Bangalore University 2008-1st Sem B.Sc Mathematics Vester B.A./, - Question Paper
VI Semester B.A.?B.Sc. Examination, June 2008
(Semester Scheme)
MATHEMATICS (Paper - VIII)
Illilllllllllllllllllllllll SM - 225
VI Semester B.A./B.Sc. Examination, June 2008 (Semester Scheme) MATHEMATICS (Paper - VIII)
Max. Marks : 90
Time: 3 Hours
Instructions : 1) Answer all questions.
2) Answers should be completely in Kannada or in English.
(15x2=30)
1) Find the locus of the point Z, satisfying ( z + i | <2.
lim [ z2 + 1
2) Evaluate z-i
3) Show that f (z) = xy + iy is not an analytic function.
4) Show that u = x2 - y2 + x + 1 is a harmonic function.
5) Show that w = ez is a conformal mapping.
6) Find the invariant points of the bilinear transformation.
7) Evaluate J (x2-iy)dz along the path y = x.
o
8) State Liouvilles theorem.
9) Evaluate z + 2i where C : | z | = 1
P.T.O.
r sin 3z
SM - 225
-2-
I (z + /) dz where C : | z | = 3
10) Evaluate
11) Prove that F {elat f (t) } = f(a+a).
12) Write the formulae for
i) Complex Fourier transform
ii) Inverse complex Fourier transform.
13) Find the Fourier sine transform of f (x) = , x > 0.
14) Find the Fourier cosine transferm of
 |
0 < x <1 x >1 |
15) Prove that Fs [f (x) ] = - a Fc [ f (x) ].
16) Write the general formula for second method.
17) Using bisection method, find a real root of f (x) = x3 - 3x - 5 = 0 between
2 and 3 in two steps.
18) Using power method, find the largest eigen value of matrix
- . Do two steps only.
dy
19) Using Eulers method, solve = x + y with the initial value y (0) = 1 for x = 0.1 in two steps.
20) Write the formula for Runge - Kutta method to solve
dy
= f (x, y) with intial conditions x = xQ, y = yD
SM - 225
-3-
II. Answer any four of the following : (4x5=20)
1) Show that Amp f -- ] = X represents a cirde.
2) Derive the necessary conditions for a function f (z) = u + iv to be analytic.
3) Of f (z) = u + iv is an analytic function and u - v = ex [ cos y - sill y], find f(z).
4) Define a bilinear transformation show that it establishes one-one correspondence from z + plane to w - plane.
5) Find the bilinear transformation which maps z = 1, i, - 1 into w = 2, i, - 2.
1
z + 2
6) Discuss the transformation w = -
III. Answer any two of the following : (2x5=10)
(2,4)
1) Evaluate J + x -* dx + (3x - y) dy along the parabola x = 2t, y = t2 + 3.
(0,3)
2) State and prove Cauchys integral formula.
f COS 71 Z
3) Evaluate J 2 _ 1 dz around the rectangle with vertias i, 2 - i, 2 + i, i.
c z 1
4) Prove that every polynomial equation of degree n > 1 with real or complex coefficients has at least one root.
IV. Answer any three : (3x5=15)
1) By using Fourier integral formula, show that
1 cos sx + s sin sx f (x) = - J 2 ds, where
10, for x < 0 f (x) = j e~x , for x > 0.
2) Find the Fourier transform of f (x) = e
SM- 225.
_4_
3) Find the Fourier sine transform of - (a > 0).
4) Find the inverse Fourier cosine transform of
sin aa
( a > 0).
2 a
5) Given Fc [e"ax] = J
2 2 n a + or
find Fs [e"ax].
(3x5=15)
V. Answer any three :
. 1) Solve for a positive root of x3 - 4x + 1 = 0 by regula - false method upto 3 decimal places.
2) Using Neuton - Raphson method, solve x3 - x2 - x - 3 = 0 in (2, 3) correct to 3 places of decimals.
3) Solve by Gauss - Seidel iteration method 3x - y + 2z = 4
x + 3y + z = -2 2x + y + 3z = 3.
4) Applying power method, find the largest eigen value of the matrix
(25 1 2
1 3 0
A =
2 0 -4
dy n
5) Using Runge-Kutta method, solve = x + yz with y 0) = 1. Compute y (0.4) by taking h = 0.2.
Illlllilllllllllllllll -5- SM - 224
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I. 15 d,$rtori tooAj. (15x2=30)
1) V (F) toodD siQd Afeixedd a. (a - P) = a. a - a. p v a e F, a , p e V
eOod
2) W = { (x, y, z) / x, y, z g Q} isow rosjr?rs4 V3 (R) traddewodD
3) S = { (1, 1), (3, 1) } isoto r!rad) V2 (R) wpsdsroAdd sodj gjseoj.
4) (1, 0, 1) ,(1, 1, 0) ,(-1, 0, -1) ToOsirteb V3 (R) steads* wdwoarte&odb gjjeeS).
5) T : V2 (R) V3 (R), T (x, y) = (y, x, x + y) od3 sids* drasreod <aodD &raeO*j.
6) T : V3 (R) V3 (R) T (x, y, z) = (x, 2y, 3z) wd's* dressodd Ireestefeb ? ,oda* erad4 Jo'cbhQcsoO.
sJQ Q> -6
7) T = V4 (R) > V2 (R) oci> d* djsssod dd, dd SLra?| 1 wfidd dzSraifii odD&SotoQ.
8) C : x = a cost, y = a sin t, o < t < Y2 ddeiS wdd J"x ridc&sb odb&SotoO.
(2, 3)
9) J xydx + x dy sSdoobFb odDkQc&e 0,0 01
10) odD rad sSfrlraer 'trfd 4 Cos 2 t i- 8 sin 2 t 1 + 16 t K arari tod ab?3
' a m J
dt>WdE> jfos&SoSCCDO.
M efc
11) joda ?>rad;& x = t, y = t2 + 1, z = 2t2 ddes3a> sfc?e> t = 0 <aod t = 1 ddrt F = 2xy i - 3xj - 5zK sooQod eSAasrf ej"oote>rt>d Sudsfc
0dD2oSC05S.
12) | } rdrdG 233o*5kot>2oSotoo. o 0
13) y2 = x x2 = y srfddwo&rW jdnbarf IJjfodfdb eyorfjtfusj erosicdjsefto ods&SCCDO.
2 3 2
14) J J J x y2 z dxdydz si sMc&jbocsb&Sc&iO. o 1 i
15) C, x2 + y2 = a2>ow drodd fyfc? djSfceccb erodc&raeftS) <!><, 2xy dx + x2 dy t35o*fi oct)&ao3j8.
16) rfStfEd OTSdoiraeflAi //r. n ds s> t3e3o>j&> odj&oSotoO. S iJoUDci)
* S *
- 1 < x < 1,-1 <y< 1, l<z< 1 qj>jdd s3b|fe|_ej
17) &e5d,5fc?a> crosrfcdroeAX) 0C yzdx + 3xdy + xydz dtfakFk odi&>otoO.
C d, x2 + y2= 1, z = 2.
V _D J 7
18) 8odi ero&&3) y Rod Ao.roftdd d& sUjoQcctorbd wo&o" <Dxdc$d?>
u) ca O -tf <r> ci
todo&O.
*2
19) J [y2 - (y')2 + 2yex ] dx ddddOri o*pr *etfdra y" - y = ex rt
XI
AioSe&xbd <aot> Sjaea .
<A u>
i) dsdop s>?|
ii) eteix&)d ;&e>rf ag*& 6ea3rteb
II. cdrascsddra 5330 s3,irtert eroe* : (4x5=20)
1) V = {a + b /, b e Q} rtn>4't' s&ab'.'rt essbrbraHBA Q s&e3 odD ;dwi rids& 3od3 irseoj.
eJ
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Attachment: |
| Earning: Approval pending. |
