Download GTU B.Tech 2020 Summer 4th Sem 2141905 Complex Variables And Numerical Methods Question Paper

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GUJARAT TECHNOLOGICAL UNIVERSITY

SEMESTER- IV EXAMINATION - SUMMER 2020

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Subject Code: 2141905 Date:28/10/2020

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.

MARKS

Q.1 (a) Verify Cauchy-Riemann equation for f(z)=cosxcosh y—isinxsinhy . 03

(b) Find all cube roots of complex number (—8i ) . 04

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(c) Evaluate ∫ dx with n=1 by (i) Trapezoidal Rule (ii) Simpson’s ⅓ Rule 07

(iii) Simpson’s ⅜ Rule.

Q.2 (a) Find the principal value of (1—i )⁴ⁱ . 03

(b) Using Parametric representation of C, evaluate ∫z dz; C is the circle 04

z=2e^it(0<t<2π)

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(c) Show that u(x,y)=2x—x³+3xy² is harmonic function and find harmonic 07

conjugate v(x,y).

OR

(c) For f(z)= z≠0 07

0 ; z=0

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Show that C-R equation is satisfied at origin but f'(0) does not exist.

Q.3 (a) Derive the Taylor series representation 03

1/(1-z) = ∑ zⁿ⁺¹ > |z - 1| < 1

(b) State Cauchy Integral formula. Use it to evaluate ∫ dz C: |z| =1. 04

cz(z² +8)

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(c) Find Laurent series representation of f(z)= 1/z²(1 +z ) 07

for (i) 0 <|z| <1 (ii) 1<|z|<∞ .

OR

Q.3 (a) Determine residue of f(z)= (3z +9)/(z²+9) at z=3i. 03

(b) Find the fixed points of the transformation w = (z—i)/(z+i) . 04

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(c) Evaluate ∫ (x dx)/(x²+1)³ using residue theorem. 07

Q.4 (a) Show that 1+Δ=E 03

(b) Find f(0.12)& f(0.26) by appropriate interpolation formula from following 04

table

(c) Determine images of Vertical and Horizontal lines under the transformation 07

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w=e^z.

OR

Q.4 (a) Using Lagrange’s formula, express the function 2/((x-1)(x-2)(x-3)) as a sum 03

of partial fractions.

(b) Find interpolating polynomial using Newton’s divided difference formula from 04

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following table

(c) Show that a function f(z)=u(x,y)+iv(x,y)is analytic in a domain D if and 07

only if v is a harmonic conjugate of u.

Q.5 (a) Use Newton-Raphson method to find positive root of sinx =1-x correct to 03

three decimal places.

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(b) Use power method to find largest eigen value and corresponding eigen vector 04

of [1 2; 2 4] correct to four decimal places.

(c) Apply Runge-Kutta fourth-order method to find y(0.2). Given that 07

dy/dx =y—x where y(0)=2 and h=0.1.

OR

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Q.5 (a) Use Secant method to find a positive root of the equation x³ +x—1= 0 correct 03

to three decimal places.

(b) Given that dy/dx =x² +y, y(0)=1. Find y(0.1) using Modified Euler’s 04

method with h= 0.05 correct to three decimal places.

(c) Solve the following liner system 07

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10x+2y+z=9

2x+20y-2z=-44

—2x+3y+10z=22

Correct to two decimal places by Gauss-Seidel method.

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