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 = ex (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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