ims4maths.com Mains Mathematics... · I NSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATTONS MATHEMATICS ry...

UPSC-CSE 2020Mains

MATHEMATICSOptional Paper-I

Solutions

(1)

-elSc-W r-I) SotgrgonS - Boeo

I NSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATIONSMATHEMATICS ry K. Venkanna

1 (o)\---,

T'S0--)

Grraidu &e I-tL V 0J qll nxh aua[ ynq7tc Jgrone, .Shoce

di.qf V i,r o- VfC[o]' L?Lce ovut R. dire Uarrrp.DS oJ

troo 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 (rr.terrol Gr,pr$on)

NJo t^.r .{C I e R

lf,rv, o-4 eV o) *t^r Jqr,,r "f ou>rr:l ond

oluynvr,\ ol &a. u-,ill h or-l(r :Trra exierno,[

[arnpo]itiorn .d Adia$d.

AXwn [v,+) {o,r) qy ahiltan

(i) "(tp e v Jtx d,p .,y e v

f*'l

tiii) {evo

tivl - 4 J)

matht'x i) *da

A,( + b"( '

Atu)

Lth/

,,tnvuk q 4 nnl,"tt'X

[vt (onyn,ialivt WlrV 4+p = pt4

A1ain $n a.,b eR

t^J[ hAVe.-

Lu) at"(tP) = a;+ bp

4,P CV

( Opoufiovr ir. rnolhrx)

ruitqviir

HIAD 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,N.9. 011"{5629987,9,1}|,9197615

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Niii ) 1."( =4 (h4ulli plicotton toi*h ,-iduli*y,Ynotnix

lpa re .

)Hevrcu V i) o_ vlc*ol

e xo, pl,l

4= f zlz ,

t lzs1s)

INSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATIONSMATHEMATICS rs K. Venkanna

HEA0 OFtlct: 25/t,olD RA,[ND[R ilAcAR IrtARr(8,otut 60 BRAxcH oFf rct: r0!D6, top tl.oon, Mutfitn ff lolvtn MtxHtR tt I{A6AR, otr}n.g.lEGl0tlAt oftlct: H. ,{0.1.10.2!r, 2N0 ftool, R00M No. 202 R.Xs{ANCHArvt',S BrUt SAppHtRI ASIiO( t{aGAi, xyo.2o. 96tr 35u52, 965266t !52.

011"rs6299tr. 99tBt97Q5wsry.ims4mrthr.com

(3)

INSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATT()NSMATHEMATICS ry K. Venkanna

1.b---------)

So.[n

tl| 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-o

T(A) = [

-)t3 )tz-[\

xqt4llj -+U"+4U

A = [U xz-lLh, )qJ

It l=H _l-l_

+

rz-xq oooo l

K" ItKr

:) A e Ntr)

)(1

)tj

l

ltr)ta

I x,-Y r

[*r*yu3

)tr -Ig

-tlx1*uu,

--o -) q=r3\,ta - \ =o

- +\ t4r3 =6

- +}{ r-t4),(4 =6

ut*ir

l[.rn )r-r

c )t2 )t4

\+ --V,Ig= kr and

)r 3

*lll- t(

Lr1(, 1

ruJ 3 ]-o'[I

I

Tul NuU Apctcc =

Av.d, hulL\ = /

t::lli ) l

ll[,A0 0rflft: 25/t,OtD R Et{otR ilAGAn MAixfi, Ofl}t}60 tMflCH Off lGt: 109106,IOP f IOOR, MUfitRrE TOWTR Mt (HtRrEtflAGAR,ottll.g. 01t.t562998r,E 99197525

I)

tt6l0!lAt 0rf lCt: tl. t{0.1.r0.23r, 2NO f r@R, RooM rO. 202 R.r5{AricHArrit,S Et ut SarpHtit AsHot( NA6Ai, l{Y0.20. 5t235uSl, 96St5GtrS2. ww.ims4mathl.com

(4)

INSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATIONSMATHEMATICS ry K. Venkanna

JheorreYn -

1 (r)r NtT) = 4

'1 tr) = 4-2 - L

rnabl Io hq[ rnc,tr"x [tLr

B Jrnnk _ ",^lrr*7

fl = [', 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'9.

t(;lo|{Jl! orftct: r{. No.r.lo.23?, 2r{o froot. R@M r{o. 202 i.[s.nilcHAr5 UW 5ArPrllRE A!HOr t{A6a& HYD-I0. !1652151152, 9652651152.

0r1.ts629$7, 9l'9919761twrw.imllmlthl.com

(5)

INSTITUTE FOR IAS/IFOS/CSIR/GATE EXAMINATIONS

MATHEMATICS rv K. Verkanna

E,v,J,u^lu- y,t-nP'*& 11"" -)

fun2x-U=

a U-0

lan 2x- .lrff^*

L+-

?L-1t/+ % j= ttpr/+

ltn zrt hdtun

Ltb

l.vt x- (+)r/+ at t*

*

firl+L

'&i'fi/-s"t*

-2 2/-

(.8)' <^ -I_L

I-P

a

HtADOfflCr 25/t, oD R f,ilOn ilAGAn M Htr,Dtllll60 cR flcH Of r$: lot 106, lDP rUrOS, Muutlltt lol lEi Mt (HtR EE llAGAR. Dtut.g. 011.0s6299t7, 9gB191625

ItGlOtlAL Olf lct: H' l{O. l'10-llr, ,2r{o fl,oo& tooM l{o.102l.rt{ mHAirS !r,w sAP?riR! ASI{O( '{16^l,

XYO"20. !r5215t152, 9651661132. ww.imstlmatts.com

(6)


1'cL

So ln

Find. otl- *h" ortrtp&I.l oJ *he Orrtue 1211fi)Y=()t-

He u dl,q 6uvrn

Cwu e (2x-3)t i(-1)

t& -zit +J

rr\-z\- 2)tJtd,+I --o'

?,ilany y = h,xtc

i1e- [rx [rnr,+ c)l -z)t*3 [mttc) tt =g

xe- Athn - Lcil- - L\( -3rnr - 3C *1 =o

tl-,?h1))r? - x_[ac +2+3rn) -3c+ 1 =6

P*? coficient of 12= o and uL--o

L-l,\-t =

t-&n'r -- ofrr= Lz

2-C+2 +3rn = g

2C+2+3xt :oL

LC + 4+3 =o--T2c+?

---6-O<

z(>

Itru v = +-+ u^ on*y'PJose,l

EryS?^ tBM pouur- e.1,u o[ $ 1ero" ru 6rL*?rploler PuoJIrL ,*D Y* qrfl '

2t{t3--o )t =-!_

C=-a+

H[AD0tflCt:25/8,OtDR,tNDtRilAGARMARI(EI,DttllI6{IBMI{CHOrtKt:10}106,TOP[U,OR,MUxxtRttrcWtRMtxHtRJEtllAGAR,Dtl.}11.9.011-05629987,91x}9197Q5RtGl0ML Off lC€; H. NO.t.l0.23r, 2N0 t(ooR, RooM NO. 202 R.x3-xANCHAJi,,S EtuE SAtPHlRt ASxox NAGAi, xYD.20. 9651t5rr52, 9652651152. ww.ims4maths.com

(7)

INSTITUTE FOR IAS/IFOS/CSIR/GATE EXAMINATIONSMATHEMATICS ry K. Venkanna

Su\

Fin4 -pre quatt'on5 ,b th. ta.r6c",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- r.t-[<< =O =]d tae-9 -O?l.rrl

. +h'oth $ive"r1 line O

ean 4 QmA

4 + l\rt'Jt aZ-9 ) 0rr-[

r+ 7) + t t-l- A) +z (1r+271 3)7rt L- o

rom" O 6t^cht* 6i""n ol\rgSoi4 f'ovt

^P ttrt tu ut+ 2_ 7-P

2-

a??tr tn6.fn + n t-

-bco

Brrurtr 6rlgsoi 4 rrtt e f"at z+

7 L*+Ltrht=r{+ {?

? L 2_

')\ +1)"+!t=t5d2- .r-L-1, 3

??

o 2_ kat

So [ror\ 7^

Lttl ) + e Gr-l)21 S[ra-'f= PlfB ? Lt

gg*att \8 [2]4)L l6x)L

zl tt (\ t 2t) k3Ot11t5t18=0

--) f [.] +3t1r-r t 2l talttr +s-

,l - EqiL+ Su = o

71

)L-- |

l\

-- !ll=l ;

l= -l i

G) = 11-26-t1-- I tqao*t"{ efr "\ry.3ent Plomc$O= z= 3

lJAI"Df t$rou! orlcr: H. No.l.lo.2rr, rNo ft@r, roou to. 202 t.rs{ r{cHArvts lur€ sApprtiE AsH(x ffacl& HyD-to.965zr5llsz, 96s265t!52. wur.ims4mrths.com

(8)

I NSTITUTE FOR IAS/IFOS/CSIR/GATE EXAMINATIONS

MATHEMATICS ry K. Venkanna

4 C..^L.^-Jz

$: kv L= i ).;.t (r-f)c/

T

oI

)c

Qtw-,^ xG 1rl)1"/,--l 1o) 1-<0a-\

('- lil Jx

r^J ) J^ ---0

+i,fg)\- t-^ '/

6' t#r"rt

4

I

r.-f Jd

)+ t;

"J -k;o

I

Jo

\

)a

B

H)rA-r1--\_>.

I

t {d"

-9),'u/,

-14-r,*

-^'L./-

I

l

:Jo

I.+lo"^

F

I

)O

J"t

\

d^

J.

"r&)(3-U-,

t-oil * t

.-- [l-

rJ(

l"

it", r"ol- b

= )fu>

--@J*

A^ -a, -[{ 'Ttr-14i'f

A

)).1-l*u

\

-") J

).

I

}\

AA&

>L-,

( a

-oI)a

)-

109106,

p

o

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(9)



h =S->Lv<-ptfur -

aA-S-YruLur..' 'te&l-

-Ar f l\

-e hdcAXK- +ht

q p.t-l,- tzt\ V|<-il""Aft

T_ t.{r

N,rYffffrlL t^-g^^tC

A

A ,t A#( ) rf-2u ,f)

(-- E 2Ltu,Jt- hll^ ..','r) tu d )

2r,tf 2Lr, T

u Cfl*)"Iu t:l

g- - lt \'t'tf t ,-, ..f tr o-)

K G- *dr .^<$t),r. t\o(i r.. uZ- L,t urf oO

\^t 'xA f rl: \l.u l\r= ' Ju _!1,f isl

f t huuT+ l-r td uAA --;-

OID RAIINDTR TIAGAR MAM([I, 0flx[60 8MI{CH Ottl(t: 101106, IoP 'U'OR,

MUIfitN,E TOWTR Mtxmn tEllA6AR, otttr.9. 011-0s52998?' 999919745

HTADOTTIG:

ttclotlAt Ortlct: H. NO'l't0'217' 2l{D fr.@R,i@M t{O. 202 R.X5{AI{CHA 'lt'SBIUT SAP?T{RI AS}IOl( TTAGAB, Hm-lo. 96t2351152, 9652661152. wu*.imslmaths.<om

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I

(10)

I NSTITUTE FOR IAS/IFOS/CS!R/GATE EXAMINATIONSMATHEMATICS ry K. Venkanna

A |t 0.,^ o'16o6o ,o..^#" 1)t .

G*e (.A) = Trya-u (r-+u'{)

=G lJ)->l'Uqrl - zu)

= nr-L.o

'/) A=ltobI

lo lolool

Y1

Yt

\s

Yt 1l-

l-- ]z

looo 10ool

2-/14t4t

"{l '( 1

Y{ I Y/q

t1l \(1

lo o,11

,h\h

8tt"hyLt

{11oto

"h eltol

+h

-Yh*'.r [1

-' 11 -t(1Y 1 -v(?

--3 /1 Yl

<- <

HIADorfl(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{. llo.l.ro-2!r, 2ro ft@t, iOOM t{O. 202 R.rS{ flCHAryr5 ltu€ SATPHRI aSHOt( ilr6^& nyo.2O. 9652tSu5t, 9652661rS2.

0lr{56299tr, 99!'8t97Atw**.imgtlmaths.com

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(11)

INSTITUTE FOR IAS/IFOS/GSIR/GATE EXAMINATIONS


(1d C)

!{

F,r4 *rr<- gurt ort o[ +le c7l\ndel uhose- 6ezro'ratu'5

o,ra<- f analte-l to +he lin e += \= : & u3hoSc

1.iAi"A uurve is *al'--V ,7-- L I

let P (1,,H,,2,) b<- % 9o\nt - Y: Qrin dei

[r.,ovr <qt "B Affi,ei t^"t'6''' P ls

B =Yi-L

L {r"A 6raratol 6ee'ts t\( ?\naa z=L rn th.

P,i"l B\'*t,[-X t

? [r,also & 3

u6

:-)-, t =lA [-' z =')_2^

-L \,

-g ),-ry')+B',')e/nof cto{ in\c'rsecL t €' 6ni

zla-o

qivcfl

)i[-'5'')Biu"n

1:T,T;;

Ltlrrr*t?)t t,

? t$rt ?-zrft ttl'nt lzr f= 36

-)3drt(1.,f+c^ltt1-2,)+9lit(12|A4)i6Y1ttz;.v)=31t?) gr'tf+ 6Yl' +7?(r+'r) +1rz't-6 )+vI'1r') tl2?r'-")l

-/ - uzr-lbzrf\trL --3L

4tr^i*3ur*s2?-L^r\tnl'u''t-eozlt\21t-zru'-t(=o(, . nqu^'.q edn 4 $\n(e't is

3il\ 1l' t st-(*t +t'f,z-ao1't rzil-2UX-l1= o ryS'

r-ivf-f H$D ottlc: 25/E, olD RAf, ilon il^GAR MAfi s, 0tul6o!JtJ!.Hl trcrotrlt otrtct: H. No.l'lo'237, 2xo tlooR, nooM ilo' 202

3MilCilOffrE:lot106,IoPr]oon,MUriltoElDwtnMtxHtREEtlAGA,R,Dttll.g.011.1s6299t7,91'99r9r62tR.I5{ rrtcHAil5 !l,Ut 5Ar?tltl ASHOI( ilAGAt, XYO.20.9652!51152,96tr66t152. wvs.imstlmaths.com

(12)


MATHEMATICS tv K. Venkanna

o

tr)

Qil(i\

o

(r

It,t

!,lv'r'r'

(il)

t')

HtAO OttlCt: 29/t, OtD RA'€T{D$ ilAGAR MARI(EI. D$lll,50 BIANcH OffKt: lott05, rcP tu)08, MUIUERIEt TOWTR MtfiHtR tE NAGA8,0ttlll'9. 01r.0561998r, 991Rt9r613

STUT

'APPHIRE

ASHOI( NAGAR,

[ourid en 'fl^. 0^ [*) = Lt- 5t +,1 t-st+e)/r^fi.(-,

rdh,l lscr^.0 0auu,

fi,J '+1"" Ltu o\*t,r -l{,P" en

Tttox

P,^rl J{^" ho,uh o'[

HA *L"' l'oiutroF D(lt& I^oca-D l'4arrr'n"^ 0ou'n*

Pt*t 'l&e l'[^' ^be^^ fl'il^ P*/(r)i'Lo

)

,[

,,J

?

b

d

l:t b}l, Jl^rb"prn

(r\ ' (f-artv) (rs-[r+t)

atyeme lolru [t) =Q

l;r) = (r* ,) (r-,) 4G-.)L,?c=\ I

u= 4 0^e (,,jhil l';'fu'7 v

rr(r\= (!**5) t*- sr+e)t (t- a?c+q)(2' -5)

f -$tc+L + z"-5x+u\\

W-torn to)/

'(,\ =Q*.5)f

Qr-u)0+ r

o['(!)

a. b l,A

d,

[z-s) (t- ro+'o)- -3

A+ bL t') = [u-a) [t- 2D+to]

fy\AYl N\A#A

+,4

C

[tr=s 1-bo-rto

Wt6,1\l r*o

tl

:, T= L l..s YY)Y1^1rt"or

'rAil.'b)=

t6-5)I 1

+4ot )c=3

ItrEI trclotlAr Orf lct: x NO'l'10'21',,2NO ft@i. R@M NO. 2o1R.r5-rAilcHAM'9xYD.2o. 9652151r52, 9652661152. ww.imrlmaths.com

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u)

[* t=! , d"(.t

(B 5 ( ls - Llotto

q)3 (-: - LiL

a,. )G\ l,s f4r^xlr^a

q-t i^ 1o

t(

v

2

)'L C'lv

/(,

t lot'{ + 3 t 50P +rqtl5 L1 3 L

H ry t.(r- #T*vzl'4 of ---

>sl30

tt6

?8"4 ( f'"41

*<u = -li{,

, + T3 Ln^tnl

1a :) 8.t (MT)tO

*tvl - nL i r'q 6 tptl,

4t")rr

=-O ,

{-l-,.'-7>O K 4uL ft<t< o

-9< I o, j'l't-*^,- .+.^

f)

d^"'{+

-41;v-YL 4:t o\

@t'a.1 't\t L

rJ rX oo

HEAD 0tIlCE: 25/8,010 RA,EtlDtR iIAGAR IUAR|(EI, DflltF6O BMflCH OfftGt: l0' 106, IOPfrooi,MUIHERtEToWIRMlfilttRtEilAGAR,OH,Ht.g.0tl-455299tr,9t9919751t1(il0ilAl OlrlCf : tl. N0.l.l0.2tr, 2No f t@n, RooM r{0. 202 l.rS{Ar{CHAA,ts STUE tarPHRt ASllOt( i{AGAn, xyo.l0. 9652tSuS2, 96526GtrS2. wum.ims4mrths.com

(14)

INSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATI()NSMATHEMATICS ry K. Venkanna

/"t t,

1=

Lp} F be G e"A {eildTo Anohorn Sror',\ft--

= t!* trtltst, "q

- r'r{'tê to"dltto. * Orn o I

\" u^11 ^P":" t

r 1

1L0,5,.1 = [a+btZcq [r,u rc) o u(r)u-Irr*

T [orb rcl =o

E tn*+ I'trumbe """1t, a $",a hr T(x. rf,,,rs)- 3q {*r- t$, \DU*} oê

h r c Auch {{^} fo,u,c)pt",[ +l",.e N,JI\ i

,, b- 6, - 3c'+ b-cJ

=) O.+b+3C=Oda-b = o

- 3o +b*c: ort31 -l o-5 t -1

e/ru Jo €dîonKq

d

b-GP'

(--+ R'- zA

Rs* Re+ 3&

0o0

abc=)

R

3I

0o

I

0

0

\8t3

-b -60oRr- 3 R3+ '1('

Ct+b+3cs o-3b-6c=Q

b+>c = O

HEAD 0f llCt: 25/8, 0lD RA,tt{otR I{MAR lt4ARrn, otl}ll'60ItGIOtlA! Otf lCt: ,1. NO.l'ro'2!r, 2N0 tt@R, nooM NO. 202

EMNCHOftl(t:10!!06,TOPr[OOR,MUIlltR,tElowtRMt (llti,Et,{AGAR,DILHI'9' 011'15629987'9!XEl97Q5

t.(s.tAtrCHA ,t5 Btut SAPPHlit ASHOI NAGAR, xY0-20. 96t2351152, 965265r152. wurw.ims4maths.com

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a@

(15)

INSTITUTE F()R IAS/IFoS/CSIR/GATE EXAMINATIONSMATHEMATICS ry K. Venkanna

l5= -zca-- -L

I.lr,ll ol"* ir 1)ven 1(r,rr,'il = f"K , -{K

,

-I

LL0" GK

'It/rr"t

'It ^ tlrll'

Irl

K

KL

?

HAD 0rflCt: 2tl8,01D iA,tilDtR t{AGAn MARXtT, Dfl}t}60ttclollAl OttlCt: H. 110.1.10.2!r, 2No rtooi. iooM NO. 202

EMricHOfflG:lOi106,IoPtlooi.MUpllRlElomnMu(lltRtttNAGAR,ottH.g. ott-ts5t99!r,9tr9r9r5tti.[s-rAt{cHAr\,rs ttut sApp}IRt asxor NAGAi, xyo-lo. 9651rsu92. 965266u52. w$ry.imr{maths.com

b*[rHa^, a-\e

(16)

!NSTITUTE FOR IAS/IFOS/CSIN/GATE EXAMIilATIOlISMATHEMATICS ry K Venkanna

3k Isuta? + = + is ons of {e. {e.r,-n- -tut3d

, +t^0, ib is oormcl tD -+t'n- f,*" {t'rbrqh

ql

Tn

?r,^r'*o

+tnrc{ore

_L\or

-ttô

oldve.rh'1.

rl

*{ -{t^q Cona io h'ro J{ar Errôro}ors

cq,1*a.tioo 4 -lf.o Plo*n- ir

*tr( t3*=q -

O.t0 +"d fl^!. linet $ inhtrectioo 4 {"i3

srnn conr ord [e.F orB 4{t{ lrru o+

[+n =O E7 I =-n

+t.a-

NOr,rr t{rq Cl.re

$are otd +(.o-

inhrrech'on bE rr.

Tg*fnn

rthet\ urc hOn a Q+tm + Bn = 0 o*d 5vnn-8n[-3lrn=o4

El.intualrt1 !. h1n^rcen -lto,k- ,^se- 6*5mn - t8. + jrrr) [- Qn+an;] =o

:* ltrnt + 3omn +6nrl --O

+ .wiy+Svtn *lrrj> = O

a (tn*o) (rrr+trh) =O

trfu{ rn =-n , 4^t @ utc E+@m n

t I-t

I5

L kl)

urf*rr rn g -[rD 't otn O t,.ee 6f l-sn=o * l= 5n

a

o*a a^rtdslt

olro eoc}r ohr

--L=!kq) t

-ttdle orL atar ct

o.f lg.lro is -tbr

=F

e

t.5 + I (-q) + (-l)1 =9 old

to +fa Stven Xr'.ûotur'

tlurqo- -t . 6 or A O -t€"' ott'" hD ?Ar'ralors

orc ?=+ =& or,.d += + -+

D$r{4.

li:f,nffitrt03'16.

116rr, no.D. lsSslltr, raraault. rrr,hulnr$lcoaSI ll

(17)

Lt,t

to2"I -l 3

+t8

\o

1

tA arnd S-t -t

[in{ AB- ll 2 {_

LL

-t{ o

L 7

l0otoo

o0

I

Fi" { ddtA) 1 4grt$)

JetLn) = llrt '101-l+t I\

I [- 8-]) - o \\t-t1) t x[a]'.'{ )

_tl+\t

I AEs

dd[h) = lB\ =-\l a

-r{ 0

6 -r 1\

\ h^E



a)

It

t\)

HtAD 0ffrct 25/t, orD RAf,ilDA ilAGAn MAirE, Dtlrll'60 IMI{GH off l(t: 10}106,IOP tutoi, NUItltt Et IOU'tR Mu(Hti,tt flAGAR, Dtttl.9. 011-1s6299t7, 9l'99r9rQ'

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ri

Fr o {-'\

Ll l;l

(18)



ui)

5o$,

SoWe $e !rto'^"[ SYSte' aB

n+Q7 =!, 2d-b+]12$ urttu

linecn QruU o'ns

tg1= l\

rt+2r-\.Iil t

o'I t Q7 =j

[ + 31 "J

U tS7. =ttl

AX =

t02. -lql

s

L3t

d)7.

j3

t\

l.l0\") [o' u-]1 o tljI -r 3 \ Iq r B l,',

$eaxrj tD cch\on f ort't

I

o

o

t

o

o

a? 3 F.a R1 lRt

I I -3 Y RtuR't 0

?_

to 3

t,tnR'-to

-3

)

?r\ttr1 z 3

-b-7.. '3

-1 - -1

=) 7-= \ rH'1, )t= I /kn.

ilt O Otftct: Zslt, OtD RAf,ilogl ilAGll MAE(E, DttXl.6O BRAI{CH off l(t: 10}106, IOP tOOi, MUI}ltR E IOWTR Mt (hliltt I{AGAR, Dttll'9'tt6tott/ltotflct: t{. ilo.t.lo.tlr. ltD tr@i, t@M r{o. l0l l.r5{ar{cHAM5 ltut SArPllRt ASHOI( r{A6At,

'lYD,l0. 96t2$U52,9651661152'

011.t56299t7, 9tr'9r975t'www.imstlmrth3.com

(19)

I NSTITUTE F()R IAS/IFOS/OSIR/GATE EXAMINATIONsMATHEMATICS T, K. Venkanna

4t FJne fi,e- ,loors of .by ptfib frrrD ut*cArtrrec- rnuhhild Ff*Ji",rld t*Uenb ean k&*,^n /D 1r" pfuloil

Qt/ol)ld /b)= 22.

Norrl fr,e equolirn of itw

trffi,fll,f* aih wrl"x

SSr 2 TLHgre

s : (t'/t)_(;r/ t)_ 2L ,

Sr: U'/*) *(t/ f) * 2\ ,

T= Axb')-UA/0- pry)A,A

.'. ftawl @ , 11e- eau"toq of rtecone of fr? Viuo 'am,,rtat

urlb, rnzrfer * U, p, y) .rb

ilEro0fflc: tl/l,OlD iAf,ilon ilACrn MArttl, Dtlfll60 llAti(HOtt(t: l0]106, top ,r00i, MUIl{auE towm MuHtn'tt titacAt, ofl,Ht.9. ot t{t6299!r, 999919rAti(ilotl^l OUICI: x. tO.l.l0.llr. ,lto ltml. l@M xO. lOt Lr!{lncxAir5 trUt !Attllll ASxOx trcAt. XyD.l0. taflrtuSl.96'166ttSl. ww.iml4natht.tom

(20)

I NSTITUTE F()R tAS/IFOS/CSIR/GATE EXAMINATIONS


( #-+ -2)(#-# -2Y

att-c 4t -z-Y.F _9L

**

N& Yoe- l<lnoul bA .if 'ttis cane tqlno,frw) Svn

of t, " ^,J* in 'it wu*t

d

b zab

6r

,L

-5-*P,

a€-

bLte*-#-') t"6 2?,-T+

+ brl =o

-2 -l=0*U;qL\)-#6 ,

{\d'- zQ!-*)z + *t=o

dL+ P\ zy(&- *) + *b'= oN

.'. Rqdd. ,locu of 'tle pbrt((,F,'D is

,rtAo0fttct: 23/t, oo i f,io:i xA6Ar i$irn, DttHl60 liArl(H 0ffkt: lot106, loP rooi, MuxlltlltE lowti Mt 0[,R !L tt^6lR, DttHl.9. 0t l {:6lrt!?, t 9ll 117615

IMS trclolot Otflc: H' tOl'10'l!?' Iro ,[email protected] llo. ml t.rt{rl{c}nMl nw rlrPHtt atxo( l{lcAt, xYo'lq t6tl!3tt3t, t63r36tltl. rxY.imtlmltht.(on

(21)



0 qLc)

-

ry

€Va otn e*tner e, U",0.rre, 4 4l,re ,lrr",rl-*9-o *. UG,^Ar+a

u *+Lx- +Z

ilt\Ael-e.o"[ t

f+*, s.,bl:iS2. : 30; u5 0}1z dz

e-cF

0 hdejLer\!^n^*A

-

T=30

9{'\

>30

LeF S.t **+ AP=* *l* se- to]

t- y3 -+

df

Fo^ l(ax ll.&" , o[F = o

dF = frr-+ zaTc[x--+ tf; + art1d6 nVz+5cr]da

)z+ 2d =O --Q).Qr, -f 3d = o

-< (3)

"eg -+ Sd = o -- L3)

r$u,Q-lrp,t6re bf r'XrL P 3**dQio f Trn \ ; []-'U * szr Jt

-ttr+ , ntsO =o

AA,5n OC= - -Z i J-3d

z

'L+ld-qd "{C c{ = 60

BBcr =60 + d= H

e = -5/2-

P"ttt^l \n

-lrb][-a J

Ll

L2a + 39 + 5t

+.r$l*

UrrU,"-e-: -t"[?] - +W" *fAr&+

= t't SoT-dr{1-*d*] ) o

U

_\A=) 0^",i.

HtAD0ttlCt: 25/E,0tD RAJT'{OTR NAGAR tluffifl, DttllISO ERAllCH oftl(t: l0!105, rcP 'OOR,

MUT}ItRItE IOWER MI'(HIR'EE IIAGAR, Dttlr.9. 011-0552998r, 91XX'l9rA5

REGI0rlA! OfFlCt: X' NO.l'10'237, 2NDfI.@R,R@M l{0. 202 R.(5{ANCHAI,5 ltut tAPPHRI ASHO( ilA6A& xYO-l0. 96t1151152, 965266rr52. ww.imtlmaths.com

(22)

INSTITUTE FOR IAS/IFoS/csIB/GATE EXAMINATIOI{sMATHEMATICS T, K. Venkanna

w-r.(or (3lr) (X dt, + :cd6 ) =6

stn(g/"X"t - ld"\

Srl'n:' 'l.. cos$A) (Sar + ,rdt; = y-6,^(v7x; ("dV-Ud") -}@a

@L *il''91 -(u*'+ -Ll *-!)r*=0

(qI2_ A,tr r- btLalx\ +, 3in(5 lr\\

I sin($ltt- r h) LCo.tU

o

!+= t /") s l"(u l")l CUlr)0r /r) &n CCU vlx\ _ Ca.s CU I x_

Tl<e-Jlr,=t +

Wn ' u^ilba t

ilz fy ,to{t^h

@tArldL

d4W = t+x

Vt rdtt .dr

tla^n0 ?,O

=\,

Zd^L

drI = rt (t*sv + V Sinrf

V9nv- Cuvoi AdL = tt)nt - CoxttL

v Cox v

(f,oxtr+ W lnvr)\r9n

) €r

V

v - b,t,

=dobtvtt.()nr - bo

d't --tH-+J d,v

f,= -bqbg('1,, ulvl

v +-1,-r,a6a

c

+

,

+

"0Da

*v Cu?= L

\Cr^(AlU

xuoomc3 t, 10lD t fron i&lt nlr.?, D8r+.0 mrr*isr& t6r. D' tloil. ri.Ett[E t wtt xtllrtSt i5rt mx-t 0rt{$rrr.rrtrt otINIS ItBlOlAtO?llC: tl. iol.lo.llr. r3 rt@t, r@M tA r0l [rl{ rcumtI'ltt ?ilrtAt Blrtltr]lDtc.Itr$tltlta'lrtt'L rrr.larlnrdtrcsn

__o

(23)


MATHEMATICS rv K' Venkanna

[,4)-ri

,"dt Lta^n*h) =ns'q$)**@tl dclr) =Y.[ &"fu-/'J {LUf")

oL t*Y % Y l" A La t")

)*X cut tV l^ q)'-7,J*"1

^U\.t-

r-A\,1

"1,i) -" Iqc

t_

Z[xr Ci"zW I

+ --Cr"tf,

HtAD 0tIlCE: 25/8, OID RA,tNDm ilAGAR M nxEI, Dtlll}6{t BMIICH Otrl€t: 109106,IOP ttOOR, MUrfi Ei'Et TOWTR Mt (HtR EE tiAGAR, Dlttll.9. 011-{3619118r, glxBl97Q5

ItGlOttAl. Otf lct: H. NO. l.lo.23r, 2NO fl,@R, RooM NO. 202 R.(5.rANCHAM',' SlUt SAPPlllRt ASHOI( ilAGAR, xYo-20. 9652351152, 9652561152' www.ims4maths.com

*rX a

(24)


MATHEMATICS rs K. Venkanna

&s b)

,w

pit..otr "\t* 0a*lloq

r*TUon"-0 '{-,./

'flu '

ec*orreA el Lhe lor^;-lr4

,b [o,r\ rJ(

CI"clr

[o ,-r)

{-t^.o {he P

oi

\,dQ, K^our '+{,^"F

r]+f*rXL+ ,dr +c[ =Q

tr'rdeli,.= Faaon oA

%uer' {{^}U{Q^u-

t",

r[ [o,4 o",A [o,-Drn

0+H +o + -{ *o[ oQrH

t+d o -O t1^ r,., (-O,r)\

h^ to,AN{rr\. ti*tlil*C)

.[ of,"

So.luit q

*f,*(6r,

Orr*) d q

{t^* d** ^z)+ [' + T u^t- o

4-lu;

c W."c.t E

+

>x-+ t(.k)-''x = o

Q= _r,-..[[Hl -,\ L

lu. u.!,. 1 t \tn 9oo

tln

?,,snt0

W,

HtAD OrflCt: 23/8, OtD RAJEI{OT3 NAGAR MAffiET, DTU}60 'RANCH

OIf I(T: lot106, roP fooR, MUrHt[rtf IOWTR MIXHTR'EE NAGAR, Durr.9. Ol1.l562gt8r, 9$Xrr9752t

tEGlOMt Otf lCt: H' NO'l'r0'2!', 2N0 ft@R, nOOttr nO. :Of f .XS{ANCHAT't's ltuE tAPPlllRt ASHOI( lilA6Ai, xYD-20. 9651 35rt5 2, 9692 661 l5 2. www.imstlmlthlcom

\o=

c

(25)

I NSTITUTE F()R IAS/IFoS/GSIR/GATE EXAMINATI()NSMATHEMATICS r!, K. Venkanna

=0- /+f -zrlr.il#il-, =e -@a?0 l,r {]**u \'**

LLr, /"r. ofu,lot^^,0 q"^nfu)h.n" 4,'t =\_,0t-t- dil,

"X{0,* \de fl**"r't' IIhI -11 =o

=l f+ L rthE-

t *+ r[hOb

)L

o0o

,l&- +LlL t'

=| "tAr.

- e&--S+,,\t =e

,: ,1 dr + t1u xl-f\ = o

t$t^ug '7t1= ,t a,^J N = $-f- \

J. )yllX

Lr- -DNl =Dy

Qxo "l-

LZ

UEA

1'\^ Nou

HEADOff|Ct:25/8,0lD RArlN0tR ttAGAR MARt(tI. Otuil.60 BMflC]t Orfl(t: 10t106, Iop f (,o8, MUIlttRJt[ IOWtn MtxH[R Et !{AGAR, o(,rf .9. 011.t55299tr, 9tt9:,19r625lEGlOilAl OftlCt: l{. 110.1.r0.2!r, 2t{O f t @t, A@M il0. 20l R,(5.(ANCHAM's BI,UT 5APPHIRE ASHOX ilA6AR, HYO.2O. 9652151r52, 965266!t52. wrw.ims4mathr.com

(26)



-ry )- r2"1.-2-,

l'\ ,*z,--t.l.I_

,16?

-Jl' 'dA'A

aI e L-Ld,"r1

<-gj-_

I

z-{n

Y*

o

O

ry d^J. Y -

x

*d* tx-

4U

,-?rt84/r-ag--

c ./u4 G^ ur4l--

J4d^

U

+ tt/

74 L C

tr)-L f Y+h C ,t

9-tv 7 C

')A. trfil

+ t-qQ-'lJ-l -+1r- (h

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ttGlollAt Olflct: ll. l{O.t't0'23r, 2lt0 ftooR, iooM no. 202 t.rs-r^ilCHA i',s BtUt tAPal{Rl AsHo( NAGAR, xvD-20. 9632tSI152, 9652651132. ww.ims4mrths.com

c

(27)

I NSTTTUTE FOR IAS/IFOS/CSIR/GATE EXAMINATIONS

MATHEMATICS rv K. Venkanna

5G)

5olY)

frt t"rro.l u"lue ,(, o',L',L is fhe veclo

? = (-url-3U*oz)i t[!ttr)*5z)r'rirrrtohot^l 't Hernce e4prcb5 n a$the

vtvnQL VIV = 0

' c-5=0U-& zO

4V

tuu-1[+v

eq"nhY Lud-S

Jvt

'( frqttnt+U +32) (

6radie,nl ,B o.

If

K

-LdL

h,e hqxe j

)q0

n-rX vi

)ail

-rrr-5f$( li{+3Jtst \r*tL6$

=i[ -5)-i("-o)tKtb+3)

+c==7 a"

5q

ft|'o ? b='3

l.lodotrr( f i' T=\-uu-3il-'T:illf

cr?6ass\ aS th 6raaio^t "t q-t

v

vQ

z )i t [-3rra

?

,il ntJ

oaU

= !-qJU\=e,7-)

t'l s*H*ile7tsz)J t [ur

D E -ttlt-ldtq1 t Q: -r*\r-\x(t {,tX,')r-o

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'{AGAR,

Dtt}1.9. 011.1562t98r, 99{D197615

ttclOrlAt OtflCt: H. NO.l'to'l!r, 2ttro fl.@& t@M nO. m2 t.r5{ariacHAl 5 BIUE SArttlit ASHOI( l{ac^t, xYD-20. 9652131132, 9652661t52. wtw.imstlmrthr.com

St,nlar( f-c,F o't q' delcrrmrne +

{1n e. vectot' il rs imotaho'na}- t v xT = 0

(28)


MATHEMATICS tv K. Verkanna

)q€

= -3rt+{ tS ( ? + ? -3"J r srX"+ sU(+ t*3

4J(

o, O, O eg% Tqoeg'o+s 4 'tlttt %*' 4

*,S )

t^re tOo SC

f,(52) ' ?'* T"+5[z'2- .i,. +a

2_ tlx (lrlt,') '{r('r,x) '

2 1t

2*t

+1,2 yo-:rb

q. '2^t''+i .1ea3r il*\rt(+5NzdAe4 o''t\7L

to rohich m9b1 ^fuxSB,nt

q , -2r^'r Zf r ZLl-3rfl turtz+ 512 + c

11',

HEADOff|G: 25/8, OID RAf,t{Dn ilAGAn MAnf,R, Dtl}l}6o ERAtlCX $f l(tr lot106,IO? tU,Oi, MUfitruE IOmn MU(H[R'tttlAGAf,, Dtr,lr.9, 0u.l56lr98r, 9t9!119761,

ItGlotl^l. Ortl(t: x' NO.l't0'21?. 2t{0 tlOO& t@M '{O'

202 r.r5{ ilCHAlr! ltttr saPP}lit A3llc l{lc^t, trYD'to. 9612!51152, 9632661152' www.imstlmrtts.com

(29)



5 td)---)

$f

ln tW*1n

u

ve,r*icq[ by 0- horuq'n1al, Johce Qctivr2 qt thr. 0*l"r

,l-r^il o.f lno nod an,t q,ua[ ,t0 hotl if,r tl-igl'*'AI ohcrt irrcJ-unahorr "t0 *he vortical- o'rLL

'*l^ruiod tu)t Q

A unifixrn J'rod ' in ue&ticnt.- Jxtrition ' coh 'rturrr

obo"f one 0f r*J rhdl and il putUd' oijtdr f'om

-Ttu $lcel .ir' *f.i I atrIern ffu-ol

(il l[t t^.,upht \,,] oj *he "rrdAt) ecln"g uuhco.U4otrrnrrflr. I

tii) Ttu itnce P[#) t-,rj*hurh,.rcJrr *hr Jto& ..ir pullrl

o.Ilidr {ro rn *l,u u fh}, CaL

(iii) *1* ruqc$ol4 R qf *hr,rv\[ A, oJ *i,.r ru&

g

(^

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Ld Jtu dru2ront br l[ po]il,on ,J qoiXbruuth-U

Th o)rdu dhc,t ,*tu jora,r Shoutrol h- .-.t n e1urilibriun,

t*^\ $vrel Ahotr[lL rt urt oL o pin* . H en.e Jl,.r

$uaction R lhodol ?eM *t'aou6h 0 , *fu poir.f of..ivrkr'lect'un oI Jhr. dig*dioh

"f *t". $nal hj qnd _!l*

Lt* {l^} ,Lngth oj nof, br zj_

.'. Ah -- Bt, = IJn AABC , G,\$= Ac

z!

l^/

= AC =LI(oJo.

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tEGlOtlAt OIf lCt: X. I{O.l'10'2t7,,2N0 f l.@R, i@M NO.102 R.IS.XANCIIA}I'5 EI'UE SAPPHIRE ASHOI( NAGAi, xYO.lo. 96513s1152, 96t266tt52. ww-ims4maths'com

(30)

INSTITUTE FOR IAS/IFOS/CSIR/GATE EXAMINATIONSMATHEMATICS ry K. Venkanna

Tvr AA0'& Sine=AOt Aor=ISing."L

ny Ynovnentt qbout d1\ffA , *tna o[9-.brric lqr'au*hl

Ino rne vr tJ "iJ { e ro . A r n ce *t.. 'fo rce ahr

eLU tri bU vrn .

-'. P-AC - t^l'A0' = Q'

Taki

ofdh

0h -l-H.r"[C,trS- t,t} .l-Sing =o2

(oJs--- Sing o\ *qy,o= ! *an +s"

$ =45',

"" Tf\ ,.ir.cl-i nah on of {tr e roJ of *he v ed, ca[

.i^ 45'.

HEADOFFICT:25/8,0lDRAJEI{oIRtlAGARMARl(tI,0EU}60Bf,ArlCHoFflCE:10}DS,IOPrOOR,MUffitnEIOlVtRMt(HtREtI{AGAR,Dtt}n.9.011"1562998r,9l}9919752tICCIOtlAL OIfICE: H. l{O.1.10-2t?. 2N0 f LOOR. RooM l{O. 201 i.l5.rANCHA,til'S ELUE SAPPHIRE ASHO( llAGAi, ,{Y0.20. 965235U52, 9652651152. ww.imr4mrthr.com

-----=:::=:--4='---'

(31)

INSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATIONSMATHEMATICS rs K. Venkanna

5 Q a U6l^r- nffd YDA Ae c ho\ *hl^.-o- portt&-r ea& o,f

r^cl^a Tn ouo&or\ +o \F of N , B ar^d C ' tl^" rucl tr

a b\oo F aF Eru' ",.,fl^

FD it- '-b a Po\nt

A ot*.] FD Bc. P"ut't_ +0..^I -lt^q- tciru--tic-q1--

AkutR

cLlhr*t* m

So{1,

Bc=b.

L"k a blou: of impuhe

rYN

aL.-au+ f , rr{nue- A 0=aA"F "? ts L & +ab + b"*ffio-d

Phe

Ptg

ffu* tD +{.o 'rod of

+t-l potnt,O Jtar

Ao=BC =b

tT +t^p- rod AL&q^ +tel-.u+bo u'-'AB=a

u)dta- [o- u-oo

C(rr\)'oB=a-b flirnp.trc +0,"- 3'6.<hr',r

b Ca-o t bUC

al- resE. Tu-rt- o$nrBd3!-

{t^l irn

t,c +t!-

+Le-

pr.tou .l.rh u bt -tt^-t veloci\ u{ -lt - potnF O a*d

uqfociF6 o{ +C.A rrd , Tl^r{adtic +C.e-

vo-locll"ior, o-+ w\o.,naq dI 4 , B o.,^..rl C t^ritl ba-

u+ bo , Q - ta-O d o,^d u - ato Ta.ego clt udff ,

Noco , f i, c-o- '{X.l_ W in no0rvre-f,rhc.trrr = Smpul*{*$r*

,n- m (.^ + b.o ) + trnftr- [a-6) d] + yn [q_cto) = p

L"u -2tcL--b)6] = P6

, t-aHr{ mowrn_r,rE o-bouf O , -{&-lt rnOrn.c-rrf .e4uoltA lro

is

r

HtAD 0rrlCt: 25/8,OlD RAtrNDtR ]{AcAn MARIEI, Dtul}50ttclollAl. Otf lct: H. t{O.1.10.23r, 2NO Ft@R, nooM ilO. 201

8MflcH 0rrl(t: 10!105, roP ttoo[, MUI]ttott rcnfiR MtxHtR,Et ilAGAR, Dtt]n.9. 0tl.ts5t99tr,9t 99l9retR.r5-rAilCHA rtS Bt ut SAppHtRt AtHot( NA6AR, XyD.2o. 9652t51152, 965255rrS2. wuil.ims{math'rom

-rn (u+bo) , 0A* - L"- (o-b).o]. oe+rnltr-aur).0C:6:) - [t^+bo)b + [., -(a-a)uil(a-b) + fta-oo)a= o

=7(a-U)u- (&-ab+fr)o=o @

Jr- @, ur = (a.b), u 1 tX-,.b+&)StrUv(llil^t in C, .u

frll:

(32)

INSTITUTE FOR IAS/IFoS/GSIR/GATE EXAMINATIONSMATHEMATICS ry K. Venkanna

-

3u- A [a-b), (a-b). '^ / ({- ab* v) = P6

" ts (a,-ab + b') - > La-ufl = P (of-ou + b") /n",

(,\ --_P t""- ab+ b')m (d*yab+ f)

K.E Sol *l' = lrP ( v * Vr) -- YrPLo+u)

tzP,(o}-c.b + b')nrl (d+ab+b9)

HEAD0ftlCt:I5l8,0tDRAtrt'IOtRNAGARMARXtT,DtUI-60BMNCllOfflCE:r0!r0S,IOPfOOS,MUrHtREIOWIRMt(HtRltEi{AGAR,Dtt}l-9,0ll.l5629l,tr,gXDrgr5ltItGtOt{A! Otf tct: H. NO.l.lO.237, 2NO tt@R. R@M NO. 20l R.(5.rANCHAM5 EI.UE SATPHIRI ASHOI( IIAGA& XYO-20. 96t1t51132, 965266t152. ws,wims4m.tht.comWHI

(33)

II{STITUTE FOB !AS/IFoS/CSIR/GATE EXAMINATIONSMATHEIUATICS ry K Venkanna

Sotvr- bd-=f^r *qf,.rA { vaAr^tfo q 4 tla'ort

* + i,-to{r') #*- t co,t-,' i gi'ir '

l. 6i*, 'l*r,.l-

!"*[ l- ,fr)VLU crtrt - c'i'I" '-O

Cp.Iri lr- \fC vc,.t..)tr- crh'( = gU

,s.itq it'+ PJ +" ( -- Ob\q @-UyYr-

R= o'e+R P, l-acir. A

f - Cl -cofr) <ffi= o 3 [^,or^qJ*A

ce f d.1-p't62-4. i.r ,\ N4+kaF U?L 4o b"UzUN.

lrl- tt4 *f G g"l''fr''',', c

b(*,{:

?"4"Uot, *

A %"bD

O.ul \l H 3' \^t,t4 1r

[-con+Ir0T^

eOl.t Crl -+

T,J ,

dK (t+c e o+cta dtu Id*,(o t+, "0- Tt c

l=* t^*l'1@

rai(.

ilr6rt0sr{.1mll Offi! ll ]r.t r$a7, AO ROOi, n00ll 15. tm r.f3{fiOlrttt ]tut ilAEri ilrDin' rSA$r52, rt2afl fl't' wr.In3larduonr

(34)

I }I STTTUTE IO R IAS/I FOS/C SI VG ATE EXAM I ilAII O 1I $

UATHEMATICS T, K Venkanna

a

\9 , Irq lCS "*'-c'lti t qL

{) rr = e.; }Qtn(tr*y9

-Lt-LL

3rin"l"

e

a(hl

J)

di.

xornr.rtdllorr33l,t

+J $*

Jrs

(35)

I N STI TU TE FO R IAS/I Fo S/C S! N/GATE EXAtt'I I 1{ATI O N SMATHEMATICS ry K Venkanna

Urt (1nr-(fl

.-\+e [r"

- -+ grarr-e.oJa + taii - t'u*,(*A'-'a'',A

-L (iazrr + J- (l+ cax) '- i-q -'rE\ '| 4' -*@* -et")e- .I (u

20\Cu'\**-43r"n ")

t [z co\ '-"r--

lla)'^)

I t' ,t. + zr+vut*he'

xcc f Sina^4 dr^0&^

us{nt/'*r!

r

+{^r

=iilli::l'"Ll.l'7

0ax!4B nrlar offl(E r6.r01rptr00i, ilunEltrtlotm Hrafiilrt ilrGri ftu]rlEloilt o;flctr lL X0.l-102tr, tXD flooi, rcorl m, r02 i.rtilllloxr3 ltut sl?nnt

^sfior f,tGlt H}}20, t65ltttt5t, rE236il5t. rrv,hnlm$hr.or

(36)



6 Cb)

t)d

[ur the vettor {umdrorlt eo *.f R ' c^rh^l^t S X

7 ., *k ,. 3 'Sthb6)t -ttr6zi

fi {ro-t to/o,o){o o'l'r)(-

.r*A fle [ottot^'\i

*--t, , h't z=fo't th. 3

iverrr Uurv e

Pd\s ,

C, uce hqxe

+ fitf r-

f"]0

tui6 1i +(K' t't

& -' i t rtj +3f,Kd,l,

A So !:' H.llT;; ? 1;' 5?"[':' I l:lya! {^c- fb'

+ t--r o%Ffru'l]

Sf,dr; ![n*)"L', = !r[o+6y

)i_ru[z_jtrortir] . [itztj+3t"dar

L

'LQ t'* 5t') tBtL *uotg ]--o t

= l€ \'- 2at6t eo f )dt

dk 7.

t))-2.'-

=[t

[tt'- ut1 + 6

(,",r ilJr)'t,

D

= t-q t 6

5

HIAD offtc: 2il& OtD R rilom ilAGAn MAIiltI, DEUIIGO 8iAilC[ Ofrt: t0]106, IOP TDOR, MUIHIiJE IOTVER Mt (mn tt llAGAR, Dtl'tl'g. 0l1.ls52r9tr,99l,l,l976tt

tcctoil/l!offtct:H.no.r.ro.2rr,2raoftoot,t@Mt{o.lo2t.r!{^rcHAr SBttESArPlliEASHOI(M6ll,XY]'20.9651r511t1,969266U52. ww.imtam.tht.Gom

I

(37)

st"i6l* line iurn\n6 (o.,o,o ) tp (l/o/o

t r ,r, J) orn4 the/n to tt ' l/l )

v{a hqre F4r -- [tc'r.t6u)i-l'luzi+e'o"r

) thc4 tD

L-rLu) t

( d;i ta;i tdzK)

= bfatr) d*- lu[(diltrolid a

Irt q demute 'tvl e s\*tJhl lrne Ju"tt

[0'0'0) t!

(r,o,b.) t\c/n alaY t' ' XfO '7--0

& r,t A"$ from otr r'

o5v,"t'6 A.A t, )6'04 dz=o

rrt cr dsnote t\e st,at3nt t':: t'^Y tl'o'b) 1r"t:"''

{vn al.lt c. 'x' t '^=)

annd Y'*d &o'n O b.' )''

,5rlonotX ".t*fl cr, drt & {<=o

Aadn,a1a*ottthe51ra6u\incjoln\'r51t''\'o)

n; rr, r r ) ^*6'i -1".1T . #-;rt,[t'-=',1' o obvio\^bld ^t'{

L

Noru Sfran' (n'o't!tr'a't In'ol. i, t"3

,e hrvL ! A'a, ' ( ta'inrtr)art-t"d"Ail raortfa<

'v-! ir cl r..ar^Ac,r5'o,'^'= ( orr^ t 1r--, ,dzz6, r^, ot'']

'l,z 0

_lu

0

ts.d u'atr"fl

'[t{ =L

ry1

r+l)z

I A'd Lcl.dr:Drd?=o

Y: otr1 I



ii

50fl;

llE DOtftC: 25/8, OlD R f, t{DtR t{AGAn MAfitI, DtUl6O TRAflCH Ofrlct: lOit05,IOP f U,OR, MUIlltRltE IOWTB Mt (HIRJtE I{AGAR. Dtl,}|.9. 0lr{562t9tr, 99',99197615

tlcloil !Orflc!: H' l{O.l'10'2!r,2ltlo ft@i, t@M r{o. 202 t.x5{AilCt{Arrr5ll'w 5ArPtrRE AtHq( ila6Ai, xYO-lo. 96t1151t52, 965266!152. ww.imsr&nrthr.com

(38)



SoP,

NOW [-

t

= S - lt'lb

Y--o.

It3

[.' ^o-il.co,dttr0rdl'o: Otn Il

toro,o)

tr7dt = \rortt'a1

z'- 6

.o dU [--' olo"f, ce,7=o)

,o,o) {p (lrlrl )

star6"{ \ine ld'^i"6

,"t*rl)2.D

20

0

= tt0r-,. I R-d x

c

Str-0\t line i

Se- equahonS

=L:3-3

otnt [0

,r|, th'

arn{ [t,t ' t) tute

1-o ' =e = ?o '-\tW)l-0 I

-t\?^^ al$\d c | *'f "l=\ ''1al{o { ' }"i+Ijt{K

-'- dt --((n i tr)dt

cdr,.6 c r n= qtiet) i-rqfi t2oPK

ut Lo,o,o) ,t=o o'm{ cLt [\'t't)'t--lt lqflz+P] d'k 2 I

! n' a'r ; I t6f+6t')- ltlt-trorj q 3

L 1--0

HtADOtflCt: 25/t,0lD RAf,NDn flAGAR MAilE, DIUI6O BRAIICH Of[l(t: 10116,IOP tOOi, llut ttn E IOmi MU(HIR tE ilAGAR, Dtl'll'g. 011.156299t7,9l'',1'919r525

IEGIOtlA! Orflct: lt. tro.r'r0'2!?, 2a{o ftool, tOM ,rO. ru2 ir5{At{Cl{aiil5 rtu SAr?Htt ASHO( t.16^& trYD.ro. l,65rtgr132, 9632661152. ww.imstlmrthr.com

(39)



4! -t -, IL 13Clea-{lil valuel. A.d? atre 5,

3,3

(o'ltii),t\i ) coSe

rebruh't tit

65t rt i8 Tro\a n all u€- ce$c$ ' lecar$ a

$ome t

A i s not G"ngenvaliv e

vxtr*o rohrch ryEamS

gath de4endr,rr'l Yeclolliad veclo'r'

-tSoA S

aJ l<

I

-.,vl ff ) a a

od a\)*

f.rf.t6 -\u1 f o.r*

t I trtr4+\ au-

at ,'u,) -,[*l' ovt) -ngf-+t))TL,

sr^lrul)

a

I 0L

t \.[+,lj'r") --L I.'

'd

ruSi - to*f -L (-6)K

+o

HtAD 0tflG: 25/t,0tD RA,tt{DtR i{AGAR MAmn, DElxl'601E6l0llAl.Olflct: x. l{O.l'10'21r,2NO [email protected] NO' 102

BnANCXOfflct:101106,IoPtt(,on,MUIlltRrEIOWIRMt (HtilEt'{A6AR,D[t}t.g. 011.0t6291,87,991'!}r9r62t

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I

o

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(40)


MATHEMATICS tv K. Verkanna

{-o' o1*1 I ) -21 AL>; S il wra- ."[ad Y"ka/-L

.DVor*-t

d+fls Stofq -l}ldt"-rn 4u" il^L veLfo{

p = ,t-.4i" "UL-i

+o'i e^ln'

,\ +ha- P^'r+- +tlq *

$a q shk,/ A;v-we/w^ ,&0 ; dr = .lJ kxfr)'il d s

LS__!

Lery \^^ +amtXGi *na 'tK^rv

*,d'Alot^.ic-q

trt

$

ato

t.o^^{

F.Ao 2

o-t i * 12i +^2b ^

gsd--aA S

^;'1- 2 =li

c

C

?.dt =

p"-

0 *t 4o-+ Y 'ed-r xz)Y

Z

'l +l-l.-Lf

{+L = 4q en

->o d* t t0 -f,) Cz"dt

\

lkO

2^> z'I+ >l )

Ic-e1

C-(

Y1c-2 r

l

Iolc d,.

\ " .2--1

,-Z*2L'----''li<.- r "t'+3

,f\

A, crg .71€I -\ d a*o ,2=bdt. - o

)

Pd {: Jo = O,HtAD 2tl8, OID RArtNoER i{AGAR MAS(tr. DtUl-50 lxAliCX OfflGt: 10!106, IOP ftOOR, MUIHtn E rcWtn Mu(HtRJtt tlA6AR, Dtllr-9. 011"0562998r,99'l'919761t

RE6lOtot OfflCI: ll' l{O.l't0'23r, 2N0 f tooR, ROOM NO. 202 t.l$.r r{cHA '15

Btut sAPP}rRt AsHol( I{AGAR, iYD-20. 965235r152, 965266t152. www.imt{maths.Gom

I

I

(41)

INSTITUTE FOR !AS/IFoS/CSIR/GATE EXAMINATIONSMATHEMATICS ry K. Venkanna

9+qJ €'dn =%

\e-ao = [ U,,)Cc+ '>

.z+4zo d"( +0

JI

[":- t#*

i +=1,

=-) Ab--o '-4_

HD

lI

-O

-^-)

I{r't

)t e-odx= o

PJlJ^

i o--(r-i -?)_il

ty

=- O'

z (t

-2x122-.=\

q"

{q

HIAD 0f llG: 25/& 0tD RAlEtlDtR ttAGAR MAiltr,ttclOttAl. Otflc!: H. ilO.r.10.23r, 2t{D rtOOR, iOOM ,{O. tol

rcP

)rLo^q %<t

(42)

INSTITUTE FOR IAS/IFoS/CSIR/GATE EXAMINATI()NSMATHEMATICS ry K. Venkanna

+tb)

9

)

HIAD OrflCt: 2518,OtD RA,EI{DIR i{AGAR MARXB, DIUF6O BRAilCH OFftCt: l0}t0G, IOp fOOR, MUTHER tE IOWtn MU(HtR tt |{AGAR, Dlufl.g, 0ll"t56t9t E?,99:Dr9?Q5

Vr.L"-

tuiq1^a-

{-rY

(=t-tu

2_

J

- [.'$o*rg -r] -r tE *' *] *"[ =?

dr${o.er-.-t'io-l t4uor'iotnr

( =?= ds r.tog (s +r)JS+z a-D\--\- 2_

= (s +r)

fl'* 'a

tru^rfor''r

to

+hi

+{

t{

Ga"n^1['--s,t'

-* [t's -s('l-A]

/rsY +r -9{

- esg * x-s

+ -ts'+t*)*-rsg=?-r

( [s +z)

utucu ls +t - !'i*tJfr;,,a*

Sot rfio,n, g* O il

s6l

U

b

I

--&s

J[

[, 2- 1 I

I r- .i s(s+z)

+t^4-

a*JV

t)

t*

**fi=e

Ha\

a

)

t:?"nrq

\a

tG,l

tqt{-t

dsd{dj

3k 9r'F = L

,nr

e_

+

Y

L^P

'il

u1-iJ

t

ve-

(

e-rn

tdd

L

uSt,(

Ptob 1".

b

sh\^)

-s'

t-e-

+

Do

u4

rd

Ir

)l

U1+ Q[=

)'d

rl

t

o 3)

dd

2s

I

i\4

Lt

&

e,l

L

U

Y

0

s

r.g

g

+

tt

,-

l"!t7

tO

t-

s

+

u

+

2

)

]

t

+

4l

s

hm

N

dd

q-ds

a

a

a=+

+

-+

Y

&

)

x

V

&

t Ltt

o

s

.Ls

lY O)

d

0s

+-

tt)

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I

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(43)



$[s+21'= c + J (,-#) ( s+.rf a,

= c+ I s"- ,l drg1-

dl

= C+ t s+ \ ls

Y

/^ s'+4Z

G* rl" ' s (s +r)L

5 csq-

+2) - ,JS+ C

S (e+"t)v €+a )"

rh=--+s (s+r)z

C_+-G+r)'

V= + C- tls G+J)

w

V t- [.1

Ltl

+ffi;+ r' [fir.]lj.l

[t+1= I + Cc-,1 ) drt- t

&ivu,4 t( (o) = t

-+ '( tot = t =

=+ l: I

I + (c*A)e-l(o)

,', C ic o.

,'. rL,-ra.-

\^-t-.1(art, Fre

\^{.t*^

"6 Co{^rt-a}^f-

tqj{ tno*( soL"f'roM Po{4ibtt d"ro.re-

c-r { Va!.r-^,c- "+C

a-:-

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(44)


MATHEMATICS rs K. Venkanna

,o

So.Ln------,

A sywi.- Jnavner-.ronK /or.e& of untfotrrvr 6uV dtodl

'i .,iuÎ uuX.ht r"r joivrh& dog't*heit, *a havr2 up vyOhe Co.hneh3,' Ohd *he Shary oj *Le 'Jguuu- ",r,S

p1.ra&reJ,

O}, n Jryht ,vtod" o.l-0n0, $r holui4ontol- ctrapna!. Find

,$* grl*t of *r"r {uy^t }ed,

A BcO r.il 0- J ?lurl jr,arne,,ros.K joa rned ,f i un ,rtodl

eoc}l ,l wl,rVl

'"lvno*h 20.,

+rrl^ $^.,- Po

H h tnd /lUV ofrf r,\ Edpefidrd

?ffi$ I.inf A 0

v

A

t..l

w ,iJ )uJ pen&d Jr*

and. C chon

!V

D

N*iu Jtnnw Jourur- Coanelr)

B,C on[ 0. A U qht o\!&

*np &lu hol.i$nrC arynolBo pm.uents *l,û ly:*ern frornCo.llop)i?. LdT U $.1 $ê.urt ir, *tu .rLoA BD.

16 JoIo1 r,odq.l"* +w 'i *hr 'bdi

AB ,Bc ,6p amd DA

Oyr be. Jokevr l^ Achrrq, at O-0

T0 $nt T tpe )halL ho.ve fu give Jhi- lylk, offiphatvrent .in ohich BD h,,,lrk chmpe 'So ;up-l-ace

,*"r su& BD bU Sr,:o ,1,m[ 0/nil oppoXrb Jor,c.l t-0) A ho r,r n ,in J+- + iryt- and- Q,Jtu rnr- gAcL /-BA C= 0 =/CAD . [Nok. JJôt ,*l^!- an6b BnC oi].[_ ery. dru.ir.

tr dbp -l.acer,rent .ih ,,:hi ch Bb .r/ J5o chanp-e , jNoo

ZLve- *|.tt lyd.* L ArnuLL Jg-rnrn$.rtrc-a[

draplacLrnlwt otrout 4C in ol',ich" g chavgU E+ Es.-T-f.-L poirit A s,unorlltr fifeJ' 0/nd &l^r po-ir.*a'B,o,D

HUO OrflG: 25/8, OID RAf,ilDn lRAl{CXOffl(t:109106,TOPfux)n,MU;lltR,t[IWVERMtl(]ltRtEllAGAR.Dtt]1.9.011.11629ttr,999919rQ3WH tEGlOfl l OtrlCt: t{. No.l.l0.2lr, 2No f l,oo1. l@M ,{O. l0l l,(5{At{CtlAAiS EtUt JAPPlltl ASHO( ,{rGAt, xm-20. 9632351152, 9652661152. wsu.imt4m.thr.com

(45)

lfir .h,rn6*hl u] $u or.o dl AB ,

,hB" "ul'u,lr $"r *B* BD

hJl hous BD = ZB0 - 2

BC,CD

= 4t"t tong

and Dn da nol

= {-0Sino,ABSine

elrl &p*h .J e ucl^, 01- *h' PirJ,l B , C and D brlo,.;

Sn _f x.d" polnf A

A 0 = 2t[nbo

tunc\ dh{ qfi, ,t C b-r.Io.,r A =2AO -- 4OC0J0

Bv ,i4tr ogurnclplu "f vintuot r-rohKr u1 hav.0lTS [4&Sino) ++ti Siraco]o) + ailS [eeCo]0)

+ t^tE Go-Gto) =o

0)L 4oTColg SO - gcWSing Se - 4Or,,..tSino Sg -4{US;.-i0

0h +o lr CoJ 0 - 4t^J Sinol So -o

0h TCol0--4HSinO -O f "..s0*oJ.

0,$L TCo)0 = 461SfnO

Cor0

B L,.I in $l"e pod}ior., oJ eqrr.t.nr b.g.i urn 0 = 4s '

,'. T =4hlton45' = 4\^)

C tl"l Jo*l 9.0dA. ,



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'APPIIRI A'HOI( r{t6a& xYD-20. lrsttstlt2, 961266U52. ws*,imstlmrlhs.com

(46)

INSTITUTE F()R IAS/IFOS/GSIR/GATE EXAMINATIONS


gSo-tug {t,,e f lotptw

L*-{'dlr{erre,^,lJc,-l cgoa.tr' on'o(\ I* dl = 6tr*f-' *"

"f, [*.').t

Xt+r) = ze+ 'f1

tt IL.| Q.t) = e-L :)

o,,rt\ .t"l- Lr Ldt

.l{,- (lrr)b = Dr r (t*.1 DP= o'b,r'),', Q= tD,to,- \- u Dr * lt = 6 e**+ s\n 3

Tl-r Ut"rr,I!*t { :

Df-5D,*6 = o

+ Ll- 3Dr-2Dr +6 -c)

-) Dr tD,- \ - LtD,-3) = o

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0.

f.t: Lf a SinZ

Q,-D [b nG

L+ 3'tnt,+Q,-D(u-r) Q)'-D(D'-3)

G &* + -rr^>r1n e

Q,-z) Lu,- 3) DrL- 5b, + 6

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Qr kD -r- 6-5D

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INSTITUTE FOR IAS/IFoS/GSIR/GATE EXAMINATIONSMATHEMATICS ry K. Venkanna

Sin ?o 5 t- D')

6L+Le- + l+bt trnL

^LY-62e-1 gnz+ GgL

5 1- ult+D) e(r-n')

: - GzeB* &_r- Dr

-- Cors> .l

' l- 5-,1 J5

- 6ze2* * \ SnnB. + CoZZ5 tlt l+ \

-g** *I9nz +cr&^lo

P.I = -cLI"Xt*Dlt+*) +j6,1\tt+rl*S"[tDt-l

lo

.-lo

t

q- C.F+P.T.

t c,,(r*f + czt *,)' - G6*rj kU*-!-t

lr" LbA qt,-At sin lo6tr-dl

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F-(nd. ,$nl- *rnu\o,I- LncI Ainyr.[o]r. Ao

d.i#ehtv,u[i- e$uut-ron g y Q- VfP=U

ilLsohiy ion ? ,

1r^|ionl oJ ,the

_ +(z-r7\ , r-;here

ey

?= t#) =-+ [?) -[u-l,''ltz-YtJ

sepuatry uqru&hld, d-x = ! 1 _t- [v{ 1s_,75'r,

r

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dr ts-y) -rG-ryltz

afmg,r .r+c = * iJt o_7lr_p Dr,lg

0L x+c = + (fll( 3) U-y)Y, + 2t3-yl',J

( 0rr Jqualirr

l-v.'l il 0. grnenol_ trohlion . ThL sinxulnctrh be So,lnl by '$r,il ,lut'

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o 0x&IY) etr,t-lhilA 0- ?uedxehC equttfron JTIfi P

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t NSTITUTE FOR IAS/IFOS/CSIR/GATE EXAMINATIONS


l.e

0t1

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sqbtl)'iuhion ol y--o and ? =o in *he Tiwn

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frXain r-!, =o Xiuu # = P =o

fubtiLult'ow a{ N.-_t and

di{fo,rth'aL uguot"ton " J A*tqttt

llrnte il-=3 tt -y

p,o in *lu Xivrn

i*,Jinttiltt" lob,tt"s*'

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ss

{ac NKflT"1 s S rF d s fr 't

E ^lrAlc t\e 5..r( e I v n

S o th St^rrF' ,t-1 (." 1'rt ,

Al )^JK Arn 4 S 5 e_

F Y t.nQ.fre f f. t= IL

G oSove t\e lru ptam e_ ?& e 5 ?

t

acr6 C $ t\ S\^rfface SaI 5 +h e C,^rQ e

TF Z h) tryl o

,i& f,, L=o

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garrorn.4'ic eofb of c'

hx Sbko$ (rrorm ure harye-

"0v = II V xg) 'nds

5.

= S F'dr = ItYit(

/z--O,o(t<zrf u\<-

, - 1*z)l -'d K] ' [i a* tjal *rr

L

=st_

trd*-t 11-r*z)d6-ailt4

f '.' ott c /= o&da=Dl

-s ! tuo +"a6 )

L

--fltgt+*.,)ontra . - ^,.-r-\ -r-aco(t tc^t^st)] Al

= fltrt^t[.-"S'$n46$\or(

= oz ( g*- - sr'Ft) dt(,

,(a-

Ia.,t

6o z(

LSi nz'laL

r.D

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INSTITUTE FOR IAS/IFoS/csIR/GATE ExAMINATIONsMATHEMATICS rs K. Venkanna

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lcce,fe,ra}''01 : o"

vdot-r\ = v

-LM.'l21- Cr\rcA( ar^& |,atl

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t 3E \r^rleelS

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INSTITUTE FOR IAS/IFOS/GSIR/GATE EXAMINATI()NS


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