JE – 849

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IV Semester B.E. (ME/EE/EC/CSE/ISE) Degree Examination, June/July 2013 (Y2K6 Scheme) ENGINEERING MATHEMATICS – IV Time : 3 Hours

Max. Marks : 100

Instructions : 1) Answer any five questions choosing atleast two from each Part. 2) All questions carry equal marks. PART – A 1. a) Show that for f(z ) =

x 2y 5 ( x + iy) ( x 4 + y10)

, z ≠ 0 and f (0) = 0 is continuous at the origin,

but derivative of f(z) at origin does not exist.

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b) State and prove Cauchy-Riemann equation in polar form.

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c) Verify that u = e2x (x cos 2y – y sin 2y) in harmonic. Find u = v (x.y) such that f(z) = u + iv is analytic.

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2. a) Evaluate the integral

1+ i

∫

( x 2 − iy ) dz along the following curves :

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z =0

i) The stright line y = x ii) The parabola y = x2. b) Verify Cauchy’s theorem for the integral of f(z ) =

1 taken over (along) the z

triangle formed by the points (1, 2), (3, 2) (1, 4). c) Evaluate

z

∫ (z 2 + 1) (z2 − 9 ) dz

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in the cases where C is the circle :

i) | z | = 2 ii) | z – 2| = 2

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P.T.O.

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3. a) Find the Taylor’s expression of f(z ) =

b) Expand the function f(z ) =

2z 3 + 1 z2 + z

about the point a = i.

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z+1 in Laurent series valid for (z + 2) (z + 3 )

i) | z | < 2 ii) 2 < | z | < 3 iii) | z | > 3

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c) Find the Taylor’s and Laurent’s series which represents the function z2 − 1 when (z + 2 ) (z + 3 )

i) | z | < 2 ii) 2 < | z | < 3 iii) | z | > 3

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4. a) Determine the poles and the residue at each pole of the function f( z ) =

z2 (z − 1)2 (z + 2 )

.

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π

dθ b) Evaluate ∫ 3 + 2 cos θ by contour integration in the complex plane. 0 ∞

c) Evaluate

∫

−∞

x 2dx ( x 2 + 1) ( x 2 + 4 )

.

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7 PART – B

5. a) Use the bisection method in four stages to find the real root of the equation xlog10 x – 10.2 = 0.

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b) Use Regulae falsi method to find the fourth root of 12 correct to three decimal places.

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c) Apply Newton-Raphson method to find an approximate root correct to three decimal places of the equation x3 – 3x – 5 = 0.

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6. a) Apply Gauss elimination method to solve the system

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5x + 3y + 7z = 4 3x + 26y + 2z = 9 7x + 2y + 10z = 5 b) Using the Gauss-Seidel method, solve the system

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20x + y – 2z = 17 3x + 2y – z = – 18 2x – 3y + 20z = 25 c) Solve the system of equations by LU decomposition method.

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4x + y + z = 4 x + 4y – 2z = 4 3x + 2y – 4z = 6 7. a) The following table gives a set of values of x and the corresponding values of y = f(x). x

0.1

0.2

0.3

0.4

y = f (x)

2.68

3.04

3.38

3.68

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Find f (0.15) using an appropriate formula. b) Using the Lagrange’s formula find the interpolating polynomial that approximates to the function described by the following table x

:

0

1

2

3

f(x) :

3

6

11

18 27

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Hence find f(0.5) and f (3.1). c) Given x:

1.0

1.2

1.4

1.6

1.8

2.0

y:

2.72

3.32

4.06

4.96

6.05

7.39

find y′ and y′′ at x = 1.2.

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8. a) By dividing the range into 6 equal parts find an approximate value of

∫

π

e sin θ

0

d θ using the Simpson’s 13 rule.

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dy = x2 + y , dx y(0) = 1 by taking h = 0.5 considering the accuracy upto two approximation in each step.

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b) Use modified Euler’s method to compute y (0.1) given that

c) Use fourth order Runge-Kutta method to find y at x = 0.1 given that dy = 3e x + 2 y , y(0) = 0 and h = 0.1. dx

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