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fu grg{ rlats Ffi.d grrfrc frg3l ffiilatt i',rg kc'g ail ir
filq 196 frn*rr{r trsFfoTdEr rardg fiiur ddrd
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lThis book has been adopted with the kind Permission of the
National Council ofEducational Research and Training' New Delhil
All rights, including those of translation, reproduction
and annotation etc., are reserved by the' Punjab Govemment.
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3 a fud6+ fud.rtr6 f_ezr'- 6 €tre f+s dvUFfdE il r*d irfura O HrrFJiM F-5 etrc 6f gm fr-a rr.ssql; *R(magn i fyi ng glass) er qd"or *a rr*3 i trt-i z 3 : i ks*ry i ser 6 gu E-{Ci r fuoQn sg fr tuHd €dr' ffi fu gfi fuaa' 6 ffiga r.r r (ii)-f€s a€i t gre, r.o ar3 z.r t fssq-o z.oos sfus 3 r fw sEl, 4{'g lrlm z.e ry3 z.z tfsge.a t acr i fq'rr'ii adds a-fre rartrfen6 ey.d.ex Tdrqd gr-dTi Eg?as irrr uTirfl! rea ?.6r 6 irarrs aiar'. gffi. i+a'z.o: g ft3nr* aiar'.4'6,-a.r'fu lffi n rra=au a's iu-d sET. @ +dtffi-ra t . I zlii; f+e )rr€qrfu3 (magnify) dde ilr I
2.7
Ee fur f€ss f€g' z.oos errdr 2.66 n+J z.ot i fte HkJ i r fuE sS
l9ffi.r4 t '5llt
4{' g frfri4r' aif. e fur g'or I rzrnre' furs adt-J ilfra t aq f+gd l' 1 3 (i) ll€rrsu-dr a? fu foJ s'!-dT r0 sirrsir s'dr'{+s
=eI F * 3tftF6^au
;; ;; ; €r- sf , tu86-4€q'rrr {da o', tu+ tu ftrrd r' I 3 (ii) f+s
ffi;# ftp,r'a;,6, fiU{ z.ee t d ts=orrs a-ee'-i. z-onr tu{ 2'662 6
fcJr; ""* i,4{'fu-4{'reifur ss z oos p5s' Qu fa*'*s'1Qu-sqrrl
.-qsEtfu{3t
J. {,t '
ftgo t.t.r
fr -a ryreu-ra frF at FJr|ggr 6rs fi furr iur^ B fi f *aret a ft fdtE g eu-c.d
i# ,..{ffi; 6 u-drqd t{'€""5 qtfafarr' (process of successive
magnification) fdilr ;Irgr u I* g" # rm kr anpr td gfug !u 5'-6 gdTrJtl T'*t" €trrdr HrJ
gFHF€ eE funst <gt #=rn"'" a frfirr eu' B Fffi (fr'
ts=otra) 6- Fulra -du 5'r; ?fi*m e' rsE o t
at0 o-a'+tfr frfirtr dw ? r-s tr-e-d-S errr<-ftrs'd e'-d fr-q
="][''#,,i;;,tua itu ;i t + 1*.1e-a'lt r raar<u-o-s nR
s.s alfr &:ti rt=u'-' 6 ="=#' i ne rsrs'd ,rr'€[do d-dn' frFflr' eq'
E;m*rrdxffiJcF6EU't
e*cr.r+fu t Fe e"o faa ryrt sans'dT' d;g}.S#ri'.ffiiffiS5#RT,,ffi
H 5
9fr"" ll : fifrl{r alr ? 5 sgrrgg Fq]-6i *q, al-as.s s.sz ltl iE s,zi e,haue-ii r
.tt
{ ?7
uffi-4ffi foJ a€ J' fd. 5 4l1i 6 a f€.€-d.d s.S? Hfus 3 r rrrarB rror Fgrdl s.zj e's.r ar3 s.+ i ks*'a ss-6 ffiTs *e3 o'tErrc O o-d
ufo* rS grr Stre t sf affi ewsf€d fu i fur aer $ eF sd'rd ftifr"{'f +s ?s fr ui- 4€ zr'<qa-s. fr* a'o fuo i€ u' fu s. : z'r3 s. s s ? fusdEnfus t r s.a7 e' Aa r{fire rfr gu ttre gs,,ttfi s.zl 7'rr3 s.rrs t lffisrdr c; grlrdr eF qdEd frfird' f+s fu ffi u' 4rJ s. :7 3 @re $ afe u',ffi fu f+gd r. 14 (iv) f+s fuu'fsl{' tom } r fort's fu9 fu s.:7, s.tttt ?5ra'v& s. r zz a e fg)r{.Er Ff, -S4 a rgq' ft ig-d 1. 1 4 (iv)l I
ftre1, FE areqds frB a's rars't &€ ar3 as'fr e'rsfe--e iu' j gF
stlg, fifiF fis s.:7 FfsJ a, ffi f€€ arfr et a-nrrs'{dg lrtfr b,Ofafur A fafud ffi <q. F{a u' I iv. t Qr g'ol e' ntrr d is' e'frE },fuJ d€' d €x H's-d. 3 fud-rd a-rs i, ffi'4r6F€ 41fr irFrfir ]qr 13
frffi{r it Hfuit ttrs' efr u' t
Ec ie gtr frd f,lrd HHY drg i-A id ffi {HE r'5 frnrt{' aq' "ugr=r 4f.'-ryr€Gt eglf5€ f€p3rd <'-61 epdqd frfoti'' 6- €ud fis A
gtd:*gtfrtpq-E ttfrm affi afi erc, 413 667rsv alaav6t et aautr i 4{qra g unf futgaw dfJ Hde w fu Jed, erysf<a itfunlr { frfuw eu' E fra 41-fu7
(fu) frg gara, ean few Tisr j 1 s6 71, fifuw 3a' e' ada frg Ed? hae fu e,nsfaa fifu>t, O ean _e, i t
alfgrrila 1.4
1. Fdrs-a ?r<qd6 {dn frF*r' aq' t :.zos g; ti- I
2. 4 EFH$-€Eq.rd.r:.d.frfrft{'h{. ? +zo 4;itt
1.5 <rysfs{ frftP,f. B ffaryqt:firedali axrgr €s, gfi- fyd rr{ ili A fu rrfuriq;'1q'irre' frg ar3 oe-r i{H qereilr (commutative), EfiftTd=rr lassociative; rrfi- €add (distributive) f6.qHr
g r-snz a-o-+rr d6 r€: 4{ff fuJ d .r+ fa d' fd i 41fr € ufd}iq irni?lr'-e 5*Si, uradB, qe' q-dt m g6r a-&B (fu.{d fu a) re d ng k ulaFv
"rfrB1fr* Ufg ird t trrrasB e-E, urz€, Era"rni g.ar E rflax'a uffcsfrfiffq. frirrs tclosed) ildar us r rue
"r* Fu + a€ u' fu +nrfuxc ;rnprrn
+ i-J4.ri ge-. a_pe-erro', Hftro.d3"r.rg esd-frfuqx.U# ""t-;ild I rra, atrffiq fifoat g- i fr-E, ure€, a.ards r,{3 qa ds E€r rrflJfuFq 6ttr
snrfr.0-re-s* , rJil-t,Et,tJj.J},rri f qa*mn ,+*,ry€-Ue am fe-o &frB fd H-A Fd u{i}iq fr flrr{r f +s uufil}iq fr nfl.r'. ffi d-,fl3 frd. t{fuiq fifrpr1"r O tr-r 4ufuiq frnr,li. a.u qre'- *a-i o,, o * go 5 ,
€e-d-rc i id 3, .,6 fra'l.rrrid}iq frfrrr t I Te 2 + .,f al3 z'E ofr us z
fu€fd Jl E-q r.rxr-g Dre 4rr€-d-$ eErlrtr€ f€E-J-d i, fuE ES foJ d.g2+Jtw3. z..6 isSdfr'olrfsnod: 2+..6 4fJ 2\6 drvqfullq
fifoasEt6''
Qwaatz:*ea?fq 2.6,4, uQ +21,p - 2 ),{ufuHqfrfifl.rrq.'uae,afrlV5
t'g' ..6 = 2.236... , Ji = r.ataz...,n = 3.l.il5... T6 I
ge zG =8.652.... + = :!: -t"E -1 r1n4 ;rFlVs u'5 x J5 5 "'-" '"'
\D + 2l -- 22.4142.... r. - 2 = l. t415...
ftT Fr&?ii )fiHrg rli-€--a1.€r€t snlf{ts rl; I fex s* fuo-lflllolq frfwfr€.d6.l
ge"arc 13:2.,0+s.6 ati vt-:,'6 SiFr
tv: (zJi + sf$fr (Jl -16) = (zJi + J7)+ (s..E -:,8)
= tz + t)."/i + (s - 3) rE = tJi + zJz
?e.trd l4 z ouE O 2"6 a's qre'a- r
i'g: 0..6x2.,6 =6x2x .,5 t",5 = 12x5=60
frrrdte rs r aJis g 2fi a'o aer a-o t
nA " [?= " 6_aJr*s _ sV3a.is __+1s6vr) r rvr =-tr= zJl
p65 @ srs'3lfr ia' ffi # a Ae d gHtE *a Fdi a' i fd FST6:
rl fta ufrt{q frfo}r 4{3 4flItd}iq fifrrm er +5,'?€ ulz€ F{ bruFoiq
xfr*,lr'ffiil,;, ftq rraffitr frfury'? a'-g Fq fr.{d 3 ffi'(non-zero) ufuiq fif!ryr er
oes-6s f,r grdras aflrtoriq ue't t
rtiir i arfr E urrHro fffuz{,.€ 6 ffie. urd!. qe' {d}g il'Fd 4rutdtiq'-' ;,f# S gf ,{ufufq frfoz'a'tr gqr ofri 3r fi'-Er uforiq f,i }aufariq
risd-r*e'ttoo ryfr ainret ucer <'rgk{ itfri?t'r'+ n <aayo &-€-c d lfrtr'+r' B
o*tl,r:a.5 q'e A':dr fu i , trq r{'|trfu-{ frft{4{'i, 3 '6 = 6 e'rrav ti,, = n a€ r, > o r ffi rfg'x' qagrrq <'R3f<d ftfo?lir€i t d s'ar ffi rna-e1 ttirs € n > offe e'RSf€d frnafr 3 t gE',ii = l' s' "'l-ss
3 b2 = a+{} b > o
ryo-srcr 1.2 f+s,4}tr AftP{'t{ H frfi{4{'
aH''3 fi f, . ftB,, ftaua'srraT=*"ryT'3. tl-dut;iJ. s6r ] I Ee ufi fu+'etn fuH {' q, fie ,' fraftdarf ussrqerrcfeq irfiflfr ]. p4rfu3fr lgeometrically) gu
616 df€"ar flier 3lge.d-rc s1fr,rx€ 4mT
M; . = 3.s a 8S q'rrs ilfrB rry-ds's )flfl}
,/i-5 q iTli'bs !u 615 *{rd I
fra fS i* eq' t trd Ffua frg e 3 :.s ffi d ge B ft-{ 6-d'gc Brur rm-o rug B lTrJ EE a, fif,F 6'5 AB = r's ffi^(€B ffJd l' I s ) | B
+ r ffi *9& 3 frg tld€ 4€ 6€ frg O c rs eQ t e'c e"iq frg d'a
Drf feF ES O . ra sg r o g iiru,o rrri oc O rrsq i€r{D-r tr6a f* rraqu{d E"€ | e< i lta rvfoo' E-s fr€ i n t€S a ii e',e' f,.€ rrrs rfldq +{dO Dqf dilsE BD= 6s ilxqda' id 3. ,( e't!'6 fu-€ 6f, fi+e rft-E q6gfi' ern3fed' ifrl{r i fta arfru'frg s s€ u'. lsp as na -- t fptr.:i i*llri fxi fu ffisa r.to fis' fgaft"r{r fnr,xrt,furyfou'frecsqfesasnc=rfucr$ Gl ye. ft? fu zrrfr xfu*.r = 3.5
A 5S dtr.. E-g tsD = J; firre'?r rffi3-Jr.t6) |
A+_ .r -__-__-_{ B
ffgo r.ro
lc
4{tr fuF lH-i f rlewmx @ox f lrge Fqr tH.q *o r*d u, r
fo,{'6 re fu f€-Jd lo Fg' a o,eo FE xxiat H-e } r
t-*a e'nraq fufl'E ' ]l pu"E 3,
fuF 6S, oc = oD = on =,Ij-l fua.et
!-d,oB = '-i t; ] -:1fer s*. udErdnx @+r aqr a-rs B, E 6-fHs-E, i,
Br), = olr, - oB, = f +l' -fi:-l)' =o' =,'\2) \2) 4
fusJrrg.6-d'FliFa rrr-.'= G ii i-rfsF dsr;' 3' Fu ssn -€e sd f{ sgr+ 4r} HH,f€81 fefi f,{6- #S } fdxd+r. e,rsf-q trfvrrc' .r> 0i uS, G di.e}ri,lrtrfrfirl{.Av, t-",i et rfe-it i{--s. v._F ot
=. rrr€ 4rfr i.u, sc 6- frfiw iw frs g:ig, s
d fr.frd rf6 stle ,rrs c' O I kd Btfrs 4r,fu r s 6 d?d Dri so O araq fur.Firz; d'F-q uu frd ;T Efip,r. iu, g e E #-s,-; (€q fr..Jd ,.rl ,s-* u,J;funrs -{€'. ; I
iftlNr yErd
tt-" ,.i (' E
qe- ar* eaat1rs d ucs' $ urelm. en irs' ar3 rorrs 3a 3, t trs',fri-, rt-q q-asriq frua6 frfrt i' ii. t s''Er a-os' o-u.A I ga6 u-e ffi rafrrepfuF;n+rsr H-gm €dE =dlt{g''lrs
ure-ro. e'ufiia ildg& -l16 61 ] z ffi fr.€tg ili fd fuJ fq ua's}|{ fifrrrrr i, fere'ura s a, 4r3
sfr fre'*' 4aa ffir' fu?r{r iedl'td .6 = z i r ar€ al* Vz+: e'!u €'*fr} r efr g# f* 41..Ftol frflzr a eei i foF srs u' = z+z i4 ?
€"ga i :, fuF PJS Vz+: = : ifernt
616 Qe'-uo-e' FF d g# V, rffis {d Ff,e i, ffii a > 0 ffi €rrsl€dfrftf4r'A ,l€ n ffc usera 4ra-s'irfirr'a ?
lra uQ a > 0 frd €rl€fEd frfu4lr t ,,tlf , Fa qstslrd nrrrs frftrzn i t se<li -- o,e€fe a" = o +B a > o t foars fu9 fd .,4, fi, do 4{'Fe f€s €-d3
f,€ fu; rI*8 " ..f " O ild-et ft-{ (radical sign) fuTr friEr } |
ua +rfr <a-oronrs'a.u fr&fos d's FdsErgE a di il'i {u ev fufrtrt'E6- €rnfr -dfr irsi1' o-a' r firsdar sxB f €€ 3rtr ffi' f€+ dv rasxrs€':'f,€ a {'.i3r u.d <'rste-d frfu}t1'r€i e fu 3 ae'i €afuqH-6rg:r€'EAEElf3r (identities) 5S, ffi (r + y) (x-y)=f -y" 6'5, ffie
r ar3 y €rrcfea" frfut'F€i' o-4., r4'rrs irdmustirs €, €, elrcftd ftfifirr+r Jsl 3E
(D J"b = J;Jb
@
25
EJ;(ii) {t =E
(J;+Jt\(Ji-Ju)="-o (iv) (a+ Jt)Q- Jo)=' - n
!-rr-J
("r (G. S)(f * Ji)= J; + Jaa + Jt" * Jurr2
lvit (Va + Vb) =a+2,lab+o
rrf'9 4* fud-6' E€EE)€rei i+rrt'-gg HF Fffil{' ? fue.d ilfrirge'-d-o-c 16 : t6 ffi f€tFr-d' 6 F<rs di:
(s.'FXz...6)-.2(J: + Jz)
ci) (5 + J5) (t - .6)
ri"r (Jr r - J7) ("/iT + .,F)
(u
(iii)
i'r:1r1 (s +"ti)(z+6)=to*s.,6+ z"E +Jx
rrr (s+ Js)(s- \6)=r' -(\6)' =25 -5=2o
cii) (.6 *Jj)' =(Ji)' +zJlJi +(Ji)' =t+zJi+t =to+2ytzt(i') (Jr r - J7) (v'ii * J7) = (Ju)' - (Jj)' =n - 7 = 4
Er€1, fu,rr-6 re fd Srd ge-d-d-d f*e ffi ari nse ,,Fir6 dir6r" sr )flir€rfuJ A fd f<,irr{ O
qfuHq frnJ"{-€+ 4{g 4afd.fiq frfurfr€+,e ia egu fEvfur€r €.'-dE,' ir alfr fux mifiur i fss.d ilde ia fu ; frnrr eq'- 3f*e Hks 3, fux ryAg'ar O E* fr nxnrs {ilg u' r al* *'F o. fa fuJ Fa.rflrFdlq i'r* ua Fd. ufdiq frfoD{'te, si fu86 Firwrr 6E ts ats, r.r*e'i t rrt€ afre fd d rrr# ffi uu e'ufuhil*ea.(rationatise) dcr Friir,'rrdqEr dt'dd' p; Fd ufu*q frfulrr. Fs ses' ild rr*e u. r ffi uS r.&€-dd=H6'aafilrEraaxrse-e} e-_ees o-d tr'ri'g €dt fu ft"6 fu+dkrfr;d€'r}| - - Y-;---
Cs.d-tr€ rz : * i'us, er ufu,hilfids da I\tz
Ii'-o''rrfr 15 O fd',.rffi E6 futmd A gu fu fuue, sr_tri di, ffrF fr
Td trd rrfuliq iriw,{. ie r r{fr fr.dg d' fd Jr. J, ufuFc i r,rrnt fuJ +1l;
H.dgd fd +6+ asurs'*rs.36E"{ j5f?tmd{nrsUst,-11 = l^tz
agfu #: r irfs sSffiefu-Offi*d3,n-e;Unrsfe'ir
r r JrJ,E t;" t- .\a \la \z
ferg,{f€E f ermntnu* ?xe6 fsdq-d-6ftira frierAlfu OTT3JZ tc- -.c,- a ,t=rdot, o HtqS u I
Iqs.d-r€ rE : 2l$ d uo e' rrisriritseg Ei'r
Ifo : ffi 6g1 rl{tr €Fu-d ftd dr* Eirqg}Grr (iv) er {d-ar aei o.r ,;$ O
2 -.ti sw Ere. tr3 sktT diro. 3, EE ft{s-e' t:
| z-Jt 2-J, ^ E......_-.-^..............=2+.,13 2-.13 4-3
€gtrfJ 19: =-. = gugE ulsncfrceg as t
v3-{5
dit:'fuFsf,
fu ;F.^,rrfi
gnt frd roqxx-s' 1iiil e' qd'o aei a' t
I
ee-ffJ 2nz -------------= t uo e'uaudra-ee airI + JlZ
:- r r (t-tJl\ 7-3,{, 7-3J,7+3J2 7+3J2 \7-3J2) 49-18 3l
s s .. J5.,. uf _ s(J5 + 6) _ 1-r'1,-6 , "l;,E= = E="GI16- 3L =17j(vs+v:l
3',rfti3
fuF sd, f,-g Fd f?trfr{ t o-a Fs €?ETUg <'s. Fe ue ire'i (fr'dS ftnt{.d-rcl fud a 'tr€-d'ie), s'@-fr-d uffi gs-fut{rr-d fio, fere' ua fr-autoio fifirtrr t, qgse d fi{fr 6 oa e' ufdriqTdd€ (rationalising the_ a-_
-
-4.,AenOmlnAIOrt lq..t' t1'V' .J I
''ffigrrrrF 1.5
L €F io' F*,rn' frfr4rEr Ed ffil.r ubriq us ari ffiar )rilfuiq ird ?
(D 2_GI
.>^li6i) (3 + J23) - Jzz 6, ::+
(rv) !t (v) 2n
2. to', ffi feltrtrd, fy$ u-{ t<,trff g ree a? r
ro (: +.6)(z+ 11; oi) (3 + S) (, - .6)
GD (6-Jt)(.6.Jt)r. go.6 qs fun fu ,T 6Fd ff{d i ti' tra oQ al,?€@ f?lr'E (lrd
nQ a) i r6rr.s r"o r@rs *s' grgr ], )ddqE:, = I t' ru-o e" ;u oftiq dter Afrryr fuq'f Fe' a fd n atrrfu*c t r ftx d S edrs- sfr fdxgg'f€g'i'i
c. frfrfrlreq. ? .,6: quon€rs. iro ffi*ar t uo s. uffrfr*es qa
r
llll'n Ji tiil fr-T tiii) uga.,5 tut Ji-
l.e <ryske frfiqlrr.e' E g* ur'g,tn f6qx
*gtrt6ftttfud{nrr*3fr?(t 172 .l7s = (ii) (5)? =
(u) =
= (iv) 7i.9r =
fiv.lrr15cd 29
ffi f,s-d' g Untg ildd A6*, 3fr d ffi we'ira laEIHi ( Laws ofexponents) e' t1iidr il=fd ,jifJ' iedr',
fi\ 172 .175 = 177 (ii) (5'?)7 = 5r1
^^ lr
6) ==z:t'
(iv) 7r.9r = 63'z)
[ftR a, z rr3 ,r, qrfFbd fiftfa{r€i ui'1 3g'6 oe fun fu u e; trFq-d
(base) rrf3 rn ui n $ urv fw (exponents; f4o'*tg'iil finoa+ E alfoitsgmfudstar'il{'s'Ee*ufii i,O dn . ai - ah+ i (i1'1 @,,'1" = o"
0ii) ;=o"'nt> z (rv, u'b"' = fuhl'
,, i t r l-" I t r cfi fsd atftiaia uffi d ild i-d. a ff 1,1 1" = t te' 3 t
I
tur ss, (ti) sEr ildi, gff 7 = n " r{'us ad xae u t qa'flfr M- tdqx'' g fd-c--stfd ur.er trE- i 4 g'dl ild rtrA d'I
fursf,9e'.rdse+d3,
(i) 17' 17-5 =t -,-i tii) (5r) 7 =5 "
t?-ro{n1 !! - = 23'' (iv) (7) I (9) r = t63t '
tr6 sg lfm fu- ffir+f' t #cE s't€ u':2 1 f'l)"(i) 2..2' (l') lFl.l
'tt(iii) 1r (iv) ,ri ,.,r;
-a
u# M- & fdx Td' #A ? fuJ 31ons 66ps i ta ur-J 4r{ f6qH' fird'e rmiro nm=ufr6';. ii il', €F FfirS f€o d ug i r*i uo, eEfe11frq.,-a'u6'3H51 €rFde-d finf4f' A-e t€: ur.3,tra rrb{q frfrf}t{'G (dr,i'
otro'}a- qe.d f, )trfr fuJ aui-fr fd fso faqx 9e + u'Er o rae oa' ffi& ur-s
drfd's
,fiq' €'r€fsd-frfor{r i'€) r rro fuua'ftsq}r' e'ds6 €E J uffi rrs ffifsarr. g u'gr a-o-a 3 ufoll'-, fuu qHv +d' r-d& i fu , Se.od- i ud o. q1i t fus str fun nsq fto atnt riv a-ea' t€-on r
4tE-srdl 1.4 f+s, 'fid" Vt O fEE s-d'rffis dl-s'4, ffrn a > 0 ft-q€E-sfd *fu?H. A
irrs9 o >o ftq<wsfu-{sfi*+n}' n F*q-6.s}rdn-1ffa; frnfi'. at
3t Vr =rie'3,H-;ifu b,,=a4jer>0-elur.s frqi € s'H'ftg. ry* V; = ,,i i _uu fts uffi3 dds crl
^lE€ryils e 6sl. VJ = .2, o r Ea nrfr a' 6 € ffirrr au ic F{A d, r
I r)'= l4rj =zr=s
= 1+,;l = 1o+;l = a
fEF E*, H'E-fg-dlJk{':H' {nr3- !d t:lrd oQ a > 0 fra <'rcfu-*frnflr.r' ] ffi. i, a1ftffia1i F-1rd6 fufir,f-€,
us fu r i fvq. fuo-r' e' ffi ua nv' Crard&J ds A,?rg r r=o 3 r g.
,,': =(*l-') =d7fEH sEt F$e ffi€ tra f6qH ta ftr& us:ira gQ , > o Fa <qsFe frftrtrrl A ,rr3 p ,rri q rrbirT frfoa{-€, 0-d si,(D (1' . aq = u +'t fiiJ ktt'\,r = at,,t
ru) !!=u'u
"re5,=m
rfr" fB ari i{n.d. E gs-d tn-d 6d fET6.i- faur+ e, qfr-o *o
QvtoE ft ! EirF dA: tt, ,i ,lI
.... 75{ rJ -7'
'11
3
(itl nt'l/'= (ag)r'
/r)"(ii) lI](iv) t:' . tzi
frfo"frWrd 3l
f,tr'
(D ,1 ,i=rl'l =21=2,=z ,', [rl]'=r:I
fl ll r 5 : r I I I
(iii) -i = 7" r' =7 t5 =7t5 (iv) l3J l75 = (l3xl7)s = 2215
73
afarr 1.6
*ll1l. u:rr dj': (i) 64, (ii) 325 (iiD t25r
*l2lZ. r|Jr A?: (r) 91 (ii.y 32s (iii) y6r (iv) 1257
- z , .... / l\t .... ,ti r r
3. FiIS 4ir: (i) ri x2i (n) | ;. | (ui) =
(iv) 7:132l.' J l r,'
1.7 Ffd-'itE
fuF tr{fof'fu f+s, 3m 0-6 ffi O*prrt e. trrfoaia *s'i ,
l. frfrlri g umq fifo'rrr fut' f,1€r a, i tu"6 : i gu fte ffi+ar s.
rE-e'G, ffi p4G q ff-rres frftm'e-ua lrr3 q +o it
2. frflr4{. " O 'rnriufo frfi*ry' fddr frier t, i @' : i g{ fts 6' ffi{,r'ffF{er i+, fifi p4f3,/ n-rrrr fifi*ar'-e' ua z€ a + o }t
3, fr-{ Vid}iq frfi*ar. e. SRHF€ fuFJrd fr'3. F-J Ie. i e')dEig }f€-+t'e-d-$ ie'ir ss fr, gJ fffwrr', fure'eprs< ferJ'-d Fs fr' }t{FrJq'ea-$A,,umqffitt
4. FE ,rnffufq fffrrq' e' €FIIF€ ar€- tl{.€d* a, ufufq fe' 3 t a's d,9o mitn fqre. em{F< f<E-g.d rrrr.s af€'-r{R?gt t, arr"rfuBc ffi tt
s. Erfrryi uf4iq 4f3 aruiorlc fifvar.e.6 E{6'' &e 3 e"rsfu-q frnffr€rs trdIfo lFrs le'tt
32 attes
r. xfi+,rr iqr A utq frg a hdrs ffd e,rsF{ frftm. ffi ir a'-s fr, q}E<wsf<i frfi{a'i frdrs frfrr,r{'err.3 Fc fu uer5r
z. i r qtotiq ? a€ s anrfurie?,8€ r+s?r3 r- s rrrufriq fffru-€t ffir{.irc'fr rsry3 f mrfuFofrfrryre'ffiar.ua i r*o tr
s
& qdrglcr €rtrskdr frftp{.et' a at3 a i Fs }6 f6rfird'FdqFrcr+ s'dTire'hr ua:
G) J,b=J;.tE @ .E=gI b .lb
<n (Ji + Jr)(Ji - JE)= " - b ca Q. S)(, - ^!-t): o, - r
(") (G* Jtl =o*2Jab+t
I - Ji-oe. G;; iua e. rrfurlrile-o-e a-ca t sf, @ )'rm h; *"-"'{di d', fftt o trS a frlra-o fifi*rr.s. uot
10. fr66Qa>0 Fcerysk{irfi{r,f'}}rd p ,r€ q ufurtefrfrryreioagi,G) e.a=a*q 6) (d)q =d,
(D 4=oo'' (iv)d{=(abfac
qinrg
ryfqrrqfe 2
2.. il€ltgEfirss1,r' fr|rBr Fs, gm ffi=* futrf,{tDri ffi A frJ, ulz€,ars. a€ cEI E''rfq'ira d-d i'& i r 9B g* ft!- d afpris rs ili i fu- furJ{'cy ffigye er dre-r;k*{il6 at3r iTi€r ttg* io futft'ri'-stfrdrf€d E6EEI1-3'e' 3 fu e' Er5ffi{fr6 f€€ $rB-dr 6- gt-a' EIrE dd
FfrgA
4r3,
(.r+1-)r=-rr +y2+ zry\
(x - )')' = rt - 2.r.r +,12
rt--r't=(r+,r)(.t-.r)fEH rlrfort'fE f€d, ra 3 uftrs'Fd ffi fdEH a ftT futrf,d Er, foF 6;
8 -Jrt€ (polynomial) fulr eg'3, 4f3 €E 6'{t frilfqJ n€.et€Rt iterminology)
e'rvfir+?s *a.ti' rEn tr+fr srd fu=edx (Remuinder Theor€rn),81-e6&J fu-edH
(FactorTheorem)u€e:due'iEr-5ffi{fr6f +sfu -a$f A'dre'*trrf irad?6 | f€F ? ftrtr€r alfr av i'o- ffiasE FEEE)€r€f €r 4€' qY F3 difetnqi er Ersd&*{dd*estrrSlrs-iisa s'e € arffir-d ildd't
2.2 fu+s€dE-flrErs 3' rrfirs' arrt q"e dd-fr fa srt g fre fuO gna' {dre *3' fr'-E i i aS€ ersftrd rig q'fre 4-A F6rgr i r alfr Vs-'O frrra' r, 1,, z, arfia au rlara
a-ai u, t forn-a ai fa z:r, :r, - r. - | r arffv* k'incr ud | fuJ E e Hrfr{,
-il:l :i
( E-d 4i-r5' ) x "r A W ira- I !€ fr6 sg fu um F{ rykdr f<ril-e-d fuq--' s' _rue
J' i' fd 1ft-q wge-) x (Fe Ys) i 4fg d# fuJ dfr fl-ct fa. Dfss d1 3 |
ufufr4ji Hffi?r gg, a1m al-ea- $- n. 2,, 1; 41tfu 6tr. gotz *€ o' r fur n€lfurird !f6 s€, r,,. tidr' I
fua 4 zrgs- e; {dre 6tir6 €r+ b{trd al-s'fs O r;dra {ir6 <.6 tftrd r+strsa Ue' i r Fd HF Hffi f+s 4r€6i e, E6 F€rr-Hx's vt afu i us r rrrae.J*u' 631 Ai si1furr ps uros- t !u f €e d* uice-crs a-fr i-g I uir-s ys i[o sS uFo-e-css ue' afrE ir
lre: ffirr' d g-s'e'g'Fa gegr € tCU- i__
ftigo z.r ) | fgre' ufuH'r{ (perirnete0 * A ? 3rtr H-de ii fu sesr e'rrfuxqr vtr 9rr€r e'i-g ue']r trR :] i:uiE g+'d ffi 3 ffia1t irfun sd fsFE
r
rtFgx'rl+ x 3 4irEr'g 12 ffia1i$ligeglsoiq --j - --'
grc' to ftqdar. -J sl- irtdHry * iian i rrfuxnr + x f;-ga :. r
l0 4ftrq=r 40 fsqd?x+ A+dr r i oie 9e. d uq.S -r
fudrsb{i i€ tiE 6-s-d z.z), s'uFdx'q 4.. fu6ni}a1i sr _ .,- -RUe' tr fur 5S r{# fuJ arC u. fq 3r+' d ffif"euirgf,s6-aao'rrfurnrs€.ciri€rit
" 1.,
frgxt<rdrlqnse'tseecrrs'arar*3i1fu ] ]
xxx=* gd-drffi?.fi'it _r: Fqarb|g*fuifiE-e pi _, , ]qi f gfr Z.r, x2 + Zr, xt - x2 + 4x + z go& id drfufu{fuim*'a-sderg-irfisr,{.dfiggfeEe**s-}'sr} :3? 2 :Hra drfus-d fu)tndiFs yE t ur.Jtr*frcsfrfrry.e-'. d aiusrfw au iF"tt"dt O Fc os <E E1RrE gtotyno,iiuts in one variabte) 6at ffi il€trd f$r.ri' ilEb{r €e'uae. fts, ss t I Qeryee i 6d, -ri - :i + 4.r + 7,vtr .r f€e Fa aSnre i r fer sfl :y, + 5y, yF
_v f ts Fd s_fl re i ar3 I + +, vsrfisfr-er+relr
Eltret + 2x E€ fqnft;'a'r: 4A 2* t:_orre i ve eerms) fr"r iler ilfuE 39., rlgrt€ 3-r,r + 5.y + 7 f+g fu6 r.re r.fiq's 3yr, 5y rgi 7 udt fr E[frE|-_JLIE --rr + 4"f + 7x - 2 ? rr€ fuq F{C j ? fur elRfe i sB us rj-dsB _t,4f ,lxrr& -2t6 |
r-ore d ota ue e' E-e Qf--E (coefficieng ire' 3 r fsF sS, -x,+4x2+jx- z 89.", gr Efgra -l t, -r, €r q|-drd 4 i, x er Qf€E 7 t u€ y g'argr4 -z
u.Jui14i
tt(o.Eaifu ,l= rt) rdt3fff;retAfu "f -"+zf*s " e'-drs'a*t?" e'qe'a_t it
fu,flra at fu z d fu ":o,re
il ryro €e z, -s, z +r'fe ares 4nre'(constant polynomrals) e gerud€ T6. I uirg alRrE 0 O furo qfle f*dr H.Eitau*, H fd #',{.frH's' f+€ gm eH fc-n'as:fl€' ifd.6f€sfu.ca u_oue Fq r{fJ His-€=u-d6 gnrd' ffie. i t
ue gfr * * !, Ji +l ar3 {y +1,'] €diti u63q fuirr* sB I d gm
H--i-faCm x+1 --x+x ' fucF{aiiEegirrerlravv x ,€rr{rJ
we -t i i"Fq gE6 frnfl{'Fs 3r fux s* fuo ffi:-q fuirnd FdqlRre rfr i r 68, J; n : g ri * 3 t gu f?y ffi+m q' rq-g' i r Ed.r e'ur.s ri{-o' ; a, ; fu F* grco frftpr dfr A s' d gm EHya a fa uf + 3 Farluue i i ;S, fua Fq slrre a-fr t r fr $ + y, Fa apnre a ? fud + F{- -
-.---a: i.. /^ ^'.\cg,Je 6dr d t Uqv r.,
i Fe r_uue Eo so "
a-?, =.
ufr norre $ ne) fr q(x)Flr r (,r), a{rfua's gura a-o r*i d', qE Jd-c a +d 3, alfr fu{ F{C ili:
p(x) =2f + 5x - 3
q(x) =f -t4y;=yr+y+ls(u\=2-u-u2+6u5
s_drre f€s frFlrs frfta,{' f+s u€i- * ffi ey * i r*d i r ge.rd€ n*iI 3, .r,50+rr4e+... +i+x+ tfeelttet,frFf+g l5l u-gir6l
qd ERfe 2x,2, 5x3, -5"r'?, y 4.r] a' u9 I * E:fr afg i fu fgq6t, E:fll-€,
f t€r aF e' is6' Fq rre i t fuea E-q tre e't slrre O F{ rd \m o n o m i a [)
fu-Jr fri€r A | (,irAftfr rge ,mono,€'rrftIEr i .8"*') t
Ud b- ffi dre Tflrd- E+ di{ 6 fou-a au €i.:p@)=x+1, q(x)=f -4 41,; =yro + l, t(u)=ytt -uzEi otc r_oue f fo feiri rre ua' I ffi f €f ote ue Ee *<s € rre ua t
fudd E ue'. €.8 s_dr5Er E e ud (binomiats) fudr ste' ] t tfrsrffi F:rE ,bi'
e'Drasi'?'l I
ls
arte3
H =gt
fu6-d f5d rfe. <-g E-_flt-er g fu6 \td ttinomiatst fdJr n.,e'Al(rierffi nve'tri' er',ffs i "kr' ) t fu6 u$)1n elln qs 9eErilai T6;
p(x)=x+x'+n'tlu) = u + u2 -2,
q(t) =1Q +x-x2,t(r')=l+y+5
^ '---- LcL --JqEitul.'-3'<r'-5rge Hgqe p$)=3x'-4x"+x +voeqlttHt<rJ.ret Hg
ue fd-d-s' a ? fuo rrg sr' i t ffi v{' fte .r sr ur'=r rifl z i I ffi gfl , s-_&re
q(v) = 5)r - 4): - o Fd } d ry 3 €q uI-3 <'5' tfE s1" i'rrs fur ue fr-€' rer ujr.r )i.rd 6 t r Fe r-at|e f+g ss- e1 rs 3' €rr urrs €.& ue i urT3 tff gvilJe e1 vrg tdegree uf rhe polynomiah fdTr frigr ir fug g*. Eltr? 3r? -4.f + x + q d ur.s r i nr3 Epnre 5r.u - 41, - o d ur-s o i tFq fi'd fr.da <aB4fs5sluedurr3 fir.auffiirgs'-d-f,€ r : tai ft3 oto apnre d urv #d' r
(i) .t'-x1 + 3 (ii) 2 - y?- 1,r + 2r'8 (iii) 2
f5 : (i) d6er Fsr 3 €q4-d s ir fexsS s4red urs s tl(ii) vserFerS€qqstr4 a irtill EB fres Fq rrg 2 a, @ x, i gu f€s ftrfi*ar e. r*e' 3 I fen 6f,
,r er urr=r *or o i t fur gf g_otre d urg o i I
gd Elrrgr pG) = 4x + 5, q(i = 2y, r(t) = t + J, rfl-3 s(1) = : - r 5 sB I AtM' f+s a* FEt in6 €ud d fi{s-S } i fso-s. f€s oiq v-atre d ur'a Fqi t Fq ufis e"& sltre O efi (inear potynomia4 v-orre fuo. rtet i rFa ogfio ey id efr E!ru-e zr - t, Ji y + t r,t=. z - a ir6 | {- d 4# * Fs Bdrrsr <rsr hfrr_o,re f-J ffie dr? rrr* FdlfttrJ' trfr g_urre dfr rB-g F{efu _$fu ' €s F{ itfr e _oue F€' €u J €q € ire i F{g il6 | fex oS .r Eeffi dd d &fr q-uue nr + b i gu e' iadr', fiie o t G b akrs ir6 D[3 a * 0i t rfu$ ll ffi B-g',.,y + b, r. Fd ifr v_+ue i r
Ec gfr fa,rr;ftrnl-s v_oue'g s9:
2/+5.5,t'+3r+n. r- trf3*t + Z*
* 3rfr fur ds ary EFdlrs A fa g3 ffi ari mi v-otre urg z e'S rs ?
rqEr i gd r_uu-E O n u61 *l' € ur.$ glnre (qua.tlratic potynomiall fdJr
SJLf'ft'-fr J7
Hier A | € urrj} q-_flre i qy ge.ildJ 5 - y,, 4y + 5y, ar3 6 -.y - ), u6 | 6fr gfrEa ys ffe s'd €u-+u rrE €"& Fd i urd r_otre q fs-u H-d-e i'i g*ai'di fu frd F{ ss- fis F{ €urf e:dre i +q ? €q : ue a--fr t i gffiqy -d €urEt Eau€- va' xd' s' g* Affi fu r f+s ffi luytfr q1ue, a;2 +
r-r + c i gu e' iian, fii a a g 41i o, b, c )ftrs ftfiflrfr€i dd I fEF =fl y f+se trT.$ Eilrtg ay2 + by + c i -arf
g'i-+dT', frA fd a +}aG a, b, c ttfiJ$Fru}}{r€lr 06 |
€-6 uf-S d "_Olfe
d fg6 u{El gubi< potynomialt fug. etet itr FgFd As ur.$ v-orre a qy ge'-d-frd 4f ,2:3 + 1, 5,r3 + -r2, 6"r. -.r, 6 - -r3 4t32f + 4f + 6-x + 7 d6 | 5.i ke'c ar.' Fa ee- €o fi3a urrit vuuei tisfa& rre i F51-e ud r €q J €q + ue i r*i Ua r ffi 6 ar, + b.r2 + rx + d t _dU
fYs ffifl{. iT. F{e. i, ffii o +ol,ti a, r, c al3 aargg frfi*a+'si rr I
E-s{g*affl{rfur,J'g t ,vlrg 2 frr3 €EslnrEttreEoror*oEd' ffi ffi ua, st- * gm fd gs Es, tqr :r,, <Er F{ s-utre f6{ FtrCi, ffiB rdf wFkqfriu{'a? €a vsrEY, urJ n <srY-flret6futgue'fu,imdEE t,
a,f + a^_tl't + ... + aI + ao
ffi oo, ,,, ar, . . ., a, z[s6 frfifl{t€i us >rr3 a, * o i t
fuix Eu su i ao=at=a2-d,= = a, = 0-e3'E-g fir.6d E-dttE (zero
polynomial) fltr€r i, filn6 6 6rs gtnrfurrfl ti€r a lfr.€d Edqe'd urs *F 3i fuorrMrs a-S it
Ee iE alfr fn-d-6 Fd s5 s?+ s-flre. s,a t{ffir6 dtJ' 3 t atfr Fa ftq g5. €a+ s-ftfE * rinrs ro rcE u' r ge.ua-e t uf, t' + y' + ryz (ffi&
-r, y )rr3 z trs it6) k6 g5' fig Fa r-urre i r H' 3{r p2 + qt" + r ( fire p.
q 7,f3 r iI6 ir6), u3 + v2 (ffi r a[3vssU5) il}€E fer5 s{'r 4€ € s6'ed s$rE T6 r fuF sd' a Er-JtrE e fexs d gr r{# qqre' f+s uf irr *ad t
ufsl,rF 2.1
1. t6 ffi f<,trtrg. f€€. futrs' -fuuE v-oue Fa ss f+s 3 3 ktrr'a-frii*-16Efos€*safu9t(i) 4f -3x+7 G) f +Ji
(v) xr" + y3 + fo?v
(r9r+
(ii) r'[+ rJz
l8
2. t6fufirr{' f++di{ f+dla er qr-qfuql
6) 2+x2+x (r)2-t+t @lr'*t (rv).'6-r-r
3. 35 urs A € uE s ),€ urs r oo t fu ud e' Fe -F* *e'ur€ fiee r
4. d ftrt E:trrre' F€' uia e$re d urs ffi r
(i) 5.li + 4f +'1x (t) 4-)?6ii; sr -.,F Gv) 3
5. iF fd b- ffi Tgu-e' E+ fd-d-J'-f*o-*' i* r-uue i, filr;'-fa€.€ rgue u{.* i ars kd-J' b6 s_urrE ur* }: -(i) r3 +.r G)r-.f @) y+f +4 (ig I +.r(v) 3t (O I (vii) 7r3
2.J slRrgr dr{tfx6-dr (de)
}6i-fisnf)rir ql-drrE EgP(x) = 5x3 -2i +3x-2
i p1r1 Fe ua rap B r i x+2. B r gfu,r{r fir+, g' H'E f}fre'}:*'
--'rli".;1i (l)'?+ 3 x (r) - 2
=4fsr ttf, lr{fr fud dfo F{e u'fd , = r u$plr; u f;s a } r
ffi si., p(o)=s(o)3-2(o),+3(o)-2
fr9fr p(-r)d-sFr*CA?
ge"a-€ 2 : ssr i fifi dri !-F Bf b' ft3 ari vgrrei e" li6 rrg' a?.:
(i);= lad p@)=5f -3-r+7 €r!-F(ii)y=2sS qic)=3y3-4y+Jrr E E-s
(iii) I = a 6t!} pg) = 4ta + 5f - p+ 6 gfi-siB: (i)p("r) =51 -3x+7
laJ|l$b{r
j = I s* q!n-E p(jl er f;o rinrs Eer i:p(l) = 5(t)r-3(l)+7
=5-3+7=9(ii) q()) = 3y) -4y + Ji),= 2 5* Elnre qC)') sr fs- qnfs UEr 3:
q(2)=3(2)1 -4(4 +Jtt =24-8+Jrt =to+Jtl(iii) p(t)=4f+5F-t?+6
r = o sS s-iluE p0) e' ir5 ilnf3 !.s Ap(a)=4aa+5a3-a2+6
lrt s-irtrE p(.r) =.r - I 89 t
ptrl fr t ? fq,lra;fu9 fu p(t) = t - t = oitfu -$fu p( t ) = o i, fsn sS 'l{ff afu F{a di fu l, $-iRfe p(r) er Fd fu.6d
iIlfo e-f' g# i{ F{C i fu 2, q(.r) er Fd fida-d i, fri* q(.r) = x - 2 I t
Fqrdg W f+s lrlfr fod dfd F{e dt-fd v_otfE p(.r) d fr.6d ffd 4.rfiT$
frforrr ' ] fux sfr pt.t = o t€ t
gm fsr d's-g ltg-u forr6 Es frryt€r i€drr fu qa,re (.r - l) d fu.€d
fuxeurE6-o ie-u'sa*aiUng dfutff'tit}|4rdqs ,r- | =[,fss6rs t= t {rrs Ee' i t se tdfr afu E-{g itr fa p(x) = o ffe v-+rre nft*ca i lrr3 r
fex ugre E!ft*dc- p("r) = 0 er Fc frs- i t feg ngt tafr d.fu r*i a'fa t ,
e1nrs -r- r *fx.eaimfsoE_flrEE!ftfrd"r- l =0erEdgE(roor)iIq- )fi\rs E!u-e s sB t d efr iF ffia A fd ffi fF.dd ffi a ? fuF
s-flre d aA fg-.fd rS i fu _$fu s* Fs r t Hs.6 3 ffi d irftpr o *o-a tng s d qng UE i r rrrrs- Fs, Fd #s fura args' r-uue e' AS furd -frtsrEc F€rs€F6Er 1fd fF.6d v_urre t s'a f+s * fuJr ileruerd'etryrd dt{ €ryd€a irfunr fr.ro soue S ft 'a fu.aa ffi i t
t9
lo d itc:J
$e.ero-e I ; grg ffi fu' - 2 af3 2 E_(RI-E r + z i fir.ao us irr rfr t
f5: irts o9ppry = x +2
3e pQ)=2+2=4, p(:-2)= -2+2=Ofsr 6*, -2 Er--iRr-s.r + 2 + fr fF.6d A, uir 2 Er-iruE r + z d fu.aa a-fr i t
Qg'usar: E3nr€ p(x)=z.r+ r dFafu.ae &-t;lrer: p(*) A fu5d t* €" -* a fuE 3-g' m{td-d-d
P(x) = o
oo-ua-es'
U- 2"r+l=05rsEr& "=-] nrae'ar
t^ftE sA. -; Elru-E zx + t * Fq fo.ra t I
L
{5 i p(r) = av a a, a + o fra ifr ulnrE ie, 3n lrt* fer p(.r) d fu.aa fui#s F{e d' i gEEra-d 4 f+€, en6 fure. dv Rs fi{E Fa-E i r ugre p1,y
d fu.<a tyc e. rrr+ t r_oue xxt*ra p(r) = 0 f t-u *a-s' r
Udp(.r) =0grmq i ax+b=O,a+O
fex E* ax =-b4fifE[r3 t= -!
h^afeE sS frdd ' = -: fr p6; * fi+.<a t, arceg hfr E!1lJ-e. .t fF-d-€. F{ !fr
atuco F truet ry# fuo afo F{a di fu fudd t, .r - I * Fa fr.6d } ry3 fras -2,.r + z
d fr.ca irQtr.tra-e s : srs de fu z at} 0 qlRfE r'- z.r etat' ffi ua r
fs: ifirsg p@) = x2 - 2x
3E p(2)=2r-4=4-4=O4r3 p(o)=o-o=o
E-dtrshd
fEF sd, 2 trr3 0 iF d aluue.r: - 1;fra1i fs66i 651)flr€t oe- 4fr ,lrarg ffirdt * fd qery1F'
r. H:d-fr -fr fd qnre d fir.ea, fr.e-o fr f,'€ r
z. o, egre d ffe fo.ra i r*dir:. ota dfi slrtre so fuec 3 fuea F* fu.ea od i r
+. Fe elnrE da{. Fd J €q fu f, r*dr+' o-s t
ffi4y7 2.2
t. 36 6-fiprr. t e_urre 5x _ 4l + f er f6 &-€ I
(i)-r=0 G)r=-l QDx=22. t6 m E:&rs'- Ed oi* {d'ue sS p(0), p(t) ari pQ)di:
(i) p(y)=y'z-y+1 (ii)p(t)=z+t+2P-f(ti) p(x) = f (iv) p(.r) = (.r - 1) (.r + l)
3. #€ da fu fu'{€ ar} *o frrrsfurrs Ffu}}n'f€s frtsg votre dar.fuuat(i) p(-r)=r.r+ l; x= -1 ft)p("r)= sx-n: x=lG) p(.r) =l- t; .r= t, -t (iv)p("r) = (.r + l) (-r - 2); x=-1,2
(v) p(x)=a2; a=g (vi)p(.r) =fu+^; *=-1
(vii)p(.r)= z*-r; r=-fi.fr (viii)p(x) =u*tt'=)4. fu- fr*,lt+ Hfuii??r+ f€€ rid xffi Fs slnre $ fr.as d'i t
(i\ p(x)=x+S (ii)p(.r)=x-5 (iii)p(x)=2x+5(r9p(.y)=3x-2 (v)p(.r)=3x (i)p?)=ax;a*O(vii)p(x) = cx + d; c + 0, c, d erpftq. irfiilt{rglOd .
2.4 snd fugdH
4€4rfigfrnr,ir+'ls 4f36 u*itg*n's!-fufrE)rrfr 15 66 6rgsrdr dde dr, s. E g atrdn 2 b€ s'* 3 r1nrs fe.3 | fr 3# H.e-g i'fd
4l
ll ,|| 1i,, J
fuF ts E fdx s-{' etxrlPr1r iTi€r A i ryfr rs 5 fuE gu f+s f?'.,fg d':
15= (2x6)+3
Ed"rfra€ilifasd: g'reo i'uiailHgE .ial* rz d;o a'sgan a-di, s. Unr3l qe'a:
12= (2x6)+O
ffie s.* d i ? fre s'dfr fu.6d a l rrlfr fuo dfo F-{i u' fu o, I z e' Fd{fd-6fu ffacrorl i ii 12. 6 gr qETI tnuttiplct i l
Eo r1rs frce. i fu ofr br* Fc E -_il.re 5 gxi r_c.re au s'dr i F{e d' i
aflQ rs J ufor' ry* ffi- fo E-eo S aHH Tfri Drs fuo 9E a? sE smdF{FdudAel
fuF 6* r{€ bifr $_flr€ 2.c + "t' +, 6 Forr$,. a'g g'dr ilfr} t
(zxr +x'!+x)*"= li tt,'r.f.t
=2t'+r+lEe gm a€ i' fu Zxr + f + xi oia ue f+s' H'c' it ftE 6* )rrfr
Lf +x2+x 6 x(2x2+x+ tli_duEgfuurg€u.tge alfr fuu affi ot fa -r rr€ 2.t' +,r + | E_iRff 2rr + r, +,r i qrds&s ira
4{i 2i + x2 +x,.r erFeqraria€ zr, +.r + I er*FqErdf,irtr-iRtg 3:i +r + I rrri.r erfdi-di€r6qlE* (3x2 + x+ 1) +x= (3r, =-r) + (x+r) + (l *.r) il4tfr afe d- fu 1 f, .r Fs dtr -a 3 n E F* r_urre r{.trs dfr J FfrE t
fuE 6S fuF Effi f€.** fo +A u' ),€ are d fu s.fr r ir fur sS:3*+x+ 1= {(3r+ t)xr} + I
Ei s.dr6g 3"r + I i rlr3 s.d r ir d grfl- fud Frs Fqe i fa "r v_oue3 rt + x + I er F{ qrad&J } ? fu _$fu s.* fF.Fd Ffr A. fex BS fud Er.-d&Ja-fi1r
ry'9 Eeaffi F*E-du-e gfd'-gn'd fu.co qnre a'ri s'rdr {&} I
Eiruilr{i
qfi-d-c 6t pG) q ,e(r) 615 SrdI da, ffi p("r) =x+3x2-lrrr3e(r)=l+-rlifc : ryfr g.ar ia d tfrHar,i. 6 fu- ffi u-df f+s arc0 o' :
urII I : drFI x+3r:- l aligrga(l +"r) 6_iruf+gf6l€di,),fdqsrrE 6gfdi'SD{r ur-3r e urae qx rttescending oiolrl Fe fgrre ir t
fun gf: STiI: 3,t' +.r -1. gtflEf : r + I
uirr 2 : r{it grfr i uffi ue 6 gr+Til avffi rrg a.g sEr fr T', ataq-J a{# 3r, 3x2
$ r a.sa-orft€o'"rl3r".WtrgtIe-]t ,fug gores er rrftrs'rre Eer ]It1-6p .1 , affi dHd d gqres i rrffi rre6rsq!-e'{de ili'u? funErsass6 s'iTH'ura ffi il', afiqg )r{* * * i6 :t615 O|-dr S-ilg Ur llf3 E!-d5-d5 .j.r. + 3x gdrTI 3,rr +,r - I fud uz'rSg -' fug ar{'r'6- sd -2r - l ring EEr i I
= 3r = ETd|€1E' g. Ufog. Ue
-r+ I
3f +3x
-2x-l
u.Ir { : ryEt :Ir6it -:* - t O 6<r gtrIirs' #i a'l s'trd gS q6flr dfrer -Z
= _ zStrrsr z F gs.d' u''qr {dd 3, H.g t _g""16E e- ,{dIB' uE-r{'r1J Ee' ilr = grdr65 gr etrdr ug
sr;1 a€qE- 6+ sH t ufir& u-g - 2r& gqq i rrffi ug.. a.s sqr fgg#ar; x.5 - 3,r',r= ire- i rfun rg'.S('r{S gr €rnirr uE - 2 iI I
q-s15 ; affi gn-gq' $ a.oreo i gFa uE615 qr-r dirE ur ry:r IsH q|--565 6gH f+€'ura -€t 61 a16qs: ffi r 1-1
g - z ar5 qrd' Tdi u.4fi Er-56s-Lr-2f;qf, h t Egura€eu'tffi- s.m i _du fis Frdi I rIRftr lrgrirl
6€r grdkiu
=3x-2
(x + t)(-2)= -2x -2
drtca
fud {ffid?r{' 4{m gg id {dg d' sJ i* fa st sH d ur-s s''+rd' €fr ulr3'6-'t= dra Ffr irs r fur rrar fie, a.a yrfr dfo' psr A al3 c-dT-{s i +J 66Fr-q yd5 dr-drfl6 qa"rs i ti€r A I
sre r: furg{'lrrsrdr65 3x-2 irrrSs.dt I itrrt'qrr{mafr}fuU-erfrHa,?r'f+su#a1-**s'it
3x-2x+1 3x2+x-l
3x2 + 3x
-Zx-l-Lr-z++
fo71.6 fuq fd 3r,+*_ 1=(.r+ t) (3x-2)+l
4ilqg gtrI = ( grtrq x ErEFrS ) + Sri5it
Fqrddfl€s,i p(.x) ,fl-3 8(r) 4{foidr-orrei.efu A;r;6frr!-r >g11)
et u{rs )r.ri g(:) + 0 i, s')rrfr zrffi s-flrg q(r) 4fi r(-r) UnfJ dd F|-{e J'ff,}f618fd
p(x)=g(x)q(x)+t\x),
ftIi r(;r) = 0 ilr r{x) d ur-S< s(x) d urrgt ffi 4itr dfo F4-A dr fu ptxt 6s(-r) 6r{' srdT ad t s-orgg q(.r) a,l3 :rr* r(x) l{t|g Uer a I
Srd& ggtr-ild f*9, gq{ Fd tlfr E_ffE fr | }?{fin-S Hffi f+s,fl€ 'lr*a+g fu v.d? rr3 g.a i ge F fts d* frtq i fr' -fr r
p(x) = 3x' +.r - I f€tl x € FElr6' 3 -l EfgS- 6rdt, Hrq tflIg Eer U:
p(-l) = 3(-l)'+ (-l) -l = I
feEEl81, p\x)=3x2+x- t 6 (.r+ l)615gBTed ?isdqlrslrg a,fuJqS ire. a i- fu s-flre ( .{ ; r ) d fu.aa arse.s - 1 3 Elue p1-r1 er !u ue' i r
,1.€lllfreyi-d@sSBt
soitil4r 45
Qe.a-ou Z : 3f - 4f - 3x -l S.r - I 615 sEI a? |
fs : ffi grar ar5 4- rt'rg trs i3t3-*-x-4
3f-4f-3x-l3/-3f-+
- x3 -3x-l- f + f+--A-r
-* + x+--4x-l-4x+4
-)
s'6fr - s t r pr6. .r - I 1fr fu.sc I t I fug sS p(r) Es' = I dtre ? Hrq t{"rsfe'A'
P(l) = 3(l)a -4(U - 3(1) - 1
=3-4-3-l= -s,i'tusdtt
grircct I p(x)=f +1 6r+ldrssEre€BU'r1-3sdd-Aris:ffig.atatg,
*-t+lx+r l7i-t'+x'
-f +l-l-t++
J+ Ix+ |
d dJ
lso=r'ryrrftu srdrfu B qq-s s'ritg q.e'sa il* fuu Fe s_otre q F* ilfi e-_du-e a'g s'dr ea ? qng seit &E it ftarec ffi afi t ? ud 4r* fer js
6- farraftrc-J fu _edH i _du f€.s sdrgai I
,'r# €e fuF fu _+ar{ e' nrs i fa fu + fe'{'-€.& fu fud fu _eo+ i-s fdg t r
s.d fu -go{, ira sQ p<rl fu + €q mi' F4 i EirrEkr u{€: ersr F4. rurre iDrSim sQ o af e.rst€dfrfonfri ripfxl 6 trfrsluE" - o 6's sn-61q}3"frr+. 3.r- Vr6fr prot ire, i t
x=vs' ira € lr) E-q m k J €q u{B- <rsr ftE r_orre t rrig }16 € fdF <(r)
g r - a 68 dr(il dtsr frrer t, g. s'arss s(r) fe' i a,r3 Sd ,(rllgr u | }}fttsr3
p(x) = (x - a) q(x) + r\x)
fu -€'fu r -, d urs I t br} r(x) d u5s .r - o d urs 3 uia i, fur uS 4r1d q€' = 0 i | fuF e' rrrge fEJ i fd 4ry ffie rmr t r frs ue fuu,rryc, t r
hsd 'ad+{f6i6f 4'y=,}l
fEF6SFE-q.do qqrsEe'irER p(.r) =.r'+ I irfl-3-r+ t =O g'UAr=-t 3t
p(-l)=(-l)3+lfsn'Prs =-1 +l
=0,
fuF sS, p(x) = (.r - a) q(x) + rHr gu f€o i x = a. gr fus xril*aa J r-E fvo qqs ig' 3.
p(a) = (a - a) q(a) + r
fur rg'EdHffiqA;Tifri,?r€'flm fu 6-$i g Fe aa *e.uee "3 uqr E-frB r
?sao-ec,fir: ffi,
/ + I - 2x'? +.r + I g r - I 615 gr-dT A- t qnrs S.* i{'!. f
p(x) = I + f -2x2 + x+ t tar3 r- t dfu.aa t itp(I) = (l)a + (lF - 2(l), + 1+ I =2
4f3,'(-:) =.(+)..(j)' -(+)-'
fsFsE:ts.d1fu=frHtrym.g /+f -2i+x+ r 4; 1.r- t; arsg.ar* Bs.dz {nrsfE }rge"d-fc to : xq a? fq slll.re qe) = 4f + +p - t - t, zt + | e' Fe U|E+ 1 r
iI', fue fu gfr eei t fa q(rl slus 2r + I u (!!-a* fuee gg i+dr Fg fd2, + I F?5 a1r; 6 aet fu t d* sd 6' TsE ?€ r us 2r + I = 0 &J 3 F|.&IgJ{ry3lgrO
I
=0fsrfsxrtd, zr+ t F3dlqnreq(r) erfuqrad&airraeeq(r;, zr+ I srF{qer tr
r{fqwn z.l
1. -ri + 3i + 3x + I 6 +6 ffif)rl+ 615 e.dr a- t s'd #a:
(i) -r+ I G)r- G) r (iv) x+n (v) 5+ b2. f - ax2 + 6x - a t, - o6rssETas Bs'*#A',3. lfs de fq 7 + 3x, 3l + 7.r e' fu ursa&s i m a-S r
2.5 lr-flfsr er qrgs&st€6
ry€ Ee "{# €hrd e ge'o-d-d l0 d Ffu$ 'g tqar.s, frceq fs'a dfrg r
/ r\fur€rcr.ufu-$FEsd al -; I =ot,fuxof zr+ r,q(r)erFaarsskgir\ z./DresrgffiEluEg(r)iI'd,
le3
q(r) = (2t + l) g(t) !€r U I
fEr b ftS tu=+dH A F{ ffi Fffi i:O-c-c.gsfu-edx: i p(.r)ufB n > I €Erfutr-tRf€i€trri,df ewskefrfifl{'i-e, 3'
(i) x - a, p(r) er fu qras&J 9e'3, i p(a) = 0 i-e ),fg
tiitptar = 0 irer i, i r - r. p(.r) g'F{ Orgdt|g ii I
fus a1-gsf€fr s.5} fs_edg Jghs{n"r3 a f,€ a,ig ffinr* furg ffiq*a'.i-rfaadryfr ffi q.i-ili€d3{dAd-d6,Rfu ,rHSr{rgErffe-frs fotefPrr fapr ar
ee'rrgu ll : FITiI a? fa r + 2:l--iRfgr.f + 3xr + 5"r + 6 rrl3 2.r + 4€r €qq[-d&Jiildfrtf5 : -2, r + z d Fq fu.aa it tm sQ
p(x) =.rr + 3-r2 + 5,r + 6 )rf3 s(,r) = 2x + 4
F, p(-2) = (-2)3 + 3(-Z), + 5(-2, + 6
=-8+ 12- l0+6=0
fsF 6f, Erc-5&J fu-9dH lFactor Theorem) t D{gEr(I x +2,x. + 5r, + 5.r + 6€rF*Ere-atrslreqrd , s(-2\= 2(-2)+4=0fF E, x + 2, k + 4 rr f d ere-6q-J e | 4.rFs- f+d, qe-6u-J f _CdH 6Er *3ffi- fr gm ffi Hrs dd F{A i', @fa u + + = z(x + 2) i I
Qguec 12:i{d x-1,4x3+3f -4x+k erFddr-d&Ji,stt etfsii-€r
fr : ffik, - t, p(x) = 4rr + 3r, - 4r + ft €r Fd Er€-6q-J a,
fEF 6dp(l)= oi.iant
E-, p(l) = 4(lf + 3(l)'?- 4(1) + n
s-dtfd4rr 19
fEH s*4ldqEr
4+3-4+k=0k=-3
lra ar# u{B z a€ ur.s 3 e qy Eflre. A EradeJ d-rd 6* d|3-6q-J fu _+dH
e"{d-drilfii'I
Jfr x' + /x + m €d& F urd e-fl .rE t Ar.-desh-d-d j' ep- j r g* iu uek$ cx + br €s fen 3f'eF d'fu ab = - G, q6-68+d-f,c ats'frr 3-ef + Lx + m = (x + a) (-t + b) {nt=r ilEafr fr I ger ry6 * * D.r + c, ffie a * 0 }r[3 a,
b, c )rttt6 TF, i sg'i € ui.* s_&rei g-de-6kJtd-d- *es d d"frr *a'& r
nq qg O ige vAfE .'x: + bx +c gr dre-6ll:rff,ile- u6:r H qar;rsr i r!vvY Y! vl
)fd aq ffi qr5i'C-J (px + e a{i (rJ + .r) ird I Jg,arl + bx + c = G)x + q) (rx + s) = pr x. + Qts + qr) x + qs
I e q-€tr f ?Edr dird i. n'9, =pr r{ers irs A I
fui 3-d', x i Eree. d 3e-s' ild6 B E-o b = ps + qr l{ftrs ug' } I
5u fr, afor rte'd gsa;t a-es ?, E & . = q, r{n1-g EE e I
fsn J fuJ rrs'd-dre. i fu r € iritfi1.rE ps.a€. qre' i-E i, fod'e'fl4665 (ps)(ar) = @r)(qs) =o.itfursf ,ax| + bx + cer Qf-6&*dit€l dir6a 6f, 4{# r 6 a1fta$a1i E frfurrE A frs a grf f+s fgrfC ui'ffi e'EI-d65 ac i..+ | fuJ is # f€ ds ge.d-d-d 13 j. suFz. f, frr+dr I
Qe.uee I : : f,q rre 6 +Fa 4l-J Erc-dhJ frgd}{ d <ef aa} ox2 + | 7 x + 5 €rEr-5-]f*f,dddal
fir t' ( Fq uE C; €s i ) : is-d )r{* },.rfrT&r.rr € Frfor{r+' d-d Fdt AS} fuF'
-G-dra5 tct
p + e = t7 >fii pq = 6 x 5 = 30 j?, 3r H'O- qts6eJ r{nl-3 A Frqra ird. I
fuFsS'r€,r{#30 e{rsd&ri'fu+n-6#irdciSI 3 :0, z 3ts,^ r: .^ ..-:- - -:- -
-.5 3 lU .nJ ) .,Cr d6 l
ffi-ikan'f+s, r'q z i 15 ai3 J p + q =17 r{qrgi-+drl
5o dltb -,
ftF 6t1 6f + l7x+ 5 = 6l + (2 + 15).r + 5
=6x2+2x+15x+5= 2-r(3x + 1) + 5(3.r + l)= (3_r + 1) (2x + 5)
fo : ' (dadq-Jfu-*ax*<oia-o&):
/ r1 5\6x2 + tjx + 5 = 6l x, +!x+:l =6 p(_r). H6 I5E | frild a >ttli b, pe)
\. 6 6)
e"hlti fu' i-J 31 6x2 + 17 x + 5 = 6(x - a) (x - b) i I fsn tt$ aD =
4{€ ,l{fr , 4rf r i rd W ffi 6 ii-d or r fuo t 1, t1, t I, t l, t r a332
Fd€t,,{. ir6rire. rf ' l=' *'tf ' )+ 5 *0ir.rd rfll =oirftxu*- '\2) 4 6\2) 6 \3/
[, . I l, p(.r) Er Fq dedtJ t r ffi e-g. aw a-oi 3fr fuo us' Fdr Ftrg A\ 3l16[ | r+ - l. pt.rl €r IE6t ilId6tGI iI I\ 2) '
fEF se1, 6x2 + ttx+s = 6[x+')[r* 5 j\. 3r\ 2)
-[:x+1)/2x+5)ll-l\ 3 /\ 2 )
= (3.r + 1) (2r + 5)
fuF gE-.'rfrJ -. ta aec * ffi e'r{frdr td?,{.e' r{s.€H.6 } r fea d, rr€'rtfrFqie 9e.il-dds$}r
Ce,-d-d-€ t{ : d!-grtrs fugdx it FU'fEgr 616 ), - 5y + 6 sr qdd&Jtmrd de I
f8' im € p0) = y2 - sy + 6 3 | EJ, i'{d p(y\ = e - a) (J - b) -e a. rfl*fr.sg u'fd fel-e'4ftr6 uE ar i?art fuE sd ab -- 6it fer s* p(y) i{rd5&J Un{s dds e 66 ryfr o d qrea&a ffi o' t
6 eErddE-J t, 2 4€ 3 T6
!€, P(2)=2'z-(5x2)+6=O
; T€illr I
o
E.lqrilryi
fsF6e1, y-2, p0){frd er-5&Jit615fr, p(3)=3'-(5 x 3)+6=0
fuF6*, y-l d l-sy+ oe'ffidOtgdt-Jlt
fEF gfr, y' - 5y + 6 = O - 2) (J - 3)
fq4r.5' fue fu fu rre -sy 6 eai d v'?- 5v + 6 €r qddk5.d *3' frr FtrEr
UI
)re€ Ee: 4{fr fi6 utrdt E-flrEi er Er.-d&Ja' ddt | fte }lairs f+d ts-qH de* rrf a gufrdfr fi io a-fr i+on r x'5' ufu - uiz J ufd Fd Ete-6tra 6-d-€'
tr-d-& t-E 3, fiFr+ fu, 3m d' f€ ari @ f*e iH t
qeqrfJ 15 : x3 - 23f + l42t - 120 Er q|e-c'&*mrd da I
ug: fm€pr*)=xj -23x2 + l42x- l20il
Ea rytr -tzo € x'd Ergr&-s' E ug' sdTr€id | ffi f€€' av Erea&s us t
+1, !2, +3, !4, it5, +6, +8, +10, +12, +15, +2O, +24, +30' 1160
FFrr frrd ",
F-E fr-d fr6-e' a fu p(r) = o 3r feg sf (' - 1), p(-r) €r FdEre-stralt
ue rffi a€ u'fe * -zz* + 142x - !2o = x3 -* -22f +22.x + l2ot - l2o
= x\(x -t) -22x(x- l)+ 120(x - 1) (fug?)
=(r-1) (f -22x+ 120) (.r-l)dEErBAl
ffi p6y g; tr - t; a'r a'ar t ;n' d $rs dfu' fir ntrer fr |
!- f - 22x + 1 20 sr O|d6&f5 fr r 3 ;{q rrE & €5 A a-o* st Arsse-s fu =*o+
b xu.Us'6'6 6fur sr pag. ] , ;o qg $ adrcr a-cs ?, n'6 fuo {trs YE
3f *-22x+120= x2-l2x-lox+120
= x(x - 12) - l0(x - 12)
= (: - 12) (.r - l0)
feg 6S, f -23*- 142x-120= (.r- l)(,r- lOX.r - 12)
5l
5lo||c-J
wfary.E 2.a
1. €F fa fu- Fg iB v_orrgr f€.+ fa"* s_due e, fu urda.&J -r + I i I
(i).t'+x2+x+l (ri)l+:j+x2+x+l(iii);1 +3"C +31 +x+ |
g)2f+7x+3Qv) 3*-x-4
(Dt-3l-9x-5
(rrrr+r"c + Jxl +x+ I (rv) ]i_l _(z* Ji\r*Jt2. Erc-6q-r fu
=edH sEr ildi i-F fu i6i ffuldi Ffui41r. fV+ did Fffifit s(x), pf'l e'F" a=-"g" a m 6-$,
(i) p(.r)=Lr3+f -Lr-l,g(x)=1a I
(ii) p(,r) = r:r + 3x2 + 3x + l, g(x) = x + 2
€tt) p(-r) = 'lf - 4l + x + 6, g(.r) = y - l3. ft er Li.T d'a rg 6 6l*p-1s sffirrr H. uid Hffi E€. (.r _ r ),p6) e'Fc qre-a&a'i'.€:
(1) p(x) = a2 a 1a 11 61) p(x) = 2x2 + kx + Jj(tii)p(;)= ki- Jix+l (rgp(.r)= kt2_3x+k
4. dg6&Jrrgrda:(1) lLx, -'1x + |(iii) 6l + 5"r - 6
5. qast-sai:(i) x3 - 2x2 -x + 2(ii) .t' + 13.x2+ 32x + 2O (iv) 2y3 + y2 -2y - |
2.6 +f, drb|sf, FdEFlrsEr
frIse{'_iH-g'Fy, gntfuorrq q*t fu.*fr drH=-{ FirErHHsr (algebraicidentity; Ed-*fr drffisd H+{d-d ffi ffi i frE-$ ld ds+ e F.A;isr as* HsF i r fired,rt rHBi- Fs, g# i-d frsdr,{, +fr drffi3d-p-d;Ge"rffrsa-a#i
IfililD{Efr | : (.r+))'=.f +zxy +fFliltlFlGlr II : : (x - y)2 = xz - Lry + y2
FkfEIHlGfr llf | * -); = (-:r +y) (.r-y)FftT8lTlGfr tV z (x + a) (x + b) = I + (a + b)x + ab
ffi d stf,-drF--Jd n-dExrrgr€r H€. w d sdJ gm +fr drbs-d
uaJu-€?ry i
f+)lffi{t a Erc-d{:J u€rr dird f€s tgit ff f,-ff r gfi M' * grfi-dg'itreSr (computations) tr d iu rgi i t
gg.dd-€ ro: ffi Filttqlrcrr d €d3 ildi i6 ffi*al'g'E!-5565 uirr
o(d :
(i) (-r + 3) (x + 3) (ii) (r - 3) (r + 5)
i-5 : (it fuR arfr roqgr3r I (x + .y)r = x2 + 2x.v + / d <dJ ild Ftri u' I fEF
FirrIFH:rr tr y = 3 iitr5 ?, n5 qnrs Ee't :
(x + 3) (r + 3) = (.r + 3)r =.r']+ 2(x)(3'l + Q\'z
=i+6x+9(ii) EiffiHlGr IV 9E (-t + (]) (.{ + b)=x2 + (a+b)x+ ar$ urqr {d6- ?' E C;
{'rrsEe'i:(,r - 3) (-r + 5) = xr + (-3 + 5)-t + (-3)(5)
=x2+2x-1.5g€.d-fc t7 : ffii q[d'6r *a& tos x 106 gr Lift r.rsr da I
ir: l05x lo6= (100+5)x(loo+6)
= (100)r + (5 + 6) (lU)) + (5 x 6) (rctfXn:fr Mrqr{di)= l00m+ I lfi) + 30
= lll30ds ii3 ig ftr{Fd er ql--665 r|3r dEE e s6, 4fr fll-d ffi'f' ilfu iisFEEE)€.e' et <-di at-* arftT FdqH!€r-+ *fr drbs{ latrqr er
dE6{dd dadt€-s dgrtrdfr g-e}''riua, ffi fu Cfr i6rfiiS€€I-d-rdfYeiuF{gitqgild-€ lE : {rd6&J tJ=tr 4d
(it 4gu'z + Toub + 25h: ,ii, 'j- " - f,
i-s : til €R gffi fgr iu F{A A fuq9a1 = (7a\1,25b. =(5D1, joab--2('7a)(5b)
.f + 2r-v' + f i ss Ei f,i fuire-q d gsa' *es 3' atfr fuo iri o' fa r=zaryir=sritRrrrHH=rr sEr air6 3, po fuo unrs fE i
49u1 +7oab + 25b. = Qa + 5b), = (7a + 5b) (7a + 5b)
5.1
(iilfYe 25*' - r'=flrl'-ft)'
4 e \2 ) t3lrasnrs ru i 6E fsrd E6-n;. *dd ?, nrO fu-u r3rs Ee'- i
T*, -t = (1,)'-f r)'4 e \2"') t:./(s ))fs y)=(t'-l) l;.-t)
0E- ita Frdfu rn' Frd?,n raexlg€' €- tr$ i qrdfgg' 6'.6 frtfos d-fu. il6 r
al€ Ee affi rorxrse' g Fa' rrg .r + 1' + z B utr a? r Drfr rakamrrs,rrq aai, (.r + ), + z): €rr r5r6re- 6i?rjr I
fm€ r+r=rAt3E,(.r+y+a)r =(t + z)2
= t2 + 2tz + i2 (EiIrIHlGIr I6Ef a-e6- B)
= (.r + l,), + 2(x + y)z + z2 (, Er f.5 yd6 B=f + 2xy +y2 +2xx+2yz+ t (FiIIfiICIT I gEt *eS g
=x2 + f +.22 + Zry + 2yz + Zzr(fe.6 gd-*s fie *a_s $fEF 6S, x'g io furfi rorrrrs' Tu-s 1rS iREEIGT V : (x +J * z)2 = x2 + lt + z, + ?-xy + 2yz + 24t
Eud : rrfr i-i is a f<,irnd g Vn is a fui,-d-d e, HWeru iro-C ir r
fo4Jr6 iee fu 1.r + y * :), t rg.a f€y Bs. <ear rre uifu-sq|-e-565rtEu5t
g-s'ird-J 19 : 13a + 4b + sc;, i @ gu g fuq I
ftt : E3 ia Hrrtr{ d gs-d. (x + y + 7)z i rs a-o-s B, E'5 frs-€- } fqx=3a,y-qbryiz=5c
fur s* rggrrs. v Brfl dir6 ?, n-d; fe-u Unrs t-s a.
5-5
(3a + 4b + 5c)2 = (3a)2 + (4b)2 + (5c)2 + 2(3a)(4b) + 2(4b)(5c) + 2(5c)(3a)
= 9a2 + 168 + 25c2 + 24ab + 4obc + 30ac
Qe"uo-e zo ; (4a -2b -3c)2 gr fur3lq d? |
u{f : HirErFHrrr v sT(Ir dir6 ?, n$ fo qus !-e'}:(4a - 2b - 3c)2 = [4a + (-2b) + (-3c)12
= (4al + (-2b)2 + (-3c)z + 2(4a)(-2b) +2(-2b)(-3c) +2(-3c')(4a)
= 16a2 + 4F + 9C - l6ab + l2bc - 24ac
gg'ao-e zl : 4* + f + t - 4ry - 2yz + 4xz er U-i'&*{ilJ d?lfif: frd 4* +f +22-4ry-2yz+4xz=(2x), +(-))r+ (z), +2(2x)(-y)
= [2r + (-]) * .,' ,*o",.',f''it€r.#'3
= (2x - y + z)2 = (2x - y + z) (2x - y + z)
Ed f{ }rifi e ur.S uer a-6 frtfoa HiItIHrGlr e'fr brfo4?-s ofrs' tl4€Ee ufr rcrrrsr 6 16 (r + v)3 i'so 3 uqr a-fr} t FB
(.r + Y)3 = (x + -v) ('r + ))'?
= (.x + yX-f + 2ry + Y2)
= x(x2 + 2ry + Y'?) + Y(x'? + 2xY + f)= x3 + 2x2y + ry2 + fY + 242 + f= r\ + 3x2y + 3xy2 + Y3
= lr + 13 + 3x"-(x + l)
fsr sS, F'q ic furfr rgornrg'fi{o-et A'
fdEXrGf \'l : 1.r + y)r = .t' + y' + 3ry (r + y)
n-fftill{&vt fes v ? x+-s. B -v iue €' mE qnrs !e' 3'
fdIIFH=tt \ ll : (r -y)3 = r! -Jr - 3ry(x -y)= 't'- 3r'J * 3tJt-{
-ge.crg 22 :i-o ffi urer I ftrerdlr€{ rsu:
(i) (3a + 4b)3 (ii) (5P - 3q)l
fs : (i) (x + r), 6.{, F3 d€ fut|Td; e.. *'3 F d' fr{s€' t td
t=3a4F. Y=4b
ofe3
fsn s*, FiIrIErGrr vt e' qd-ar *a-a €, x5; qnrs Ee'3,(3a + 4b)3 = (3a)3 + (4b)3 + 3(3a)(4b)(3a + 4b)
= 27 a3 + 64b3 + lO8a'zb + l44ab2
(ii) (,x -.r)r 615 ft.f f,B futm{ * gro-a'*ea t E-O fHs-e' i fdx=5p|{i y=3q
EiIETFH=rr VrI g,r r{q-in dirs 3, x-E fgu wlf3 t!-g },(5p - 3q)3 = (5p), - (3q)3 - 3(sp)(3q)(5p - 3q)
= 125p3 - 27 q3 - 225p2q + l35pq2
ge.ffct 23 : Ffr FfirxH<rr Er {A-dr ilsA, t6 ffifiat f€€' ate e'F5 rrs'dd:
(i) (lo4)r
f6:ti)F&(ii) (999)3
( 104)3 = (100 + 4)3
= (100)3 + (4)3 + 3(100X4)(100 + 4)
(FdrrH)t-JI vI Er {q-dr ild6 g= 1000000+64+ 124800
= 1124864(ii)Ee
(999)r= 11669- 1Y
= (1000)r _ (rf _ 3(l000xlxl000 _ l)(HirrlElcrr vII sr qd.o *ec ?r
= 1000000000-l _2gg7(fi
genree z,r : ri.tr + 27r.r * :r*,r ] ;]I'fi..-* *,io' ffi iE t€tid- O fEE 3rg,- fuforrr, ; EEe, a,
-- -'t2rt, + 1:yy + 3(4t)(3y) + 3(2x)(9yz)
= (2x)1 + (3y)3 + 3(zx)reD + 3(Zt)(3y),= (Zr + 3y)r CrfiIFltgr VI Er VA_df d_dd B)= (2x + 3y1 ex + Zy.S (2r + 3v)
E-+r$tii
E-(r+), +z)(x2 +)P+ * -,ry -)z -?r) er f<Fgrd dil5 ?, n-E Ere-a-go fuxgu f€e{n1-s t-e'ix(f + f + z' - ry - yz - u) + y(f + y'z + z' - ry - yz - u)
+z(f+f+z'-ry-yz-zt)=f +ry2+xz2-x2y-ryZ-zl + ly +.y3 + yz2 - xy2 -fz-ry2
+ fz + fz + z3 - xyz - yz2 - xz'
=x3 +y3+z'-3ry2 (Fd5-d6' €)fsx 6f , x-g to fzrrfr rouxrs. t1'trs i1-d ] t
FiIIIEIGF VIII : ri +y1 + f -3xyz=(x +y + 7) (xz +y2 + z'- ry -yz-zt)qe.d-fc 25 : 8.f + f + 27 z' - lSxyz er Er--5&*{d€ d? l
fir, EBg.t'+_v3+2723_lgxyz
= (2x)3 + )) + (32)3 - 3(2x)(y)(32)
= (2x + y + 3z)[(2x)' + v2 + (32)2 - (2r)Cv) - (vX3z) - (2r)(32)l
= (2x + y + 1z) (4f + y2 + 922 - 2xy - 3yz - 6xz)
arfumrr z.q
1. ga$ rasxrJ. e' {dal ildi i6 H Orasao' E f;s us' a?:
(i) (.r + 4) (x + 10) (n) (.r + 8) (x - l0) (iii) (3x + a) (3x - 5)
3.,, 3.(iv)(y'z+ :)0'-:) (v) B-2xt(3+2x)
z. fiqfr are' fr3 fr6r t6: ffif4{r El--56{'r a f;g rj-Jr da:
(i) 103 x 107 (ii) 95 x 96 (iii) 104 x 96
s. g{+ FdEErrs' e' qfr'dr ilfii td ffiff. e' qra;ffi{da da:
(i) 9l+ 6ry+y' (tiaf-aY+r (OO-*a. g*S rosrrs' e' qfrar {d& A-d' ffiadt- f€+ dtd e' fuF3rd da:
(i) (x+2y+42)2 (i) (2-r-y+z)'z (O (-2x+ 3Y+22)2
(iv) (3a-7b - c'), (v) (-2x + 5y -32)'? (o Ito"
-
",
- tf'
is
s. q!---6&*{d€da:(i) 4f + 9y2 + 1672 + 12ry - 24yz - l6xz
(i) 2f+y, +822 - zJiry + 4^fzyz-8xz6. id ffi ure. g- Hqsgrr f+s ffi
(i) (2r + 1)3 6) e.a-3b)3f3 I' I r lr
fiii) L;'*,1 t*) L'-;rl7. g** ra:xrrs' e' q$-dr a-d* i6 ffi+ar. t !.o d-i:
(t (99)3 0D (102)3 GD (998)3
E. t6 ffv,1.r, f+g ot* E, qr_dffi{ds d?:(i) 8ar + b3 + l2n2b + 6ab2 G) 8a3 - b3 _ I2a2b + 6ab?
(iD 27 - 125a3 - l35a + 225a2 (iv)64a3 _ 27b3 _ l44a2b + l08ab2
(vt 27pt - "l - 1r, - \,.2t62'4
9. tE d?:(i).t' + y3 = (x + y) (l - ry + yr) (ii) -rr - y, = (x -y) (x2 + xy + y2)
to. +6 firnrln- f€€' ote e' qrasffi*re a?:(i) 27y3 + 12523 (ii) 64m3 _ 343n3 [ffis: Aq qr6- 9]
ff.
12. glda?: f +y3+23-3xyz= -L ("* y+ z) l(r - y), + (y - z), +(z- r)r)13. i .r +y + z = Oi g'futg fu :i +y3 + z3 = 3ryz it14. ur€r S )rfFg ara6' ati H i-6 ffifr,n f€+ ote u, f;u rrs, a!:
(i) (-12)3+(7)3+(5)3 (ii) (28)3+(-15)3+(_l3fls. d qy 4{rb e ts-dds ff3 dre il6 | did d 6-s'f }rri ffi 6S
fi-g-€futrtrdH.:
U=kf65: 25d2 - 35a+ 12 +3d65: 35yz + l3y -12
(i) (ii)
!;.J{t4b{i
16. urd-€i (cuboids), if,Ti A lfrfe-Jd to;r-ffi aE uo i Hn{i 6dt Fd€Hrg-*frusi
4{'frg6': 3i - l2x 4f'fE-Jd: l2kyl + 6ky - 20k
(i) (iD
2.7 frtd *EftE ,rrtalr'fu fis, i6 ffi E{iJ4ri e' rrrfo,ira frs' }:
r. ffi ss <.s. EUrrs p(r) i-6 fu&'=au e. x Ee fa dla drfEfud f€rtrffi 3:
p(x) = aaf +a, /-' + ...+ a;f + arx+ao,
ffid ao, a' ar,. . ., a, }RI|f il?f rr|] a,*}ll on u,.a,,...,d" {}f€tf: /,r,"f,...,r ? qe-q us'rrr3 r 6 €1flJ-e d rqrg fuTr iri€€ | ai* ar.a,,f,t.....on.ffi a, + O. 5 V-tRIe 4xt g. rfg fuOt grg. i t
2. ff4. us <re v_Oue O F6..6i| fuar siarr ]13. A us. <8 aloue O e u-S kJr fri€r 3l4. fgd rrer <6 vflre 6- fva.rS kJr FJ€r Al
s. ffi ul-s e€ egre O ilfi s:du-e kJr Fre? tlo. € ur.g <6 v-ore 6 € urr$ F_flre qftr€ uar7. B6 u{s <"& r_ore $ u.$ ulowe affi ustr. e'rsk{ trfoaf' .a', rt-_Jrf€ p(r) d fr"{ f1-.€-d ffi A, i p(o) = 0 i€ r
e. ffi srr, f+s ilA-d irfi r-ure f Fd Htr€ fu.e-o !-g }tFq i-e fir.era)rfirs s!|lJg d dS fF.dd 6-s ir6}'€ otq e.rsFq frn{4' fu3d EaIIEdtrqfir.aoirslt
lo. edfr-&sx, i ptxr, Fa ? €q s. Fq i gaqa urg e's'-e!nrs A'€, Dr3
49 6 ilfr v-orre r - a 6'6 9k *3t i€ g'gd p<al ire't t
1r. i p(a) = 0i'€,3r.r- a EEtr€ p(x) e' F* grea&s ie'tal3 ir-a,p(x)e'Er€-dc-J i-+' 3r p(c) = a igr]1
12. (x + y + z)2 = f + y2 + z2 + 24 + 2yz + ?u13. ("r+ Y)' =t' 1f + 3rY(x +Y)
14. (x-Y)3=f -f -3xY(x - Y\
15. f +f + z3 -3xyz= (x + y + z) (x.2 + y2 + z'z - xy -yz- u)
r,{foadu3
ffiH)tr6rff'Hrei,,\\'h:rt s tlrt' frn)(l ol Mcrcitlor's North Polcs and Lquators, l ropics, Zones andMelitlirrr I irresl So lhc lJcllmarr would cry: and crew woull rsply . They aretrrcrely r'ortr r. rrtrrrrritl sigtts!'
{ )rdded g Q .ril qge' rrg g-r.rq iu., gtfs ug-, g-tBltr afiHtsry,r du' ftE di tffi- ii r fux stt n?qi6 Ce6r urfuD{rd€oT. ,irg af<q en - €*Jd fE3r ieil','.fuT q-r6 uinrdBrgf€6 ua " rr
LEWIS CARROLI4 The Hunting of the Snark
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