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

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

Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ?III (Old) EXAMINATION ? WINTER 2019
Subject Code: 130001 Date: 22/11/2019

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 Fourier series to represent
2
2
) (
?
?
?
?
?
? ?
?
x
x f
?
in the interval 0 < x < ? 2
.


07
(b) Using Laplace Transform solve the given IVP.
1 ) 0 ( ' , 1 ) 0 ( , ' ' ? ? ? ? y y t y y .
07

Q.2 (a) Solve by Method of Variation of Parameters. . 2 sec 4 ' ' x y y ? ?

07
(b)
Solve 1 4 3 36 ' ) 2 3 ( 3 ' ' ) 2 3 (
2 2
? ? ? ? ? ? ? x x y y x y x .
07
OR
(b) Obtain series solution of . 0 ' ' ? ?xy y 07

Q.3 (a)
(1) Solve linear differential equation
2
2 2
x
e xy
dx
dy
?
? ? .

03

(2) Solve . ) (log log
2
2
y
x
y
x
x
y
dx
dy
? ?


04
(b) Using method of undermined co-efficient method to solve
. 3 25 10 ' 2 ' '
2
? ? ? ? x y y y
07
OR
Q.3 (a)
(1) Solve 0 ) 1 ( ) 1 (
2 2 2 2
? ? ? ? ? ? xdy xy y x ydx xy y x .
03

(2) Solve 0 ) ( ) 1 (
1
tan 2
? ? ? ?
?
dy e x dx y
y
.

04
(b) Using method of undermined co-efficient method to
solve x e x y y
x
2 sin 9 ' '
2
? ? ? ? .
07

Q.4
(a) (1) Find the inverse Laplace Transform of
) 3 )( 2 )( 1 (
16 3 5
2
? ? ?
? ?
s s s
s s
.
03



(2)Using convolution theorem, find the inverse Laplace transform of
2 2
) 5 4 (
2
? ?
?
s s
s
.




04
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Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ?III (Old) EXAMINATION ? WINTER 2019
Subject Code: 130001 Date: 22/11/2019

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 Fourier series to represent
2
2
) (
?
?
?
?
?
? ?
?
x
x f
?
in the interval 0 < x < ? 2
.


07
(b) Using Laplace Transform solve the given IVP.
1 ) 0 ( ' , 1 ) 0 ( , ' ' ? ? ? ? y y t y y .
07

Q.2 (a) Solve by Method of Variation of Parameters. . 2 sec 4 ' ' x y y ? ?

07
(b)
Solve 1 4 3 36 ' ) 2 3 ( 3 ' ' ) 2 3 (
2 2
? ? ? ? ? ? ? x x y y x y x .
07
OR
(b) Obtain series solution of . 0 ' ' ? ?xy y 07

Q.3 (a)
(1) Solve linear differential equation
2
2 2
x
e xy
dx
dy
?
? ? .

03

(2) Solve . ) (log log
2
2
y
x
y
x
x
y
dx
dy
? ?


04
(b) Using method of undermined co-efficient method to solve
. 3 25 10 ' 2 ' '
2
? ? ? ? x y y y
07
OR
Q.3 (a)
(1) Solve 0 ) 1 ( ) 1 (
2 2 2 2
? ? ? ? ? ? xdy xy y x ydx xy y x .
03

(2) Solve 0 ) ( ) 1 (
1
tan 2
? ? ? ?
?
dy e x dx y
y
.

04
(b) Using method of undermined co-efficient method to
solve x e x y y
x
2 sin 9 ' '
2
? ? ? ? .
07

Q.4
(a) (1) Find the inverse Laplace Transform of
) 3 )( 2 )( 1 (
16 3 5
2
? ? ?
? ?
s s s
s s
.
03



(2)Using convolution theorem, find the inverse Laplace transform of
2 2
) 5 4 (
2
? ?
?
s s
s
.




04

(b)
(1) Find the Fourier Transform of the function
2
) (
ax
e x f
?
? .
04

(2) Evaluate dx e x
ax n
?
?
?
0
.
03
Q.4 (a)
(1) By using first shifting theorem, obtain the value of } ) 1 {(
2 t
e t L ? .

03

(2) Find
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
b s
a s
L log
1
.


04
(b)
(1)Find the Fourier Cosine transform of the function
?
?
?
?
? ?
?
a x
a x k
x f
0
0
) (
.

04



(2) Evaluate
?
2
0
cot
?
? ?d .
03

Q.5
(a)
Prove that (1) x
x
x J sin
2
) (
2
1
?
? (2) x
x
x J cos
2
) (
2
1
?
?
?



07
(b) A taut string of length l has its ends 0 ? x and l x ? fixed. The mid-point is
stretched to a small height and released from rest at time 0 ? t . Find the
displacement ) , ( t x u .
07
OR

Q.5
(a)
Show that
?
?
?
1
1
, 0 ) ( ) ( dx x P x P
n m
if n m ? and
?
?
?
?
?
1
1
2
....) 1 , 0 (
1 2
2
) ( n
n
dx x P
n
if
. n m ? (m, n being integers)


07
(b) Solve 0 ? ?
yy xx
u u which satisfies the boundary condition o y a u y u ? ? ) , ( ) , 0 (
for b y ? ? 0 and ) ( ) 0 , ( , 0 ) , ( x f x u b x u ? ? for . 0 a x ? ?
07

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