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

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

LIBRARy
V
IN

CHIKODI

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

15MAT21
z
c,

4

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

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

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

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

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

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

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

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

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

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

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

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

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

Module-2
O ,
3
a. Solve x
3

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

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

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

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

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

0

? =
..m. -
to

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

.
] =
CA rZ
E -
3

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

c
3

C?
c

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

?-
4 7d
2

'el i

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

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

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

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

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

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

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

u

O >
3 P.
b. Solve -

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

-t=1

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

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

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

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

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

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

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

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

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

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

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

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

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

O,

,
2
u

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

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

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

2 u LI
.. (05 Marks)
,
FirstRanker.com - FirstRanker's Choice
b.

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

dy
Solve xy
dx -(x dx
USN
LIBRARy

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

V
IN

CHIKODI
15MAT21

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

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)

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

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

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

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

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

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

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

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

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

O ,
3
a. Solve x
3
y"

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

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

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

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

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

? =
..m. -
to
.

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

] =
CA rZ
E -
3
c

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

3

C?
c
?-

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

4 7d
2

'el i
j

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

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

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

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

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

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

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

O >
3 P.
b. Solve -
t

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

ax
2

-t=1
t ct

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

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

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

z 00
b
3
b.
4

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

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

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

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

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

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

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

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

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

,
2
u
n

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

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

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

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

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

dO
fl
VSin 0
0
(05 Marks)

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

15MAT21
32
u

c. Find the sOlution of one dimensional heat equation C

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

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

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

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

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

, 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) ?

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

rn
im+n
(05 Marks)
OR
t 2

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

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

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

evaluate
.
1((
2
+ y

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

2 of 2
LIBRARY
FirstRanker.com - FirstRanker's Choice