This download link is referred from the post: GTU B.Tech 2020 Winter Question Papers || Gujarat Technological University
GUJARAT TECHNOLOGICAL UNIVERSITY
BE- SEMESTER-IV (NEW) EXAMINATION - WINTER 2020
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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:
- Attempt any THREE questions from Q.1 to Q.6.
- Q.7 is compulsory.
- Make suitable assumptions wherever necessary.
- Figures to the right indicate full marks.
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MARKS
Q.1 (a) Separate real and imaginary parts of f(z) = ez2, and also prove that it is analytic everywhere. 03
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(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
20x+y—2z=17
3x + 20y —z=-18
2x — 3y + 20z =25
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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
- |z| =2
- |z|=1/2
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(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 - π/2) 03
(b) Find the center and radius of convergence of the power series ∑n=0∞ (n+20)nzn. 04
(c) Find the Laurent’s series expansion of f(z) = 1/(z2-3z+2) in the region 07
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- 1<|z|<2
- |z|>2
Q.4 (a) Find the Maclaurin’s series of f(z)=sin2z 03
(b) Find all values of z such that ez =1+ i 04
(c) Evaluate § (sinz)/(z2-4) dz counterclockwise around C: |z| = 5/2 07
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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
X | 10 | 11 | 13 | 17 |
---|---|---|---|---|
f(x) | 2.3026 | 2.3979 | 2.5649 | 2.8332 |
(c) Evaluate ∫00.6 dx/(1+x) by Simpson’s 3/8 rule dividing into 6 intervals and hence calculate loge 2. Also, find the error involved in the process. 07
Q.6 (a) Approximate the root of the equation ex —2cosx =0, by three iterations of Newton Raphson method, taking initial approximation as X = 2. 03
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(b) Find an approximate value of f(3.6) using Newton’s backward difference formula from the following data: 04
x | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|
f(x) | -5 | 1 | 9 | 25 | 55 |
(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
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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
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This download link is referred from the post: GTU B.Tech 2020 Winter Question Papers || Gujarat Technological University