Download GTU (Gujarat Technological University) BE/BTech (Bachelor of Engineering / Bachelor of Technology) 2018 Winter 3rd Sem Old 130001 Mathematics Iii Previous Question Paper
1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER ?III (OLD) EXAMINATION ? WINTER 2018
Subject Code:130001 Date:17/11/2018
Subject Name:Mathematics-III
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)
(1) Solve ) 1 2 ( ) 2 ( ) 1 (
3
? ? ? ? ? x x x
dx
dy
x x
03
(2) Solve 0 ) 2 ( ) 2 (
2 2 2 2
? ? ? ? xdy y x dx y y x
04
(b)
Find the power series solution of the equation 0 2 4
2
2
? ? ? y
dx
dy
dx
y d
x about
0 ? x .
07
Q.2 (a)
(1) Solve x e y D
x
2 sin ) 4 (
2
? ? ? .
(2) Solve
x
x
e
e
y
dx
dy
dx
y d
?
? ? ?
1
2 3
2
2
by using method of variation of parameter.
03
04
(b)
Solve
x
e y
dx
dy
x
dx
y d
x ? ? ? 2 4
2
2
2
07
OR
(b)
Find series solution of the differential equation 0 ) 1 (
2
? ? ? ? ? ? ? y y x y x .
07
Q.3
(a) (1) Show that
?
? ?
1
0
3 2
60
1
) 1 ( dx x x .
(2) Prove that ? ?
) ( 1
) (
x n
n
n
n
J x x J x
dx
d
?
?
03
04
(b)
Find the half range cosine series for
?
?
?
? ? ?
? ?
?
l x l x l k
l x kx
x f
2 / ; ) (
2 / 0 ;
) (
Also Prove that
8
) 1 2 (
1
2
1
2
?
?
?
?
?
? n n
07
OR
Q.3 (a) Find the Fourier series of x x x f sin ) ( ? in the interval ) , ( ? ? ? . Hence, deduce that
. . .
7 5
1
5 3
1
3 1
1
4
1
?
?
?
?
?
?
?
? ?
07
(b)
(1) Evaluate
?
1
0
5 5
) (log dx x x
(2) Find half range Fourier sine series of the function x x f ? ? ? ) ( for ? ? ? x 0 .
03
04
Q.4 (a)
(1) Find the Laplace transform of the function t t f sin ) ( ? .
03
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1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER ?III (OLD) EXAMINATION ? WINTER 2018
Subject Code:130001 Date:17/11/2018
Subject Name:Mathematics-III
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)
(1) Solve ) 1 2 ( ) 2 ( ) 1 (
3
? ? ? ? ? x x x
dx
dy
x x
03
(2) Solve 0 ) 2 ( ) 2 (
2 2 2 2
? ? ? ? xdy y x dx y y x
04
(b)
Find the power series solution of the equation 0 2 4
2
2
? ? ? y
dx
dy
dx
y d
x about
0 ? x .
07
Q.2 (a)
(1) Solve x e y D
x
2 sin ) 4 (
2
? ? ? .
(2) Solve
x
x
e
e
y
dx
dy
dx
y d
?
? ? ?
1
2 3
2
2
by using method of variation of parameter.
03
04
(b)
Solve
x
e y
dx
dy
x
dx
y d
x ? ? ? 2 4
2
2
2
07
OR
(b)
Find series solution of the differential equation 0 ) 1 (
2
? ? ? ? ? ? ? y y x y x .
07
Q.3
(a) (1) Show that
?
? ?
1
0
3 2
60
1
) 1 ( dx x x .
(2) Prove that ? ?
) ( 1
) (
x n
n
n
n
J x x J x
dx
d
?
?
03
04
(b)
Find the half range cosine series for
?
?
?
? ? ?
? ?
?
l x l x l k
l x kx
x f
2 / ; ) (
2 / 0 ;
) (
Also Prove that
8
) 1 2 (
1
2
1
2
?
?
?
?
?
? n n
07
OR
Q.3 (a) Find the Fourier series of x x x f sin ) ( ? in the interval ) , ( ? ? ? . Hence, deduce that
. . .
7 5
1
5 3
1
3 1
1
4
1
?
?
?
?
?
?
?
? ?
07
(b)
(1) Evaluate
?
1
0
5 5
) (log dx x x
(2) Find half range Fourier sine series of the function x x f ? ? ? ) ( for ? ? ? x 0 .
03
04
Q.4 (a)
(1) Find the Laplace transform of the function t t f sin ) ( ? .
03
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2
(2) Find the inverse Laplace transform of the function
20 4
4 6
) (
2
? ?
?
?
s s
s
s F .
04
(b) Solve the differential equation using Laplace Transformation method
t t y
dt
y d
cos
2
2
? ? , Given that 0 ) 0 ( , 0 ) 0 ( ? ? ? y y , . 0 ? t
07
OR
Q.4 (a) (1) Find the Laplace transform of the function t t t f cos ) ( ?
(2) Find the inverse Laplace transform of the function ?
?
?
?
?
?
? ?
2
1
1 log ) (
s
s F
03
04
(b) Define Convolution theorem for Laplace transform. Using Convolution
theorem to find Laplace inverse of the function
? ? ? ?
2 2 2 2
2
) (
b s a s
s
s F
? ?
?
07
Q.5 (a)
(1) Form the partial differential equation of ? ? 0 ,
2
? ? ? ? z y x z xy f .
(2) Solve y x q z x p z y ? ? ? ? ? ) ( ) ( .
03
04
(b)
Solve by the method of separation of variables 0 2
2
2
?
?
?
?
?
?
?
?
?
y
z
x
z
x
z
07
OR
Q.5 (a)
Using method of separation of variables, solve , 3 4 u
y
u
x
u
?
?
?
?
?
?
given that
y y
e e y u
5
3 ) , 0 (
? ?
? ? .
07
(b)
(1) Solve ? ? 0 2 1
2 2
? ? ? ? dy xy dx y x
(2) Solve 8 4 ) 2 3 (
2 2 3
? ? ? ? ? x x y D D D by using method of undetermined
coefficients.
03
04
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This post was last modified on 20 February 2020