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Time: 3 hrs.

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3a.b.

c.

4a.

Fourth Semester B.E. Degree Examination, Dec.2013/Jan.2ol4Engineering Mathematies - lV

Max. Marks:100

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O6MAT41

series up to fourlh

(06 Marks)

(07 Marks)

(06 Marks)

(07 Marks)

Marks)

Marks)

Marks)

Note: Answer FIVEfull questions, selectingat least TWO questionsfrom euch part.

PART _ A

b.

L.

b.

c.

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qO5.v>'1-bo-io0o=o-Btr>=oVL0

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Applying Taylor's series method, find'y'at x:0.1 by considering the

{=*+v2-v(0):1.degree term. Given O, 1.

Find y at x: 0.2 using Runge Kutta

step size 0.2.

x 0 0.1 0.2 0.3

! I I .1 169 1.2733 t.5049

fourth order method. Given {y - y-*, y(0) : 1 and

dx y+x(07 Marks)

dvUse Milne's predictor corrector formulae to find y(0.4). Given # = ,,, + y). with

(Use comector formula twice.)

/ ;) :2 \If(z) in a regular tunction of 'z' show that I + * *l t ffrl l' = 4lf'@)l' .

[ 6"' Ay' )'

tction f(z) :u * iv where , = ;n,l#;r 2-Findthebilineartransformationwhichmapsthepoints z:1, i, -1 intothepoint w:2,i,-2.Also find the invariant points of the transformation. (07 Marks)

:,". ":11:* au*ly's theorem under complex values tunctiol tl.:rl",t"r. (06 Marks)

Expand f1z; = in Laurent's series valid tbr (i) lzl < 1, (ii) I <lzl<2.10t Marks)lz-r)(z-2)

Evaluate l r"ot1., dz where 'c' is lz - 1l : 1 using Cauchy's residue theorem. (07 Marks)! (r-Y,)'

12 r

Solve 14* r9+ y =0 using the method ofsolution in seriesdx' dx

1 d'.)With usual notation prove that P, (") = fr# (x' - 1)n .

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Express xo - 3x' + x in terms of Legendre's polynomials.

PART _ B5 a. Fit a curve in the form y : a t U* T .7Uy-tn method of least squares for the data given

ffifrx.-/9tN;e NAg82/

x 1 1.5 2 2.5 J 3.5 4

v 1.1 1.3 t.6 2 2.7 3.4 4.1

below:

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lnes o

x 1 2aJ 4 5

v 2 5aJ 8 7

5 b. Obtain the two li ssions for the data given below:

and hence findffirrelation between x and y'

7a.

b.

Find the (@ distributions of x and Y,

The contents of the three wns are as given below:

Urn 1: 1 white ball,2 red balls, 3 greenballs'

lJrn2:2 white, 1 red and 1 greenball'

Urn 3: 4 white, 5 red and 3 green balls' .1 - - ---Two balls are drawn from i randomly chosen urn and are found to the one white and one

;;r. ;;;i;rr. p."u"unirv that the bails so drawn came from the third urn' (07 Marks)

FindthemeanandvarianceofPoissondistribution(06Marks)In a certairr town the duration of a shower is exponentially distributed with mean 5 minutes'

#"; ilil pr"u"il1iry that ashower will last for (i) 10 minutes or more' (ii) less than 10

ffi;;,'iiid;i;#10 and 12 minutes? . .. .. (07 Marks)

The marks of 1000 students in an examination foilows a normal distribution with mean 70

and standard deviation of 5. Find the expected number.of, students, whose marks will be:

i) less than 65 ii) more than z5 iil) between 65 and75

iCiu", A(z:1):0.34i31'. (07 Marks)

A coin is tossed 1000 times and head turned up 540 times. Test the hypothesis that the coin

is an unbiased one. r 1 L- 1. (06 Marks)

A sample of 200 tyres is taken from 11o1. The mean life of tyres is found to be 40000 kms

with standard a.riuiio,.or izoo kms. Is it reasonable to assume the mean life of tltes in the

lot as 41000 kms? irro .uuurish 9-5% confidence limits with in which the mean life of tyres

in the lot is expected to lie. . r ,i , . 1 L- -. t. (07 Marks)

Ten individuals are chosen at random from a population and the heights are found to the in

inches 63,63,66",;,";;,*ei, io,70,7l i"d lt lg-tfre light of this data discuss the

suggestion that the *"u, height in the population is 66 inches' If the population mean is

unknown, obtain q-0%-co"fiae"nce limits foithis mean. (07 Marks)

O6MAT41

(07 Marks)

c.

c.

c.

8 a. The joint distribution for two random variables x arid y is given below:

(ii) covariance of x and Y.(06 Marks)

b. Find the unique fixed probability vector of the regular stochastic matrix

[o % Y^1

- I I 107Marks)P =l y y, o | \- "-.---'

[o I o-l

A software .ngin.e, goes to his work place every day by motor bike or by car' he never goes

by bike on two .orrrJ*ir. days, but if he goes by car on a day then he is equally likely to

go by car or by bik;;;the next day. Find t[e transition matrix for the chain of the mode of

transport. If car is used on the first day of a week find the probability that (i) bike is used, (ii)

;;;'";, o, ,r,. 5,n ;;. (07 Marks)

{<r,<***. ^J]a

c.

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