Download GTU BE/B.Tech 2018 Winter 4th Sem New 2141905 Complex Variables And Numerical Methods Question Paper

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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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1. Attempt all questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.

Q1

1. (a) Find the roots of the equation Z² + 2iz + (2—4i)=0 [03]

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2. (b) Show that f(z) = z Re(z) is differentiable only at z = 0 and f'(0) = 0. [04]
3. (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

1. (a) Evaluate ?_c (2+i) z² dz along the line joining the points(0,0)and (2,1). [03]
2. (b) Determine the mobius transformation that maps z₁ =0, z₂ =1, z₃ = 8 onto w₁ = —1, w₂ = —i, w₃ = 1 respectively. [04]
3. (c) Prove that the n^th roots of unity are in geometric progression. Also show that their sum is zero. [07]
   OR

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4. (c) Verify that C-R equation are satisfied at z=0 for the function
   f (z) = { z² / z¯ if z?0
   0 if z=0

Q3

1. (a) Evaluate ?_c [z³ / (z—6i)²] dz, where C: |z| = 2. [03]

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2. (b) Find the radius of convergence of ? (2n+5) / (2ⁿ) (z+2i)ⁿ [04]
3. (c) Using the residue theorem, evaluate ?₀^2p d? / (5-3sin?) [07]
   OR
4. (c) Expand f (z) = (z-1) / (z²+3z+2) in a Taylor’s series about the point z = 0. [07]

Q3

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1. (a) Check whether f(z) = sinz is analytic or not. If analytic find its derivative. [03]
2. (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

1. (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
2. (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
3. (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
4. (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

1. (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
2. (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]
3. (c) Use power method to find the largest of Eigen values of the matrix A = [[4, 2], [1, 3]] [07]
   OR

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4. (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]

Q.5

1. (a) Prove that kD = log(1 + ?) [03]
2. (b) Evaluate I = ?_-1¹ 1/(1+x²) dx by one point, two point and three point Gaussian formula. [04]
3. (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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