Time Allowed: 3 hours
Maximum Marks: 300
Caudidates should attempt Ouesboo Nos l and 5 which are compulsory. and any or the relll!llmng
queslions selecting at Jettsl one question from each .section.
PAPER II
SECTION A
I Answer any FIVE of the following: (5x 12=60)

Let Ru be the set of all real numbers except zero. Define a binary operation " on R0 as: a*b b I a I den01es absolute value of a. Does form a group? Examine.
Suppose that there is a positive even integer n such that an a for all the elements of some ring R. show that a a for all a belongs r and a b for all a,b belongs to R
For show
Let
H.ne N 1 ·" e N fu{r. . 11 e N
Find denved set ofT also find Supr•mum and grea1est uumber
IC f R Js continuous and
f(xl
for E n • tllCll sho\ lh!ll x IT
lor all " e 0
find 1be residue of
cot =coth . ai r. 11

Find 1he dual of the folio" lng It near proyamrnlng problem:
Max.. Z
such lbal
3x1 SR
4.t; .rJ 2
2,r1 3x:-·li 5
u
12)

(hi
12)


Let 0 and G be IWO groups and fe( 0 -.G be a homomorphism. For element a. 0
prove U1a1
3.
4.
Ker llrlnndl KUbJlTf>Up llfG.
IS)
l.,ot R l>e u ring "iU1 tltll If the prl)duct ofhny two non Yt:rl) i1 lt:tcl 1•rovc 111at
ab 1 bn • L
Wholhcr hns OLbi>'C property JlOl C.'tplain. II in inle!lf1ll domain?

Dtseuss U1t conviltgenee of
1 ·3·5x' ........ x o.
l 2·4 2·'l·6
c 15)
Show tlutlluaeri"" L i.• equivalent to
11 n" 1
-2I 1• ..

Prove thai Dllmain can b" l:mbedded w a

Shov I hat MY maximn l ideal in the commutntivu ring of polynomials over • fi eld the
pnnoipol genernted by on irreducibla polynomial,

Let fbe • eontJnuous function on [0.1). Using first Mcon V>lu" lltcorcm qn lntcgmtion, prove
that
Po'Qvctlthl thescN

A {0.1 B 0. !Ire equivnlern
Provo that


H.valujlle
" tr " j l
s in
I II:
• log
where C i& the cin::lol71 Smte the the<:>rerns in "valuating nh()veintegrnl.



I
Let be nn entire I k 11. lllr some positive con, tonl k unJ all z.
Show tlut l'rzl n'L2 for some cousl3nt tL.

problem

Av:oil:thiliY
F. 2 3 3 2 5 50
3 2 2 3 •I .to
.3 5 .j 2 I
4 2 2 I 2 2 30
Demand 30 50 20 .w 10
by finding lhe inlti.•l solution by MAtrill Mlnimn metbod.

SECTION B
5. thllol•'lng:

7.
Pind the $olutio:n of Ute parlin I ,litlerentin .
.:qunlicm

and also ftnd th.:.pnrtlculor solution whloh through I be llnos" I. y 0.
general solution ol'thc pnr1iclc d.it:l'c:rentoal t:quation

where D

il

12)
Find tbc root cusx=O Do Utree
it=llqm.
12)
Sbtd tlte lll'inclplc of dlUIUty
in Boolailn algcbt'll and lito dunLoflho Boolton L'Xprcssion5
· f)·(f and X .Y=O

Represent
8 iu NOR In NOR log.lc network

A hoord placed on • •moolh horironlal plane and a boy round the edge of it al
a uniform rate. What is the motion of the centre of Ute bD:ttd Explain. What lt3Jlpens if the
liL15S of beard and aroc:qunl

II'Utt: n:looity polenti• l tlu[d .:.tan·• )· x2 ).l 11teu 'tltllt

strcrun 1m. eo. lie (Itt Ih e· surf•a ces 1 y·' c bl :.m Q et
.ronl
12)
Find the s teody st.lre tcmper•wte distribution in • Utin pbte bopndt<l by lite lines
x 0. x n. y 0 and y h The edge!! x 0. 11 u and y 0 are kept at tempernture zero
while: the edge y kept at IOO"C.

Find COOIJ>Icte •ingolnr integ.rnl!t of
pq tl
using ooethod.
USl
11tc Ji.lllnwing values of Ute function tin x 1 cos x art
" 10" 20" 30°

1·1 585 L·2S 17 I·.BGO
" lruct the qunclmlic inte, olating polynomi:ol thnt lits the dMa 1-fence
culclll•te j ).. Cumpare wilh exact value.
Apply method to z.from
-12
6.r y 11 -33
32
with initllsl values (4·67. 7·62. 9·05). Carry out computations for two ilcratioos.
15)
Draw n 11!1V cl1:1rt for equation tl Clrre<U to live decimal plnces hy Ne"1Qn·
R:lphson mt!hl)d.


uniform rod of length 21 has middle point fixed and • m is attochcd
to of its The rocl when in n liorizonL'll position $et rotating abt,ul 3 vertical
lhrough its centre with an angular velocity J¥. Sha\ thar the hea vy end ol'che rod will
falt till Ote inclinolioo of iJIC> rod to the vcrtiOlll is co.4 .fi

Let we lluid till• the region " O(right of 2d plnne), l.et A 'ource .lt Ytl 3nd e<lu•l
Lot tltc pressure llc pressure lit inl'init)• i.1:. Show thut
l""lUIIliJJ! o.n tlu: 1Juund3ry

p bdng the tletu!il) <Yf the fluid.