Download GTU B.Tech 2020 Winter 4th Sem 2141905 Complex Variables And Numerical Methods Question Paper

FirstRanker.com

GUJARAT TECHNOLOGICAL UNIVERSITY

BE- SEMESTER-IV (NEW) EXAMINATION - WINTER 2020

--- Content provided by FirstRanker.com ---

Subject Code:2141905 Date:09/02/2021

Subject Name:Complex Variables and Numerical Methods

Time:02:30 PM TO 04:30 PM Total Marks:47

Instructions:

1. Attempt any THREE questions from Q.1 to Q.6.

--- Content provided by​ FirstRanker.com ---

2. Q.7 is compulsory.
3. Make suitable assumptions wherever necessary.
4. Figures to the right indicate full marks.

MARKS

Q.1 (a) Separate real and imaginary parts of f(z) = e^z2, and also prove that it is analytic everywhere. 03

--- Content provided by⁠ FirstRanker.com ---

(b) Use De Moiver’s theorem and find 4^th root of unity in the complex plane. 04

(c) Use Gauss-Jacobi method to determine roots of the following equations 07

20x+y—2z=17

3x + 20y —z=-18

2x — 3y + 20z =25

--- Content provided by⁠ FirstRanker.com ---

Q.2 (a) Evaluate the following integral along the curve 03

z(t) =t +it²

?₀²⁺⁴ⁱ Re(z)dz

(b) Evaluate § (5z-2)/(z(z-1)) dz where C is the circle 04

1. |z| =2

--- Content provided by‍ FirstRanker.com ---

2. |z|=1/2

(c) Verify that u = x² — y²+y is harmonic in the whole complex plane and finds it’s conjugate harmonic function v. 07

Q.3 (a) Obtain the Taylor’s series of f(z) = sin z in powers of (Z - p/2) 03

(b) Find the center and radius of convergence of the power series ?_n=0⁸ (n+20)ⁿzⁿ. 04

(c) Find the Laurent’s series expansion of f(z) = 1/(z²-3z+2) in the region 07

--- Content provided by‍ FirstRanker.com ---

1. 1<|z|<2
2. |z|>2

Q.4 (a) Find the Maclaurin’s series of f(z)=sin²z 03

(b) Find all values of z such that e^z =1+ i 04

(c) Evaluate § (sinz)/(z²-4) dz counterclockwise around C: |z| = 5/2 07

--- Content provided by‍ FirstRanker.com ---

Q.5 (a) Use Bisection method to find the real root of 03

x³ —4x —9 = 0. (Do 4 iterations)

(b) Using Newton’s divided difference interpolation formula, compute f(10.5) from the following data: 04

(c) Evaluate ?₀^0.6 dx/(1+x) by Simpson’s 3/8 rule dividing into 6 intervals and hence calculate log_e 2. Also, find the error involved in the process. 07

Q.6 (a) Approximate the root of the equation e^x —2cosx =0, by three iterations of Newton Raphson method, taking initial approximation as X = 2. 03

--- Content provided by​ FirstRanker.com ---

(b) Find an approximate value of f(3.6) using Newton’s backward difference formula from the following data: 04

(c) Using power method, determine the largest eigenvalue and the corresponding eigenvector of the matrix A = [[2, -1, 0], [-1, 2, -1], [0, -1, 2]], taking initial eigenvector x₀ = [1, 0, 0]. 07

Q.7 Using three point Gaussian formula evaluate the following integral and compare with the exact value. 05

?_-1¹ dx/(1+ x²)

OR

--- Content provided by‍ FirstRanker.com ---

Q.7 Solve the following system of linear equations using Gauss Elimination Method. 05

x+y+z=9; 2x—-3y+4z=13; 3x +4y + 5z = 40

FirstRanker.com