Download VTU BE 2020 Jan Question Paper 15 Scheme 15MAT21 Engineering Mathematics II First And Second Semester

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V
IN

CHIKODI

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15MAT21
z
c,

4

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1,
6
Second Semester B.E. Degree Examination, Dec;z_to
Engineering Mathematics - II
an.2020

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'EL cz
1 a. Solve = Cosh (2x -1) + by inverse differential operators method. (06 Marks)
dx"
b. Solve (D
3

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- 1)y = 3 Cos 2x by inverse differential operators method. (05 Marks)
c. Solve (D
2
+ a
2

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) y = Sec (ax) by the method of variation of parameters. (05 Marks)
OR
2 a. Solve (D
2
- 2D + 5) y= e`

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x
Sin x by inverse differential operator method. (06 Marks)
b. Solve (D
3
+ D

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2
+ 4D + 4) y = x
2
- 4x - 6 by inverse differential operator method. (05 Marks)
? 1.)

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Module-2
O ,
3
a. Solve x
3

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y"
?
+3x
2
y" + xy' + 8y =65 Cos (log x)

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- :,:_
(06 Marks)
ti -
)2
:.=

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0

? =
..m. -
to

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.
] =
CA rZ
E -
3

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c
3

C?
c

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?-
4 7d
2

'el i

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j
tr)
-,,-
Given z = y
2

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+ 2f (
1
+ lo (06 Marks) 0
? g yj -
.-- c

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c to
x
15 -
c... ?
6,2

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u

O >
3 P.
b. Solve -

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t
COS X , given that u = 0 when t = 0 and ?
ax at at
at' ax2
OR

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6 a. Obtain partial differential equation of
f (x
-
+ 2yz, y
-

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+ 2zx) = O.
(06 Marks)
Time: 3 hrs.
Max. Marks: 80
Note:

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Answer any FIVEfullquestions, choosing ONEfullquestion from each module.
Module-I
? dy
- + y- ) ? + xy = 0 (05 Marks)
substitution X = x2, Y = i. (05 Marks) CA

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Ft
c. Solve the equation (px -y) (py +x) =2p by reducing into Cla raut's form taking the
t:1
a
cd

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' -e' ..
m ^-
OR
17J 7:1
^N2

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Solve ? = a
-)
z , given that when x 0, z = 0 and
CZ 0 Z
b.

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? = a sin y. (05 Marks)
ax
2

-t=1

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t ct
i
oo
.;:: + c. Solve y" - 2y' +3y = x
2

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- Cos x by the method of undetermined coefficients. (05 Marks)
z 00
b
3
b.

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4
a. Solve (2x1)
2
y"+(2x -1)y' -2y =8x
2

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-2x +3 (06 Marks)
7,c
Solve y =2px +p
2
yby solving for 'x'. (05 Marks)

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? E.:
.
Find the general and singular solution of equation xp
2
--py + kp + a = 0. (05 Marks)

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Module-3
5 a. Obtain partial differential equation by eliminating arbitrary function.
au
= 0 at x - (05 Marks O. )
c...

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O,

,
2
u

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n
2.,
... ,-..;
C. Derive one dimensional wave equation =
C

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2 u LI
.. (05 Marks)
,
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b.

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dy
Solve xy
dx -(x dx
USN
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V
IN

CHIKODI
15MAT21

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z
c,

4
1,

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

6
Second Semester B.E. Degree Examination, Dec;z_to
Engineering Mathematics - II
an.2020
'EL cz

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

1 a. Solve = Cosh (2x -1) + by inverse differential operators method. (06 Marks)
dx"
b. Solve (D
3
- 1)y = 3 Cos 2x by inverse differential operators method. (05 Marks)

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c. Solve (D
2
+ a
2
) y = Sec (ax) by the method of variation of parameters. (05 Marks)

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OR
2 a. Solve (D
2
- 2D + 5) y= e`
x

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Sin x by inverse differential operator method. (06 Marks)
b. Solve (D
3
+ D
2

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+ 4D + 4) y = x
2
- 4x - 6 by inverse differential operator method. (05 Marks)
? 1.)
Module-2

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O ,
3
a. Solve x
3
y"

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?
+3x
2
y" + xy' + 8y =65 Cos (log x)
- :,:_

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(06 Marks)
ti -
)2
:.=
0

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? =
..m. -
to
.

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] =
CA rZ
E -
3
c

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3

C?
c
?-

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4 7d
2

'el i
j

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tr)
-,,-
Given z = y
2
+ 2f (

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1
+ lo (06 Marks) 0
? g yj -
.-- c
c to

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x
15 -
c... ?
6,2
u

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O >
3 P.
b. Solve -
t

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COS X , given that u = 0 when t = 0 and ?
ax at at
at' ax2
OR
6 a. Obtain partial differential equation of

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f (x
-
+ 2yz, y
-
+ 2zx) = O.

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(06 Marks)
Time: 3 hrs.
Max. Marks: 80
Note:
Answer any FIVEfullquestions, choosing ONEfullquestion from each module.

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Module-I
? dy
- + y- ) ? + xy = 0 (05 Marks)
substitution X = x2, Y = i. (05 Marks) CA
Ft

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c. Solve the equation (px -y) (py +x) =2p by reducing into Cla raut's form taking the
t:1
a
cd
' -e' ..

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m ^-
OR
17J 7:1
^N2
Solve ? = a

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-)
z , given that when x 0, z = 0 and
CZ 0 Z
b.
? = a sin y. (05 Marks)

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ax
2

-t=1
t ct

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i
oo
.;:: + c. Solve y" - 2y' +3y = x
2
- Cos x by the method of undetermined coefficients. (05 Marks)

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z 00
b
3
b.
4

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a. Solve (2x1)
2
y"+(2x -1)y' -2y =8x
2
-2x +3 (06 Marks)

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7,c
Solve y =2px +p
2
yby solving for 'x'. (05 Marks)
? E.:

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.
Find the general and singular solution of equation xp
2
--py + kp + a = 0. (05 Marks)
Module-3

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5 a. Obtain partial differential equation by eliminating arbitrary function.
au
= 0 at x - (05 Marks O. )
c...
O,

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,
2
u
n

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2.,
... ,-..;
C. Derive one dimensional wave equation =
C
2 u LI

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.. (05 Marks)
,
c. Evaluate f.
v
Sin 0 dO using Beta and Gamma functions.

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dO
fl
VSin 0
0
(05 Marks)

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15MAT21
32
u

c. Find the sOlution of one dimensional heat equation C

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-
at ax
Module-4
(05 Marks)
1-

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x dx dy dx
7 a. Evaluate (06 Marks)
0 0
(1 + x + y + z)
3

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, x
b. Evaluate integral xy dy dx by changing the order of integration. (05 Marks)
x
e. Obtain the relation between Beta and Gamma function in the form 13(m, n) ?

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rn
im+n
(05 Marks)
OR
t 2

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8 a. Evaluate JJ e x *'
2)
dxdy by changing into polar co-ordinates. (06 Marks)
0 0
b. If A is the area of rectangular region bounded by the lines x = 0, x = 1, y = 0, y = 2 then

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evaluate
.
1((
2
+ y

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2
)dA (05 Marks)
A
' 2 2
Module-5

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9
a.
Find Laplace transition of i) t
2
e

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2t
ii)
? c
(06 Marks)
b.

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If a periodic function of period 2a is defined by f(t) =
t if 0 < t < a }
1
2a ? t if a < t < 2a
Then show

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Solve y"(t)
, ,
that Lif(t)i=
+ 4yr(t) + 4y(t)
I

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--.-, tan h(Ls). (05 Marks,?
s
-
2
= e' with y(0) = 0 y

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1
(0) = 0. Using Laplace transform. (05 Marks)
OR
7s
10 a- Find LI (06 Marks)

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(4 s
-
+ 4s +9)
b. Find using convolution theorem. (05 Marks)
(s ?1)(s

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-
+ )
c- Express the following function interms of Heaviside unit step function and hence its
Laplace transistor f(
I t 0 < t 2

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4t, t > 2
\f.
. . o
""
(05 Marks)

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