GUJARAT TECHNOLOGICAL UNIVERSITY
BE- SEMESTER-IV (NEW) EXAMINATION - WINTER 2020
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Subject Code:2140001 Date:09/02/2021
Subject Name:Mathematics-4
Time:02:30 PM TO 04:30 PM Total Marks:47
Instructions:
- Attempt any THREE questions from Q.1 to Q.6.
- Q.7 is compulsory.
- Make suitable assumptions wherever necessary.
- Figures to the right indicate full marks.
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MARKS
Q.1 (a) Define following terms : (a) Analytic function (b) continuous function 03
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(b) Determine the bilinear transformation which maps the points z=2, 1, 0 into the 04 points w=1, 0, i respectively.
(c) Use Gauss-elimination method (with Partial Pivoting) to obtain the solution of the 07 system
2x+2y+z=6, 4x+2y+3z=4, x+y+z=0
Q.2 (a) Using the C-R equations, show that f(z) =z2 is analytic everywhere. 03
(b) Evaluate ?C (z2 / (z(z-2))) dz, where C is the circle |z| =3. 04
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(c) Show that u(x, y)=x2 —y2 is Harmonic. Find the corresponding analytic function f(z)=u+iv. 07
Q.3 (a) Expand f(z)=ez in a Taylor series about z =0. 03
(b) Determine the residues of f(z) = 1/((z+1)(z+2)) at each of its poles in the finite z plane. 04
(c) Determine the Laurent series expansion of f(z) = 1/((z-1)(z-2)) valid for 07
(a) |z|<1 (b) 1<|z|<2
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Q.4 (a) Check whether the function f(z)=z+1 is analytic or not at any point. 03
(b) Find the radius of convergence of S (zn / n!), n=0 to infinity. 04
(c) Using Residue theorem, evaluate ? (z2-4)/(z-1)2(z+2) dz where C is circle |z|=3 07
Q.5 (a) Perform five iterations of Bisection method to find the real root of equation x2—x—1=0. 03
(b) Solve the given System of Linear equations by using Gauss Elimination method: 04
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x+3y+2z=5, 2x+4y—-6z=—4, x+5y+3z=10
(c) Use second order Runge-Kutta method to solve dy/dx = x+y2, y(0)=1 and find y(0.2) with h =0.1 07
Q.6 (a) Perform three iteration of secant method to find approximate root of equation x3 +x2 -3x-3=0. 03
(b) Use Euler’s method to solve dy/dx =x+2y, y(1)=1. Hence find y(1.5) with h=0.1. 04
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(c) Using Lagrange’s interpolating polynomial, find f(10) from the given data: 07
| x | 5 | 6 | 9 | 11 |
|---|---|---|---|---|
| f(x) | 12 | 13 | 14 | 16 |
Q.7 Find a real root of x3+x—1=0, correct to two decimal places using Newton-Raphson method. 05
OR
Construct an Interpolating polynomial which takes the following values : 05
| x | 1 | 2 | 7 | 8 |
|---|---|---|---|---|
| y | 1 | 5 | 5 | 4 |
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