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
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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
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