Download VTU ((Visvesvaraya Technological University) B.E/B-Tech 2019 July ( Bachelor of Engineering) First & Second Semester (1st Semester & 2nd Semester) 15 Scheme 15MAT31 Engineering Mathematics III Question Paper
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.
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CruKom
RV
15MAT31
C
0
Envrt
Third Semester S.E. Degree Examination, June/July 2019
Engineering Mathematics - III
Time: 3 hrs.
Note: Answer any FIVE full questions, choosing
ONE full question from each module.
Module-1
1 a. Obtain the Fourier series for the function :
? 7t in-Tc
x in 0 < x < TE
Max. Marks: 80
Hence deduce that
I
7c
2
?
(
2n-1 r 8
(08 Marks)
b. Express y as a Fourier series up to the second harmonics, given :
x 0 IX 2% Tr
4% 5
% 2n
y 1.98 1.30 1.05 1.30 -0.88 -0.25 1.98
(08 Marks)
OR
2 a. Obtain the Fourier series for the function A x) = 2x - x
2
in 0 x 5 2. (08 Marks)
b.
Obtain the constant term and the first two coefficients in the only Fourier cosine series for
given data :
x 0 1 2 3 4 5
y 4 8 15 7 6 2
Module-2
(08 Marks)
a. Find the Fourier transform of .
e
-ax
(06 Marks)
b. Find Fourier the sine transform of , a > 0 . (05 Marks)
x
c.
Obtain the z - transform of sin nO and cos nO (05 Marks)
OR
a. Find the inverse cosine transform of F(a) =
- a, 0 < a <1
0, a >1
Hence evaluate
0
b. Find inverse Z -
c. Solve the difference
z - transforms.
sin
21
. s
dt .
6
Yri
i + 9y1 = 2
11
with
(06 Marks)
(05 Marks)
y
0
= 0, y, = 0 , using
(05 Marks)
t
3z
2
+ 2z
transtbrm of
(5z -1)(5z + 2)
equation
Yn+2
3
4
1 of 3
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Societ
y
.
?
/.
CruKom
RV
15MAT31
C
0
Envrt
Third Semester S.E. Degree Examination, June/July 2019
Engineering Mathematics - III
Time: 3 hrs.
Note: Answer any FIVE full questions, choosing
ONE full question from each module.
Module-1
1 a. Obtain the Fourier series for the function :
? 7t in-Tc
x in 0 < x < TE
Max. Marks: 80
Hence deduce that
I
7c
2
?
(
2n-1 r 8
(08 Marks)
b. Express y as a Fourier series up to the second harmonics, given :
x 0 IX 2% Tr
4% 5
% 2n
y 1.98 1.30 1.05 1.30 -0.88 -0.25 1.98
(08 Marks)
OR
2 a. Obtain the Fourier series for the function A x) = 2x - x
2
in 0 x 5 2. (08 Marks)
b.
Obtain the constant term and the first two coefficients in the only Fourier cosine series for
given data :
x 0 1 2 3 4 5
y 4 8 15 7 6 2
Module-2
(08 Marks)
a. Find the Fourier transform of .
e
-ax
(06 Marks)
b. Find Fourier the sine transform of , a > 0 . (05 Marks)
x
c.
Obtain the z - transform of sin nO and cos nO (05 Marks)
OR
a. Find the inverse cosine transform of F(a) =
- a, 0 < a <1
0, a >1
Hence evaluate
0
b. Find inverse Z -
c. Solve the difference
z - transforms.
sin
21
. s
dt .
6
Yri
i + 9y1 = 2
11
with
(06 Marks)
(05 Marks)
y
0
= 0, y, = 0 , using
(05 Marks)
t
3z
2
+ 2z
transtbrm of
(5z -1)(5z + 2)
equation
Yn+2
3
4
1 of 3
15MAT,
Module-3
5 a. Find the lines of regression and the coefficient of correlation for the data :
x 1 2 3 4 5 6 7
y 9 8 10 12 11 13 14
(06 Marks)
b.
Fit a second degree polynomial to the data :
x 0 1 2 3 4
y 1 1.8 1.3 2.5 6.3
(05 Marks)
c- Find the real root of the equation x sin x + cos x = 0 near x = rc, by using Newton ? Raphson
method upto four decimal places. (05 Marks)
OR
6 a. in a partially destroyed laboratory record, only the lines of regression of y on x and x on y
are available as 4x ? 5y + 33 = 0 and 20x ? 9y = 107 respectively. Calculate x, y and the
coefficient of correlation between x and y. (06 Marks)
b.
Fit a curve of the type y = ae
bx
to the data :
x 5 15 20 30 35 40
y 10 14 25 40 50 62
(05 Marks)
C.
Solve cos x = 3x ? 1 by using Regula ? Falsi method correct upto three decimal places,
(Carryout two approximations). (05 Marks)
Module-4
7 a. Give f(40) = 184, f(50) = 204, 1(60) = 226, I(70) = 250, f(80) = 276, f(90) = 304. Find f(38)
using Newton's forward interpolation formula. (06 Marks)
b. Find the interpolating polynomial for the data :
x 0 1 2 5
Y
2 3 12 147
By using Lagrange's interpolating formula. (05 Marks)
0.3
c.
Use Simpson's 8th rule to evaluate (1 ?8)(
3
)
2
dx considering 3 equal intervals.
0
(05 Marks)?
OR
8 a. The area of a circle (A) corresponding to diameter (D) is given below :
D 80 85 90 95 100
A 5026 5674 6362 7088 7854
Find the area corresponding to diameter 105, using an appropriate interpolation formula.
(06 Marks)
b.
Given the values :
x 5 7 11 13 17
g ) 150 392 1452 2366 5202
Evaluate f(9) using Newton's divided difference formula. (05 Marks)
c. Evaluate f
`
dx by Weddle's rule taking seven ordinates. (05 Marks)
0 1+ x
.,,,
......0:... 3oc
ito
,
t)
,
: 2 of 3
FirstRanker.com - FirstRanker's Choice
Societ
y
.
?
/.
CruKom
RV
15MAT31
C
0
Envrt
Third Semester S.E. Degree Examination, June/July 2019
Engineering Mathematics - III
Time: 3 hrs.
Note: Answer any FIVE full questions, choosing
ONE full question from each module.
Module-1
1 a. Obtain the Fourier series for the function :
? 7t in-Tc
x in 0 < x < TE
Max. Marks: 80
Hence deduce that
I
7c
2
?
(
2n-1 r 8
(08 Marks)
b. Express y as a Fourier series up to the second harmonics, given :
x 0 IX 2% Tr
4% 5
% 2n
y 1.98 1.30 1.05 1.30 -0.88 -0.25 1.98
(08 Marks)
OR
2 a. Obtain the Fourier series for the function A x) = 2x - x
2
in 0 x 5 2. (08 Marks)
b.
Obtain the constant term and the first two coefficients in the only Fourier cosine series for
given data :
x 0 1 2 3 4 5
y 4 8 15 7 6 2
Module-2
(08 Marks)
a. Find the Fourier transform of .
e
-ax
(06 Marks)
b. Find Fourier the sine transform of , a > 0 . (05 Marks)
x
c.
Obtain the z - transform of sin nO and cos nO (05 Marks)
OR
a. Find the inverse cosine transform of F(a) =
- a, 0 < a <1
0, a >1
Hence evaluate
0
b. Find inverse Z -
c. Solve the difference
z - transforms.
sin
21
. s
dt .
6
Yri
i + 9y1 = 2
11
with
(06 Marks)
(05 Marks)
y
0
= 0, y, = 0 , using
(05 Marks)
t
3z
2
+ 2z
transtbrm of
(5z -1)(5z + 2)
equation
Yn+2
3
4
1 of 3
15MAT,
Module-3
5 a. Find the lines of regression and the coefficient of correlation for the data :
x 1 2 3 4 5 6 7
y 9 8 10 12 11 13 14
(06 Marks)
b.
Fit a second degree polynomial to the data :
x 0 1 2 3 4
y 1 1.8 1.3 2.5 6.3
(05 Marks)
c- Find the real root of the equation x sin x + cos x = 0 near x = rc, by using Newton ? Raphson
method upto four decimal places. (05 Marks)
OR
6 a. in a partially destroyed laboratory record, only the lines of regression of y on x and x on y
are available as 4x ? 5y + 33 = 0 and 20x ? 9y = 107 respectively. Calculate x, y and the
coefficient of correlation between x and y. (06 Marks)
b.
Fit a curve of the type y = ae
bx
to the data :
x 5 15 20 30 35 40
y 10 14 25 40 50 62
(05 Marks)
C.
Solve cos x = 3x ? 1 by using Regula ? Falsi method correct upto three decimal places,
(Carryout two approximations). (05 Marks)
Module-4
7 a. Give f(40) = 184, f(50) = 204, 1(60) = 226, I(70) = 250, f(80) = 276, f(90) = 304. Find f(38)
using Newton's forward interpolation formula. (06 Marks)
b. Find the interpolating polynomial for the data :
x 0 1 2 5
Y
2 3 12 147
By using Lagrange's interpolating formula. (05 Marks)
0.3
c.
Use Simpson's 8th rule to evaluate (1 ?8)(
3
)
2
dx considering 3 equal intervals.
0
(05 Marks)?
OR
8 a. The area of a circle (A) corresponding to diameter (D) is given below :
D 80 85 90 95 100
A 5026 5674 6362 7088 7854
Find the area corresponding to diameter 105, using an appropriate interpolation formula.
(06 Marks)
b.
Given the values :
x 5 7 11 13 17
g ) 150 392 1452 2366 5202
Evaluate f(9) using Newton's divided difference formula. (05 Marks)
c. Evaluate f
`
dx by Weddle's rule taking seven ordinates. (05 Marks)
0 1+ x
.,,,
......0:... 3oc
ito
,
t)
,
: 2 of 3
15MAT31
Module-5
a. Using Green's theorem, evaluate (2x
2
y
2
)dx + (x
2
+ y
2
)dy where C is the triangle
formed by the lines x = 0, y = 0 and x + y 1. (06 Marks)
b. Verify Stoke's theorem for f = (2x ? y)i ? yz
2
j ? y zk for the upper half of the sphere
x
2
+ y
2
+ z
2
=
1
. (05 Marks)
c. Find the extermal of the functional r { y
2
+ (y
1
)
2
+ 2ye
x
}clx (05 Marks)
x
i
OR
10 a_ Using Gauss divergence theorem, evaluate f f ds , where f = 4xzi ? y
2
j + yzk and s is
the surface of the cube bounded by x ? - - - - - - - - - - - - - - - - - - - - - - 0, x ? 1, y ? 0, y ? 1, z ? 0, z ? 1. (05 Marks)
b. A heavy cable hangs freely under the gravity between two fixed points. Show that the shape
of the cable is a Catenary_ (06 Marks)
7t
/
C.
Find the extermal of the functional .11
1 2
? y
2
+ 4y cos x Idx , give that y = 0 = y(7t/2).
0
(05 Marks)
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This post was last modified on 01 January 2020