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Time: 3 hrs.
2a.
b.
c.
3a.
b.
1 ',u, Find the Fourier series
*2@ril \'- I
B 3(2n-1)r'
fi*.{''{,1
1OMAT31
Marks:100
,,,,,,,,
hence deduce that
,,, (06 Marks)
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most closely to the
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Third Semester B.E. Degree Examination, Dec.2013 lJan.2Dl4Engineering Mathematics - lll
Max.Note: Answer FIVEfull questions, selecting
at least TWO questions from each partPART _ A
expansion of the function f(x) = lxl in Gn,x),(JooLo.
6
o!
oL
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lFh
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4l!
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5.e>\ (tsao o-c or.)
o=o.Btr>=o5!
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b. Obtainthehalf-rangecosineseriesforthefunction, f(x)=(*-1)'intheinterval 0<x<1
ura n.n.. show that n' = r{} * } * +. } (07 Marks)u'3'5')c. Compute the constant term and first two harmonics of the Fouripr series of (x) given by,
Obtain the Fourier cosine tru for* of f(x) = -L.l+x'h- *'for lxl < I
Find the Fourier translorm of f r x) = i -
I o forlxl >t
SFind the inverse Fourier sine transform of ,;*Obtain the various possible solutions of two dimensional Laplace's equation, u,* *u,, = 0
by the method of separation of variables. (07 Marks)r2 r2. _.. o-u o-u
Solve the one-dimensional wave equation, C' j=-, 0 < x < / under the following' 6x' A',\ ,, ,\ UnX
conditions (i) u(0, t) : u(1, 0 : 0 (ii) u(x, 0) : ; where u6 is constant
(iii) ;(x.0) = 0.
c."' Obtain the D'Almbert's solution of the
u(x. 0) : (x) una $1*.0) = 0a"'
(07 Marks)
wave equation u,, - C'r** subject to the conditions
4 a. Findthe best values of a, b, c, if the equation y=a+bx+cx2 is to fitfo llo wing observations.
Solve the following by graphical method to maximize z = 50x + 60y subject to the
constraints, 2x+3y<1500, 3x+2y <1500, 0( x<400 and 0<y<400. (06Marks)
By using Simplex method, maximize P=4xr -2xr-x, subject to the constraints,
xr +x2*X: (3,2xr+2xr*X., ( 4,xt-x, <0, X,)0 and xz )0. (07Marks)
x 0 fiJ
2n.,,.
J
.,.....fi 4x;J
5n;J
2n
f(x) 1.0 r.4 1.9 1.7 ,,,1 .5 t.2 1.0 CENfRALLIBRAflY
x 1 2 -) 4 5
v 10 t2 13 t6 t9
c.
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1OMAT31PART _ B
Using Newton-Raphson method, find a real root of xsin x + cosx = 0 nearer to n, carryout5a.
b.three iterations upto 4-decimals places.Find the largest eigen value and the corresponding eigen vector of the matrix,
lz -1 oll-, 2 -11
Io -r ,)By using the power method by taking the initial vector as [t t t]'
Solve the following system of equations by Relaxation method:l}x+y+z=31 ; 2x+8y-z=24; 3x+4y+l0z=58
carryout 5 -iterations.(07 Marks)
(06 Marks)
lassified below,
(07 Marks)
rule by taking 6 - equal strips and hence
(07 Marks)
(06 Marks)
(07 Marks)
c.
6 a. A
7a.
sulvey conoucteo m a slum locallty reveals tne lollowlns mlormatron as cIncome per dav in Ruoees 'x' Under 10 10 -20 20-30 30-40 40-50Numbers of persons 'y' 20 45 115 210 115
Estimate the probable number of persons in the income group 2A b 25.
diflerence formula.x 2 4 5 6 8 10
f(x) t0 96 t96 350 868 1746
nducted in a slum locali ls the followins info
. Estlmate tne pr0DaDle numDel oI persons rn tne mcome group lU to"/.5. (07 Marks)b. Determine (x) as a polynomials in x for the data given below by using the Newton's divided
c. Evaluate [. +dx by using Simpson'sjl+x H'
c.
deduce an approximate value of log. 2. (06 Marks)^ ) ^)
Solve the wave equation, + = 4+, subject to u(0, t) : 0, u(4, t) : 0, u1(x, 0) : 0 andol- dx'u(x. 0) : x(4 - x) by taking h : l. K : 0.j upto 4-steps. (07 Marks)
Solve numerically the equatio" + =* subject to the conditions. u(0. t):0: utl. t.1.'A0x'
t ) 0 and u(x,0): s.innr, 0< x <1, carryout the computation fortwo levels taking h =1l.
and K: + . (07 Marks)36
Solve u,o * u, = 0 in the following square region with the boundary conditions as
indicated in the Fig. Q7 (c). (06 Marks)
Fig. Q7 (c)
0000Find the z-transform o[ (i) sinh n0 (ii) coshnO
Find the inverse z-transform or" 222 +32' (z+2)@-a)
Solve the difference equation, y n+2 * 6y,*r * 9y , - )nwith yo = yr = 0 by using z-transform.
*rk*rk>k
tt1
?n
8a.b.
c.
500 I 00 100 50
(xr) n-
(07 Marks)
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