Seat No.:
Enrolment No.
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GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER- IV (New) EXAMINATION - WINTER 2019
Subject Code: 2140001 Date: 07/12/2019
Subject Name: Mathematics-4
Time: 10:30 AM TO 01:30 PM Total Marks: 70
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Instructions:
- Attempt all questions.
- Make suitable assumptions wherever necessary.
- Figures to the right indicate full marks.
Q-1 (a) Find the principal argument of z = 1-7i
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(b) Show that f(z) = xy / (x2+y2) if (x,y) ? (0,0) =0 if (x,y) =(0,0) Show that f(z) is not continuous at the origin.
(c) Solve the following system of linear equations by Gauss-elimination method. x +y+2z=9, 2x —3y+4z=13, 3x+4y+5z=40.
OR
(a) Check whether the function f (z) = Z is analytic or not?
(b) Show that u(x, y) = 2x — x3 + 3xy2 is harmonic in some domain and find a harmonic conjugate v(x, y).
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(c) Determine the mobius transformation that maps z1 = 0, z2 = 1, z3 = 8 onto w1 = —1, w2 = —i, w3 =1 respectively.
Q-2 (a) Find real and imaginary parts of (—1 — i)7 + (—1 + i)7
(b) Prove that ?C sin3z / (z+p) dz = 2pi, where C is the circle |z| = 5.
(c) Expand f (z) = 1-e2z / z3 in Laurent’s series about z = 0 and identify singularity.
OR
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(a) Use residues to evaluate ?08 x2dx / ((x2+1)(x2+4))
(b) Find the radius of convergence of ?n=18 (n+1 / n+2) (1 + z/n)n2 zn
(c) Evaluate ?C (x2 — iy2)dz along the parabola y = 2x2 from (1,2) to (2,8).
Q-3 (a) Expand f (Z) = 1 / ((z+2)(z+4)) valid for the regions (i) |z| < 2, (ii) 2 < |z| < 4, (iii) |z| > 4.
(b) Prove that : ?lnf(x) = ln [1 + ?f(x) / f(x))]
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OR
(a) Find a real root of the equation x3 + 4x2 — 1 = 0 by using bisection method correct up to two decimal places.
(c) Determine the interpolating polynomial of degree three using Lagrange’s interpolation formula for the table below.
| X | -1 | 0 | 1 | 2 |
|---|---|---|---|---|
| y | 2 | 1 | 0 | -1 |
Q-4 (a) Use trapezoidal rule to estimate ?01.2 ex3 dx using a strip of width 0.2.
(b) Evaluate I = ?-11 dt / (1+t2) by one point, Gaussian formula.
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(c) Solve the following equations by Gauss-Seidel method correct up to two decimal places 20x + 2y +z = 30, x — 40y —3z=-75, 2x —y + 10z = 30.
OR
Q-5 (a) Compute cosh(0.56) using Newton’s forward difference formula for the following table.
| X | 0.5 | 0.6 | 0.7 | 0.8 |
|---|---|---|---|---|
| F(X) | 1.127626 | 1.185465 | 1.255169 | 1.337435 |
(b) Using Newton’s divided difference interpolation formula compute f (9.2) from the following data.
| X | 8.0 | 9.0 | 9.5 | 11.0 |
|---|---|---|---|---|
| f(x) | 2.079442 | 2.197225 | 2.251292 | 2.397895 |
(c) Using improved Euler’s method, solve y’ = 1 — y with the initial condition y(0) = 0 and tabulate the solutions at x = 0.1, 0.2. compare the answer with the exact solution.
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OR
(a) Using N-R method find an iterative formula to find vN (where N is positive number) and hence find v5.
(b) Evaluate the integral ?11.2 x loge x dx, using Simpson’s 1/3rd rule.
(c) Find the largest eigen value and the corresponding eigenvector for A = [1 6 1] [1 2 0] [0 0 3]
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