Engg Mathe 3 June 2013
Uploaded by
Prasad C MEngg Mathe 3 June 2013
Uploaded by
Prasad C Ml'
1
USN C-
1OMAT31
(07 Marks)
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.SlnX-XCOSX 1I
l-o*=
'
;x+
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Third Semester B.E. Degree Examination, June/July 2013
o
E
Yor
tg
-=i
o- 6-
oi
6.!l
ooo
'b=
a8
-i
ti
o
z
E
Ix,
if,0
I -^
[r-x,
tl
')
irom the
Find the Fourier cosine and sine transform of(x)
:
xe-ul, where a
>
0.
constant.
4 a. Using method of least ol least souare. rt a
x I 2 3 4 5
v
0.5 2 4.5 8 t2.5
b.
[/!-^,-rr
c. Obtain the constant term and coefficients of first.cosine and sine terms in the expansion ofy
Ix,
if 0<x<fl
b.FindthehalfrangeFouriersineserieso[f1x1=]
if
f
<x<r
lollowinp. table:
x 0 60" 120' 180: 240" 3000 3600
v
'7.9
7.2 3.6 0.5
''-
0r9 6.8 7.9
2
a. Find the Fourier transform of
r(*l={u'-x''
lxl
<a
and hence deduce
|
0.
lxl>
a
b.
c.
3a.
b.
Find the inverse Fourier transform of e-s" .
Obtain the various possible solutions of one dimensional heat equation ut
:
c2 u,* by the
method of separation of variables.
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A tightly stretched string of length .[ with fixed ends is initially in equilibrium
position. It is
/\
set ro vibrate by giving each point a velocity U. talIf
.,J
. Find the displacement u(x. t).
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c. Solve u*,.
+ uyy:0 given u(x,0) = 0, u(x, 1) - 0, u(1, y) = 0 and u(0, y) = u0, where u0 isa
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.
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Time: 3 hrs.
Max. Marks:100
Notel Answer FIVE
full
questions' selecting
ot least TlltO questions
from
each part.
PART
-
A
I x. if 0<x<n
I
a. Obtain the Fourier series expansion ol i(x)={^
"'
and hence deduce
[2r-x.
il n<x<Ztr
Engineering
Mathematics
- Ill
n'l11
thal
-=
. *... * . *.........
8l'l-5'
x+y>4, x+3y>6
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t
5a.
b.
1OMAT31
PART
-
B
Using Newton-Raphson method find a real root of x + logrox
:
3.375 near 2.9, corrected to
3-decimal places. (07 Marks)
Solve the following system ofequations by relaxation method:
12x+y +z--31, 2x+8y-z=24, 3x+4y+102=58 (07 Marks)
Find the largest eigen value and corresponding eigen vector of following matrix A by power
method
A
us;x.01:
tl,
....
a. In the
giG;i*
o- teffn- Imd,
lzs t 21
=l
, 3 o I
l,
o
-4]
0, 0]r as the initial eigen vector. (06 Marks)
below, the values ofy are conse
irst and tenth terms ofthe series.
x 5 '4 5 6 7 8 9
v
4.8 8.4 t4.5 23.6 36.2 52.8 73.9
an interpolating
polynomial for the data Construct
difference formula.
consecutive terms ofseries of which 23.6 is the
ieries.
.,..
.'
:.
.
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-fs] i:',-
given below using Newton's divided
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I
c. Evaluate
[.
j
.
a* by Weddte's rule taking 7-ordinates and hence find log.2. (06 Marks)
jl+x'
or,
^
7 a. Solve the wave equation utt =.'{ux, subject.tq u(0, t)
:
0; u(4, t)
:
0; u(x, 0)
:
0;
u(x, 0)
:
x(4
-
x) by taking h
i.l,
k: 0.5 uptofour steps. (07 Marks)
^^)
b. Solve numerically the equation
+=+
subjecttotheconditionsu(0,0:0:u(1,1),t>0
dl dx'
and u(x. 0) = sin nx.0 S x
<
l. Carryout compulations for two levels taking h -
/,
*d
k
:
/to
. (07
Marks)
c. Solve the ellibtic eouation u*. + u,., = 0 for the followins souare mesh with houndarv vahres Solve the elliltic equation u*, + uyy = 0 for the following square mesh with boundary values
as showt,!!r'Fig.Q7(c). (06 Marks)
'
u
5oo looo
5oo
u
f-T t- r- l
,"""1
l', l.u 1,, 1,"^"
,*J--P]-lrf,o.,o
Fig'Q7(c)
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8 a. Find the z-transform of: i) sinhn0; ii) coshn0.
b. Obtain the inverse z-transform of
8z'
(22-1)(42-1)
c. Solve the following difference equation using z-transforms:
yn+zl2yn+t + yn: n with ye: yr
:0
2 of2
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