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

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GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER- IV (Old) EXAMINATION — WINTER 2019

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Subject Code: 140001 Date: 07/12/2019

Subject Name: Mathematics-IV

Time: 10:30 AM TO 01:30 PM

Instructions:

Total Marks: 70

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

Q.1 (a) Find the fifth root of unity. 07

(b) Define interpolation. Using Lagrange’s interpolation, find y(2). 07

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x -1 0 1 3

y 2 1 0 -1

Q.2 (a) Define bilinear transformation. Find bilinear transformation which maps points 07

0,1, 8 into points —1, —i, 1 respectively.

(b) Expand (z/(z-1)(z+2)) in Laurent series in region (1) 1 < |z| < 2 (ii) |z| > 2. 07

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OR

Q.2 (a) Prove that tanh^-1z = (1/2) log ((1+z)/(1-z)) 07

Q.3 (a) State Cauchy’s integral formula. Evaluate ? (3z²+z)/(z-1) dz ; where |z—1|=1. 07

(b) Define (i) Analytic function (ii) Harmonic function. 07

If f(z) = u(x,y) + iv(x,y) is analytic and u(x,y) = y³ — 3x²y, then find v(x,y).

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OR

Q.3 (a) Derive Cauchy — Riemann equations in polar form. 07

(b) Evaluate ? (exp( -pz/2))/(z²+4) dz ; where C is the boundary of square with vertices 07

0,1,1 + i,i with counterclockwise direction.

Q.4 (a) Prove that (i) (1+?)(1—-?)=1 (ii) E = e^hD 07

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(b) Using Runge-Kutta 4^th order method find y(0.1). 07

dy/dx =x²+y² ; y(0)=1

OR

Q.4 (a) Find Newton’s forward interpolating polynomial and hence find y(5). 07

x 4 6 8 10

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y 1 3 8 16

(b) Evaluate ?₀⁶ f(x) dx (take h =1) using 07

(i) Trapezoidal rule (ii) Simpson’s 1/3 rule and (iii) Simpson’s 3/8 rule.

Q.5 (a) Set up Newton iteration for computing square root of given positive number 07

N and find v2.

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(b) Evaluate ?₀¹ dx/(1+x) using 2 point and 3 point Gaussian integration. 07

OR

Q.5 (a) Find a root of x³ —4x — 9 = 0 correct upto three decimal places using 07

Bisection method.

(b) Solve by Gauss-Siedel method correct upto 3 decimal places. 07

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27x + 6y —z =285

6x + 15y +2z=72

x+y+54z =110

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