QP Code: 24/U/T/314
Undergraduate Degree Programme ( Under CBCS )
Term End (Theory) Examination June-2024 (SEM-VI)
MATHEMATICS : DSEC
Discipline Specific
Integral Transform
DS-MT-41
Time : 2 hours Full Marks : 50
š[¹[³t¡ * ™=à™= l¡üv¡ì¹¹ \>¸ [¤ìÅÈ ³èº¸ ëƒ*Úà Òì¤ ¡ú "Ç¡‡ý¡ ¤à>à>, "š[¹ZáÄt¡à &¤} "š[¹ÍHþà¹
ÒÑzàÛ¡ì¹¹ ëÛ¡ìy >´¬¹ ëA¡ìi¡ ë>*Úà Òì¤ ¡ú l¡üšàì”z šøìÅ¥¹ ³èº¸³à> Îè[W¡t¡ "àìá ¡ú
Scientific calculator ¤¸¤Ò๠A¡¹à A¡ìk¡à¹®¡àì¤ [>[ȇý¡¡ú
Special credit will be given for precise and correct answer. Marks will be
deducted for spelling mistakes, untidiness and illegible handwriting. The
figures in the margin indicate full marks.
Use of scientific calculator is strictly prohibited.
[¤®¡àK - A¡
GROUP - A
1. ë™ ëA¡àì>à W¡à¹[i¡ šøìÅ¥¹ l¡üv¡¹ [ƒ> –
Answer any four of the following questions : 3 4 = 12
a) [ , ] "”z¹àìº f "ìšÛ¡ìA¡¹ Fourier series [i¡ [>o¢Ú A¡¹ç¡>, ë™Jàì>
f ( x ) x , x ¡ú
Find the Fourier series of the function f defined by
f ( x ) x , x on [ , ] .
b) &A¡[i¡ "ìšÛ¡A¡ f ( x ) -&¹ Fourier cosine series Òº f (x ) fn cosn x ú
n 0
f ( x ) cos 4 x Òìº, ( f f ) -&¹ ³à> [>o¢Ú A¡¹ç¡>¡ú
4 5
The Fourier cosine series of a function f ( x ) is given by
f (x ) f cos n x .
n
n 0
If f ( x ) cos 4 x , then find the value of ( f f ) .
4 5
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QP Code: 24/U/T/314 2
1 2
x
c) e 2 -&¹ Fourier transform [>o¢Ú A¡¹ç¡> ¡ú
1 2
x
Find the Fourier transform of e 2 .
d) f (t ) -&¹ Fourier transform [>o¢Ú A¡¹ç¡>, ë™Jàì>
1, 0 t l
f (t )
0, t l
Find the Fourier transform of f (t ) , where
1, 0 t l
f (t )
0, t l
4t
e) f (t ) e cos h ( 2t ) -&¹ Laplace transform [>o¢Ú A¡¹ç¡>¡ú
Find the Laplace transform of f (t ) e 4t cos h ( 2t )
sin ( 2t )
f) f (t ) -&¹ Laplace transform [>o¢Ú A¡¹ç¡>¡ú
t
sin ( 2t )
Find the Laplace transform of f ( t ) .
t
1
g) F ( p ) -&¹ inverse Laplace transform [>o¢Ú A¡¹ç¡>¡ú
p ( p2 1 )
1
Find the inverse Laplace transform of F ( p ) .
p ( p2 1 )
1
e p
h) Z -&¹ ³à> [>o¢Ú A¡¹ç¡>¡ ë™Jàì> Z 1 inverse Laplace
( p 4 )2
transform operator Îè[W¡t¡ A¡ì¹ ú
e p
1
Find Z , where Z 1 denotes the inverse Laplace
( p 4 )2
transform operator.
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3 QP Code: 24/U/T/314
[¤®¡àK - J
GROUP - B
2. ë™ ëA¡àì>à [t¡>[i¡ šøìÅ¥¹ l¡üv¡¹ [ƒ> –
Answer any three of the following questions : 6 3 = 18
a) [ , ] "”z¹àìº f (x ) -&¹ Fourier series [>o¢Ú A¡¹ç¡>, ë™Jàì>
x, x 0
f ( x )
x, 0 x
2 1 1
t¡à¹š¹ ëƒJà> ë™ 1 2 2 ...
8 3 5
Find the Fourier series of f ( x ) on [ , ] , where
x, x 0
f ( x )
x, 0 x
2 1 1
Hence deduce that 1 2 2 ... 4+2
8 3 5
b) ™[ƒ "ìšÛ¡A¡ f (x ) x , 0 x 2 -&¹ Fourier series [¤Ñzõ[t¡
4 ( 1 )n 1 n x
x
n
sin
2
ÒÚ, t¡àÒìº Parseval's identity ¤¸¤Ò๠A¡ì¹
n 1
1 2
ëƒJà> ë™ 2
¡ú
8
n 0 ( 2n 1 )
If the function f (x ) x , 0 x 2 has the Fourier series expansion
4 ( 1 )n 1 n x
x
n
sin
2
,
n 1
then, using Parseval's identity, deduce that
1 2
2
8
.
n 0 ( 2n 1 )
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QP Code: 24/U/T/314 4
c) ëA¡àì>à "ìšÛ¡A¡ f (t ) -&¹ Fourier transform F { f ( t ) } F ( ) ‡à¹à Îè[W¡t¡
Òìº &¤} ' a ' &A¡[i¡ ë™ìA¡àì>à ¤àÑz¤ Î}J¸à Òìº, šø³ào A¡¹ç¡> ë™
1
(i) F { f ( t ) cos at } [ F ( a ) F ( a )]
2
1
(ii) F { f ( t ) sin at } [ F ( a ) F ( a ) ]
2
If Fourier transform of f ( t ) is denoted by F { f ( t ) } F ( ) and ' a ' is
any real number, then prove that
1
(i) F { f ( t ) cos at } [ F ( a ) F ( a )]
2
1
(ii) F { f ( t ) sin at } [ F ( a ) F ( a ) ] 3+3
2
d) "ìšÛ¡A¡ f -&¹ Fourier transform [>o¢Ú A¡¹ç¡>, ë™Jàì>
1 x 2, | x | 1
f (x )
0, | x | 1
&¤} t¡à¹š¹ ëƒJà> ë™
x cos x sin x x 3
0 x 3
cos dx
2 16
Find the Fourier transform of the function f defined by
1 x 2, | x | 1
f (x )
0, | x | 1
and hence show that
x cos x sin x x 3
0 x 3
cos dx
2 16
. 3+3
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5 QP Code: 24/U/T/314
1
1
e) Convolution theorem ¤¸¤Ò๠A¡ì¹ L [>o¢Ú A¡¹ç¡>¡ú
p ( p 1 )3
Using convolution theorem, find L1
1
.
p ( p 1 )3
f) f ( t ) t 2 sin2 ( 3t ) -&¹ Laplace transform [>o¢Ú A¡¹ç¡>¡ú
Find the Laplace transform of f ( t ) t 2 sin2 ( 3t ) .
[¤®¡àK - K
GROUP - C
3. ë™ ëA¡àì>à ƒå[i¡ šøìÅ¥¹ l¡üv¡¹ [ƒ> –
Answer any two of the following questions : 10 2 = 20
a) 2 š™¢àÚ [¤[ÅÊ¡ &A¡[i¡ š™¢àÚyû¡[³A¡ "ìšÛ¡A¡ f-&¹ Fourier series
0, x a
[>o¢Ú A¡¹ç¡>, ë™Jàì> f (x ) 7, a x b
0, b x
x 4 a [¤@ƒåìt¡ ëÅøoã[i¡¹ ë™àKó¡º [>o¢Ú A¡¹ç¡> &¤} t¡à¹š¹ ëƒJà> 뙡
sin [ n ( b a ) ] b a
n
2
ú
n 1
Find the Fourier series of the periodic function f with period 2,
where
0, x a
f (x ) 7, a x b
0, b x
Find the sum of the series at x 4 a hence deduce that
sin [ n ( b a ) ] b a
n
2
. 5+2+3
n 1
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QP Code: 24/U/T/314 6
b) Fourier transform ¤¸¤Ò๠A¡ì¹ [>ì´•¹ t¡àšãÚ Î³ãA¡¹o ( Åt¢¡ÎÒ ) γà‹à> A¡¹ç¡> –
u 2u
2 , 0 x , t 0
t x 2
Åt¢¡ ÎàìšìÛ¡ –
(i) u ( 0, t ) 0, t 0
(ii) u ( x , 0 ) e x , 0 x
(iii) u ( x, t ) "इý¡¡ú
Use Fourier transform to solve the heat equation
u 2u
2 , 0 x , t 0
t x 2
With the conditions :
(i) u ( 0, t ) 0, t 0
x
(ii) u ( x, 0 ) e , 0x
(iii) u ( x , t ) is bounded.
c) Laplace transform ¤¸¤Ò๠A¡ì¹ [>ì´•¹ šøà=[³A¡ ³à> γθà[i¡ γà‹à> A¡¹ç¡> –
/
/
t
y ( t ) 3y ( t ) 2y (t ) e ; y (1 ) 0, y (1 ) 0
Using Laplace transform, solve the following initial value problem :
/
/
y ( t ) 3y ( t ) 2y (t ) e t ; y (1 ) 0, y (1 ) 0 .
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7 QP Code: 24/U/T/314
d) f "ìšÛ¡ìA¡¹ Fourier transform [>o¢Ú A¡¹ç¡>, ë™Jàì>
0, t 0
f ( t ) at
e , t 0 (a 0 )
t¡à¹š¹ convolution theorem-&¹ ÎàÒà외
1
1
F [>o¢Ú A¡¹ç¡>¡ú
12 7 i 2
Find the Fourier transform of the function f, where
0, t 0
f ( t ) at
e , t 0 (a 0 )
Hence using convolution theorem, find
1
1
F . 4+6
12 7 i 2
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