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GUJARAT TECHNOLOGICAL UNIVERSITY
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SEMESTER-IV(NEW) — EXAMINATION - SUMMER 2019
Subject Code:2140001 Date:09/05/2019
Subject Name: Mathematics-4
Time:02:30 PM TO 05:30 PM Total Marks: 70
Instructions:
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- Attempt all questions.
- Make suitable assumptions wherever necessary.
- Figures to the right indicate full marks.
Q1 (a) Find the principal argument of z = v3 + i. 03
(b) Check whether the following functions are analytic or not at any point: 04
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- f(z) = x² + ixy
- f(z) = z¯
(c) (i) Expand f(z) = z cos(1/z) in Laurent’s series near z = 0 and identify the singularity. 07
(ii) Show that if ¢ is any nth root of unity other than unity itself, than 1 + c + c² +...+ cn-1 = 0.
Q.2 (a) Find and sketch the image of the region |z| < 1 under the transformation 2z — i. 03
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(b) Show that the function u(x,y) = y³ — 3x²y is harmonic in some domain D and find its conjugate v(x, y). 04
(c) Find the Mobius transformation that maps the points z = 1, i, —1 into the points w = i, 0, —i. Hence find the image of |z| = 1. 07
OR
(c) Evaluate the integral ? Re(z¯)dz, where C is the boundary of the square with vertices 0, 1, 1+i, i in clockwise direction. 07
Q3 (a) Evaluate ? (x + iy)dz along the path y = x². 03
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(b) Find the residue at each pole of f(z) = zez / (z²+1) 04
(c) Expand f(z) = 1/(z(z-1)(z-2)) in Laurent’s series in the region 07
- |z| < 1
- 1 < |z| < 2
- |z| > 2.
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OR
Q (a) Write the Cauchy integral formula and using it evaluate ? cos(z)dz / (z(z+1)), where C is the circle |z| = 4. 03
(b) Evaluate ? z²dz / (z(z+1)(z²-3)), where C is the circle |z| = 2. 04
(c) Using the residue theorem, evaluate ? dx / (3-3sin?) from 0 to 2p. 07
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Q.4 (a) Find the positive root of the equation 2 sinx — x = 0 using bisection method in six stages. 03
(b) Solve the following system of equations by Gauss Seidel method: 04
28x + 4y — z = 32; 2x + 17y + 4z = 35; x + 3y + 10z = 24
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Correct up to two decimal places.
(c) Using the power method find the largest eigenvalue of the matrix 07
1 -3 2
4 4 -1
6 3 5
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OR
Q.4 (a) Use the secant method in three stages to find the root of the equation cosx — xex = 0. 03
(b) Find an approximate value of f(3.6)using Newton’s backward difference formula from the following data: 04
X 0 1 2 3 4
f(x) -5 1 9 25 55
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(c) Using Lagrange’s interpolation formula find y when x = 5 from the following table: 07
X 1 2 3 4 7
y 2 4 8 16 128
Q.5 (a) Use Simpson’s 1/3 rule to evaluate ? e-x²dx from 1 to 2. Take h = 0.25. 03
(b) Use Gauss elimination method to solve the system of equations 04
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2x1 + 4x2 — 6x3 = —4; x1 + 5x2 + 3x3 = 10; x1 + 3x2 + 2x3 = 5.
(c) Derive Euler’s formula to solve the initial value problem dy/dx = f(x,y); y(x0) = y0. Find y(0.1) for dy/dx = x² + y, where y(0) = 1 using improved Euler’s'method. Take h = 0.05. 07
OR
Q.5 (a) Find the real root of the equation x³ — 9x + 1 = 0 up to five decimal places by the Newton-Raphson’s method. Take x0 = 3. 03
(b) Find f(15)from the following table using Newton’s divided difference formula: 04
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X 4 5 7 10 11 13
f(x) 48 100 294 900 1210 2028
(c) Apply fourth order Runge-Kutta method to find y(0.1)and y(0.2) for the differential equation dy/dx = 3x + ½y, y(0)=1. Take h=0.1. 07
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