INSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATIONS MATHEMATICS wK. Venkanna [ETT a Salqwap of rep G. Tan weNgna’ | Aa [Ter W te o Sulawap grep CG. They w Qari is a normal Seabarocep of CG. set: Ler a= ghgt aud be gig! be hwo alenante of Khan [ a08'- fa! neu | att = ghg'(ghg')! = gh tay Ki ) = gag! git? = gh gq”! Noe Wye ond His o kabpoup of G. tne bijou. Shea from CF) above abt =9 (bk! de guy! Hence gue lea subgroup of G tov oul qea Since Ka interseetion of Sobqroupe ig a ubogroup , Wis a Subqwup G. Le 1eG,wew tan wegug! vgea. voc have to chew hat awe gut gec , which in tum will yield Hat avoir ew. Wee qeq and tar at Aapgnte Mak ant egug!. thea nor! = ghg' for dome hey hut G1 wig = hey =e (qa) eu fer Ye tig. They qeay. Hence in order bo show thar -awet equq! fr a given Geo, Sat wo weed to find YeG Auch that Te BA ZLREY scconacorcr evo ansuy no rcon-roorwo mos sence Auta soo aca DIO SETI TENSE. ewesINSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATIONS MATHEMATICS wK. Venkanna Since 4 = a (09), Wwe Cau choote y= x" So there enists YEG Auch that 474° Sime YEG, we have wey yt and we phy! far came her. Her wor! = a (yy! )i" = ayhyetct = (m4) b(ayy! =qhy! € qug! Sica gee wo arbitra ; aot! © guy! for all ges. dan, WS a norval Gakorup of G. BAY FEY cconar corn vososm, movioon, nota ma nxssanchan'’ ue swme Ssosceh nbs” MUEEHSG sestais. ww eamINSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATIONS MATHEMATICS wK. Venkanna Atopy Mt Re PS 2) Javexy aud Jat ¢ be Hee wneppiing thar mes [2 4] to ab . O) stew frat g Ba homomorphism + Gy Petermitne too Eernel of p- Gi) hew that R/rerg & Komorphic +o Z- GM tek gi RZ huek tone afta] eet we hawt — a > arty b+ ‘) a(t +02 2))= [oo ae = (ata) — (+4) = we “Hla bi) g(ocjre(4 a) gp homomonent ed fe) J /elee ok ooce # ERR) = arb =)» ab= wa a=b marge f(e 2) faex for each aezg 4g ACK gum tw OCA) = * @ & onto. By Furdamental tadere ef Hormomorphitm. BA ZLREY scconacorcr evo ansuy no rcon-roorwo mos sence Auta soo aca DIO SETI TENSE. ewesINSTITUTE FOR 1AS/IFoS/CSIR/GATE EXAMINATIONS MATHEMATICS 5K. Venkanna tae Hayes eee here e128 nie shat -f ig Rremann regs mal oud ftoy=o. Prove ; Weegee et Net --- SAm 2 Peyy =} HD nar ni = ° Ata > a eee = %> s1eh =O ,w= oO => fay is bounded al continuout on [0:1] encept a Ms points ity er %-- the Sek of re of — digembinuity of fon [or']] is 1%) hy BG ob which hos Om Leite point Qe Cre et poiots OF — digconhnuily — OF fon [on] bos o Ainite no.of Gimit poiots 2, Hig integrable on Co. oO ‘waar 57 OW IAENOE NAGAR MAT DEG 6a BIANGNOFFE 10-1 oP DOR MURAER TOWER MOOT AGAR, TLR. OLAS, ORT ecionat Onc: WN. 10597, NO FLOOR ROOM NO. 202 RAS ARCANE PPR NAGAR HPO SSRIS, SEIETIS. went conLA) INSTITUTE FOR 1AS/IFoS/CSIR/GATE EXAMINATIONS MATHEMATICS wK. Venkanna Aids} O se i tuedem ad for Comedy wdealic Rormuta BG evalurte fre lla oS \ a de ay ae da. yatey jaa-te % Gare Given that as da. (te a. jaine Yee fevre at ee anckghts nt =. Comparing Ata gen inbegeat ost is “4 we dere fen do {B33 analytic te WHE. Sewt Can ope! tee subagrnt Bo cmuta. “ 3 feuds ant fcr) lee © a8 Ine Gin o te ar" ya 4 =sI4 we have Yar-r-tl= F ka civck with ate ab es ti amd Radia, Ty fe ak te OE 1, wo 2-1-0 ee sl 7 mtu) =P R= A wy bey ined C7 ” ber feta ge waist & CLanty matte ar every qo ait tt, and er C. - | tea = {[A& de hur Bat =) a awe $C) = arta es See. ‘gi ERD OFCE 15/8, OW RAENDER HAGA WARE DED 6 BRANCH OFFICE 10506, TOP FLOOR MUGIENEE TOWER MONDE NAGHR, DENS. aH 2581765 IONAL OFFICE M 40140237, 20D FLOOR, OOM NO 202RIESAANCHAN'S LE SAPPHIRE ASHOK NAGAR HYDID SHSPISIIS, 6SIGGIS2. wma40D INSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATIONS MATHEMATICS wK. Venkanna Sol Ave the dual of tte LP problem: minx] 214 Sarhg Susjak to te. tonsreiatt + Qa 4o4%L4 Sag 722, Sou AWLATAGET MAURO SC, WMT On tag TS un veshtcted + late Ha Venfabte Ag Ts unvehtoed Tr Sign rHe Fuco Lp pavlem COM be tmosfocmd Rn be Stendord + gt pied fore wy Subs MS gs MAME rele s aro, WZ Garfoe Stadard 0 Detomay » sas fram AL Hany BAW Cg HE > Latfured 40 Lao Seroi ahd t ermeBar-§ (rere IS -d Ba amet Cagh-a') <3 y mtu enr -#Oag meds 3 mravara g Cag) ac Lea MS saa! Zo. i dual ofc stve Yerdaed peal th, fin ety = HRW AT Co! tot Dae Soy Sul jel totta noted a) pape we say 2b) 42 lug! tay! Ja Wy Z-E , tee: ‘i opal ee ety 4 og otal 4 Weg z-3 ME Re eats Say = DAABWAAW, > Sua, AF Cro! tol!) 4bwg2-4 “ ae Sw AWA WOT -S Sey — Flay)! Jb wy ‘ “3 Lon Ses) Peja? Uy lap, , Wz 70 wey Wy zd ond —_ to, os Unvesricred , BAY FEY cconar corn vososm, movioon, nota ma nxssanchan'’ ue swme Ssosceh nbs” MUEEHSG sestais. ww eamINSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATIONS MATHEMATICS », K. Venkanna Let M be othe def Of aU 3X3 rmatricer Of i : aoo othe folowing form: ~ S boca where a,b, C & % - Show that with étandlaxd matrix addition and muttiplication (over h)Mwa commuta- time xing - Find aul uke idempotent elements | M. 0 0 | 546 CEB a ° a @ a, 0 het rele an 5 4G, i Aysb 4 Be Solution grey Matrix Ma |O Oo b oO 9, —[% oo fa 00 fe ° Arte a a, 0 0, 0 ai % br % i bak, at F% 4 © 0 4,00 Gita. © O76; =}0 % 0 04,0 © att 2 b, Co Ma} |b, er % bth, FS Ate 2Gt Since Ot4, =9279, bees Ot Hf As 049451 Baty hci and addition of tw? is 4 WSS 2 Hence, M us commutaHve un Giandard metix addition. For _Madaix Muttiplicection @,0 07/400} fA, 9 O Ayan =]o a offo 40 }=|0 aa © b, a Nf fb, & 4% bi449)b, oan aa, FA Kay 120 re ccs even ncaa ancy eta tec Ome 15-5, TOP LOOK MOREE: TOWER MOGIEUEE WAGAR OBIS, CUL0 Ce) Cy) Let cay) > 0:0) og ai ‘a 4'to) a ee am = tae . spe da on™. « Pleo) dort not enist- 4 10) . does vol — dfFerentionle ar (0.0 . £2) he given fusoieo ‘rie not othe Given Stabmenr Is tas e 4 sanefian CR egpaations ott hovgh digferantiable of (0,0) — = ADOT 90 ROR GARE IRENE OD MITRE EMER RMT GAR ORI, GPE RITES stoma rr: NO 30337, 290 GDR, NOON. 22 RAS-EANOUN MAAR, wr SSPISLISR ELEGISD. emINSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATIONS MATHEMATICS wK. Venkanna he Integra) -feurchion F(R) Sobishier everqarhere the Inequality LR)[< All” where Aad Kare pative Constonts. Prove Haat 22) 18 a polyvomtol of degren rel Crcaeding K. GAY Sieg fa) Ie oxalic in tha Flaite port af Hr plane, -Hadafhe by Sgt tarde Peoye SF g.ar , whre l2leR np” trole Now, if man Hea) = MO) °" we Thea (rep) , thy by Couche inequay we have \on| < MeL bral 9 © AUSIK) — Since mta)= Head] alee » esha 3} £ ae = Ayh Here Of yo , the right hand fund 1D BED, Sate nrk. ta. Qy=d Aor nk, Le. all tee. Cofffedents On for which nok beema Rev. OPQ) = ag tayd 4a,2% 4 --- 40,28 tohick te a pobpromial Hf degree BBS nO ETO NAR AEB OUNCES OD MURR TONER MRREIEMAGHS DIR OREN SITS IONAL OFFICE M WO140237, 210 FLOOR, OOM NO 202RISAAACHAN'S LE SAPPHIRE ASHOK NAGAR WYDID SPL, 666152.INSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATIONS MATHEMATICS wK. Venkanna 3)| dy fin a wing R , with Unity (xy ear y” forall a. 4eQ TF tay chow Hat Q If Commutative. QL) chou Prat Ake Tg R Of Meo) Vatiud Continuous functions 8 foi] has Perv divisors. sot ch we have (myrery™ VIER —O Reslacing y by WHER WO, we ge faCyro Tee 4” =e (aya y= er Cray) => (aya) Cage) =O CP 2gF Ey (uy Cry ya Fa (ey r™= Wy bay + a” —@) =r Gy Gy) a 4 ctayea” = Gy fH avyia™ => Mryjag a(ry) any Cr Lec eece inR,+)) Sy zyx gaye an’y Spayacayv ayee (ecr)—@ Replace aA bY AHER IN®, Gey y Cat Hy => (asi (yxty) =@4+) (ry+y) =p Uy + Ay yay = ay Fayrry ty =¥ Yreay VTMER Crlee we Rel mMQ@,+)) Rig & Commubative wg - (i) Comtider Hie tere $e) = Lin fa ) , Itna™ Late tou |= be 7 lor ys _% 4 Vw thoy dy | (4m) (120). dx : Em” - ax (amr) ry For mart. torr OY \ F 9 =p lewd => wee >. 2 Cte”) 20% ho 4. Gre) Gow - Unt ay Qan')* Lani (ta wet) nt (lat) ~ yawa™)$ ot| RUD ey, a” ~ Cry - z nate iy =” | Te mattinuus worhou xe and marin] Volus # 4 a ~ ar = ah BAY FEY cconar corn vososm, movioon, nota ma nxssanchan'’ ue swme Ssosceh nbs” MUEEHSG sestais. ww eamINSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATIONS MATHEMATICS wK. Venkanna ‘ cman, atw-Fiar} = x Ma = rae ag Hamo—Fool = eat aba s tL aL N-r 0d aw? I Houee peseH: pete) Wher Age Mag & she lst + GaHuUhe Be genes A WUuRe = HUSH sp hee cl4 Bas aH 4 26G, HAG. The tonverte of tHe atove weed rox be trues . 1 — = andes TAD OFC 2, 1D BAER NAGAR WARE DLN ANH OFIE 05106, TOP FLOOR, MUNHEWE TOWER MURMERIEWAGAR DEMS. OUL- Ss os h To : Using Newton's poseeond inden otahion fprmata, » we ge gas = Yuot PAYae + PPD ary, + pler) (P39) A®yy, 3 BAY FEY cconar corn vososm, movioon, nota ma nxssanchan'’ ue swme Ssosceh nbs” MUEEHSG sestais. ww eamINSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATIONS MATHEMATICS wK. Venkanna = 4, + oS xya + 0-S(-0-5) x9 + 0-S(-0-S)A9 2 6 x-as)+ os(ro-OMS)C2S) . 35 24 = 3lt+ 21—- |-12S— 1-Séas — 144s = 47-87 , 61 simblibicatin The numben of shidents with monks Jers tan 4s d+ 47-87 je 48. But she numben + students wilh monks dos tan Yo js 3). Hence dhe number oh shidends acting manles bahseen Yo and 4s = 4e@-3| BA ZLREY scconacorcr evo ansuy no rcon-roorwo mos sence Auta soo aca DIO SETI TENSE. ewesINSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATIONS MATHEMATICS »,K. Venkanna Using Gauss Seidel iterative method and tthe starting sourtion %)=%2=%,=0 deleamine the Solution 4 dhe following eyatem of equations in use iteration, }0%,-%, —X%3= 8 My -% + OXWeIoO, 5 UHL Omaha = 12 Solution day y tn equation can be Aewritten a— A, = BAA 2 OB +OID+ONG 10 wD (2 -%Mi- Xe - DONT — OND —|- Ke = (O-Y%+X2 = 14 - OFM FON to Also given that = %y=%A,=%y =O ond use meed to find Abe olution of he Ayatom 4 equation in tio ateration, Cteration -41 Taking ; UW =%, =%MZzO HM) = OB FOIKO4O1XO = og coe 2— O1K0.%§ —OlXO = Leda, Hy = t- O1xX0-8 +041xtI2 = 1.032, After pirat Steration ; = 0-8 Me = ASD Ma, = 1-032. EAD OFC: OLD RARNDER HAGA MARKT DEIR BANG OFCE 105 6 TOP FLOOR MURHENEE TOWER MURHUEE RAGAN OBA, 2087, 01075 REGINA FC: NO-10337, 24D FLOOR, ROOM WO 202 BLUE SAPPHINEASMOK HAG, HYDZE, 95235152, 965266152. wwowmedmathscomINSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATIONS MATHEMATICS wy K. Venkanna eration -2 aaa = Gs tN2 5 Wye 4032 %G = = 10152 Me = 42 — OAK 10152 — O1% 1032 OR +> ONKINID + OVKVOBD = £2 — 0-10152 —o:lo32 = 019953 1- O-1KIOIS2 + O0-1% 0:9953 L - 010152 + 0-09953 = 0-998o4, % = 0:9953 %z = 0-99801 EAD OFC: 15/8, OW PAEIOEN AR (MARNE DEIN BRANCH OFC 105106, TOP ROOR MURVENEE TOWER WAAR Dea, OLE AECIBT,oRBLTORS RelONAL OFF: H NO 30-227, 2N0100R, ROOM NO. 02 RAS 4ANCHAM'S BLUE SAPPHIRE ASHOK AGAR HYD-20.SES2S1I52, ES26SIS2, wormlmAmathomINSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATIONS MATHEMATICS wK. Venkanna 52) | Prove that te mecemary and Culticient Condition tat vorten hints be ak wight to the Atreamblueg oe wv =H (28, B42) , where H oud fore Functions of 7,4 2,6. ls The differantial eqpabions of streamlines ound vortor linet are spectively. du. dy, de ea —é ad dt - dy Ge — 4 & © Daud © win “terse arthogovally iE UE +0, +¥G=0 oO _ yy (du _ dup wy i) 5 = 1% a )4u(5 #2) w0(St- 5) But Hale ie “HL Condition tal tudx + vdy 4 ode fe perfed differential Spades Udy tude =pdy cy 3 = (% y da tahagt Hae) hig => Ujywe on(st it) —=——_ BA ZLREY scconacorcr evo ansuy no rcon-roorwo mos sence Auta soo aca DIO SETI TENSE. ewesINSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATIONS MATHEMATICS »K. Venkanna Elo Fea a partiak Afferotial equation by tren Cbc fom ye Wy Bor ot We SP! Give tar ate TU ar — @) e 2 wegen Dffvertintey © wrt mond Yr Q an | 2 Wee acater FO —8 av te > od 4 HOA) Ch+be Bod -é& pte %y 1 stfferentinbing @ ort nao Qerh ye howe - . ea OR 4 oe =O -—Q ’ Glin e 0 —©) fon @, c= 6) ” pectin tris vla oh @ at Hey by owe oats . ht @8) 4 P59 —© Lutte. hes OG, ay +1By-2% =O pifferntiating gy pentbay 9 ho yt] BA DLBEY scconacorcr evo ansny norton nooo mons sence At aon soko hacam DIO SETI, TENSE. owesINSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATIONS MATHEMATICS wK. Venkanna GUD Reduce Ate eauotion Yr Cary) La ats 0 bo CanoSeal fourm dnd hente ed Vs Jerexal Sdudton- SINE] Given Ya tnayySane=o 90 tompaseng (1) with byt Set Te 4 Cmiy,2 Py) =O hee PeySenty aud Tom So-tat Suet « (nay eunys @-yd70 for wEY. ans 40D ts lyperbolte. The A-Awadeahe canahion CA SAET=EO vedutey to Yara Cary)At AE O (or) CYyAtan( att) =o So-ttak Ae ay «Ran ty Lomyponding Carel eauations are Stven by Ste ad Ay (44) =0 Smtesrabing Hye gree ak WZ =O Qn order to reduute Oe (1) to F Canont tol form . use Uhoote - us Yr ad ve gy of —-® “pe Ot -B% Ou 4 Bt Sv Fi. Bu Be “av On = (FE er B)O-® yo 2k 28% Qu 4 Ot W_ 23% ay2*), uy) ‘ay Oe By av 94 Bu “0 eign ew Nos027, OMOOR OOW NO OER HaNCHANS NUE OPEL SDK MAG HYOGO MEETS NEED. mcaINSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATIONS MATHEMATICS wy K. Venkanna ee ve 229 Now, 423% = 9 f arr $(E3 2) Oy im; Qu Oe aed A Bats Bufo 2(B)- PLS) S(F) “18 (7-3 a (eyerZ (eye) - o(h ( a Ee my 2ueB (2 x | ae Urapiies ve a ton a at Tt a Dut a an tage ) 46 2( wer 212) 2 (a mga yf R (Bist 2(2 ne at 9 ov 5) ON Mt 4 ay2t y ovr Se 7 Bua ee aINSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATIONS MATHEMATICS »K. Venkanna ’ 3; % fo te dh a 3h, 13e bas Sou 4 ve OV Bio $23 « 2(®)-2 wey 2%) ops) ay an BF (93) 220g G) ar\a (2 av ary af a ~ +2 ( Bu “ont sox (3 aN ot) av + 3( 8 nl 3 Jt. uy 3h _ ° 7 Jom ar FO a “Bt (wg 2h 99 Se Quy ° b® ufing 5) (6) ey tn) we ser - 4 By ame aoe & Ot” WN ae Cw] FE ae - -ow ~ ae) or 43 sy % tt wre wins wn + : By Ly a Seo (9 drag- trash BEd Sa 2 a ar) Bro PB ea BE RRR a, CU CHT Fo MO TUR MICAS RD. STE OTS IONAL OREM. HO.40237,2NDFIOOR, HOMO 202RA'SHANCHAN'S LUE SAPP SHOR NAGAR DZD S235, 6S2GE1S2. wowsINSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATIONS MATHEMATICS. K. Venkanna to Se gud xo oon se 42% 20-9(t) guav ON NOV ON Cr Wo aud Yds U byw) WOrts Ha reaBeed CommBeal foc of 10 olution of- (E) motitelytons, ol tthy ot Bry v tse get gu WB) Porro wm Good cola 022, od 0's Bay + Ww redute G) Soto Brear earokton uth Conttenk (oe fetdeukt ; we tee ne Yodarty x ad 0} gollsos lee We e® ad Weed Sodtah te logu pYo logy b® Ju ne 9A, aA = Yay Ban @) vedn 0y4o trol xo} ase (9 pl (oar) r=0 S43 general Soluson 4 da LY BLL oul gw, Copy’ + g, ese’) (or Aedlip, woryylud =r ¥, CDAD, tol oud. WA ont ax rary unetlent . eS TO FS orc ec anes cna our BENENOFFEE 16105 1> FOR, MONERTE TOWER MNTERTE NAGAR OLS TINT STATES LAA recionat once n013023, 2N0FL0OR ROOK NO 202 KANCHAW'SBLLE SAPPHIRE ASHOKMAGAR, WDD 95255152 ES1US}S2. wvwimsemats comINSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATIONS MATHEMATICS jy K. Venkanna ANT ate points of trisection Of O Shiv are puted arise thro dettance h on Oppotite Sides of the foktton of aqyilieiam, Quad he 4 Ye volosed fom teat. Derive On axprenion for wha Sty UE Ary Aub hipaa Bme aud hows hor “ha middle poinr of Ha Shing Mpeasnei of a,c otD (4), cot, -h) ) wt, rept Quitesd deflection oS a y ° Equation of oP is & o=b (a-ovie us he Equation OF Ac and CD repectively ae get u Brus Gate) 4 0(010) @ud U-(ch)=_O- Cb) (xn) 3h- Qh wu = WOa-3d) => —T “the Sunpined deflection ig Giver by Hed = 2 Entel St aa o— fei whe Ene SP Fey ROT dy, Sf _ ADO, OD AOR A WANT, Dre HOUT TP TL0O8 EN TONER MRNEHT NAR DNA, WED TOS CIT Fed scot crct'n wonosay, on noosa et sane LUE SaPnO ASHORAGA, DU SSRILIS2, EIESISE, weroINSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATIONS MATHEMATICS »K. Venkanna Tha decplacemerk 4 (Ak) Sf aig Pole of Hee shi is Giveu oq Councdory Conditions Y(Ot)= Y (38,t) 0 .s° Wt, whey osasd hGk2) whan 1<2228 t Tnitial Gondi ties Y (4,0) = AA-3) eta 28ene at q ond . Na? —o we hove | Suntr 3K —® Diffeeratig @) fatialyy towk t, we a 1) 5.03 2b: = j- En ME tin Rh ARE rn COOGEE f Boat | “ee fe ‘putting te tn Oxr® sting, Bat © wd @ vv Fr Oz aa) = = 3 Fy te fin B29 cey®) ico nei 2 Thee 4. Nic You.t) = = { Ey Ch AESE + En bin “ay t Fn ae j ro) Hoot dx=0 Oz yoore4ey = 2 eat if where Ene 2 fer Hin TO de —® ° 3 em = [ For) %o2™, AT ty ° DY Fa toc Do wasn a, De vencn 1 Wt, f MNTE OWER ARTEAGA DRA. ON TS BAD Fed sions orcs: n.was0.207, 20 008, noo WO. 202 RS * = [ee yar 6p ste 4) = (844 +86] [0,6 ave swall) G7 bea the totat Kinetic energy and w He coor Beneton Of He Ayptem, then Toke of WAG + ke bf Wd BC . o =Pemgtty om mere, [+f SO eva, ] a gel Eee th rer Jrgm [Ae drre Ge ha289)) 2 Ie =r (MERE) Sy vesn ovr, Ou mNENOER AGAR WARE De BEANE ORR: 10 10, TOP ROOR MAREE TOWER MUDKEREE NAGAR, DEDNA.OL-SSIDO, S557 Bed secon orc ho..-227, 200 L00R, ROOM WO. 22 RAS HANCHANTS BU SAPPHIRE ASHOKNAGAR MYDUD. RDI, SILI. wor itadmath comINSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATIONS MATHEMATICS jy K. Venkanna ond we MgYa,+ MGYa, +e = mg [yteno + hate +4 lead Jeo \ J = ymgh (scot + cot (2 8 se ogeegt oven ied [s at SF apash -te8 (tere +94) —O Eqpatiout Oand@® Cou be aa 4 s+or CUT =O hare ton eqpoitiont roe get w (8d+ dy] -0 -EMPEES MMOD = -Hrgte (si (go tacje +30 =o Eliminating > between (errs taprracy—ap" J =0 ey (40% ued + 2aC™)O =O ton DE we pertods of nova} ebciltanous 2 in am ta sdustion of © , must be o = Aos(nt +e) Dee we aud DNS onto. Supedhasing w@, co (Cant aden? ate Or? wpa “uae” Hote =0 [Greed wre wee Teor Qt ye (+4) 4 (c= 4n) 84D: 6 34+ 2 ~ fF "RAO OFT 17, CAD RACE AGAR HAART, DAN-4D BLANC OFC 106 TOP GOR MERE TOWER MOVEIE NAGAR, DOMIA. 180207, SIVTES ew OFF: A NOS-202%7, 28D DOR ROOM WO. 272 RES-XANCHANS BLU SAPPHIRE ASHOK NAGAR DID SEZISLIS, SULT. wom.INSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATIONS MATHEMATICS 5K. Venkanna A Sphere Bf vadius a aud mow M roll down arm floxe tnetinad ar Ou a ty “he horitorlal. oF x be the distowe Of the point of Contact of the sphere from a frud point on he plone ,-Pind “Whe accetombion by tkug Hamilton’ eQualon- gains Lek a Sphere Of mactiut o aud ma M Wil down a i 1 plane inclined at On ong ra stovting initially -for aRiued Piet 0 Of He Pant. Sh time t, Gr cthe Sphere wt! down odie oud Vere Lat it tam inca hore ie HO i * LX wh A “= OW= arc AB =a8- So that Lead BT oud Vo ove he Wiuatic ound Poratiat enargict of | in Sphere, “Hey T= Mee” +g Mae = ME SMa arte tui &i ea Af rove 7 Br (Gn te A) aud V= —Ngol ce > f o LeT-¥ = {Me + NQe Bye Hore % is tho Only qeneratiteal Couctivale . wb eat -&MA @ TO Bure L deer nok Contain Lanphicity , = Tavs mee Maa Buc 5 ., => He YM Chk) mgt = 5 - trom OINSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATIONS MATHEMATICS »yK. Venkanna Hence Be hoo Hamiltoult equawout are, bs ~ 24 2 Mg tna OH) pom 4b aah —t) Pifermatiakivg Un) and weg (Hi) wwe et Eating chick giver Hee Required acceleration.IMS INSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATIONS MATHEMATICS »yK. Venkanna Qusa:8coy Show that =f ca) id 2 Postible foun for te velocity potential for an uncompacssible fluid motion. Sf othe fleece velocity F-7 0 warn, find che derpaces 4 covitant sped, Aolution - given, b =xfa) @ ga-76 = -T [xfer] ¥= “TIO Vx4x Vf] ——@ Now, = rary? 2? 2 PR( BM) aoe => OR x ox z A Ainsley « aye enol we wos yu = ['Gi)*I&)+«()]* vt «Fel Fe vf) = of) (2%, +i) (*%ay)* @ f'y(24,) vfey =i fe tO (th, )r fas) vf) = 1 pay Chetiye Rey = Lone “@s5 qa Furi (Ato? —© For & fossible avetion Of Qn uncomprcssible fluid, we have — ‘EAD OFFICE: 17 OLD RAIENDER NAGAR MARAE, EDW-G BRIE OFC 15-05 TOP FLOOR, MUKHERIE TOWER MUDIERIEENAGAR DEIN) OL SGTOM7, SUATORS REGIONAL OFPCE:.NO-3027, 200 POON, ROOM WO. 22 RX'SHAMCHANTS BLUE SAPPHIRE ASHOK NAGAR VDDD, 9a2IS52 SESUGLISE. wars matheINSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATIONS MATHEMATICS ,;,K. Venkanna wy ro or W(-9b)=0 > vo=0 2 [& +i da [xfei] =o ,(vsingO) 6) 2 Now, alr tal]= 213 txfen4] = 2. [ fo +% BEt] on ou 2 Pu fej] = OF 4 2b yx PHA) = 298. y Su pplttel] = BF + BE +n Pte BE x Mew Alxos 2 spl) x ye BD [pn] = x BF az og2 -. © becomes 22 4n(M MH, FE) 9 —@ Dar Oy? 4 ; — vsi & = F'O%) vsing® EAD GFRCE 257, OLD RARNOER NAGA MARKET, DEL, MANGH OFC 105 106, OP GOR, MUHEREE TOWER MUDHEREE NAGAR DAL. OLTASSIONT,SOUTATOS REGIONAL OFC: H.WO-6297, 240 FLOOR, ROOM NO, 20 A'SKANCHAN’S BUI SAPPHIRE ASHOK NAGAR FWD 20, 95235152 SES2GLIS2. nrc iarithoconINSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATIONS MATHEMATICS», K. Venkanna ond DM. £', ater xg —_@ az A a3 Adding. @ © ond @ ,wx get: OE OMe ig = febagieat] pt page Dae dy?” az® Of MH VE 8 Spike ont oye oer + af aX» oe 3 a St OF OF Be owl a 2 ol ps pat Toye" oer tf af egtten vsing and @ , @ acduces 20 (Je > Fla df! Lo aA >it ee fk Anteqrating, log f1 + 4 Log = bog ey fo that olf = art] Tteyrating ©), fe (Gp) P46 ® FA Kt OF 278, OD RNR nan tana Oa a, ANC OFC 0-5, To? FIDO, MUNTGUEE TONER MODVEREE WAAR BELA G1-SFIDTSRDITEIS iedmathcm RESIONAL OFFICE: NOB, 20D FLOOR, ROOM NO. 202 RXCSKANCIAMTS BLE SAPPHIRE ASHOK AGAR, HY0-20.9EERSLI52,SESDEGIS2. welINSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATIONS MATHEMATICS »,K. Venkanna Cz, being an aabitrany, conskant Suu bibubing, tbe values Of fland f from @ &@ sn ©, we get ye-|t (¢H/g,3)- 297" = (%5)% —© Given, that G20 % A>%, perce (8) shows that C, = from (iS) 5 a = et L(7 22) Now; = 4G © S(7- BE), ar — 3x8 RE (?- Pu ee Bie pel ee aa it * e ane (4 -& a dao = =, (ame 3x2) Hence, the dequiced duscfaces & condtant speed asce Ye Constant con oe J gad) = Constent 9a q7= (224327) 478 © constant required Souction FS ore 275, ow aseNoER AGAR Maa, LNs MANCH OFC: 1516, T0P FLOOR MOREE TOWER MUNEREE AGAR DELS OLTGESS6, OISTEDS Bod recion cence: 90.10237, 260 FLOOR, ROOM WO, 22 KS FANCHAMTS BLUE SAPPHIRE ASHOK MAGAR, HYDE 9652353352, SS2G6L152, wemwimstraths cm