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c. Given 9 = *, * y' ; y(0) = l, y(0. I ) = l. I 169, y(0.2) = 1.2773, y(o.3)= 1.5049, find y(0.4)dx

correct to three decimal places, using the Milne's predictor-corrector method. Apply thecorrector formula twice. (07 Marks)

2 a. Employing the Picard's method, obtain the second order approximate solution of thefollowing problem at x = 0.2,

b.

c.

dv dz--:-=x+yzidx'dxUsing the Runge-Kuttagiven condition:

method, solve the following

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problem |{*:'9-Oy =0, y(0) = l, y'(0) = 0.1. Given y(0.1) = 1.03995,ox ox

y'(0.1) = 0.6955, y(0.2) = 1. 138036, y'(0.2) = 1.258, y(0.3) = 1.29865, y'(0.3) = 1.373.(07 Marks)

3 a. Derive Cauchy-Riemann equations in polar form. (06 Marks)/^) ^)\b. Iff(zrisaregulartuncrionolz.provethat l+-+lF,rl'= +lt'r"[ . r0TMarks)' [d*' ay' )' 'r I ''

c.Ifw=$+iyrepresentsthecomplexpotentialforanelectricfieldandy=x2_y2x' + y'

determine the function $. Also find the complex potential as a function of z. (07 Marks)

Engineering Mathematics - lV

Note: Answer FIVE fult questions, selecting A*at least TWO qaestions from each part. i

PART - AUsing the Taylor's method, find the third order approximar" ,otutiorr.,]---$yffi"problem 9 = *'y + I . with y(0) = 0. Consider terms upto foufih degree. (06 Marks)

b. Solve the differential equation I = -rr' under the initial condition y(O) = 2, by using theox

modified Euler's method, at the points x = 0.1 and x = 0.2. Take the step size h = 0.1 and

=y+zx; y(0)= I, z(0)=-1.

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= i, zz = l, zt = -l on to the(07 Marks)

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kr4 a. Discuss the translormalion ol-w =z-i.

b. Find the bilinear transformation that transforms the points z;

points wt = l, wu = 0, w,l = co respectively.

c. Evaluate Jsin nz2 +cosrz2

dz where c is the circle lzl = 3, using Cauchy's integral formula.

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PART -B5 a. obtain the solution of x2y'+ xy'+ (x2 - nt)y = 0 in terms ofJn(x) and J n(x)'

b' E*p..r, f(x) = 1a + 3x3 - x2 + 5x-2 in terms of Legendre polynomials'+l 2q. Prove that IP.

(x) . P,, (x)dx = ,;]1 , m = n.

a. From five positive and seven negative numbers, five numbers are chosen at random and

multiplied. what is the probability that the product is a (i) negative numbel and (ii) positive(06 Marks)

1, rtut =1, ,,o ,,. B) =+, rind P(A/B), P(B/A),number?

b' lt a and B are two events with P(A) =

Ptatgl. Pr slAt ana PrA/Br.

taller than 1.8 m, what is the probability that the student is a girl?

c. In a certain college, 4a/o of boy students arild lo/o of gill students are taller than 1.8 m.

Furthermore , 600/o of the students are girls. If a student is selected at random and is found

find: i) P(x < l), (ir P(l < x < 2), (iiD P(x<2), iv) P(x > l), (v) P(x > 2)' (06 Marks)

b. Obtain the mean and standard deviation of binomial distribution. (07 Marks)

c. In an examination 77o of students score less than 357o marks and 897o of students score less

than 60Va marks. Find the mean and standard deviation if the marks are normally distributed.

It is given that P(0 < z < 1.2263) = 0.39 and P(0 < z < l'47 5'7) = 0'43' (07 Marks)

a. A random variable x has the density tun.tion pt*r -{Kl ' :t'.". Evaluate K. and' I 0. elsewhere

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a. A random sample of400 items chosen from an infinite population is found to have a mean

of 82 and a st;ndard deviation of 18. Find the 95Vo confidence limits for the mean of the

population from which the sample is drawn. (06 Marks)

U. inite past, a machine has produced washers having a thickness of 0.50 mm. To determine

whether the machine is in proper working ordel, a sample of l0 washers is chosen for which

the mean thickness is found as 0.53 mm with standard deviation 0.03 mm. Test the

hypothesis that the machine is in proper working order, using a level of significance of(i) 0.05 and (ii) 0.01 . (07 Marks)

c. Genetic theory states that children having one parent of blood tlpe M and the other of blood

tlpe N will aiways be one of the thrse tlpes M, MN, N and that the proportions of these

tftes will on u., ou.rug. be | :2 : l. A repofi states that out of 300 children having one M

parent and one N parent, 3Oo/o were found to be of tlpe M, 459o of type MN and the

remainder of type N. Test the theory by 1' lchi square) test. (07 Marks)

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Fourth Semester B.E. Degree Examination, Jlune 2Ol2Graph Theory and Gombinatorics

Time: 3 hrs. Max. Marks: I 00Note: Answer FIVE full questions, selecting

at least TWO questions from each part.

PART - AI a. Show that the max number of edges in a simple graph with n vertices is n(n - I )/2 (06 Marks)b. How many veftices will the following graphs have if they contain :

i) l6 edges and all vertices ofdegree 4.ii) l2 edges, 6 vertices of degree 3, and other veftices of degree less than 3. (08 Marks)

c. Prove that number of vertices of odd degree in a graph is even. (06 Marks)

2 a. Determine the number of vertices, number of edges and the number of regions for each ofthe planar graph in rhe Fig.Q.2(a)(i) and Fig.e.2(a)(ii) and also verifu that Eulers rheoremfor connected planar graph is satisfied. (06 Marks)

b. Fina a Hamilton cycle, if exisrs, for each of the graphs shoin in Fig.e.2(b)(i) andFig.Q.2(b)(ii). If graph has no Hamilton cycle, determine whether it has a Hamilton path.

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Detemine the chromatic polynomials P(G",7,)following graphs Fig.Q.2(c)(i) and Fig.Q.2(c)(ii).

Fie.Q.2(b)(ii)and chromatic numbers of each of the

(08 Marks)*

3 a. Show that for every tree T = (V, E) if lvl>2, then T hasat leasttwo pendent vertices.

((M Marks)

Fig.Q.2(a)(i) Fig.Q.2(a)(ii)

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vertices of degree 4. Find the number of pendent veftices in T. (06 Marks)c. What is a prefix code? Construct an optimal prefix code for the symbols a, b, c, .... i, j that

occurin agiven sample with respect to frequencies 78, 16,30,35, I25, 31,20,50,80,03.( 10 Marks)

a. Explain Prim's algorithm for finding shortest spanning tree ofa weighted graph. (08 Marks)b. Define the following with respect to bipartite graph G = (V, E) :

i) Complete matching ; ii) Maximal matcingii) Deficiency of graph C. (06 Marks)

c. Define : i) Cutset ; ii) Edge-connectivity ; iii) Vertex - connectivity with one example foreach. (06 Marks)

PART - B5 a. State two basic counting principles. Give on example for each.

b. A committee of 4 is to be chosen out of 6 Englishman, 5 Frenchmen and 4committee is to contain one of each nationalityi) In how many ways can it be done?ii) In how many arrangements will a paticular Indian be?

2, 3, or 5.

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Determine the coefficient of x8 in(x - 3Xx - 2)'?

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(08 Marks)c. Consider the following program segment, where i,j and k are integer variables

fori:=lto20doforj:=ltoidofor k := I toj doprint(i *j + k).Detemine how many times the print( ) statement is executed in above program. (06 Marhs)

6 a. In how many ways can these integers 1,2,3,....., l0 be arranged in a line so that no eveninteger is in its natural position. (06 Marks)

b. Determine the number of positive integers n where I < n < 100 and n is not divisible by(08 Marks)

c. A board consists of the shaded part as in the Fig.Q.6(c). Find its Rook polynomial. (06 Marks)

Find the generating function for each of the following sequence :

it 0,2,6, 12,20. 30.42.......ii) 8.26.s4.92........Find a formula to express 0r + lloperator.

(08 Marks)

(06 Marks)+ 22 + ..... + ,l as a function of n, using summation

(06 Marks)

Solve the relation Fn*2 = Fn+r + Fn, where n 2 0 and Fo = 0, Ft= 1. (06 Marks)Using generating function, solve yn+:-4yn+r +3y,=0 given thaty6= 2.yt=4. (08 Marks)Find the generated solution of S(K)-35 (K- l)-45 (K-2)=4K where K>2.

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at least TWO questions from each part.

PART - AI a. If tr(n) e 0 (gr (n)) and t2 (n) e 0(gz (n)), prove that

rr(n) + r2(n) e 0(max {gr(n), g:(n)}).b. If M(n) denotes the number of moves in tower of Hanoi puzzle when n disks are involved,

give a recurrence relation for M(n) and solve this recurrence relation. (07 Marks)

c. Give an algorithm for selection sort. If C(n) denotes the number of times the algorithm is

executed (n denotes input size), obtain an expression for C(n). (07 Marks)

2 a. Assuming that n is a power of 2, solve the recurence relation T(n) = 2T(nl2) + 2. Take

T(2) = 1 216 111; - O. (0s Marks)

b. Ifne[2k-1,21), prove that binary search algorithm makes at most K element comparisons fora successful search and either K - I or K comparisons for an unsuccessful search. (06 Marks)

(05 Marks)c.d.

Give an algorithm for merge sort.Consider the numbers given below. Show how partitioning algorithm of quick sort will place

106 in its correct position. Show all the steps clearly.106 17 128 134 141 91 84 63 42.

3 a. Let J be a set of K jobs and o= ir, iz, i:,......., i1 be a permutation ofjobs inJ such that

dir < di: < ....... 3 dir,. Prove that J is a feasible solution if and only if the jobs in J can be

. processed in the order o without violating any deadline. (07 Marks)tree for the weighted graph

(07 Marks)b. Using Prim's algorithm, determine minimum cost spanning

shown below, fig.Q.3(b):

e+

Fie.Q.3(b)c. In the weighted digraph given below, fig.Q.3(c) determine the shortest paths from vertex I

to all other vertices. (06 Marks)

Obtain the shortestbelow; fig.Q.4(a)

paths faom every vertex to every other

10cs43

vertex in the diagraph given(10 Marks)

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Using Warshall's algorithm, obtain the transitive closure of the matrix given below:[o r o o'r

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PART-BShow how insertion sort algorithm arranges the following members in increasing order.61 28 9 85 34. (06 Marks)

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c.

6a.

Give algorithms for the following:i) Comparison counting; ii) Distribution counting. (08 Marks)

Define the following: i) Tractable problems; ii) Class P; iii) Class Np; iv) polynomialreduction; v) NP complete problems. (05 Marks)

b. State subset sum problem. Using back tracking, obtain a solution to the subset sum problem' by taking s = [6, 8, 2, 14] and d = '16. (07 Marks)c. Explain approximation algorithms for NP - hard problems in general. Also discussapproximation algorithms for knapsack problem. (08 Marks)

7 a. What is prefix computation problem? Give the algorithms for prefix computation which uses.. ... ni) n processors; ii)

- processors. Obtain the time complexities ofthese algorithms.

tog n(10 Marks)

b. For an n x n matrix M with nonnegative integer coefficients, define ft and give analgorithm for computing ,I . Prove that ft can be computed from an n x n matrix M in0(log n) time using n3*' common CRCW PRAM processors for any fixed e > 0. (10 Marks)

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Write short notes on:Traveling salesperson problem.Input enhancement in string matching.Decision trees.Challenges of numerical algorithms.

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the unix file(06 Marks)

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Fourth Semester B.E. Degree Examination, Jurl,e?Ol2Unix and Shell Programming

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PART - A1 a. With neat diagram, explain the architecture of unix operating system.

b. With the help of a diagram, explain the parent-child relationship. Explainsystem.

c. Explain the following with examples:i) Absolute and relative path names ii) Internal and extemal commands.

2 a. A file's current permissions are rw_r_xr__. Specify the chmod expression required tochange them for the following:

3 a. Explain the thee standard files with respect to unix operaring syste;lff. ; - , . (06 Marks)

b. Expiain the mechanism of process creation. . '* ,.. --.- : ^ r. (06 Marks)

c. Explain the following commands with an example, , '1,' - '\-

i) Running jobs in bickground ii) Execute Lter. '',i "' -,.,,., ,' ; fua uarks)

:. ''4 a. Explain the Iollowing environment variables with examples: (.1**". . .Z

DSHELL ii)LOGNAME ii|PATH \1-!:i*' (06Marks)

b. Differentiate between hard link and soft link in unix with examples. (06 Marks)

c. Explain the following commands with example:

using both the relative and absolute methods of assigning permissions.

b. Explain briefly the file attributes listed using ls - I command.

c. What are the different modes of Vi editor? Explain with a diagram.

i) tail ii) paste iii) tr iv) pr

PART - B5 a. With suitable examples, explain the grep command and its various options.

b. Explain the line addressing and context addressing in sed with examples.

c. Explain the different ways of using test statements, with examples.

6 a. What is shell programming? Write a shell program that will do the following tasks in order:Clear the screenPrint the current directoryDisplay current login users.

b. Explain the shell features of 'while' and 'for' with syntax.

c. What is the 'exit' status of a command and where is it stores?

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a. What is AWK? Explain any three built in functions in AWK. (07 Marks)

b. Write an AWK sequence to find IIRA, DA and Netpay of an employee, where DA is 25Vo ofbasic, HRA is 50% basic and netpay is the sum of HRA, DA and basic pay.

c. Explain the list and anays in PERL.

8 a. Explain the following storing handling functions of PERL with examples:i) length ii) index iii) substr iv) reverse (08 Marks)

b. Write a PERL pro$am to pdnt numbers that are accepted from keyboard using while and

(07 Marks)

(06 Marks)

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array construct.

c. Explain the following in PERL with examples.i) fore each loping constnrct ii) join

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Fourth Semester B.E. Degree Examination, Jlurrre 2O1^2

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Note: Answer FIVE full questions, selectingat least TWO questions from each part.

PART-A

Describe the memory map of a PC system, with a neat diagram. (0E Marks)Explain the flags of 8086 processor using suitable examples. (06 Marks)Draw_and explain the programming model of the g0g6 through the coRE-2 microprocessorincluding the 64-bit extensions. (06 Marks)

What are the advantages of memory paging? Illustrate the conceptdiagram.Discuss the following addressing modes with examples:i) Directii) Regisrer indirectii, Base plus indexiv) Immediatev) Scaled indexed.

Describe the following instruction with suitable examples:i) PUSH ii) MUL iii) IN iv) AAA.Write an ALP using 8086 instructions to generate and add the first l0save the numbers and result in memory location Num and Sum.Bring out the importance of XLAT instruction using a suitable program.

write an ALP using 8086 instructions to count the numbers of zeros in a given g bit numberand store_the result in memory location ,Res,.

(08 Marks)Explain the following assembler directives: i) Assume; ii) proc; iii) Ends; iv) DB.

Briefly explain any four bit test instructions.

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PART - B

Explain public and extrn directives of assembler and write ALp to read data throughk-eytoard using external procedure and save the keycode in public data segment. (08 Marks)write a c program that uses '-asm' function to display strings on output device. (06 Marks)Explain with neat diagram clock generator IC82g4. (06 Marks)

Explain in brief the funcrions of 8086 pins: it MN/MX: ii) ALE; iii) NMI; iv) Ready;v) Reset; vi) BHE-. (06 Marks)Describe demultiplexing of multiplexed AD bus with neat diagram. (06 Marks)With neat timing diagram, explain memory read cycle. (08 Marks)

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Interface 512 KB RAM to 8088 MP using 64 KB RAM using 3:8 decoder with startingaddress of memory as 80000H. Clearly mention decoding logic and memory map. (08 Marks)

Explain memory bank selection in 8086 and mention the number of memory bank in80x86 MPs. (06 Marks)

Differentiate between memory mapped I/O and I/O mapped I/O (isolated I/O). (06 Marks)

Interface 8 digit seven segment LED display to 8088 MP through 8255 PPI. Writeinitialization sequence for 8255 with all port as output ports in mode 0 and address of deviceis FFOOh- (08 Marks)

Explain control work format for IC 8254 and interface IC to 8086 MP to generate squarewave of 100 kHz using counter 0 write an ALP for the same. Assume clock frequency ofl0 MHz.Explain intemrpt vector table in brief.

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.of size, speed and cost per bit in the hierarchy.What are static and dynamic RAMS? Explain.Define memory latency and bandwidth of a synchronous DRAM memory unit.Define and explain the following :

ii) Memory access timeiv) Memory cycle time.

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8a.b.

Draw the block diagram of virtual memory organization.Explain the virtual - memory address translation with a neat diagram.Discuss the organization and accessing of data on a disk.A 3.5 inch (diameter) high-capacity, high-data-rate-disk has the following parameters. Thereare 20 data-recording surfaces with 15000 tracks per surface. There is an average of400sectors per track and each sector contains 512 bytes of data. Calculate the total capacity offormatted disk. (02 Marks)

Discuss Booth algorithm by multiplying the numbers -13 and +l l.Explain bit pair recoding technique by multiplying the numbers +13 and -6.Explain restoring division using a 4-bit example.

Draw the three-bus organization of the data path and describe in detail.With a neat sketch, explain the basic organization of a micro programmed control unit.

(10 Marks)

Gomputer Organization

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Define and explain briefly the following : \',.{ .*r';1": " ' j "'i) Clock rate ii) CISCl) ( toct( rate I) L tSL ri.., ,!.,;,iii) RISC iv) Basic performance equation. -

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Describe the basic functional units of a computer with a neat diagram. !---. - - '-' (08 Marksr1)@

What are generic addressing modes? Write down the assembler syntax and addressingfunction. (I0 Marks)Explain the following with reference to subroutines :

i) Subroutine nesting and the processor stack ; ii) Parameter passing ;

What is an exception? Describe the different kinds of exception.How do you enable and disable interrupts?Define privileged instruction and explain how privilege exception occurs.

Explain the use of a PCI bus in a computer system with a neat sketch.Draw the diagram of universal setial bus tree structure.Define the following : i) Serial port

iii) SCSI

i) Static memoriesiii) p414

PART-B5 a. Draw a neat block diagram of memory hierarchy in a computer system. Discuss the variation

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Time: 3 hrs. Max. Marks:100Note: Answer FIVE full questions, selecting

atleast TWO questions frorn each part.

PART - A

1 a. What is performance measurement? Explain the overall SPEC rating lor the computer in a

program suite. (04 Marks)

b. List and explain the technological features and devices improvement made during differentgenerations of computers. (08 Marks)

c. Mention four types of operations to be performed by instructions in a computer. Explain

with basic types of instruction formats to carry out C <- [A] + [B]. (08 Marks)

2 a. Define an addressing mode. Explain the following addressing modes, with example :

immediate, indirect, index, relative and auto increment. (10 Marks)

b. What is a stack frame? Explain a commonly used layout for information in a subroutinestack frame.

c. Explain shift and rotate operations, with example.

3 a. In a situation where multiple devices capable of initiating interrupts are connected to

processor. explain rhe implementation of interrupt priority, using individual TNTER and

INTA and a common INTR iin. to all devices. (10 Marks)

b. Define the terms 'cycle stealing' and 'block mode'. (02 Marks)

c. What is bus arbitration? Explain the different approaches to bus arbitration. (08 Marks)

4 a. Explain with a neat blockkeyboard to a processor.

b. Briefly discuss the main phases involved in the operation of SCSIc. Explain the tree structure of USB with split bus operation.

PART-B

5 a. Explain the intemal organization of a 16 megabit DRAM chip,

b. With a block diagram, explain the direct and set associativemain memory.

c. Describe the principles of magnetic disk.

6 a. Explain with figure the design and working of a 16-bit carry - look - ahead adder built form4-bit adders. (06 Marks)

b. Explain Booth algorithm. Apply Booth algorithm to multiply the signed numbers +[3 and

-6.c. Differentiate between restoring and non-restoring division.(10 Marks)(04 Marks)

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7 a. List out the actions needed to execute the instruction add(Rr), Rr. Write and explainsequence of control steps for the execution of the same. (10 Marks)

b. With a neat block diagram, explain hardwired control unit. Show the generation Zin andBndcontrol signals. (10 Marks)

8 a. State Amdahl's law. Suppose a program runs in 100 sec on a computer with multiplyoperation responsible for 80 sec of this time, how much it requires to improve the speed ofmultiplication, if the program has to run 5 times faster? Justify your answerExplain the classic organization of a shared - memory multiprocessor.

(06 Marks)(06 Marks)b.

c. What is hardware multithreading. Explain the different approaches to hardwaremultithreading. (08 Marks)

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