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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