Download GTU B.Tech 2020 Winter 4th Sem 2140001 Mathematics 4 Question Paper

Download GTU (Gujarat Technological University Ahmedabad) B.Tech/BE (Bachelor of Technology/ Bachelor of Engineering) 2020 Winter 4th Sem 2140001 Mathematics 4 Previous Question Paper

Seat No.: ________
Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE- SEMESTER?IV (NEW) EXAMINATION ? WINTER 2020
Subject Code:2140001 Date:09/02/2021
Subject Name:Mathematics-4
Time:02:30 PM TO 04:30 PM Total Marks:47
Instructions:
1. Attempt any THREE questions from Q.1 to Q.6.

2. Q.7 is compulsory.
3. Make suitable assumptions wherever necessary.

4. Figures to the right indicate full marks.


MARKS
Q.1 (a) Define following terms : (a) Analytic function (b) continuous function
03

(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
2 x 2 y z 6 ,
4 x 2 y 3 z 4 ,
x y z 0


Q.2 (a) Using the C-R equations, show that
3
f (z) z is analytic everywhere.
03

(b)
5 z 2
04
Evaluate
d z
, where C is the circle | z| 3.
z (z 1
)
C

(c) Show that
2
2
u(x, y) x y is Harmonic. Find the corresponding analytic function
07
f (z) u i v .




Q.3 (a) Expand ( )
z
f z e in a Taylor series about z 0 .
03

(b)
(z 3)
04
Determine the residues of f (z)
at each of its poles in the finite z
(z 1)(z 2)
plane.

(c)
1
1
07
Determine the Laurent series expansion of f (z)
valid for
(z 1
)
(z 2)
(a) | z| 1 (b) 1 | z| 2




Q.4 (a) Check whether the function f (z) z 1 is analytic or not at any point.
03

(b)
n
04
Find the radius of convergence of the
z
n 0
n !

(c)
2
07
Using Residue theorem, evaluate
z
d z
where C is circle | z | 3
2
(z 1
) (z 2)
C



Q.5 (a) Perform five iterations of Bisection method to find the real root of equation
03
3
x x 1
0 .

(b) Solve the given System of Linear equations by using Gauss Elimination method: 04
x 3 y 2 z 5, 2 x 4 y 6 z 4
, x 5 y 3z 10

(c)
dy
07
Use second order Runge-Kutta method to solve
2
x y , y(0) 1 and find
dx
y(0.2) with h 0.1
1

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Q.6 (a) Perform three iteration of secant method to find approximate root of equation
03
3
2
x x 3x30.

(b)
dy
04
Use Euler's method to solve
x 2 y , y(1) 1. Hence find y(1.5) with
dx
h 0.1.

(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 3
x x 1 0 , correct to two decimal places using Newton-
05
Raphson method.


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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2

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