Download VTU BE 2020 Jan ME Question Paper 17 Scheme 3rd Sem 17MAT31 Engineering Mathematics III

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Third Semester B.E. Degree Examination, Dec.2019/Jan.2020
Engineering Mathematics - III
Time: 3 hrs. Max. Marks: 100

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Note: Answer any FIVE full questions, choosing ONE full question from each module.
Module-1
1 a.
Find the Fourier series expansion of f(x) = x -- x
2

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in (?it, n), hence deduce that
rc
-
1 1 I 1
12 = 1

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2
+ 2
2 4-
3
2

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+ 4
2
+ ?
(08 Marks)
b. Find the half range cosine series for the function f(x) = _( x 1)

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2
in 0 < x < 1. (06 Marks)
c. Express y as a Fourier series upto first harmonics given
x 0 60
0

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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 :

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1 +
4x
in
-3
? < x 0

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3 2
1 -
4x 3
in0
3 2

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'
Hence deduce that Tr
1
? = ?+ ? + +
8 12

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1

3'
1
5-

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x in 0 < x < y
2
TC ? x in V < x <
2
Show that the half range sine series as

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4
f(x) = ? [sin x ,
3
4
5

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2
sin 3x sin 5x
Obtain the Fourier series upto first harmonics given :
x 0 1 2 3 4 5 6
y 9 18 24 28 26 20 9

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(06 Marks)
Module-2
3 a. Find the complex Fourier transform of the function :
1 for I a
f(x) and hence evaluate f

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sin x
dx .
0 for lxl>a
Find the Fourier cosine transform of e
-ax

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.
Solve by using z transforms u
n
,, ? 4u
n

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= 0 given that uo = 0 and u
1
= 2.
f(x) =
b.

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if f(x)
(08 Nlarks)
(06 'Marks)
c.
=X, .

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b.
c.
(08 Marks)
(06 Marks)
(06 Marks)

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17MAT31
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Third Semester B.E. Degree Examination, Dec.2019/Jan.2020
Engineering Mathematics - III

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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 Fourier series expansion of f(x) = x -- x

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2
in (?it, n), hence deduce that
rc
-
1 1 I 1

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12 = 1
2
+ 2
2 4-
3

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2
+ 4
2
+ ?
(08 Marks)

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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

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0
120? 180? 240? 300?
y 7.9 7.2 3.6 0.5 0.9 6.8
(06 Marks)
OR

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2 a. Obtain the Fourier series for the function :
1 +
4x
in
-3

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? < x 0
3 2
1 -
4x 3
in0

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3 2
'
Hence deduce that Tr
1
? = ?+ ? + +

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8 12
1

3'
1

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5-
x in 0 < x < y
2
TC ? x in V < x <
2

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Show that the half range sine series as
4
f(x) = ? [sin x ,
3
4

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5
2
sin 3x sin 5x
Obtain the Fourier series upto first harmonics given :
x 0 1 2 3 4 5 6

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y 9 18 24 28 26 20 9
(06 Marks)
Module-2
3 a. Find the complex Fourier transform of the function :
1 for I a

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f(x) and hence evaluate f
sin x
dx .
0 for lxl>a
Find the Fourier cosine transform of e

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-ax
.
Solve by using z transforms u
n
,, ? 4u

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n
= 0 given that uo = 0 and u
1
= 2.
f(x) =

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b.
if f(x)
(08 Nlarks)
(06 'Marks)
c.

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=X, .
b.
c.
(08 Marks)
(06 Marks)

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(06 Marks)
171\
OR
4 a. Find the Fourier sine and Cosine transforms of :
x 0 < x < 2

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f(x) =
0 elsewhere
b. Find the Z ? transform of : i) n
2
ii) ne

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ar
'.
c. Obtain the inverse Z ? transform of 2z
2
+3z

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(z+ 2)(z ?4)
(08 Marks)
(06 Marks)
(06 Marks)
Module-3

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5 a. Obtain the lines of regression and hence find the co-efficient 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.

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Fit a parabola y = ax
-
bx + c in the least square sense for the data :
x 1 2 3 4 5
y 10 12 13 16 19

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(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)

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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 co-efficient of correlation. Find o if 6
x = 3.
(08 Marks)

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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

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(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)
Mod u le-4
7 a. From the following table find the number of students who have obtained marks :

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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

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following data :
x 4 5 7 10 11 13
f(x) 48 100 294 900 1210 2028
and hence find f(8).
C.

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Evaluate
dx
taking seven ordinates by applying Simpson s 78 rule.
o 1+x

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(08 Marks)
(06 Marks)
2 of 3
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17MAT31

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0
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.

--- Content provided by‌ FirstRanker.com ---

Module-1
1 a.
Find the Fourier series expansion of f(x) = x -- x
2
in (?it, n), hence deduce that

--- Content provided by‍ FirstRanker.com ---

rc
-
1 1 I 1
12 = 1
2

--- Content provided by​ FirstRanker.com ---

+ 2
2 4-
3
2
+ 4

--- Content provided by‌ FirstRanker.com ---

2
+ ?
(08 Marks)
b. Find the half range cosine series for the function f(x) = _( x 1)
2

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in 0 < x < 1. (06 Marks)
c. Express y as a Fourier series upto first harmonics given
x 0 60
0
120? 180? 240? 300?

--- Content provided by FirstRanker.com ---

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 :
1 +

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4x
in
-3
? < x 0
3 2

--- Content provided by FirstRanker.com ---

1 -
4x 3
in0
3 2
'

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Hence deduce that Tr
1
? = ?+ ? + +
8 12
1

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3'
1
5-
x in 0 < x < y

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2
TC ? x in V < x <
2
Show that the half range sine series as
4

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f(x) = ? [sin x ,
3
4
5
2

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sin 3x sin 5x
Obtain the Fourier series upto first harmonics given :
x 0 1 2 3 4 5 6
y 9 18 24 28 26 20 9
(06 Marks)

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Module-2
3 a. Find the complex Fourier transform of the function :
1 for I a
f(x) and hence evaluate f
sin x

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dx .
0 for lxl>a
Find the Fourier cosine transform of e
-ax
.

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Solve by using z transforms u
n
,, ? 4u
n
= 0 given that uo = 0 and u

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1
= 2.
f(x) =
b.
if f(x)

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(08 Nlarks)
(06 'Marks)
c.
=X, .
b.

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c.
(08 Marks)
(06 Marks)
(06 Marks)
171\

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OR
4 a. Find the Fourier sine and Cosine transforms of :
x 0 < x < 2
f(x) =
0 elsewhere

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b. Find the Z ? transform of : i) n
2
ii) ne
ar
'.

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c. Obtain the inverse Z ? transform of 2z
2
+3z
(z+ 2)(z ?4)
(08 Marks)

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(06 Marks)
(06 Marks)
Module-3
5 a. Obtain the lines of regression and hence find the co-efficient of correlation for the data :
x 1 3 4 2 5 8 9 10 13 15

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y 8 6 10 8 12 16 16 10 32 32
(08 Marks)
b.
Fit a parabola y = ax
-

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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

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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,

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mean of y's and the co-efficient of correlation. Find o if 6
x = 3.
(08 Marks)
b. Fit an exponential curve of the form y = ae
bx

--- Content provided by‌ FirstRanker.com ---

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

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correct to four decimal places. (06 Marks)
Mod u le-4
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

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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

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f(x) 48 100 294 900 1210 2028
and hence find f(8).
C.

Evaluate

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dx
taking seven ordinates by applying Simpson s 78 rule.
o 1+x
(08 Marks)
(06 Marks)

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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

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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)

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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
(06 Marks)
0.6

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c.
Use Simpson's
3
rd
rule to find j e'dx by taking 6 sub-intervals.

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U
(06 Marks)
Module-5
9 a.
Verify Green's theorem in the plane for

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4),(3x
2
8y
2
)dx

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+ (4y ? 6xy)dy where C is the
closed curve bounded by y =-Fc and y = x
2
. (08 Marks)

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b. Evaluate xydx + xy'dy by Stoke's theorem where C is the square in the x ? y plane with
vertices (1, 0)(-1, 0)(0, l)(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

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10 a. If F = 2xy + yz
-
X7 k and S is the rectangular parallelepiped bounded by x = 0, y = 0,
z = 0, x 2, y 1, = 3 evaluate
n ds

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(08 Marks)
b.
?
d
Derive Euler's equation in the standard form viz ?

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a
y
dx
of
a

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y
'
= 0 . (06 Marks)

It

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c.
Find the external of the functional 1=
6
1
.

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? y
12
2y sin x )dx under the end conditions
y(0) = y(n/2) = 0. (06 Marks)
3 of 3

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