Download VTU B-Tech/B.E 2019 June-July 1st And 2nd Semester 18 Scheme 18MAT21 Advanced Calculus and Numerical Methods Question Paper

Download VTU ((Visvesvaraya Technological University) B.E/B-Tech 2019 July ( Bachelor of Engineering) First & Second Semester (1st Semester & 2nd Semester) 18 Scheme 18MAT21 Advanced Calculus and Numerical Methods Question Paper

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USN
18MAT21
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01

Second Semester B.E. Degree Examination, June/July 2019
Advanced Calculus and Numerical Methods
Time: 3 hrs. Max. Marks: 100
-.:0
to ii OR
? oc
.
?
N
2 a. Find the total work done in moving a particle in the force field F = 3xyi ? 5z] + 10xk
-
c +
E) e along the curve x = t
2
+1, y = 2t
2
, z = t
3
from t = 1 to t = 2. (06 Marks)
cu 0
b. Using Green's theorem, evaluate
f
(xy + y
2
)dx + x
2
dy , where C is bounded by y = x and y = x
2
. (07 Marks)
0.)
z -
.?. . 0 C
C4 ,E
z
;

,...:
2 0-
c. Using Divergence theorem, evaluate if ds , where F = (x
2
? yz)i + (y
2
? )3 + (z
2
? xy)k
f, c,
71
.
8

? -----
? "c1 taken over the rectangular parallelepiped 0 x a, 0 5_ y 5_ b, 0 < z < C.
to =
(07 Marks)
czt ms
ii ?
+
_
4)y
=
e
2x
. ?
3
,. ..?
4 a. Solve (D
2
? 4D + sin x. (06 Marks)
b. Solve (x+1 )
2
y" + (x+1)y' + y = 2sin[log
e
(x+1)]
1-. cL)
(07 Marks)
0
,
,-,-
.--
C.
The current i and the charge q in a series containing an inductance L, capacitance C,
?
-r:$
0
.
17. 1 :f
emf E, satisfy the differential equation L ?
d2q
+ ?
q
= E , Express q and i interms of 't'
= u
z 8 dt
2
C
a E-
>
0
u
given that L, C, E are constants and the value of i and q are both zero initially. (07 Marks)
+.,
O <
,-,' N Module-3
.1- 2.
5 a. Form the partial differential equation by elimination of arbitrary function from
0
z
(0(x + y + z, x
2
+ y
2
+ z
2
) = 0 (06 Marks)
c7i
i-.
O b. Solve ? cos(2x + 3y)
(07 Marks)
a
8
x 2
3
z
a
y

'
Note: Answer any FIVE full questions, choosing ONE full question from each module.
6
U
c
cz
.
Module-1
s
-
a
P 1 a. If F = V(x3 + y
3
+ z
3
? 3xyz), find div F and curl f . (06 Marks)
cA
-0
b. Find the angle between the surfaces x
2
+ y
2
+ z
2
= 9 and z = x
2
+ y
2
? 3 at the point
,,,
(2, ?I, 2).
L-
(07 Marks)
L-
C)
,
C, c.
Find the value of a, b, c such that f = (axy + bz
3
)i + (3x
2
? CZ)
^
j + (3)(Z
2
? y)k
ui
c)

th
ct ?
=
is irrotational, also find the scalar potential (I) such that F =V4). (07 Marks)
i c5
in
e
?
*7
1
.

Module-2
ct
3 a. Solve (D
2
? 3D + 2)y = 2x
2
+ sin 2x. (06 Marks)
b. Solve (D
2
+1)y = sec x by the method of variation of parameter. ( 07 Marks)
c. Solve x
2
y" ? 4xy' + 6y = cos(2 logx) (07 Marks)
Fz
d
OR
c.
Derive one dimensional heat equation in the standard form as
au
? = C
2 a2u
, (07 Marks)
et ex-
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. so -
3.
4
,
F
' I h.
l
.all
USN
18MAT21
50-citify
,
*

ctisK.0
01

Second Semester B.E. Degree Examination, June/July 2019
Advanced Calculus and Numerical Methods
Time: 3 hrs. Max. Marks: 100
-.:0
to ii OR
? oc
.
?
N
2 a. Find the total work done in moving a particle in the force field F = 3xyi ? 5z] + 10xk
-
c +
E) e along the curve x = t
2
+1, y = 2t
2
, z = t
3
from t = 1 to t = 2. (06 Marks)
cu 0
b. Using Green's theorem, evaluate
f
(xy + y
2
)dx + x
2
dy , where C is bounded by y = x and y = x
2
. (07 Marks)
0.)
z -
.?. . 0 C
C4 ,E
z
;

,...:
2 0-
c. Using Divergence theorem, evaluate if ds , where F = (x
2
? yz)i + (y
2
? )3 + (z
2
? xy)k
f, c,
71
.
8

? -----
? "c1 taken over the rectangular parallelepiped 0 x a, 0 5_ y 5_ b, 0 < z < C.
to =
(07 Marks)
czt ms
ii ?
+
_
4)y
=
e
2x
. ?
3
,. ..?
4 a. Solve (D
2
? 4D + sin x. (06 Marks)
b. Solve (x+1 )
2
y" + (x+1)y' + y = 2sin[log
e
(x+1)]
1-. cL)
(07 Marks)
0
,
,-,-
.--
C.
The current i and the charge q in a series containing an inductance L, capacitance C,
?
-r:$
0
.
17. 1 :f
emf E, satisfy the differential equation L ?
d2q
+ ?
q
= E , Express q and i interms of 't'
= u
z 8 dt
2
C
a E-
>
0
u
given that L, C, E are constants and the value of i and q are both zero initially. (07 Marks)
+.,
O <
,-,' N Module-3
.1- 2.
5 a. Form the partial differential equation by elimination of arbitrary function from
0
z
(0(x + y + z, x
2
+ y
2
+ z
2
) = 0 (06 Marks)
c7i
i-.
O b. Solve ? cos(2x + 3y)
(07 Marks)
a
8
x 2
3
z
a
y

'
Note: Answer any FIVE full questions, choosing ONE full question from each module.
6
U
c
cz
.
Module-1
s
-
a
P 1 a. If F = V(x3 + y
3
+ z
3
? 3xyz), find div F and curl f . (06 Marks)
cA
-0
b. Find the angle between the surfaces x
2
+ y
2
+ z
2
= 9 and z = x
2
+ y
2
? 3 at the point
,,,
(2, ?I, 2).
L-
(07 Marks)
L-
C)
,
C, c.
Find the value of a, b, c such that f = (axy + bz
3
)i + (3x
2
? CZ)
^
j + (3)(Z
2
? y)k
ui
c)

th
ct ?
=
is irrotational, also find the scalar potential (I) such that F =V4). (07 Marks)
i c5
in
e
?
*7
1
.

Module-2
ct
3 a. Solve (D
2
? 3D + 2)y = 2x
2
+ sin 2x. (06 Marks)
b. Solve (D
2
+1)y = sec x by the method of variation of parameter. ( 07 Marks)
c. Solve x
2
y" ? 4xy' + 6y = cos(2 logx) (07 Marks)
Fz
d
OR
c.
Derive one dimensional heat equation in the standard form as
au
? = C
2 a2u
, (07 Marks)
et ex-
I
1
dx using Simpson's ? rule by taking 7 ordinates.
+ x
-
8
e. Evaluate
3
th
(07 Marks)
18MAT21
OR
6 a. Solve
a-
z
oz
ax ax
+ z = 0 such that z = ey where x = 0 and ? =1 when x = 0.
ax
Solve (mz ? ny) ?
az
+ (nx ? e )
az
= ey mx
ax av
c. Find all possible solutions of one dimensional wave equation
method of separation of variables.
Module-4
' (n +1)"
n
7 a.
Discuss the nature of the series 1 x .
n-.1 n
n+I

b.
(06 Marks)
(07 Marks)
using the
(07 Marks)
(06 Marks)
a'u
= c'
a'u
ate ax'
b. With usual notation prove that J
i:2
(x) = ?
2
? sin x (07 Marks)
nx
c. if x
3
+ 2x
2
? x + 1 = aP3 + bP2 el
)
! + dPo , find a, b, c and d using Legendre's polynomial.
(07 Marks, _
OR
8 a. Discuss the nature of the series
X X
2
X
--
1.2 3.4 3.4
(06 Marks)
h.
Obtain the series solution of Legendre's differential equation in terms of P
n
(x)
(1 ? x
2
)y" ? 2xy' + n(n+l)y = 0 (07 Marks)
c. Express x
4
? 3x
2
+ x interms of Legendre's polynomial. (07 Marks)
Module-5
9 a. Find the real root of the equation xsinx + cosx = 0 near x = TC using Newton-Raphson
method. Carry out 3 iterations. (06 Marks)
b. From the following data, find the number of students who have obtained (i) less than 45
marks (ii)between 40 and 45 marks.
Marks 30 ? 40 40 ? 50 50 ? 60 60 ? 70 70 - 80
No. of Students 31 42 51 35 31
(07 Marks)
10 a.
b.
OR
Find the real root of the equation x
login
x = L2 which
Regula-Falsi method.
Using Lagrange's int o
lies between 2 and 3 using
(06 Marks)
iven data:
(07 Marks)
x 0 1 2 5
y 2 3 12 147
5.2
c.
Evaluate i
s
log
e
x dx using Weddle's rule by taking six equal parts. (07 Marks)
4
2 of 2
? et
fr
. N
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