Download VTU BE 2020 Jan [folder1] 3rd Sem 17MAT31 Engineering Mathematics III Question Paper

Download Visvesvaraya Technological University (VTU) BE-B.Tech (Bachelor of Engineering/ Bachelor of Technology) 2020 January [folder1] 3rd Sem 17MAT31 Engineering Mathematics III Previous Question Paper

This post was last modified on 28 February 2020

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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 in (-7T, 7). hence deduce that
-

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(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 :

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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 :

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f(x) =
. -3
1+
4x

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? m ? < x 0
3 2
4x
3
1 in 0 x <

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3 2

n
2

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12
1
1
2

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1
-
7"
2
-

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1
3
2

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1
4
-

it' 1 1

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Hence deduce that ?=--,+?, + ?
1
, +---.
8 1
_

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3
,
5
.
x in 071

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/
2
IT ? x in r 2
Show that the half range sine series as
[

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f(x)
. _4
sin x
sin 3x sin 5x -
n

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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
Module-2

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3 a. Find the complex Fourier transform of the function :
1 for i sn x
f(x) = and hence evaluate f

dx .

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0 for lxj>a
? 0
x
Find the Fourier cosine transform of e
x

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.
Solve by using z - transforms u
n
+2 - 4un = 0 given that u0 = 0 and ul = 2.
b. If f(x)

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3
2
5 2
b.
c.

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(08 Marks)
(06 Marks)
(06 Marks)
(08 Marks)
(06 Marks)

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(06 Marks)
1 of 3
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? MEE
17MAT31

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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.

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

Module-1
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)

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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? 120? 180? 240? 300?
y 7.9 7.2 3.6 0.5 0.9 6.8

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(06 Marks)
OR
2 a. Obtain the Fourier series for the function :
f(x) =
. -3

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1+
4x

? m ? < x 0
3 2

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4x
3
1 in 0 x <
3 2

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n
2

12
1

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1
2

1
-

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7"
2
-

1

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3
2

1
4

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-

it' 1 1
Hence deduce that ?=--,+?, + ?
1

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, +---.
8 1
_
3
,

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5
.
x in 071
/
2

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IT ? x in r 2
Show that the half range sine series as
[
f(x)
. _4

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sin x
sin 3x sin 5x -
n
c.
Obtain the Fourier series upto first harmonics given :

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x 0 1 2 3 4 5 6
y 9 18 24 28 26 20 9
Module-2
3 a. Find the complex Fourier transform of the function :
1 for i sn x

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f(x) = and hence evaluate f

dx .
0 for lxj>a
? 0

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x
Find the Fourier cosine transform of e
x
.
Solve by using z - transforms u

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n
+2 - 4un = 0 given that u0 = 0 and ul = 2.
b. If f(x)
3
2

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5 2
b.
c.
(08 Marks)
(06 Marks)

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(06 Marks)
(08 Marks)
(06 Marks)
(06 Marks)
1 of 3

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17
OR
4 a. Find the Fourier sine and Cosine transforms of :
f
{x 0 < x < 2

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(x) =
0 elsewhere .
(08 Marks)
b. Find the Z ? transform of : i) n
2

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ii) ne'. (06 Marks)
c. Obtain the inverse Z ? transform of
2z
2
+3z

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(z + 2)(z ? 4)
(06 Marks)
Module-3
5 a. Obtain the lines of regression and hence find the co-efficient of correlation for the data :

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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

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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)

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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

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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 a
y
if a
= 3.

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(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

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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)
Module-4

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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)

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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).

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Evaluate
i dx
taking seven ordinates by applying Simpson's %
th
rule. (06 Marks)

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6
1
1 + x
(08 Marks)
c.

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2 of 3
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? MEE
17MAT31
USN

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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.
Module-1

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

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

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

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)

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OR
2 a. Obtain the Fourier series for the function :
f(x) =
. -3
1+

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4x

? m ? < x 0
3 2
4x

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3
1 in 0 x <
3 2

n

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2

12
1
1

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2

1
-
7"

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2
-

1
3

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2

1
4
-

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

it' 1 1
Hence deduce that ?=--,+?, + ?
1
, +---.

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8 1
_
3
,
5

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.
x in 071
/
2
IT ? x in r 2

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

Show that the half range sine series as
[
f(x)
. _4
sin x

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

sin 3x sin 5x -
n
c.
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
Module-2
3 a. Find the complex Fourier transform of the function :
1 for i sn x
f(x) = and hence evaluate f

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

dx .
0 for lxj>a
? 0
x

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

Find the Fourier cosine transform of e
x
.
Solve by using z - transforms u
n

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+2 - 4un = 0 given that u0 = 0 and ul = 2.
b. If f(x)
3
2
5 2

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

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(08 Marks)
(06 Marks)
(06 Marks)
1 of 3
17

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

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0 elsewhere .
(08 Marks)
b. Find the Z ? transform of : i) n
2
ii) ne'. (06 Marks)

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

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(z + 2)(z ? 4)
(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
2

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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

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

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,

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

mean of y's and the co-efficient of correlation. Find a
y
if a
= 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

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

(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)
Module-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

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

following data :
x 4 5 7 10 11 13
f(x) 48 100 294 900 1210 2028
and hence find g8).
Evaluate

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

i dx
taking seven ordinates by applying Simpson's %
th
rule. (06 Marks)
6

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1
1 + x
(08 Marks)
c.
2 of 3

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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

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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 :

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x 2 10 17
y 1 3 4
0.6
c.
Use Simpson's

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3
rd
ruleto find I e
-
x

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-
dx by taking 6 sub-intervals.
(06 Marks)
(06 Marks)
Module-5

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9 a. Verify Green's theorem in the plane for ili
c
(3x
2
? 8y

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

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b.
Evaluate
e
i xydx + xy
2

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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

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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 ?

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F. n us ?
Derive Euler's equation in the standard form viz ?
of
?
[ af

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a
y
dx ay'
= 0 .
n.

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2

Find the external of the functional I = J(y
2
? y'

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2
? 2y sin x )dx under the end
(08 Marks)
(06 Marks)
conditions

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y(0) = y(n/2) = O. (06 Marks)
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This post was last modified on 28 February 2020

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