Download GTU BE/B.Tech 2019 Summer 3rd Sem Old 130001 Mathematics Iii Question Paper

Download GTU (Gujarat Technological University) BE/BTech (Bachelor of Engineering / Bachelor of Technology) 2019 Summer 3rd Sem Old 130001 Mathematics Iii Previous Question Paper

1
Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ? III(OLD) EXAMINATION ? SUMMER 2019
Subject Code: 130001 Date:30/05/2019

Subject Name: Mathematics-III

Time: 02:30 PM TO 05:30 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)
Obtain series solution of 0
2
2
? ? y
dx
y d
.
07
(b) Attempt any two of the following. 07
1) ? ?
y y y
x x
dx
dy
cos sin
1 log 2
?
?
?

2) 2 2
y x ydx xdy ? ? ?

3)
x x y
dx
dy
cos cot ? ?

4)
? ? p xp y ? ? sin . Where,
dx
dy
p ? .


Q.2 (a)
Obtain the Frobenius series solution of ? ? 0 1 3 2
2 ' " 2
? ? ? ? y x xy y x .
07
(b) Attempt any two of the following. 07
1)
0 36 13
2
2
4
4
? ? ? y
dx
y d
dx
y d


2)
? ? 5 2 3
2
? ? ? y D D .

3)
x y y y 2 cos 2
' "
? ? ?

4)
Solve by Method of variation of parameters. ax y a y tan
2 "
? ? .

OR
(b) Attempt any two of the following. 07
1)
. log 3
2
2
2
2
x x y
dx
dy
x
dx
y d
x ? ? ?

2)
Using method of undetermined multipliers solve
2 "
8 4 x y y ? ? .

3)
Using method of undetermined multipliers solve
x
e y y y ? ? ? 2 3
' "
.

4)
Prove that
? ?
? ?
1
0
log
1
?
?
?
?
? c x
c
c
c ?
dx
c
x



Q.3 (a)
Define Laplace Transformation of a function ? ? t f and using it obtain ? ? at L sin
and ? ?
n
t L .
07
(b) Attempt any two of the following. 07
1)
Find the Laplace transform of t t t cosh 5 2 sin
3
? ? .

2)
Evaluate ? ? t t t L 3 sin 2 sin sin .

3)
Evaluate ? ? ? ? t t e L
t
4 sin 3 4 cos
3
?
?
.

4)
Evaluate
? ? ? ?
?
?
?
?
?
?
? ?
?
1 1
1
2
1
s s
L .

OR
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1
Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ? III(OLD) EXAMINATION ? SUMMER 2019
Subject Code: 130001 Date:30/05/2019

Subject Name: Mathematics-III

Time: 02:30 PM TO 05:30 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)
Obtain series solution of 0
2
2
? ? y
dx
y d
.
07
(b) Attempt any two of the following. 07
1) ? ?
y y y
x x
dx
dy
cos sin
1 log 2
?
?
?

2) 2 2
y x ydx xdy ? ? ?

3)
x x y
dx
dy
cos cot ? ?

4)
? ? p xp y ? ? sin . Where,
dx
dy
p ? .


Q.2 (a)
Obtain the Frobenius series solution of ? ? 0 1 3 2
2 ' " 2
? ? ? ? y x xy y x .
07
(b) Attempt any two of the following. 07
1)
0 36 13
2
2
4
4
? ? ? y
dx
y d
dx
y d


2)
? ? 5 2 3
2
? ? ? y D D .

3)
x y y y 2 cos 2
' "
? ? ?

4)
Solve by Method of variation of parameters. ax y a y tan
2 "
? ? .

OR
(b) Attempt any two of the following. 07
1)
. log 3
2
2
2
2
x x y
dx
dy
x
dx
y d
x ? ? ?

2)
Using method of undetermined multipliers solve
2 "
8 4 x y y ? ? .

3)
Using method of undetermined multipliers solve
x
e y y y ? ? ? 2 3
' "
.

4)
Prove that
? ?
? ?
1
0
log
1
?
?
?
?
? c x
c
c
c ?
dx
c
x



Q.3 (a)
Define Laplace Transformation of a function ? ? t f and using it obtain ? ? at L sin
and ? ?
n
t L .
07
(b) Attempt any two of the following. 07
1)
Find the Laplace transform of t t t cosh 5 2 sin
3
? ? .

2)
Evaluate ? ? t t t L 3 sin 2 sin sin .

3)
Evaluate ? ? ? ? t t e L
t
4 sin 3 4 cos
3
?
?
.

4)
Evaluate
? ? ? ?
?
?
?
?
?
?
? ?
?
1 1
1
2
1
s s
L .

OR
2
Q.3 (a) Define periodic function and obtain the Laplace transformation of periodic
function having fundamental period p ,
07
(b) Attempt any two of the following. 07
1)
Evaluate
?
?
?
?
?
? ?
t
t
L
2 cos 1
.

2)
Evaluate ? ?
t
e t L
3 3 ?
.

3)
Using convolution theorem evaluate
? ?
?
?
?
?
?
?
?
?
2 2 2
1
1
a s s
L .

4) Using Laplace transform technique solve the following IVP.
? ? ? ? 0 0 , 1 0 sin 4
' "
? ? ? ? y y t y y .


Q.4 (a)
Obtain Fourier series for the function ? ?
2
x x x f ? ? over ? ? ? ? ? x and hence
show that .
12
1
2
1
2
?
?
?
?
? n
n

07
(b) Attempt any one of the following. 07
1)
Obtain the Fourier series for the function ? ?
2
x x f ? , ? ? ? ? ? x . Hence
show that
6
1
2
1
2
?
?
?
?
? n
n
.

2)
Find the Fourier series to represent the function ? ? x f given by
? ?
?
?
?
? ?
? ?
?
. 2 x for x - 2
. x 0 for
? ? ?
? x
x f Hence show that
? ? 8 1 2
1
2
1
2
?
?
?
?
?
? n
n
.

3) Obtain half-range cosine series for the function
? ?
?
?
?
?
?
? ?
? ?
?
. x
2
for x -
.
2
x 0 for
?
?
?
?
x
x f

OR
Q.4 (a)
Expand ? ?
x
e x f
?
? as a Fourier series in the interval ? ? l l, ? .
07
(b) Attempt any two of the following. 07
1)
Express the following function as Fourier integral ? ?
?
?
?
?
?
?
?
?
1 x for 0
1 x for 1
x f ,
Hence evaluate a) ?
?
? ?
d
x
?
?
0

cos sin
and b) dx
x
x
?
?
0

sin
.

2)
Show that ? ? ? ? ? ?. x J x x J x
dx
d
n
n
n
n
?
?

3) Show that ? ? ? ? ? ? ? ? ? ? x nP x P n x xP n
n n n 1 1
1 1 2
? ?
? ? ? ? .

Q.5 (a) Solve the one-dimensional wave equation together with following initial &
boundary conditions.
2
2
2
2
2
x
u
c
t
u
?
?
?
?
?
. Where,
?
T
c ?
2
.
? ? ? ? 0 t , 0 , , 0 ? ? ? ? t l u t u
? ? ? ? ? ? ? ? l x x g x u x f x u
t
? ? ? ? ? 0 , 0 , and 0 ,
07
(b) Attempt any two of the following. 07
1) z yq xp 3 ? ?
2)
? ? ? ? ? ? mx ly q lz nx p ny mz ? ? ? ? ?

3)
? ? xz xyq p z y x 2 2
2 2 2
? ? ? ?

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1
Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ? III(OLD) EXAMINATION ? SUMMER 2019
Subject Code: 130001 Date:30/05/2019

Subject Name: Mathematics-III

Time: 02:30 PM TO 05:30 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)
Obtain series solution of 0
2
2
? ? y
dx
y d
.
07
(b) Attempt any two of the following. 07
1) ? ?
y y y
x x
dx
dy
cos sin
1 log 2
?
?
?

2) 2 2
y x ydx xdy ? ? ?

3)
x x y
dx
dy
cos cot ? ?

4)
? ? p xp y ? ? sin . Where,
dx
dy
p ? .


Q.2 (a)
Obtain the Frobenius series solution of ? ? 0 1 3 2
2 ' " 2
? ? ? ? y x xy y x .
07
(b) Attempt any two of the following. 07
1)
0 36 13
2
2
4
4
? ? ? y
dx
y d
dx
y d


2)
? ? 5 2 3
2
? ? ? y D D .

3)
x y y y 2 cos 2
' "
? ? ?

4)
Solve by Method of variation of parameters. ax y a y tan
2 "
? ? .

OR
(b) Attempt any two of the following. 07
1)
. log 3
2
2
2
2
x x y
dx
dy
x
dx
y d
x ? ? ?

2)
Using method of undetermined multipliers solve
2 "
8 4 x y y ? ? .

3)
Using method of undetermined multipliers solve
x
e y y y ? ? ? 2 3
' "
.

4)
Prove that
? ?
? ?
1
0
log
1
?
?
?
?
? c x
c
c
c ?
dx
c
x



Q.3 (a)
Define Laplace Transformation of a function ? ? t f and using it obtain ? ? at L sin
and ? ?
n
t L .
07
(b) Attempt any two of the following. 07
1)
Find the Laplace transform of t t t cosh 5 2 sin
3
? ? .

2)
Evaluate ? ? t t t L 3 sin 2 sin sin .

3)
Evaluate ? ? ? ? t t e L
t
4 sin 3 4 cos
3
?
?
.

4)
Evaluate
? ? ? ?
?
?
?
?
?
?
? ?
?
1 1
1
2
1
s s
L .

OR
2
Q.3 (a) Define periodic function and obtain the Laplace transformation of periodic
function having fundamental period p ,
07
(b) Attempt any two of the following. 07
1)
Evaluate
?
?
?
?
?
? ?
t
t
L
2 cos 1
.

2)
Evaluate ? ?
t
e t L
3 3 ?
.

3)
Using convolution theorem evaluate
? ?
?
?
?
?
?
?
?
?
2 2 2
1
1
a s s
L .

4) Using Laplace transform technique solve the following IVP.
? ? ? ? 0 0 , 1 0 sin 4
' "
? ? ? ? y y t y y .


Q.4 (a)
Obtain Fourier series for the function ? ?
2
x x x f ? ? over ? ? ? ? ? x and hence
show that .
12
1
2
1
2
?
?
?
?
? n
n

07
(b) Attempt any one of the following. 07
1)
Obtain the Fourier series for the function ? ?
2
x x f ? , ? ? ? ? ? x . Hence
show that
6
1
2
1
2
?
?
?
?
? n
n
.

2)
Find the Fourier series to represent the function ? ? x f given by
? ?
?
?
?
? ?
? ?
?
. 2 x for x - 2
. x 0 for
? ? ?
? x
x f Hence show that
? ? 8 1 2
1
2
1
2
?
?
?
?
?
? n
n
.

3) Obtain half-range cosine series for the function
? ?
?
?
?
?
?
? ?
? ?
?
. x
2
for x -
.
2
x 0 for
?
?
?
?
x
x f

OR
Q.4 (a)
Expand ? ?
x
e x f
?
? as a Fourier series in the interval ? ? l l, ? .
07
(b) Attempt any two of the following. 07
1)
Express the following function as Fourier integral ? ?
?
?
?
?
?
?
?
?
1 x for 0
1 x for 1
x f ,
Hence evaluate a) ?
?
? ?
d
x
?
?
0

cos sin
and b) dx
x
x
?
?
0

sin
.

2)
Show that ? ? ? ? ? ?. x J x x J x
dx
d
n
n
n
n
?
?

3) Show that ? ? ? ? ? ? ? ? ? ? x nP x P n x xP n
n n n 1 1
1 1 2
? ?
? ? ? ? .

Q.5 (a) Solve the one-dimensional wave equation together with following initial &
boundary conditions.
2
2
2
2
2
x
u
c
t
u
?
?
?
?
?
. Where,
?
T
c ?
2
.
? ? ? ? 0 t , 0 , , 0 ? ? ? ? t l u t u
? ? ? ? ? ? ? ? l x x g x u x f x u
t
? ? ? ? ? 0 , 0 , and 0 ,
07
(b) Attempt any two of the following. 07
1) z yq xp 3 ? ?
2)
? ? ? ? ? ? mx ly q lz nx p ny mz ? ? ? ? ?

3)
? ? xz xyq p z y x 2 2
2 2 2
? ? ? ?

3
OR

Q.5 (a) A homogenous rod of conducting material of length 100 cm. has its ends kept
at zero temperature and the temperature initially is
? ?
?
?
?
? ? ?
? ?
?
100 50 for 100
50 0 for
0 ,
x x
x x
x u . Find the temperature ? ? t x u , at any time t,
at a distance x.
07
(b) Attempt any two of the following. 07
1) 2 2
1 q p qy px z ? ? ? ? ?

2)
y x q p ? ? ?
2 2


3)
2 2
q p z ? ?


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