DEPARTMENT OF MATHEMATICS

B. TECH END SEMESTER EXAMINATIONS, JUNE 2024

II SEMESTER (COMMON TO ALL BRANCHES)
ENGINEERING MATHEMATICS – II (U23MATC02)
Answer Key
Q. Marks
Key Points
No. Allotted
PART A (10 2 = 20 Marks)
Answer all the Questions
Define Fourier series.
Solution:
If f(x) is a periodic function and satisfies Dirichlet’s condition,then it can be represented by an
Q.1 a ∞ 2 Marks
infinite series is called Fourier series, f(x) = 0 +∑ ( an cosnx+b n sin nx ) , where a 0 , an , ∧b nare called
2 n =1
Fourier coefficients.
Find b n in the expansion of x 2 as a Fourier series in the interval (−π , π ¿ .

Solution:
2
Given f(x) = x
Q.2 2 Marks
f ( x) ( x) 2  x 2  f ( x)
 f(x) is even function

 The Fourier coefficient bn = 0 .

State first shifting property of fourier transforms.
Solution:
Q.3 F[f(x-a)] = e ias F( s) 2 Marks
F[e iax f (x )¿=F [s+ a]

State change of scale property of Fourier Transforms.

Solution:
For any non zero real a, (i.e. a>0 )
(i) F[f(ax)] = F
1
a
s
a ()
Q.4 2 Marks
(ii)
1
F s [ f ( ax ) ] = F s
a
s
a ()
(iii) F c [ f ( ax ) ] =
1
a
Fc()
s
a

Q.5 Find the Laplace transform of e−t sin 2t 2 Marks

Solution:
L¿ sin 2t ) = ¿ ¿
 2
=
( s +1 )2+ 4

State Convolution theorem of Laplace transforms.

Solution:
Q.6 2 Marks
If f(t) and g(t) are functions defined for t≥ 0 , then L[f(t) *g(t)] = L[f(t)] L[g(t)]

Find L
−1
( s +16
1
)
2

Q.7
Solution: 2 Marks
L
−1
( 2
1
s +16
1
= sin 4 t
4 )
Find L
−1
( s+1
2
(s+ 1) −4 )
Q.8 −t −1
Solution:e L
−t
s
(s) −4 ( 2 ) 2 Marks

= e cos 2t
Find Z ( a n U(n) )
Solution:
∞
Z ( a n U(n) ) =∑ a z
n −n

n=0
Q.9 2 Marks
()
2
a a
=1+ + +…
z z
z
=
z−a

Q.10
Find Z
−1
( z +3z ) . 2 Marks
Solution: Z (
z +3 )
z
−1 n
=(−3 )
PART B (5 5 = 25 Marks)
Answer ALL the Questions.
Q.11 Find the cosine series of f(x) = x in ( 0 , π ). 5 Marks
Solution:
a ∞
f ( x )= 0 + ∑ an cos nx
2 n=1
π
a 0 = 2 ∫ f ( x ) dx
π 0
π
a n = 2 ∫ f ( x ) cos nx dx
π 0
π
a 0 = 2 ∫ f ( x ) dx
π 0
π
2
= ∫ x dx
π 0
 =π
π
a n = 2 ∫ f ( x ) cos nx dx
π 0
π
2
=
π 0
∫ x cos nx dx
2
= 2 [ (−1 ) −1 ]
n

n π
a ∞
f ( x )= 0 + ∑ an cos nx
2 n=1
∞
π 2
¿ + ∑ 2 [ (−1 ) −1 ] cos nx
n

2 n=1 n π

Find the Fourier transform of f(x) given by f(x) given by {

f ( x )= 1 for|x| ≤a
0 for|x|≥ a

Solution:

The Fourier transform of is given by

Here,

=
Q.12 5 Marks

= =

= =

Q.13 5 Marks

(e )
−at −bt
−e
Find L
t
Solution:
 [ ]
∞
f (t )
L =∫ F ( s ) ds
t s

( )=∫ L (e
−at −bt ∞
e −e −at −bt
) ds
L −e
t s

∫ [ s+1 a − s+1 b ] ds
∞
=
s
∞
= [ log ( s+ a )−log (s +b) ] s

= log 1- log ( s+s+ ab )

= log (
s+ a )
s+ b

Using Convolution theorem find L

−1

(( s
s +1 )
2 2
) .

Solution:
L
−1

( s
( s +1 )
2 2
=L
) (
−1
2
1
. 2
s
s +1 s +1 )
¿ L−1 2
1
s +1 [ ] [ ]
∗L−1 2
1
s +1
¿ sint∗cost
t

∫ f ( u ) g ( t−u ) du=f ( t )∗g (t)

Q.14 0
t 5 Marks
=∫ sin u cos ( t−u ) du
0
t
1
¿ ∫ ¿¿
20

¿
1
2 [
t sint−
cost cos ⁡(−t)
2
+
2 ]
t sint
¿
2

Q.15 5 Marks

( z ( z −1 )
)
2
−1
Evaluate Z 2
( z 2+ 1 )
Solution:
z ( z 2−1 )
Let u ( z )= 2
( z 2 +1 )
u ( z ) ( z 2−1 )
= 2 2
z ( z + 1)
( z −1 ) Az+ B Cz+ D
2

2
= 2 + 2
( z 2+1 ) z +1 ( z 2 +1 )
 ( z 2−1 ) =( Az + B ) ( z2 +1 ) +Cz + D
A=0 , C= 0, D= -2 , B=1
u (z ) 1 2
= 2 −
z z +1 ( z +1 )
2 2

u ( n ) =z
−1
[ ] [
2
z
z +1
−z
−1 2z
( z 2 +1 )
2
]
nπ sin π (n−1)
¿ sin −2 n
2 2

PART C (3 10 = 30 Marks)
Answer any THREE Questions
Q.16 Find the Fourier series of f(x) = x ( π−x ¿ in (0, 2 π ¿ 10 Marks
Solution:
a0 ∞
f ( x )= + ∑ ( an cos nx +b n sin nx )
2 n=1
2π
a 0 = 1 ∫ f ( x ) dx
π 0
2π
1
= ∫ x ( π−x ) dx
π 0
2
a0 = - 2 π
3
2π
a n = 1 ∫ f ( x ) cos nx dx
π 0

2π
1
= ∫ x ( π−x ) cos nx dx
π 0

−4
a n= 2
n
2π
b n = 1 ∫ f ( x ) sin nx dx
π 0

2π
1
= ∫ x ( π−x ) sin nx dx
π 0

2π
b n=
n

a0 ∞
f ( x )= + ∑ ( a cos nx +b n sin nx )
2 n=1 n
 ∞
−π 2
f ( x )= +∑ ¿ ¿
3 n=1
∞ 2
x dx
Evaluate ∫ using fourier transforms.
0 ( a + x 2 ) ( b 2+ x 2 )
2

Solution:
Parseval’s identity is
∞ ∞

∫ f ( x ) g ( x ) dx =∫ F s [ f ( x ) ] F s [g ( x ) ]ds---------------------------- (1)
0 0
−ax
Let f ( x )=e

√
∞
2
F s [ f ( x ) ]= ∫ f ( x ) sinsx dx
π 0

√
∞
2
¿ ∫ e−ax sin sx dx
π 0

¿
√[ ] 2 s
π s + a2
2

Let g ( x )=e−bx

√
∞
2
F s [ f ( x ) ]= ∫ f ( x ) sinsx dx
π 0

√
∞
2
Q.17 ¿ ∫ e−bx sin sx dx
π 0
10 Marks

¿
√[ ]
2
π s +b 2
2
s

−ax −bx − ( a+b ) x

f ( x ) g ( x ) =e e =e
∞
=∫ f ( x ) g ( x ) dx
0
∞
¿∫ e
− ( a+b ) x
dx
0
1
=
a+b
∞ 2
1 2 s
Eqn (1) becomes = ∫ 2 2 ds
a+b π 0 ( s +a ) + ( s 2 +b2 )
∞ 2
∫ 2 2 s 2 2 ds= 2(a+b)
π
0 ( s +a )+ ( s +b )
∞ 2
x dx π
S is a dummy variable → ∫ 2 2 2 2 =
0 (a + x ) (b + x ) 2(a+ b)
 {
T
E , 0 ≤t ≤
2
Find the Laplace transform of f(t) = and f (t+T) = f(t)
T
−E , ≤ t ≤ T
2
Solution:
Laplace transform of the periodic function is given by

T
1
−sT ∫
L [ f ( t ) ]=
−st
e f ( t ) dt
1−e 0
Period T= T

[ ]
T
2 T
1
Q.18 ¿
1−e
−sT ∫e −st
E dt +∫ e (−E)dt
−st 10 Marks
0 T
2

[( ) ( )]
− sT T −sT T
E e 2 e
¿ −sT
−
1−e −s 0 −s T
2

[ ]
− sT
E 1
¿ −sT
1+ e−sT −2e 2
s 1−e

=
E
s
tanh
sT
4 ( )
Q.19 Solve y ' ' −3 y ' +2 y=4 t +e 3t , y ( 0 ) =1 , y ' ( 0 )=−1 by Laplace transform method. 10 Marks
Solution:
Given :
'' ' 3t
y −3 y +2 y=4 t +e

Taking laplace transform on both sides we get,

' 3t
L( y ¿¿ ' ' )−3 L( y )+2 L( y)=4 L(t )+ L( e ) ¿
¿
¿
4 1
L y ( t ) [ s −3 s +2 ] = 2 +
2
+ s−4
s s−3
2 3 2
4 ( s−3 ) + s + s ( s−3 )−4 s (s−3)
L y ( t )=
s 2 ( s−3 ) ( s−1 ) (s−2)
2 3 2
4 ( s−3 ) + s + s ( s−3 )−4 s (s−3) A B C D E
2
= + 2+ + +
s ( s−3 )( s−1 ) (s−2) s s s−3 s−1 s−2

1 1
A= 3, B= 2, C= , D = - , E = -2
2 2

( ) ( )
1 −1
−1
y ( t ) =L ( 3s )+ L ( s2 )+ L
−1
2
−1 2
s−3
+L
−1 2
s−1
+L
−1 −2
s−2 ( )
 1 3t 1 t 2t
y ( t ) =3+2 t+ e − e −2 e
2 2

Solve y n +2−4 y n+1 +4 y n=0 , y 1= 0 and y 0=1 , by using Z−transform .

Solution:
Given : y n +2−4 y n+1 +4 y n=0
Taking Z transform on both sides we get
Z( y n +2 ¿−4 Z ¿)
[ z y ( z )−z y ( 0 )−z y ( 1 ) ]−4 [ z y ( z )−z y ( 0 ) ]+ 4 y ( z )=0
2 2

[ z 2 y ( z )−z 2−0 ]−4 [ z y ( z )−z ]+ 4 y ( z )=0

y ( z ) [ z 2−4 z+ 4 ] =z 2−4 z
y (z) z−4
=
z ( z−2 )2
z−4 A B
= +
Q.20 ( z−2 ) z−2 ( z−2 )2
2

A= 1; B= -2
y (z) 1 −2
= +
z z−2 ( z−2 )2

y ( n )=z ( ) (
−1 z
z−2
−z
−1
) 2z
( z−2 )2
z az
z ( a )= ; z (n a )=
n n
z−a ( z −a )2
n
y ( n )=2 - 2n n

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Staff Incharge HOD Dean Academics

(M. Guna) (Dr. T. Gayathri) (Dr. S. Anbumalar)