Download VTU BE 2020 Jan Question Paper 18 Scheme 18MAT21 Advanced Calculus and Numerical Methods First And Second Semester

Download Visvesvaraya Technological University (VTU) BE/B.Tech First And Second Semester (1st sem and 2nd sem) 2019-2020 Jan ( Bachelor of Engineering) 18 Scheme 18MAT21 Advanced Calculus and Numerical Methods Previous Question Paper

LIBRARY
18MAT21
(13,
Second Semester B.E. Degree Examination, Dec.2019/Jan.2020
Advanced Calculus and Numerical Methods
USN
Time: 3 hrs. Max. Marks: 100
Note: Answer any FIVE full questions, choosing ONE full question from each module.
Module-1
1 a. Find the directional derivative of(= 4xz
3
? 3x
2
y
2
z at (2, -1, 2) along 2i 3j + 6k . Marks)
b. If f = V(x
3
y +7.
3
?3xyz ) find di v f and curl f (07 Marks)
c. Find the constants a and b such that F = (axy + z
3
)i + (3x' ? z)j+ (bx y)fc is irrotational.
Also find a scalar potential 4:1 if F = V. (07 Marks)
OR
2
a-
.

If = xyi + yz j + zxk evaluate F.dr where C is the curve represented by x = t,. y = t
-
,
1.
b. Using Stoke's theorem Evaluate F.dr if F = (x
2
+3/
2
)i ?2xy j taken
bounded by x = 0, x a, y 0, y = b_
c. Using divergence theorem, evaluate S
5
F.11 ds if F
(06 Marks)
round the rectangle
(07 Marks)
2 ?
yz)i y zx)j + (z
2
? xy)lc
2. Any reveal
taken around 0 < x 1, 0 < y < 1, 0 < z < 1. (07 Marks)
Module-2
3
a. Solve (4D
4
-8D
3
-7D
3
+ I 1 D + 6)y = () (06 Marks)
b.
Solve (D
2
+ 4D +3)y = e' (07 Marks)
c. Using the method of variation of parameter solve y" + 4y = tan2x. (07 Marks)
OR
Solve (D
3
? 1)y = 3 cos2x (06 Marks)
Solve x
2
y" Sxyr + 8y= 2 lOgx (07 Marks)
The differential equation of a simple pendulum is
d'x
+ W
o
x = F
0
Sinnt , where W
0
and F
()
dt`
2
5
a.
b.
c.
are constants. Also initially x = 0,
dx
= 0 solve it.
dt
Module-3
(
?+ log y
x
and z = 0,
(07 Marks)
(06 Marks)
when y is odd multiple
(07 Marks)
(07 Marks)
Find the PDE by eliminating the function from z = y
-
+ 2f
a
2 z
Gz
= 0 Solve = sin x sin y given = ?2 sin y , when x
Nay

of
2
Derive one-dimensional wave equation in usual notations.
4 a.
b.
C.
1 of 2
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LIBRARY
18MAT21
(13,
Second Semester B.E. Degree Examination, Dec.2019/Jan.2020
Advanced Calculus and Numerical Methods
USN
Time: 3 hrs. Max. Marks: 100
Note: Answer any FIVE full questions, choosing ONE full question from each module.
Module-1
1 a. Find the directional derivative of(= 4xz
3
? 3x
2
y
2
z at (2, -1, 2) along 2i 3j + 6k . Marks)
b. If f = V(x
3
y +7.
3
?3xyz ) find di v f and curl f (07 Marks)
c. Find the constants a and b such that F = (axy + z
3
)i + (3x' ? z)j+ (bx y)fc is irrotational.
Also find a scalar potential 4:1 if F = V. (07 Marks)
OR
2
a-
.

If = xyi + yz j + zxk evaluate F.dr where C is the curve represented by x = t,. y = t
-
,
1.
b. Using Stoke's theorem Evaluate F.dr if F = (x
2
+3/
2
)i ?2xy j taken
bounded by x = 0, x a, y 0, y = b_
c. Using divergence theorem, evaluate S
5
F.11 ds if F
(06 Marks)
round the rectangle
(07 Marks)
2 ?
yz)i y zx)j + (z
2
? xy)lc
2. Any reveal
taken around 0 < x 1, 0 < y < 1, 0 < z < 1. (07 Marks)
Module-2
3
a. Solve (4D
4
-8D
3
-7D
3
+ I 1 D + 6)y = () (06 Marks)
b.
Solve (D
2
+ 4D +3)y = e' (07 Marks)
c. Using the method of variation of parameter solve y" + 4y = tan2x. (07 Marks)
OR
Solve (D
3
? 1)y = 3 cos2x (06 Marks)
Solve x
2
y" Sxyr + 8y= 2 lOgx (07 Marks)
The differential equation of a simple pendulum is
d'x
+ W
o
x = F
0
Sinnt , where W
0
and F
()
dt`
2
5
a.
b.
c.
are constants. Also initially x = 0,
dx
= 0 solve it.
dt
Module-3
(
?+ log y
x
and z = 0,
(07 Marks)
(06 Marks)
when y is odd multiple
(07 Marks)
(07 Marks)
Find the PDE by eliminating the function from z = y
-
+ 2f
a
2 z
Gz
= 0 Solve = sin x sin y given = ?2 sin y , when x
Nay

of
2
Derive one-dimensional wave equation in usual notations.
4 a.
b.
C.
1 of 2
9 a. Using Newton's forward (06 Marks)
x 40 50 60 70 80 90
f(x) 184 204 226 250 276 304
18MAT21
OR
,
6
a.
Solve T
a2z
x2
= a
-
z given that when x = 0 ?
az
= a sin y and z = 0. (06 Marks)
ax
b. Solve x(y ? z) p + y (z ? x) q = z (x? y). (07 Marks)
c. Find all possible solution of U
t
= C
-
U?, one dimensional heat equation by variable separable
method. (07 Marks)
Module-4
7 a. Test for convergence for
2
1
31 4
1

(06 Marks)
2
-
3
-
4
2

Find the series solution of Legendre differential equation
(1 ? x
2
)y" - 2xy' + n(n + 1) = 0 leading to P?(x). (07 Marks)
Prove the orthogonality property of Bessel's function as
$ x j? (ax) j
n
((3x)dx = 0 a # 1 (07 Marks)
0
OR
8 a. Test for convergence for
3/ 2
x-1 1 \
--F--
-
(06 Marks)
) A
in
b. Find the series solution of Bessel differential equation x
2
y" + xy' + (n
2
? x
2
) y = 0 Leading to
(x) (07 Marks)
c. Express the polynomial x
3
+ 2x
2
? 4x + 5 interms of Legendre polynomials. (07 Marks)
Module-5
c.
b. Find the real root of the equation xlog
w
x =1.2 by Regula falsi method between 2 and 3
(Three iterations). (07 Marks)
5.2
c. Evaluate log x dx by Weddle's rule considering six intervals. (07 Marks)
4
OR
10 a Find t19) from the data by Newton's divided difference formula:
x 5 7 11 13 17
f(x) 150 392 1452 2366 5202
(06 Marks)
b. Using Newton ? Raphson method, find the real root of the equation x sin x + cosx = 0 near
X (07 Marks)
c. By using Simpson's
6
rule, evaluate
j
, by considering seven ordinates. (07 Marks)
0
1 + x
-
2 of 2
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