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Download VTU BE 2020 Jan [folder1] 3rd Sem 15MAT31 Engineering Mathematics III Question Paper

Download Visvesvaraya Technological University (VTU) BE-B.Tech (Bachelor of Engineering/ Bachelor of Technology) 2020 January [folder1] 3rd Sem 15MAT31 Engineering Mathematics III Previous Question Paper

This post was last modified on 28 February 2020

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IgrAT;
of
Third Semester B.F. Degree Examination, Dec..241*Jan.2020
Engineering Mathematics ? III
Time: 3 hrs. Max. Marks: 80

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Note: Answer any FIVE full questions, choosing ONE fill! question from each module.
Module- I
Obtain the Fourier expansion of the function f(x) = x over the interval (-7c, rc). Deduce that
1 1 1
? = 1 ?

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4 3 5 7
(08 Marks)
The following table gives the variations of a periodic current A over a certain period T:
CS
E.

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15MAT31
USN
b.
1 for
0 for

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

?
sin x
dx
Hence evaluate j
f(x)=

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

x
ix
>a
(06 Marks)
t (sec) 0 T/6 T13 T/2 2T/3 5T/6 T

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A (amp) 1.98 1.30 1.05 1.30 -0.88 -0.25 1.98
Show that there is a direct current part of 0.75amp in the variable current and obtain the
amplitude of the first harmonic. (08 Marks)
OR
2 a.

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.
Obtain the Fourier series for the function f(x) = 0 x <
b. Represent the function
x, for 0 < x < it/2
f(x)

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

"7 12 for rc/2(06 Marks)
in a half range Fourier sine series. (05 Marks)
C. Determine the constant term and the first cosine and sine terms of the Fourier series
expansion of y from t
(05 Marks)

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Module-2
3 'Find the complex Fourier transform of the function
xc' 0 45 90 135 180 225 270 315
y 2 3/2 1 1/2 0 1/2 1 3/2
b.

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If u(z)
*3z +12
show that u
o
= 0 u

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1
= 0 = 2 11. (05 Marks)
(7
-1
)

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c. Obtain the Fourier cosine:transform of the function
4x, 0 < x
.
<1
f(x) = 4 x, I< x <4

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

(05 Marks)
0 X > 4
I
;
)

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".
c.
1 of 3
FirstRanker.com - FirstRanker's Choice
0

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

Cip
(.1
-1

IgrAT;

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

of
Third Semester B.F. Degree Examination, Dec..241*Jan.2020
Engineering Mathematics ? III
Time: 3 hrs. Max. Marks: 80
Note: Answer any FIVE full questions, choosing ONE fill! question from each module.

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

Module- I
Obtain the Fourier expansion of the function f(x) = x over the interval (-7c, rc). Deduce that
1 1 1
? = 1 ?
4 3 5 7

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

(08 Marks)
The following table gives the variations of a periodic current A over a certain period T:
CS
E.
15MAT31

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

USN
b.
1 for
0 for
?

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

sin x
dx
Hence evaluate j
f(x)=
x

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

ix
>a
(06 Marks)
t (sec) 0 T/6 T13 T/2 2T/3 5T/6 T
A (amp) 1.98 1.30 1.05 1.30 -0.88 -0.25 1.98

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

Show that there is a direct current part of 0.75amp in the variable current and obtain the
amplitude of the first harmonic. (08 Marks)
OR
2 a.
.

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

Obtain the Fourier series for the function f(x) = 0 x <
b. Represent the function
x, for 0 < x < it/2
f(x)
"7 12 for rc/2(06 Marks)

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

in a half range Fourier sine series. (05 Marks)
C. Determine the constant term and the first cosine and sine terms of the Fourier series
expansion of y from t
(05 Marks)
Module-2

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

3 'Find the complex Fourier transform of the function
xc' 0 45 90 135 180 225 270 315
y 2 3/2 1 1/2 0 1/2 1 3/2
b.
If u(z)

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*3z +12
show that u
o
= 0 u
1

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

= 0 = 2 11. (05 Marks)
(7
-1
)
c. Obtain the Fourier cosine:transform of the function

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

4x, 0 < x
.
<1
f(x) = 4 x, I< x <4
(05 Marks)

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

0 X > 4
I
;
)
".

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

c.
1 of 3
b. Find the Fourier sine transform of f(x) =
OR
4 a. Obtain the Z-transform of cosnO and sinnO.

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15
(06 Mark
and hence evaluate
f
x sin mx

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dx m > 0.
1 + x
-
0
(05 Marks)

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c.
Solve by using Z-transforms y,,, + 2v
+ yn n with yo = 0 =
yi.
(05 Marks)

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Module-3
5 a. Fit a second degree parabola y = ax' + bx + c in the least square sense for the following data
and hence estimate y at x = 6. (06 Marks)
b. 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

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y 8 6 10 8 12 16 16 10 32 32
(05 Marks'
c? Use Newton-Raphson method to find a real root of xsinx + cosx = 0 near x = rt. Carryout the
upto four decimal places of accuracy. (05 Marks)
OR

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6 a. Show that a real root of the equation tanx + tanhx = 0 lies between 2 and 3. Then apply the
Regula Falsi method to find third approximation. (06 Marks)
b. Compute the coefficient of con
-
elation and the equation of the lines of regression for the

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data:
x 1 2 3 4 5 6 7
y 9 8 10 12 11 13 14
(05 Marks)
c.

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Fit a curve of the form y = ae
bx
for the data:
x 0 2 4
y 8.12 10 31.82

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(05 Marks)
Module-4
7 a. From the following table find the number of students who have obtained:
i) Less than 45 marks
ii) Between 40 and 45 marks.

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2 3 4 5
y 10 12 13 16 19
Marks 30-40 40-50 50-60 60-70 70-80
Number of students 31 42 51 35 31
(06 Marks)

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b.
Construct the interpolating polgnomial for the data given below using Newton's general
interpolation formula for divided differences and hence find y at x = 3.
x 2 4 5 6 8 10
y 10 96 196 350 868 1746

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(05 Marks)
r

C. Evaluate dx by Weddle's rule. Taking seven ordinates. Hence find log
e

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2. (05 Marks)
1 +
2 of 3
-
A4vC

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

?
c)r)
?7,
(;
LIBRARy

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

-c!h
0
FirstRanker.com - FirstRanker's Choice
0
Cip

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

(.1
-1

IgrAT;
of

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

Third Semester B.F. Degree Examination, Dec..241*Jan.2020
Engineering Mathematics ? III
Time: 3 hrs. Max. Marks: 80
Note: Answer any FIVE full questions, choosing ONE fill! question from each module.
Module- I

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

Obtain the Fourier expansion of the function f(x) = x over the interval (-7c, rc). Deduce that
1 1 1
? = 1 ?
4 3 5 7
(08 Marks)

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

The following table gives the variations of a periodic current A over a certain period T:
CS
E.
15MAT31
USN

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

b.
1 for
0 for
?
sin x

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

dx
Hence evaluate j
f(x)=
x
ix

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

>a
(06 Marks)
t (sec) 0 T/6 T13 T/2 2T/3 5T/6 T
A (amp) 1.98 1.30 1.05 1.30 -0.88 -0.25 1.98
Show that there is a direct current part of 0.75amp in the variable current and obtain the

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

amplitude of the first harmonic. (08 Marks)
OR
2 a.
.
Obtain the Fourier series for the function f(x) = 0 x <

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

b. Represent the function
x, for 0 < x < it/2
f(x)
"7 12 for rc/2(06 Marks)
in a half range Fourier sine series. (05 Marks)

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

C. Determine the constant term and the first cosine and sine terms of the Fourier series
expansion of y from t
(05 Marks)
Module-2
3 'Find the complex Fourier transform of the function

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

xc' 0 45 90 135 180 225 270 315
y 2 3/2 1 1/2 0 1/2 1 3/2
b.
If u(z)
*3z +12

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

show that u
o
= 0 u
1
= 0 = 2 11. (05 Marks)

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

(7
-1
)
c. Obtain the Fourier cosine:transform of the function
4x, 0 < x

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

.
<1
f(x) = 4 x, I< x <4
(05 Marks)
0 X > 4

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

I
;
)
".
c.

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

1 of 3
b. Find the Fourier sine transform of f(x) =
OR
4 a. Obtain the Z-transform of cosnO and sinnO.
15

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

(06 Mark
and hence evaluate
f
x sin mx
dx m > 0.

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

1 + x
-
0
(05 Marks)
c.

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

Solve by using Z-transforms y,,, + 2v
+ yn n with yo = 0 =
yi.
(05 Marks)
Module-3

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

5 a. Fit a second degree parabola y = ax' + bx + c in the least square sense for the following data
and hence estimate y at x = 6. (06 Marks)
b. 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

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

(05 Marks'
c? Use Newton-Raphson method to find a real root of xsinx + cosx = 0 near x = rt. Carryout the
upto four decimal places of accuracy. (05 Marks)
OR
6 a. Show that a real root of the equation tanx + tanhx = 0 lies between 2 and 3. Then apply the

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

Regula Falsi method to find third approximation. (06 Marks)
b. Compute the coefficient of con
-
elation and the equation of the lines of regression for the
data:

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

x 1 2 3 4 5 6 7
y 9 8 10 12 11 13 14
(05 Marks)
c.
Fit a curve of the form y = ae

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

bx
for the data:
x 0 2 4
y 8.12 10 31.82
(05 Marks)

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

Module-4
7 a. From the following table find the number of students who have obtained:
i) Less than 45 marks
ii) Between 40 and 45 marks.
2 3 4 5

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y 10 12 13 16 19
Marks 30-40 40-50 50-60 60-70 70-80
Number of students 31 42 51 35 31
(06 Marks)
b.

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Construct the interpolating polgnomial for the data given below using Newton's general
interpolation formula for divided differences and hence find y at x = 3.
x 2 4 5 6 8 10
y 10 96 196 350 868 1746
(05 Marks)

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r

C. Evaluate dx by Weddle's rule. Taking seven ordinates. Hence find log
e
2. (05 Marks)

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

1 +
2 of 3
-
A4vC
?

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

c)r)
?7,
(;
LIBRARy
-c!h

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0
15MAT31
OR
8 a. Use Lagrange's interpolation formula to find f(4) given below. (06 Marks)
6

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f(x) -4 mi 14 158
4
b.
Use Simpson's 3/8
11

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' rule to evaluate I e"dx
C.
The area of a circle (A) corresponding to diatneter (D) is given by
D 80 85 90 95 100
A 5026 5674 6362 7088 7854

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(05 Marks)
Find the area corresponding to diameter 105 using an appropriate interpolation formula.
(05 Marks)
Module-5
9 a. Evaluate Green's theorem for (1),.(xy + y

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

2
) dx + x
2
dy where c is the closed curve of the region
bounded by y = x and y (06 Marks)

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b. Find the extrema! of the functional f (x
2
+3/
2
+ 2y

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

2
+ 2xy)dx (05 Marks)
z,
c. Varity Stoke's theorem for F = (2x ? y) yz
2 .

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? y`z k where S is the upper half surface of
the sphere x
2
+
y

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+ z
-
= I C is its boundary. (05 Marks)
OR
of af

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10 a. Derive Euler's equation in the standard form
d
0 .
dx , )
(06 Marks)

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0y1
b. If F = 2xyi +3,
2
41+ xzk and S is the rectangular parallelepiped bounded by x ? 0, y = 0,
z = 0, x = 2, y = 1, z = 3. Evaluate if F.fids (05 Marks)

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c. Prove that the shortest distance between two points in a plane is along the straight line
joining them or prove that the geodesics on a plane are straight lines. (05 Marks)
3 of 3
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This post was last modified on 28 February 2020