Download GTU BE/B.Tech 2019 Summer 4th Sem Old 140001 Mathematics Iv Question Paper

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Subject Code:140001

GUJARAT TECHNOLOGICAL UNIVERSITY

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SEMESTER-IV(OLD) - EXAMINATION - SUMMER 2019

Subject Name: Mathematics-1V

Time:02:30 PM TO 05:30 PM

Instructions:

1. Attempt all questions.

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

Q1 (a) Find all roots of 3/8i. 07

(b) 1) Find real and imaginary part of f' (Z ) =z" +4z. Also, calculate the value of f at z=1+i. 04

2) Show that /(2)= [Im(z); z?0 / 0; z=0] is not continuous at the origin. 03

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Q2 (a) Find the image of the region ‘Z’ <1 under the transformation W=2z—1. Sketch the region and its image. 07

(b) Show that U(X, y ) =2x-X³ +3xy² is harmonic in some domain D and find a harmonic conjugate of u(x, Y). 07

OR

Q3 (a) If f(2) is an analytic function of z, show that (?²/?x² + ?²/?y²)|F'(z)|² = 4|F'(z)|² 07

(b) Evaluate ?(0 to 2+i) z²dz along the line y=x/2. 07

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Q3 (a) Evaluate:

1. ? dz/z, over the contour c, where c is the circle |z| = 1.
2. ? z/(1-z)³ dz, counterclockwise over C, where C:|z| = 2
3. ? e^z/(z+1)³ dz, counterclockwise over C, where C:|z| = 2

07

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OR

(b) Determine the Laurent series expansion of f(z) = e^z/z² valid for a) |z| <1 b) 1<|z|<3 07

Q4 (a) Using Newton’s divided difference formula, compute f(10.5) from the following data: 07

X: 10 11 13 17

f(x): 2.3026 2.3979 2.5649 2.8332

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(b) Find a real root of the equation x³ + 4x - 1 = 0, lies between 0 and 1 by using bisection method correct to decimal places. 07

OR

Q4 (a) Solve the following system of equation using partial pivoting by Gauss Elimination method. 07

8x₁ + 2x₃ = -7

3x₁ + 5x₂ + 2x₃ = 8

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6x₁ + 2x₂ + 8x₃ = 26

(b) Solve the following system of equations by using Gauss-Seidel method. 07

10x+y+z=6; x+10y+Z=6; x+y+102=6

Q.5 (a) Using the power method, find the largest eigenvalue of the matrix 07

A=|2 -1 0 / -1 2 -1 / 0 -1 2|

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(b) Apply Runge-Kutta fourth order method to find an approximation value of y when x=0.1 in step of 0.1 if dy/dx =x+y², y(0)=1 07

OR

Q5 (a) Evaluate the integral ?(0 to 1) 1/(1+x²) dx, by Gauss three point quadrature formula. 07

(b) Solve the differential equation dy/dx +xy=0; y(0)=1, from x=0 to x=0.25 using Euler’s method taking step size 0.05. 07

Evaluate ?(1 to 2) dx/x with n=6 by using Simpson’s 3/8 rule and hence calculate ln 2. 07

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Date:09/05/2019

Total Marks: 70

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