Download GTU BE/B.Tech 2019 Winter 4th Sem New 2141905 Complex Variables And Numerical Methods Question Paper

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GUJARAT TECHNOLOGICAL UNIVERSITY

SEMESTER- IV (New) EXAMINATION — WINTER 2019

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Subject Code: 2141905 Date: 07/12/2019

Subject Name: Complex Variables and Numerical Methods

Time: 10:30 AM TO 01:30 PM Total Marks: 70

Instructions:

1. Attempt all questions.

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2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.

Q-1

(a) Determine whether the function $f\left(z\right)=\{\begin{array}{ll}\frac{z^4+3iz-2}{z+i}& z\ne -i\\ 5& z=-i\end{array}$ is continuous? Can the function be redefined to make it continuous at Z = —i? [03]

(b) State De Moivre’s Theorem. Find the roots of the equation Z⁴+1=0. [04]

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(c) Solve the following system of equations using Gauss Seidel Method correct to four decimal places. [07]

30x —2y+3z=75; 2x+2y+18z=30; x + 17y — 2z = 48

Q-2

(a) Check whether the function f (z) = e^x is entire or not. Also find derivative of f(z). [03]

(b) Find the bilinear transformation which maps z = 1, 0, —1 into the points W = i, 1. [04]

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(c) Using the Residue Theorem Evaluate, ∫ 0 2π dθ / (5 + 3sinθ) [07]

OR

(c) Show that the function u(x,y) = 3x²y + 2x² — y³ — 2y² is harmonic. Find the conjugate harmonic function v and express u + iv as analytic function of z [07]

Q-3

(a) Evaluate ∮_c (z²+1)/ (z-1) dZ if c is the circle of unit radius with centre at z = 1. [03]

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(b) Find the real part and imaginary part of (1/z). [04]

(c) Evaluate ∫ f(z)dz where f(z) is defined by [07]

f(z) = { 1 when y < 0; 4 when y > 0 }

And C is the arc from z = —1 —i to z =1+ i along the curve y = x³.

OR

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(c) Find the type of singularity of the function f(z) = e^2z / (z²(z-1)⁴) [07]

Q-4

(a) Expand f(Z) = 1/((Z-1)(Z-2)) valid for region [03]

(i) |z| < 1 (ii) 1 < |z| < 2 (iii) |z| > 2

(b) Use Euler’s Method, find y(0.2) given that dy/dx = x - y² ; y(0)=1 take h = 0.1 [04]

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(c) Evaluate √8 to two decimal places by Newton’s iterative formula. [07]

OR

(a) Solve the following system of equation using Gauss Elimination Method [03]

x+y+z=7, 3x+3y+4z=24; 2x+y+3z=16

(b) Use Secant Method to find the root of f(x) = xlog₁₀x —1.9 =0 [04]

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(c) Using Newton’s Divided Differences formula to find a polynomial function, satisfying the following data. [07]

Q-5

(a) Evaluate ∫₁² 1/x dx by using Gaussian formula for n =2 and n = 3 [03]

(b) Use fourth order Range-Kutta method to compute y(0.2) and y(0.4) given that dy/dx = y - x² ; y(0) = 1 [04]

(c) Find the dominant Eigen value of A = [[4, -5], [-3, 4]] by Power Method and the corresponding Eigen vector. [07]

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OR

(a) State Trapezoidal Rule and evaluate ∫₀¹ e^-x2 dx using it with n = 10 [03]

(b) Use Lagrange’s formula to fit a polynomial to the data [04]

(c) Apply improved Euler’s method to solve the initial value problem y' = x +y with y(0) = 0 choosing h = 0.2 and compute y₁, y₂, ..., y₅. [07]

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