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Download GTU BE/B.Tech 2019 Summer 4th Sem New 2141905 Complex Variables And Numerical Methods Question Paper

Download GTU (Gujarat Technological University) BE/BTech (Bachelor of Engineering / Bachelor of Technology) 2019 Summer 4th Sem New 2141905 Complex Variables And Numerical Methods Previous Question Paper

This post was last modified on 20 February 2020

This download link is referred from the post: GTU BE 2019 Summer Question Papers || Gujarat Technological University


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GUJARAT TECHNOLOGICAL UNIVERSITY

SEMESTER-IV(NEW) — EXAMINATION - SUMMER 2019

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Subject Code:2141905 Date:09/05/2019

Subject Name: Complex Variables and Numerical Methods

Time:02:30 PM TO 05:30 PM Total Marks: 70

Instructions:

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

Q.1 (a) State De’Movier’s. Find arg [i/(-2-2i)]. 03

(b) Define the operators A,V and E. Prove that EV= A. 04

(c) State Cauchy — Riemann Equations. Show that 07

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(1) f(z) = sin z is everywhere analytic

(i1) f(z) = xy + iy is nowhere analytic.

Q.2 (a) State the formula for sin-1z. Find sin-1(-i) 03

(b) Find analytic function f(z) =u +iv, if u=2x(1-y). 04

(c) Classify the singularities of the analytic function. 07

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In each of the following case, identify the singular point and its type with justification.

(i) z/(z4+1) (ii) sin(1/z) (iii) (1 —coshz)/z5

OR

(c) Use residues to evaluate the improper integral: 07

0 x2 dx / (x2+ 1)(x2+4)

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Q.3 (a) Evaluate the integral ∫C z̄ dz when C is the right-hand half z=2e(-π/2 < θ < π/2), of the circle |z| = 2 from — 2i to 2i. 03

(b) Show that the mapping by w = 1/z transforms circles and lines into circles and lines. 04

(c) Give two Laurent series expansions in powers of z for the function f(z) = 1/[z(1 —2z)] and specify the regions in which those expansions are valid. 07

OR

Q.3 (a) Find the bilinear transformation which transforms z1 = ∞, z2 =1, z3 = 0 into w1 =0, w2=1, w3 = ∞ 03

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(b) Determine and sketch the region : 04

(i) 0 < arg z < π/4, (ii) | z-3 | > 4,

Which of them are domains?

(c) State the Cauchy’s Integral Formula and its extension. Hence evaluate integral ∫C (7z2-2z+5)/(z+1)3 dz, where C is circle |z - i|=2 07

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Q.4 (a) Evaluate ∫37 x2log x dx taking four sub-intervals by trapezoidal rule. 03

(b) Apply Bisection method to find a real root of the equation 2x4 — 5x + 1 =0 correct to 2 decimal places. 04

(c) Given f(1)=22, f(2)=30, f(4)=82, f(7)=106, f(12)=206, find f(8) using Lagrange’s interpolation formula. 07

OR

Q.4 (a) The velocity of a car (running on a straight road) at intervals of 2 minutes are given below. 03

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Time (in min.): 0 2 4 6 8 10 12

Velocity (in km/hr): 0 22 30 27 18 7 0

Apply Simpson’s 1/3rd rule to find the distance covered by the car.

(b) Newton Raphson method find a root of the equation xsinx +cosx = 0 correct to four decimal places (taking initial guess X0 = π). 04

(c) State Striling’s Interpolation formula. 07

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Interpolate by means of Gauss’ forward formula the population for the year 1936 given the following data:

Year: 1901 1911 1921 1931 1941 1951

Population (1000s): 12 15 20 27 39 52

Q.5 (a) Using secant method find a real root of the equation x4 — 5x + 1 = 0 up to three iterations. 03

(b) Use Runge-Kutta method to solve y’ = xy, y(0)=1, for x = 0.2, with h=0.1. 04

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(c) Using Gauss-Seidel method to solve the following system correct to 3 decimal places: 83x + 11y —4z =95, 7x + 52y + 13z =104, 3x + 8y + 29z =71. 07

OR

Q.5 (a) Solve xlog10x =1.2 by Regula Falsi method correct to two decimal places. 03

(b) Solve y’=1-y, y(0)=0 in [0, 0.3] by modified Euler’s method taking h=0.1. 04

(c) Find the largest eigenvalue and corresponding eigenvector using power method, for 07

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A = | 4 4 2 |

| 1 4 1 |, taking X0 = (1, 1, 1)T.

| 2 1 3 |

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This download link is referred from the post: GTU BE 2019 Summer Question Papers || Gujarat Technological University