Download Visvesvaraya Technological University (VTU) BE ( Bachelor of Engineering) ME (Mechanical Engineering) 2017 Scheme 2020 January Previous Question Paper 3rd Sem 17MAT31 Engineering Mathematics III
17MAT31
USN
Third Semester B.E. Degree Examination, Dec.2019/Jan.2020
Engineering Mathematics  III
Time: 3 hrs. Max. Marks: 100
Note: Answer any FIVE full questions, choosing ONE full question from each module.
Module1
1 a. Find the Fourier series expansion of f(x) = x in (7T, 7). hence deduce that

(08 Marks)
b. Find the half range cosine series for the function f(x) = ( x  1)
2
in 0 < x < 1. (06 Marks)
c. Express y as a Fourier series upto first harmonics given :
x 0 60? 120? 180? 240? 300?
y 7.9 7.2 3.6 0.5 0.9 6.8
(06 Marks)
OR
2 a. Obtain the Fourier series for the function :
f(x) =
. 3
1+
4x
? m ? < x 0
3 2
4x
3
1 in 0 x <
3 2
n
2
12
1
1
2
1

7"
2

1
3
2
1
4

it' 1 1
Hence deduce that ?=,+?, + ?
1
, +.
8 1
_
3
,
5
.
x in 0
/
2
IT ? x in r
Show that the half range sine series as
[
f(x)
. _4
sin x
sin 3x sin 5x 
n
c.
Obtain the Fourier series upto first harmonics given :
x 0 1 2 3 4 5 6
y 9 18 24 28 26 20 9
Module2
3 a. Find the complex Fourier transform of the function :
1 for i sn x
f(x) = and hence evaluate f
dx .
0 for lxj>a
? 0
x
Find the Fourier cosine transform of e
x
.
Solve by using z  transforms u
n
+2  4un = 0 given that u0 = 0 and ul = 2.
b. If f(x)
3
2
5 2
b.
c.
(08 Marks)
(06 Marks)
(06 Marks)
(08 Marks)
(06 Marks)
(06 Marks)
1 of 3
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17MAT31
USN
Third Semester B.E. Degree Examination, Dec.2019/Jan.2020
Engineering Mathematics  III
Time: 3 hrs. Max. Marks: 100
Note: Answer any FIVE full questions, choosing ONE full question from each module.
Module1
1 a. Find the Fourier series expansion of f(x) = x in (7T, 7). hence deduce that

(08 Marks)
b. Find the half range cosine series for the function f(x) = ( x  1)
2
in 0 < x < 1. (06 Marks)
c. Express y as a Fourier series upto first harmonics given :
x 0 60? 120? 180? 240? 300?
y 7.9 7.2 3.6 0.5 0.9 6.8
(06 Marks)
OR
2 a. Obtain the Fourier series for the function :
f(x) =
. 3
1+
4x
? m ? < x 0
3 2
4x
3
1 in 0 x <
3 2
n
2
12
1
1
2
1

7"
2

1
3
2
1
4

it' 1 1
Hence deduce that ?=,+?, + ?
1
, +.
8 1
_
3
,
5
.
x in 0
/
2
IT ? x in r
Show that the half range sine series as
[
f(x)
. _4
sin x
sin 3x sin 5x 
n
c.
Obtain the Fourier series upto first harmonics given :
x 0 1 2 3 4 5 6
y 9 18 24 28 26 20 9
Module2
3 a. Find the complex Fourier transform of the function :
1 for i sn x
f(x) = and hence evaluate f
dx .
0 for lxj>a
? 0
x
Find the Fourier cosine transform of e
x
.
Solve by using z  transforms u
n
+2  4un = 0 given that u0 = 0 and ul = 2.
b. If f(x)
3
2
5 2
b.
c.
(08 Marks)
(06 Marks)
(06 Marks)
(08 Marks)
(06 Marks)
(06 Marks)
1 of 3
17
OR
4 a. Find the Fourier sine and Cosine transforms of :
f
{x 0 < x < 2
(x) =
0 elsewhere .
(08 Marks)
b. Find the Z ? transform of : i) n
2
ii) ne'. (06 Marks)
c. Obtain the inverse Z ? transform of
2z
2
+3z
(z + 2)(z ? 4)
(06 Marks)
Module3
5 a. Obtain the lines of regression and hence find the coefficient of correlation for the data :
x 1 3 4 2 5 8 9 10 13 15
y 8 6 10 8 12 16 16 10 32 32
(08 Marks)
b.
Fit a parabola y = ax
2
+ bx + c in the least square sense for the data :
x 1 2 3 4 5
y 10 12 13 16 19
(06 Marks)
C. Find the root of the equation xe
x
? cosx = 0 by Regula ? Falsi method correct to three
decimal places in (0, 1). (06 Marks)
OR
6 a. If 8x ? lOy + 66 = 0 and 40x ? 18y = 214 are the two regression lines, find the mean of x's,
mean of y's and the coefficient of correlation. Find a
y
if a
= 3.
(08 Marks)
b. Fit an exponential curve of the form y = ae
bx
by the method of least squares for the data :
No. of petals 5 6 7 8 9 10
No. of flowers 133 55 23 7 2 2
(06 Marks)
c. Using Newton?Raphson method, find the root that lies near x = 4.5 of the equation tanx = x
correct to four decimal places. (06 Marks)
Module4
7 a. From the following table find the number of students who have obtained marks :
i) less than 45 ii) between 40 and 45.
Marks 30 ? 40 40 ? 50 50 ? 60 60 ? 70 70 ? 80
No. of students 31 42 51 35 31
(06 Marks)
b. Using Newton's divided difference formula construct an interpolating polynomial for the
following data :
x 4 5 7 10 11 13
f(x) 48 100 294 900 1210 2028
and hence find g8).
Evaluate
i dx
taking seven ordinates by applying Simpson's %
th
rule. (06 Marks)
6
1
1 + x
(08 Marks)
c.
2 of 3
FirstRanker.com  FirstRanker's Choice
? MEE
17MAT31
USN
Third Semester B.E. Degree Examination, Dec.2019/Jan.2020
Engineering Mathematics  III
Time: 3 hrs. Max. Marks: 100
Note: Answer any FIVE full questions, choosing ONE full question from each module.
Module1
1 a. Find the Fourier series expansion of f(x) = x in (7T, 7). hence deduce that

(08 Marks)
b. Find the half range cosine series for the function f(x) = ( x  1)
2
in 0 < x < 1. (06 Marks)
c. Express y as a Fourier series upto first harmonics given :
x 0 60? 120? 180? 240? 300?
y 7.9 7.2 3.6 0.5 0.9 6.8
(06 Marks)
OR
2 a. Obtain the Fourier series for the function :
f(x) =
. 3
1+
4x
? m ? < x 0
3 2
4x
3
1 in 0 x <
3 2
n
2
12
1
1
2
1

7"
2

1
3
2
1
4

it' 1 1
Hence deduce that ?=,+?, + ?
1
, +.
8 1
_
3
,
5
.
x in 0
/
2
IT ? x in r
Show that the half range sine series as
[
f(x)
. _4
sin x
sin 3x sin 5x 
n
c.
Obtain the Fourier series upto first harmonics given :
x 0 1 2 3 4 5 6
y 9 18 24 28 26 20 9
Module2
3 a. Find the complex Fourier transform of the function :
1 for i sn x
f(x) = and hence evaluate f
dx .
0 for lxj>a
? 0
x
Find the Fourier cosine transform of e
x
.
Solve by using z  transforms u
n
+2  4un = 0 given that u0 = 0 and ul = 2.
b. If f(x)
3
2
5 2
b.
c.
(08 Marks)
(06 Marks)
(06 Marks)
(08 Marks)
(06 Marks)
(06 Marks)
1 of 3
17
OR
4 a. Find the Fourier sine and Cosine transforms of :
f
{x 0 < x < 2
(x) =
0 elsewhere .
(08 Marks)
b. Find the Z ? transform of : i) n
2
ii) ne'. (06 Marks)
c. Obtain the inverse Z ? transform of
2z
2
+3z
(z + 2)(z ? 4)
(06 Marks)
Module3
5 a. Obtain the lines of regression and hence find the coefficient of correlation for the data :
x 1 3 4 2 5 8 9 10 13 15
y 8 6 10 8 12 16 16 10 32 32
(08 Marks)
b.
Fit a parabola y = ax
2
+ bx + c in the least square sense for the data :
x 1 2 3 4 5
y 10 12 13 16 19
(06 Marks)
C. Find the root of the equation xe
x
? cosx = 0 by Regula ? Falsi method correct to three
decimal places in (0, 1). (06 Marks)
OR
6 a. If 8x ? lOy + 66 = 0 and 40x ? 18y = 214 are the two regression lines, find the mean of x's,
mean of y's and the coefficient of correlation. Find a
y
if a
= 3.
(08 Marks)
b. Fit an exponential curve of the form y = ae
bx
by the method of least squares for the data :
No. of petals 5 6 7 8 9 10
No. of flowers 133 55 23 7 2 2
(06 Marks)
c. Using Newton?Raphson method, find the root that lies near x = 4.5 of the equation tanx = x
correct to four decimal places. (06 Marks)
Module4
7 a. From the following table find the number of students who have obtained marks :
i) less than 45 ii) between 40 and 45.
Marks 30 ? 40 40 ? 50 50 ? 60 60 ? 70 70 ? 80
No. of students 31 42 51 35 31
(06 Marks)
b. Using Newton's divided difference formula construct an interpolating polynomial for the
following data :
x 4 5 7 10 11 13
f(x) 48 100 294 900 1210 2028
and hence find g8).
Evaluate
i dx
taking seven ordinates by applying Simpson's %
th
rule. (06 Marks)
6
1
1 + x
(08 Marks)
c.
2 of 3
17MAT31
OR
8 a. In a table given below, the values of y are consecutive terms of a series of which 23.6 is the
6
th
term. Find the first and tenth terms of the series by Newton's formulas.
x 3 4 5 6 7 8 9
y 4.8 8.4 14.5 23.6 36.2 52.8 73.9
(08 Marks)
b. Fit an interpolating polynomial of the form x = f(y) for data and hence find x(5) given :
x 2 10 17
y 1 3 4
0.6
c.
Use Simpson's
3
rd
ruleto find I e

x

dx by taking 6 subintervals.
(06 Marks)
(06 Marks)
Module5
9 a. Verify Green's theorem in the plane for ili
c
(3x
2
? 8y
2
)dx + (4y ? 6xy)dy where C is the
closed curve bounded by y = srx and y = x
2
. (08 Marks)
b.
Evaluate
e
i xydx + xy
2
dy by Stoke's theorem where C is the square in the x ? y plane with
vertices (1, 0)(1, 0)(0, 1)(0, ?1). (06 Marks)
c. Prove that Catenary is the curve which when rotated about a line generates a surface of
minimum area. (06 Marks)
OR
10 a. If F = 2xyl+ yz
2
xz k and S is the rectangular parallelepiped bounded by x = 0, y = 0,
c.
z = 0, x = 2, y = 1, z = 3 evaluate ^ A ?
F. n us ?
Derive Euler's equation in the standard form viz ?
of
?
[ af
a
y
dx ay'
= 0 .
n.
2
Find the external of the functional I = J(y
2
? y'
2
? 2y sin x )dx under the end
(08 Marks)
(06 Marks)
conditions
y(0) = y(n/2) = O. (06 Marks)
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This post was last modified on 02 March 2020