Download Visvesvaraya Technological University (VTU) BE ( Bachelor of Engineering) Civil Engineering 15 Scheme 2020 January Previous Question Paper 3rd Sem 15MAT31 Engineering Mathematics III
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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
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)
The following table gives the variations of a periodic current A over a certain period T:
CS
E.
15MAT31
USN
b.
1 for
0 for
?
sin x
dx
Hence evaluate j
f(x)=
x
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
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.
.
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
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
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)
*3z +12
show that u
o
= 0 u
1
= 0 = 2 11. (05 Marks)
(7
-1
)
c. Obtain the Fourier cosine:transform of the function
4x, 0 < x
.
<1
f(x) = 4 x, I< x <4
(05 Marks)
0 X > 4
I
;
)
".
c.
1 of 3
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Cip
(.1
-1
IgrAT;
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.
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
(08 Marks)
The following table gives the variations of a periodic current A over a certain period T:
CS
E.
15MAT31
USN
b.
1 for
0 for
?
sin x
dx
Hence evaluate j
f(x)=
x
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
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.
.
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
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
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)
*3z +12
show that u
o
= 0 u
1
= 0 = 2 11. (05 Marks)
(7
-1
)
c. Obtain the Fourier cosine:transform of the function
4x, 0 < x
.
<1
f(x) = 4 x, I< x <4
(05 Marks)
0 X > 4
I
;
)
".
c.
1 of 3
b. Find the Fourier sine transform of f(x) =
OR
4 a. Obtain the Z-transform of cosnO and sinnO.
15
(06 Mark
and hence evaluate
f
x sin mx
dx m > 0.
1 + x
-
0
(05 Marks)
c.
Solve by using Z-transforms y,,, + 2v
+ yn n with yo = 0 =
yi.
(05 Marks)
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
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
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
data:
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
bx
for the data:
x 0 2 4
y 8.12 10 31.82
(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.
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)
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
(05 Marks)
r
C. Evaluate dx by Weddle's rule. Taking seven ordinates. Hence find log
e
2. (05 Marks)
1 +
2 of 3
-
A4vC
?
c)r)
?7,
(;
LIBRARy
-c!h
0
FirstRanker.com - FirstRanker's Choice
0
Cip
(.1
-1
IgrAT;
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.
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
(08 Marks)
The following table gives the variations of a periodic current A over a certain period T:
CS
E.
15MAT31
USN
b.
1 for
0 for
?
sin x
dx
Hence evaluate j
f(x)=
x
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
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.
.
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
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
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)
*3z +12
show that u
o
= 0 u
1
= 0 = 2 11. (05 Marks)
(7
-1
)
c. Obtain the Fourier cosine:transform of the function
4x, 0 < x
.
<1
f(x) = 4 x, I< x <4
(05 Marks)
0 X > 4
I
;
)
".
c.
1 of 3
b. Find the Fourier sine transform of f(x) =
OR
4 a. Obtain the Z-transform of cosnO and sinnO.
15
(06 Mark
and hence evaluate
f
x sin mx
dx m > 0.
1 + x
-
0
(05 Marks)
c.
Solve by using Z-transforms y,,, + 2v
+ yn n with yo = 0 =
yi.
(05 Marks)
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
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
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
data:
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
bx
for the data:
x 0 2 4
y 8.12 10 31.82
(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.
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)
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
(05 Marks)
r
C. Evaluate dx by Weddle's rule. Taking seven ordinates. Hence find log
e
2. (05 Marks)
1 +
2 of 3
-
A4vC
?
c)r)
?7,
(;
LIBRARy
-c!h
0
15MAT31
OR
8 a. Use Lagrange's interpolation formula to find f(4) given below. (06 Marks)
6
f(x) -4 mi 14 158
4
b.
Use Simpson's 3/8
11
' 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
(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
2
) dx + x
2
dy where c is the closed curve of the region
bounded by y = x and y (06 Marks)
b. Find the extrema! of the functional f (x
2
+3/
2
+ 2y
2
+ 2xy)dx (05 Marks)
z,
c. Varity Stoke's theorem for F = (2x ? y) yz
2 .
? y`z k where S is the upper half surface of
the sphere x
2
+
y
+ z
-
= I C is its boundary. (05 Marks)
OR
of af
10 a. Derive Euler's equation in the standard form
d
0 .
dx , )
(06 Marks)
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)
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)
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This post was last modified on 02 March 2020