Download GTU BE/B.Tech 2019 Winter 4th Sem New 2141905 Complex Variables And Numerical Methods Question Paper

Download GTU (Gujarat Technological University) BE/BTech (Bachelor of Engineering / Bachelor of Technology) 2019 Winter 4th Sem New 2141905 Complex Variables And Numerical Methods Previous Question Paper

Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ? IV (New) EXAMINATION ? WINTER 2019
Subject Code: 2141905 Date: 07/12/2019

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.

Q-1 (a)
Determine whether the function {
? 2
+3? ?2
? +? ; ? ? ? 5 ; ? = ? is continuous?
Can the function be redefined to make it continuous at ? = ? ?
03
(b)
State De Moivre?s Theorem. Find the roots of the equation ? 4
+ 1 = 0.
04
(c) Solve the following system of equations using Gauss Seidel Method correct to
four decimal places.
30? ? 2? + 3? = 75; 2? + 2? + 18? = 30; ? + 17? ? 2? = 48
07

Q-2 (a)
Check whether the function ? (? ) = ? ? 2
is entire or not. Also find derivative of
? (? ).
03
(b)
Find the bilinear transformation which maps ? = 1, 0, ?1 into the points ? =
? , ?, 1.
04
(c)
Using the Residue Theorem Evaluate, ?
?
5?3? 2? 0

07
OR
(c) Show that the function ? (? , ? ) = 3? 2
? + 2? 2
? ? 3
? 2? 2
is harmonic. Find the
conjugate harmonic function ? and express ? + ? as analytic function of z
07

Q-3 (a)
Evaluate ?
? 2
+1
? 2
?1
?
? if ? is the circle of unit radius with centre at ? = 1.
03
(b)
Find the real part and imaginary part of ? ?
04
(c) Evaluate ? ? (? )? where ? (? ) is defined by
? (? ) = {
1 ? ? ? < 0
4? ? ? ? > 0

And C is the arc from ? = ?1 ? ? to ? = 1 + ? along the curve ? = ? 3
.
07



OR

Q-3 (a)
Find the type of singularity of the function ? (? ) =
? 2? (? ?1)
4

03
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Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ? IV (New) EXAMINATION ? WINTER 2019
Subject Code: 2141905 Date: 07/12/2019

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.

Q-1 (a)
Determine whether the function {
? 2
+3? ?2
? +? ; ? ? ? 5 ; ? = ? is continuous?
Can the function be redefined to make it continuous at ? = ? ?
03
(b)
State De Moivre?s Theorem. Find the roots of the equation ? 4
+ 1 = 0.
04
(c) Solve the following system of equations using Gauss Seidel Method correct to
four decimal places.
30? ? 2? + 3? = 75; 2? + 2? + 18? = 30; ? + 17? ? 2? = 48
07

Q-2 (a)
Check whether the function ? (? ) = ? ? 2
is entire or not. Also find derivative of
? (? ).
03
(b)
Find the bilinear transformation which maps ? = 1, 0, ?1 into the points ? =
? , ?, 1.
04
(c)
Using the Residue Theorem Evaluate, ?
?
5?3? 2? 0

07
OR
(c) Show that the function ? (? , ? ) = 3? 2
? + 2? 2
? ? 3
? 2? 2
is harmonic. Find the
conjugate harmonic function ? and express ? + ? as analytic function of z
07

Q-3 (a)
Evaluate ?
? 2
+1
? 2
?1
?
? if ? is the circle of unit radius with centre at ? = 1.
03
(b)
Find the real part and imaginary part of ? ?
04
(c) Evaluate ? ? (? )? where ? (? ) is defined by
? (? ) = {
1 ? ? ? < 0
4? ? ? ? > 0

And C is the arc from ? = ?1 ? ? to ? = 1 + ? along the curve ? = ? 3
.
07



OR

Q-3 (a)
Find the type of singularity of the function ? (? ) =
? 2? (? ?1)
4

03
(b)
Find and Sketch the region of an infinite strip 1 < ? < 2 under the
transformation ? =
1
?
04
(c)
Expand ? (? ) =
1
(? ?1)(? ?2)
valid for region
(i) |? | < 1 (ii) 1 < |? | < 2 (iii) |? | > 2
07



Q-4 (a)
Use Euler?s Method, find ? (0.2) given that
?
?
= ? ? ? 2
; ? (0) = 1
take ? = 0.1
03

(b)
Evaluate ?8 to two decimal places by Newton?s iterative formula.
04

(c) Determine the polynomial by Newton?s forward difference formula from the
following table
? 0 1 2 3 4 5
? -10 -8 -8 -4 10 40

07

OR

Q-4 (a) Solve the following system ofequation using Gauss Elimination Method
? + ? + ? = 7; 3? + 3? + 4? = 24; 2? + ? + 3? = 16
03
(b) Use Secant Method to find the root of ? (? ) = ? ? 10
? ? 1.9 = 0 04
(c) Using Newton?s Divided Differences formula to find a polynomial function,
satisfying the following data.
? -4 -1 0 2 5
? (? ) 1245 33 5 9 1335

07



Q-5 (a)
Evaluate ?
?
1+? 2
1
?1
by using Gaussian formula for ? = 2 and ? = 3
03
(b) Use fourth order Range-Kutta method to compute ? (0.2) and ? (0.4) given that
?
?
= ? ?
2? ? ; ? (0) = 1.
04
(c)
Find the dominant Eigen value of ? = [
3 ?5
?2 4
] by Power Method and the
corresponding Eigen vector.
07
OR
Q-5 (a)
State Trapezoidal Rule and evaluate ? ? ? 1
0
? using it with ? = 10
03
(b) Use Lagrange?s formula to fit a polynomial to the data
? -1 0 2 3
? 8 3 1 12

04
(c) Apply improved Euler?s method to solve the initial value problem
? ?
= ? + ? with ? (0) = 0 choosing ? = 0.2 and compute ? 1
, ? 2
, ? , ? 5
.
07
*********
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