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.
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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@
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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
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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
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(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
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MCA (Revised)
Term-End Examination
June, 2OO7
MCSE-004 : NUMERICAL ANDSTATISTICAL COMPUTING
Time : 3 hours Maximum Marks : 700
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.
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(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
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(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-
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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
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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
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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 :
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(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
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(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 .
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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
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' :(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
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MCA (Revised)
Term-End Examination
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.
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(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
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(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
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(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
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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.
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(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
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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
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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
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MCA (Revised)
Term-End Examination
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
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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 ?
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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
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(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.
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(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 -
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rl(of-.@O
MCA (Revised)
Term-End Examination
|une, 2009
MCSE-OO4 : NUMERICAL AND STATISTICALCOMPUTING
Time : 3 hours Maximum Marks : 100
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.
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(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
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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
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(") 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
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(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.
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I MCSE-004 I
MCA (Revised)
Term-End Examination
December, 2009
MCSE-004 : NUMERICAL AND STATISTICALCOMPUTING
Time : 3 hours Maximum Marks : 100
Note : Question number 1 is compulsory. Attempt any three
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
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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
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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 ?
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(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 ?
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1MCSE-004 No. of Printed Pages : 4
MCA (Revised)
Term-End Examination
June, 2010
MCSE-004 : NUMERICAL AND STATISTICALCOMPUTING
Time : 3 hours Maximum Marks : 100
Note : Question number 1 is compulsory. Attempt any three
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.
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(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.
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(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.
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(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.
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No. of Printed Pages : 4 MCSE-004
MCA (Revised) O c\1 Term-End Examination O
June, 2011
MCSE-004 : NUMERICAL AND STATISTICAL COMPUTING
Time : 3 hours Maximum Marks : 100
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
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(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)
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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
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(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.
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