL(1) because:

A virtual memory system uses First In First Out (FIFO) page replacement policy and allocates a fixed number of frames to a process. Consider the following statements: P: Increasing the number of page frames allocated to a process sometimes increases the page fault rate. Q: Some programs do not exhibit locality of reference. Which one of the following is TRUE?

Consider the following C code segment: int Is Prime (n) { int i, n; for (i = 2; i <= sqrt (n) ; i ++) if (n% i == 0) { print f (“ Not Prime n”); return 0; } return 1; } Let T (n) denote the number of times the for loop is executed by the program on input n. Which of the following is TRUE?

Two processes, P1 and P2, need to access a critical section of code. Consider the following synchronization construct used by the processes:

Here, wants1 and wants2 are shared variables, which are initialized to false. Which one of the following statements is TRUE about the above construct?

There are n stations in a slotted LAN. Each station attempts to transmit with a probability p in each time slot. What is the probability that ONLY one station transmits in a given time slot?

The message 11001001 is to be transmitted using the CRC polynomial x^{3} + 1 to protect it from errors. The message that should be transmitted is:

An operating system uses shortest remaining time first (SRT) process scheduling algorithm. Consider the arrival times and execution times for the following processes:

Process Execution time Arrival time P1 20 0 P2 25 15 P3 10 30 P4 15 45

What is the total waiting time for process P2?

In a simplified computer the instructions are:
OP R_{j} , R_{i} - Performs R_{j} OP R_{i} and stores the result in register . R_{i}
OP m, R_{i} - Performs val i OP R_{i} and stores the result in R_{i}. val denotes the content of memory location m.
MOV m, R_{i} - Moves the content of memory location m to register R_{i}
MOV R_{i}^{ }, m - Moves the content of register R_{i} to memory location m.
The computer has only to registers, and OP is either ADD or SUB. Consider the following basic block:
t_{1} = a + b
t_{2} = c + d
t_{3} = e - t_{2}
t_{4} = t_{1} - b
Assume that all operands are initially in memory. The final value of the computation should be in memory. What is the minimum number of MOV instructions in the code generated for this basic block?

The address of a class B host is to be split into subnets with a 6-bit subnet number. What is the maximum number of subnets and the maximum number of hosts in each subnet?

In a token ring network the transmission speed is10^{7} bps and the propagation speed is 200 metres/$\mu$s. The 1-bit delay in this network is equivalent to:

The distance between two stations M and N is L kilometers. All frames are K bits long. The propagation delay per kilometer is t seconds. Let R bits/second be the channel capacity. Assuming that processing delay is negligible, the minimum number of bits for the sequence number field in a frame for maximum utilization, when the sliding window protocol is used, is:

Suppose the letters a, b, c, d, e, f have probabilities $ \dfrac{1}{2},\dfrac{1}{4},\dfrac{1}{8},\dfrac{1}{16},\dfrac{1}{32},\dfrac{1}{32}$ respectively.

Which of the following is the Huffman code for the letter a, b, c, d, e, f?

Match the following:

(P) | SMTP | (1) | Application layer |

(Q) | BGP | (2) | Transport layer |

(R) | TCP | (3) | Data link layer |

(S) | PPP | (4) | Network layer |

(5) | Physical layer |

Suppose the letters a, b, c, d, e, f have probabilities $ \dfrac{1}{2},\dfrac{1}{4},\dfrac{1}{8},\dfrac{1}{16},\dfrac{1}{32},\dfrac{1}{32}$ respectively.

What is the average length of the correct answer to Q.?

A process has been allocated 3 page frames. Assume that none of the pages of the process are available in the memory initially. The process makes the following sequence of page references (reference string): **1, 2, 1, 3, 7, 4, 5, 6, 3, 1**

If optimal page replacement policy is used, how many page faults occur for the above reference string?

A process has been allocated 3 page frames. Assume that none of the pages of the process are available in the memory initially. The process makes the following sequence of page references (reference string): **1, 2, 1, 3, 7, 4, 5, 6, 3, 1**

Least Recently Used (LRU) page replacement policy is a practical approximation to optimal page replacement. For the above reference string, how many more page faults occur with LRU than with the optimal page replacement policy?

Consider the CFG with {S, A, B} as the non-terminal alphabet, {a, b} as the terminal alphabet, S as the start symbol and the following set of production rules:

S$ \rightarrow$ a B S $ \rightarrow$bA B $ \rightarrow$b A$ \rightarrow$ a B $ \rightarrow$bS A $ \rightarrow$aS B $ \rightarrow$aBB S$ \rightarrow$ bAA

Which of the following strings is generated by the grammar?

Suppose that a robot is placed on the Cartesian plane. At each step it is allowed to move either one unit up or one unit right, i.e., if it is at (i, j) then it can move to either (i + 1, j) or (i, j + 1).

Suppose that the robot is not allowed to traverse the line segment from (4, 4) to (5, 4). With this constraint, how many distinct paths are there for the robot to reach (10, 10) starting from (0, 0)?

Suppose that a robot is placed on the Cartesian plane. At each step it is allowed to move either one unit up or one unit right, i.e., if it is at (i, j) then it can move to either (i + 1, j) or (i, j + 1).

How many distinct paths are there for the robot to reach the point (10,10) starting from the initial position (0,0)?

Consider the CFG with {S, A, B} as the non-terminal alphabet, {a, b} as the terminal alphabet, S as the start symbol and the following set of production rules:

S ^{$\rightarrow$} aB S ^{$\rightarrow$ }bA
B ^{$\rightarrow$ }b A ^{$\rightarrow$} a
B ^{$\rightarrow$ }bS A ^{$\rightarrow$ }aS
B ^{$\rightarrow$ }aBB S ^{$\rightarrow$} bAA

How many derivation trees are there?

Consider a 4-way set associative cache consisting of 128 lines with a line size of 64 words. The CPU generates a 20-bit address of a word in main memory. The number of bits in the TAG, LINE and WORD fields are respectively:

Consider a disk pack with 16 surfaces, 128 tracks per surface and 256 sectors per track. 512 bytes of data are stored in a bit serial manner in a sector. The capacity of the disk pack and the number of bits required to specify a particular sector in the disk are respectively:

Consider a machine with a byte addressable main memory of 2^{16} bytes. Assume that a direct mapped data cache consisting of 32 lines of 64 bytes each is used in the system. A 50 × 50 two-dimensional array of bytes is stored in the main memory starting from memory location 1100H. Assume that the data cache is initially empty. The complete array is accessed twice. Assume that the contents of the data cache do not change in between the two accesses.

Which of the following lines of the data cache will be replaced by new blocks in accessing the array for the second time?

Consider a pipelined processor with the following four stages: IF: Instruction Fetch ID: Instruction Decode and Operand Fetch EX: Execute WB: Write Back The IF, ID and WB stages take one clock cycle each to complete the operation. The number of clock cycles for the EX stage depends on the instruction. The ADD and SUB instructions need 1 clock cycle and the MUL instruction needs 3 clock cycles in the EX stage. Operand forwarding is used in the pipelined processor. What is the number of clock cycles taken to complete the following sequence of instructions? ADD R2, R1, R0 R2 $\rightarrow$ R1 + R0 MUL R4, R3, R2 R4$\rightarrow$ R3 * R2 SUB R6, R5, R4 R6 $\rightarrow$R5 - R4

Consider a machine with a byte addressable main memory of 2^{16} bytes. Assume that a direct mapped data cache consisting of 32 lines of 64 bytes each is used in the system. A 50 × 50 two-dimensional array of bytes is stored in the main memory starting from memory location 1100H. Assume that the data cache is initially empty. The complete array is accessed twice. Assume that the contents of the data cache do not change in between the two accesses.

How many data cache misses will occur in total?

Consider the following program segment. Here R1, R2 and R3 are the general purpose registers.

Instruction |
Operation |
Instruction size (no. of words) |

MOV R1, (3000) | R1 _ m[3000] | 2 |

LOOP: MOV R2, (R3) | R2 _ M[R3] | 1 |

ADD R2, R1 | R2 _ R1 + R2 | 1 |

MOV (R3), R2 | M[R3] _ R2 | 1 |

INC R3 | R3 _ R3 + 1 | 1 |

DEC R1 | R1 _ R1 - 1 | 1 |

BNZ LOOP | Branch on not zero | 2 |

HALT | Stop | 1 |

Assume that the content of memory location 3000 is 10 and the content of the register R3 is 2000. The content of each of the memory locations from 2000 to 2010 is 100. The program is loaded from the memory location 1000. All the numbers are in decimal.

Assume that the memory is word addressable. The number of memory references for accessing the data in executing the program completely is:

Consider the following program segment. Here R1, R2 and R3 are the general purpose registers.

Instruction |
Operation |
Instruction size (no. of words) |

MOV R1, (3000) | R1 _ m[3000] | 2 |

LOOP: MOV R2, (R3) | R2 _ M[R3] | 1 |

ADD R2, R1 | R2 _ R1 + R2 | 1 |

MOV (R3), R2 | M[R3] _ R2 | 1 |

INC R3 | R3 _ R3 + 1 | 1 |

DEC R1 | R1 _ R1 - 1 | 1 |

BNZ LOOP | Branch on not zero | 2 |

HALT | Stop | 1 |

Assume that the content of memory location 3000 is 10 and the content of the register R3 is 2000. The content of each of the memory locations from 2000 to 2010 is 100. The program is loaded from the memory location 1000. All the numbers are in decimal.

Assume that the memory is word addressable. After the execution of this program, the content of memory location 2010 is:

Consider the following program segment. Here R1, R2 and R3 are the general purpose registers.

Instruction |
Operation |
Instruction size (no. of words) |

MOV R1, (3000) | R1 _ m[3000] | 2 |

LOOP: MOV R2, (R3) | R2 _ M[R3] | 1 |

ADD R2, R1 | R2 _ R1 + R2 | 1 |

MOV (R3), R2 | M[R3] _ R2 | 1 |

INC R3 | R3 _ R3 + 1 | 1 |

DEC R1 | R1 _ R1 - 1 | 1 |

BNZ LOOP | Branch on not zero | 2 |

HALT | Stop | 1 |

Assume that the content of memory location 3000 is 10 and the content of the register R3 is 2000. The content of each of the memory locations from 2000 to 2010 is 100. The program is loaded from the memory location 1000. All the numbers are in decimal.

Assume that the memory is byte addressable and the word size is 32 bits. If an interrupt occurs during the execution of the instruction “INC R3”, what return address will be pushed on to the stack?

Consider the following C function:

```
int f(int n) {
static int r = 0;
if (n <= 0) return 1;
if (n > 3) {
r = n;
return f(n - 2) + 2;
}
return f(n - 1) + r;
}
```

What is the value of f (5)?

The following postfix expression with single digit operands is evaluated using a stack: 8 2 3 $\land$/ 2 3 * + 5 1 * -

Note that $\land$ is the exponentiation operator. The top two elements of the stack after the first * is evaluated are:

Consider the following segment of C-code:

```
int j, n;
j = 1;
while (j <= n)
j = j*2
```

The number of comparisons made in the execution of the loop for any n > 0 is:

Consider the following C program segment where CellNode represents a node in a binary tree:

```
Struct Cell Node {
Struct Cell Node * left child;
Int element;
Struct Cell Node * right Child;
};
Int Get Value(struct Cell Node * ptr) {
Int Value = 0;
if (ptr! = NULL)
if ((ptr - > left child == NULL) &&
(ptr - > right Child == NULL))
Value = 1;
else
Value = value + GetValue(ptr - > left Child) + Get Value(ptr - > right child);
}
return (value);
}
```

The value returned by GetValue when a pointer to the root of a binary tree is passed as its argument is:

Information about a collection of students is given by the relation studinfo (studId, name, sex). The relation enroll (studId, courseId) gives which student has enrolled for (or taken) what course(s). Assume that every course is taken by at least one male and at least one female student. What does the following relational algebra expression represent?

$\pi_{courceId}\left(\left(\pi_{\text{studId}}\left(\sigma_{sex="female"}\left(\text{studInfo}\right)\right) \times \pi_{courseId}\left(\text{enroll}\right)\right) -\text{enroll}\right)$

Consider the table employee (empId, name, department, salary) and the two queries Q_{1}, Q_{2} below.
Assuming that department 5 has more than one employee, and we want to find the employees who get higher salary in department 5, which one of the statements is TRUE for any arbitrary employee table?

```
$Q_1$ : Select e. empId
From employee
Where not exists
(Select * From employee s where s. department = “5” and s. salary >= e. salary)
$Q_2$ : Select e. empId
From employee e
Where e. salary > any
(select distinct salary From employee s where s. Where s. department = “5”)
```

Consider the following schedules involving two transactions. Which one of the following statements is TRUE?

S_{1} : r_{1} (X); r_{1} (Y); r_{2} (X); r_{2} (Y); w_{2} (Y); W_{1} (X)
S_{2} : r_{1} (X); r_{2} (Y); r_{2} (X); w_{2} (Y); r_{1} (Y); W_{1} (X)

Consider the relation employee (name, sex, supervisor Name) with name as the key; supervisor Name gives the name of the supervisor of the employees under consideration. What does the following Tuple Relational Calculus query produce?
{
e. name| employee (e) ^{$\land$}
^{$\forall X$}[^{$\neg$}employee (x) ^{$\lor$ }x.supervisorName ^{$\ne$}e.name ^{$\lor$} x.sex = “male”]
}

Which one of the following statements is FALSE?

The order of a leaf node in a B^{+} - tree is the maximum number of (value, data record pointer) pairs it can hold. Given that the block size is 1K bytes, data record pointer is 7 bytes long, the value field is 9 bytes long and a block pointer is 6 bytes long, what is the order of the leaf node?

Which of the following problems is undecidable?

Which of the following is TRUE?

Consider the following Finite State Automaton:

The language accepted by this automaton is given by the regular expression

The language {0^{i} 21^{i} | i $\ge$0} over the alphabet {0, 1, 2} is:

A minimum state deterministic finite automaton accepting the language L = {w | w $\in${0, 1}*, number of 0s and 1s in w are divisible by 3 and 5, respectively} has

Consider the following Finite State Automaton:

The minimum state automaton equivalent to the above FSA has the following number of states

Which of the following languages is regular?

Consider the set $S = \left\{ a , b , c , d \right\}$ . Consider the following 4 partitions $\pi_1,\pi_2,\pi_3,\pi_4$ on $S : \pi_1 = \left\{\overline{abcd}\right\} , \pi_2 = \left\{\overline{ab}, \overline{cd}\right\}, \pi_3 = \left\{\overline{abc}, \overline{d}\right\}, \pi_4 = \left\{\bar{a}, \bar{b}, \bar{c}, \bar{d}\right\}$. Let $\prec$ be the partial order on the set of partitions $S' = \{\pi_1,\pi_2,\pi_3,\pi_4\}$ defined as follows: $ \pi_i \prec \pi_j$ if and only if $\pi_i \text{ refines }\pi_j $. The poset diagram for $ (S',\prec)$ is: