Download JNTUA B.Tech 1-2 R15 2016 Dec Supple 15A54201 Mathematics II Question Paper

--- Content provided by‍ FirstRanker.com ---

(Common to all)
Time: 3 hours Max. Marks: 70
PART ? A
(Compulsory Question)

--- Content provided by‍ FirstRanker.com ---

*****
1 Answer the following: (10 X 02 = 20 Marks)
(a) Write the conditions for existence of Laplace transform of a function.
(b) Define Unit Impulse function.
(c) Write Dirichlet conditions for Fourier series.

--- Content provided by​ FirstRanker.com ---

(d) Write the Parseval?s formula for Fourier series.
(e) Write the complex form of Fourier integral.
(f) Write any two properties of Fourier transform.

(g) What are the assumptions to be made for one dimensional wave equation?

--- Content provided by​ FirstRanker.com ---

(h) What do you mean by steady state and transient state?

(i)
Find the Z-transform of
1

--- Content provided by‌ FirstRanker.com ---

.
n


(j) Find

--- Content provided by⁠ FirstRanker.com ---

2
1
2
2
.

--- Content provided by FirstRanker.com ---

( 1)
zz
Z
z
?

--- Content provided by‍ FirstRanker.com ---

? ?
?
?
?

--- Content provided by‍ FirstRanker.com ---

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

UNIT ? I

--- Content provided by FirstRanker.com ---

2 (a) Find the Laplace transform of ( ) 1 1 , 0. ft t t t = ?+ + ?
(b) Use Laplace transform to evaluate
0
sin

--- Content provided by⁠ FirstRanker.com ---

.
t
et
L dt
t

--- Content provided by​ FirstRanker.com ---

?
?
?
?
?

--- Content provided by⁠ FirstRanker.com ---

?
?
?

OR

--- Content provided by​ FirstRanker.com ---

3 (a) Apply Convolution theorem to evaluate
1
2 22
.
()

--- Content provided by​ FirstRanker.com ---

s
L
sa
?
?

--- Content provided by​ FirstRanker.com ---

?
?
+ ?
?

--- Content provided by‍ FirstRanker.com ---

(b) Solve 2 cos , (0) 1. ty y y t y ? ? + += =

UNIT ? II

4

--- Content provided by‌ FirstRanker.com ---

Find the Fourier series for
2
() 1 fx x x = ++ in ( , ). ? ? Hence deduce that
2

--- Content provided by‍ FirstRanker.com ---

2 22
11 1
..... .
6
12 3

--- Content provided by​ FirstRanker.com ---

?
+ + + =

OR
5 (a) Expand ( ) cos ,0 fx x x ? = << in a Fourier Sine series.

--- Content provided by‌ FirstRanker.com ---

(b)
Find the complex form of the Fourier series of ( ) in [ 1,1].
x
fx e
?

--- Content provided by‍ FirstRanker.com ---

= ?

--- Content provided by‍ FirstRanker.com ---

Contd. in page 2

--- Content provided by FirstRanker.com ---

Page 1 of 2
R15

--- Content provided by​ FirstRanker.com ---

FirstRanker.com - FirstRanker's Choice

Code: 15A54201

B.Tech I Year II Semester (R15) Supplementary Examinations December 2016

--- Content provided by​ FirstRanker.com ---

MATHEMATICS ? II
(Common to all)
Time: 3 hours Max. Marks: 70
PART ? A
(Compulsory Question)

--- Content provided by​ FirstRanker.com ---

*****
1 Answer the following: (10 X 02 = 20 Marks)
(a) Write the conditions for existence of Laplace transform of a function.
(b) Define Unit Impulse function.

--- Content provided by‍ FirstRanker.com ---

(c) Write Dirichlet conditions for Fourier series.
(d) Write the Parseval?s formula for Fourier series.
(e) Write the complex form of Fourier integral.
(f) Write any two properties of Fourier transform.

--- Content provided by‌ FirstRanker.com ---

(g) What are the assumptions to be made for one dimensional wave equation?
(h) What do you mean by steady state and transient state?

(i)
Find the Z-transform of

--- Content provided by FirstRanker.com ---

1
.
n

--- Content provided by​ FirstRanker.com ---

(j) Find
2
1
2
2

--- Content provided by‌ FirstRanker.com ---

.
( 1)
zz
Z
z

--- Content provided by‌ FirstRanker.com ---

?
? ?
?
?
?

--- Content provided by‍ FirstRanker.com ---

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

--- Content provided by‌ FirstRanker.com ---

UNIT ? I

2 (a) Find the Laplace transform of ( ) 1 1 , 0. ft t t t = ?+ + ?
(b) Use Laplace transform to evaluate
0

--- Content provided by⁠ FirstRanker.com ---

sin
.
t
et
L dt

--- Content provided by​ FirstRanker.com ---

t
?
?
?
?

--- Content provided by FirstRanker.com ---

?
?
?
?

--- Content provided by‍ FirstRanker.com ---

OR
3 (a) Apply Convolution theorem to evaluate
1
2 22
.

--- Content provided by⁠ FirstRanker.com ---

()
s
L
sa
?

--- Content provided by‌ FirstRanker.com ---

?
?
?
+ ?
?

--- Content provided by⁠ FirstRanker.com ---

(b) Solve 2 cos , (0) 1. ty y y t y ? ? + += =

UNIT ? II

--- Content provided by‍ FirstRanker.com ---

4

Find the Fourier series for
2
() 1 fx x x = ++ in ( , ). ? ? Hence deduce that

--- Content provided by‍ FirstRanker.com ---

2
2 22
11 1
..... .
6

--- Content provided by​ FirstRanker.com ---

12 3
?
+ + + =

OR

--- Content provided by FirstRanker.com ---

5 (a) Expand ( ) cos ,0 fx x x ? = << in a Fourier Sine series.
(b)
Find the complex form of the Fourier series of ( ) in [ 1,1].
x
fx e

--- Content provided by​ FirstRanker.com ---

?
= ?

--- Content provided by⁠ FirstRanker.com ---

Contd. in page 2

--- Content provided by FirstRanker.com ---

Page 1 of 2

--- Content provided by‍ FirstRanker.com ---

R15

Code: 15A54201

--- Content provided by⁠ FirstRanker.com ---

--- Content provided by​ FirstRanker.com ---

UNIT ? III

6 (a) Find Fourier cosine transform of .

(b) Find Fourier transform of

--- Content provided by‍ FirstRanker.com ---

2
1, 1
() .
0 , 1
x x

--- Content provided by⁠ FirstRanker.com ---

fx
x
?
? ?
?

--- Content provided by‌ FirstRanker.com ---

=
?
>
?
?

--- Content provided by​ FirstRanker.com ---

OR
7 (a)
Find Fourier sine transform of .
ax

--- Content provided by‌ FirstRanker.com ---

e
x
?

(b) Find the Finite Fourier sine and cosine transform of ( ) 2 ,0 4. fx x x = <<

--- Content provided by​ FirstRanker.com ---

UNIT ? IV

8 (a) Form the partial differential equation by eliminating the arbitrary functions f and g from:
(2 ) (3 ). Z f x y g x y = + + ?

--- Content provided by‍ FirstRanker.com ---

(b) Solve by using the method of separation of variables the equation 2 3 0.
zz
x y
xy

--- Content provided by‌ FirstRanker.com ---

?
?=
?

OR

--- Content provided by‌ FirstRanker.com ---

9 A rod of length 20 cm has its ends A and B kept at temperature respectively until steady
state conditions prevail. If the temperature at each end is then suddenly reduced to and maintained
so, find the temperature distribution at a distance x from A at time t.

UNIT ? V

--- Content provided by​ FirstRanker.com ---

10 (a) If
2
4
2 5 14

--- Content provided by‌ FirstRanker.com ---

()
( 1)
zz
UZ
z

--- Content provided by​ FirstRanker.com ---

++
=
?
then find
2

--- Content provided by​ FirstRanker.com ---

U and
3
. U
(b) Use convolution theorem to evaluate
2

--- Content provided by FirstRanker.com ---

1
.
( )( )
z
Z

--- Content provided by‌ FirstRanker.com ---

z az b
?
?
?
?

--- Content provided by⁠ FirstRanker.com ---

?
?
?

OR

--- Content provided by⁠ FirstRanker.com ---

11

Use Z-transform to solve:
21
2 3 5.

--- Content provided by FirstRanker.com ---

n nn
y y yn
++
? += +

--- Content provided by FirstRanker.com ---

*****

--- Content provided by⁠ FirstRanker.com ---

--- Content provided by‌ FirstRanker.com ---

--- Content provided by‌ FirstRanker.com ---

--- Content provided by‌ FirstRanker.com ---

Page 2 of 2
R15
FirstRanker.com - FirstRanker's Choice

--- Content provided by​ FirstRanker.com ---