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Code: 15A54201
B.Tech I Year II Semester (R15) Supplementary Examinations December 2019
MATHEMATICS ? II
(Common to all )
Time: 3 hours Max. Marks: 70
PART ? A
(Compulsory Question)
*****
1 Answer the following: (10 X 02 = 20 Marks)
(a) State and prove first shifting theorem.
(b) Evaluate ? ? ? ? 2 ? sin ? ? ? ?
0
.
(c) If ? ( ? ) = ? + ? 3
? ( ? ? , ? ), find the Euler?s coefficients a
0
,a
n
.
(d) State the conditions for f(x) to have Fourier series expansion.
(e) Find the Fourier sine transform of the function ? ( ? ) = 5 ? ? 2 ? + 2 ? ? 5 ? .
(f) Find the Fourier transform of the function ? ( ? ) = ?
? 2
, | ? | ? ? 0, | ? | > ? .
(g) Write down possible solutions of the Laplace equation.
(h) A rod 20 cms long has its ends A and B kept at 30
o
C and 70
o
C respectively until steady state is
prevailed. Determine the steady state temperature of the rod.
(i) Find ? ?
1
? ? 1
?.
(j) State final value theorem on Z-transform.
PART ? B
(Answer all five units, 5 X 10 = 50 Marks)
UNIT ? I
2 (a) Use convolution theorem to find ? ? 1
?
1
? 2
( ? + 1)
2
?.
(b) Find the Laplace transform of the square wave function of period ? defined as:
? ( ? ) = ?
1, ? ? 0 < ? < ? /2
?1, ? ?
? 2
< ? < ?
OR
3 Solve ? ?
+ 2 ? ?
? ? ?
? 2 ? = 0 given y(0) = 0, ? ?
(0) = 0 and ? ?
?(0) = 6.
UNIT ? II
4 Find the complex form of the Fourier series of ? ( ? ) = ? ? ? ? ? 1 ? ? ? 1
OR
5
Obtain the Half Range cosine series of f(x) = x in 0
1
4
+
1
3
4
+
1
5
4
+ ? =
? 4
96
.
UNIT ? III
6 Find the Fourier transform of ? ( ? ) = ?
1, | ? | < 1
0, | ? | > 1
. Hence evaluate ?
s in ? ? ? ? ?
0
.
OR
7 (a) Find the Fourier sine transform of
? ? ? ? ? .
(b)
Using Parseval?s identity, evaluate ?
? 2
?
( ? 2
+ 1)
2
?
0
.
Contd. in page 2
Page 1 of 2
R15
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Code: 15A54201
B.Tech I Year II Semester (R15) Supplementary Examinations December 2019
MATHEMATICS ? II
(Common to all )
Time: 3 hours Max. Marks: 70
PART ? A
(Compulsory Question)
*****
1 Answer the following: (10 X 02 = 20 Marks)
(a) State and prove first shifting theorem.
(b) Evaluate ? ? ? ? 2 ? sin ? ? ? ?
0
.
(c) If ? ( ? ) = ? + ? 3
? ( ? ? , ? ), find the Euler?s coefficients a
0
,a
n
.
(d) State the conditions for f(x) to have Fourier series expansion.
(e) Find the Fourier sine transform of the function ? ( ? ) = 5 ? ? 2 ? + 2 ? ? 5 ? .
(f) Find the Fourier transform of the function ? ( ? ) = ?
? 2
, | ? | ? ? 0, | ? | > ? .
(g) Write down possible solutions of the Laplace equation.
(h) A rod 20 cms long has its ends A and B kept at 30
o
C and 70
o
C respectively until steady state is
prevailed. Determine the steady state temperature of the rod.
(i) Find ? ?
1
? ? 1
?.
(j) State final value theorem on Z-transform.
PART ? B
(Answer all five units, 5 X 10 = 50 Marks)
UNIT ? I
2 (a) Use convolution theorem to find ? ? 1
?
1
? 2
( ? + 1)
2
?.
(b) Find the Laplace transform of the square wave function of period ? defined as:
? ( ? ) = ?
1, ? ? 0 < ? < ? /2
?1, ? ?
? 2
< ? < ?
OR
3 Solve ? ?
+ 2 ? ?
? ? ?
? 2 ? = 0 given y(0) = 0, ? ?
(0) = 0 and ? ?
?(0) = 6.
UNIT ? II
4 Find the complex form of the Fourier series of ? ( ? ) = ? ? ? ? ? 1 ? ? ? 1
OR
5
Obtain the Half Range cosine series of f(x) = x in 0
1
4
+
1
3
4
+
1
5
4
+ ? =
? 4
96
.
UNIT ? III
6 Find the Fourier transform of ? ( ? ) = ?
1, | ? | < 1
0, | ? | > 1
. Hence evaluate ?
s in ? ? ? ? ?
0
.
OR
7 (a) Find the Fourier sine transform of
? ? ? ? ? .
(b)
Using Parseval?s identity, evaluate ?
? 2
?
( ? 2
+ 1)
2
?
0
.
Contd. in page 2
Page 1 of 2
R15
Code: 15A54201
UNIT ? IV
8 A tightly stretched string of length ? with fixed end is initially in its equilibrium position. It is set
vibrating by giving each point a velocity ? 0
sin
3
( ? / ? ). Determine the displacement function y(x, t).
OR
9 An infinitely long plane uniform plate is bounded by two parallel edges and an end at right angles to
them. The breadth is ?; this end is maintained at a temperature ? 0
at all points and other edges are
at zero temperature. Determine the temperature at any point of the plate in the steady state.
UNIT ? V
10 (a) Using convolution theorem, find the inverse Z-transform of
? 2
( ? ? 1)( ? ? 3)
.
(b) Find ? ? 1
2 ? ( ? ? 1)( ? 2
+ 1)
, by partial fraction method.
OR
11 Solve: ? ? + 2
+ 6 ? ? + 1
+ 9 ? ? = 2
? ? ? ? ? 0
= 0 ? ? ? 1
= 0, using Z-transforms.
*****
Page 2 of 2
R15
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This post was last modified on 11 September 2020