Download GTU BE/B.Tech 2019 Winter 3rd Sem New 3130005 Complex Variables And Partial Differential Equations Question Paper

Download GTU (Gujarat Technological University) BE/BTech (Bachelor of Engineering / Bachelor of Technology) 2019 Winter 3rd Sem New 3130005 Complex Variables And Partial Differential Equations Previous Question Paper

Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ? III (New) EXAMINATION ? WINTER 2019
Subject Code: 3130005 Date: 26/11/2019

Subject Name: Complex Variables and Partial Differential Equations
Time: 02:30 PM TO 05:00 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)
Find the real and imaginary parts of f(z) =
3i
2+3i
.
03
(b) State De-Movire?s formula and hence evaluate
(1 + ?? ?3)
100
+ (1 ? ?? ?3)
100
.
04
(c) Define harmonic function. Show that ?? (?? , ?? ) = sinh?? sin?? is harmonic
function, find its harmonic conjugate ?? (?? , ?? ).

07
Q.2 (a) Determine the Mobius transformation which maps ?? 1
= 0, ?? 2
= 1, ?? 3
= ?
into ?? 1
= ?1, ?? 2
= ??? , ?? 3
= 1.
03
(b)
Define ???????? , prove that ?? ?? = ?? ?(4?? +1)
?? 2
.
04
(c)
Expand ?? (?? ) =
1
(?? ?1)(?? +2)
valid for the region
(i) |?? | < 1 (ii) 1 < |?? | < 2 (iii) |?? | > 2 .
07
OR
(c)
Find the image of the infinite strips (i)
1
4
? ?? ?
1
2
(ii) 0 < ?? <
1
2
under the
transformation =
1
?? . Show the region graphically.
07
Q.3 (a) Evaluate ?
(?? ? ?? + ?? ?? 2
)????
?? where ?? is a straight line from ?? = 0 to ?? =
1 + ?? .
03
(b) Check whether the following functions are analytic or not at any point,
(i) ?? (?? ) = 3?? + ?? + ?? (3?? ? ?? ) (ii) ?? (?? ) = ?? 3
2
?
.
04
(c)
Using residue theorem, evaluate ?
????
(?? 2
+1)
2
?
0
.
07
OR
Q.3 (a)
Expand Laurent series of ?? (?? ) =
1??? ?? ?? at ?? = 0 and identify the
singularity.
03
(b) If ?? (?? ) = ?? + ???? , is an analytic function , prove that
(
?? 2
?? ?? 2
+
?? 2
?? ?? 2
) |?????? (?? )|
2
= 2|?? ?(?? )|
2
.
04
(c) Evaluate the following:
i. ?
?? +3
?? ?1

?? ???? where ?? is the circle (a) |?? | = 2 (b) |?? | =
1
2
.
ii. ?
sin ?? (?? ?
?? 4
)
3

?? ???? where ?? is the circle |?? | = 1 .
07
FirstRanker.com - FirstRanker's Choice
Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ? III (New) EXAMINATION ? WINTER 2019
Subject Code: 3130005 Date: 26/11/2019

Subject Name: Complex Variables and Partial Differential Equations
Time: 02:30 PM TO 05:00 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)
Find the real and imaginary parts of f(z) =
3i
2+3i
.
03
(b) State De-Movire?s formula and hence evaluate
(1 + ?? ?3)
100
+ (1 ? ?? ?3)
100
.
04
(c) Define harmonic function. Show that ?? (?? , ?? ) = sinh?? sin?? is harmonic
function, find its harmonic conjugate ?? (?? , ?? ).

07
Q.2 (a) Determine the Mobius transformation which maps ?? 1
= 0, ?? 2
= 1, ?? 3
= ?
into ?? 1
= ?1, ?? 2
= ??? , ?? 3
= 1.
03
(b)
Define ???????? , prove that ?? ?? = ?? ?(4?? +1)
?? 2
.
04
(c)
Expand ?? (?? ) =
1
(?? ?1)(?? +2)
valid for the region
(i) |?? | < 1 (ii) 1 < |?? | < 2 (iii) |?? | > 2 .
07
OR
(c)
Find the image of the infinite strips (i)
1
4
? ?? ?
1
2
(ii) 0 < ?? <
1
2
under the
transformation =
1
?? . Show the region graphically.
07
Q.3 (a) Evaluate ?
(?? ? ?? + ?? ?? 2
)????
?? where ?? is a straight line from ?? = 0 to ?? =
1 + ?? .
03
(b) Check whether the following functions are analytic or not at any point,
(i) ?? (?? ) = 3?? + ?? + ?? (3?? ? ?? ) (ii) ?? (?? ) = ?? 3
2
?
.
04
(c)
Using residue theorem, evaluate ?
????
(?? 2
+1)
2
?
0
.
07
OR
Q.3 (a)
Expand Laurent series of ?? (?? ) =
1??? ?? ?? at ?? = 0 and identify the
singularity.
03
(b) If ?? (?? ) = ?? + ???? , is an analytic function , prove that
(
?? 2
?? ?? 2
+
?? 2
?? ?? 2
) |?????? (?? )|
2
= 2|?? ?(?? )|
2
.
04
(c) Evaluate the following:
i. ?
?? +3
?? ?1

?? ???? where ?? is the circle (a) |?? | = 2 (b) |?? | =
1
2
.
ii. ?
sin ?? (?? ?
?? 4
)
3

?? ???? where ?? is the circle |?? | = 1 .
07

********
Q.4 (a)
Evaluate ? ???? (?? )???? 2+4?? 0
along the curve ?? (?? ) = ?? + ?? ?? 2
.
03
(b) Solve ?? 2
?? + ?? 2
?? = (?? + ?? )?? . 04
(c)
Solve the equation
????
????
= ?? ?? 2
?? ?? ?? 2
for the condition of heat along rod without
radiation subject to the conditions (i)
????
????
= 0 for ?? = 0 and ?? = ?? ;
(ii) ?? = ???? ? ?? 2
at ?? = 0 for all ?? .
07
OR
Q.4 (a)
Solve
?? 2
?? ?? ?? 2
+ 2
?? 2
?? ???????? +
?? 2
?? ?? ?? 2

= ?? 2?? +3?? .
03
(b) Solve ???? + ???? = ???? using Charpit?s method. 04
(c) Find the general solution of partial differential equation ?? ????
= 9?? ?? using
method of separation of variables.
07
Q.5 (a)
Using method of separation of variables, solve
????
????
= 2
????
????
+ ?? .
03
(b) Solve ?? (???? ? ???? ) = ?? 2
? ?? 2
. 04
(c) A string of length ?? = ?? has its ends fixed at ?? = 0 and ?? = ?? . At time ?? =
0, the string is given a shape defined by ?? (?? ) = 50?? (?? ? ?? ) , then it is
released. Find the deflection of the string at any time t.
07
OR
Q.5 (a) Solve ?? 3
+ ?? 3
= ?? + ?? . 03
(b) Find the temperature in the thin metal rod of length ?? with both the ends
insulated and initial temperature is sin
????
?? ?
.
04
(c) Derive the one dimensional wave equation that governs small vibration of an
elastic string . Also state physical assumptions that you make for the system.
07
FirstRanker.com - FirstRanker's Choice