### Computer Science (GATE Exam) 2003 - Previous Question Paper Solution

Description: GATE Exam Previous Year Question Paper Solution Computer Science(CS) - 2003 | |

Number of Questions: 90 | |

Created by: Aliensbrain Bot | |

Tags: Computer Science GATE CS Previous Year Paper |

Let A be a sequence of 8 distinct integers sorted in ascending order. How many distinct pairs of sequences, B and C are there such that (i) each is sorted in ascending order, (ii) B has 5 and C has 3 elements, and (iii) the result of merging B and C gives A?

n couples are invited to a party with the condition that every husband should be accompanied by his wife. However, a wife need not be accompanied by her husband. The number of different gatherings possible at the party is

Let $P(E)$ denote the probability of the event $E$. Given $P(A) = 1$, $P(B) = 1/2$, the values of $P(A\mid B)$ and $P(B\mid A)$ respectively are

Let G be an arbitrary graph with n nodes and k components. If a vertex is removed from G, the number of components in the resultant graph must necessarily lie between

Assuming all numbers are in 2's complement representation, which of the following numbers is divisible by 11111011?

Which of the following suffices to convert an arbitrary CFG to an LL(1) grammar?

In a bottom-up evaluation of a syntax directed definition, inherited attributes can

Consider the set $ \sum^* $ of all strings over the alphabet $ \sum $ = {0, 1}. $ \sum^* $ with the concatenation operator for strings

Assume that the SLR parser for a grammar G has n_{1} states and the LALR parser for that grammar has n_{2} states. Which of the following statements about the relationship between n_{1} and n_{2 }is true?

In a heap with n elements with the smallest element at the root, the 7th smallest element can be found in time

Which of the following statements is FALSE?

Consider the following three claims
I. (n + k)^{m} = O (n^{m}) where k and m are constants
II. 2^{n+1} = O(2^{n})
III. 2^{2n+1} = O(2^{n})

Which of these claims are correct?

In a system with 32 bit virtual addresses and 1KB page size, use of one-level page tables for virtual to physical address translation is not practical because of

Which of the following functionalities must be implemented by a transport protocol over and above the network protocol?

How many perfect matching are there in a complete graph of 6 vertices?

Consider the following formula and its two interpretations $I_1$ and $I_2$.

$\alpha: (\forall x)\left[P_x \Leftrightarrow (\forall y)\left[Q_{xy} \Leftrightarrow \neg Q_{yy} \right]\right] \Rightarrow (\forall x)\left[\neg P_x\right]$

$I_1$ : Domain: the set of natural numbers

$P_x$ = 'x is a prime number' $Q_{xy}$ = 'y divides x' $I_2$ : same as $I_1$ except that $P_x$ = 'x is a composite number'.

Which of the following statements is true?

The usual O(n^{2}) implementation of Insertion Sort to sort an array uses linear search to identify the position where an element is to be inserted into the already sorted part of the array. If, instead, we use binary search to identify the position, the worst case running time will

Using a larger block size in a fixed block size file system leads to

Which of the following is a valid first order formula? (Here $\alpha$and $\beta$ are first order formulae with x as their only free variable)

$m$ identical balls are to be placed in $n$ distinct bags. You are given that $m \geq kn$, where $k$ is a natural number $\geq 1$. In how many ways can the balls be placed in the bags if each bag must contain at least $k$ balls?

Consider the following graph

Among the following sequences I a b e g h f II a b f e h g III a b f h g e IV a f g h b e

Which are depth first traversals of the above graph?

Let (S,$\le$) be a partial order with two minimal elements a and b, and a maximum element c. Let P: S {True, False} be a predicate defined on S. Suppose that P(a) = True, P(b) = False and P(x)$\Rightarrow$P(y) for all x, y $\epsilon$S satisfying x$\le$y,

Where $\Rightarrow$stands for logical implication. Which of the following statements CANNOT be true?

A graph $G=(V,E)$ satisfies $|E| \leq 3 |V| - 6$. The min-degree of $G$ is defined as $ min_{v\in V} \left \{ degree(V) \right \}$. Therefore, min-degree of $G$ cannot be

A piecewise linear function f(x) is plotted using thick solid lines in the figure below (the plot is drawn to scale).

If we use the Newton-Raphson method to find the roots of f(x) =0 using x0, x1, and x2 respectively as initial guesses, the roots obtained would be

Which of the following assertions is FALSE about the Internet Protocol (IP)?

Let f : A $\rightarrow$ B be an injective (one-to-one) function. Define g : 2^{A} $\rightarrow$ 2^{B} as:
g(C) = {f(x) | x $\epsilon$ C}, for all subsets C of A.
Define h : 2^{B$\rightarrow$}2^{A} as: h(D) = {x | x $\epsilon$ A, f(x) $\epsilon$ D}, for all subsets D of B.

Which of the following statements is always true?

Let $\Sigma = \left\{a, b, c, d, e\right\}$ be an alphabet. We define an encoding scheme as follows:

$g(a) = 3, g(b) = 5, g(c) = 7, g(d) = 9, g(e) = 11$.

Let $p_i$ denote the i-th prime number $\left(p_1 = 2\right)$.

For a non-empty string $s=a_1 \dots a_n$, where each $a_i \in \Sigma$, define $f(s)= \Pi^n_{i=1}P_i^{g(a_i)}$.

For a non-empty sequence$\left \langle s_j, \dots,s_n\right \rangle$ of stings from $\Sigma^+$, define $h\left(\left \langle s_i \dots s_n\right \rangle\right)=\Pi^n_{i=1}P_i^{f\left(s_i\right)}$

Which of the following numbers is the encoding, $h$, of a non-empty sequence of strings?

he sum of the number of times each literal appears in the expression. For example, the literal count of (xy + xz') is 4. What are the minimum possible literal counts of the product-of-sum and sum-of product representations respectively of the function given by the following Karnaugh map? Here, X denotes “don't care”

Consider the following circuit composed of XOR gates and non-inverting buffers

The non-inverting buffers have delays d1 = 2 ns and d2 = 4 ns as shown in the figure. Both XOR gates and all wires have zero delay. Assume that all gate inputs, outputs and wires are stable at logic level 0 at time

- If the following waveform is applied at input A, how many transition(s) (change of logic levels) Occur (s) at B during the interval from 0 to 10 ns?

Consider the grammar shown below. S $\rightarrow$ C C C $\rightarrow$ c C | d This grammar is

Consider the following system of linear equations $$\left( \begin{array}{ccc} 2 & 1 & -4 \\ 4 & 3 & -12 \\ 1 & 2 & -8 \end{array} \right) \left( \begin{array}{ccc} x \\ y \\ z \end{array} \right) = \left( \begin{array}{ccc} \alpha \\ 5 \\ 7 \end{array} \right)$$ Notice that the second and the third columns of the coefficient matrix are linearly dependent. For how many values of $\alpha$, does this system of equations have infinitely many solutions?

Consider the translation scheme shown below. S $\rightarrow$T R R $\rightarrow$ + T {print('+');} R|$\epsilon$ T $\rightarrow$ num {print (num.val);} Here num is a token that represents an integer and num.val represents the corresponding integer value. For an input string '9 + 5 + 2', this translation scheme will print

In a permutation a_{1} ... a_{n}, of n distinct integers, an inversion is a pair (a_{i}, a_{j}) such that i < j and ai > aj.

If all permutations are equally likely, what is the expected number of inversions in a randomly chosen permutation of 1. . . n?

Consider the syntax directed definition shown below,
S $\rightarrow$ id : = E {gen (id.place = E.place;);}
E $\rightarrow$ E_{1} + E_{2} {t = newtemp( );
gen(t = E1.place + E2.place;);
E.place = t;}
E $\rightarrow$ id {E.place = id.place;}

Here, gen is a function that generates the output code, and newtemp is a function that returns the name of a new temporary variable on every call. Assume that ti's are the temporary variable names generated by newtemp. For the statement 'X : = Y + Z', the 3-address code sequence generated by this definition is

Consider the grammar shown below S $\rightarrow$i E t S S' | a S' $\rightarrow$ e S |$\epsilon$ E $\rightarrow$ b In the predictive parse table, M, of this grammar, the entries M[S' , e] and M[ S' ,$] respectively area

A program consists of two modules executed sequentially. Let $f_1(t)$ and $f_2(t)$ respectively denote the probability density functions of time taken to execute the two modules. The probability density function of the overall time taken to execute the program is given by

Consider the following logic program P

$\begin{align*} A(x) &\gets B(x,y), C(y) \ &\gets B(x,x) \end{align*}$

Which of the following first order sentences is equivalent to P?

The following program fragment is written in a programming language that allows global variables and does not allow nested declarations of functions. global int i = 100, j = 5; void P(x) { int i = 10; print(x + 10); i = 200; j = 20; print (x); } main() {P(i + j);}

If the programming language uses static scoping and call by need parameter passing mechanism, the values printed by the above program are

In a permutation a_{1} ... a_{n}, of n distinct integers, an inversion is a pair (a_{i}, a_{j}) such that i < j and ai > aj.

What would be the worst case time complexity of the Insertion Sort algorithm, if the inputs are restricted to permutations of 1. . . n with at most n inversions?

The following program fragment is written in a programming language that allows global variables and does not allow nested declarations of functions. global int i = 100, j = 5; void P(x) { int i = 10; print(x + 10); i = 200; j = 20; print (x); } main() {P(i + j);}

If the programming language uses dynamic scoping and call by name parameter passing mechanism, the values printed by the above program are

Let $G= (V,E)$ be a directed graph with $n$ vertices. A path from $v_i$ to $v_j$ in $G$ is a sequence of vertices ($v_{i},v_{i+1}, \dots , v_j$) such that $(v_k, v_k+1) \in E$ for all $k$ in $i$ through $j-1$. A simple path is a path in which no vertex appears more than once.

Let $A$ be an $n \times n$ array initialized as follows.

$$A[j,k] = \begin{cases} 1 \text { if } (j,k) \in E \\ 0 \text{ otherwise} \end{cases}$$

Consider the following algorithm.

```
for i=1 to n
for j=1 to n
for k=1 to n
A[j,k] = max(A[j,k], A[j,i] + A[i,k]);
```

Which of the following statements is necessarily true for all j and k after termination of the above algorithm?

What is the weight of a minimum spanning tree of the following graph?

Let G = (V,E) be an undirected graph with a subgraph G1 = (V1, E1). Weights are assigned to edges of G as follows.

$$G= \begin{cases} 0 \text { if } e \in E_1 \\ 1 \text{ otherwise} \end{cases}$$

A single-source shortest path algorithm is executed on the weighted graph (V,E,w) with an arbitrary vertex
v_{1} of V_{1} as the source. Which of the following can always be inferred from the path costs computed?

Consider the set {a, b, c} with binary operators + and × defined as follows

+ | a | b | c |
---|---|---|---|

a | b | a | c |

b | a | b | c |

c | a | c | b |

* | a | b | c |
---|---|---|---|

a | a | b | c |

b | b | c | a |

c | c | c | b |

For example, a + c = c, c + a = a, c ×b = c and b × c = a. Given the following set of equations: (a × x) + (a × y) = c (b × x) + (c × y) = c

The number of solution(s) (i.e., pair(s) (x, y) that satisfy the equations) is

A uni-processor computer system only has two processes, both of which alternate 10 ms CPU bursts with 90 ms I/O bursts. Both the processes were created at nearly the same time. The I/O of both processes can proceed in parallel. Which of the following scheduling strategies will result in the least CPU utilization (over a long period of time) for this system?

The following are the starting and ending times of activities A, B, C, D, E, F, G and H respectively in
chronological order: “a_{s} b_{s} c_{s} a_{e} d_{s} c_{e} e_{s} f_{s} b_{e} d_{e} g_{s} e_{e} f_{e} h_{s} g_{e} h_{e}”. Here, x_{s} denotes the starting time and x_{e}
denotes the ending time of activity X. We need to schedule the activities in a set of rooms available to us.
An activity can be scheduled in a room only if the room is reserved for the activity for its entire duration.
What is the minimum number of rooms required?

Which of the following is NOT an advantage of using shared, dynamically linked libraries as opposed to using statically linked libraries?

A processor uses 2-level page tables for virtual to physical address translation. Page tables for both levels are stored in the main memory. Virtual and physical addresses are both 32 bits wide. The memory is byte addressable. For virtual to physical address translation, the 10 most significant bits of the virtual address are used as index into the first level page table while the next 10 bits are used as index into the second level page table. The 12 least significant bits of the virtual address are used as offset within the page.Assume that the page table entries in both levels of page tables are 4 bytes wide. Further, the processor has a translation look-aside buffer (TLB), with a hit rate of 96%. The TLB caches recently used virtual page numbers and the corresponding physical page numbers. The processor also has a physically addressed cache with a hit rate of 90%. Main memory access time is 10 ns, cache access time is 1 ns, and TLB access time is also 1 ns.

Assuming that no page faults occur, the average time taken to access a virtual address is approximately (to the nearest 0.5 ns)

A 2 km long broadcast LAN has 10^{7} bps bandwidth and uses CSMA/CD. The signal travels along the wire at 2 x 10^{8} m/s. What is the minimum packet size that can be used on this network?

A processor uses 2-level page tables for virtual to physical address translation. Page tables for both levels are stored in the main memory. Virtual and physical addresses are both 32 bits wide. The memory is byte addressable. For virtual to physical address translation, the 10 most significant bits of the virtual address are used as index into the first level page table while the next 10 bits are used as index into the second level page table. The 12 least significant bits of the virtual address are used as offset within the page.Assume that the page table entries in both levels of page tables are 4 bytes wide. Further, the processor has a translation look-aside buffer (TLB), with a hit rate of 96%. The TLB caches recently used virtual page numbers and the corresponding physical page numbers. The processor also has a physically addressed cache with a hit rate of 90%. Main memory access time is 10 ns, cache access time is 1 ns, and TLB access time is also 1 ns.

Suppose a process has only the following pages in its virtual address space: two contiguous code pages starting at virtual address 0x00000000, two contiguous data pages starting at virtual address 0×00400000, and a stack page starting at virtual address 0×FFFFF000. The amount of memory required for storing the page tables of this process is

Host A is sending data to host B over a full duplex link. A and B are using the sliding window protocol for flow control. The send and receive window sizes are 5 packets each. Data packets (sent only from A to B) are all 1000 bytes long and the transmission time for such a packet is 50 micro second. Acknowledgement packets (sent only from B to A) are very small and require negligible transmission time. The propagation delay over the link is 200 micro second. What is the maximum achievable throughput in this communication?

The following resolution rule is used in logic programming. Derive clause (P$\lor$Q) from clauses (P $\lor$ R), (Q$\lor$$\neg$R) Which of the following statements related to this rule is FALSE?

Suppose we want to synchronize two concurrent processes P and Q using binary semaphores S and T. The codes for the processes P and Q are shown below. Process P: Process Q: while (1) { while (1) { W: Y: print '0'; print '1'; print '0'; print '1'; X: Z: } } Synchronization statements can be inserted only at points W, X, Y and Z

Which of the following will always lead to an output starting with '001100110011'?

Suppose we want to synchronize two concurrent processes P and Q using binary semaphores S and T. The codes for the processes P and Q are shown below. Process P: Process Q: while (1) { while (1) { W: Y: print '0'; print '1'; print '0'; print '1'; X: Z: } } Synchronization statements can be inserted only at points W, X, Y, and Z

Which of the following will ensure that the output string never contains a substring of the form 01n0 or 10n1 where n is odd?

The subnet mask for a particular network is 255.255.31.0. Which of the following pairs of IP addresses could belong to this network?

Consider the following class definitions in a hypothetical Object Oriented language that supports inheritance and uses dynamic binding. The language should not be assumed to be either Java or C++, though the syntax is similar.

Class P { Class Q subclass of P { void f(int i) { void f(int i) { print(i); print(2*i); } } } } Now consider the following program fragment: Px = new Q() Qy = new Q(); Pz = new Q(); x.f(1); ((P)y).f(1); z.f(1); Here ((P)y) denotes a typecast of y to P. The output produced by executing the above program fragment will be

Consider an array multiplier for multiplying two n bit numbers. If each gate in the circuit has a unit delay, the total delay of the multiplier is

For a pipelined CPU with a single ALU, consider the following situations I. The j + 1-st instruction uses the result of the j-th instruction as an operand II. The execution of a conditional jump instruction III. The j-th and j + 1-st instructions require the ALU at the same time

Which of the above can cause a hazard?

Consider the ALU shown below

If the operands are in 2's complement representation, which of the following operations can be performed by suitably setting the control lines K and C0 only (+ and - denote addition and subtraction respectively)?

Consider the following assembly language program for a hypothetical processor
A, B, and C are 8 bit registers. The meanings of various instructions are shown as comments.
MOV B, #0 ; B ^{$\leftarrow$ }0
MOV C, #8 ; C ^{$\leftarrow$ }8
Z: CMP C, #0 ; Compare C with 0
JZX ; Jump to X if zero flag is set
SUB C, #1 ; C ^{$\leftarrow$} C - 1
RRC A, #1 ; right rotate A through carry by one bit. Thus:
; if the initial values of A and the carry flag are a_{7}… a_{0} and
; instruction will be c_{0}a_{7}…a_{1} and a_{0} respectively
JCY ; jump to Y if carry flag is set
JMP Z ; jump to Z
Y: ADD B, #1 ; B ^{$\leftarrow$} B + 1
X: If the initial value of register A is A0, the value of register B after the program execution will be

Consider the following assembly language program for a hypothetical processor
A, B, and C are 8 bit registers. The meanings of various instructions are shown as comments.
MOV B, #0 ; B $\leftarrow$0
MOV C, #8 ; C$\leftarrow$8
Z: CMP C, #0 ; Compare C with 0
JZX ; Jump to X if zero flag is set
SUB C, #1 ; C $\leftarrow$ C -1
RRC A, #1 ; right rotate A through carry by one bit. Thus:
; if the initial values of A and the carry flag are a_{7}… a_{0} and
; instruction will be c_{0}a_{7}…a_{1} and a_{0} respectively
JCY ; jump to Y if carry flag is set
JMP Z ; jump to Z
Y: ADD B, #1 ; B $\leftarrow$ B + 1
X:

Which of the following instructions when inserted at location X will ensure that the value of register A after program execution is the same as its initial value?

Consider the following C function. float f,(float x, int y) { float p, s; int i; for (s=1,p=1,i=1; i

Assume the following C variable declaration

int * A[10], B[10][10]; of the following expressions

I. A[2]

II. A[2][3]

III. B[1] IV. B[2][3]

Which will not give compile-time errors if used as left hand sides of assignment statements in a C program?

Suppose the numbers 7, 5, 1, 8, 3, 6, 0, 9, 4, 2 are inserted in that order into an initially empty binary search tree. The binary search tree uses the usual ordering on natural numbers. What is the in-order traversal sequence of the resultant tree?

Let T(n) be the number of different binary search trees on n distinct elements. Then T(n) = $\sum_{k -1}^n T(k -1)(x) $, where x is

Consider the following 2-3-4 tree (i.e., B-tree with a minimum degree of two) in which each data item is a letter. The usual alphabetical ordering of letters is used in constructing the tree.

What is the result of inserting G in the above tree?

A data structure is required for storing a set of integers such that each of the following operations can be done in O (log n) time, where n is the number of elements in the set. I. Deletion of the smallest element II. Insertion of an element if it is not already present in the set Which of the following data structures can be used for this purpose?

Let S be a stack of size n $\ge$1. Starting with the empty stack, suppose we push the first n natural numbers in sequence, and then perform n pop operations Assume that Push and Pop operations take X seconds each, and Y seconds elapse between the end of one such stack operation and the start of the next operation. For m $\ge$1, define the stack-life of m as the time elapsed from the end of Push(m) to the start of the pop operation that removes m from S. The average stack-life of an element of this stack is

Consider the C program shown below. #include <stdio.h> #define print(x) print f(”%d “, x) int x; void Q(int z) { z + = x; print (z); } void P(int *y) { int x = *y+2; Q(x); *y = x-1; Print (x); } Main (void) { x = 5; P (&x) Print (x); } The output of this program is

Consider the function f defined below. struct item { int data; struct item * next; }; int f(struct item *p) { return ((p == NULL) || (p ->next == NULL) || ((p->data <= p -> next -> data) && f(p-> next))); } For a given linked list p, the function f returns 1 if and only if

In the following C program fragment, j, k, n and TwoLog_n are integer variables, and A is an array of
integers. The variable n is initialized to an integer $\ge$3, and TwoLog_n is initialized to the value of 2
*$\lceil log_2(n) \rceil$

for (k = 3; k < = n; k++)
A [k] = 0;
for (k = 2; k < = TwoLog_n ; k++)
for (j = k+1; j < = n ; j++)
A[j] = A[j] || (j% k);
for (j = 3; j < = n ; j++)
if (!A[j]) print f(”%d ”,j);
The set of numbers printed by this program fragment is

Consider three data items D1, D2 and D3, and the following execution schedule of transactions T1, T2 and T3. In the diagram, R(D) and W(D) denote the actions reading and writing in the data item D respectively.

Which of the following statements is correct?

Consider the following SQL query.
select distinct a_{1}, a_{2}, ..., an
from r_{1}, r_{2}, ..., rm
where P

For an arbitrary predicate P, this query is equivalent to which of the following relational algebra expressions?

Consider the set of relations shown below and the SQL query that follows. Students: (Roll_number, Name, Date_of_birth) Courses: (Course number, Course_name, Instructor) Grades: (Roll_number, Course_number, Grade) select distinct Name from Students, Courses, Grades where Students. Roll_number = Grades.Roll_number and Courses.Instructor = Korth and Courses.Course_number = Grades.Course_number and Grades.grade = A Which of the following sets is computed by the above query?

Which of the following scenarios may leave the transaction vulnerable to dirty reads, phantom reads, etc. in a database system?

Consider the following functional dependencies in a database.

Date_of_Birth ^{$\rightarrow$} Age Age ^{$\rightarrow$} Eligibility
Name ^{$\rightarrow$} Roll_number Roll_number ^{$\rightarrow$} Name
Course_number ^{$\rightarrow$} Course_name Course_number ^{$\rightarrow$} Instructor
(Roll_number, Course_number) ^{$\rightarrow$} Grade

The relation (Roll_number, Name, Date_of_Birth, Age) is

The regular expression 0*(10*)* denotes the same set as

If the strings of a language L can be effectively enumerated in lexicographic (i.e., alphabetic) order, which of the following statements is true?

Nobody knows yet if P = NP. Consider the language L defined as follows.

$$L = \begin{cases} (0 +1)^* \text { if } (P = NP) \in E \\ \phi \text{ otherwise} \end{cases}$$

Which of the following statements is true?

Consider the following deterministic finite state automaton M.

Let S denote the set of seven bit binary strings in which the first, the fourth, and the last bits are 1. The number of strings in S that are accepted by M is

Ram and Shyam have been asked to show that a certain problem ^{$\prod$}is NP-complete. Ram shows a polynomial time reduction from the 3-SAT problem to^{$\prod$}, and Shyam shows a polynomial time reduction from ^{$\prod$}to 3-SAT. Which of the following can be inferred from these reductions?

A single tape Turing Machine M has two states q0 and q1, of which q0 is the starting state. The tape alphabet of M is {0, 1, B} and its input alphabet is {0,1}. The symbol B is the blank symbol used to indicate end of an input string. The transition function of M is described in the following table.

0 | 1 | B | |
---|---|---|---|

q0 | q1, 1, R | q1, 1, R | Halt |

q1 | q1, 1, R | q0, 1, L | q0, B, L |

The table is interpreted as illustrated below. The entry (q1, 1, R) in row q0 and column 1 signifies that if M is in state q0 and reads 1 on the current tape square, then it writes 1 on the same tape square, moves its tape head one position to the right and transitions to state q1.

Which of the following statements is true about M?

Let G = ({S},{a,b},R,S) be a context free grammar where the rule set R is S $\rightarrow$ a S b | S S | $\epsilon$ Which of the following statements is true?

Consider the NFA M shown below.

Let the language accepted by M be L. Let L_{1} be the language accepted by the NFA M_{1} obtained by changing the accepting state of M to a non-accepting state and by changing the non-accepting states of M to accepting states. Which of the following statements is true?

Consider two languages $L_1$ and $L_2$, each over the alphabet $\Sigma$.

Let $f: \Sigma \to \Sigma$ be a polynomial time, computable bijection, such that:

$$\forall x: \Bigl(x \in L_1 \iff f(x) \in L_2\Bigr )$$

Further, let $f^{-1}$ also be polynomial time computable.

Which of the following canNOT be true?

Define languages L0 and L1 as follows:
L_{0} = {<M, w, 0> | M halts on w}
L_{1} = {<M, w, 1> | M does not halt on w}
Here <M, w, i> is a triplet, whose first component, M, is an encoding of a Turing
Machine, second component, w, is a string, and third component, i, is a bit.
Let L = L_{0$\cup$}L_{1}. Which of the following is true?

The following is a scheme for floating point number representation using 16 bits.

Bit Position | 15 | 14 .... 9 | 8 ...... 0 |
---|---|---|---|

s | e | m | |

Sign | Exponent | Mantissa |

Let s, e, and m be the numbers represented in binary in the sign, exponent, and mantissa fields respectively. Then the floating point number represented is:

$$\begin{cases}(-1)^s \left(1+m \times 2^{-9}\right) 2^{e-31}, & \text{ if the exponent } \neq 111111 \\ 0, & \text{ otherwise} \end{cases}$$

What is the maximum difference between two successive real numbers representable in this system?

A 1-input, 2-output synchronous sequential circuit behaves as follows: Let zk, nk denote the number of 0's and 1's respectively in initial k bits of the input (zk+nk=k). The circuit outputs 00 until one of the following conditions holds. zk - nk=2. In this case, the output at the k-th and all subsequent clock ticks is 10. nk - zk = 2. In this case, the output at the k-th and all subsequent clock ticks is 01.

What is the minimum number of states required in the state transition graph of the above circuit?

The cube root of a natural number n is defined as the largest natural number m such that m^{3} $\le$n. The complexity of computing the cube root of n (n is represented in binary notation) is

Consider the following recurrence relation

$T(1)=1$

$T(n+1) = T(n)+\lfloor \sqrt{n+1} \rfloor$ for all $n \geq 1$

The value of $T(m^2)$ for $m \geq 1$ is