Download VTU BE 2020 Jan CSE Question Paper 18 Scheme 3rd Sem 18MAT31 Transform Calculus, Fourier Series and Numerical Techniques

Download Visvesvaraya Technological University (VTU) BE ( Bachelor of Engineering) CSE 2018 Scheme 2020 January Previous Question Paper 3rd Sem 18MAT31 Transform Calculus, Fourier Series and Numerical Techniques

as
s
\
?jtanh ?
s , 2
c. Employ Laplace transform to solve
d'y dy
,

= 0, y(0) = y, (o) = 3.
dt
2
dt
OR
2
a. Find (i)
J
s
-
?3s+ 4}
s3
(
(ii) cot
-
'
2
Find the inverse Laplace transform of.
I
s(s
-
+1)
b.

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?

USN

18MAT31

Third Semester B.E. Degree Examination, Dec.2019/Jan.2020
Transform Calculus, Fourier Series and Numerical
Techniques
Time: 3 hrs. Max. Marks: 100
Note: Answer any FIVE full questions, choosing ONE full question from each module.
Module-1
1 a.
b.
Find the Laplace
(
4t+5
\2

(0
wave function
transform
(
sin 20
2

of:
(iii) t cos at .
with period 2a defined by f(t) = { f(t)
(10 Marks)
_ < 1 0 t
a
_
.
. Show that _ l
ast<2a
(i)
e
21
J
The square
(i ii)
(s + 2)(s + 3)
using convolution theorem.
(05 Marks)
(05 Marks)
(10 Marks)
(05 Marks)
c. Express f(t) =-
transformation.
2 if 0 < t <1
if l< t <-
11
in terms of unit step function and hence find its Laplace
2 2
cost t >
2
(05 Marks)
Module-2
3
a.
Obtain the Fourier series of f(x) =
2 -2 < x < 0

x 0 < x <2
b. Find the half range cosine series of, f(x) = (x + 1) in the interval 0 x
c. Express f(x) = x
2
as a Fourier series of period 27 in the interval 0 < x < 2rc .
(08 Marks)
(06 Marks)
(06 Marks)
1 of 3
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as
s
\
?jtanh ?
s , 2
c. Employ Laplace transform to solve
d'y dy
,

= 0, y(0) = y, (o) = 3.
dt
2
dt
OR
2
a. Find (i)
J
s
-
?3s+ 4}
s3
(
(ii) cot
-
'
2
Find the inverse Laplace transform of.
I
s(s
-
+1)
b.

trb
I* LIBRARY *
CHIKODI
?

USN

18MAT31

Third Semester B.E. Degree Examination, Dec.2019/Jan.2020
Transform Calculus, Fourier Series and Numerical
Techniques
Time: 3 hrs. Max. Marks: 100
Note: Answer any FIVE full questions, choosing ONE full question from each module.
Module-1
1 a.
b.
Find the Laplace
(
4t+5
\2

(0
wave function
transform
(
sin 20
2

of:
(iii) t cos at .
with period 2a defined by f(t) = { f(t)
(10 Marks)
_ < 1 0 t
a
_
.
. Show that _ l
ast<2a
(i)
e
21
J
The square
(i ii)
(s + 2)(s + 3)
using convolution theorem.
(05 Marks)
(05 Marks)
(10 Marks)
(05 Marks)
c. Express f(t) =-
transformation.
2 if 0 < t <1
if l< t <-
11
in terms of unit step function and hence find its Laplace
2 2
cost t >
2
(05 Marks)
Module-2
3
a.
Obtain the Fourier series of f(x) =
2 -2 < x < 0

x 0 < x <2
b. Find the half range cosine series of, f(x) = (x + 1) in the interval 0 x
c. Express f(x) = x
2
as a Fourier series of period 27 in the interval 0 < x < 2rc .
(08 Marks)
(06 Marks)
(06 Marks)
1 of 3
x 4 4.1 4.2 4.3
y 1 1.0049 1.0097 1.0143
2 of 3
18111A\
OR
4 a. Compute the first two harmonics of the Fourier Series of f(x) given the following table :
x? 0 60? 120? 180? 240? 300?
y 7.9 7.2 3.6 0.5 0.9 6.8
(08 Marks)
(06 Marks) b. Find the half range size series of e
x
in the interval 0 < x
TC
2

C. Obtain the Fourier series of f(x) =
X
valid in the interval (?t n) (06 Marks)
12 4
Module-3
5 a. Find the Infinite Fourier transform of e . (07 Marks)
b. Find the Fourier cosine transform of f(x) = e
-2
" + 4e
-3
" . (06 Marks)
c. Solve u?,
2
?3u,,_, + 2u,, = 3" , given u, = u, = 0 . (07 Marks)
6 a.
OR
.?
If f(x) =
)
11 for Ix
,
a
, fi nd the infinite transform of f(x) and hence evaluate dx .
(0 for Ix' > a "
b.
e,
Obtain the Z-transform of cosh nO and sinh ne .
4z
2
?2z
Find the inverse Z-transform of ,
z
-
?5z
-
+8z-4
Module-4
(07 Marks,_
(06 Marks)
(07 Marks)
7
a. Solve ?
dy
= e' ? y , y(0) = 2 using Taylor's Series method upto 4
th
degree terms and find
dx
the value of y(1.1). (07 Marks)
b. Use Runge-Kutta method of fourth order to solve ?
dy
+ y = 2x at x = 1.1 given y(1) = 3
dx
(06 Marks)
Apply Milne's predictor-corrector formulae to compute y(0.4) given ?
dy
= 2ex y , with
dx
(07 Marks)
x 0 0.1 0.2 0.3
y 2.4 2.473 3.129 4.059
OR
8 a. Given ?
dy
= x + sin y y(0) = 1. Compute y(0.4) with h = 0.2 using Euler's modified
dx
(07 Marks)
Apply Runge-Kutta fourth order method, to find y(0.1) with h = 0.1 given ?
dy
+ y + xy
2
= 0 ;
dx
y(0) = I. (06 Marks)
dy
dx
(Take h 0.1)
c.
method.
b.
C.
Using Adams-Bashforth
-
method, find y(4.4) given 5x = 2 with
FirstRanker.com - FirstRanker's Choice
as
s
\
?jtanh ?
s , 2
c. Employ Laplace transform to solve
d'y dy
,

= 0, y(0) = y, (o) = 3.
dt
2
dt
OR
2
a. Find (i)
J
s
-
?3s+ 4}
s3
(
(ii) cot
-
'
2
Find the inverse Laplace transform of.
I
s(s
-
+1)
b.

trb
I* LIBRARY *
CHIKODI
?

USN

18MAT31

Third Semester B.E. Degree Examination, Dec.2019/Jan.2020
Transform Calculus, Fourier Series and Numerical
Techniques
Time: 3 hrs. Max. Marks: 100
Note: Answer any FIVE full questions, choosing ONE full question from each module.
Module-1
1 a.
b.
Find the Laplace
(
4t+5
\2

(0
wave function
transform
(
sin 20
2

of:
(iii) t cos at .
with period 2a defined by f(t) = { f(t)
(10 Marks)
_ < 1 0 t
a
_
.
. Show that _ l
ast<2a
(i)
e
21
J
The square
(i ii)
(s + 2)(s + 3)
using convolution theorem.
(05 Marks)
(05 Marks)
(10 Marks)
(05 Marks)
c. Express f(t) =-
transformation.
2 if 0 < t <1
if l< t <-
11
in terms of unit step function and hence find its Laplace
2 2
cost t >
2
(05 Marks)
Module-2
3
a.
Obtain the Fourier series of f(x) =
2 -2 < x < 0

x 0 < x <2
b. Find the half range cosine series of, f(x) = (x + 1) in the interval 0 x
c. Express f(x) = x
2
as a Fourier series of period 27 in the interval 0 < x < 2rc .
(08 Marks)
(06 Marks)
(06 Marks)
1 of 3
x 4 4.1 4.2 4.3
y 1 1.0049 1.0097 1.0143
2 of 3
18111A\
OR
4 a. Compute the first two harmonics of the Fourier Series of f(x) given the following table :
x? 0 60? 120? 180? 240? 300?
y 7.9 7.2 3.6 0.5 0.9 6.8
(08 Marks)
(06 Marks) b. Find the half range size series of e
x
in the interval 0 < x
TC
2

C. Obtain the Fourier series of f(x) =
X
valid in the interval (?t n) (06 Marks)
12 4
Module-3
5 a. Find the Infinite Fourier transform of e . (07 Marks)
b. Find the Fourier cosine transform of f(x) = e
-2
" + 4e
-3
" . (06 Marks)
c. Solve u?,
2
?3u,,_, + 2u,, = 3" , given u, = u, = 0 . (07 Marks)
6 a.
OR
.?
If f(x) =
)
11 for Ix
,
a
, fi nd the infinite transform of f(x) and hence evaluate dx .
(0 for Ix' > a "
b.
e,
Obtain the Z-transform of cosh nO and sinh ne .
4z
2
?2z
Find the inverse Z-transform of ,
z
-
?5z
-
+8z-4
Module-4
(07 Marks,_
(06 Marks)
(07 Marks)
7
a. Solve ?
dy
= e' ? y , y(0) = 2 using Taylor's Series method upto 4
th
degree terms and find
dx
the value of y(1.1). (07 Marks)
b. Use Runge-Kutta method of fourth order to solve ?
dy
+ y = 2x at x = 1.1 given y(1) = 3
dx
(06 Marks)
Apply Milne's predictor-corrector formulae to compute y(0.4) given ?
dy
= 2ex y , with
dx
(07 Marks)
x 0 0.1 0.2 0.3
y 2.4 2.473 3.129 4.059
OR
8 a. Given ?
dy
= x + sin y y(0) = 1. Compute y(0.4) with h = 0.2 using Euler's modified
dx
(07 Marks)
Apply Runge-Kutta fourth order method, to find y(0.1) with h = 0.1 given ?
dy
+ y + xy
2
= 0 ;
dx
y(0) = I. (06 Marks)
dy
dx
(Take h 0.1)
c.
method.
b.
C.
Using Adams-Bashforth
-
method, find y(4.4) given 5x = 2 with
18MAT31
9
a.
b.
Module-5
(
Solve by Runge Kutta method
d'y = xdy
? - y
,
for x = 0.2 correct 4 decimal places,
dx
2
dx
using initial conditions y(0) = 1, y'(0) = 0, h = 0.2. (07 Marks)
of
Derive Euler's equation in the standard ? d = 0. (06 Marks)
y dx cry
c. Find the extramal of the functional, +(y') + 2ye'dx (07 Marks)
OR
10
a.
Apply Milne
d y
s predictor corrector method to compute
y d
and the following table
dx dx
of initial values:
x 0 0.1 0.2 0.3
Y
1 1.1.1.03 1.2427 1.3990
y' 1 1.2103 1.4427 1.6990
(07 Marks)
b-
Find the extramal for the functional. ?
y
f 2
- 2y sin x ix ; y(0) ? yr =1.
c. Prove that geodesics of a plane surface are straight lines.
(06 Marks)
(07 Marks)

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