Download VTU ((Visvesvaraya Technological University) B.E/B-Tech 2019 July ( Bachelor of Engineering) First & Second Semester (1st Semester & 2nd Semester) 17 Scheme 17MAT31 Engineering Mathematics III Question Paper
Third Semester B.E. Degree Examination, June/July 2019
Engineering Mathematics - ill
Time: 3 hrs.
Max. Marks: 100
Note: Answer any FIVE full questions, choosing ONE full question from each module.
Module-1
1 a.
Obtain the fourier series of the function f(x) = x ? x
2
in
?
TC X 7 and
(08 Marks)
b. Obtain the Half Range Fourier cosine series for the 1(x) = sin x in [0, rt].
(06 Marks)
c. Obtain the constant term and the coefficients of first sine and cosine terms in the fourier
expansion of y given
x: 0 1 2 3 4 5
y:
9 18 24 28 26 20
(06 Marks)
OR
Tr
hence deduce
12
=
1 1
? 1?+?
1
2
2 3
1
+
4
2
2
a. Obtain the
-
Fourier series of gx)
it 1 1 1
4 3 5 7
in [0, 27] and hence deduce that
(08 Marks)
7t
2
b. Find the fourier half range cosine series of the function f(x) = 2x ? x
2
in [0, 3]. (06 Marks)
C. Expre
x: 0 _ 30 60 90 120 150 180 210 240 270 300 330
y : 1.8 1,1 0.30 0.16 1.5 1.3 2.16 1.25 1.3 1.52 1.76 2.0
(06 Marks)
Module-2
3 a. Find the fourier transform of 1
--
(x) =
x a
0
x I > a
and hence deduce
i
a
snx?xcosx
x'
dx =
4
b. Find the fourier sine transform of
(08 Marks)
a
ix
and hence evaluate
.1 X sin x
; a > 0 (06 Marks)
+ x
-
Important Note :
C. Obtain the z-transform of cosnO and sin no.
OR
4 a. Find the fourier transform of f(x)
b. Find the fourier cosine transform of gx) where
x ; 0 < x <1
f(x) = ? x ; 1 < x < 2
x > 2
(06 Marks)
(08 Marks)
(06 Marks)
I oft
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17MAT31
Third Semester B.E. Degree Examination, June/July 2019
Engineering Mathematics - ill
Time: 3 hrs.
Max. Marks: 100
Note: Answer any FIVE full questions, choosing ONE full question from each module.
Module-1
1 a.
Obtain the fourier series of the function f(x) = x ? x
2
in
?
TC X 7 and
(08 Marks)
b. Obtain the Half Range Fourier cosine series for the 1(x) = sin x in [0, rt].
(06 Marks)
c. Obtain the constant term and the coefficients of first sine and cosine terms in the fourier
expansion of y given
x: 0 1 2 3 4 5
y:
9 18 24 28 26 20
(06 Marks)
OR
Tr
hence deduce
12
=
1 1
? 1?+?
1
2
2 3
1
+
4
2
2
a. Obtain the
-
Fourier series of gx)
it 1 1 1
4 3 5 7
in [0, 27] and hence deduce that
(08 Marks)
7t
2
b. Find the fourier half range cosine series of the function f(x) = 2x ? x
2
in [0, 3]. (06 Marks)
C. Expre
x: 0 _ 30 60 90 120 150 180 210 240 270 300 330
y : 1.8 1,1 0.30 0.16 1.5 1.3 2.16 1.25 1.3 1.52 1.76 2.0
(06 Marks)
Module-2
3 a. Find the fourier transform of 1
--
(x) =
x a
0
x I > a
and hence deduce
i
a
snx?xcosx
x'
dx =
4
b. Find the fourier sine transform of
(08 Marks)
a
ix
and hence evaluate
.1 X sin x
; a > 0 (06 Marks)
+ x
-
Important Note :
C. Obtain the z-transform of cosnO and sin no.
OR
4 a. Find the fourier transform of f(x)
b. Find the fourier cosine transform of gx) where
x ; 0 < x <1
f(x) = ? x ; 1 < x < 2
x > 2
(06 Marks)
(08 Marks)
(06 Marks)
I oft
17MAT31
c. Solve Li
n o + 6u,i-h1+ 9u,, =
with u
0
= ul- 0 using z-transform. (06 Marks)
Module-3
5 a. Fit a straight line y = ax + b for the following data by the method of least squares.
x : 0 3 4 6 8 9 11 14
y
. 1 2 4 4 5 ill 8 9
(08 Marks)
b. Calculate the coefficient of correlation for the data:
x: 92 89 87 86 83 77 70 63 53 50
y : 86 83 91 77 68 85 54 82 37 57
(06 Marks)
c. Compute the real root of xlogi
o
x - 1.2 - 0 by the method of false position. Carry out 3
iterations in (2, 3y (06 Marks)
OR
6 a. Fit a second degree parabola to the following data y = a + bx + cx
2
.
x: 1 1.5 2 2.5 3 3.5
y: 1.1 1.3 1.6 2 2.7 3 A 4.1
(08 Marks)
c.
b. If 0 is the angle between two regression lines, show that
1 -r
-
cf,a,
tan 0 = , ? ; explain significance of r = 0 and r ?1.
r 4- Cr
Using Newton Raphson method, find the real root of the equation 3x = cos
x
o
= 0.5. Carry out 3 iterations.
(06 Marks)
x + 1 near
(06 Marks)
Module-4
7 a. From the following table, estimate the number of students who obtained marks between
40 and 45.
Marks : - 30 - 40 40 - 50 50 - 60 60 - 70 70 - 80
No. of students 31 42 51 35 31
(08 Marks)
b. Use Newton's dividend formula to find f(9) for the data:
x : 5 7 11 13 17
f(x) : 150 392 1452 2366 5202
(06 Marks)
c.
8 a.
Find the approximate value
6 equal parts.
The area A of a circle of
of
diameter
z 2
Simpson's
-
1 rd
rule
3
the following values:
by dividing
rr
0,
(06
into
Marks)
Vcos 0 dO by
OR
d is given for
80 85 90 95 100
a 5026 5674 6362 7088 7854
Calculate the area of circle of diameter 105 by Newton's backward formula. (08 Marks)
b.
Using Lagrange's interpolation formula to find the polynomial which passes through the
points (0, -12), (1, 0), (3, 6), (4, 12). (06 Marks)
5 2
C.
Evaluate f log
e
x dx taking 6 equal parts by applying Weddle's rule. (06 Marks)
4
is 1
? F
...'
Ci4iNtIni
isa
'
4
??
FirstRanker.com - FirstRanker's Choice
17MAT31
Third Semester B.E. Degree Examination, June/July 2019
Engineering Mathematics - ill
Time: 3 hrs.
Max. Marks: 100
Note: Answer any FIVE full questions, choosing ONE full question from each module.
Module-1
1 a.
Obtain the fourier series of the function f(x) = x ? x
2
in
?
TC X 7 and
(08 Marks)
b. Obtain the Half Range Fourier cosine series for the 1(x) = sin x in [0, rt].
(06 Marks)
c. Obtain the constant term and the coefficients of first sine and cosine terms in the fourier
expansion of y given
x: 0 1 2 3 4 5
y:
9 18 24 28 26 20
(06 Marks)
OR
Tr
hence deduce
12
=
1 1
? 1?+?
1
2
2 3
1
+
4
2
2
a. Obtain the
-
Fourier series of gx)
it 1 1 1
4 3 5 7
in [0, 27] and hence deduce that
(08 Marks)
7t
2
b. Find the fourier half range cosine series of the function f(x) = 2x ? x
2
in [0, 3]. (06 Marks)
C. Expre
x: 0 _ 30 60 90 120 150 180 210 240 270 300 330
y : 1.8 1,1 0.30 0.16 1.5 1.3 2.16 1.25 1.3 1.52 1.76 2.0
(06 Marks)
Module-2
3 a. Find the fourier transform of 1
--
(x) =
x a
0
x I > a
and hence deduce
i
a
snx?xcosx
x'
dx =
4
b. Find the fourier sine transform of
(08 Marks)
a
ix
and hence evaluate
.1 X sin x
; a > 0 (06 Marks)
+ x
-
Important Note :
C. Obtain the z-transform of cosnO and sin no.
OR
4 a. Find the fourier transform of f(x)
b. Find the fourier cosine transform of gx) where
x ; 0 < x <1
f(x) = ? x ; 1 < x < 2
x > 2
(06 Marks)
(08 Marks)
(06 Marks)
I oft
17MAT31
c. Solve Li
n o + 6u,i-h1+ 9u,, =
with u
0
= ul- 0 using z-transform. (06 Marks)
Module-3
5 a. Fit a straight line y = ax + b for the following data by the method of least squares.
x : 0 3 4 6 8 9 11 14
y
. 1 2 4 4 5 ill 8 9
(08 Marks)
b. Calculate the coefficient of correlation for the data:
x: 92 89 87 86 83 77 70 63 53 50
y : 86 83 91 77 68 85 54 82 37 57
(06 Marks)
c. Compute the real root of xlogi
o
x - 1.2 - 0 by the method of false position. Carry out 3
iterations in (2, 3y (06 Marks)
OR
6 a. Fit a second degree parabola to the following data y = a + bx + cx
2
.
x: 1 1.5 2 2.5 3 3.5
y: 1.1 1.3 1.6 2 2.7 3 A 4.1
(08 Marks)
c.
b. If 0 is the angle between two regression lines, show that
1 -r
-
cf,a,
tan 0 = , ? ; explain significance of r = 0 and r ?1.
r 4- Cr
Using Newton Raphson method, find the real root of the equation 3x = cos
x
o
= 0.5. Carry out 3 iterations.
(06 Marks)
x + 1 near
(06 Marks)
Module-4
7 a. From the following table, estimate the number of students who obtained marks between
40 and 45.
Marks : - 30 - 40 40 - 50 50 - 60 60 - 70 70 - 80
No. of students 31 42 51 35 31
(08 Marks)
b. Use Newton's dividend formula to find f(9) for the data:
x : 5 7 11 13 17
f(x) : 150 392 1452 2366 5202
(06 Marks)
c.
8 a.
Find the approximate value
6 equal parts.
The area A of a circle of
of
diameter
z 2
Simpson's
-
1 rd
rule
3
the following values:
by dividing
rr
0,
(06
into
Marks)
Vcos 0 dO by
OR
d is given for
80 85 90 95 100
a 5026 5674 6362 7088 7854
Calculate the area of circle of diameter 105 by Newton's backward formula. (08 Marks)
b.
Using Lagrange's interpolation formula to find the polynomial which passes through the
points (0, -12), (1, 0), (3, 6), (4, 12). (06 Marks)
5 2
C.
Evaluate f log
e
x dx taking 6 equal parts by applying Weddle's rule. (06 Marks)
4
is 1
? F
...'
Ci4iNtIni
isa
'
4
??
17MAT31
Module-5
9 a. If F = 3xyi ? y
-
j , evaluate f F.dr where 'C' is arc of parabola y = 2x
2
from (0, 0) to (1, 2)
(08 Marks)
b. Evaluate by Stokes theorem
(sin zdx ? cos x dy + sin y dz), where C is the boundary of the rectangle 0 < x < ;
0 y 1, z= 3 (06 Marks)
x =
C.
Prove that the necessary condition thr the I = if(x,y,y')dx to be extremum is
af d
= 0 (06 Marks)
ay dx ay
OR
10 a. Using Green's theorem evaluate f(3x
2
8y' )1x + (4.y ? 6xy)dy , where C is the boundary of
the region bounded by the lines x = 0, y = 0, x + y = 1. (08 Marks)
(,
Find the external value of
.
1
.
[( y')
2
?
2
+4ycosxidx . Given that y(0) = 0, y = O.
b.
1)
(06 Marks)
c. Prove that the shortest distance between two points in a plane is along a straight line joining
them. (06 Marks)
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This post was last modified on 01 January 2020