Download AKTU B-Tech 2nd Sem 2017-2018 RAS203 Engineering Mathematics II Question Paper

Download AKTU (Dr. A.P.J. Abdul Kalam Technical University (AKTU), formerly Uttar Pradesh Technical University (UPTU)) B-Tech 2nd Semester (Second Semester) 2017-2018 RAS203 Engineering Mathematics II Question Paper

Printed Pages: 02 Sub Code: RAS203
Paper Id: 1 9 9 2 2 3 Roll No.

B. TECH
(SEM-II) THEORY EXAMINATION 2017-18
ENGINEERING MATHEMATICS - II
Time: 3 Hours Total Marks: 70
Note: Attempt all Sections. If require any missing data, then choose suitably.

SECTION A
1. Attempt all questions in brief. 2 x 7 = 14
(a) Determine the differential equation whose set of independent solutions is ? ?
x x x
e x xe e
2
, , .
(b) Solve:
x
e y D
?
? ? 2 ) 1 (
3
.
(c) Prove that: ) ( ) 1 ( ) ( x P x P
n
n
n
? ? ? .
(d) Find inverse Laplace transform of
5 4
8
2
? ?
?
s s
s
.
(e) If ??
s
e
t F L
s / 1
) (
?
? , find ? ? ) 3 ( t F e L
t ?
.
(f) Solve: 0 ) 5 4 (
2
? ? ? ? z D D , where
y
D
x
D
?
?
?
?
?
? ' , .
(g) Classify the equation: 0 ) 1 ( 2
2
? ? ? ?
yy xy xx
z y z x z .
SECTION B
2. Attempt any three of the following: 7 x 3 = 21
(a) Solve
2
(2 4) cos sincos3.
x
DDye x x x ?? ? ?
(b) Prove that:
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ?
?
x
x
x
x
x
x
x J
cos 3
sin
3 2
) (
2
2
2 / 5
?
.
(c) Draw the graph and find the Laplace transform of the triangular wave function of period ? 2 given by

?
?
?
? ? ?
? ?
?
? ? ?
?
2 , 2
0 ,
) (
t t
t t
t F .?
(d) Obtain half range cosine series for
x
e
?
the function
2, 0 1
()
2(2 ),1 2
tt
ft
tt
? ? ?
?
?
? ??
?

(e) Solve by method of separation of variables: u
x
u
t
u
2 ?
?
?
?
?
?
;
x x
e e x u
4
6 10 ) 0 , (
? ?
? ? .
SECTION C
3. Attempt any one part of the following: 7 x 1 = 7
(a) Solve the simultaneous differential equations:
y x
dt
dx
dt
x d
? ? ? 4 4
2
2
?
and
t
e x y
dt
dy
dt
y d
16 25 4 4
2
2
? ? ? ? .
(b) Use variation of parameter method to solve the differential equation
x
e x y y x y x
2 2
? ? ? ? ? ? .

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Printed Pages: 02 Sub Code: RAS203
Paper Id: 1 9 9 2 2 3 Roll No.

B. TECH
(SEM-II) THEORY EXAMINATION 2017-18
ENGINEERING MATHEMATICS - II
Time: 3 Hours Total Marks: 70
Note: Attempt all Sections. If require any missing data, then choose suitably.

SECTION A
1. Attempt all questions in brief. 2 x 7 = 14
(a) Determine the differential equation whose set of independent solutions is ? ?
x x x
e x xe e
2
, , .
(b) Solve:
x
e y D
?
? ? 2 ) 1 (
3
.
(c) Prove that: ) ( ) 1 ( ) ( x P x P
n
n
n
? ? ? .
(d) Find inverse Laplace transform of
5 4
8
2
? ?
?
s s
s
.
(e) If ??
s
e
t F L
s / 1
) (
?
? , find ? ? ) 3 ( t F e L
t ?
.
(f) Solve: 0 ) 5 4 (
2
? ? ? ? z D D , where
y
D
x
D
?
?
?
?
?
? ' , .
(g) Classify the equation: 0 ) 1 ( 2
2
? ? ? ?
yy xy xx
z y z x z .
SECTION B
2. Attempt any three of the following: 7 x 3 = 21
(a) Solve
2
(2 4) cos sincos3.
x
DDye x x x ?? ? ?
(b) Prove that:
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ?
?
x
x
x
x
x
x
x J
cos 3
sin
3 2
) (
2
2
2 / 5
?
.
(c) Draw the graph and find the Laplace transform of the triangular wave function of period ? 2 given by

?
?
?
? ? ?
? ?
?
? ? ?
?
2 , 2
0 ,
) (
t t
t t
t F .?
(d) Obtain half range cosine series for
x
e
?
the function
2, 0 1
()
2(2 ),1 2
tt
ft
tt
? ? ?
?
?
? ??
?

(e) Solve by method of separation of variables: u
x
u
t
u
2 ?
?
?
?
?
?
;
x x
e e x u
4
6 10 ) 0 , (
? ?
? ? .
SECTION C
3. Attempt any one part of the following: 7 x 1 = 7
(a) Solve the simultaneous differential equations:
y x
dt
dx
dt
x d
? ? ? 4 4
2
2
?
and
t
e x y
dt
dy
dt
y d
16 25 4 4
2
2
? ? ? ? .
(b) Use variation of parameter method to solve the differential equation
x
e x y y x y x
2 2
? ? ? ? ? ? .

4. Attempt any one part of the following: 7 x 1 = 7
(a) State and prove Rodrigue?s formula for Legendre?s polynomial.
(b) Solve in series: 0 3 ) 1 ( ) 1 ( 2 ? ? ? ? ? ? ? ? y y x y x x .
5. Attempt any one part of the following: 7 x 1 = 7
(a) State convolution theorem and hence find inverse Laplace transform of
) )( (
2 2 2 2
2
b s a s
s
? ?
.
(b) Solve the following differential equation using Laplace transform
t
e t y
dt
dy
dt
y d
dt
y d
2
2
2
3
3
3 3 ? ? ? ?
?
where 0 ) 0 ( , 1 ) 0 ( ? ? ? y y ?and? 2 ) 0 ( ? ? ? ? y .

6. Attempt any one part of the following: 7 x 1 = 7
(a) Obtain Fourier series for the function
?
?
?
? ? ?
? ? ?
?
?
?
x x
x x
x f
0 ,
0 ,
) ( and hence show that
8
......
5
1
3
1
1
1
2
2 2 2
?
? ? ? ? .
(b) Solve the linear partial differential equation: y x
y x
z
x
z
2 cos sin 2
2
2
2
?
? ?
?
?
?
?
.

7. Attempt any one part of the following: 7 x 1 = 7
(a) A string is stretched and fastened to two points l apart. Motion is started by displacing the string
in the form
l
x
A y
?
sin ? from which it is released at time 0 ? t . Find the displacement of any
point at a distance x from one end at any time t.
??

(b) A rectangular plate with insulated surfaces is 8 cm wide and so long compared to its width that it
may be considered infinite in length without introducing an appreciable error. If the temperature
along one short edge 0 ? y
?
is given by 8 0 ,
8
sin 100 ) 0 , ( ? ? ? x
x
x u
?
?

while the two long edges 0 ? x
?
and 8 ? x
?
as well as the other short edge are kept at C
?
0 . Find the
temperature ) , ( y x u
?
at any point in steady state.
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