Rs-10-..;/~•..•--=~.=~__ z_I~.U

-. 03

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NORTH MAHAIU~.HTRA UNIVERSITY,. JAJLGAON.

Syllabm fur F.Y.B.Sc.

MATHli.1\'IA TICS.

( W.d. Acd. 'I'f. 2002 - 2003)

llyU!!m )'01" F,Y. B.s".

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-~~~'~~~'=======~lNorth Mabl~:l!!btra Unive.'Si!y,Ja1llaon. I,

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~l!tbemlltit!.J:Yilli Efk~li'Om A.d. Yr. 2002-2003 '

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Paper1 _ Calculml CodeNo. :

i'Papern:l(A) : Geomdry~ldhl~bril. Code'No. :

OR

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P3perJII(H) : "OnpItThlo'''YROOAIgebnt

1,

Code No. :

(8 Period) (7 M,arlai)

(6 Period) (8 M trIB)

(6Perlod) (10 10k"",)(6 Period) (10 r'lfarll,s)

North Mtih!!!'!~lltra Universi{y. Jalga.Z!!!JCJl~Matbelri!!tig .

~labwi withllmitat~ !!Id!cvpe(withetrKt tromAaL "To 2002-10'llliP.~~J~I ':CALCl~U~

1. Sequem-~s. (81"eriod)(lOMwil:s)InleIVa\$. Bowided ~nd un1>oUnCt{' s: I::or rwJnumbers. Leas! ~'PJXll'bound and am tl:!'t lower

boUnd. Clmverg.;:ncc'o'l"a lleql1"n~e[>oa lnit}MotiVarionof E- N detinitkln.Alg1lI:lrll of limitsofa '~sequcn" (stlten",epts QJliy).M( nOtouk' 6tq .!ences. C~mvergence ofmor.otonk: ~". (~-£ence of the s«J,\lett"J (I + _~._)llt /TI.:.u ••nbcr 'c'. Conv~1gttICcob~lri:,:Il«~

"1. Series ,'-1 . n P \ - (8 Period)(10~(arU)Convergence ofl~ (fnon.n;gatil c.\.ef r s. OIWh)"g!Jm«al principle of conwrgencc. O:>tnpari-IlOJltCo'it COll\~.encc ofihi lIelies1-"':':-' I' '" R I)' A.lembCrt',jnNOteet. (without proof Caudly'!root test. (WithlilltprOO;). d' •

3. lmIe~nnimrt(Jrom Ill.

,L' Hospit;~" Rules. (wiihO\ll Pr:101t)4. Continuity ,Offunn ions,

"omtinUityof afutloooo ofrWllurnt. on.PropCltiesora cllntiniJ(rUSfu11etion ~f\d01<<dlllld boUnded~.

(i) Bo.md:~. -"(ii) Attalns llldIOU1d8.(m) lo.lt11nediatt:v.dUotJte.~

L'nifonn CoritimJily,5. MeanVWue'Ibeonms. (10Perioo) (10Marks)

, ,DiJfeientiability ill a fur ,caon:, CantirNity all;! difIeruliiability~ Rolle \l 'I'hc:Uem.~.~ Mean VW: TbeOrr<:m.C md'r I SMi.Wt Value 1'heorcm:

,.' ' .', '..' ,-'

6. Su«essiveDi.trerelltiadoll. (8Perlod)(10r,bm)The nlh &rlIla~ of ~.OO'\Csta3lbtO o:"fitctions.l..eibnitz '8thoortm.

7. Taylor;" Theorem, Maclalu'in's, 'nll'rorem.& 'Inte~tlmt.

,hitq;taiiOO bypartlal fi-Jic!iOllB~Dcnomillntorimcl\~ i) Loeal" .Ion<,,::peared,li) IJncanepeated

1E iiI Q\,~(,IJ<Ili,;non-rcpeatoo factors (mly

9. Infegrirlion of imlti.:m~ al&ehr-ai,;functions of ole foJ1J1,(9 Period) (10 ;\'liuv). ..

,

i)' me",J (px+qJJax+~

~j .'~"q)J ll«2+bx .•.c dx

'1 I '(2.; 1

'('14 I

10. Rrductlon Fonttt'ilte.

i) J _x2::..!.._ dx~lth2+1

(8 Pf'rtod)(10 MMiIs~

ROOuclioofonnulaeiiT:

.,f smLl )'; r.osJl x d!!, J "inm ~ l'OS!!X i'l,

'Il'2 (\'1tlmutp1"!{'f)f cosnxd'l

o. ~lnnx1---

n. Appllcatlon ofInlegntJor.,Recti6calion An:a \tune Slrtac~A~la

'!lnJl:

(8Period)(10 Marlm)nh~'OI}' is 1lI~tJ_'qlfdoo)

PAPER - II: M \TlUCES AND DIFFERENTL\.L EQUATIONS

1. Adjoint-andmvtl'lwoh matrix. (12P~OdK)(HiMarb)Tran~pose of a malri". SyrlRlI:tric lUldskm s:mmetric rn.1mcei. Ad;oil11of a matrix. InIitr5e of I!.fll.JIri'i. Exi~lell'e and ~ emm o:firMn'l ill ~_mattU ProptrriC!l ofin\llll.lIe.

Z. RankofllMatrix (l2Periods)(t5Mark5i. .E1W1lmlary lr;mfonn.1linm Equivak:nr 1lliI1rico~Ble:mentary malric:cs. )'t.:d: (If a Ill;IIrix Imiaimce ofrank lmder ekmenlal)' tNnsi:urmtkm.. Reduclioo of a matrix 10 ilJIoorr13I form, Non-sillgular maW.a.~a product oiE-!IUI1;iccs. Rank 0' product ci twQ~.

3. !,)'Stem ofUnearelluat om;. (W Periods) (t6l'\1arks)Consi!rtencyandsdutioo01~lOOOU~ ardflOO.~LD lint;ar~.

4. E1~enV>lIUl~.cd j~igellVtcttl-1'l'I (10Pt'liods) (i6 :Marb)Eigenvalues and dger. ,;ech" of a ;qoore rr,.atri1. Ot.tractemtic equatioll of a:malrix CilyIey-HamiIIonThemm (stalCrl);mt 1.'IIy) Jfl.d -.,-erifil;atioo, 11IWTlled a miltrix by Il'1ing Cayley. Hiniilion Them:m.,

5. Di~ermt1a1Equarion.)ffln;t6n!;~rmul flnt degree. (10Periods) (20MaJ'tJ)Homl'F-eotJ~ cquati<ALN Iftohom J@:meOl~tXIuabon. Eu..:l o:quation. futegrll.tingfactor. Uncal' dif-fmzllial~cn Be,moulli '5di/faull'ialox;iLrticn.

6. llitl'erentbtl Equatloh ,If'flrst onlet' nnd higherdegrer. (10Periods)(15Marks)SdwbJeforp,y,x. Clairmfl equatiJn •

7. AppiicatVon ofDiffeftntlfli Eltmrtioos (ll Perlods)(15Marb)Orthogottalttajed:ories, Singular Solutions Envelopes.

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PAPER - III (A): (:::;;OMETRY At"iD ALGEBRA

1. Cf)-()rdillBW i~'pal,"c. (8 PerilJdsl (: ,\l.lIrk"Co-ilrdir,~Ic.'lofu pout in ,pax. C' l:l.1g: ofoJigin, Di.~1anceIonncla. Section f,mnu}:. [>ir~iiNlcORil\e<;81d d:irec.Uonr.n;os 1)[,lline., Ang t b.~iOOlllWO lin(~, Pro~ lion of a line sogme ot

2. PL'UIC (tz Ptriods), 15Mll!'k~)Eqt:.atiOl1'lof ~il;ane. fI.I>gId~1Netn m ,)) dm~s. ~c of a prlin' ft<lmthe plane: ]Ji~lllIce bet\\'tcrJ.l\>n para!t,lplaHe~; S)'l'r~mof j,lauu .

3. J..inl' (12 Pa-Iods) d 5 !\-1arli!l)r:.quariorg of a lille, Di~iancc of a p nil! Ir lfll a line, Angle bell',e~ line and plane, C<lIlalief Iiocs,f>kew lin,:!:.

4. Sphertl (U Ptr-iods). 15Matks)EquatklJ'! of a:phe:re, Ta:l&,,11pk~; lll,l <ollilition oflmgtncy, Se.;tion of a ~phereby a pIme,Inrelprct:uio'l o"lhee.: !l.1t\''\\'!SH~ =) ~11S~U;=O; Relati~~pc:oollsmtwosphere; (:oodltiot,-of1.'l1hog!ma1ity,

S Uivlslbl!ityd inttgen;, (UPeriods) 1~Marks)Naturnl r urn1::lCn.Pe:1t n',' a-.iolTO'l.V-'en (Il\erinlPrinci[ie. (stal'~mt onIy).Priru:ipk of n .atl~. . .1~.

Di\cL"tlJiJiLv of integer<; ;md tho:;(=DiWlion l~<Jritbn(wil 'toLIIPC( of)Q.e.D. n,d LClI.tEl:;;Iide."IlI algor'run ( ,lilhout proal:) l~\Il; fa,:;torilialion lheorem(wi heLIIproof)

6, Congmellct' cla.'l."~1:'artitkm •••f:l. iC:'Ii) F.qui1,'l'\(;lK; l relati. In.F.quivale1l ;c c1 ;<;'llo1t!lOOIeffi.(i) Clm!"U'--rJC';n::lati, m (moo lIlo n ) rim lh,OI:(>11!IlUi) Pno;;rtiCl; ofrel.i :.ue classesCom{lO!'lilion w::>leHl7cl.lrun', mCI!l1:m

7. Compl~lnumbenA1gwm ('f oorrpkx n: tmb-.:n;Geometdc- r<'!'n:renf;:.!ionof rom! k,rumt>el'!'.Modu,o; • I\mplitude r01lll of a cor tp',,,: J;lt n!oe, (Polar fontl)r~larIn::cpalitiu;

& De - MI'h'i'C':< TIHOrE!lll.

D••..:!doi,r"'~ tJ",orom for Tali,m:tl D.d'e",;.n-nill m(~; vf llaity a..,.jlheirj:; lOlneri .:a!~\t>.lf~IaIiO'l.ntll roo!s nf a CNDpJe:; -.:.,;'01:'" :

4

(to Periods) JI'Marks)

(Hl Periods) (lOMflks)

(to P~riods)(1.-5Mnrks)

.. - ..•..•.------ ----""---- ... - --

110 PeHods) (1II_1\1arI{,;

PAI.!ill.m (Bl : Gr:al1:J THEORY A,1\(J).li,GFDRA-------------1. Grapft,j: 00 Perloos)(10MaI"ks)

JJdinihcn, 6im[:c Gl'oIpb, Ha, vi'lhak.l3,lemfllll. OOrnorphlsm Q:'G,aplls, TYJX'8of graph ••Opel"a_liuMon~m.

2. ConnC4':tedGillpfu ; (8 Pcrioos)(l11 Mal'ks)Wan., mil, fl

liI1;, :;ycle, tciraJj~), Gl!1ro<:ll~ &; discOlI!ltCie4gl'aJiI, (..'utwnice;. and COfl1leew.~.-

3. TItilS: (i Periods) (10 ~fark~)DdInilio'l at«1 fl )oer, ~~01 tn eli, Di~l,n.:o: &: Cenlres:n ~ tree, R, lOl;d ;111<1Biliary trees. :;panl'lilgt1\les, KnI4<('l'~ )'gornJ nn for ~~orte:;l ~li11omg tn:e/l, weighte<l gr.rpt

4. Eukrill:llantl) bm Jtollht1 Gr.a,J.'t: (I)Periods) (5 Marks)K 'o'lligs:'C1S~ ."'-w bI idg~ prI Nero 1:1~<:I1m !rail. Euler.lll gr.-pt, EIllr.iltonilln ~lh. HmnilloniwteycIe.Ha'l1:1tori~'1~-II. Tra\A:Billg~t911\lnprohlen

5. PlltftlJrandrm••1(; "'lJI,~'s' (SPl:riods)(5Marks)P1alIar gr'qJh PL<>leOr Ifh, F,u);:o 's !h(,)'ttI, for piallf.r !Tolph.KWlJI'i'l1i '!tWo grap\18, Gt:omebtl1-lldUlliCokJrilw ()f " ~ll'" I

6. Matrixi{epJ;'~nlltal-ioll ula Gn-'f1h: (~Periods)(~Mark11TheAl1ja m:rlT;rm,;!ldJllciloncel are ptt~

7. Ditedl'llr:"':t1!;(f);grvrh/ (J Pe-riods)(5 Mark'!)Dtfinifion. fll.1esr';e, Oillietrre, t'fa \;:1t~;.rile At!iacerq lind Jnc 'dence Matrn: of a digr;lj~ l)"J)tl:ol'digrap~,

8. Oh1!!ihiliryof irttt'l !I'S. (12PerkHls)(ISMuk~l~lItllrahmll:belll. ~':'1l11li .n:::r.'Ilt~Well<: akringprlnciplc. (~mcll; Ollly),Principleof mat~<J;f1cluclim, Di..wJ:"itty, inl''9ClI;:n1l llle01t!rll':liWrion~d\l11 .••ilf!,-:JfpilJOl)

G.CD. ani: l.<:'~-l. }::tlcldCillIa~~C!rthn(l'Ij(bUll'roof) L'nique: faet(,ris,llion lho:.«cm (1ViliIoutproof)

9. Congl"Je.lct clIJ'IlleS ''8 Pnlods) (10 Marli:!l)Patdion 01.~ 9ft (il Equi ,''l!e-r;eerciali<m, Eq Jiwlllnce claiJot'll 'boon ttl.Ii) ('1JJ\flrl.QlCe'-eil/ioll mlx!n1"m):m.i'ftt.ll'Cm!l (Iii) hopertiet ofro;iducc.llSSCllCOmj'losilit"l ~bb ;oCl1'liCt's <hcc>rem

lO. Complex numn. rs,~of;ompb llllmlml{~uic . rcpn-S1'nuti(n 01 co;rll'lle:i:r, ,mb=.Modul~ - .\Mpum IrfuJ n cf"c )ffiphJ!~11l1'ber"FolarfollniTriqulllr inequUl-'li

11.lk-l\;ohl~" T !Cor'm. (10PmodS)(l5MJlrkll)T )e..MOM~ > thooi\ :tI[()1 ra'julla indi~::f'n"1l"'toots (.fum'y ,.nJ til ~r g.:on ",met,1 inluiOll'/l'lion.,!"'roof'; ofacompt ;.;mnn!:>e;',

IMP NoW-- nl>.l\'h'ibla,:e ofm./tswU;cbal~acwrding,othcpalter::n 0fthe ~/iOllpaperwilli sIit\!u: Hril'1i< f'1l.

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,North Mahar(J.sl!tra University, Jalga0!l

E Y B,'lc. M!,tI!oma!lcsSyIlllbwnrith ltmM11U and s£ope (wlt!l effect I'l'oai Ac'd.Yr. 200z-2003)

PAPEJ~I: CALCUUJS

. .(8Period)(l~Ualils)

•

1.02

1.03

1.041.051.061.07l.(I~1.>)9

1.101.11

2.052.0,;

1.00 Sequelll'CS.1.01 BOll~dedarrdunb:'unJedsels.

Definauorn - 'l P?er be uni J~":r b-mnd. I3:)unaed sets.

LeaHUp!XT Boun~. Gre,k:stl (lwe ':;k'UlldDc1initiotu II1tde..~'rnples.Se.quen~~ - I:efin'Jior . l.irT it,,f, inC'jW.-rJ::C.

Tr.oomn.:onEnito,"equ=.(liichO'lf 1'l'tXlt;

CllIJV;nr-;r.;e of .;:'l'.lePc{ I -D, fu Iill1. lS!'r'Jr<.""rtic:s •.of C<lIl\ <,'!!C1l! ~eqUCitC';<!.Gronetric 'JeqllCfI ;0:&.

M;;r ('lome f'l.:qu~.',;,,",,.Bou:~jcd~:~ellc",.SuI"'!in:tr:1 ,mil" in~;mwn ;),~,1:''tI 'J, ':1(~Moc (,[prr1cbOJ,-:J~:d~~qu :::;ceilTh<;; nlW1.l?';r,c~PT, "~ t;:a: lIe Sl:<f.m: ~ej { -"I . - --\ (II- --;,.-) is com-l:q'en! a K it ~:mVCrg.:510 theron.bel' e,wh:re 2 <: e~ 3.

2.COInilnlk~l:"~. (8Peri<)d)(10l\1arIa)2.01 Ccr.n~'I"gCn<;eof3 :li:s. t'dil!il OJ':.5. <n

~,~n-l.2.02 G<:o!1lelri:;e;i~. • Prav.', il11lf'h~ .~e X'letri:C series _ K .~.•., .,.j}ron"l}tpmfiJ if "i ana ii)non-CQ1IWTgentiflr1~l

2.03 Thelm:ms 1m~n")Ig~nc('ofi>l Ii!",.. -

(wifhoutp-<);):f) 0 ':l."P,ris. J{lTe l (1'0:'!lC,utproof)2.04 Hyp:rhormonic~-ncs'J[ (l-foe;~.

Pro'>e !lu! !lw aen 'J.I 1 1 '1:J;- "~,~ ~,.........."c'.-

,rf [p zP 3TJ <jl' ,J

rom"rge~:fp > 1and di'eTE"" if P ;tD' iJ..l!llW:.i'S Ra::io T<:lt (willI{ mIll ))f)

Cau<:IJY'sj"(~Jttest (i,ie,m t jlfO< ,r.

c

c

,

3.00 lndetcnnina~eFOims.3.01 L'Hc~1aI9111ies(statnmtOllY).3.02 Ex;;rnpl~c~.l!::'~~)lms'

(8 Period) (J [\I[m'b)

oo

4.00 Contirmity4.01 Ddinilion{;ra1in;~cL"(:,-' ~ :,~e: dt t) r~

(6 Period) (81\ £1U'b)

•

-----. .

4.02 ClIciILsl..Y fundion - _1dinili<:ort4.03 One-sidedlirnit -Rigl f hand lmit. L;'JihiJldIimit.4.04 JJefirQ.Jn of ;I limit o,:fl:x) as :I:tinm 10 JO.,4.05 Thcorertu! OIlalwlR illirnilIl (wilhot~ prllof)4.06 ~Lilyofa;lllX:liqnal;lpJint

Dc1inililJn. '.=";' de11li1:ion4.07 Continuity "f ,1 jilneti<" OIlan inllr:r\aJ.

4.08 D».;ontinuity and its \ "ooll'! t~petI.R',lDO''lIbk- ••1IdirroOOY;lb/lll!ist'JnlirJuitial.4.09 I'mpeI1;~ offtu'l;liom .;cnlinOOI1t <It,I point (wilbOUl proof)4.10 Proper1iee of furl;oom oonllnuous 'ner a ::IO!1Cdintm.'lll.

(I) Ewry cOlllir~UO\Il;functio,l on cJtt,oo ,1Jllit>ounded interval is }(~mded.(i) E-.ery comir>uous hmo.'1io 1on clvt--m ilIld bOlUlded interval allains ill bolllidF..(ii) Hl(~) is cootmUD II;Oil [a ,bJ and ,ij I):]b) have oIJPOfiite ~igru, then f[x) '" 0 fot l!lm~

_lE'I ii,bJ (WIthOUT proof)(iv) (Int:nnediaiev.ahe \hetO<:m). I,l'.--nlt aevnlinuousfunctioo,m [a,b), then f

.Ii$In1CllI eve!'}' valuo: bern'em f(i: ~-l1Ilif(b)4.11 UnifulTJlCOlllinuiJy. r 'l:finitioll

s.oo Mt'aIl \1l1ue llIeorenn. (WPcriod)(lO Marb)5.01 Difftrtntiabilio/Jlldd :rivab11i'Yof a f:alClim

Detiniti'Jl1lI• Dcliw!ll e. Rlgh, hand a~.Ilefi hand derivalive.5.02 If a funr.~ion fill deriv ibk rot x, then f '~~r)nlintlOU9 at ll, Conve~.sm Rolle's '!lIeonn GCOIrt~ fnlcqm:.wtioll.5.04 ~'sM"qValilelheorem.

GeomCilrical Jnt."rpref ltioo.5.05 Clruchy'sMean ValutTh:pn:m,

6.00 Soccessh--ernrrel':llti.dion. (8Period)(10Marks)6.01 SUC:celII'iwOeD,1ltM.~,6.02 l1JeJl1hderival:i~of:OIlW8tndanl:f,lllchWl. <l(•••b). (ax+bl'" .x"', l/(ax+b)

Iag(IlJ;;+bj, eUn(aHb), c08lax--b), ,ax,:i«bx+c), ellXoos(bx-k).6.03 Lelboitt',ThcOl'mt

7.00 Taylor'. aDdMaclaurin's The"Mnel>, (6 Period) (10 Mllrb) _./7.0J Taylor'Mtheore1n "it!' Ugr/llJgc 's Corn,oj Tm'Iilinder.7.02 MIc1lIurill'slI\e1>rm1''litbLagrangr'~lbtnof ~maindtJr.7.03 T8}Iot'. lhooreJ'l wit! ,eauch. ••'~fOlm ,lfr;rntrtdQ-.7.04 Mac1aurin's1he"ran~iihCal1chy'~fmn.)fKmaillder.7,OS Power 1'::riCH-:1IJ:Jl\SiI'illHif c'Kmc: clelllmlary funct»u.~.

ell, lIirtx, C08X,l!-'-X)' 1, log(l ~x) -if;lo , 17.06 f'.urntlIes.

1.00 Iatelration(Dellnttf: and IlIdefLn1fe) (l5PerlodX20 Mark!!)R,OJ Melhodofpartialbtion<..

(i) lJmomillillor ca: .trlining ilDD'"rer.eall>J IiIlearfK:tm! only.(u)Denommawr con :a:iningl~ Iinearfacltm,(iiI;Denmninll1ol" COl"tmmg :me 1.f1el~ard OIle im:Wcib1e quaotario,; factorooly.

,

,

',') J r--IP"-.,l..j ;,x2-rbx+c.Jx

f x2, 1. i) --,.....::- d{

x4+ J '

9.00 Reduction fQrmulae.9.01 'RedUdi"",wm11l1.lt"Or"

(widwut "lOOt)~.02 J?-eductiorofornwlatbl"Q '"i) !silnlXq<s'xw~

"RedUCtionJolUlUlafor f .i'l;Ili:-~.., ax.n>!

, ~nx

,~l'erlod){lOMar"s)

(II Peri,lld)(10 Marlij)

,

:[0.00 Applicatlon~ oflilltegratlon.(Theol)' isnol ::(pect~)

10.01 Rct:t:ificatkr" •10.02 Area.10.03 \bitIl:ne.10.04 SmfEear~a.

~~. " •• """""-""'..............,"""'"""~ '" v- ~~~.""'''''~''', ••_ •• ,._ •

aPAPER -II: MA1R[CES ANI!.D L~.Yf:RENT!Al EQUATIONS,

1.00 Adjoint and IIlVers£o<lfa matrb. (t;: P~rl"dli) (15 MarkJ!,J.()l Dcfiniti'lllS oflrm'lpne. Or 1111alrb ,syn netri.) :\Od*,"'.~~Itrrle1ric mlJrice>.1.02 . I)c:fmiti(mll of lllinor and c(jfu'*)f <J:"Jn( I, rr.er;t il[a fI1Jtrix.l.m Dektnlin.rt of i matro. sin,guL1f anc f,)tl S;11,'ou'armatri<;e;;.1.04 RelalC'd,roperlies ofd~iermiJ :lUll();-~m ] ri';,1.05 Adjointormitr;x.1.06 Theore1r,,: I) A(~dJ\)=.(:djA) \ ~I\IJ_ 2) (adJ'\""'(adjA'1

3) ,adj(A'Jr=(atbBHI4iAI 4) ladJt\1 o~it:itl.5) adj(~(.[A)"I,~n':!A 6) .adjr'kA)" Ie'.' (adiAl

1.07 DmnitiolillfIn\=,-,fa!natri~1.08 Noomary.mdsll.fficielllcondjjonf, ,rm leJlC('"finvtn~oh ma:ft~(~ith proof>LQ9' .Uniquenf$S oflrMmlt: nf a mat, ill ("' 'f!; 1'1r,(lj I1.10 CO!fllXIta1on ofinvers, byusiTlg~ 1dioLl'l)f~ matrix .

•

,.'--------------------------- -~~~~,

1.11 I~ ofi'lversecfam>trill:.1)(.AB:.1~),.l A-I Z' Aryl ~(A.lf

1-1",

AQ' tH 01

2.00 Rankoh matriL mPcrlod5){lS:Har'b)2.01 E',cmenl.vyTn::d'orn1iticm. Equ \ DmI, matrices, Ekmmtn)' m;trice.~,2,02 Theof1:tu(ooI) ;tai«lIe:llanc l'lllIrllim)

o(AB)c'(m\.)l\ '~';,ElH::r(AB)"" A(oE; . ~ I~ECT

2.03 TII:Xmt (~ lroof'!o A"'E,\ O"ilI.lla ..•.•'".~,E (Tis,SCT

2.04 l1~lOI'mI{wllk;':oot)nl<-dll~l!lt oCat: ettlt1etWlfy mllr ~ ill af!ClmiClIt3ry 1n,'tID,oJ'thi' 'l&n\(ltype.

2.05 l),linitiOIS Ii"!l 11ITl&1fV,.min )f of 1 !\1.-tbix,Rank r:f a m.lt!h.2.00'i TI:«ll'Cm lWirh )'>:1011Ilval1a" lC~(f fn under clement'lY lmtvlft-rmaaon~2.07 DdDli.onofNo'1IIIforn2.08 TIa:oJCfIll(Wilh [I"OOt)

1) Rodoclion 01'.1matr X IOJ :oiltl!<llm

2) E,UIrmioc oj' non"'llil t&UIw nlJlt oj lei 1:'&Q !FUcl\ th,Bl;PAC) ,,; [If 1]o <1

3) Non.OgWllffllalrix 1lII1J r<>\1<cltfEk:melllary matices

[0"

4) PA'" ;OJ2.09 R.tkofll..e Pro<'wtTIKorcm(~1itI1)J.I)CfJ

p(.{B) :nninfpl 'lJ.!•.-{Bl}

3.00 S,...oJ'Lioou EqllltJOll'l. (to :Periods)(lll Mu1.s)J.nl [)eiTitioo andre ,=1Iti nan' filOln~oc OOIII!andn(T.-homog~(l\l'H'YStel.1'lII of

• ~uequ.llioM.~,A.'('0, AX:~B.J.02 Th,l(Jrtm (rnly SI'_lI:m~'Illl

A~stern-\..'{rl!iHon;ktt:n dd,\ "dABlJ.OJ Wor1dtli!rlit: for ,d\'ing <\X=:) tl '-3

4.00 Ei!efl vltllies ~id EJrm VKfof'll no Pmods) tl0 Ml:rk$)4.01 Delntion'llU1d jJ 1l6lnl1llJll!oL.iga- "a1llt~••nd Eigen \«~,'.,

COlIl<id«,(,nly2J 2matrce811 ill' l'4letl'WCtors.4.02 DcIinilionandeF.l~ .JI\ch 'tlC\lt islic'JqIliItkm (1(& Imlrk4.03 CI)'L-y. Hamilto-,l Theoj"\:m(K.III,.:u,~,1onJ)'!4.04 vcn:fication ofOl)ler. Hmilo Gn ~ORlI\.

4.05 IrM~of.lmlllri,byuSl~ (,,,Aly HamillonThoorem.E.'tllllt):k;.

- -

,

,5.00 'DJ&1raItiol Equ..&lluf61~ t or der lllid lil'llt degree. (20Periodll)(2 )II1arli!)

5.01 !fum .'go:nell!; Diff, rentid Fql aliell i::nfinition andmcthxiof ilOlvingit5.02 NOO-,1om~,neoIl'>i)ift'e! :11'm 1'f1'.laSon5.03 E:xacIEquation; DeliniUu-i.

5,04 NecC:l'J<ll)'a ,d suffi.:itnl' 0':d: ti '" (with proal) f[lf Ibe .:quati-:mMdt ~ Ndy. 0 tobe txac!.

5.05 l>.IDnilioouJ~lle~'ingff "jIlt'.

5.06 RtJ1esf01"fin:ling iF :with :l!OO~ fllrm~ equation Md.'f.+Nd)" ~Oif~ ltecqual'iooBIHmOpl~(ll':ii) tt,c equ.H.ionis {f!he VPI::I fj (xy)Jx +xf2l'xy)dy" 0

.~M $N. ..ill) ••••• - ••.• Ii ~ fw,(wmotxal1JneE~' &." ..•.•• " .0.

NIpr ,');\.1

'ill) i! a ~uurj"n ofy ~opcfI; 0-_. -----.-'".- .

"JF by:n~pecrion is no It eX! ~ re; I.

5.07 jinead)ifre.cnlilJE{,lWiol dyDt>finiJionill: d method of ",',£,~:,h~ 1;""." diff~nlial ~'!U'>tion.- -,-" " I'}"'00 ~~_,x

. Whcr., P &; () •••~ 1bJ,..tioJ: " ,\f:: 11Im'J. A~oeX'l'npks 0'1 thw<. "'lUlitioo-of the: ~1re

- ~. +.PJi. Q. antcxa!TIJlfS t< :h>e<:JIeto lineardilferenliJI equaclionbysut:.!ti\utk n

'" !Jl.oulllbe Wen.5.08 Hemollli'aOC:~()l\

M:t!l(<! off;(.i\inI; Bnnoli!i '"l.' l'~liOn

~"+Py'~P',lInco}Jlf. .';1, dxA1lloexampi"son!tll- eqlldi<~ ,,fth;,type be 1;:+ Px =0 ~ be taken,

6.00 Dtft'erentbll ElJuaHonof flrs, uJ'{ ( r Ilnd htgher de,.ree r(:t,y,p) '" 0(10 PeriOLis) (H Mllt"Ia-)

6.01 SolvaNefor]16.02 SQ1vilUefOI':'6.03 Salvai,iefur;;6.04 CbiraJll'seIF:.mon mIl menlld J~nhingit.6.05 Eqnaliom ,ecfudl¥<:to t ::Iaira iI'; Cplillionhy proper subslitulians (subllitutiom re pj A;r)

7.00 Applkl'ltioR ofDiJferentiffi E 11"0:iO'I' (l;t Pmodll)(t! MlU1is)7.lJl ' ( Iitb.O!!fJllal {Ia.illCtori :£7.02 DiffeJ~11tiaie.luatk'l'l "ffan i1.l ",I lW\"eS.

7.03 To find the l1!lbop:Ql1IllTlti, cL)ri;,.. { h f~miIy of CUIVI>awhenitl cqUltion is in 0 tocartesian form . The1h1Iowin,~fmJ Ii<s (t,-,-,n~betaken

i; F~mily o;"~land.1ld cirdc.; it) J'amilyof gener.J citclelli) Fanill} ofparabol~\ iI'} Fmnifyofc:Ik' \) F/lmitrofcul'vmruo;;;bns

G•

.,~"y=o;;; ,

10

,

•

7.().~ Si.-:;~lulillll

i) p.<i,crimm.l;lIfrc,ItOl uf'!(~,):p)~Oi) c-di;crlmin,llIlr~aMI ,JfK"J,p)~Oiii) I:>cfit ,tiono! Ut'gU :If so \ lion of ainqXe diflen'11ti;d equatione

7.0~ Er'\e\op<~ffam,lr of ,II"V(" simplt;axarrlfllesiII'~~:pected ....•••.~~~J_~ ...•...•-'0"- ~~ ~ ~"~ ~~~~ ••••...~~~~"""'" ~~~.~

PAPER III (A): (;E;O\lETRY ANDAUiEBRA

1.00 Co-cnlhJlltes in "pIlCE', (8 PtriOOll)(5 Marks)1.0 I COOroinlk!l fa poi It in f:);) C<:

1.02 Chal\!c,.Jfoql1n1.OJ DislIrl c.eFoi'11Ula1.04 3oction form;lll and ;oroU]I"e.~h

il !l.fid Poim. i.l Cenlnid of run ~e iii)Cenlr:Jid oftetrabcdron1.05 Direllti<'lo CO'inc, an,! dirt. tj, JIl r.1li<l6of aline; lder\tilie!\,1.06 Angk:bctwe.'ltwoHnc&1,07 1'roj«1im of ;llinell::gnteI i.

(12 Periods) (,,5 Mar"") Y2.00 Plane2.01 G_iI1Equ,lionoflpbt~_Z.o:! Equationsof c.oordin31e p11l1eJ.ud p1anCl!paralclt{) ~ JUnCli.2.03 intcrcl'ptfiJrf'.2.04 NomLll.l'o£l1l2.05 J>oini..JiJocti 'I ratio Jorm.2.06 l'!ane pasainj; hrougIt thrc !poinl!.'2.07 Reduclionol:~ equakn IJ!he-

i) ~lrorm \ irI~foon2.08 ,Wgic ~twcnnt\llorlaIlC82.ll9 DMWl<:t ofa )lo.lffr:m tt~l'l:n,.210 TJiBuncebetl'~ tw J par.iI~ll.lIleS2.11 ~~ofpln:lC8(u'-Av' C).

•

3.00 LIne (U Perloos)( .SMlr~l<) ../

3.01 Linealtlmtne«ionofm" lun::a3.02~'ymn;ttricetl~ofa'I)..3.03 ~ClllI of j liIk tIl.oogtlHO pMts.

, 3.04 'Ii'mdonnatt'llfrom U11I)ltnet ;:to ~ynundri<:fur.n.3.05 pQSili(mofllleandl'laoc3.06 AngIl:be~",llir1ea Idpl; ill_3.07 Distance of a pow iT lItlU1J In:.3.(llI Coplarlef tin, ..,3.09 ';ofinllhep.,intljfh.fmc :tiXll.'frN(l~.).10 1.•cnalh Allde :iuation , of tl c lho~1M.diIltatlcc bctf,uuJ:cwlil",.

•

4.00 Sphete4.01 StandaruCljI.JdiOll oJ'~~,4.02 Cenln -R,adiu<fonn.4.03 DiamflerfiJl,'l

11

(12 Period!) (,5 Marts)

,

,

.'

•

c

4.04 Spher"pas!in.gthrnJgJ>focl'poil~'4.0~ TnteIli~clion)f a Jin. and Sf.here.4.06 Tangmlline mdbi''i!"entpl;m 1<,t1,~!~'1~re4.07 COlldiliOllofl.JJ'~h.Y.4.OIl Sec/iol} ofa Iphere t>" a pI.a,,, E{ 111'1< f • "f I ~irtk4.09 Intap.''l.'tatiOll oflh,-equari;lns ~ ~ ,,:; • c ~)\d S + i\.U '" ~.4, J(l Relati,~,PO!li,.ion~ 'J.' tWo sp It:re.l .

i)Noo-inlen;oc:ingii' lnrcrse ~tiJll!:i) Ii uctin:l'Pl=(in~~\ e"tlmJally),4.11 CillIdij:,Ollof \'!1hogu\.llit~'.

5.00 Divifllrilit), of Inleger~. (12 ~rtods) (15Malk(S.fJ1 Natw.:al \IUffi\);n;.5/)2 Peano": I){jOIl~,S.M WellQT,kting prulci:p ". (slah.men!)l ly5.(14 Princlpl: of m.atbr:nj~licilliild .uiol ,5.(15 ExampJ,~~on p:incipk of mat1,etna1.:< 1il\i~;li(m5,06 Divi!il1iI1y-~finitim-~,07lfaiball<jhlclhen~lc(wirilpro< f)5,011 lfalb ll,d IIIc [hen a ! (b 0:0; (wi! 'lrl'uf)5.0:; !fa iblllitl a Ic then a: {bx + "y) wIn:

.'I: and Ynr; any integcJq wilt prod)5.10 If al b ard a I bdQtt"\1.will!eger{ (Ilit l[n:xlf{S.lI If a Ib all db! a 1111:'11a ,-.t b (\vi1hpc of,5.12 Examplei- i)JJ'a,b.;:Z,a t> mdb'~Olto""lh3t lal5: Ibl

ii)lfo~ «b andbl, sijoHILlll/1 "'05.13 DivmollAIg",itIlI11 (wit~out pre of)5.14 Greatest G1mllIllJl DiVIlOl'. De ffnlti( ~5.15 , Theorem: Any!wonol zeroh,tllgel;, -al~[b !trw it unique g.Cd,.l!Klit'llll beoxpr.$Sed in lhc frnn tllil +nb. Ivhm:; 11,11G Z ('I tb(JUtproof)5.16 L:asr Clrzunon lfulcq,k -Ddinilioo5.17 £Uctidean Aigorllhm(w:,thoulPlOOI)5.111 ExampksonG.C'.D.5.19 Prime IIIId Comp)'!ire nllmb<.'l'S5.20 UniqueFa~~iOll 'f11:orem(vithQ,! :J:l'•.f)

(i.OO COIJIf1IenCe{llllMt! (.10 JPeliods) (10 Marl(s)6.0~ PartiIionoLi,noo-empl\'set6.02 I\quiyaIeQc~ relatilms -D ,finitior6.03 Eumpb(>J, iIll ef[u1\-"aJeJterela;ioruI6.04 EquivalmC!cL1~f&_ Dol:liaitioll6.05 .ExanipIes en equi'r.a!enc( cLv;.'le~6.06 "l1Joo:RmLM'-'b:lomd'quh-a/" .cen la_iol Qll3liCtSanda, hFS llien

(iJ a e ["1 (ijJlnJ"'[bJifa.."'ld(ttl~i',,_bIii) aqy two equr••.a/CllC~clas.-~HlI'C,i~II:l{~ioint ofldctt!ieat

6.07~'cla!!~s them"'n - F.:'"~.(>.,lr\;a<llI<;,dation 011OIlIOl:l-CmptySllt AiDduces a f7/Il1itio.nof A am: conV\ rsel,\'"HI)' p ll1;li(1 of /\ defines an equivalent:tl rdalion lJtlA (with P((lllf)

"

•.

~"'W" . -

(10 Pertod~) (15 ~ [arb)

•

Ii.G8 CoI\!:,-uIDh:f mOOu .J m - D .:f:r:ti~.K5.09 Proofsof

(A) IfIl ••bfmo,!m]t1l.micH ale, l

~) (a+ci.,,{b., cHi'HI r) alld!i) a, ••bc~mndm)(8) Ifa~b{m{l'lm)andc••~lm).lJ!lllhen

(i) (a+ch(h..-d) 'IfJ)drl) (ii) (a-e) •.(Io.dj (modm;(ii) oW"'!.! (modi i)

6,}!) DdiJ:iWX\lH.f Ii)addihvnn xu].. 'II iIlId (i) muhiplk:aOOn odulo . ,.6.H Composilioatable," . m m m".6.12 ProofofFelImat's 'heo:JterI5.13 Exarnpl:!I0.. F=, l'!lb,!>!'ern

7.00 CompleultlmooJ"S (10PeMoos){toM iii ••}

'7.01 NotationanllrmTi:lO"L'!gy7.02 Equr/;tyofc"oJnph nlrnler1l7.03 .-'\dditiOll, SllblTlk:l<lll, .\1u dplic ,ti 1II l:Kl Dr.isioo of c.ornp,le,>;:rwmb-n7.04 Comrie:': CJl1iuCll': Numl>e:rs7.0S Prop()&ld.on. IfZ, atd 7-:,« tw ) ,.oll:>1<;X lI\Itnbers 1hcn

- -- --(l)Z,+~ ZI+::' (i) ~1-:!:I"'Z,-4 {it) Zl~=l,l

. 1'7-, _ Zj(M.7~.'- :z:-7.06 Gto: ,llibit :'teprt"S:1tati0J' oro '"iJll :;:Numl:crs.7,07 ModlJus-AnrplkuJe fut'm of a ( :O!llJ):XNlIltIber( ••.pcb! Colm)7,08 ThecHl,:tIIl (onMlKJl1us IlDlI Am l1illk e

6) Z,.;- Z I ' iZ, 1+ iZ,1 "fZ 'Z'1EC

Z' 'zl'z' d •••. ,.'~" .••.••z,+O_e(i) 'Z, I"", L -II an , ,-" -.,-"

(il) If ;';I'Z;"C.lben \.~. ~ ~t~ ./

13

----_.----~._-

(I.) 4dlfitiOflof a V<:rtexr. <;lini j.)1l'i 8:. ilfustralion

Grnphs :'J .01 DeJinilions, Simpk grnph, Tmill:a!',4:dil~tI)grapm1,02 D~gr;,tlll,Degree 0:";1Vert,,,

LerrmL11.021 H311l13bakmglemr);l w d'proofandwrificali(lQ'Throl't'DlU~221b numb-lf of, 41 , ~'li-~~~(\IIl1'I,;.xl'\\1ngodidegr«l) inagaphi~

alw<l!""Sewn (with IJro!ll)1.03 S~apI~ S;:xmnin~subgr<pb,I:' ,{ n" il'Ill)f I1'ldlJcedSu'w:I(lh, Edge -1nduretJ subulIph wi1hill.,llrali,ID1.04 IsomrrphisJII of""" !Jfaph- , E.xa11l'le, )f l'l'Jl.l101plJic grap.I]~.1.05 T)pC'iufg""pll6

Cont~Mc gr.qlh, rc ,1.Ilar gr!ph, 1,il'!lf j e :!l"ph (A biwaph), CQlllplt,!C.~oartitegrap 1,

Null gmph n ••finil:innaitfId ~IlJ'lll!1 oj t1:clJ I t'J)3"fllrophs.J.06 OpeDtionsangrapl~

(a) Remo'!'"of~"."""".(d) Additi,)~.d'anedge

1.07 l.JDb,lJf~"'lJhs1.011 fnN'",lClio:noflJl'lP'"l.lr GWll({graphs1.10 J>rDclm:1of p:r;qms1.11~'=1,ofaglaph1.l? ThcRingsumofgmplll1.13 Fusion m\oCllicCli :Celinitior11& iJ ,II!1r. b[Jf,

1.00

2.00 Conned:ed Grapt'5 . {8 Perlodsh1l'1 Marts;UH WIIIb,PillhsattdCicles (Circuit ,J_Jdinidoll84ndexarnpks2,02 COIII\Il'Jkdg,aph!J, ilicollUl~ted gl"lp ~ C"OInponen(ofag.r.lp!J. IDustrations

Therotm2.021 I.dO bt a sin pl~! r:lph 'mm pvmicu, q e.jgellandk componen sth.enp-k~'I ~112(p-k)(P'"k+;) Hllhproof)

Corohry2.021 J..eIG ~ a cf'nne,tll'l grnr b wi 1 pwrtico, q edaes thenp-l,q5l!2p(p-l)Theorem2.1l22 ,lgroph Gloll ip]J1it~iranJonly itG ha~llC'lmd"yclcs(withpro<:f)

2.03 ~ andBrid~ De:initiCl~ 311iI illumliOWJ f"

2.04 Conn.,x;lNty: Vlllte"oonnr~ivit; (l-«lJ, dcfWrlon&dWllplCll, edgH;OIUtocdvity (:.{O»).definition an.f o.'\IllIJp1~TI\Wt'Olll2Jm Fo:' any ga\it G, K(G: _"-\(:) ;15(0) \1i1l.e1l' Ii(O)=.minirnnmdegr..'C mil vm1c:l.inG (witbpoot)'Verification )f1l1oorem b} =tpI:tI

3.110 Trees (8 Periods) 001I1arks)3.01. Tm..>s:De6nitionalld =;Jles

Tbeorem 3.M 1 :A graph, 1 0'= (' t,E): u. ifeeit" and 0I1Iy.if any two dStin<;I'.C1ticel1 a 't<

jilini& by iJ unique palh ( with !¥'Oaf)Theonm 3.{112:J....:IG be;l grar ~ '.wn p ~rtices then tbcf?lk>Wingitlllemenl_~

14

-

3.'\: equiv Iier>..f

iHi is a II~Ii) ( ill ;l.1.yeW,;an i"j( ,(; ,k. ofedgtls 1..'1G, ~ II-Iilil , • is c<:mle.::" oj1111.1c G) ,. n-l I ~ it:\ ,,,"oo{)

Ib.~m:m. d3 F.\Q} non. in.i, I 'l"Ul,withI\o\'Qorl1)('l'Ilvertict~)haiiatJcllSlw. )P~tkntver1l.~(mthiI')f '';3 02 Ceam:s

F,c<,t21trici,yof J 'mt,x, ,entn tldiul;,nddiamererofag.nq:h, Dcinition 'I"~thilluHa- lb~em J.ll A ;J'el;ItirHlith. r, lI1' J1"!I>.UClrltrel (mm proof13 ill Spinm,mg'l",,,,,,,,, C)efuUli Jt1

TI",orem J:ll ~WI~ C(nne~~' g Jp!,har. aspar>-11~tr~ j\\i'h ~ro(ll)33)4 FumLwnem:alciJ\ '.Ill:!!(C)clC!l)

3.11S FwiJanwriUllOUl'~ De, initio I <of, l.l!t'tll, DeftnitiGli uf}un.ltttental CUIlIetv,ithiiIn,trawl\. Cul8~lrnnllf G( 1";(G})3.06 Roolled<lIld Bi'm 'flre<:S.Jdin W<lt mu,IULnllS

Th,rnml.61 p, 'we t!U' in a ;it,ll!" tro'l: •••ilh n. vill1iresi) lllC' nurl00T ulvt:rtia!! s om: iJ) JJ - ('14 11'2, where pi rwmhenfptltl~ll vern '"

iii)q~p-l. wlxn q if;f11mb<:o0 "n!:KlPIn, eUlVl'ltices (with Twaf)3.07 Knlfbl'g ,ilgcrit!"" for '10l'tt., i' pa ning lfIi,e

We~:hted;:rnph (1efinw'1fI) II d '1'" I ~ltl, !:.~liOfl OJa1g,Jrilhn:with uample

4.00 EUlarflflund Hamfltfjnlangr ~pluJ -6P~r'l(ltls)(.5 Mllrks)4.01 K('.$lK'-m 1trid~PI' ,I:-en,'El:jllainatiO!!)402 Eu1:::rillll,graplw.

Euleriantnil [k initatiOl off. dl'll;,11graph wi!h"xamr,kTh,;ocem 4.21 L 1 G he Iun !~~ttj~lIfoh Vlitbp 'micl's (jl ~ 3), 1hrnpn:,vclha tit:

tbIIoIWJg8tarem~nlllar,.<!o:J.ihWllent( Ag; I JbG l.E~ @TJI'n i1)Everywr I:>;OfGiSohven doigl.'Il(\ ii) " 11edgl'> of G ,lOP ,ti.ione..i mfo disj(>im oycles.403 Hall'jfoniJn.£.I'IPh

Hanifonlanpa4l, Hanjr,.,n\an ~.;Ie Ilionnlrontan~(DdlllllallOll), The Travd!ln~lIIllesman :J,l)ble\n (tXill.auW.m4.04JlIu~taltdt:.=jj~~,

5.00 Plallar fllldDud grll'h~ (SPeriods) (5Mllr"")5.01 Pla--m:gllI)hand1Jl.itl'egal'ht 106Dkn&,rQttPI~)

Tht:urem :;,(,11( :OO's fhro: ill" 1) (~'h proof)If ( i ill a C"nne<;l :d plillt. gt'llf ~ vi;b p vntices and q ~d,~es th~ I' .q + r'" 2 , W}~!l, r

••~ 'Ifoo;s orGExIUnple~ Oil . i.e!.G b~ a 2. .:(ltl' e;txl planar gmph wirh p \ertlces iIlI.d ~ ed8"" It P

,jtb<lni~1)Wtlli1lq'Jp-65,02 hur.:OW8kI'SflVr grarhs K" at ~ K,

fu:,u~ ~ 021. ',1lO"'t1af K. it. on l )llalUrgrapilExonnple 5.m21 ,~Ci lw a 2. XnJ ~;tld pLmargrapb without a tlian21e. liG has ~.

v..~ and Q edges Ib,'!\Q,,', p. 4utmple ~_023~hllWthll. K, «tr ow~ki'~8<:Wndgraph)is not a planargrapb

S 03 GeoJl1l<llTKaldui::0efinitioo & a~,., tim5.(14 (Q;euringolagnphDetllitia a~da'IJ.':Ir'Cions

.IS

(~ PerlOlb) (~ Marla),

I

'0I

•

o

Thilure.1l 5.D4:. (l~itb.oJt pH 'lJl ••. -:O:lJl~ted grapll G 11two cok"U'li"le if ;ll\d 011'yifGi.-Iabipartilegnph .

'c orolkn 5.4) ("W I pro 'f) pI C\It: itA! ifT .is1/moe ''lith \, (Tj ~ 2 then lI"ec ill 2.~hnmL1r" (i.e. >J(T)'" !}

6.0(1 Matrfx repm~nUltJ."1 of a grap 11;.01 TIi~A,jj",,~n~' 1.,rStrix, Ddil ilion I :i\lmpl<'l!l& proptniil86.02 TlI~Illcidcnoo~'l'l1ri~.Detin l:ii~~o:;;ml)ln& p1ope,tk!l

7.11»Din'clfljgrllpi'jsiDij:rRpll) (3Periods)(5MlJ'b)7.01 Digrapir; Deb~ilioo, e>ampl;& !JIrtd,:gJ.~ & ioil~ if I/~_

. 7,02 11.." :"'dJaceno:y M"atrix "f a dJ '!!'aj 'J" ~:"\llIple,s7.03 ll" fncid"nce /I. lim" oJ'di>JgI,'Iffl, ~ ~ rIll)Na7.04 BaLmced ru..,"tllp"" R~g,,'~J<l"g;~p 1

8.00 Dhitdlo;iJlty "rud~!l"el"JJ. ~U~5)(1.;Marlt,.,8.01 NliluralnlllllOOn .•

. IJ.Ol Pe1UlO'Sa;UOlll'i.

t.o~ Wdl C5"lkling principle. (;1M'" "'-,Il,,,~)).8,04 PrilcipIe {f maU'I'>'m~ria'lindr :too8.1\5 ]:O"'~uJl!Cs0fI./"'Ycip1c: of IliIIth :nat ~ II indllCIio"U16 Thi'ibilil}'-Defill1tiorJ8.07 L.~alb andbi c lJ,~fltIl C(ividrpwr H1108 Ifa Iband a: c (Len a' (!o"tC): wil q)lexof)

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8, J 0 If a: b <llld a Ibe:j or eve!" mleJgu, (wi.ttJ proof)8.11 If ••.rbomdbl~rl-.,,,;l.= :':b(\'ithJ~.,d)

8,12 -E:l:amp!ClI" i)I!d. bE?. n b,nlt :.(is'towtfu1! Ja"j.£Jblii)Ji,;s ,I <: b Iud ,: 1d,ow Illata "-0

8.13 DNlIionNsoridu~_(wilbcutp1: ill)8.14 ~1eSI'O:mrnoll mise '. ~ :iriti. ~I8.lS .Theorem: An:yt1;Dnon a:ro n tCgc~ a ar,d bha\-'<:a unique .~c.dMdit ~anbe Q(jl:U';ied

in t/Jd(J(/!J rna .•.lib, wh(l'e m, 1 i ,~(V"i1IwU1proof)8.16 l.t:ait Com!non MtdtipJe..I Iefini i",8.17 El.IcP.<bln illiori!f !11("'iIt llUtp, o(1) .8.18 fu:alnpl~(nG.C»,8.19 PrirrI{'and(:()fl~,~nul1lbm!8.~O Unir.lllc Pal:IOrlsmi'mThe, orom, 'w:tb Kit <TIlOt)

9.00 COllll"ll'JlI"eCil:SSI!!> (IiPeriods)(IO M&rfc;J9.01 P~rtilimo1ilnon 'l:lTlptpel9.u2 Equii.!IImc<, rcl.ai:i,ns. Delinitio-)9.03- Exarnplcs on an cr'40.a!ell ;e rolli, ~9.04 Equl~'a1enc.-,cla5S,~_DdiHition9.05 Exar1pks 'on eqw,,.J~ct ,'~ ;9.06 TI'C{uemL~t '-'btand lX,uival:n;e n,latioo on a set Si':I)d a, btoS 111<m

6) a ~ tal (j) fa] ~ih]i'andon1yifn_b

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(il) If ll'z,,€c,th~n' l.:;l and arg Z, ~I1'gZI-ar!!z,,~ 1,;[ Zl

10.01J(l.(I2

10.0310.041O.0S

10.0610.0710.08

(ill allyr.voe~ tb.~ un:eithei' disjoim oridellliC319.07 E.~:nc. c13S'iClJ!beaMl.I> a:~, qujv;:kinee I"( Iatioa ,m a ncn- cmptylK,tA inducu a

piIltIIUl of A andconYenl:fy ~~ry pilrtitiol of A ddilJe$an cquivalenCere1alioDoo A!"11b pr\lof l

9.01l (\If1!l'UlDOl'rlllJduhm. i)t::fillir: ( n

9.09 Ptoo1i1tf(.\) Ifa ••b(m,l(Jm)tb:nforalt-:=Z6) (/l+c)a(b+cl(locn,) and (iI) a1:~bc(modm)

(B) If a" b(modm)/lIllle ~ ( (~Klfm) llli;n(i) (a+ t):qb+d) (1lt.~ 11') (I) (.l-e)~(b.G) (modm)(ii) at;~bd {mooII,)

9.W Dlbtsio18.>f (i)adliti:m rr(~10m •• l (i)ntUllipi'alilnmoolJionlin Z.9.11 C"mplliitionlables'.l.lL ProoJo{FcntA!'.1beorem9.13 Eumplisml'c:nnat'sTheultm

10.041 COIIlpie(llltlllbin

(i) Ifl),l-:EC,tht<[lill'l) 2:1z.\"2;,110.09 CkomttJic ReplnClItati(Jn Jf lie .lUTS,Il~differlUCI,lhe P"~:IucI aod quotient of

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11.001>01MQ~'~ 'n..'OrfII1

H.01 De Moivle', Thwrem IDI"1at< IllIIil odi:a (with p wi)11.02 nlo Rt'Obi crfunity~dthti"~O.1Ktr cal re~ition.11.03 n" R<0t6 'Jfa CompIeKNllm~:1"11.M Examplc'onabovetopi;J.

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NORTH MAHARASHTRA UNIVERSITY. JALGOAN .

Weie:htage of Marks fOT F.Y.B.Sc. Mathematics.

Paper-l: CALCULUS.

l.

Topic.

1 Sequence2 Series3 Indetermmate forms4 Continuity5. Mean Value Theorems6 SuccessiveDifferentiation7 Taylor's and Maclaurin's Thim8, Integration<). Reduction Formulae10. Application of Integration

Weie:htage.

08 Marks.085/608121108101118

Paper-II: MATRICES ANDDlFFERNTJAL EQUATIONS.

1,J4.5.

6

7.

Adjoint and InverseRank of MatrixSystem of Linear EquationsEigen Values & Eigcn VectorsDifferential Equations offirst order & frrst degreenitf'. eqns. ofT'l order &higher degreeApplication diff Equations

16161206

22

1018

PAPER-III (A}GEOMETRY & ALGEBRA.

I. Co-ordinatesin space 06.., Plane 10:;. L,ne 164. Sphere 18

5. DiVisibility of Integers: 136. Congruence Classes II7. Complex Numbers 148, De-Moivre's Theorem. 12

PAP~wR-III CIDGRAPH THEORY ANn

ALGEBRA.I. Gmphs 082. Connected graphs . 083, Trees : 144. Eulerian & Hamiltonian

graph 055, Planer & Duel graphs: 086. Matrix Representation: 067 Directed b.-raphs 068, Divisibility oflntegres: 139, Congruence Classes 11IO.Complex Numbers 1411.De-Moivre's Theorem: 12

Note: Subject to change of 10% variations of marks in the question paper.