GTU Complex Variable & Numerical Methods - May 2016 Exam Question Paper

GTU Mechanical Engineering (Semester 4)
Complex Variable & Numerical Methods
May 2016

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

Short Questions

1(a) Express

$\sqrt{3}-i$ into polar form

1 M

1(b) Evaluate Δ cos x.

1 M

1(c) Evaluate

$\lim_{z\rightarrow i}\dfrac{z-i}{z^2+1}$

1 M

1(d) Find the radius of convergence for the series

$\sum ^{\infty}_{n=0}Z^n$

1 M

1(e) Write formula for Simpson's 3/8 rule.

1 M

1(f) Find the fixed points of

$w=\dfrac{z-1}{z+1}$

1 M

1(g) Give the names of two iterative methods for the solution of system of linear equations.

1 M

1(h) State the theorem, ' Cauchy's Integral Formula'.

1 M

1(i) Find the pole and its order for

$f(z)=\dfrac{e^z-1}{z^3}$

1 M

1(j) Find the third divided difference with arguments 2, 4, 9, 10 of the function f(x) = x³ - 2x.

1 M

1(k) Find Res (f(z), 1) for

$f(z)=\dfrac{1}{z(z-1)}$

1 M

1(l) Find the interval for x³ - x ' 11 = 0 in which the root lies.

1 M

1(m) State DeMoivre's Theorem.

1 M

1(n) Write iterative formula to find

$\sqrt{7}$ using Newton-Raphson method.

1 M

2(a) Find all the values of

$\left ( \dfrac{1}{2}+\dfrac{\sqrt{3}}{2}i \right )^{3/4}.$

3 M

2(b) Show that the function f(z) = xy + iy is continuous everywhere but is not analytic.

4 M

Solve any question from Q.2(c) & Q.2(d)

2(c)(i) If u = e^x (x cos y - y sin y), find the analytic function f(z).

3 M

2(c)(ii) Find the value of

$\int _0^{2+i}(\overline{z})^2 dz,$ along the real axis from 0 to 2 and then vertically from 2 to 2 + i.

4 M

2(d)(i) If f(z) is a regular function of z, prove that

$\left ( \dfrac{\partial ^2}{\partial x^2}+\dfrac{\partial ^2}{\partial y^2} \right )|f(z)|^2=4|f'(z)|^2.$

3 M

2(d)(ii Prove tha

$\sinh^{-1}x=\log \left \{ x+\sqrt{x+1} \right \}$

4 M

Solve any three question from Q.3(a), Q.3(b), Q.3(c) & Q.3(d), Q.3(e), Q.3(f)

3(a) Evaluate

$\int _c \dfrac{e^{2z}}{(z+1)^3}dz$ , where C : 4x² + 9y² = 16 using residue theorem.

3 M

3(b) Find the bilinear transformation which transforms z = 2, 1, 0 into w= 1, 0, i.

4 M

3(c) Expand

$\dfrac{1}{z(z^2-3z+2)}$ about z = 0, for the regions (i) 0 < |z| < 1 (ii) 1 < |z| < 2 (iii) |z| > 2.

7 M

3(d) Evaluate

$\oint _c \dfrac{z-1}{(z+1)^2(z-2)}dz,$ where C is the circle |z-i| = 2.

3 M

3(e) Find the image of |z ' 3i| = 3 under the mapping

$w=\dfrac{1}{z}.$

4 M

3(f) Evaluate

$P.V.\int ^{\infty}_{-\infty}\dfrac{x\cos x}{x^2+9}dx$ .

7 M

Solve any three question from Q.4(a), Q.4(b), Q.4(c) & Q.4(d), Q.4(e), Q.4(f)

4(a) Using Newton's divided difference formula, find a polynomial function satisfying the following data:

x	-4	-1	0	2	5
f(x)	1245	33	5	9	1335

3 M

4(b) The table below gives the values of function y=tanx. Obtain the value of tan(0.40) using Newton's backward interpolation.

x	0.10	0.15	0.20	0.25	0.30
y=tanx	0.1003	0.1511	0.2027	0.2553	0.3093

4 M

4(c) Use the power method to find the largest eigen value and corresponding eigen vector of the matrix

$A=\begin{bmatrix} 1 & 6 & 1\\ 1 & 2 & 0\\ 0 & 0 & 3 \end{bmatrix}$ .

7 M

4(d) Find the root of the equation x² - 4x ' 10 = 0 correct to three decimal places by using bisection method.

3 M

4(e) Compute f(4) from the tabular value given

x	2	3	5	7
f(x)	0.1506	0.3001	0.4517	0.6259

Using Lagrange interpolating polynomial.

4 M

4(f) Solve the following system of equations by Gauss ' Jordan method:
10x + y + z = 12, 2x + 10y + z = 13, x + y + 5z = 7.

7 M

Solve any three question from Q.5(a), Q.5(b), Q.5(c) & Q.5(d), Q.5(e), Q.5(f)

5(a) The velocity v of a particle at distance s from point on its path is given by the following table:

s(meter)	0	10	20	30	40	50	60
v(meter/Sec)	47	58	64	65	61	52	38

Find the time taken to travel 60 meter, using Simpson's 1/3 rule. (Use

$v=\dfrac{ds}{dt}.$

3 M

5(b) Use the method of Regula Falsi to find the root of x = e^-x correct to three decimal places.

4 M

5(c) Use fourth order Runge Kutta method to find the value of y at x = 1 given that

$y'=\dfrac{y-x}{y+x}$ such that y(0) = 1. (Take h = 0.5)

7 M

5(d) Use Gauss Seidel method to solve: 83x + 11y ' 4z = 95, 7x + 52y + 13z = 104, 3x + 8y + 29z = 71.

3 M

5(e) Evaluate the integral

$\int _{-2}^6(1+x^2)^{3/2}dx$ by the Gaussain formula for n = 3.

4 M

5(f) Using Euler's method solve for y at x = 0.1 from

$\dfrac{dy}{dx}=x+y+xy ,$ y(0) = 1, in five steps.

7 M

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