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I.Lin"qr rAlsebro Irf atrr"ce s
Pc.o P-e-"'l-ie-s oF Deterr"inofe :l--Lt *oo \,otoq - ec colr+ms oE a nna-tzix qde jdernlrql , lLen
*he d efe,r,minsfe is Zero
o lv_
2-fL*roo-troo-q. oft, -Ao|t+m:qs -[E- A rr+,qh-ix a?re- i-t "U-.og.+c
A. tb *hrrep roto-q Or- eolqmrn-q oF q En-a#X e-lqo@
*L3L_L
n o^"&+.t"g+hq- Sqrrr orr diPFerErrce ob trno e-.terqen1( r $o.o{t- oorn --be
__a___:
qlerory
o_
g2-z
r - -QzaL=QAg- q33
'LLo uce.nr -fri q"nlgL.l qr m afrix CLr m) ? --
Ait *l+e- elernents a-hove ,l he b{nci p*.1a
cLio-6on.l ls co.qed \^is (erooo
All *he e-lenneyr*,s he-.loo +hp ffni$sf, -eh-dlqg"r*Lji--r€-rco + is eo.ued u^
h..to+-e : -LF,-q rnc.h,{", -fo_ d+g.r t",o"r ++""tS"tq" f t*pf.t@ -fee d"+erhnb.f, t@gg-P$oCgel olteraon*l qene-{ ,-
L. -L
q2- 5} c2-
R + Rr - Rr -
ecL
: f er-u) Cu-c)C e-q)
l$sos-l?cle---
-> D^ -) o-t- o2-hL o I alho h -c rl-h n1- - fn-k) ( S-c O I hl-r]. Lc(L
3't:t
ab c
Too cs-l qroros q r. ?d",olfccr,t
bc q bc I
cq b cq1
e(b c- abl
clicu DlAFrY55i6lGMA L.CONI 978423.i98i
:fq=kt. s: ll*46 ; -t*.S
" -5XIJ
is n rt Sx a llq (xI-
- h6-- a hc'
+ L;, tl , -'r-LQrt/u __arrlb 11b-
Rt lfr-f4t-/ c LIc
x/u *LlaLl" L l"
Ir4 g_
CrbC
Qz*Cr , QL*cr -
!t]1.
= c.bc (\* {+Io*t)
rtb tT T+C
l171
-e){2(t-6)
ek Al+ern'fe t'ne{nnd -to [frrd del-e'rholnoJz l-GnIB f..,r gxa .'nrtdin'-) CShc".+ct,t nno+h"d-)-
o19-
Cr) tc*i{" +ne -eirsLffi +r"e ;dS!t--s!fu.
C2) ri'h"* Md *he Prnduc-l- oLt,e,g dieflo,n.J qmd s.,l"h-"^!,1- -5, 6, + Cqgorno.J ,
A :o{- q+o--t2-*o-l * -L ll
--n
-- c ouc-d ( r+ h ++*+t-) =
-e C8ts
j 2- -2-
-r 3 o
o -L I
sA-r : Adi a
A
-'. A-\ -- . I - I "to.d _ b c [
_ c-bl*l
rt! El
Aotrc.ho:nglry +hs elsrner,* of tsldjogonel qud chorrgizrl
-8 I
r=tpoj : coll
--ltno.r sE qn e,{..n"'f s1-ft^, p - irr :o -S7u,q:rl rm.fA.t--lhe de*srrdnq4
!E .tbe-s+rq'rp St'h melax lo -ll)-hie-h *he-E ol o(hcl +l^e--co11 oF
- *he 9q-z$sdqf-ejenegf, L,sr -fo- bp de-.le fe-d'
* dllAlkrtr"tc. Clhorfcr'.f ) rnel-hod --1"-
g!4 A-) I -*Z
2gl_t
-Q_-t,c5+g loe tst +€o- -QoL)'o*+',. I<fuu{ s9 de- an "l IfoL -L b&
@tej{s{=&g --'b^illI!tt-ontd - 1.
o+9 +L*\a--c -r = ''L
co_s 19 Sin
:S'il0-
- -&r*!lter- oderoF=lx3 or- sr ots E----
Ad'.r * : frrauslose oF c-o!sc*o]G fd^le$x
@n--9,une,l-&- Dl1 mrror
r 2_ 3rl
51-+
t2_JI
6
@ -fhen n';+<. +de Rest -l-l- of, dekrrrlh*f" o.t' +oP'
@ fuen Lorle .iLe- -ls+ {oo s:ocos on *he oloor:r a?{e oF
ept_: pLolr 2- o t-
0 -- Ther. -1.".,8e *oo t'oro S ,su ece ssi ve-l B 'crfi d Pfr''d +heleJtrrni
-----.ncro+ Co-b * b c.t in hzr' dirl- et-,d +hen to#te hLrezdcq&rl-
zz
2- o- -3I L -2_ | L
9 -z I 9-z_l z ,-z I L
-t 3 *t 3
A = +3+o-.4tot2=t
I
2- 6l
EJ2-
I6z-
I
uD
l+ 1 i\\23lrl{2
Sr9:2tfl
I
2131
A L2:+€ I ls-s -+-t2-_ _ =_[_
-,s-!rog
os0
R A M l( AMA L. C H O U D H AR y 5 5 i @ G Nr A r L C! Vr 978. 2 -l b !r 3.l
: $ !a S----Cos€
t i:-i.ell--16lao:soF o:-d,er (rrt) qre (e-lrQ+hq
- *|.,ex-e L--e4t-ecs-L brne -AoL{gn! nr?no?r o@ ; i5 e"isf=
-r+t c*,f.q +t. r"" e--IqcHr-qa-d-i^--4emoJ<-{=b6
&AJ- r
\)_=_ ooo
- @--rf- ll is - -q-orm - u=eva ,nnery1x--;*len+
SGI 7,, J-
-2- I
-Z -3 I
Ll
> Sltr Fr e.:< F --e---Srar9 Cos 0
6nL o o*C\
Oo
J _ l..lin w Rq",K oF@zb ,
L3) "af i br{-l,e?nx x , {*o?fl
qr'tr r\o! - Zero hq*H"wf+ nnc.*air or lderrl.6
ts t'rn q=F"x oF oc{<r
J (r.) =r)ex l- s(rd :ar S{rs) : B r sC 1:,) r s
8_ _\ .S (aLl_za_5_ -6_
l2 z+ 's6
Alloqef ene norr <e*o *hen8(r:+,)_>2-
-Ck o.j-l <ez-o ,_ *\.e',a, ( nl
lz? o f 1'l 2k+
Oheek *he-f<]- or-c.lenr , ftq Norr -z.eru.'. _-S(A\: Z
l=ffj@c}.e"lq oorcla-s zsr-n, *heo ele@ -
Rg={te* +e"-}-jcel:z
q (A) < 3
tr_tn_n _<ere_-: S (a) =3
3
( (n)s g
Rs BRrIZrx;
Non Zera- of order 3x3---
to I e-l(v> \r 2 s[ +
I s r i l.r,
L E L--rb l,- z + z ls 2 o -r[ -l a s tLr l:!_5tl-Ll lr 3 + r-l
4 = XE[_Zsrro
I 3 --9--:-1
+--5r4s &-1--5- lL :/
A : Zeryo (edleqs*-+eauaqr
NoLo- ,
cher.rS .frr 2- or"le^- ,
0..lon {ev-o
{ Crrl =-L
I -1 L,
tof
s2-o
4-!: I -o-l,.tx1
2^ --3- 4 -_-:
-l * t-.t o
fo"Jl
lo-(ts) I r 3 \ 3
I
la s 2 3l =>3(A)<sll3 q ll3x+
llp O Chee-h &?r nr z oyd e.L__-
__ : Non <e A', 3eh
R+- Cr?++ Bz+Rg)t23
Ooo
2L
Rz * Dt't '
I 3 -2
-jrrn ft'Roe g e,.l] is
O+xr',r +hen t +ho.^ it is djPgaeutL *.. eor" Pqte 3o cno lng
-
'rt .t ej+h"r bU- errnJ<.rekr\nees 6r qlg +u9^n3 '
!r be Ze ro
Rrzllti
, 5(q)<{-
R -- rn r pRz-- 3 R,-, Aq .CArl flrf- RaJ
- ,f.te,LD- ,-^ Lar,a* _+-u--
_lq_ a s= -L ls
' I (n) < g
ChecK_fur_:.+
I -l -2 -+a 3 :L -I
l-- : l- =')-oSo q 3 1o
onr,n, I\.A-.rOlfFAqrq-l l. i --V iTcZ)\6c81
ej.c.', la\qr., ^n
fr q.J giuen sU.+** oF e1*,0r]c-Fbr)-tC1 z = drO:rgbz: lQ:-Z = dr@rdg
( cseP+icie*t
nqe.l'"tu)
-- hos I so\".
- tn O'nsisie,.'f SUAttro -AetL nn S o\*,
(h p're-G--9tr-3{ae-}j {CA ) : N-lo' oF UnKno".r.,-re*ffi@__ce ((+e) :((Ar- No. bF lqt]oollrs
re;sc+l-IUor^ +1"^ gts+.^', ,
t^
OMq hqr rno 3o\*'a*14 +o b e Nr\ cons\ ste-"{
--- rlnen +t^o (dslerrL-a--g*fo ia--h-M
-
A,"nd tao"rjro J'rr,F-n.Hz- rno " sP Ssf---$:r- l'noro {'uqrn -gqll
I3
I, Tne h o. oF Sol*.s +o1r
|Iff + L =3*he S 6<+enn oF eq".
x +21 +3z-: {-xt tSttl_= 6
@o u@-L .O2. @e-sottl +F - eil4rrr-q--fts*orr hq"e rr&+re sd* 6 tloL:nilr Le
ex E !uCed ']eQqdl,@eiakteol*,]Nse:G +e lA8l L hn"teqsle- nq..f,i2l .thtiulqs$ Col* mqs+llehelr*de{
(-Dp*rt tryc'F.e e.lennacf, 'rrero ln-lo,a+ col* ) -*---- -@ --tp,-'-Ag' is s q , q* -oJ#x , *ievr lq c* -Lr th , nn q st' Je-
Tn cllLqJ-asL.
- @ m++e lqs{- col* i. --erc-l qd e d,'u-a-freseu *fug =.td.trl:Lhsrcm(- -@--IF-: lu AR Ro{rrr l< ( .
-=s!-f
e - A - '--- ( Rs.lorn €le )
.'. _grcluole \qr.* colY3r q
q
Rn * Rr , Rq- R,
AJ_?_3: ?
'' u lr.rs .t F S o tljs---fl qi' qr{ e
_ ConqicBrf---2----2t * 2-J + 37 *- 2
Lx-*z : 3
sol* )
: I I *'2 3
e o -3 1 l-- 3tc{srn5le
_L=l_ol <e ro
R+ + zls
Lrnlgue s *,\ *
ll
{o --g 3
\l
lLr6Tt3)+Lz---)r* )-J = 1
loy+32z_ -Lax -3y*Z = S-
A r14SoJ '' J- Ag = S,+qqlrln "fix
lr-ri &nd o d"d )Q- Lp- -9L-3
_o_
g__z
lo q -2-'l *t -3
Rr* 2Rq , Re * ERq
32--1 2 o'1
--- {+e e S- rri*h h-,, lns r--i$- freselol--,*. Frfil : \
O,lo 3 -)'L -g -r f
R',-3Rr Rq - LR'
t-1€-L. Chec,.l< *1ne Co6s
' efeney 06 *tre
4lc *LJ {-6 z: Sx 1- J: 3Z = -llrx *'?Jtgr: z*l
S3s*c,rrn ok e?"
-l 3
Rr-+-Rr , Rs*-RrI -8 I
S o ogb o o( 3X
@ jqc--e -Ld-e -qLff .rrl
X4 :!
lA\ s, e :o
[1_e-$o_o4_$o:r_ s o I v ir'.rg _gy s+ep _ ok € ?" Jno,.tr'S ons_ -S _o_L: [: __
O )c-:-A *r B@ <rornrr)e-r's fu..le .
)c : A_L_rjy__: Az x = A.zA A A
OXlo
ta(3t' tr Y')qt-'
--llJ-32: -lrlq: 3
-J r:lI+-5-32. = -r-
(fnrc-e hre- h,q"e- dtsc-rlgsn4 fr,e cc'r") s isfrc.ncff o h o, U Reo{S;s{e,nrr o b ?4,^
. o. S o\* o-l ocoy Re-*l h o -J
3ol' : feel ; I
CHOUDH,ARYS51@GMA i COV 97842Jb:lti
bg1_ar]<-_rr o , oF -s o_ll.
Let Z =A, fohere k e R, l{
) - tx : -L
1= l<
t:-Iftr ol l$ vo.lres -uE lri tJe' lL hqra -di![-. -so l* errnql- tLe.ssSolt qrp im Eni*er -- - --
-2, Ch rck lfe-,fo-nsis*enqrJ
3r-\ *z = A
-t +J - z = 9
l-l__q_ qoo
_r_2
_L_6 ,
,]
Clr,
/Al u*, d n{f +n) : 4
3(AB) / (tel--
E+ 5 + a_;_6I + zJtAz = lo)c{-13 l.Az =v
lem oF@
-l
Ru*I
J- -l -l Io I o lgo 2- o lso 3 | lq
l'rCql Noq-
sol* Cb)
Insl:un?e1ue Sol* Cc) Iflore *t",ort ona So\*
I qs* Crol* li. ekc lt^"{t"fSx,t '
l. 1
)1. I
I
3
,\
I1o
1'r.L2-lt 2-
-Rr-lle_3
B-e =
__t
I
,\ -3
Cqqp ff-P X*
Ssge= 1-- ---,Lp )-3 =o ',=-.L:lp_ :_o
i{Ag)-;-J.[aL 2- <s,t'\orre fhql cne Rol* , iA-)-:,:*-:r-lq
OocalTl- ? -Ip .\^q *o
J (+B) = g (A) .:Uojque (o l- lE ) :L.3-
4,'TI + ?r ? )_z : B
2:c* 3.t * ),2 : lrPol- ?,- A8--: L? i
es qnd- Li \reolorq f --
1-L*f,'q''Le--.pne g+lre, m@ urrn cr! l'b.
-- I to sol* -iF
^:g- - xf to
2-g
& l\rq nneler ' ' lB
qlr^re tl^oofs o F the ta er ro Cs.fu eC drqra c leu f, -. lics 1oof if, - L**.sd roots or H5;ertr re,-{ tre on Pnof<rr
Yo] ti t' I
fre" y@lael-oF ..- = I XrI&
hlhfch .e+={d"rr s+er+. ss,.U-sEas *he'rnot-a{- err ,
[n - f fJr --eo.ma-spmdftr:r
e+nVe c-to^r of *tne nn otat x
.l.tat1 l:iS elttal {o
.\oA{s
r{ T
O rixe--pro.o
+o I ts --ol e-+enrroi"nqrr+:
---€tr,= G b/--_'-"@re @,Lr3
AT: A
.\ : l,r
e--S-- Dg-en--\rc'Ju: oS skero -Stsioxae]^'ic -lo*,tlX Cq"ll -r pt t^relJ irnafi inqr3--OS_-!e-lpS:
(ar:-A)
l5A: o -r
1r ot A'=forl, l-t o I
A7: l; j/4T:-4
---@ o're--E-f-er @ lo,aer A- es tr*Pgr {-*t*HL
{i %+sl eJ@4- E'e.ol rqlqqsrz sl
*a-L * f5.n-- YaJ t -es ,
_ A_:- l* 3 , 6 &€ern' hq.lqes
- * T" pio{-g+* -ure,Ltres . N-errea Cso w o*f }he Lgs4q oP: tqsHx
- -l-r.en *Le Vdller grl tct-ot',g'
r3z\
A-A:It,
): l,e-{. F$.d \hsrdsen -trelre oE sa=sLs---_-
--- l..to aee4 jq I"k" *I-rkng&g Fod,,* [-I
3e- I lg 2- I
I I-'l--23
II
2oz"slg{c-l{:-*- s,s
a lt, s'
'-;- 4n q Shot^l d lhe" ec^te oP Qoo,
: L- c---u-I
4xr + Arg; o--*--ru" mgj+ t""-+ {oo trnkrrnu;rt
-'. r r {- r a : o ---
i.l- cen, he sol,reol ort L,qi[@T r ? - I" '&nlS hh€n ellrof, +o Zero-Ar-h-ile-
Xr
srrl-rl
l-l , *rJ T
)c:f-f l 0rLl ,-,]tLI,l
-*-=' r 2=L I UXer,-Vq.f,qes
Ve"lors oF *l,p-_ nnsL{x_,
crgonoL@
>-6) :-o
Ttr A = 6
\-, :, I li',]
LI
(tr)
-)cr {- 9:cz :oXt =4xz-Tr :xzt{l-l!
[.+,-rf]
-0
o . --Ti-e--ehqrqo{qrisffr eolt olli-c--mohjx :
- @ g, r, B -G_t, ", t+ L@ _adrlrQCl r , 4r j_
2_
A =__l_q8 .l_qd_t_4J_:_LB +28 * tq3
"-''- A : o, ' . Avrs Prnd tret eh ould h e ze-a
Q0 1, Lz, 4 (Il_O*2,4- (cr -r ,1,4 (_dl_,2+2r2
4. rnc,-hi
-(q-r- 9, 3 +si , d,=-[ --- ---- (h ) *(Lsi_g-rL_a. L
--cc)--3ti.-l.i*s4i-- L{d) s* -t + ei , *t -:itn ++fs fl^e,
the "ti*$*r"l
tf vo-lt,-agle.rnentc .
Coro e o.Ns o "f qrnq nn c.l-eh e-! l,c i++'
L *q 3
-6 T -q -6
9_ L S
LZqld *r'e E€", re c*o 'r ?qe8s Fi"nd
1T" Xr."n I,ta -+ e-
I
lo -11__ 2_: r +_:_4- 4_____
- jorlL-9-o-lI: - *"j eillUqgoosfgqmcd-
Lr I
5- 6xg =O --
(x r + aAz {- j)c" :o
S{r : * LXz*![t : fCr
-"12+gxg=o
2- --s-
23
:l l1\ ="
?,2_rl
0
frt):r -elolo
:6 rr+--
*lLXL**2JQ--:-O
U rC, = 2d3 R A M K A Nr A L. a rl O U D n A RY5 5 1 @ C M Ar L.CO M 97 8 4 2 3t 9E,-)
.;u _:_ :(s _2- tl
- -6tr -Lz(z) -t- 3 ( rr) : 6
6
Ia : 3r Iq 3r2_ 0r lz-
ll 6(
Por ):-)x
Srcr L2x*=*2_x3:oK x, * Lrz-
_ 2ct. : ler,1 q-
L\J, fionn the hn.,trfx '1"' =
I I L
to o
,\ : l,t-13
One ol *he E.v , tchi ch on e k- ec"ogr der leo si l:vl -
*re Ev
Col {{, r, tJ 1 u-(tr) [ r, r-+Lkf+;-r , ,-J' rdr f s_+tI
_-tr*1!C3:Q __ Da-}tr +XL=oTz- =- 2-X-:
oL3
1[r = )ca
nr Y_ t l-I:--a^T r - 'rL2'
-
lu--2_t
-- Lx, , r, , {.] . I r,-r, rJ&. The re-o-l rn- n rns+;x ,_
t /..1
i_:J
Qr3-_- :
= - (.j!2-
Qa t Gar- Oe: - ().rvr
et
Qn, Qy''z- Q'n,r ; Grrn
11 03nn
2"s-l_ennq H^
)L-B\ - 2-S -9--
ffiil:l =*-Rrr + Qrc., : o
Ir:XzX -g 2-
D(t - !C2-
I -2_
)C
- 1)CrFUIc :O2)c' = 9Y,)Ct : )q
lo-z} *s-^+-A2-- q :gl)*-lA+-6:o
S, Tt,e
sg'*t-
f-o-rlz'rI
fn**rfx safrE e-l-jt oon €Lurat*zr$lte{
2L\a,e+ \qce > ,oi+r'''
t" ='lq + 6a:0checr '
, as ,sl - \ 3s p-Q 1 * \ e o I =\, 8l lt lql lo'J-L.:_H :S_ = R, H.S
Or , -6-L-;, lA - AL6: a-r ::1nn---!a1t-l64-l : -lf *A
T:rLl-l-u- r -l-he *he rn*l = ?(gt o,r Ct t r,rl Ccl - ro Gt - m- _
R.ople"ce ' r' hJ *tn s t_ r,a0_ A-re --r1- : -nn3rrn-\:-nhl Ia- = A-t
pq-l * _ rnl1t LrJ 'lo' tp a e x-? -,,ho c,f Jx to i*n e hor e ?* ' -
>"+ 2'+al*-l :o --'{-hetr P-l = ?
L. lf *lee Follo$i nS rre-Pyeqelh +Lq ery- oP a{ l-fme +k'n +he /hL--
(o) o,o (b) 3, d Cc) +,3 Cdt +,d-
Cot f}+P+r (h) PL..t_pfur \k{--(-PtfP+2-1) (aJ )rLP>fP+lJ
8r t-rJ :- 3 2-
P*+ tt". o?tiorr 1, +h. zVnCh"ck LHs: RHS
6x +o + 4J -g:-- r) =-o
.o 5-;-tS a.trt Tfl Coa"rsr\+<l,rf sys+<.ro
Rq.rrn oF'ff'l+iH 3en Ve cfq6 oP *h e roc4?-tx.
Rn M KAM A L.CTT OU D HAR\ 55 1 G,GM A t CON4 978,i23a:!i:l l
Ar :gposs ihle
Gi\ et'n
oto\t\rL-
z.h'kOF ecy
(qr 13. tnhich
q 3x q tnotJx cnd I\- *he", *he hlgh" s1- P,
GrL (c)goP *he Pott oo\n3 1\
1s o q\o s D
looe-
I cb) \ol Gr ]I lEi tL -1r\ Il--- 1
'n i- -- --l
furrs ?- A P\T : -
e I
od) I
o L-1
t_l
- e " Tt^,e T.orw:x-e qL G nt]o+t{- ,
Ar,s:.-
oL
2Q+k- Rmntr op
J' Tbc Lcnn
of Non -Zer-o e,lerne,,.f, fn *rre dic.3o,-,J
.-hnix
0 lrola
r t*) {_3
--lnl zxr*o
** A+n ;--?
0r-rh : 2-,1 a zl : 'I6(oLs
(s)la (brez (_e)eq .-fu)/rE
dektr- rn jrre.,\t *hr* r-u-si-
Pouoe'r 'n' rn.{st be t,c\-i }k$ ,'-
lsL. _lgfl- 23 I*\
--__________-___-- _
- e [ro:\
3ot2L:t_3 3
L -1L
-9oJ- I =-
l'lL
-ir [ +dj (nqia; | =
=(
Adinln-l
l*l n-t ;
rt-l
2-J
lnl(''-l]' I
I,-ktlal be- q gxs rno#x eLrsd lB' E j-r,,n ad+ 'ql qqrd 16:&e
- lBL:6q +herr-$e lnlrj__91_L*L- B= Adj_a
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Tt = 1 - l__:_JqSzoGuo TF - ; 4o >ctr +- q -xh t* er rcTfgl+----. .- .[_cr,nn yL--
-' t*our _x i4__:tr ]
ErQrl C?z
, r9J--p rbz =?)AL &JEr,lef* nof PossiU\e'
-(0 tF K= #z31|Y:J-= !'z i' hqm de8o-ee: Lx.'+y
NPTg t-#(.)
TFZ( Al*crrn"*tve)li n of oJgebz'c'fc
n(") ti *l3elr€^l'c arnq
Y ' *her
qtrdt.1OT) lrt
Lqf, S)c qrdhonno sneoqs oP de-saT€e
t()! Re nn orre Log- &ronn Qqes*r1oo .
S Cz) = e3e *]+51 Rlr*o'no''fl"'o*"'t1o'T'""'l':"
- S_t-e) tsirr r:.x:tf: - -Yt E 2=l :--L2qJ_ _
- r . rc zx *- 5 ZJ -" -J':Sfin< - 4ero3 */7,-:Cos z
xI<_+-- trla:JAJC a3
[tora 'rt
x-y6
bx -ql_ - -- sqc3z -----eesz
APP@€r-Le n-$eqresq- t -'
qrn{-}-, tLeq-L9-c)<,L-J10! =lDzt ---Eror aJ:
'rr -l
b r/r aZ^ z_
b3
-QrD - lp
-\) )CZx -t_\zv : ?____-.--'
No+t=:4'^ J -fr' rs fit-nr..lq {'s c.PPd Cq.ble c.,lizeClJ i F 'Z' is q
5Z
Alsebre*Jc
&pc-1bqs.nofle:^
- *aem *r4-4 ftcal)z- L5=)+ z- ?
---i )c:-- alqqJ a5L
S o l- tr : t(+-rlxz-Lt:l x Z-
oJ
x+! D xL__ *_ b
---&l.-lt ( t=-Dz:0
_ *hen,xla< +-z:c''nz.J & =-r@___ bxl D)ca) ay'
+--cpe$o+ion-oF Et*lerb - i*#yb e-cq qse +e.n Fqmc
*-*rrern- Nel
, '- A \terrn#ve_ hne*hoC_ -:j ___xe+ ds--= ctT, e z
Dx DJ
_-- X E24, +ag . | * yA.1 : 2*Co-t2tlZbre' ,b x i: q:_ a)C
--*-4-J-L.1 -A-<- = tlerszz - l) :. a_a) 5' 6rro3 83
I-bi .+lE:!.!-a = LClss-La * I) x oa,
-llqL - -!4al Dr
.'. e1* CLI , ,
3€
- -&qdlEt*r+"1"
a (: cos tz -t) [= jg+: UlL )r a:Jf3t. ' [z
LCoS 2a Sizn ZZ\=$in4L Jr (zCosez *t)x sio2Z
slz', *z * Sln za4 - I€ z; Ez si", ! tU - y,-+q,-' (+)l+$een-tg *here is lr,,,unrx'e- furc* *hesJeSrel o+ orogte L<u*
pgmc- erc-irepl--Sxisfs'
x h< .+ 5 dZ -;-zzDr
--5-. T-P z : Lx3+f)dlb f,Here cd-ss cI -cx-Lqr:,en+ie,{x zx t#-Lt?iL fuqeaf. <eo-c, :
ich, \n:g-@ 4r,,e- Forrrqor d++ler" y : F (>c) or r : S (Yl efc rc r"Ur4-9tra-
I. v:qd-_'l-t'rlcdr : er'r {- b
Iqf
aJ
dry : 2adxt
9*3# fh p..o i" tn o df gereulfu-1p L!rh! dedrrehve-og €xrquf
wt-<rmc ,
--- fo-J Rr"oc* uthi-ch-li -rn of €xl]_.srt js sqr,4 +o te !{{ltmrrlci*-n+"nrc:l
,l-o- G >e 1+ ztuelr + bg;L+-O
- $* t..rh L"r b.,rer Frrrcl -&- {loflrcrl $nc-A:{g-en-@
Fsrn{l
sv+ o'o s ho1* cl*t frirmuJ e +o Ffnd I 3+ olttd rc^f-irre*
b-Edq bxE-r
b_)
.t, - b FDX
bF ( lhrc tahlr]- -bu
L. z = FC:c,J) tl.'eo_ffoslP : Q4 - ct,=D<
Brc E: 3vl
C=,rt iouo"
-4'dxe a,t t
fujrqre shorler$ furnn*,{q *o l}joC Mo P ol.rt 1vr Pl ie-it F.ro.* i
( Pq'6-h"c!jqr R^raortrq)-
- l F c4x' f- zhcc5 + b3|- : I C:{'} '--,
d5= hLclb-qra Lhre*rrr& er.nn pl et .
-
l. L:cz+ yr5 J- g g-e_t *CI:ar,h:3 .Zhz_!j_,
;: - ;ll^3tL
--Q:l , b:L-, Jr =.lL
Cr-x t_aj
*-1/ u k 8L-:-_r---;-r
55 ftrHe--Coyr -+ :f, oSc"e@LC"-t--f-r ave -' ?1r* 1rs [..-l otr - ze,co *-he.q tg eC"rn rnahe i+ L otr '\ aPFt3 *+re Fo-rn q q , b.,t Lf *t"r e
Corrc* f's Ze'r-o , *t,,'erl N"1-o'+PUc.'t(e'
C:ct- +l
, .' h :z !'l
S s--l+ thor*cq frtrot,^,{q (s*eodqr.d fo"rr)"1o.{ ) +o Mrd zn"l{ea.h"+,rr":
tF'qxz*zhr3 *bJt+ 2-g{ + 2-B +c:-o*hen,
doH a qbc *Z-
,.j
dx'? Ch-:c + bg+fJ3-Exsorrel-
ot = 2 - h : 3, C t l,h =L :--3-:-l JttyPE'f :l ""*''1'r ioN'r e724/"'''r?'1
Er_Uemg E1
- l. y'-*-eLJ qleo .: E
---qsLS*ffn dqr+ .Srrqq^lq-
{ r.t _c_ryf
A,JTF 'z' fq q &nnc--. irr .rc'qr""-l 'y) qr ltqte +rye
7: FI
-lz
- fo.{-o-] olq,a-rve.l-1're oF o -eoP,'lPcsfu ApC-----
- tP <: f {r-y] Ai *ct) , ) z h(t)--.J{en:tte lpt,{
dervo+,ve oL Z rr$ '+- (s dennl.d' bg qli, cx,d l*
d< : i:--qtdt
eL lJ-g"c.,"q Jt b e tcl-.st "-{s ri'tqhlre- q-E :lri:e----.=--
in d"Per+{e,..ct- r,qdo-Lle 'C em{!-:-- - -- -i\/+ - lP--q- r- P -(T-, J ,*)Y'--'*rn eq d!- * l*q
c{t brc. .ty +-'h !l---d3- -.{t .ba qt
*y
s6PggHs-nu: -l. Lt eI sfln: {z ^ o
o{t
: ex slnybatI3:,. = e*{gs1 . qlre =-l*, -dy :2tal qt ,t {t
__ gi = _er sho} T-co-s;,{r*L
='( -f s i,ny + z t'c,osyJ -
:r2 a6L -1
I = e,t ,y = C,tcos ?f ,z : eztsim 3t - -= -- d.g ?3"{t
s!f-l.- q{ e"t)i.L )y+ Coss 3 inBat
, Q: teQ-', dJl = Be-4t'
dt
-
' . gz+-ecr'.?+-sivrttJ . Lelajl:lfofe;--e.*tai-U = gzteot,_*'
*'. ( d,.{ I : 2- pzt' cos LJ--L-qL*( *t c in a-t* )
\ {t/t:o -
L
=_2tQ_X epSs _ l
z 2x lxfn')
qt
L-
s+-s'V::c3-J.e'
T =f,Y=t-,2= la5t
:/I
id-q\\ "lr )t=t
- ---gs]:-h u: t3- t, El^St L l.'-eLo!* :L{ a gc
t
;_6x32 i]j z-
-6, CF F--=Jx
--J1r4.o }3i =z c..l r->2 r U :JErc_ay _ '
--
-f rn e *''od j -(c\* S- D-Lr:e+Tl e dcrll = n.xH ]
b5
Aq_: (J-) )c'
e2F __- utbxay - +
I rq:-lNq-l- qkqL= q] t"g q\J
bx
q"F = e_y_trlS*_rq3 +_Dx'a3 r J l: e-x lx o 4-.t_
AI
--g' {e--q : eryl -J:!rian----r}r1-- b xd 5bz.,
..X'J L(,
/ a)ct--\\J
@Lqil.4
e'l'-J '
y=,[: x'-.e""+
( d,,r\ ;61 r\ Atlt>t _ 6xzr
Sol * xYz bq
seb'q z aTYz ( >.y *+z)
o*5L fn?. ! Lr!*z+!] +xyelY'[tv *E)
&. IF-r : ycos tr, ! = r sirn \iIten;Cn a:e +e + ?d:c? D 5t
Dt ay
aLlL
-Ci €g* = 1
x o\J
Sol*B:@__ ?-Cos S {
r\ t9= ler6-/ V ' ,
-
)C
G) -'.3s - I x :L0x t+yu )c-)&
1 0s : -YDr x'*--?-
_ ?r.- Lx_r+,r-\z-
- llt, aLg : :a:cJ
D")C
a'F :lJ(rz*y) - yx:-yDrqy Lx_'+v-r1r z-.,
ar'"t1--*-lr, cos)9 -a.Lrble J ,. cr<lg - si.nb
: aos rg
DrcajDzi€*r f tt qlva.*lq'f'z-J
aL 2*x_
*r2- c-os 4-By\
= - C-o-s z0
sy
tV2.-
RA N4 (A\4AL.Cf r O U Drl ARYS 5 1 O CM Al t.CCI\4 97842 3693 1
--o-4i:niHm-l:___ t^rrqvm \.oJqr_l!J_
0E ct QorrH:ootg &{c* E(Xl {r+eonques +o t} Cvo{"ua- Va\r-t e qn - +,jlnet lreL[,tg- [. c*l-{ e-d Nt Otr
_- -
[P o___C d eerael Es_Jq a C e3-+{, Fn
-- ---vs=Lqe -enloLj+1.,cq jrD-sreqses
-:fh"+: rc<tlqe t -qq4|ed- *
-aG-+1^.F\ t/gf-utlc'-
N-alt-i"-
Lo fhe mqi[L_chod -xoraran q Ssc-grs o l*er"no{-tve{'_Euln<* Corr, he\r€ -(pvrrb-l hnx i In-urn clnC Seve ro,{ rn,n-
vel,Ue t '
cfue- h,"lrt Vo-lctL
---!---,{Le Jegs* f,"r'irn 5,uerlue. ls- -Qg.Uqd {r" n G )o ho,,l fn'r-l^r+mr-q-ol
- $r qaruersul hr# a'.'d thq hbhqrL rnazrw vc.lue--k- -
-@ho{- hqx* o-r--q!rrrer.*] hnar* -fu+ ^2-IoeYintq 0:r trn'wrirnq :- ' -
F 'c1):0
-lae fu"nd-
6o
Is TP :c ? .o\,P"'.("() ) o ', F- is rnit q*lc
= 4
oo. lq in vo) ue = F (d.' )
.'. l..,qqj Vol qe = F (P)3:--CPre:1.
F' ti ,s .'i d -Jo_ he- slgAonqq ,
e-Ip- p"] ( Y ) + o , +t".n -'[-L1s sqb to be the ln ete .+f o^n
polmt,
I' FCx) -f {
--SsIf - o d(d')=rr (t+logx.'tF'(v) =rx( I+log.,c] _F" (x) = r"u :L +(t+IoSx) (rt(t+tojx.t ___ .
-
Xffi_-=-ff i
F" (:rj : r II-{-+r. K ( t +l os:ct'No,r,
l='e.):ra(t+losxJ =o-l t+losx .-n
' ,l osar :-t
r is--:Hrq -+:tlrsr *-e q'I-s i. +ve ,
ooo ) oi
- " F ls 'rnim o l_- >c =.t.ehnlr kqrqe =+t+L{+}f:w
F (x) = 1A. hqs it< hiqx af: : e-
rqirVo"l
Fn clx\f c.l.qe
of
-le
qe
/e-
xLx gle .
6l
% +C>c) &s+
Sot*+ tr:x+CI_g:L
P'(x) : J x"t +-lbsrr r / * )x rC u \ y")
-f (r) = j.o [r- t.s d-Fl-tu-l-=--r3- ( 1- I o trl S
TE 12- \- rc )
-Iuug*
r ,, ( t-log>c): oT"
l- loSr : oII oS e- r- = -l- ----Y : EJ
no, x- : e.
- NJotP,
-gtr(e) = o--{ (.e--r-- dl-.t^O," F ls tnad o*l(:el4q4r-rlc.f ue :-F Le): logs--+-
ee
$-l -i-:- "'. ) : LoJr-rrIL -
! trlogrr;-Lo5fl
e
?C:q
61
I .e{ +-{\ =)JtC
'-*, ::be?e:.-ltaea zrce-T +2 P-.r
-', -F" (*l -: e-x- fr gE r4:)-!|,rt,], _
_o -c(( :r::c1 =o_
-'. DC :._ S 12- _
A+ rL = o
- P" (oJ =e -o (z*-qlq):2>o
- Lis h',?^n tT se =oAtr: )
Q:
Fr]. nnqrecfr -)g FCic,r::r3 e-t ha" iA finq- otl_= -
6. Fcxt : r3ff_
-'- la" Cx) = 6:c+ 6
d3krp_La_,',
Frcr, =r' -g-2-- -t3 :o '
. 5cz
19* / - o
yL= I
ll-2o
63qs oP' l" nnia alx =:L
A+ )c= -lF t'(-t)
)
6 (*l) + aGt )"
-l z-< o
'. tr it -lrt-qx of x : .-fF-e4f Lqeqs :c- 16-rlrl r L c-o,Nujn)tq-l_re * C -/ qt+-b- RA M
oo, o t cf l o u D H r R! -1 5 1 ai G M A I t.ao f'',r 9 7 84 2 i59 8 i
- -E)qx rJ = c-+:J@srY" Lescf yofrreo&-,3rorx +- qs,:al+ 2:[.?
i +\-* ? t z*!a?+s'
- 9 - <--- ^ - ?
___- : gcos's +.*_c..sp - 3,? sL g_ta_
!_.lS_ CoqD :J-&iL e+2-_2- 2_
--- rnox.\rc4qe: " -J +*f- Fi'.,d *t"e h+ 'oE.1-1." eo_l-e_cf }rqx_val'-.t+tr @
1r, q sp hp re oP rc4ai-r cf- t -qn1a -
-- hL+* : tL -
V:__l=rr LLh5
=-I Ct-n)crrh)--V =T ( r+r-.-
^--h*)3\
\r
t"=q (-z-Gh) 6g(o
I b'.LU-l =o(an-r) (h+t) : o
3h=1.h :J
a.J
He_!r+ : Lfi
=4lt-3"
in q sPhe.re -o-e- J.qdl&
.ou V: trr:-n
---ry tel *lh
tku-: sf1;P
- ht ='-Lql-3
I H . l'orn,{o' fot' ql hlorln a SP h€rL '
p
/'njo.r'6o$
6s:l+ z=FCr,:)P=e-,- ,97=Dz. tT-_-a'< iS:& 4. =a?-
fort' Q nnqxirno.{' or' rslininnq r k1' t P' t O q^d ?-o:-f +n{ 79rt &rnev ln :rqo{)- { so uoe hq"e:{roo
}rqry''d ) rsplrre {qre ag-s ,:qrnd S"J re]- -tetr-{Xe.n^gle:=
r !) __r4 !:f -Y.:l r !c =L - - :
- F.-om?leJ-
-+ Pq +aL':efa ejfioe-J"r_l:o oy 9_:o +tlaen 4Le- eqa:h',&',n.4 +o c*u."]@; lt-r -.-lr"*he- e?-
e?t i,rqo4J
f :--lccr-2e S z-
)aY"i], Yre xo, 5o)-+tx+JJ iQb-l?)- -'- s\e+onqg P-oioE on citesl
- '4oiaIl'r,? Oote c Poirt {x--Js) ,H',a *ue VrJq€J oL.I, S qrr!- ]6--
eus_e_f i *
- Se-o 4l+e &ros6. oL+aiy,-q irs ntiri.d*l+qt-tarnf^'" fqtrrn -lrdLre : Le&rfd
IFtt-s':-Qqo4 r<O4ht M Erne.. ajg"irr" f t= msx l- qt- 1l4.i- golxt
Cqstflt- - -lF a-t : sa(-o ,_t!rr
Se -e!I* -,- ltqs no ' ex.lermrp ' Vnr tqe -NdJ&q5-tni11
'-
- hcf -xroy*-,''
-Ert --q1:-othq++q &r,se-f* d-cstrlklll qn"l n!*C Eu,nrhtr--
o .o\:owr*r$q$q--
--+ t0heu -iTrn-- Arns -og+iercr' lms |Qi.-/.qqx '_-tr 1q-.foo=o31i olttnen \ul *+.a. Vo-lttEs 'rn f =o rffsr{ } = O #hon Chocse -he
C" rr""f aons ,
*fh-oU\enns t -_f " -fi,.-R,",c*Ftx,J). d+st l-s:c ;t:*hos
66
l
\_t@ rn'rn * a.f (-g,o) (bl r,n.rt* c+ eSp)Ca) !to !!:lare rrduat$3*o) CC, lro,are
r\;r5_e| q*
P:3! = art:$ s n =OE .: 2]Br aJ'
^6:- aFO:cL ar0J
NLq $,
-.- 2o Tr,e Func* FG,y) :t-rL-J- L"..r? . -_fA) hnir,*o:1" (rc,o) (.b) nnqddf Lqd
(e) No €xtQremqrn o*Co,oJ edr_trtpn_e_
O:< ,AJ Dra y
_ S : o _r ^r L_-L
B- !F F (:<,r)-;-rB l*33 -€r j ho* '- :-- fQ) rcrinw t-f C t , tJ G) For' * c-T (I,I )
Cc J rn i n* qf, Cr*r,t Ca I rqc,\.t *c{ (r, *t )
--&S:= 3r-
S=-Q F :: --3"1.
blp = 4DebYa: dr'
aJ]_*', r* *-CL
= q6--g =t7- >o
-'.'Y:6> o
\, F(.>c, l) -x-2+5,+r5 *-x-_qytE*Cq ) r",ia qf L2,-j) Cb) nr,qx tt Lt,-s)
u-@ rnln at C-z t 3) (dr rnot"' of (*r-r s)
e7
Y'- 3:- a2bx- E :CaRi fV
KAMAL.cH ou DHARYS s 1 @ G M Al L.coM 9784236e81
f : a'c -Lbb'
S: -It e Rt"c* FCx ) = 4r+ffiJ 2-8r --f418 hqs t T(o r nr-,,i^,.,8 (5 r rna. -€ tCa) frn?n I o (4 r fuqx lo
3o/w3- F=3e --8"c-,* , ? =5=C = la-y-*&x cJ
)'=&ap, =R S=alF',:o ,t==azc z-lrBx-L D>cr 1 dJ'
}-t *,(L.= ?6-o )Ot
33
[22>o
C. F = sio x +-sinJ +sin (.€) Aq c 2
(qJ tn?,,, .& (b ) lnq. - E (c) rniin-@!,.,G (d) rha":, 3.13-z- 2- *T 2*
: 4--;.- = lo t,
-, <-
sol*9- P= * =cosr-t-eos(--rt) 6a
A ArYV - Dtr :e".ry+cosCr+3]D9
Y = OtE' - -e?r,r - (Shf. rgr)arP
t )c zY
Ftr f"rg - *t^e t.c. lc/e o F ryr frr 'p) "e4 *-
s hovtcc"r,C-ns2-:f : Crrrl '- t-.*-'*JL=,?:--a'rr O
Cos 2x : Cqr CTr*:c\ 7
3r:7f "-,rJ(o
J:7-/s c eee (l8o-&l =--c-osroIt
t = Dltr-: -sfn) -sit, ( v+y l
D e Fi nite qod trn pro Per r"l+e J"..{ t -
e-v"n
' ra ,iP F('q-rt '*F(,.,t
r-rJ-z) ( n
67,
k =-r , iP 'rr g oddk : I , iP 'rr' t't ev'e.nn ,
2-
Prn,n rn-:) - - - (rr -l) Cu-U:*x-r
F(-r) =FCr
k "-q, iP bo*h rn cmd ?1 Qrq Q-wtt '
Jo ( rr.,+r.) ( rn ,r-tr *:- ) ,- * .-..-
- zz,( t dxrsinh(i))tm*
lo.lo' j,o dr = ahtl
tr+.L
11 .- -L-dr -: (. og tr Jr==-
I
c*3,
I
Fcx
^l ,x
P(x
}q'
--l
,C
Jz.T-rc ch o os e F (lc ) , 'r^.nnern be n 1_he tn3 e".o/ ,
Tv
L A TEr r ,. l
-) €x p o4e1fi'e{ .., I \-=Tnrtrse bg *lgeiraic TriSooomefu
2egx4r ;d lz tJ
-r.-l "*;; d,, ; t tr'*r**,, - -
*a ui._Jr = e-a l*.-.?r-+6r -()
# rihp &bove +"<ltlc="tl-i iL -lhe torrn o P -!r*: (D"tlh oal --Poflen nf e '-*4u\ Jh-e-- so/* ciil ,bp
eI qn4 *hpn r sd -l: tt crrts#-t* .
r.€q'cdll : g ji xlc4z-
Jtr T", *A e- G A cv:e- Fm h /-e_rns- e h qs -!o nna _&:lsu +*hp- r =fe -
J<- ; ct *he #mp s"l* wi rl' be- o-\w rnu-l* Ptap=s{ tJ -re k*m dtdqe{fu,* Cr,i,s*e,rf ir ern /* ing{r x- &
- dr' FFs-anJi *-4 Atr eondgo1.
=rf t*s,c *9
,-Lbyt-dt : r, I r r"s x:'-,2- t .* 4
:CL - 2=X
{._ : p ?{ [ n."s hr t b ril _fr _J
Q'lb- e )
q3qL
1sR-o b\ e.rrns : -
"-L 4x, er dx
t lrxu,
Loq3 t an ^
: b,t ( r-h"b"
o ga+ €-r la-1412
IF'rx)1;S P1111li-
* 1*,n-te - +em-L | '
| -l o -rr t --L*)I
n+l
7,t.o ,a'(tct) f'.G 4xlt--)L zJ? +F*,
. ( Ecr; dx = b-q
-J Fc* )+ F ( o +a-:c; 2-
-l- L c-J
(lL L,
Q:xlx1 ----x iFx
-vLl I * >c'\L'2-)o
- I \ --(o-o|z) I r ,;) t zJ
L:c )o
?s'-.'. Bol-l- Above &.ob 'hq-r ln*.Ser,
Cx+t) = -r t l.r(x(-J
,J - o (rl(I-
lrtrlrl-r{
RA M KA M A L.CH O U D H A RYS 5 1 @ G M A I L.CO M 9f 842369 8L
:ct\-
oLeo-hf*
l+x:F c*r : I'
t t-(-rl I +v'
=7
t^fhe *her Rrnr - li ever or n "l 4 '
71
sirr3x C-os'{ r dx ,I-q
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Lv<n o d4
1-7
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p(x)=.qir,, 3x co.rYr_f' r-t) =si',r3(-x) cosv(-r,)
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L $r., s
fx3xl lrPro4ue-./at oy
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TF*+.en {-> Sfurg,
l'rxL *hB-no JT* I =- {*rtn I
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tr(r): slnax,F (rr-:ct, sln3(T-rc\ - sin. 3x =F(x t
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t-\i yer, qzt* & +" A,w
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*tr Arhenerer S? qql-€ rcol h +r.er-a Tr h o+h nLn - 9 rr4 dan;' c+yrr -h €-l I ra i an'l'P Y ar.t h'-+t nn ?Ou fn !'l A
d>r - (-L) )( dx..fl' L l-1arz-
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l-
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: Cosho CoSh (-,c)'- l- m ". Ccsho:-g',Ls-
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r)f z:c I , | +df ( rr -L, 3)
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( z, -t-, s)\'' N :- V-d = 2(*1'*-r--j+LK) = -i,tr-j #rK
lV Al z^,t tTi rT 3
7*v- r ?o.l!-- I +
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3
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el-fr" ot_ i+zj rr-r,snl":- VF : (7te.l i+ (r*-z,tj t(J{-rl/<
yF : ,i +i +-,qR
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(1,*t,3)
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8<
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o -- vg -- zi-l-i -r
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=t1JNHY
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lql
t( o.l. r)t
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J =xr+-Jr+<r=.q
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O.,tr : V
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tnoJ iat*-rs"c* oylhoJ o nn!.(J al' ( L , *L t u)
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