This download link is referred from the post: GTU BE/B.Tech 2018 Winter Question Papers || Gujarat Technological University
Seat No.:
GUJARAT TECHNOLOGICAL UNIVERSITY
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BE - SEMESTER-1V (NEW) EXAMINATION - WINTER 2018
Subject Code:2141905
Subject Name:Complex Variables and Numerical Methods
Time: 02:30 PM TO 05:30 PM
Instructions:
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- Attempt all questions.
- Make suitable assumptions wherever necessary.
- Figures to the right indicate full marks.
Q1
- (a) Find the roots of the equation Z2 + 2iz + (2—4i)=0 [03]
- (b) Show that f(z) = z Re(z) is differentiable only at z = 0 and f'(0) = 0. [04]
- (c) Solve the following system of equation by Gauss-Seidal method correct to three decimal places. [07]
2x + y + 54z = 110
27x + 6y — z = 285
6x + 15y + 2z = 72
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Q2
- (a) Evaluate ∫c (2+i) z2 dz along the line joining the points(0,0)and (2,1). [03]
- (b) Determine the mobius transformation that maps z1 =0, z2 =1, z3 = ∞ onto w1 = —1, w2 = —i, w3 = 1 respectively. [04]
- (c) Prove that the nth roots of unity are in geometric progression. Also show that their sum is zero. [07]
OR - (c) Verify that C-R equation are satisfied at z=0 for the function
f (z) = { z2 / z̄ if z≠0
0 if z=0
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Q3
- (a) Evaluate ∫c [z3 / (z—6i)2] dz, where C: |z| = 2. [03]
- (b) Find the radius of convergence of ∑ (2n+5) / (2n) (z+2i)n [04]
- (c) Using the residue theorem, evaluate ∫02π dθ / (5-3sinθ) [07]
OR - (c) Expand f (z) = (z-1) / (z2+3z+2) in a Taylor’s series about the point z = 0. [07]
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Q3
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- (a) Check whether f(z) = sinz is analytic or not. If analytic find its derivative. [03]
- (b) Evaluate ∫c (z2—z+z-1) / (z2-z) dz counter clockwise around C, where C is |z| =1 and |z| = 3. [04]
Date:22/11/2018
Total Marks: 70
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Q4
- (a) Using Newton’s forward formula , find the value of f(1.6) if [03]
X 1 1.4 1.8 2.2
f(x) 3.49 4.82 5.96 6.5 - (b) Find the Lagrange interpolating polynomial from the following data [04]
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X 0 1 4 5
f(x) 1 3 24 39 - (c) Find a root of x4 — x3 + 10x+7 = 0 correct to three decimal places between a = —2 and b = —1 by Newton-Raphson method. [07]
OR - (c) Solve the system of equation by Gauss elimination method. [07]
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x+y+z=9
2x—3y+4z=13
3x+4y+5z=40
Q.5
- (a) Compute f(8) from the following values using Newton’s Divided difference formula [03]
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X 4 5 7 10 11 13
f(x) 48 100 294 900 1210 2028 - (b) Evaluate ∫61 1/(1+x) dx ,taking h =1 and using Simpson’s 1/3 rule. Hence obtain approximate value of log 7. [04]
- (c) Use power method to find the largest of Eigen values of the matrix A = [[4, 2], [1, 3]] [07]
OR - (c) Use Euler’s method to obtain an approximate value of y(0.4) for the differential equation y’ = x +y, y(0) = 1 with h=0.1. [07]
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Q.5
- (a) Prove that kD = log(1 + Δ) [03]
- (b) Evaluate I = ∫-11 1/(1+x2) dx by one point, two point and three point Gaussian formula. [04]
- (c) Determine y(0.1), y(0.2) correct upto four decimal places by fourth order Runge-Kutta method from dy/dx =2x+y, y(0)=1 [07]
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This download link is referred from the post: GTU BE/B.Tech 2018 Winter Question Papers || Gujarat Technological University
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