Download GTU BE/B.Tech 2018 Winter 4th Sem New 2140001 Mathematics 4 Question Paper

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GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER-IV (NEW) EXAMINATION - WINTER 2018

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Subject Code: 2140001

Subject Name: Mathematics-4

Time: 02:30 PM TO 05:30 PM

Date: 22/11/2018

Total Marks: 70

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Instructions:

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

Q.1

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(a) Find the complex conjugate of \frac{1}{z}. 03

(b) Find the locus of z given by |z| = 1. 04

(c) Show that u = y³ — 3x²y is a harmonic function. Also find its harmonic conjugate. 07

Q.2

(a) Determine the region in the z-plane represented by 1 < |z — 2| < 3. 03

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(b) Show that \frac{1+2z}{z^2+2z^3} = \frac{1}{z} + \frac{1}{2} - 1 + z - z^2 + ... in 0 < |z| < 1. 04

(c) Find the roots common to the equation z⁴ + 1 = 0 and z⁶ — i = 0. 07

OR

(c) Evaluate ?_c zdz along the straight line joining z = 1 — i to z = 3 + 2i. 07

Q.3

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(a) Expand f(z) = \frac{i}{z} as a Taylor’s series about the point z₀ = 1. Also determine the region of convergence and radius of convergence. 03

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

(c) Evaluate ?₀^2p \frac{cos2?}{5+4cos?} d?. 07

OR

(c) Determine and sketch the image of |z| = 1 under the transformation w = z + \frac{i}{2}. 07

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Q.4

(a) Determine the poles of the equation f(z) = \frac{2}{(z-1)^2(z+2)} and residue at each pole. 03

(b) Evaluate ?_c Re(z²)dz , where C is the boundary of the square with vertices 0, i, 1+ i, 1 in the clockwise direction. 04

Q.4

(a) Given

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(b) Solve the following system of equation using Gauss Elimination method with partial pivoting 04

x + y + z = 7

3x + 3y + 4z = 24

2x + y + 3z = 16

(c) Find the values of y for x = 21 and x = 28 from the following data. 07

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OR

Q.4 (a) Find the largest eigenvalue and corresponding eigen vector for A = \begin{bmatrix} 2 & 5 \\ 1 & 4 \end{bmatrix} 03

(b) Find the positive root of x = cosx correct upto 3 decimal places, using N-R method. 04

(c) Solve the following system by Gauss-Jacobi method. 07

27x + 6y — z = 285

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6x + 15y + 2z = 72

x + y + 54z = 110

Q.5 (a) Evaluate ?²cos2x 03

(b) Express the function \frac{x}{(x-1)(x-2)(x-3)} as a sum of partial fraction, using Largrange’s formula. 04

(c) Find the value of y for \frac{dy}{dx} = x+y; y(0) = 1, when x = 0.1, 0.2 with step size h = 0.05. Also compare with analytic solution. 07

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OR

Q.5 (a) Find a root of the equation x³ — x — 11 = 0, using the bisection method up to fourth approximation. 03

(b) From the following table, find f(x) using Newton’s divided difference formula 04

(c) Determine the largest eigenvalue and the corresponding eigenvector of the matrix A = \begin{bmatrix} 4 & 4 & 2 \\ 4 & 4 & 1 \\ 2 & 1 & 8 \end{bmatrix} 07

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