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

Download GTU (Gujarat Technological University Ahmedabad) B.Tech/BE (Bachelor of Technology/ Bachelor of Engineering) 2020 Winter 4th Sem 2141905 Complex Variables And Numerical Methods Previous Question Paper

Seat No.: ________
Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE- SEMESTER?IV (NEW) EXAMINATION ? WINTER 2020
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.
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 () = (+2), and also prove that
03
it is analytic everywhere.

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

(c) Use Gauss-Jacobi method to determine roots of the following equations
07
20 + - 2 = 17
3 + 20 - = -18
2 - 3 + 20 = 25


Q.2 (a) Evaluate the following integral along the curve
03
() = + 2
2+4
()
0

(b) Evaluate cos
where C is the circle
04
-1
1) || = 2 2) || = 1/2

(c) Verify that = 2 - 2 - is harmonic in the whole complex plane
07
and finds it's conjugate harmonic function .
Q.3 (a) Obtain the Taylor's series of
() = sin in powers of ( - ).
03
4

(b) Find the center and radius of convergence of the power series 04
( + 2).
=0

(c) Find the Laurent's series expansion of
1
() =
in the region
07
(+1)(-2)
1) 1 < || < 2 2) || > 2




Q.4 (a) Find the Maclaurin's series of ()=2
03

(b) Find all values of such that = 1 +
04

(c) Evaluate cos
counterclockwise around C: || = 5
07
2-4
2
/
Q.5 (a) Use Bisection method to find the real root of
03
3 - 4 - 9 = 0 . (Do 4 iterations)

(b) Using Newton's divided difference interpolation formula, compute
04
(10.5) from the following data:
x
10
11
13
17
()
2.3026
2.3979
2.5649
2.8332
1

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(c) Use Simpson's 3/8 rule and evaluate the following integral taking n=6,
07
and hence calculate log 2. Also, find the error involved in the pross.
3
1 +
0



Q.6 (a) Approximate the root of the equation - 2 cos = 0, by three
03
iterations of Newton Raphson method, taking initial approximation as
0 = 2.

(b) Find an approximate value of (3.6) using Newton's backward
04
difference formula from the following data:
0
1
2
3
4
()
-5
1
9
25
55

(c) Using power method, determine the largest eigenvalue and the
07
2
-1
0
corresponding eigenvector of the matrix = [-1
2
-1] , taking
0
-1
2
1
initial eigenvector 0 = [0].
0
Q.7

Using three point Gaussian formula evaluate the following integral and
05
compare with the exact value.
1
1 + 2
-1


OR
Q.7

Solve the following system of linear equations using Gauss Elimination
05
Method.
+ + = 9; 2 - 3 + 4 = 13; 3 + 4 + 5 = 40

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2

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