Download GTU BE/B.Tech 2018 Winter 4th Sem Old 140001 Mathematics Iv Question Paper

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GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER-IV (OLD) EXAMINATION - WINTER 2018

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Subject Code:140001 Date: 22/11/2018
Subject Name: Mathematics-1V
Time: 02:30 PM TO 05:30 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) (i) Define harmonic function and show that f(x,y) =x²—y² is harmonic. 03
(ii) Show that lim_(x,y)->(0,0) x²y / (x⁴+y²) does not exist. 04

(b) Apply Gauss-Seidel method to solve the equations 07

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20x+y—2z=17, 3x+ 20y —z = —18, 2x — 3y + 20z = 25.

Q.2 (a) (i) State Trapezoidal rule, Simpson’s 1/3^rd rule and Simpson 3/8^th rule. 03
(ii) Find the positive root of x³ — x — 11 = 0 correct to the three decimal places by Bisection method. 04

(b) Obtain approximate value of y at x = 0.2 for the differential equation 07
dy/dx =2y + 3e^x, y(0) = 0, using Taylor’s method. Also compare the obtained numerical solution with the exact solution.

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OR

(b) Using Runge-Kutta method of fourth order, solve dy/dx = (x²-y²)/(x²+y²) with y(0) = 1 at x = 0.2. 07

Q.3 (a) (i) Evaluate ?₀¹ e^-x2 dx using Gauss quadrature formula of three points. 03
(ii) Using usual notations, show that ?= E-1, ?= E^1/2?= dE^1/2 and (E^1/2+ E^-1/2)y_1/2 =2µy_1/2 04

(b) Using Lagrange’s interpolation formula, find the value of y when x = 10, if the following values of x and y are given: 07

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x	5	6	9	11
y	12	13	14	16

OR

Q.3 (a) (i) Define entire function and show that f(z) = e^z is entire function. 03
(ii) Find harmonic conjugate of u(x,y) = 2x — x³ + 3xy². 04

(b) Using Newton forward interpolation formula, find the value of f(1.6) if the following values of x and f(x) are given: 07

x	1	1.4	1.8	2.2
F(x)	3.49	4.82	5.96	6.65

Q.4 (a) (i) Evaluate ?_c 1/z dz, where c is a circle |z| = 1/2. 03

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(ii) Determine and sketch the image of |z| =1 under the transformation w = z + i. 04

(b) Find Maclaurin’s series expansion of the function f(z) = sin²z. 07

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Q.4 (a) (i) Evaluate ?_c 1/(z-1) dz, where c is a circle |z| = 2. 03
(ii) Find bilinear transformation that maps the points -1, 0, 1 onto -1, -i,1 respectively. 04

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(b) Expand f(z) = 1/((z+1)(z+3)) in Laurent’s series valid for |z| < 1 and 1 < |z| < 3. 07

Q.5 (a) (i) Find an upper bound for the absolute value of the integral ?_c e^-z2dz, where C is the line segment joining the points (0,0) and (1, 1+i). 03
(ii) Determine the poles of the function f(z) = 1/((z²+1)²) and the residue at each poles. 04

(b) Evaluate ?_-8⁸ dx / ((x²+1) (x²+2x+2)). 07

OR

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Q.5 (a) (i) State (1) Rouche’s Theorem (ii) Liouville’s Theorem 03
(ii) Evaluate ?_c (z-e^lnz) dz, where C is the square with vertices at ±1, ±i. 04

(b) Evaluate ?₀^2p d? / (2+cos?)². 07

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