UPSC-CSE 2020

1 UPSC-CSE 2020 Mains MATHEMATICSO ptional Paper-I Solutions(1)-elSc-Wr-I) SotgrgonS- BoeoI NSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATIONSMATHEMATICS ry K. Venkanna1 (o)\---,T'S0--)Grraidu &e I-tL V 0J qll nxh aua[ ynq7tc Jgrone, . V i,r o- VfC[o]' L?Lce ovut R. dire oJtroo diftrnif Ly,2- rna,ic Eguore}.Lnt V be Su JiJ C etL hxh ,r,r^t nayc /'qt\ail)L$ "(,p V , *l^.ry1 Jurn 0j qlt urototr [^A colurnn,,\-j d 0^d p . u, i ll- br- kr ar\ d. Kz CoJlurne)AXwn )um of ,Jwat and- cohnn rf rnfrix dl|u uill be K1 *Kz + 4+p e V ( Gr,pr$on)NJo t^.r .{C I e Rlf,rv, o-4 eV o) *t^r Jqr,,r "f ou>rr:l ondoluynvr,\ ol &a. u-,ill h or-l(r :Trra exierno,[[arnpo]itiorn.]]]

2 D Adia$ [v,+) {o,r) qy ahiltan(i) "(tp e v Jtx d,p .,y e vf*'ltiii) {evotivl - 4 J)matht'x i) *daA,( + b"( 'Atu)Lth/,,tnvuk q 4 nnl,"tt'X[vt (onyn,ialivt WlrV 4+p = pt4A1ain $n a.,b eRt^J[ ) at"(tP) = a;+ bp4,P CV( Opoufiovr ir. rnolhrx)ruitqviirHIAD otflc[: 25/8, OID R ,tND$ IIAGAR tttARxtT, Dtt]ll'60 BMIICH oftrct: 10]106, ToP f toOR, MUrHlR tt rmR MU(]ltR tE NAGAR, Dtl, 011"{5629987,9,1}|,9197615 REGl0ilAt Off lCt: H. , 2ND rtooR, RooM ,{O. l0l {ANCHA ,t't Bl,Ul tAPPt{it ASHOI( NAGAR, trY0-20. 9652151152, 9652661!52. ws, \IUEI(2)Niii ) 1."( =4 (h4ulli plicottontoi*h ,-iduli*y,Ynotnixlpa re .)Hevrcu V i) o_ vlc*ole xo, pl,l4= f zlz,tlzs1s)INSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATIONSMATHEMATICS rs K.]}}}}}

3 VenkannaHEA0 OFtlct: 25/t,olD RA,[ND[R ilAcAR IrtARr(8,otut 60 BRAxcH oFf rct: r0!D6, top , Mutfitn ff lolvtn MtxHtR tt I{A6AR, otr} oftlct: H. ,{ !r, 2N0 ftool, R00M No. 202 {ANCHArvt',S BrUt SAppHtRI ASIiO( t{aGAi, 96tr 35u52, 965266t ! "rs6299tr. (3)INSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATT()NSMATHEMATICS ry K. )So.[ntl| MztR) b{ il* vrc}0,r. )foce ,j 0ll_ 2yz tucLl-r'ufihrce) ' Ld B = [, -,1] ' supP]s T,N'tr(R)+[r,(R.)d.) a rineo,t ,*huvi,rlFri*r;' ,{,y.{& uyT{A) = BAFinA *[" &tavrK o"a- vrupi*q o$ f . Find a y*qb,xA t^.,hiclr Tno?t fu *t^{ n*7 *-ot-*tYPo"L-oT(A) = [-)t3 )tz-[\xqt4llj -+U"+4UA = [U xz-lLh, )qJItl=H_l-l_+rz-xqoooolK"ItKr:) A e Ntr))(1)tjlltr)taI x,-Y r[*r*yu3)tr -Ig-tlx1*uu,--o -) q=r3\,ta - \ =o- +\ t4r3 =6- +}{ r-t4),(4 =6ut*irl[.]]]]]]]]]}}}}}}}

4 Rn)r-rc )t2)t4\+ --V,Ig= kr and)r3*lll- t(Lr1(, 1ruJ3 ]-o'[IITul NuU Apctcc = , hulL\ = /t::lli)lll[,A0 0rflft: 25/t,OtD R Et{otR ilAGAn MAixfi, Ofl}t}60 tMflCHOff lGt: 109106,IOP f IOOR, MUfitRrE TOWTRMt (HtRrEtflAGAR, ,E 99197525I)tt6l0!lAt 0rf lCt: tl. t{ , 2NO f r@R, RooM rO. {AricHArrit,S Et ut SarpHtit AsHot( NA6Ai,l{ 5t235uSl, 96St5 GtrS2. (4)INSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATIONSMATHEMATICS ry K. VenkannaJheorreYn -1 (r)r NtT) = 4'1 tr) = 4-2 - Lrnabl Io hq[ rnc,tr"x [tLrBJrnnk _ ",^lrr*7fl = [', J]Y,ur\HIAD 0tftCt: i5l8,010 RArEt{Dn iAcAR IttAR[[, Otl]l[60 lS tlCH OFFKE: t0]105, mP tuni, MU[rlER tt lowtR Mt (HtR tE iIAGAR' 0tt]n' (;lo|{Jl!]]]]}}}}

5 Orftct: r{. , 2r{o froot. R@M r{o. 202 i.[ UW 5 ArPrllRE A!HOr t{A6a& HYD-I0. !1652151152, $7, 9l' (5)INSTITUTE FOR IAS/IFOS/CSIR/GATE EXAMINATIONSMATHEMATICS rv K. VerkannaE,v,J,u^lu- y,t-nP'*&11"" -)fun2x-U= ^*L+-?L-1t/+%j=ttpr/+ltn x-(+)r/+ at t**firl+L'&i'fi/-s"t*-22/-(.8)'<^ -I_LI-PaHtADOfflCr 25/t,oD R f,ilOn ilAGAn M Htr,Dtllll60 cR flcHOf r$: lot 106, lDP rUrOS, Muutlltt lol lEi Mt (HtR EE , 9gB191625 ItGlOtlAL Olf lct: H' l{O. l'10-llr,,2r{o fl,oo& tooM l{ { mHAirS!r,w sAP?riR! ASI{O( '{16^l, XYO"20. !r5215t152, (6)INSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATIONSMATHEMATICS ry K. Venkanna1'cLSo lnFind.)]}}}}}}}}}}

6 Otl- *h" ortrtp& oJ *he Orrtue 1211fi)Y=()t-He u dl,q 6uvrn Cwu e (2x-3)ti(-1)t& -zit +Jrr\-z\- 2)tJtd,+I --o'?,ilany y = h,xtci1e- [rx [rnr,+ c)l -z)t*3 [mttc) tt =gxe- Athn - Lcil- - L\( -3rnr - 3C *1 =otl-,?h1))r? - x_[ac +2+3rn) -3c+ 1 =6P*? coficient of 12= o and uL--oL-l,\-t =t-&n'r -- ofrr= Lz2-C+2 +3rn = g2C+2+3xt :oLLC + 4+3 =o--T2c+?---6-O<z(>Itru v = +-+ u^ on*y'PJose,lEryS?^ tBM pouur- ,u o[ $ 1ero" ru 6rL*?rploler PuoJIrL ,*D Y* qrfl '2t{t3--o )t =-!_C=-a+H[AD0tflCt:25/8,OtDR,tNDtRilAGA RMARI(EI,DttllI6{IBMI{CHOrtKt:10}106,TOP [U,OR,MUxxtRttrcWtRMtxHtRJEtllAGAR,Dtl.} ,91x}9197Q5 RtGl0ML Off lC ; H.]]]]]]]

7 , 2N0 t(ooR, RooM NO. 202 ,,S EtuE SAtPHlRt ASxox NAGAi, 9651t5rr52, 9652651152. (7)INSTITUTE FOR IAS/IFOS/CSIR/GATE EXAMINATIONSMATHEMATICS ry K. VenkannaSu\Fin4 -pre quatt'on5 ,b th. ",t ?larne to -llre-eJ\\?SDid rf + ,fn Zt= 2* r^rt^i% ?ossF$ ltarof,tr$e \lne 1-3-< = 0 = 1-A+11-9 ?.tirem li"r c- [<< =O =]d tae-9 -O? . +h'oth $ive"r1 lineOean 4 QmA4 + l\rt'Jt aZ-9 )0rr-[r+ 7) + t t-l- A) +z (1r+2713)7rtL-orom" O 6t^cht* 6i""n ol\rgSoi4 f'ovt^P ttrt tuut+2_7-P2-a??tr +nt--bcoBrrurtr 6rlgsoi 4rrtt e f"at z+7L*+Ltrht=r{+ {??L2_')\+1)"+!t=t5d2- .r-L-1, 3??o2_katSo [ror\7^Lttl ) +e Gr-l)21 S[ra-'f= PlfB?]]]}}

8 Ltgg*att \8 [2]4)Ll6x)Lzl tt (\ t 2t)k3Ot11t5t18=0--)f [.] +3t1r-r t 2l talttr +s-,l- EqiL+ Su = o71)L-- |l\-- !ll=l ;l= -l iG) = 11-26-t1-- Itqao*t"{ efr "\ Plomc$O= z= 3lJAI"Df t$rou! orlcr: H. , rNo ft@r, roou to. 202 { r{cHArvts lur sApprtiE AsH(x ffacl& , 96s265t!52. (8)I NSTITUTE FOR IAS/IFOS/CSIR/GATE EXAMINATIONSMATHEMATICS ry K. ^L.^-Jz$: kv L= i ).;.t (r-f)c/ToI)cQtw-,^ xG 1rl)1"/,--l 1o) 1-<0a-\('- lil Jxr^J ) J^ ---0+i,fg)\- t-^ '/6' t#r" Jd)+ t;"J -k;oIJo\)aBH)rA-r1--\_>.It {d"-9),'u/,-14-r,*-^' :JoI.+lo"^FI)OJ"t\d^J."r&)(3-U-,t-oil * [l-rJ(l"it", r"ol- b= )fu>--@J*A^ -a, -[{ 'Ttr-14i'fA)).]]}}}}}

9 1-l*u\-") J).I}\AA&>L-,(a-oI)a)-109106,poHt D Otrlct: 2518, otD RA,t,{Dm rAGAnIVIARXN, DTU}60 BRAflCXREGl0llAl. OfftCt: lt. !?, 2NO f , RooM NO. 202 ASr{Ot( [{AGAR, } 9552351152,9652 65u5?. ^., *@(9)I NSTITUTE FOR IAS/IFOS/CSIR/GATE EXAMINATIONSMATHEMATICS ry K. Venkannah =S->Lv<-ptfur ' 'te&l--Arfl\-e hdcAXK-+ ,- tzt\V|<-il""AftT_t.{rN,rYffffrlLt^-g^^tC AA,tA#() rf-2u,f)(--E2 Ltu,Jt- hll^ ..','r) tu d)2r,tf2Lr,Tu Cfl*)"Iut:lg- - lt \'t'tf t ,-,..ftro-)K G- *dr.^<$t), \o(i uZ- L,turfoO\^t 'xAfrl:\ \r= 'Ju _!1,fislfthuuT+l-r td uAA--;-OID RAIINDTRTIAGAR MAM([I,0flx[608MI{CH Ottl(t: 101106,IoP 'U'OR, MUIfitN,ETOWTR Mtxmn tEllA6AR, 011-0s52998?]]]}}

10 ' 999919745 HTADOTTIG:ttclotlAt Ortlct: H. NO'l't0'217' 2l{Dfr.@ 202 {AI{CHA 'lt'SBIUT SAP?T{RIAS}IOl( TTAGAB,Hm-lo. 96t2351152, *.imslmaths.<omo{Fr>A**{*I(10)I NSTITUTE FOR IAS/IFOS/CS!R/GATE EXAMINATIONSMATHEMATICS ry K. VenkannaA |t 0.,^ o'16o6o, ^#"1)t .G*e (.A) = Trya-u (r-+u'{)=G lJ)->l'Uqrl - zu)= '/)A=ltobIlo loloolY1Yt\sYt1l- l--]zlooo 10ool2-/14t4t"{l '( 1Y{ I Y/qt1l \(1lo o,11,h\h8tt"hyLt{11oto"h eltol+h-Yh*'.r [1-' 11 -t(1Y 1 -v(?--3 /1 Yl<-<HIAD orfl(t:ztlqOO RAf,t{on ilAGAn MAlxr,0tlllt60 SRA,{CHOfff,t: l0}106,lopfloon, Muptti,EIOlt/Ei MtrHti,tE,IAGAR,Ottt|.9,l$loll^l OfftCt: t{. !r, 2ro ft@t, iOOM t{O.))}}}}}}}}}}}}