MATHEMATICS 2001

PAPER II

SECTION A

I. Anempt any fiVe of the following:

LeT K be a fIeld and G be a finiTe subgroup of THe mJI Ilplicalll'e group ofnont.ero elemencs of
Show !hal G is a cyclic group.
12)
Prove rhac lhc polrnomial .... where p is a prime number. is ble over
lhe tield ra tional numbers.
12)
I
• Ill! r"
Show chat J · dt exisls
sinl!l
if.and only ifm< n
12)
f L I oL L .. 1 I 0 men prm•e mal I . · • H n
I.
I
Prove thai lhe Riemann function de6ned by q converges for Re z I and

converges wlifonnly Re z 2 c 1vhere s o IS arbiltruy smalL
Compuce all basic f-eas ible solutions of the linear programming problem
MttY z 2x; 3.r0

.rulyec/ tQ 2.r, 3x1 8
.r, 2.•,
.r,
And hence indicate the optimal solution
Lec N be a llOrmal subgroup of a group G. Show d1at GIN is abelian if and only if for all
.LJ' eG .. :vz·•y· l EN
12)
12)

lfR is a commulative nng with w1it elemenl and M is an1deal of R. tl1en show thai M is a
ma,imal ideal ofR if and only 1f RIM is a field.

Prove chat Cinile exlension of a field is an algebraic e.'tension. Given an example 10
sho11 that the converge is not lrue,

·
4.
A fis defined in U1e lntervnl ns
=ll q',when pl q
J I ./.when q
Whore q are relatively prime integers.
o foratl other valuesof x.
fs f R.ictnann intcgra'bfo'? Justify your 1lnsWer.
(101
Cl•) Show tbal Ll yz zx has a maximum value when three 'llrbbles ru'e connected by
the rcl"Uon by cz Land b. c are positive! consl3.nt satisfying lltc condition 2(ab tbe
(o· -..c·)

Evaluate


JJJ taken U1rougb out the region




Pinel l.he Laurent series fm- the fun ction in IJ z l Ising this exponslon.
that! exp( cosO)cos(sin 0 nO)dO t
Show that J" dx
•
4 1• v 2
(.Ising duality or otherwise soh·c the liner programming prohlem
l•linimize 1Sx, 1
Subjeelln 2.r1
3., • 3
0

(lSI
A manufacturer has at DeU1i. Kolknlll and (1,cnnai. Tiu:"c ""ntOJ'lo
have avnilnbk 30. 50. and tmits of bls producl His four ret nil outlds Un>
following number of un it..: A. 30; 13. C. 60; 40:
1 he transponaliun oost per unil in n1pees he.tweeu each ceutre nml outlet is giveu in
follO<m g l;lb l
Distribution Rclail outlets
Centres
J B c D
Delhi JO 7 3 6
Kolknta I 6 7 3
Chen113i 7 4 5 3
Determine the minimum transportation cost.

SECTION a
5. Attempt any five oft he fo llowing:
Find thecnmpleteinlegral offhe partial diHe ren lial equa tion
g·I 3.. sx-I y·'

Find the general integral oftheoquatino
y Cl= I cy • Ll·-my

6.
7.
Show that lhe error associated with line:ll' interpolation of using ordinol.es at
Xu and x1 with "" x is not lflrger in onagnilodo then 1/SM1(x1 where
max Jlf n .:c ·'1·

show lhut if f(x dl.
Thc (11JIIC3linn C<)tl'Cl!ponding to ijnear inttll)('Jzotion in ·'V " X s .(j C.1nnot
.
2 2nl'
Civcu AJJ'1
Show lhnt Jl.C .C B .
(121
(iii tbe ore• of tho triangle h>ving s of lengtlo! 6.Ji.12.6..fi. units u1 bitwy
numbt1'
• 6 12)
l)etepnine the mlomenl of inertial of a uniform hemisphere "bout <lf •ymmetry and
about an axis lar 10 tile lXis of .Bymmetry and through centre of Ue

If the velocity distribution of an incompressible Huid at the point (x. is given by


I J.v= 3y= J . d . •• k L lh • . . . Lheu c1·em1mc Wa parameter !IUt;Jl at il ts posst c motton..
r r
Hcnoc rind its' polontinl.

Prove thallbr the equation
1. px- ltY 0
the clumn:Lcristic strip• nrc giv<>11
1 1
. .
Ce /II
11(t (AC rBD)
+De"
Whort 0 und a ao-e Mhilr3ry con.sL:mL li11d tho vallJcs ohtst nrbi trory
il'tbe intogr•l surtooc. pas$t>Slhrougb the littt z y.


Write clown the system of e<tnaloons for obta ining the geneml equat ion of
ortloogonal to tlte family given by

!-inlve the eq11a!ic>n
... ...1 ... ...
r J I r.x- . fly
hy re(luclng it to the cqu:ltion with ..
10)

osing Go us. Seidel itcr:oti1 e method ond the starting solution ·'J in" the
solution of the follow system of equ;otion in two iteration.•
10.'1 Xl 0
10"! 12
x1 .t, • lOx, 10
Compare the opprox.im:.lc solution with the exact •ulution .




Write a computer program in BASIC to evall"Ue the polynomiol
Oo·r " -..2 .... " ... ft,.
for values of x 1.2(!l.2)2.0.
15)
Find 1l1e volues of U1e tl()·v•lued variables A. B. C lllltl D by solving tile set of
situultaneous cqu•lioM
B
A.B A.C
A. B • A. C. D c· . D
IS l
Find the equation of motion lbr a particle of m which is to move on the
surface • cone of sem i-vertica l angle« and whic.h is s ubjcctt:d to a gravitaJional l'orce.

Show thnl tl1e velocity distrjbution in axiJJI flow of viscous incompressible j]uid along u pipe
of cross--sectlon. rndii fl is given by
r1 rf loQ .
.Jp log( r1