Download JNTUA B.Tech 1-2 R15 2019 Dec Supple 15A54201 Mathematics II Question Paper

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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) State and prove first shifting theorem.

(b) Evaluate ? ? ? ? 2 ? sin ? ? ? ?
0
.

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(c) If ? ( ? ) = ? + ? 3
? ( ? ? , ? ), find the Euler?s coefficients a
0
,a
n

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.
(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 ? ( ? ) = ?

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? 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

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C and 70
o
C respectively until steady state is
prevailed. Determine the steady state temperature of the rod.

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(i) Find ? ?
1
? ? 1
?.
(j) State final value theorem on Z-transform.

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

UNIT ? I

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2 (a) Use convolution theorem to find ? ? 1
?
1
? 2

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( ? + 1)
2
?.
(b) Find the Laplace transform of the square wave function of period ? defined as:
? ( ? ) = ?

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1, ? ? 0 < ? < ? /2
?1, ? ?
? 2
< ? < ?
OR

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3 Solve ? ?
+ 2 ? ?
? ? ?
? 2 ? = 0 given y(0) = 0, ? ?
(0) = 0 and ? ?

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?(0) = 6.

UNIT ? II

4 Find the complex form of the Fourier series of ? ( ? ) = ? ? ? ? ? 1 ? ? ? 1

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OR
5

Obtain the Half Range cosine series of f(x) = x in 01
1

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4
+
1
3
4

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+
1
5
4
+ ? =

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? 4
96
.

UNIT ? III

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6 Find the Fourier transform of ? ( ? ) = ?
1, | ? | < 1
0, | ? | > 1