GTU Complex Variable & Numerical Methods - May 2015 Exam Question Paper

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GTU Mechanical Engineering (Semester 4)
Complex Variable & Numerical Methods
May 2015

Total marks: --
Total time: --

INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

Do as Direct:

1 (a) Solve the equation z²+(2i-3)z+5-i=0.

2 M

1 (b) Discuss the differentiability of f(z)=x²+iy².

2 M

1 (c) Discuss the continuity of

$f(z)= \left\{\begin{matrix} \frac {\bar z}{0}; & z \ne 0 \\0; &z=0 \end{matrix}\right.$

2 M

1 (d) Find the image of infinite strip 0≤x≤1. Under the transformation ω=iz+1. Sketch the region in ω-plane.

2 M

1 (e) Find the second divided difference for the argument x=1, 2, 5 and 7 for the function f(x)=x².

2 M

1 (f) Evaluate

$\oint_c \dfrac {e^z}{(z+i)}dz,$ where C:|z-1|=1.

2 M

1 (g) Expand

$f(z)= \dfrac {z-\sin z} {z^2}$ at z=0, classify the singular point z=0.

2 M

2 (a) (i) Find the analytic function f(z)=u+iv, if u-v=e^x (cos y-sin y).

4 M

2 (a) (ii) If f(z)=u+iv is analytic in domain D then prove that:

$\left ( \dfrac {\partial ^2}{\partial x^2}+ \dfrac {\partial ^2}{\partial y^2} \right )\Big \vert Re (f(z))\Big \vert^2 = 2 \Big \vert f'(z) \Big \vert ^2$

3 M

Answer any one question from Q2 (b) & Q2 (c)

2 (b) Using theory of residue, evaluate

$\int^\infty_{-\infty} \dfrac {\cos x}{x^2+1} dx$

7 M

2 (c) Expand

$f(z)=\dfrac {1}{(z+1)(z-2)$ in Laurent's series in the region (i) |z|<1 (ii) 1<|z|<2 (iii) |z|>2.

7 M

Answer any two question from Q3 (a), (b) & Q3 (c), (d)

3 (a) (i) Evaluate

$\int_c \overline{z}dz$ where C is along the sides of triangle having vertices z=0.1,i.

4 M

3 (a) (ii) Determine bilinear transformation which maps the points z=0, i, 1 into ω=i,-1,∞.

3 M

3 (b) (i) Determine the points where

$\omega = \left ( z+ \dfrac {1}{z} \right )$ is not conformal mapping. Also Find the image of circle |z|=2 under the transformation

$\omega = \left ( z+ \dfrac {1}{z} \right )$

4 M

3 (b) (ii) Find the radius of convergence of

$\sum^\infty_{n=1} \left ( \dfrac {6n+1}{2n+5} \right )^2 (z-2i)^{n}$

3 M

3 (c) (i) Using residue theorem, evaluate

$\int_c \dfrac {e^z+z}{z^3-z}dz, \ where \ C:\ |z|=\dfrac {\pi}{2}$

4 M

3 (c) (ii) State Cauchy integral theorem. Evaluate

$\int_c \left (\dfrac {3}{z-i} - \dfrac {6}{(z-i)^z} \right )dz, \ where \ C:|z|=2.$

3 M

3 (d) (i) If

$x+\dfrac {1}{x} =2 \cos \theta,$ prove that

$(i) \ x^n+\dfrac {1}{x^n} = 2 \cos \ n\theta \ and \\ \dfrac {x^{2n}+1}{x^{2n-1}+x} = \dfrac {\cos n \theta}{\cos (n-1)\theta}$

4 M

3 (d) (ii) Determine and sketch the image of region 0≤x≤1, 0≤y≤π under the transformation ω=e^z.

3 M

Answer any two question from Q4 (a), (b) & Q4 (c), (b)

4 (a) (i) Construct Newton's forward interpolation polynomial for the following data:
Hene y for x=5.

x:	4	6	8	10
y:	1	3	8	16

4 M

4 (a) (ii) Evaluate

$\int^1_0 e^{-3x^2} dx$ by using Gaussian Quadrature formula with n=3.

3 M

4 (b) Using the power method, find the largest eigen value of the

$A=\begin{bmatrix} 2 &-1 &0 \\-1 &2 &-1 \\0 &-1 &2 \end{bmatrix}$

7 M

4 (c) (i) Find the real root of the equation x

$\log_{10}^x$ =1.2 by Regula false method.

4 M

4 (c) (ii) A river is 80 meters wide. The depth 'd' in meter at a distance x meters from one bank is given by the following table. Calculate the area of cross section of the river using Simpon's

$\dfrac {1}{3}rd$ rule.

x:	0	10	20	30	40	50	60	70	80
y:	0	4	7	9	12	15	14	4	8

3 M

4 (d) Use Largrange's method to find polynomial of degree three for the data
hence find the value of x=2.

x:	-1	0	1	3
y:	2	1	0	-1

7 M

Answer any two question from Q5(a), (b), (c) & Q5 (d), (e), (f)

5 (a) State diagonal dominant property. Using Gauss-Seided method to solve
6x+y+z=105, 4x+8y+3z=155, 5x+4y-10z=65.

7 M

5 (b) Using the Runge- Kutta method of fourth order, Solve 10

$\dfrac {dy}{dx}=x^2+y^2,$ y(0)=1 at x=0.2 and x=0.4 taking h=0.1.

7 M

5 (c) Derive Euler's formula for initial value problem

$\dfrac {dy}{dx}=f(x,y); \ y(x_0)=y_0$ Hence. Use it find the value of y for

$\dfrac {dy}{dx}= x+y; \ y(0)=1$ when x=0.1, 0.2 with step size h=0.05. Also Compare with analytic solution.

7 M

5 (d) (i) Use Gauss elimination method to solve the equation.
x+4y-z=-5, x+y-6z=-12, 3x-y-z=4.

4 M

5 (d) (ii) Use Newton- Raphson method, derive the iteration formula for

$\sqrt{N}$ . Also find

$\sqrt{28}$ .

3 M

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