Download JNTUA B.Tech 1-2 R15 2017 Nov Supple 15A54201 Mathematics II Question Paper

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B.Tech I Year II Semester (R15) Supplementary Examinations November 2017
MATHEMATICS ? II
(Common to all)
Time: 3 hours Max. Marks: 70
PART ? A

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(Compulsory Question)

*****
1 Answer the following: (10 X 02 = 20 Marks)

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(a)
Find inverse transform of
) 4 (
1
+ s s

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(b) Define unit impulse function of Laplace transform.

(c) If x
4

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in (-1, 1) then find the Fourier coefficient of

(d) Obtain the Fourier series for the function = x in the interval

(e)

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Find the Fourier cosine transform of .

(f) Find the Fourier sine transform of
x
e

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?
.

(g) Solve p
2

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-q
2
= x-y.
(h) Solve q
2

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= z
2
p
2
(1 - p

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2
).

(i) Find
(j) Find Z transform of n

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3
.

PART ? B
(Answer all five units, 5 X 10 = 50 Marks)

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UNIT ? I

2 (a)
Apply Convolution theorem , Evaluate

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?
?
?
?
?

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?
?
?
+ +
?

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) )( (
1
1
b s a s
L

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(b) Solve 0 ) 0 ( , 2 ) ( , 0
2
2
= ? = = + + y o y xy
dx

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dy
dx
y d
x using Laplace transform method.
OR

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3
Define Periodic function. Evaluate
?
?
?

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?
?
?
?
?

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?
? t t
dt
t
t e

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t L
0
sin

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UNIT ? II

4 Expand as a Fourier series in the interval (0, 2 ? ).
OR
5

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Obtain a half range sine series for the function: =
?
?
?

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?
?
?
?
? ? ?

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? ? ?
1
2
1
,

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4
3
2
1
0 ,

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4
1
x x
x x

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Contd. in page 2

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Page 1 of 2


R15

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Code: 15A54201

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B.Tech I Year II Semester (R15) Supplementary Examinations November 2017
MATHEMATICS ? II
(Common to all)
Time: 3 hours Max. Marks: 70

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PART ? A
(Compulsory Question)

*****
1 Answer the following: (10 X 02 = 20 Marks)

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(a)
Find inverse transform of
) 4 (
1

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+ s s

(b) Define unit impulse function of Laplace transform.

(c) If x

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4
in (-1, 1) then find the Fourier coefficient of

(d) Obtain the Fourier series for the function = x in the interval

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(e)
Find the Fourier cosine transform of .

(f) Find the Fourier sine transform of
x

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e
?
.

(g) Solve p

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2
-q
2
= x-y.
(h) Solve q

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2
= z
2
p
2

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(1 - p
2
).

(i) Find

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(j) Find Z transform of n
3
.

PART ? B

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(Answer all five units, 5 X 10 = 50 Marks)

UNIT ? I

2 (a)

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Apply Convolution theorem , Evaluate
?
?
?
?

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?
?
?
?
+ +

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?
) )( (
1
1
b s a s

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L
(b) Solve 0 ) 0 ( , 2 ) ( , 0
2
2
= ? = = + + y o y xy

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dx
dy
dx
y d
x using Laplace transform method.

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OR
3
Define Periodic function. Evaluate
?
?

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?
?
?
?
?

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?
?
? t t
dt
t

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t e
t L
0
sin

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UNIT ? II

4 Expand as a Fourier series in the interval (0, 2 ? ).
OR

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5

Obtain a half range sine series for the function: =
?
?

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?
?
?
?
?

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? ? ?
? ? ?
1
2
1

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,
4
3
2
1

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0 ,
4
1
x x
x x

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Contd. in page 2

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Page 1 of 2

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R15


Code: 15A54201

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UNIT ? III

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6

(a)
Find the Fourier transform of
?

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?
?
?
?
>

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? ?
1 , 0
1 , 1
2
x

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x x

Hence evaluate

dx

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x
x
x x x
?
?

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?
?
?
? ?
?

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?
2
cos
sin cos
0

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3

(b) Write the conditions of Parseval?s identity for Fourier transforms.
OR
7

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Find the Fourier sine transform of
( )
2 2
1

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a x x +


UNIT ? IV

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8 (a)
Form the partial differential equation ( ) ( ) x y f x y f z 3 2
2 1
? + + =

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by eliminating the arbitrary function.
(b) Use the method of separation of variables, solve u
t
u
x

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u
+
?
?
=

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?
?
2 where u(x, 0) = 6e
-3x
.

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OR

9

Solve the Laplace equation 0

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2
2
2
2
=

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?
?
+
?
?

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y
u
x
u
subject to the conditions and

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?
?
?
?
?

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?
l
x n ?

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UNIT ? V

10 (a) Find the Z-transformation of .
(b) If U(z)=
4

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2
) 1 (
12 3 2
?
+ +

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z
z z
, evaluate
3 2
,U U using Initial Value Theorem.

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OR
11

Solve the differential equation 5 3 2
1 2

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+ = + ?
+ +
n u u u
n n n
using Z-transforms.

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*****

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Page 2 of 2
R15
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