Download VTU B-Tech/B.E 2019 June-July 1st And 2nd Semester 2015 June-July 14MAT21 Engineering Mathematics II Question Paper

Download VTU ((Visvesvaraya Technological University) B.E/B-Tech 2019 July ( Bachelor of Engineering) First & Second Semester (1st Semester & 2nd Semester) 2015 June-July 14MAT21 Engineering Mathematics II Question Paper

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14MAT21
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5.
.,1
O E
PART ? A
4
d
4
?
,
d
3
u d
z
u du

O
-0 1 a. Solve 4
44
4 ?4 ' ? 23 +12-. + 36y = 0.
t
dx dx dx dx
c\
Coi
day
cl
2
y dy
/01:1
b. Solve
cilc
+ 6?
dx2
+11?
dx
+ 6y = e
x
+1 using inverse diffeNntiff operator method.
O A
c
4
L
'
''' 3
(07 Marks)
g cr
z '
1
C. Solve (D
2
? 2D)y -- e
x
sinx using method of undete
O
on
0 00
ii
coe dents.
}
(07 Marks)
tz
2 a. Solve (4D
4
? 8D
3
? 7D
2
+ 11D + 6)y = 0.
r
) ''' li Ask %
b. Solve (D
2
+ 4.)y ? x
2
+ e' using inverse differk4tfal operator method. (07 Marks)
Marks)
O 4
8
c. Solve (1D
2
? 2D +.2)y = e
x
tan x using method of variation of parameters. (07 Marks)
= .
1
* ..
-
4
PRT ? B
.5 ,S
(;)
.
. cid
,

? ..,
cl I 3 a. Solve ?
dx
? 7x + y = 0, ? ? ?5y = 0 .
4.---t, d
y
,
,---s,
-a 8
?
dt dt
c
,,:\,
b.

N
aive
x
2
?
d'y
+ 4x ?
dy
S.)
e
x
. (07 Marks)
dx
2
d ..,..
1

Solve y ? 2px +.' p by solving for x. (07 Marks)
d

2
y
C.) __I'
Vi 4?)
4 a. Solve (1,4
1
?
x)2
?
dx 2
+ (1 + x)
dy
+ y = 2 sin (log(1 + x)) .
dx
(06 Marks)
0
USN
Second Semester B.E. Degree Examination, June/July 2015
O Engineering Mathematics - II
Max. MarlreVd
.
0
-

Note: Answer any FIVE full questions, selecting ONE full question from f
4..,
91 Part.
Time: 3 hrs.
(06 Marks)
(06 Marks)
-5 is
1
1
"
.

, a
dy
?
dx
=
x y
A
. e -- by solving for P.
O?
dx dy ? ?y x
00

0
-f, .11.r t
1/4
te Solve (px ? y) (py + x) = a
2
p by reducing to Clairaut's form.
8 .
o
rts, PART ? C
? ?g
0 44 5 a. From the function f(x
2
+ y
2
, Z ? xy) = 0 form the partial differential equation.
b. Derive one dimensional wave equation as
I 2-x
c. Evaluate f xydydx by changing the order of integration.
x
2

1 oft
0
0
z
,9
2
1.1 2 7
2
11
=
at2
E
aX
(07 Marks)
(07 Marks)
(06 Marks)
(07 Marks)
(07 Marks)
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RpF
k
0
0
14MAT21
0
0
O
5.
.,1
O E
PART ? A
4
d
4
?
,
d
3
u d
z
u du

O
-0 1 a. Solve 4
44
4 ?4 ' ? 23 +12-. + 36y = 0.
t
dx dx dx dx
c\
Coi
day
cl
2
y dy
/01:1
b. Solve
cilc
+ 6?
dx2
+11?
dx
+ 6y = e
x
+1 using inverse diffeNntiff operator method.
O A
c
4
L
'
''' 3
(07 Marks)
g cr
z '
1
C. Solve (D
2
? 2D)y -- e
x
sinx using method of undete
O
on
0 00
ii
coe dents.
}
(07 Marks)
tz
2 a. Solve (4D
4
? 8D
3
? 7D
2
+ 11D + 6)y = 0.
r
) ''' li Ask %
b. Solve (D
2
+ 4.)y ? x
2
+ e' using inverse differk4tfal operator method. (07 Marks)
Marks)
O 4
8
c. Solve (1D
2
? 2D +.2)y = e
x
tan x using method of variation of parameters. (07 Marks)
= .
1
* ..
-
4
PRT ? B
.5 ,S
(;)
.
. cid
,

? ..,
cl I 3 a. Solve ?
dx
? 7x + y = 0, ? ? ?5y = 0 .
4.---t, d
y
,
,---s,
-a 8
?
dt dt
c
,,:\,
b.

N
aive
x
2
?
d'y
+ 4x ?
dy
S.)
e
x
. (07 Marks)
dx
2
d ..,..
1

Solve y ? 2px +.' p by solving for x. (07 Marks)
d

2
y
C.) __I'
Vi 4?)
4 a. Solve (1,4
1
?
x)2
?
dx 2
+ (1 + x)
dy
+ y = 2 sin (log(1 + x)) .
dx
(06 Marks)
0
USN
Second Semester B.E. Degree Examination, June/July 2015
O Engineering Mathematics - II
Max. MarlreVd
.
0
-

Note: Answer any FIVE full questions, selecting ONE full question from f
4..,
91 Part.
Time: 3 hrs.
(06 Marks)
(06 Marks)
-5 is
1
1
"
.

, a
dy
?
dx
=
x y
A
. e -- by solving for P.
O?
dx dy ? ?y x
00

0
-f, .11.r t
1/4
te Solve (px ? y) (py + x) = a
2
p by reducing to Clairaut's form.
8 .
o
rts, PART ? C
? ?g
0 44 5 a. From the function f(x
2
+ y
2
, Z ? xy) = 0 form the partial differential equation.
b. Derive one dimensional wave equation as
I 2-x
c. Evaluate f xydydx by changing the order of integration.
x
2

1 oft
0
0
z
,9
2
1.1 2 7
2
11
=
at2
E
aX
(07 Marks)
(07 Marks)
(06 Marks)
(07 Marks)
(07 Marks)

14MAT21
2 a u
a1.1
6 a. Solve ? = sinxsiny for which ? = ?2siny when x ? 0 and u = 0 when y is an odd
&ay
ay
multiple of
2

?.
a
u
02
u.

b. Derive one dimensional heat equation as ? = c
2
?
ax
2
.
at
1 a xi-.
c. Evaluate
I
J
J
(x + y + z)dyclxdz
-I 0 x-z
(06 Marks)
10
rni
(07 Marks)
(07 Marks)
PART ?D
7 a. Find the area between the parabolas? = 4ax and x
2
= 4ay using do teg;ral. (06 Marks)
dx

b. Evaluate using beta and gamma functions. Or Marks)
0
Express the vector Zi ? 2xj + yk in cylindrical coordinates. c. (07 Marks)
n/2 -02
, \
' '
S
V
I
%
1 I
V
II4? W

b. Evaluate f .Ni n 9 de x
d8
1
o
0 Vsin0
us' beta and gamma functions.
c. Express the vector field 2yi ? zj in spherical polar coordinate system. (07 Marks)
1 , 4
66
0 1 c
PART ?E
Vi
I
an
e
at
? e
-at
. Find the
l
Laplace trans eOZN)fte
-4t
sin3t and
%.
t
(06 Marks)
b. Express f(t) in to
,
unit step function and find its Laplace transform given that
i
t
2
, 0 itiZ2
4t,
L
re)< t < 4 . (07 Marks)
114
\ t > 4
{( 1)
2
9)
1
using convolution theorem.
C.
(07 Marks)
find L{ fW} .
8 a. Find the volume of the solid bounded by the planes x ? 0, y = 0, x + y + z = 1 and z = 0
using triple integral. (06 Marks)
(07 Marks)
f(t)
-
10-
-
k, a. A periodic function f(t) with period 2 is defined by f(t)
t, 02 ?t, 1b. Find L
-1
{
3s
2
+ 4s +8
5s ? 2
+
0

)1.
(06 Marks)
(07 Marks)

c. Solve using Laplace transform method
d2y

?
T
+ 2
dy
+ y = te with y(0) = 1, y
1
(0) = -2.
dt dt
(07 Marks)
2 oft
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This post was last modified on 01 January 2020