MCSE-(XH@ : NUMERICAL AND STATISTICAL COMPUTING · 4. (a) A farmer buys a quantity of cabbirge...

MCSE-oo4@

MCA (Revised)

Term-End Examination

June,2OOT

MCSE-(XH@ : NUMERICAL ANDSTATISTICAL COMPUTING

Time : 3 hours Maximum Marks : 700

Note : Question number 1 is compulsory. Attempt any

three questions from the rest. Use of calculators

is allowed.

l. (a) Solve the quadratic equation *2 + 9.9 x 1 : 0

using two decimal digit arithmetic with rounding. 5

(b) Obtain the positive root of the equation yz 1 : 0

by Regula Falsi method. 5

(c) Solve the following system of equations :

*1 + xZ + x3 = 3

4*, 3*, + 4*, : 8

9*, + 3*, + 4*, = 7

by Gauss Elimination method. 6

MCSE-oo4@ P.T.O.

Downloaded from www.paraman.in

x k l 0 1 2 5

f k : 2 3 L2 L47

(e)

(f)

(d) Find the interpolating polynomial that fits the data

given below :

using the lagrange interpolation formula. 6

Solve the initial value problem u' : - 2tu2 with

u(0) : 1 and h : 0'2 on the interval 10, U, by using

fourth order classical Rung e - Kutta method . 8

In partially destroyed laboratory record of an analysis

of correlation data, the following results only are

legible I

V a r i a n c e o f x : 9

Regression equations : 8x - 10y + 66 : 0

4 0 x - 1 8 y - 2 L 4 : 0

What were

(i) the mean values of x and y ?

(ii) the correlation coefficients between x and y ? 6

Suppose that the amount of time one spends in a

bank to withdraw cash from an evening counter

is exponentially distributed with mean 10 minutes i.e

l, : L/L}. What is the probability that the customer

will spend more than fifteen minutes at the

counter ? 4

(g)

MCSE-oo4@


2. (a) Find the smallest positive.root correct upto 3 decimal

places for the equation *7 + 9x5 - 13x - 17 = 0 byi

h using Newton-Raphson method. Give any two

S drawbacks of Newton-Raphson method.

(b) . Solve the following system of linear equations by using

E, Jacobi's method and perform three iterations : 6l

2 * r - x 2 + x 3 = - 1

x r + 2 x , * 3 = 6

x1 - x2 + 2x, = -3

F (c) Find the rnlue of e correct to three decimal places. 6

3. (a) What is the interpolating polynomial for

! f(x) : x2 + sin ID( through (0, 0h (1, 1); (2, 4) ?'.'::!-

what is the error when x : L/2 ? what is the

maximum error ?

5.2(b) Calculate the value of the integral I log x dx

t-

by using Simpson's L/3 rule and Simpson's 3/8

rule . 70

(c) Write short notes on :

(i) Euler's method

(ii) Runge-Kutta method

6

MCSE-oo4@ P.T.O.


4. (a) A farmer buys a quantity of cabbirge seeds from acompany that claims that approximately 90% of theseeds will germinate if planted properly. If four seedsare planted, what is the probability that exactly two i

will germinate ? 6

Find F-1.

5. (a) Solve the following system of linear equations byusing Gauss-Seidel method :

2x. - X^ * X^ = -1L Z J

x . + 2 x ^ - x ^ : 6' l z J

x L - x z + 2 x a : - 3

Perform 3 iterations. 6

I

(b) Evaluate the inteqral t = |. ,&- J 1 + x0

using Gauss - Legendre three point formula. 6

(b) Show that the moment generating function of arandom variable X which is chi-squ are distributedwith v degrees of f,reedom is M(0 : (1 2t7-'tz. I

(c) Let X have the Weibull distribution with followingprobability density function :

( ^ - - u 1

f / . , \ , | " re- l ' *oxcr- l i fx>or[x, : 1t 0 i f x < O

MCSE-oo4o


(c) Differentiate between (any twol r'-

(i) Discrete Random Variables and Continuous

Random Variables

(ii) Linear Regression and Non-Linear Regression

(iii) Direct methods and lterative methods of root

finding

MCSE-oo4@ 2,000



MCA (Revised)


June, 2OO7

MCSE-004 : NUMERICAL ANDSTATISTICAL COMPUTING


Note : Question number I is compulsory. Attempt anythree questions from the rest. Use of colculatorsfs allowed.

l . (a) Let a : 0 .4! ; b= 0.36; c : 0 .70,

p r o v e ( a - b ) * g !4c c c

(b) Use Regula-Falsi method to compute the positiveroot of x3 - 3x - 5 = 0. perform two iterations. 6

(c) Solve the following system of linear equations : 6

*l + *Z + *3 : 6

3 * r + 3 * r + 4 x r = 2 0

2 * r + x z + 3 * r : 1 3

by using Gauss Elimination method.

MCSE-OO4 P.T.O.


(e)

(f)

(d) Find the Lagrange interpolating polynomial of

degree 2, approximating function y = ln (x)

x 2.0 2 ,5 3 .0

In (x) 0.69315 0.916929 1.09861

and hence estimate the value of In (2'7').

Apply Runge-Kutta method of fourth order and solve

dY Yz *2 ,, , i* l- , , , lA\ - 1 rf v : o,T = # w i t h V ( 0 ) : 1 a t x : O ' 2 .ctx y' + xo

Calls at a particular call centre occur at an average

rate of 8 calls per 10 minutes. Suppose that the

operator leaves his position for a 5 minutes coff,ee

break. What is the chance that exactly one call

comes in while the oPerator is awaY ?

What is a Residual plot ? Briefly describe the utility

and disadvantages of Resdiual plots.

Solve by Jacobi's iteration method, the equations

2 0 x 1 Y 2 z : L 7

3 x + Z A y z : - 1 8

2x 3Y + 202: 25

Perform 3 iterations.

(g)

2. (a)

(b) Given the following system of linear equations'

determine the value of each of the variables using

LU-decomposition method.

6 * , 2 * z : 1 4

9*, xz + x3 : 2I

3*, 7*z + 5*, : 9

MCSE-OO4


(c) Use Secant method to find the roots of the equation

f(x) : 0'5e* - 5x + 2, cotrect to two decimal places' 6

5.2f

3. (a) Calculate the value of the integral I log x dxto

by using Trapezoidal rule and Weddle's rule ? 10

(b) From the following data estimate the value of f(2'251

using forward difference formula

x f(x)

0

0.5

1 . 0

1 . 5

2 .0

2 .5

1 . 0

3.625

7.000

11 .875

19.000

29.r25

(c) Solve the initial value problem

dY + zv: 3e4td t '

to compute approximation for y(0'1), y(0'2l- by using

Euler's method with h : 0'J-

MCSE-OO4 P.T.O.


4. (a) If a bank receives on an average l, : 6 bad chequesper duy, what is the probability that it will receive4 bad cheques on any given day ?

Use Acceptance Rejection method to generate arandom deviate from gamma density function. The

gamma density function with the shape param eter a.,given as

( 1f(x) :

]rt xo-l e-x if x

|. 0 otherwise

A chemical engineer is investigating the effect ofprocess operating temperature on product yield. Thestudy results in the following data :

Temperature ("C) (X) Yield (o/ol (Y)

100

1 1 0

L20

130

L40

150

160

L70

180

190

45

51

54

6 T

66

70

74

78

85

89

Determine the Goodnesscomment on whether thethe data or not.

to fit param eter 'R' andpredicted line fits well into

(b)

1 0(c)

5

MCSE-OO4


5- (a)

(b)

(c)

Using Runge-Kutta method of fourth order,'solve fov

y(0'1) given that y' : xy + x2, y(0) : 1.

Find a real root of the equation *3 - x - 1 : 0 usingBisection method. Perform three iterations

A survey was conducted to relate the time required todeliver a proper presentation on a topic, to theperformance of the student with the score he/shereceives. The collected data is given below :

Hours x Score y

0.50 57

0.75 64

1.00 59

L .25 68

1 .50 74

r . 7 5 76

2.00 79

2.25 83

2.50 85

2.75, 86

3'00 88

3.25' 89

3.50 90

3 . 7 5 94

4.00 96

Find the regression equation that will predict a

student's score, if we know how many hours the

student studied. Hence predict the score when a

student had studied for 0.85 hours.

MCSE-OO4 7,000



TI MCSE-O,_ _ |

Term-End Examinatiotl

December, 2OO7

MCSE-004 : NUMERICAL ANDSTATISTICAL COMPUTING

Time : 3 hours Maximum Marks ; 100

Note : Question number 1 is

three quesfions fromis o l lowed.

compulsory. AttemPt anY

the rest. LJse of calculstors

l . (a)

MCSE-OO4

(b)

Fvaluate the sum S : .6 + .6 + J7 to 4

significant digits and find its absolute and relative

errors

Find the root of the equation 2x = cos x + 3 correct

to three decimal places.

Use the Newton - Raphson method to find a root of

the equatior, *3 - 2x - 5 : 0

(c)

P . T . O :


(d)' Use Lagrange's interpolation formula to find the

value of sin(n/61 given y - sin x. 6

X 0 n/4 n/2

Y : s l n x 0 0.707rL 1 . 0

Determine the value of y when x - 0'1. Given that

y(0) : 1 'and y' : *2 + y. Use Euler's method. 6

Determine the constants a and b by the method of

least squares such that y : unb* fits the following

data : 7

X 2 4 6 8 1 0

v 4.077 1 1 . 0 8 4 30.L28 81.897 222.62

(S) A car hire firm has two cars which it hires out day

by day. The number of demands for a car on each

day is distributed as Poisson variate with mean 1'5 .

Calculate the proportion of days on which I

(i) neither car is used

(ii) some demand is refused

2. (a) What are the two pitfalls of the Gauss Elimination

Method ?t

MCSE-OO4

(e)

(f)

2


(b)

(c) Use secant method to find the roots of the

f(x) : 0'5 e* 5x + 2

Solve the following system,Method :

2x + y + z 10

3 x + 2 y + 3 z : ' 1 8

x + 4 y + 9 2 : L 6

Find the minimum number1F d x

e v a l u a t e l - w r t h a nJ 1 + x0

the Simpson rule

using Gauss Elimination

of intervals required. to

accuracy 10-6, by using

equation 7

3. (a)

MCSE.OO4

1. l d x(b) EvaluateJ

, . "0

using composite trapezoidal rule with r| : 2 and 4.

(c) Solve the initial value problem

d Y : y x w i t h y ( 0 ) : z a n d h : 0 . 1

dx

Using fourth order classical Runge - Kutta Method,

find y(0'1) and v9.zl correct to four decimal places. I

P . T . O .


4. (a) Show that the moment generating function of a

random variable X which is chi-squ are distributed

with v degrees of freedom is

M(t) : (1 2t7^'/z

(b) An irregular six faced die is thrown and the

expectation that in 10 throws it will give five even

numbers is twice the expectation that it will give

four even numbers. How many times in 10000 sets

of 10 throws would you expect it to give no even

number ? 6

(c) Write short notes on :

(i) Acceptance Rejection Method

(ii) Non-linear regression

' 5. (a) The population of a town in the decennial census was

as given below :

Year 1891 1901 1 9 1 1 192I 1931

Population : y(in thousands)

46 66 8 1 93 1 0 1

Estimate the population for the year 1895 using

forward difference table 6

MCSE-OO4


' :(b) A chemical engineer is investigating the effect of :

process operqting temperature on product yield. The

study results in the following data :

Tem ("C) (X) Yield o/o (Yl

100 45

1 1 0 5 1

r20 54

130 6T

140 66

150 70

160 74

L70 78

180 85

190 89

Determine the Goodness to fit parameter 'R' and

comment on whether the predicted line fits well into

the data or not. .t I

6(c) Define

(i) Absolute and Relative Errors

(ii) Bisection Method

5,OOOMCSE-OO4



MCA (Revised)


June, 2OO8

MCSE-(XI4 : NUMERICAI ANDSTATISTICAL COMPUTING

TIme : 3 hours Maxtmum Mqrks : 7O0

Note : Questton number 7 is compulsory. Ataempt ony

. three guestlons from the rest- Us ol calculatorsis allowed..

1. (a) Find the value of e correct to three decimal places. 6

(b) Add 0.2315 x 102 and 0'9443 x 102 ming conceptof normalized floatlng point. 4

(c) Solw x3 + 2x2 + lhx- 20 = O by Newton-Raphsonmethod.

(d) Estimate the approximate derivatirre off(x) = x3 at x - 2lor h = 0'01, 0'05 and O'1 using(i) the first order forward dlllerence quofent.(ii) the first oder backward difference quotient. I

P.T.O.


(e) Estimate y(0 4) by the Classical Runge-Kutta method

when y'(x) : *2 * y2, y(0) = 0 and h = 0'2. 6

(0 Using the followtng data obtain two regressionequations : 5

x t l 6 2 7 2 6 2 3 2 8 2 4 l 7 2 2 2 7

y: 33 38 50 39 52 47 35 43 47(g) Suppose that an airplane engine will fail, when in

flight, with probability (1 - p) independently from

engine to engine. Suppose that the airplane will

make a successful flight if at least 50 percent of its

engines remain op€rative. For what values of p is a

fow-engine plane preferable to a two-engine plane ? 5

2. (a) Sollre the following system of equations 8

x + y - 2 = 0

- x + 3 Y = l

x - 2 z = - 3

by Jacobi Method, both direcdy and in Matrix form.

Assume the initial solution vector as

IO 8 0 .8 2 .1 r .(b) Write short notes on the lollor*'ing : 4

(i) Gauss Elimirntion method

(ir) hrler's method

(c) Find a root of the equation x3 -x -4:0 between

1 and 2 to three places of decimal by bisection

method. I

3. (a) Find the real root of the equation x3 - 9c + I = 0

by Regula-Falsi methoo. I

MCS E-OO4


(b) Find the minirnum number of interrrals required to1

evaluate [ :dx with an accuracy o{ 10-6, fu uslngJ 1 + x0

the Simpson's nrle. 6

6 .(c) Evaluate the inteoral [

* = bu Weddle's Rule.-

| l + x ' 6

4. (a) Find the form of the function given by : 6

(b) Prove the following property by Cote's number : 6

nF r - " - rL " k - '

k = 0

l -

(c) Evaluate the integral of f _ t d* by using

J l + x

Trapezoidal nrle. I

5. (a) It is given ttnt 3% of the electric br.rlbs

manufactured by a company are defectir.re. Using

Poisson distribution, find the probability that a

sample of 100 bulbs will mntain no defective bulb.

G i v e n t h a t e 3 - 0 . 0 5 . 6

(bl Give an algorithm for simr.rlating a random variable

having density function 5

( x ) = 2 0 x ( 1 - x ) 3 ; 0 < x < 1 .

MCSE-OO4 3 P.T.O.

x 5 2 1

(x) 3 72 l 5 -21


(c) Equations of two lines of regression are :4 x + 3 y + 7 = 0 a n d 3 x + 4 y + 8 = 0 .

F i n d :

(i) mean of x and mean of y

(ii) regresslon coefficient of b, and b*u

(lii) conelation coeftident between x and y 9

4MCSE-004 10,ooo


McsE-ooro

MCA (Revised)

Term-End Examinatlon

June, 2OO8

MCSE-004@ : NUMERICAL ANDSTATISTICAL COMPUTING

Time : 3 hours Moxlmum Morks : 70O

Note : Quesilon number 7 is compulsory- Attempt onythree questions from the rest. Use ol calculotorsls ollowed.

l. (a) If 0'333 is the approximate value of l,/3, findabsolute, relatir.re and percentage errors. 6

(b) Add 0'1234 x 10-3 and 0'5678 x 10-3 uslngconcept of normabzed floating point. 4

(c) Aie the lollowing matdces diagonally dominant ? 6

| 2 -5.81 34.l lr24 s 56'lt t t lo = l * 4 3 1 l B = 1 2 3 5 s u ll1z3 16 u L% 34 rzeJ

MCSE-OO4@- 1 P.T.O.


(d)

(e)

(b)

(c)

b)

Compute the approxinate derirntives of f(x) : xzat x = 0 5 for the increasing value of h from 0'01 to0'03 with a step size of 0.005 using : (i) first orderforward difference model (ii) first order backwarddiflerence model.

Use the Classical Runge-Kutta method to estimatey(0 5) of the following equations with h = 0.25 :

tut = * * y ,

y { u ) - l

(0 Fit a straight line to the following datd by the methodof least square :

Suppose a book of 585 pages contains 43typographical enors. If these errors are randofi y

distributed throughout the book, whai is theprobability that 10 pages, sel€cted at random, will beIree {rom errors ?

(IJse e-o'zgs = o.4z9s)

Find the real root of the equation log x - cos x : 0correct to three places of decimal byNeMon-Raphson method.

Write short note on :

(i) LU Decomposition method

(ii) Gaus-Seidelmethod

Find the root of the equation *3 - * - 1 = 0 lyrnsbetween 1 and 2 by Bisection method.

2. lal

I

4

X 0 I 2 3 A

I l 8 3.3 4 .5 6.3

MCSE-oO4o


3. (a) Find a real root of the equation xex - 3 = 0, usingRegtrla-Falsi method, correct to three decimal places.

(b) Evaluate the integrd I e* dx by Simpson's 1/3mle. 6

5-2

Evaluate the tntegnl I log x dx by Weddle's Rule.

Compute f'(2'0) from the followlng data ol value

(c)

4. (a)

Prove the following property of Cote's number :

a n - a n-k - " n - k

Calcr.rlate the approximate value of the integral*12

I sin x dx bv usino TraDezoidal Rule.

0

5. (a) Let X be the number of times that a fair coin,

flipped 40 times,. lands heads. Find the probability

tlnt X = 20. Use the normal approximation and

then compa.re it to the exact solution.

(b) Give an algorithm for simulating a random variable

using acceptance-rejection method.

(b)

(c)

I

1 .8 1 . 9 2 0 z ' I

f(x) 6.05 6'69 7 .eo 8 .17

MCSE-004o P . T . O .


O Find the iwo lines of regression from the following

da ta :

Age of husband : 25, 22, 28,26,35,20,22,40 ,20 , 18 .

Age of wife : 18, 15, 20, 17,22, 14, 16,27,75, 14.

Hence estimate

(i) the age of husband when the age of wife is 19,

and

(ii) the age of wife when the age of husband is 30.70

McsE-oo4 @ 4 3,O00


MCA (Revised)


December,2008

MCSE - 004: NUMERICAL AND STATISTICALCOMPUTING

Time : 3 hours Maximum Marks : 100sl:@(.o(f

Note : Question Number 7

Questions fror.n the

is compulsory. Attanpt any thrgerest. Use of Calculator is allorned.

(u) If 0.333 is the approximate value of L/3,

find absolute, relative and percentage elTors.

(b) Find the value of z (the number of term

required) in the expansion of ex, such that

their sum yields the value correct to 8

decimal places at r:1,.

(.) Find the root of the equationxe{: cosr using

the Regula-Falsi method correct to four

decimal places.

(d) Find the polynomial function/(x) given that

/ (o) :2 ' f (1) :3 ' f (2) :L2 and " f (3) :3s.Hence find /(5) using Lagrange's

interpolation formula


t a .

11 dx(e) Evaluate J, ,*,

bY using

(t) TraPezoidal rule

(ii) SimPsort's L/3 raLe

(0 The probability that an evening college

student will graduate is 0'4' Determine the

probability that out of 5 students (i) none

(it) one and (iii) atleast one will be graduate'

(g) The following data about the sales and

advertisement expenditure of a firm is given

below:

Sales(in crores of Rg)

Advertisement

expenditure(in crores of Rs)

Means 40 b

itandard deviations 10 1.5

Coefficient of Correlation (r) = 0'9

(0 Estimate the likely sales fora proposed

advertisement expenditure of Rs' 10'

crores.

(ii) IAtrhat should be the advertisements

expenditure, if the firm ProPoses a

sales target of 60 ctores of ruPees ?


2. (a) By using the Bisection method, find an

approximate root of the equation ,irr r= Ir

that l ies between x:L and r = 1..5(measured in radians). Carry out

computation upto 5th stage.

(b) Estimate the number of students, who

obtained less than 45 marks from the

following using Newton's Forward

Difference :

Marks 0 - 4 0 40-50 50-50 60 -70 70-80No. of Students 31 42 51 35 31

Municipal Corporation installed ?000 bulbs

in the streets. If these bulbs have an average

life of 1,000 burning hours, with a standard

deviation of 200 hours, what number of

(c)

(a)3.

bulbs might be excepted to fail in first 700

burning hours ?

Solve the equation :

x l+x2*x3:6

3rt * 3x2-l4x3:20

2xr* xr* 3rr:13

using Gauss Elimination method.

Given Z 1.00 t_25 1.50Probability 0.159 0.106 0.067



(b) Evaluate the integra U: f !.a, using"1, I+r'

the Gauss-Legendre 1-point, 2-point and3-point Quadrature rules. Compare with

the exact solution I:Tan-l(+)+I4

(c) It is known from the past experience that ina certain plant there are on an average 4accidents per month. Find the probabilify

that in a given year there will be less than 4accidents.

(a) Apply LU decomposition metho4 to solvethe following equations :

Axr* x2* xg=3

xr*4xr*2xu:g

Zxr+ xr+Sxr:4

(b) Usrng Runge-Kutta method of fourth order

- kt u2 -x2solve 1i:* with y(0) = L at x = 0.2

dx y'+xc.

and r = 0.4.

(c) A randomvariable'X is defined as the sumof faces when a pair of dice is thrown. Findthe expected value of.'K.


(u) Find a real root of the equation 3r: cos.r * 1r

using Newton-Raphson method.

&) Solve the following differential equation by

Euley's method

dv v-x;;: i i*'slven Y (o) :1

Find'y' approximately for r:0.L in five

steps.

(c) Write a short notes on :

(i) Chi-Square Distribution.

(ii) Least Squares Estimation.

- o O o -


rl(of-.@O

MCA (Revised)


|une, 2009

MCSE-OO4 : NUMERICAL AND STATISTICALCOMPUTING


Note : Question number 7 is compulsory. Attempt any three

fro* the rest. Use of calculator is allowed.

1. (u) Differentiate between absolute, relative

and percentage error with an example.

(b) Obtain the positive root of the equation

x2-'1.-0 by Regula Falsi method.

(.) Apply Gauss - Elimination method to solve

the following sets of equation

x + 4 y - z : - S

x + y - 6 2 : - " 1 2

3 x - y - z - - 4 .

MCSE-004 P.T.O.


(d) Find Newtons Forward dif ference

interpolating polynomial for the following

data :

x 0.1 0.2 0.3 0.4 0.5

f ( x ) I .40 1.56 1.76 2.00 2.28

(e) Calculate the value of integral :

6 al a xJ . - , . , a v0 ^ ' ^

(i) Simpson's 1f ru\e.

(ii) Simpson's 3/g rule.

(0 In partially destroyed laboratory record ofan analysis of correlation data, thefollowing results only are legible.

Variance of x-9.

Regression equation : 8x - 10y + 66 : A

\rVhat are (i) Mean^::::':::::'

(i1) Correlation coefficient

between x and y.

(iii) Standard deviation of y.

MCSE-004


2. (") Solve the following system of equation by

Jacobi's method.

x + V - z : 0

- x * 3 r t - 2

x - 2 2 - 3

(b) Use False Position method to find real rootof 13 - 4x - 9:0. Correct to 3-decimalplaces .

(c) Explain the pitfalls of Gaussmethod.

3. (u) Evaluate the missing term in the following :

x. 100 101 LA2 103 n4tog (x ) 2.00 2.0043 7 2.0L282.0L70

ur. )

Evalute the integral J@'+x+2)dx using0

Trapezoidal rule, with h:L.0.

Elimination

(b)

MCSE-OO4



(") A missile is launched from a ground station.

The acceleration during its first 80 seconds

of f l ight, as recorded, is given in the

following table :

f (s) 0 1 0 20 30 40 50 60

a (m/sz) 30 31,.6333.34 35.4737.7540.33 43.25

r(s) : 70 80

(u)4 .

(b)

a(ml s2) 46.69 50.67

Compute the velocity of missile when

f : 80s, using simpson'" 1/3 rd rule.

du v- tGiven ;: ,*

With initial condition y:1- at t:0.

Find y approximately at x - 0.1 in five steps,

using Euler's Method.

Solve the following differential equation

fur;-

t*y, with init ial condition y(0):1.,

using Fourth order Runge-Kutta method

from f :0 to f -0.4 taking h-0.1.

Explain the effect of round off error in

scientific calculations.

(.)

L 0

t

MCSE-004 P.T.O.


(u)5 .

(b)

(c)

If a bank receives on an average tr:6 badcheques per dap what is the probability thatit will receive 4 bad cheques on any givenduy.

What is the utility of residual plot ? I,l/hatare its disadvantages ?

A farmer buys a quantity of cabbage seedsfrom a company that claims thatapproximately 90% of the seeds wil lgerminate if planted properly. If four seedsare planted, what is the probability thatexactly two will germinate.

- o O o -

MCSE-004


I MCSE-004 I

MCA (Revised)


December, 2009

MCSE-004 : NUMERICAL AND STATISTICALCOMPUTING



from the rest. Use of calculator is allowed.

1. (a) Explain truncation error. Show that 2+6a(b—c) � ab— ac, where :

a = .05555 El

b =.4545 El

c = .4535 El

Use bisection Method to find a root of the 8equation x3 — 4x — 9 =0

Go upto 5 - iteration only.

Use Gauss - Elimination method to solve the 8following system of equations :

x1 + x2 + x3 =3

4X1 + 3X2 + 4X3 = 8

9x1 + 3x2 + 4x3 = 7

MCSE-004

1 P.T.O.


Evaluate f(15), Given the following table of 8values :

x - 10 20 30 40 50

f(x) - 46 66 81 93 101

Calculate the value of the integral, 8

5.2f log x dx by.4

Trapezoidal rule.

Waddles' rule.

*(Take h =0.2).

2. (a) Find a root (correct to three decimal place) 8of x3 — 5x +3 =0 by Newton-Raphsonmethod.

Use Jacobi's method to solve the equation : 8

20x+y-2z=17

3x +20y — z = —18

2x —3y +20z =25

Explain the bisection method. 4

MCSE-004 2


3. (a) Given : 5+3

(x)=sin(x)

f (0.1) = 0.09983, f (0.2) = 0.19867

Use method of Lagrange's interpolation tofind f(0.16). Find error in f(0.16).

Evaluate . dx Use Gauss-Legendre three 81+x0

point formula.

Explain initial value problem with an 4example.

6

4. (a) Evaluate 5[ 2 + sin (2 .J) ] dx using 10

simpsons' rule with 11 points.

(b) Solve the initial value problem 10

u1 = — 2tu2 with u(0) =1 and h= 0.2 on theinterval [0,1]. Use Fourth order classicalRunge Kutta method.

5. (a) A farmer buys a quantity of cabbage seeds 8from a company that claims thatapproximately 90% of the seeds willgerminate if planted properly. If four seedsare planted, what is the probability thatexactly two will germinate ?

MCSE-004

3 P.T.O.


(b) In a partially destroyed laboratory record

8of an analysis of correlation data, thefollowing results only are legible.

Variance of x = 9

Regression equation :

8x —10y + 66 = 0

40x — 18y — 214 = 0

What are

Mean value of x and y.

Correlation coefficient betweenx and y.

(iii) Standard deviation of y.

(c) Suppose that the amount of time one spends 4in a bank to withdraw cash from an eveningcounter is exponentially distributed with

mean ten minutes, that is X = 110. What is

the probability that the customer will spendmore than 15 minutes in the counter ?

- o 0 o -

MCSE-004 4


1MCSE-004 No. of Printed Pages : 4

MCA (Revised)


June, 2010

MCSE-004 : NUMERICAL AND STATISTICALCOMPUTING



questions from the rest. Use of calculator is allowed.

1. (a) Estimate the relative error in z = x -y when 6x= 0.1234 x 104 and y = 0.1232 x 10 4 asstored in a system with four-digit mantissa.

2Show that the series e x = 1 + x + —x + 52!

becomes unstable when x = - 10.Find the root of the equation x x + x - 4 =0 6

using the Newton-Raphson method correctto four decimal places.

(d) The observed values of a function are 7respectively 168, 120, 72 and 63 at the fourpositions 3, 7, 9 and 10 of the independentvariable. What is the best estimate you cangive of the value of the function at theposition 6 of the .independent variable.Apply Lagrange's formula.

MCSE-004

1 P.T.O.


(e) The table gives the distance in nautical miles 8of the visible horizon for the given heightsin feet above the earth's surface :

x = height 100 150 200 250 300 350 400

y = distance 10.63 13.03 15.04 16.81 18.42 19.90 21.27

Find the value of y when x = 410 usingNewton's Backward Interpolation formula.

(f) Five men in a group of 20 are graduates. 8If 3 men are picked out of 20 at random

(i) what is the probability that all aregraduates and (ii) what is the probability ofat least one being graduate ?

2. (a) Find the root of the equation x ex = cos x 7using the secant method correct to fourdecimal places.

2Evaluate f 1 log x by Trapezoidal rule. 6

A book contains 100 misprints distributed 7

randomly throughout its 100 pages. Whatis the probability that a page observed atrandom contains atleast two misprints.

3. (a) Solve the system of equations : 104x1+x2+x3=2xi +5x2 +2x3 = —6

+ 2x2 + 3x3 —4Using Jacobi iteration method.

MCSE-004 2


(b) Use Euler method to solve numerically the 10

initial value problem.v' = - 2t v2, v (0) =1with h = 0.2 and 0.1 on the interval [0, 1].

OR

A sample of 100 dry battery cells tested to 10

find the length of life produced the followingresults :

X =12 hours, a =3 hoursAssuming the data to be normallydistributed, what percentage of battery cellsare expected to have life :

More than 15 hoursBetween 10 and 14 hours

[ Given Z : 2.5 2 1 0.67Area : 0.4938 0.4772 0.3413 0.2487

4. (a) Show that the LU decomposition method 10

fails to solve the system of equations :1 1 - 1 x1 22 2 5 x2 = - 3

3 2 - 3 x3 6

Exact solution is x1 = 1, x2 =0, x3 = -1.OR

Apply Runge-Kutta method to findapproximate value of y for x = 0.2, in steps

of 0.1, if d y = x + y2, given that y =1d x

where x =0.

MCSE-004 3 P.T.O.


(b) A problem in statistics is given to five 10students A, B, C, D and E. Their chances of

solving it are —1 , —1 , —1 , —1 and 1 —. What is2 3 4 5 6

the probability that the problem will besolved ?

5. (a) Perform five iterations of the bisection 6method to obtain the smallest positive rootof the equation f (x)= x3 —5x +1=0.

(b) With the help of Newton's forward 7difference interpolation formula obtain theinterpolating polynomial satisfying the data.

x 1 2 3 4

f (x ) 26 18 4 1

If a point x =5, f (x) = 26, is added to abovedata, will the interpolation polynomialchange ? Explain.

(c) What is a random variable ? Write down 7the expression which define Binomial,Poisson and Normal probability distribution.Give two physical situation illustrating apoisson random variable.







No. of Printed Pages : 4 MCSE-004

MCA (Revised) O c\1 Term-End Examination O

June, 2011

MCSE-004 : NUMERICAL AND STATISTICAL COMPUTING


Note : Question No.1 is compulsory. Attempt any three from the rest. Use of calculator is allowed.

1. (a) Define Absolute Error, Relative Error and 3+5 Percentage Error. Show that

(a — b) a b # — — — , where :

c c c a =0.41, b=0.36 and c = 0.70

(b) Find the real root of the equation 8 x3 — 2x — 5=0 using Bisection Method. Upto four iterations only.

(c) Solve by Jacobi's method the following 8 system of linear equations. 2x1 — x2 + x3 = —1 X1 + 2X2 - X3 = 6 x1 — X2 ± 2X3 -3 Upto 3 - iterations only

MCSE-004 1 P.T.O.


(d) Write down the polynomial of lowest degree 8 which satisfies the following set of numbers, using the forward difference polynomial.

x 0 1 2 3 4 5 6 7 f(x ) 0 7 26 63 124 125 342 511

(e) Evaluate 8

1 1

1+x —dx correct to 3 decimal places

0

Simpson's rule

(h =0.125)

Explain the cases where Newton's method 4 fail.

Find a real root of the equation 8

f(x) = x3 – x –1 =0

Up to four iterations only.

(c) Use Gauss - Seidel Method to solve the 8 equation :

x+y–z=0

–x+3y=2

x – 2z = –3

Initial solution vector is [0.8 0.8 2.11T.

Upto 3 - iterations only.

by

(i)

MCSE-004 2


3. (a) The population of a town in the decennial census was as given below Estimate the population for the year 1895.

Year : x 1891 1901 1911 1921 1931

Population : y (in Thousands)

46 66 81 93 101

6 Evaluate [2 + sin(2J )dx using

1 simpson's rule with 5 points.

Explain Euler's Method for solving an ordinary differential equation.

4. (a) Solve the initial value problem d y = 1 + y2 d x

where y = 0 when x=0 using Fourth order classical Runge-Kutta Method. Also find y(0.2), y(0.4)

2 2xdx (b) Evaluate the integral I =

1+x4 using 10

Gauss - Legendre 1 - point, 2 - point and 3 - point quadrature rules. Compare with the exact solution.

5. (a) A box contains 6 red, 4 white and 5 black 8 balls. A person draws 4 balls from the box at random. Find the probability that among the balls drawn there is at least one ball of each color.

8

8

4

10

MCSE-004 3 P.T.O.


(b) Find the most likely price in Bombay 8 corresponding to the price of Rs. 70 at Calcutta from the following

Calcutta Bombay Av. Price 65 67 Standard Deviation 2.5 3.5

Corelation Co - efficient between the prices of commodities in the two cities is 0.8.

(c) Ten coins are thrown simultaneously. Find 4 the probability of getting at least seven heads.

MCSE-004 4


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