Download GTU B.Tech 2020 Summer 4th Sem 2141905 Complex Variables And Numerical Methods Question Paper

Download GTU (Gujarat Technological University Ahmedabad) B.Tech/BE (Bachelor of Technology/ Bachelor of Engineering) 2020 Summer 4th Sem 2141905 Complex Variables And Numerical Methods Previous Question Paper

Seat No.: ________
Enrolment No.___________


GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER? IV EXAMINATION ? SUMMER 2020
Subject Code: 2141905 Date:28/10/2020
Subject Name: COMPLEX VARIABLES AND NUMERICAL METHODS
Time: 10:30 AM TO 01:30 PM Total Marks: 70
Instructions:
1. Attempt all questions.

2. Make suitable assumptions wherever necessary.

3. Figures to the right indicate full marks.



MARKS
Q.1 (a) Verify Cauchy-Riemann equation for f (z) cos x cosh y i sin x sinh y .
03

(b) Find all cube roots of complex number 8
i .
04

6
(c)
1
1
Evaluate
dx
with h 1 by (i) Trapezoidal Rule (ii) Simpson's Rule
2
07
1 x
3
0
3
(iii) Simpson's Rule.
8


Q.2 (a) Find the principal value of 4
1
i
i .
03

(b)
z 2
Using Parametric representation of C, evaluate
dz
; C is the circle
04
C
z
2 i
z
e 0 2

(c) Show that
3
2
u( ,
x y) 2x x 3xy is harmonic function and find harmonic
07
conjugate v(x, y) .


OR


z2
(c)
07
For
; z 0
f (z) z
,
0
; z 0
Show that C-R equation is satisfied at origin but '
f (0) does not exist.
Q.3 (a) Derive the Taylor series representation
03
1
z in
z i
.
1 z
n (1
i
;
2
n 1
0
)

(b)
cos z
State Cauchy Integral formula. Use it to evaluate
dz
; C: z 1.
2
04
C z(z 8)

(c)
1
Find Laurent series representation of f (z)
z
2
1 z
07
for (i) 0 < z 1 (ii) 1 z .


OR

Q.3
3
(a)
3z 2
Determine residue of f (z)
at z 3i .
2
03
z 9

(b)
z 1
Find the fixed points of the transformation w
.
z 1
04
1

---

2
(c)
x dx
Evaluate


using residues.
07
2
x
1 2
x 4
0
Q.4 (a) Show that 1
hD
e
03

(b) Find f (0.12) & f (0.26) by appropriate interpolation formula from following
04
table
x
0.10
0.15
0.20
0.25
0.30
f (x)
0.1003
0.1511
0.2027
0.2553
0.3093

(c) Determine images of Vertical and Horizontal lines under the transformation
07
z
w e .

OR

Q.4
2
(a)
3x x 1
Using Lagrange's formula, express the function
as a sum
03
(x 1)(x 2)(x 3)
of partial fractions.

(b) Find interpolating polynomial using Newton's divided difference formula from
04
following table
x
0
1
4
5
7
f (x)
6
3
138
369
1611

(c) Show that a function f (z) u(x, y) iv(x, y) is analytic in a domain D if and
07
only if v is a harmonic conjugate of u.
Q.5 (a) Use Newton-Raphson method to find positive root of sin x 1 x correct to
03
three decimal places.

(b) Use power method to find largest eigen value and corresponding eigen vector
04
1
2
of
correct to four decimal places.
3 4

(c) Apply Runge-Kutta fourth-order method to find y(0.2) . Given that
07
dy
y x
dx
where y(0) 2 and h 0.1 .


OR
Q.5 (a) Use Secant method to find a positive root of the equation 3
x x 1 0 correct
03
to three decimal places.
(b)
dy
04
Given that
2
x ;
y
y(0) 1. Find y(0.1) using Modified Euler's
dx
method with h 0.05 correct to three decimal places.
(c) Solve the following liner system
07
10x 2y z
9
2x 20y 2z 4
4
2
x 3y 10z 22
Correct to two decimal places by Gauss-Seidel method.

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