02b Pure Mathematics 1 - October 2021 Mark Scheme
November 13, 2022 | Author: Anonymous | Category: N/A
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Bdrg Scfnbn (Xnsults)
Actahnr 8:84 Zndrsak Nmnxcnl Oktnrkdtoakdl D Lnvnl Ok Zurn Bdtfnbdtocs Z4 (UBD44) Zdpnr :4
Nmnxcnl dkm H]NC Tudloiocdtoaks
Nmnxcnl dkm H]NC qudloiocdtoaks drn dwdrmnm hy Zndrsak, tfn ^G‟s ldrenst dwdrmoke hamy. Un pravomn d womn rdken ai qudloiocdtoaks okclumoke dcdmnboc, vacdtoakdl, accupdtoakdl dkm spncoioc praerdbbns iar nbplaynrs. Iar iurtfnr okiarbdtoak vosot aur qudloiocdtoaks wnhsotns dt www.nmnxcnl.cab ar www.htnc.ca.ug www.nmnxcnl.cab www.htnc.ca.ug.. Dltnrkdtovnly, yau cdk ent ok taucf wotf us usoke tfn mntdols ak aur caktdct us pden dt dt www.nmnxcnl.cab/caktdctus. www.nmnxcnl.cab/caktdctus .
Zndrsak0 fnlpoke pnapln praernss, nvnrywfnrn Zndrsak dsporns ta hn tfn warlm‟s lndmoke lndrkoke cabpdky. Aur dob os ta fnlp nvnryakn praernss ok tfnor lovns tfrauef nmucdtoak. Un hnlonvn ok nvnry gokm ai lndrkoke, iar dll gokms ai pnapln, wfnrnvnr tfny drn ok tfn warlm. Un‟vn hnnk okvalvnm ok nmucdtoak iar avnr 4>: yndrs, dkm hy wargoke dcrass 5: cauktrons, ok 4:: ldkeudens ldkeudens,, wn fdvn huolt dk oktnrkdtoakdl rnputdtoak iar aur cabbotbnkt ta foef stdkmdrms dkm rdosoke dcfonvnbnkt tfrauef okkavdtoak ok nmucdtoak. Iokm aut barn dhaut faw wn cdk fnlp yau dkm yaur stumnkts dt0 www.pndrsak.cab/ug www.pndrsak.cab/ug
Actahnr 8:84 Tunstoak Zdpnr Lae Kubhnr Z999=>D Zuhlocdtoaks Camn UBD44Q:4Q844:Q UBD44Q:4Q844:QBS BS Dll tfn bdtnrodl ok tfos puhlocdtoak os capyroeft © Zndrsak Nmucdtoak Ltm 8:84
Euomdkcn Enknrdl Bdrgoke Euomdkcn
•
Dll
cdkmomdtns
bust
rncnovn
tfn
sdbn
trndtbnkt.. Nxdboknrs bust trndtbnkt bust bdrg tfn iorst cdkmomdtn cdkmomdtn ok nxdctly tfn sdbn wdy ds tfny bdrg tfn ldst. •
Bdrg scfnbns sfaulm hn dpplonm pasotovnly. Cdkmomdtns bust hn rnwdrmnm iar wfdt tfny fdvn sfawk tfny cdk ma rdtfnr tfdk pnkdlosnm iar abossoaks.
•
Nxdboknrs sfaulm bdrg dccarmoke ta tfn bdrg scfnbn kat dccarmoke ta tfnor pnrcnptoak ai wfnrn tfn erdmn haukmdrons bdy lon.
•
]fnrn os ka cnoloke ak dcfonvnbnkt. Dll bdrgs ak tfn bdrg scfnbn sfaulm hn usnm dppraprodtnly.
•
Dll tfn bdrgs ak tfn bdrg scfnbn drn mnsoeknm ta hn dwdrmnm. Nxdboknrs sfaulm dlwdys dwdrm iull bdrgs oi mnsnrvnm, o.n. oi tfn dkswnr bdtcfns tfn bdrg scfnbn. Nxdboknrs sfaulm dlsa hn hn prnpdrnm ta dwdrm znra bdrgs oi tfn cdkmomdtn‟s rnspaksn os kat wartfy ai crnmot dccarmoke ta tfn bdrg scfnbn.
•
Ufnrn sabn `umenbnkt os rnquornm, bdrg scfnbns woll pravomn tfn prokcoplns hy wfocf bdrgs woll hn dwdrmnm dkm nxnbploiocdtoak bdy hn lobotnm.
•
Ufnk nxdboknrs drn ok mauht rnedrmoke tfn dpplocdtoak ai tfn bdrg scfnbn ta d cdkmomdtn‟s rnspaksn, tfn tndb lndmnr bust hn caksultnm.
•
Crassnm aut warg sfaulm hn bdrgnm ^KLNSS tfn cdkmomdtn fds rnpldcnm ot wotf dk dltnrkdtovn rnspaksn.
ZNDXSAK NMNVCNL ODL BD]FNBD]OCS
Enknrdl Okstructoaks iar Bdrgoke
4. ]fn tatdl kubhnr ai bdrgs iar tfn pdpnr os 5>.
8. ]fn Zndrsak Bdtfnbdtocs bdrg scfnbns scfnbns usn tfn t fn iallawoke typns ai bdrgs0
B bdrgs0 Bntfam bdrgs drn dwdrmnm iar ‖gkawoke d bntfam dkm dttnbptoke ta dpply ot‟,
•
uklnss atfnrwosn okmocdtnm. o kmocdtnm. D bdrgs0 Dccurdcy bdrgs cdk akly hn dwdrmnm oi tfn rnlnvdkt bntfam (B) bdrgs fdvn
•
hnnk ndrknm. ndrknm. •
H bdrgs drn ukcakmotoakdl dccurdcy bdrgs (okmnpnkmnkt (okmnpnkmnkt ai B bdrgs)
•
Bdrgs sfaulm kat hn suhmovomnm.
3. Dhhrnvodtoaks
]fnsn drn sabn ai tfn trdmotoakdl bdrgoke dhhrnvodtoaks tfdt woll dppndr ok tfn bdrg scfnbns dkm cdk hn usnm oi yau drn usoke tfn dkkatdtoak idcoloty ak nZNK. •
ham – hnkniot ai mauht mauht
•
it – iallaw tfrauef
•
tfn sybhal
•
cda – carrnct dkswnr akly
•
csa - carrnct salutoak akly. ]fnrn ]fnrn bust hn ka nrrars ok tfos pdrt ai tfn qunstoak ta ahtdok tfos bdrg
•
osw – oekarn suhsnqunkt wargoke
•
dwrt – dkswnrs wfocf raukm ta
•
SC0 spncodl cdsn
•
an – ar nquovdlnkt (dkm dppraprodtn)
•
m… ar mnp – mnpnkmnkt
•
okmnp – okmnpnkmnkt
•
mp mncobdl pldcns
•
si soekoiocdkt ioeurns
•
ar it woll hn usnm iar carrnct it
]fn ddkswnr kswnr os proktnm ak tfn pdpnr ar de- dkswnr eovnk
ar m… ]fn sncakm bdrg os mnpnkmnkt ak edokoke tfn iorst bdrg
•
=. Dll D bdrgs drn ‖carrnct dkswnr akly‟ (cda.), uklnss sfawk, iar nxdbpln, ds D4 it ta okmocdtn tfdt prnvoaus wrake wargoke os ta hn iallawnm tfrauef. Ditnr d bosrndm fawnvnr, tfn suhsnqunkt D bdrgs diinctnm drn trndtnm ds D it, hut bdkoinstly dhsurm dkswnrs sfaulm knvnr hn dwdrmnm D bdrgs.
>. Iar bosrndmoke wfocf mans kat dltnr tfn cfdrdctnr ai d qunstoak ar bdtnrodlly sobploiy ot, mnmuct twa irab dky D ar H bdrgs edoknm, ok tfdt t fdt pdrt ai tfn qunstoak diinctnm. Oi yau drn usoke tfn dkkatdtoak idcoloty ak nZNK, okmocdtn tfos dctoak hy ‖BX‟ ok tfn hamy ai tfn scropt.
9. Oi d cdkmomdtn bdgns barn tfdk akn dttnbpt dt dky qunstoak0
•
Oi dll hut akn dttnbpt os crassnm aut, bdrg tfn dttnbpt wfocf os KA] crassnm aut.
•
Oi notfnr dll dttnbpts drn crassnm aut ar kakn drn crassnm aut, bdrg dll tfn dttnbpts dkm scarn tfn foefnst sokeln dttnbpt.
5. Oekarn wrake wargoke ar okcarrnct stdtnbnkts iallawoke d carrnct dkswnr.
6. Bdrgs iar ndcf qunstoak drn scarnm hy clocgoke ok tfn bdrgoke eroms tfdt dppndr hnlaw ndcf stumnkt rnspaksn ak nZNK. ]fn bdxobub bdrg dllacdtoak iar ndcf qunstoak/pdrt qunstoak(otnb) os snt aut ok tfn bdrgoke erom dkm yau sfaulm dllacdtn d scarn ai ‖:‟ ar ‖4‟ iar ndcf bdrg, ar “trdot‒, ds sfawk0
: dB dD
4 ◁
◁
hB4
◁
hD4
◁
hH
◁
hB8
◁
hD8
◁
?. Hn cdrniul wfnk scaroke d rnspaksn tfdt os notfnr dll carrnct ar dll okcarrnct. Ot os vnry ndsy ta clocg mawk tfn ‖:‟ calubk wfnk ot wds bndkt ta hn ‖4‟ dkm dll carrnct.
Enknrdl Zrokcoplns iar Carn Bdtfnbdtocs Bdrgoke Bdrgoke (Hut katn tfdt spncoioc bdrg scfnbns bdy sabntobns avnrromn tfnsn enknrdl prokcoplns).
Bntfam bdrg iar salvoke 3 tnrb qudmrdtoc0 4. Idctarosdtoak
( x 8 + hx + c) 2 ( x + p )( x + q), wfnrn pq 2 c , lndmoke ta x 2 … 8 (dx + hx + c) 2 (bx + p)(kx + q ), wfnrn pq 2 c dkm bk 2 d , lndmoke ta x 2
… 8. Iarbuld
Dttnbpt ta usn tfn carrnct iarbuld (wotf vdluns iar d, h dkm c). 3. Cabplntoke tfn squdrn 8
h 8 Salvoke x + hx + c 2 : 0 x ´ ´ q ´ c 2 :, q ≪ : , lndmoke ta x 2 … 8 Bntfam bdrgs iar moiinrnktodtoak dkm oktnerdtoak0 4. Moiinrnktodtoak k ∘4 Zawnr ai dt lndst akn tnrb mncrndsn mncrndsnm m hy 4. ( x k ↘ x )
8. Oktnerdtoak k +4 Zawnr ai dt lndst akn tnrb okcrndsnm hy 4. ( x k ↘ x )
^sn ai d iarbuld
Ufnrn d bntfam okvalvns usoke d iarbuld tfdt fds hnnk lndrkt, tfn dmvocn eovnk ok rncnkt nxdboknrs‟ rnparts os tfdt tfn iarbuld sfaulm hn quatnm iorst. iorst . Karbdl bdrgoke bdrgoke pracnmurn os ds iallaws0 Bntfam bdrg iar quatoke d carrnct iarbuld dkm dttnbptoke ta usn ot, nvnk oi tfnrn drn sbdll nrrars ok tfn suhstotutoak ai vdluns. Ufnrn tfn iarbuld os kat quatnm, tfn t fn bntfam bdrg cdk hn edoknm hy obplocdtoak irab carrnct wargoke wotf vdluns, hut bdy hn last oi tfnrn os dky bostdgn ok tfn wargoke. Nxdct dkswnrs
Nxdboknrs‟ rnparts fdvn nbpfdsosnm tfdt wfnrn, iar nxdbpln, dk nxdct dkswnr os dsgnm iar, ar wargoke wotf surms os clndrly rnquornm, bdrgs woll karbdlly hn last oi tfn cdkmomdtn rnsarts ta usoke raukmnm mncobdls.
Tunstoak Kubhnr
Scfnbn
4.
∧
4
3
4
4 8
2 3 x + x + 3
Dpplons
∧
∘3
=
4 4 3 x x 48 x + ∘ = mx 2 48 Õ + Õ 8 x 8 ∘ Õ = 9 8 ∘3 9 x 8 x 3
=
B4
Bdrgs
x mx ↘ x k
k +4
4
x
∘3
B4
+c
D4D4D4D4
8
(>) (> bdrgs)
iar dt lndst akn okmnx. 4
]fn okmnx bust hn pracnssnm sa dllaw iar x ↘ x 3
D4 D4
4 4 4 ∘3 Akn carrnct tnrb sobploionm ar + c. Laag iar akn ai 3 x , + x 8 , + x ar tfn + c. 3 8 ]wa carrnct tnrbs sobploionm ar akn carrnct sobploionm wotf + c. =
4 3
4 8
4
∘3 + x 8 , + x , + c
]frnn carrnct tnrbs sobploionm ar twa carrnct sobploionm wotf + c. =
Laag iar tfrnn ai 3 x , D4
ar
4 4 8 ∘3 ↘ x ar ↘ x = x x
=
Laag iar twa ai 3 x , D4
=
4 3
4
4 8
+ x 8 , + x ∘3 , + c
4 4 4 ∘3 3 x + x 8 + x + c dll carrnct dkm sobploionm dkm ak akn lokn. 3 8 =
Dllaw sobploionm nquovdlnkts sucf ds
4 ∘3 4 4 4 x 8 ↝ x dkm x 8 3 3
↝
4 8 x
3
Dwdrm akcn d carrnct nxprnssoak os snnk dkm osw hut oi tfnrn os dky dmmotoakdl/okcarrnc dmmotoakdl/okcarrnctt katdtoak dkm ka carrnct nxprnssoak fds hnnk snnk ak ots awk, wotffalm tfn iokdl bdrg. N.e.
∧
3 x + =
4 48 4 ∘3 4 4 4 ∘3 = x + x + c mx , 3 x + x 8 + x + c 2 : 3 8 3 8
Tunstoak Kubhnr 8.
Scfnbn
Bdrgs
m y 2 4> x = + 48 x 8 ∘ 4 m x 4> x = + 48 x 8 ∘ 4 2 8 ⇘ 4> x = + 48 x 8 ∘ 3 2 : 8 8 ⇘ ∘ + 2
y 2 3x + = x ∘ x + > ⇘ >
3
3(> x 4)( x 4) 4 a.n. ⇘ x 2 ´
: a.n
>
B4 D4 mB4 mmB4 D4 (>) (> bdrgs)
B4
D4
↘ x k ∘4 iar akn carrnct pawnr > = 3 8 Dllaw iar x ↘ x ar x ↘ x ar x ↘ 4 m y = 8 m x 2 4> x + 48 x ∘ 4 wfocf bdy hn lnit uksobploionm. @ust laag iar d carrnct Dttnbpts ta moiinrnktodtn wotf x
k
nxprnssoak o.n. ka knnm ta snn mB4
Snts tfnor
m y 2 ... m x
m y 2 8 dkm callncts tnrbs ta akn somn ta ahtdok d 3]T ok x 8 m x
Mnpnkms ak iorst bntfam bdrg. 8
mmB4 Carrnct dttnbpt ta salvn salvn 3]T ok x . ]fos bdy hn hy idctarosoke, usoke tfn qudmrdtoc
iarbuld, ar cabplntoke tfn squdrn (snn enknrdl euomdkcn). ]fn dttnbpt ta idctarosn bust hn caksostnkt wotf wotf tfnor 3]T. ]fn carrnct qudmrdtoc wotf tfn carrnct dkswnrs `ust wrottnk mawk scarns B: Bust hn salvoke iar x iar x8 kat kat x x ta ta ahtdok dt lndst akn vdlun iar x iar x8. Mnpnkms ak hatf prnvoaus bntfam bdrgs.
D4
x 2 ´ 4 ar nxdct nquovdlnkt sucf ds ´ > , ´ 3 dkm osw akcn tfn carrnct dkswnrs > 4> > drn snnk. 4 os D:. Bust snn hatf vdluns sa x 2 > Oekarn dky dttnbpts ta iokm tfn y tfn y caarmokdtns.
Tunstoak Kubhnr
Scfnbn
3.(o)
Bdrgs
3 3 ; = ⇘ 3 x ; = x 8 ⇘ x ( = x ∘ 3 ) < : ⇘ :, x = : < x <
H4
3 =
B4 D4
3( x + >) y ∘ : 2 3(
H4
N.e. y < 8 x ∘ >:, y ; 3x + ''4> ''
B4
N.e. y < 8 x ∘ >:, y ; 3x + 4>, x < ∘>
D4
(oo)
8
8
(3)
(3) (9 bdrgs)
(o)
3 =
H4
Iar tfn twa crotocdl vdluns : dkm
B4
Cfaasns tfn oksomn rneoak iar tfnor crotocdl vdluns
D4
: < x <
3 =
Dwdrm iar nxdct nquovdlnkts sucf ds x ; : dkm x <
3 ar =
3 x 0 x ; : ∯ x < = (Katn tfny bdy mnmucn tfdt x ; : ds
3 3 ; = tfnk salvn ta iokm x < wfocf cabhoknm = x
3 ) = ........................................................................ ...................................... .................................................................... ............................................................ ................................................ ...................... eovns : < x <
H40 Sgntcfns HA]F erdpfs. Bdy akly snn roeft fdm hrdkcf ai fypnrhald.
y 2 =
B40 Cfaasns tfn oksomn rneoak hntwnnk : dkm 3 tfnor salutoak ta 2 =
A
x
D40 : < x <
3 =
.......................................................................................................... ........................................................................ ............................................................ ................................................ ...................... Spncodl Cdsn wfocf os vnry cabbak0 3 H40 Stdtns x < akly = ......................................................................... ....................................... ................................................................... ........................................................... ................................................ ......................
(oo) H4
Carrnct nqudtoak iar l N.e. y ∘ : 2 3( 3( x + >) . ]fos bdy hn obplonm hy n.e. soeft ai y ai y ; 3 x + x + 4> ar n.e. y n.e. y 2 2 3 x + x + g dkm dkm g 2 2 4>
B4
]wa ai y < 8 x ∘ >:, y ; 3x + "4>", x < d wfnrn ∘ > d 9.> Iallaw tfrauef tfnor strdoeft lokn pravomnm ot fds d erdmonkt ai 3 wotf d kubnrocdl “4>‒. 8
‛
‛
8
Dlsa dllaw twa ai y 8 x ∘8 >:, y 3x + 4>, x Dlsa dllaw 3 x + 4> < y < 8 x ∘ >: ar 3x + 4> ‛
…
d wfn8rn ∘ > d 9.> y 8 x ∘ >: ds 8 oknqudlotons.
‛
‛
‛
‛
‛
Ma kat dllaw oknqudlotons ok tnrbs ai X ai X n.e. n.e. X < 8 x ∘ >:, X ; 3x +4 4> >. ]fos scarns B:. 8
Iully mniokns rneoak. N.e. y < 8 x ∘ >:, y ; 3x + 4>, x < d wfnrn ∘> 8
D4
8 x ∘ >:, y 8
3x + 4 4> >, x
‛
d
‛
9.>
d wfnrn ∘> d 9.> Oi snt katdtoak os usnm, tfnk tfny bust usn “ ∯‒ hntwnnk dky ai tfnor oknqudlotons o knqudlotons rdtfnr tfdk “∠‒. Cakmakn dttnbpts ds lake ds tfn oktnktoak os clndr. N.e. Dlsa dllaw y
‛
{ x, y ∆ 0 y < 8 x
8
…
‛
Katn rnedrmoke rnedrmoke caksos caksostnkcy tnkcy iar tfn D4 oi > os usnm0
wotf x < < ∘> y < 8 x ∘ >:, y ; 3x + 4> bust ea wotf x 8
Oi
8 x ∘ >:, y 8
‛
∘> < d
‛
‛
‛
∘ >: ∯ y ; 3x + 4> ∯ x < d} , { x, y ∆ 0 y < 8 x 8 ∘ >:, y ; 3x + 4>, x < d}
drn dccnptdhln.
y
3x + 4 4> > bust ea wotf x wotf x
…
9.> os usnm tfnk x < d ar x
‛
‛
∘>
d os dccnptdhln. dccnptdhln.
Tunstoak Kubhnr
Scfnbn
=.(d)(o)
( ?:, ∘4)
(oo)
88>
Bdrgs H4 H4 H4 (3)
(h)
Akn ai Hatf
∘4 < p < : , p 2 4 ∘4 < p < : , p 2 4
B4 D4 (8) (> bdrgs)
(d) (o) H4
46: :, ∘4) Akn caarmokdtn carrnct ok tfn carrnct pasotoak. N.e (46 Dwdrm iar x 2 ?: ar y 2 ∘ 4 . Cakmakn x 2 ?:º dkm cakmakn iar
H4
46: iar ?: 8
Iully carrnct ( ?:, ∘4) wotf ar wotfaut hrdcgnts. Dwdrm iar x 2 ?: dkm y 2 ∘ 4 . Cakmakn x 2 ?:º dkm cakmakn iar Spncodl Cdsn0 Oi tfny eovn
hut n.e.
( ∘4, ?: ) ar
46: iar ?: 8
ϊ rdtfnr tfdk ( ?:, ∘4) dllaw H4H: ∘4, 8
( ∘4, 46: ) scarns H:H: (8 nrrars)
Spncodl cdsn0 Oi tfn ?: dkm ∘4 drn clndrly okmocdtnm ak tfn dxns iar T ak tfn sgntcf, scarn H4H:
(d)(oo) H4 88> Cakmakn 88>º. Dlsa dllaw (88>, :) ar (88>º, :) ar 88> ar 88>º bdrgnm dt X dt X ak ak tfn moderdb hut oi tfnrn os dky dbhoeuoty, tfn hamy ai tfn scropt tdgns prncnmnkcn. Iar cdkmomdtns wfa usn rdmodks ok pdrt (d), pnkdlosn tfos akcn ak tfn iorst accurrnkcn sa n.e.
(h) B4
(d)(o)
ϊ 8 , ∘4
(d)(o)
∘4, ϊ 8
(d)(oo)
>ϊ scarns H4H: H4 =
(d)(oo) >ϊ scarns H:H: H4 (8 nrrars)
=
Akn ai tfn ''carrnct'' pdor, hut it ak “∘4‒ irab (d).
46:, ∘8 ) B4 cdk hn dwdrmnm iar notfnr Iar nxdbpln oi tfny hnlonvn tfdt (d)(o) os (46:
∘8 < p < : ar p 2 8 Cakmakn tfn usn ai y rdtfnr tfdk p iar tfos bntfam bdrg n.e. cakmakn
∘4 < y < : ar
y 2 4 D4
Iully carrnct ok tnrbs ai p ai p
D soekoiocdkt kubhnr ai cdkmomdtns drn wrotoke tfnor dkswnrs wotfok tfn qunstoak sa hn surn ta cfncg dkswnrs dppndroke tfnrn.
Tunstoak Kubhnr >(d)
Scfnbn
Bdrgs
3 y ∘ 8 x 2 3: ⇘ b 2
y 2 ∘
8 3
H4
3 3 3 y ∘ : 2∘ ( x ∘ 8= 8=), y 2 ∘ x + 39, 8 y + 3x 2 5 58 8, 8 8 8 x ∘ 8=
Iull bntfam ta iokm akn ca-armokdtn ai Z ai Z N.e. N.e. Salvns Caarmokdtns ai Z ai Z (48,46 )
8
x + 4: 2 ∘
3
3
( x ∘ 8= )
8
B4 D4it B4 D4 (>)
(h)
B4 (H4 ak NZNK)
y 2 :, 3 y ∘ 8 x 2 3: ⇘ x 2 ... Drnd HZD Drnd HZD 2 2
4 Õ ( 8= + ''4>' >''') Õ ''46'' 6'' 2 3>4 8
mB4 D4 csa (3) (6 bdrgs)
(d) H4 B4
8 snnk ar obplonm 3 Iull bntfam iar nqudtoak ai l 8 . ]fos okv okvalvns alvns dk nqudtoak usoke tfn paokt ( 8=,: ) Erdmonkt ai l 4 2
wotf d erdmonkt usoke tfn t fn knedtovn rncopracdl rncopracdl ai tfnor
8 . Oi tfny usn y usn y 2 2 bx bx + + c tfnk 3
bust rndcf ds idr ds c 2 … D4it
Carrnct nqudtoak ai karbdl ok dky iarb n.e. y 2 ∘ katdtoak fnrn dkm dllaw n.e. l8
3 ( x ∘ 8=) hut cakmakn paar 8
3 8
2 ∘ ( x ∘ 8=) pravomnm tfos os usnm dppraprodtnly ta
salvn sobultdknausly. Iallaw tfrauef tfnor knedtovn rncopracdl rncopracdl erdmonkt. B4
Iull bntfam ta iokm akn caarmokdtn ai Z ai Z Katn tfdt wn dllaw dllaw d cdlculdtar cdlculdtar ta hn usnm fnrn fnrn n.e.
D4
3 y ∘ 8 x 2 3:, 8 y + 3x 2 58 ⇘ x 2 ... ar y 2 ... Caarmokdtns ai Z ai Z (48, 46 ) ar x ar x 2 2 48, y 48, y 2 2 46
Dlt (d)
Katn tfdt tfn iorst tfrnn bdrgs cdk cdk hn scarnm scarnm vod H4
Dttnbpts l 8 vod 8 y + 3x 2 c
B4
Iull bntfam iar nqudtoak ai l 8 . ]fos okvalvns dk nqudtoak usoke tfn paokt ( 8=,: ) dkm
8 y + 3 x 2 c D4
8 y + 3x 2 58
(h) B4(H4 ak NZNK)0 ^sns y ^sns y 2 2 : ok l 4 ok dk dttnbpt ta iokm tfn tfn x x caarmokdtn caarmokdtn ai H ai H.. Cfncgoke dky wargoke iar tfos bdrg hut ot bdy hn obplonm hy x hy x 2 2 ´4> mB4 Carrnct dttnbpt dt drnd HZD drnd HZD usoke tfnor ∘4> dkm 46. ]fos rnquorns
4 4>''' ) Õ ''46'' 46'' ar nquovdlnkt Õ ( 8= + ''4>' 8 4 carrnct warg wotf tfnor vdluns n.e. 8 Õ Ar n.e. “sfanldcn bntfam‒
8
8
( 8= ∘ 48 )
+
46
8
8 Õ
(48 + 4> )
+
46
48 8 ∘4 > 4 4 ∘ 4> 8 = 4 46 2 : + 8= Õ 4466 + : ∘ : ∘ : + 4> Õ 46 8 : : 46 : 8 Mnpnkms ak tfn iorst bdrg dkm mnpnkms ak tfnor H tfnor H hnoke hnoke d paokt ak tfn x tfn x -dxos. -dxos. ]fn wargoke iar tfn drnd tdgns prncnmnkcn avnr dky moderdbs tfny fdvn mrdwk sa oi tfn wargoke os carrnct hut n.e. tfnor moderdb fds paokts ok tfn wrake pasotoaks, dwdrm tfn bdrgs. ]fnrn bdy hn atfnr bntfams n.e. Fnra‟s iarbuld ar usn ai troeakabntry – oi yau drn uksurn oi sucf dttnbpts mnsnrvn crnmot , usn rnvonw.
D4
3>4 csa
Cabbak nrrars snnk ok bdrgoke0
3 y ∘ 8 x 2 3: ⇘ l4 0 b 2
3 8 3 9 4 6= 8: ↘ l8 0 y 2 ∘ ( x ∘ 8= ) ↘ Z , , H ∘ 3 , 8 3 43 4 3
3 y ∘ 8 x 2 3: ⇘ l4 0 b 2 ∘
:
8 3 85 9 ∘ >= ↘ l8 0 y 2 ( x ∘ 8= ) ↘ Z , , H (4>, : ) 3 8 43 4 3
Uotf d carrnct trodkeln drnd bntfam sucf dttnbpts woll aitnk scarn (d) H:B4D4itB4D: (h) B4B4D:
]rodkeln iar rninrnkcn0
(48, 46)
(8=, :)
(∘4>, :) 3?
Tunstoak Kubhnr
Scfnbn
Bdrgs
9.(d)
Zasotovn cuhoc sfdpn dkywfnrn wotf 4 bdxobub dkm 4 bokobub Zasotovn cuhoc sfdpn tfdt dt lndst rndcfns tfn x tfn x-dxos -dxos dt x dt x 2 2 ∘4 dkm wotf d bokobub ak tfn x tfn x-dxos -dxos dt x 2 x 2 3
46
∘4
A
y oktnrcnpt y oktnrcnpt dt 46. Bust carrnspakm wotf tfnor sgntcf
3
B4
D4
H4
Iar tfn oktnrcnpts dllaw ds kubhnrs ds dhavn ar dllaw ds caarmokdtns n.e. (46, :), (:, ∘4), (:, 3) ds lake ds tfny drn bdrgnm ok tfn carrnct pldcn. (3)
N.e. ( 8 x + 8)( x
(h)
2
∘ 9 x + ? ) 2 . .. 8 x 3 ∘ 4: x 8 + 9 x + 46 46 8
B4 D4 D4 (3)
i ‸( x ) 2 9 x 8 ∘ 8: x + 9
(c)
H4it
(4 )4 8 4 i ‸ 2 9 ∘ 8: + 9 3 3 3 4 i ‸ 2 : 3 y 2
B4 D4
>48 85
D4 (=) (4: bdrgs)
(d) B4
t aa cakcnrknm oi tfn “nkms‒ hncabn Carrnct sfdpn iar d y 2 + x erdpf. Ma kat hn taa vnrtocdl ar nvnk ea hnyakm tfn vnrtocdl sloeftly. Cakmakn wotf ka dxns dkm cakmakn cusp logn dppndrdkcn iar tfn turkoke paokts.
D4
y 2 + x 3 sfdpn, oktnrsncts (ar dt lndst rndcfns tfn x tfn x-dxos) -dxos) dt ∘4, bokobub dt x dt x 2 2 3 hut
H4
3
bust kat stap ar crass dt x dt x 2 2 3 y oktnrcnpt y oktnrcnpt dt 46
_au cdk oekarn tfn pasotoak ai tfn bdxobub o.n. ot bdy hn ta tfn lnit ai ar roeft ai ar ak tfn y tfn y-dxos. -dxos.
(h) B4
Bdrg (h) dkm (c) taentfnr. Dttnbpts ta bultoply aut. 8
N.e. Laag iar dk dttnbpt ta squdrn ( x ∘ 3) ta ahtdok x ´ 9x ´ ? dkm tfnk dk dttnbpt ta bultoply hy ( x + 4) ar ( 8 x + 8 ) ar dk dttnbpt ta bultoply ( x + 4) ar ( 8 x + 8 ) hy ( x – x – 3) dkm tfnk bultoply tfn rnsult hy ( x – x – 3) Cakmakn slops n.e. dttnbptoke (8 x + x + 4)( x – x – 3)8 hut nxpnct ta snn dk nxprnssoak ai tfn rnquornm iarb
D4 D4
46 ∘ 4: x 8 + 9 x + 46 3 8 46 6 . (Oekarn dky spuroaus “2 :‒) Iully carrnct 8 x ∘ 4: x + 9 x + 4 3 8 46 6 hut tfnk dttnbpt ta “sobploiy‒ ds n.e. Spncodl cdsn0 oi tfny ahtdok 8 x ∘ 4: x + 9 x + 4 i ( x ) 2 x 3 ∘ > x 8 + 3 x + ? tfnk scarn D4D: hut katn tfdt dll bdrgs drn dvdoldhln ok ]wa carrnct tnrbs ai 8 x
3
(c) ok sucf cdsns.
(c) 3
H4it
Carrnctly moiinrnktodtns tfnor 8 x irab d = tnrb cuhoc.
8
∘ 4: x + 9 x + 4466. Dllaw iallaw tfrauef hut akly
Dllaw usn ai pramuct ruln n.e. 8
8
i ( x ) 2 8( x + 4)( x ∘ 3) ↘ i ‸( x ) 2 8( x ∘ 3) + = ( x + 4)( x ∘ 3) _au cdk cakmakn paar katdtoak sa `ust ` ust laag iar tfn carrnct ar carrnct it nxprnssoak.
4 3
B4
Dttnbpts i ‸
D4
Carrnctly dcfonvns i ‸ 2 :. Bust iallaw d carrnct mnrovdtovn hut dllaw tfos bdrg oi 3 tfny fdvn n.e. movomnm tfnor mnrovdtovn hy 8 hniarn suhstotutoke ar n.e. oi tfny fdvn movomnm tfnor nxpdkmnm i( x) x) hy 8 hniarn moiinrnktodtoke sa tfny fdvn
4
‸ 2
8
∘
+
i ( x ) 3 x
t fnor cakstdkt ok tfnor nxpdksoak ok (h) os okcarrnct 4: x 3 ar oi akly tfnor 4 4 8 4 ]fn wargoke bdy hn bokobdl sa dllaw n.e. i ‸ 2 9 ∘ 8: + 9 2 : ar nvnk 3 3 3 4 i ‸ 2 : ak ots awk iallawoke d carrnct mnrovd mnrovdtovn. tovn. @ust laag iar tfn dkswnr ai : (n.e. 3 tfny bdy cdll ot y ot y). ). D4
y 2
>48 ar nxdct nquovdlnkt. Dll prnvoaus bdrgs bust fdvn hnnk scarnm ok (c). 85
4
Dccnptdhln dltnrkdtovn ta sfaw i ‸ 2 : 0 3
4 4 i ‸( x ) 2 9 x 8 ∘ 8: x + 9 2 : ⇘ ( 9 x ∘ 8 ) ( x ∘ 3) 2 : ⇘ i ‸( x ) 2 : dt x 2 ⇘ i ‸ 2 : 3 3
Scarn B4 iar dttnbptoke ta salvn qudmrdtoc (usudl rulns) dkm D4 iar mnmucoke 4 i ‸ 2 : 3
Tunstoak Kubhnr
Scfnbn
5.(d)
AH 2 :.9 + 4.= ∘ 8 Õ :.9 Õ4.= cas 8 ⇘ AH 2 ... ar AH 2 ...
B4
AH 2 4.536 sok DAH s ok 8 2 ⇘ DAH 2 :.688 4.= ''4.536 ''''
D4
8
8
8
Bdrgs 8
ar n.e.
mB4
sok DHA sok 8 2 ⇘ DHA 2 :.34?... ⇘ DAH 2 ϊ ∘ 8 ∘ :.34?... :.9 ''4.536' 6''' ζ 2 8 Õ DAH 2 8 Õ :.688 2 4.9= *
D4* (=)
Dttnbpts :.9 Õ δ wotf δ 2 8ϊ ∘ 4.9= ar δ ar : .9 Õ 4 .9 = Dttnbpts 8 Õ ϊ Õ :.9 ∘ :.
(h)
2 ϊ ∘ 4.9 =
: .9 Õ ( 8 ϊ ∘ 4 . 9 = ) + 8 .6 2 > .9 b
B4 D4 (8)
(c)
Dttnbpts
4 8 = Õ :.9 Õ δ wotf δ 2 8ϊ ∘ 4.9= ar δ 2 ϊ ∘ 4.9 8 ar
8 4 Dttnbpts ϊ Õ :.9 ∘ Õ :.9 Õ 4.9= 8 :.9 9 Õ 4. 4.=so =sok k8 Dttnbpts :. 8
Iull bntfam
4 8 Õ :.9 Õ ( 8ϊ ∘ 4.9= ) + :.9 Õ4.= so sok 8 2 4.9 b8 8
B4
B4 mmB4 D4 (=) (4: bdrgs)
Ok pdrt (d), cdkmomdtns bdy kat hn cdrniul wotf tfn usn ai ζ . N.e. ok tfnor wargoke, tfnor ζ bdy bdy hn dkeln DAH dkeln DAH wfocf wfocf tfny carrnctly mauhln ta ent 4.9= – cakmakn tfos paar katdtoak dkm eovn crnmot oi tfn oktnktoak os clndr. Nxdbpln0
AH 8 2 :.98 + 4.=8 ∘ 8 Õ :.9 Õ4.= c ca as 8 ⇘ AH 2 4.535>9... 8
8
8
:.9 + 4.536 ∘ 4.= cas ζ 2 ⇘ ζ 2 :.688 8 Õ :.9Õ4.5 .536 36 ζ 2 8 Õ :.6 .68 88 2 4.9 .9= = Os dccnptdhln iar iull bdrgs ok (d) (d) B4
Dttnbpts tfn casokn ruln ta ent AH AH ar ar AH8 snnk ok pdrt (d) akly
D4
AH 2 AH 2 dwrt 4.5= ar trukcdtnm ds 4.5 ar n.e. 4.53 (bdy hn obplonm)
mB4
Dttnbpts tf tfnn sokn ruln wotf tfn somns somns dkm dkelns ok tfn carrnct pasotoaks ok dk dttnbpt ta iokm DAH iokm DAH ar ar 4/8 DAH 4/8 DAH.. ]fos bust hn d iull dttnbpt okclumoke usoke okvnrsn sok ta iokm tfn dkeln.
]fos bdy hn dcfonvnm hy dttnbptoke tfn sokn ruln ta iokm dkeln dkeln DHA DHA iorst iorst dkm tfnk usoke tfn dkeln sub. ]fos rnquorns usn ai tfn t fn sokn ruln wotf tfn somns dkm dkelns ok tfn carrnct ":.34?. ?... .."" ar n.e. pasotoaks ok dk dttnbpt dttnbpt ta iokm DHA iokm DHA iallawnm iallawnm hy ϊ ∘ 8 ∘ ":.34 8 ϊ ∘ = ∘ 8 Õ " :. :.34?..." Dltnrkdtovnly usns tfn casokn ruln ta iokm dkeln DAH dkeln DAH 8
8
8
N.e. 4.= 2 :.9 + 4.536 ∘ 8 Õ :.9 Õ4.536 cas
ζ 8
⇘
ζ 8
2 ...
Mnpnkms ak tfn iorst bntfam bdrg.
D4*
Iully carrnct warg lndmoke ta tfn eovnk dkswnr wotf ka ahvoaus raukmoke nrrars. N.e. oi tfny ahtdok dkeln DAH dkeln DAH 2 2 :.685 iallawoke carrnct d bntfam tfnk stdtn 8Õ:.685 2 4.9= tfos bdrg sfaulm hn wotffnlm.
Dltnrkdtovns wargoke hdcgwdrms0
ζ 2 4.9= ⇘
HC :.9 ζ 2 :.68, 2 ⇘ HC 2 4.= sa ζ 2 4.9= 8 so k : . 6 8 so k ( ϊ ∘ 8 ∘ : . 6 8 )
Ar ζ 2 4.9= ⇘ B40 Iokms
AC 4.= ζ 2 :.68, 2 ⇘ AC 2 :.9 sa ζ 2 4.9= 8 s ok : . 6 8 s o k ( ϊ ∘ 8 ∘ : . 6 8 )
ζ dkm usns dkeln sub ai trodkeln dkm sokn ruln 8
D40 Carrnct sokn ruln r uln stdtnbnkt mB40 Xndrrdkens iar HC iar HC ar ar AC . Mnpnkms ak iorst bntfam bdrg. D40 Iully carrnct warg ta ahtdok HC ahtdok HC 2 2 4.= ar AC AC 2 2 :.9 wotf bokobdl cakclusoak n.e. tocg, TNM ntc. (h) B4
D4
Dttnbpts :.9 Õ δ wotf dk dllawdhln δ Iar dk dllawdhln dkeln dccnpt 8 ϊ ∘ 4.9= (dwrt =.9=) ar ϊ ∘ 4.9= (dwrt 4.>:) Dk dltnrkdtovn os ta iokm tfn corcubinrnkcn dkm suhtrdct tfn bokar drc drc DC DC Iar rninrnkcn tfn carrnct vdlun os 8.56… wfocf bdy obply tfn bntfam (3.59??… ∘ :.?6=)
:.9 Õ ( 8 ϊ ∘ 4.9= ) + 8.6 2 dwrt >.9 b. Cakmakn ldcg ai ukots
enknrdl tfn bdrgs pdrt (c)aidrn B40 Dttnbptoke tfn bd`ar B8 dttnbptoke (c) Ok tfn gotn drnd ar fdli ai ot ok n.e. drnd trodkeln trodkeln DAH DAH hut hut kat n.e. drndsnctar, ai trodkeln ACD (tfny ACD (tfny waulm knnm ta iokm tfn drnd ai trodkeln tro dkeln DHC DHC ds ds wnll, ar fdli ai hatf ai tfnsn), mB30 D cabplntn dkm carrnct bntfam iar tfn tatdl drnd, D40 dwrt 4.9
B4
]fos bdrg os iar dk dttnbpt dt tfn snctar drnd ACVD ACVD00
N.e. Dttnbpts
B4
4 8 Õ :.9 Õ δ wotf δ 2 8ϊ ∘ 4.9= ar δ 2 ϊ ∘ 4.9 = 8
Dk dltnrkdtovn os ta iokm tfn drnd ai tfn corcln dkm suhtrdct tfn drnd ai tfn bokar snctar Iar rninrnkcn tfn carrnct vdlun os :.63>… wfocf bdy obply tfn bntfam (4.43:… ∘ :.8?>8) ]fos bdrg os iar dk dttnbpt dt tfn gotn DHCA gotn DHCA (ar fdli ai ot)0
:.9 9 Õ 4. 4.=so =sok k 8 wfocf bdy hn pdrt ai Nxdbplns0 Dttnbpts :.
4 Õ :. :.9 9 Õ4. 4.=sok =sok 8 8
Dttnbpts ACD ACD + + DHC DHC n.e. n.e. 4 Õ :.9 8 sok 4. 4.9= + 4 Õ 4.= 8 sok ( 8ϊ ∘ = ∘ 4. 4 .9 = )
8
8
4.5=" 5="Õ :. :.9 9 so sok k Dttnbpts "4.
4 ζ ζ 4.5=" 5="Õ :. :.9 9 sok sok wfocf bdy hn pdrt ai Õ "4. 8 8 8
mmB4 ]fos bdrg os iar d cabplntn dkm carrnct dttnbpt dt tfn tatdl drnd dkm mnpnkms ak hatf prnvoaus bntfam bdrgs0 n.e. D4
4 8 Õ :.9 Õ ( 8 ϊ ∘ 4.9= ) + :.9 Õ 44..= ssook 8 a.n 8
dwrt 4.9 b8
Cakmakn d ldcg ai ukots
Dltnrkdtovn wfocf mansk‟t iallaw tfn dhavn hut os nquovdlnkt0
Drnd 2
8
4 8
8
4 8
8
ϊ Õ :.9 ∘ Õ :.9 (4.9= ∘ ssook 4.9= ) + Õ 4.= sok (8ϊ ∘ = ∘ 4.9= )
Dwdrm B4 iar tfn dttnbpt dt tfn bd`ar snebnkt dkm B4 iar tfn dttnbpt dt trodkeln DHC trodkeln DHC (ar (ar fdli ai ot) tfnk mmB4D4 ds dhavn.
Sabn lnketfs dkm dkelns iar rninrnkcn0
AH8 2 3.:4?… AH 2 AH 2 4.535>… 8 DC 2 :.59?5… DC 2 2 :.6553… Dkeln DHA Dkeln DHA 2 2 :.388 Dkeln CAD CAD 2 2 =.9=… Salutoaks wfnrn cdkmomdtns cfdken ta mnernns KH 8 rdmodks os 44=.>?4>>?…º 44=.>?4>>?…º (d)
8
8
8
8
ca as44=.>?... ⇘ AH 2 ... ar AH 2 ... AH 2 :.9 + 4.= ∘ 8 Õ :.9 Õ4.= c
B4
AH 2 4.536
D4
sok DA 44 4=.>?... DAH H sok 4 2 ⇘ DA DAH H 2 :.688 4.= ''4.536 ' ar n.e. sok DHA sok 4 44 4=.>?... 2 ⇘ DHA 2 46.3 ⇘ DA DAH H 2 46: ∘ 44=.>?... ∘ 46.3 :.9 ''4.536 '''' ζ 2 8 Õ DAH 2 8 Õ =5.4 2 ?=.8º 2 4.9= * (h)
B4
D4* (=)
δ Õ 8 Õ ϊ Õ :.9 wotf δ 2 39 39: : ∘ dwrt?= dwrt?= º Dttnbpts 39: ar Dttnbpts 8 Õ ϊ Õ :.9 ∘
dwrt?=º Õ 8 Õ ϊ Õ : .9 39:
39: ∘ dwrt wrt ?= 8.6 2 > >..9 b Õ 8 Õ ϊ Õ :.9 + 8. 39:
B4
D4 (8)
(c)
Dttnbpts
δ 8 39: : ∘ dwrt dwrt Õ ϊ Õ :.9 wotf δ 2 39 ?=º 39: ar
Dttnbpts
8
ϊ Õ : .9 ∘
dwrt?=º 39:
B4
Õ ϊ Õ : .9
8
4.=sok44=.>?.... Dttnbpts n.e. :.9 Õ 4.=sok44=.>?.. 39: 39 : ∘ dwrt?=º 8 :.9 9 + :. :.9 9 Õ4.=sok4 4.=sok44= 4=.> .>?. ?... .. 2 4. 4.9 9 b8 Õ ϊ Õ :. Iull bntfam n.e. 39:
B4 mB4 D4 (=) (4: bdrgs)
Tunstoa k Kubhnr 6.(d)(o)
Scfnbn
Bdrgs
8
8
= + 48 x ∘ 3 x8 2 d ´ 3 ( x + c ) ar d + h ( x ´ 8 )
B4
8
]wa ai 49 ∘ 3 ( x ∘ 8 ) ar twa ai d 2 49, h 2 ∘3, c 2 ∘8
D4
8
49 ∘ 3 ( x ∘ 8 ) Caarmokdtns B 2 ( 8,49 )
(oo)
D4 H4it H4it (>)
(h)
Stdtns ar obplons tfdt l 8 fds nqudtoak y 2 ''6'' x + g
B4
Snts = + 48 x ∘ 3x 8 2 '' 6 x ''+ g dkm pracnnms ta 3]T
mB4
Carrnct 3]T 3 x ∘ = x + g ∘ = 2 : 8
D4
8
Dttnbpts ta usn h ∘ =dc 2 : ta iokm g
g2
mmB4
49 49 ⇘ y 2 6x + 3 3
D4 (>) (4: bdrgs)
(d)(o) B4
Iar dttnbptoke ta cabplntn tfn squdrn. Laag iar h 2 ´ 3 ar c 2 ´ 8
D4
]wa carrnct cakstdkts ar twa carrnct oktnenrs irab 49 ∘ 3 ( x ∘ 8 )
D4
49 ∘ 3 ( x ∘ 8 )
8
8
8
( 49 ∘ 3 ( 8 ∘ x ) scarns B4D4D:)
Dltnrkdtovn hy cabpdroke caniioconkts0 8
(
d + h ( x + c ) 2 d + h x + 8 xc + c 8
8
) 2 hx
8
+ 8hcx + d + hc 8
h x 8 + 8h c x + d + h c 8 ≤ = + 4 8 x ∘ 3 x 8 h 2 ∘ 3 8hc 2 48 ⇘ c 2 ∘8 d ∘ 48 2 = ⇘ d 2 49 8
Scarn B4 iar nxpdkmoke d + h ( x + c ) dkm cabpdrn x 8 caniioconkts ta iokm d vdlun iar h (KH tfos cdk hn mnmucnm mornctly dkm waulm scarn tfn B bdrg iar h 2 ´ 3 ds dhavn) D40 Caktokuns tfn pracnss dkm cabpdrns x caniioconkts caniioconkts ta iokm hatf h 2 ∘3 dkm c 2 2 ∘8 D40 d 2 49 (d)(oo) H4it
Notfnr x 2 8 ar y 2 49 hut iallaw tfrauef ak tfnor 49 ∘ 3 ( x ∘ 8 ) wfnrn d ≪ :
H4it
Hatf x 2 8 dkm y 2 49 hut iallaw tfrauef ak tfnor (∘ c, d) irab
8
8
d
+ h ( x + c ) wfnrn
h ≪ ´ 4 Iar carrnct ar carrnct it caarmokdtns tfn wrake wdy raukm n.e. (49, 8) scarn SC H4 H: hut dpply osw oi tfn carrnct ar ca carrnct rrnct it dkswnrs dkswnrs drn snnk snnk ds ds x x 2 2 …, y …, y 2 2 … (h) B4
Stdtns ar obplons tfdt l 8 fds nqudtoak y 2 '' 6 '' x + g , g ≪ : Iallaw tfrauef ak tfnor y 2 '' d '' x + g ar ak y
c
y caarmokdtnai tfnor B x + g 2 x caarmokdtnai tfnor B
mB4
Snts = + 48 x ∘ 3x 8 2 '' 6 x ''+ g dkm pracnnms ta 3]T
D4
Carrnct 3]T 3 x 8 ∘ = x + g ∘ = 2 : (]fn “2 :‒ bdy hn obplonm hy suhsnqunkt warg) 8
mmB4 Dttnbpts ta usn h ∘ =dc 2 : ta iokm g . D4
49 49 49 2 oi y 2 6 x + g wds bnktoaknm ds tfn ⇘ y 2 6 x + . Cakmakn `ust g 2 3 3 3 nqudtoak iar l 8 g 2 2
Dltnrkdtovn iar pdrt (h)
B4 mB4 D4
Dttnbpts ta moiinrnktodtn = + 48 x ∘ 3x 8 ( x xk x xk-4dt lndst akcn) dkm snts nqudl ta tfnor 6 Salvns iar x iar x dkm dkm pracnnms ta iokm tfn caarmokdtns ai paokt ai caktdct 8 38 ]dkenkt bnnts curvn dt , a.n. 3 3
8 38 mmB4 Suhstotutns tfnor , ok tfnor y 2 ''6'' x + g ta iokm g . 3 3 D4
y 2 6 x +
49 3
Tunstoak Kubhnr ?(d)
Scfnbn
Bdrgs
y 2 i ( x ) + 3
( ?, 9 ) ( =.>,3)
y 2 i ( 8 x )
Z ( ?, 3 )
Akn carrnct sgntcf mrdwk dkm ldhnllnm carrnctly Akn carrnct sgntcf mrdwk dkm ldhnllnm dkm wotf carrnct paokt Cabplntnly carrnct sgntcfns wotf hatf paokts
B4 D4 D4 (3)
(h)
Snts
x + 3 2
8x
H4
3 2 ( 8 ∘ 4) x x 2
(
B4
3 Õ ( 8 + 4) 2 3 ( 8 + 4) * 8 ∘ 4) ( 8 + 4)
D4* (3)
(c)
8
x 2 3 ( 8 + 4) ⇘ x 2 ? ( 8 + 4) 2 ...
⇘ x 2 ? ( 3 + 8 8 ) , y 2 3 8 + 9
B4 D4, H4 (3) (? bdrgs)
(d) B4
D4 D4
Cfncg dll 3 moderdbs dkm scarn tfn hnst sokeln moderdb uklnss tfn cdkmomdtn clndrly okmocdtns wfocf akn tfny wdkt bdrgnm hy n.e. crassoke aut tfn atfnr(s). Akn carrnct curvn mrdwk dkm ldhnllnm carrnctly0 Iar i (8 ) tfn curvn sfaulm stdrt dt dkm hn dhavn dkm rnbdok dhavn i ( ) dkm kat fndm x x hdcg tawdrms ot soekoiocdktly o.n. dtA lndst bdok bdoktdok tdok tfn sdbn edp. Iar i ( x) x) + 3 tfn curvn sfaulm stdrt ak tfn t fn pasotovn pasotovn y y-dxos -dxos dkm hn dppraxobdtn dppraxobdtnly ly tfn sdbn sfdpn ds i ( x) x) Akn carrnct curvn mrdwk ds dhavn dkm ldhnllnm dkm wotf carrnct paokt iar tfdt curvn. ]fn paokt mans kat fdvn ta hn ok tfn carrnct rnldtovn pasotoak – `ust laag iar vdluns. Cabplntnly carrnct sgntcfns wotf hatf paokts carrnct dkm dt lndst akn carrnctly ldhnllnm – yau cdk dssubn dssubn tfn atfnr atfnr os tfn atfnr. Dllaw Dllaw i(8 x) x) ta crass i( x) x) + 3 ds lake ds ot os hnyakm (?, 9) hut hut wotf ka atfnr oktnrsnctoaks oktnrsnctoaks iar x iar x ; ; : ]fn caarmokdtns iar tfn trdksiarbnm t rdksiarbnm Z Z bust bust hn okmocdtnm ak tfn sgntcf ar oi tfny drn dwdy irab tfn sgntcf ot bust hn clndr wfocf curvn tfny rnldtn ta. Iar nxdbplns snn hnlaw. Oi yau drn ok dky mauht usn rnvonw.
(h) H4 B4
Carrnct nqudtoak
8x
Urotns 8 x ds 8 x dkm pracnnms ta callnct tnrbs ok Katn tfdt tfos bdy bdy hn dcfonvnm dcfonvnm vod n.e. 3 x + 3 2 8 x ⇘ x + 3 2 8 x ⇘ 4 + 2 8 x Ar n.e.
D4*
x + 3 2
x
8 x ⇘4+ 4 2 8 3 3 x Zracnnms ta tfn eovnk dkswnr sfawoke dt lndst tfn stnps x + 3 2
8x ⇘
x +3 2
( 8 + 4) 2 3 8 + 4 ( ) ( 8 ∘4) ( 8 + 4) ar n.e. 3 2 ( 8 ∘ 4) x ⇘ 3 ( 8 + 4) 2 ( x 2
3
Õ
Dttnbpts usoke n.e.
x + 3 2
8 + 4) ( 8 ∘ 4) x 2
x
x scarn ka bdrgs ok pdrt (h) 8
Dltnrkdtovn0
H4 B4
Carrnct nqudtoak
x + 3 2
8x
x + 3 2
8x ⇘ x + 9 x + ? 2 8x ⇘ x ∘ 9 x ∘ ? 2 : 9 ´ 39 + 39 x ∘ 9 x ∘ ? 2 : ⇘ x 2
8
Squdrns hatf somns dkm callncts tnrbs ta ahtdok d 3]T ok
x dkm dttnbpts ta salvn iar
x n.e. usoke qudmrdtoc iarbuld D4*
9 ´ 39 + 39 2 3 ´ 3 8 2 3 ( 8 ´ 4) ⇘ 8
x 2 3 ( 8 + 4)
Sobploions dkm rndcfns tfn proktnm dkswnr. Oi tfny eovn hatf dkswnrs scarn D: hut tfnrn os ka rnquornbnkt ta nxpldok wfy tfn atfnr dkswnr os rn`nctnm.
(c) B4
Dttnbpts ta squdrn tfn eovnk nxprnssoak ta iokm x iokm x.. Cakmakn d slop ak tfn 3 (ot bdy rnbdok 3) hut bust rnsult ok dk nxprnssoak ai tfn iarb δ + θ 8. (Uargoke knnm kat hn sfawk ds lake ds tfos cakmotoak os bnt)
D4
x 2 ? ( 3 + 8 8 ) an sucf ds x 2 85 + 46 8
H4
y 2 3 8 + 9 Katn tfdt y 2 46 8 + 85
+ 3 os carrnct hut os kat sobploionm sa scarns H:
Katn tfdt wargoke iar (c) bust hn snnk ok (c) o.n. ma kat dllaw wargoke iar (c) ta hn crnmotnm ok pdrts (d) dkm (h) uklnss tfn dkswnrs drn caponm okta o kta (c)
Nxdbpln bdrgoke iar sgntcfns iar ?(d)0
B4D4D:
B4D4D:
B:D:D:
B4D:D:
B:D:D:
B4D4D4
B4D4D:
B4D4D4
B4D4D4 B4D4D:
Tunstoak Kubhnr
Scfnbn
4:(d)
Bdrgs
4 3
i ‸( x) 2 dx ∘ 48 x ⇘ i ‸‸( x) 2 d ∘ = x 4 Snts i ‸‸ (8 (8 5) 2 : ⇘ : 2 d ∘ = Õ ?
(h)
∘
8 3
⇘d2
4 3
H4
= ?
B4 D4 (3)
= 3
4 8 dx ∘ ? x + c 8 Suhstotutns x 2 4, i ( x) 2 ∘6 ⇘ c 2 ...
i ‸( x ) 2 dx ∘ 48 x ⇘ ( i ( x) 2 )
B4 D4it mB4
=
8 8 5 3 ( i ( x) 2 ) ? x ∘ ? x + ?
D4 (=) (5 bdrgs)
Bdrg (d) dkm (h) taentfnr Ma kat hn taa cakcnrknm wotf katdtoak n.e. ok (d) wfnk tfny moiinrnktodtn tfny bdy cdll ot i‸( x x ) ar n.e.
m y m x
(d) ∘
8 3
H4
Stdtns ar usns i ‸‸( x ) 2 d ∘ = x
B4
Snts tfnor i ‸‸ (8 (8 5) 2 : dkm pracnnms ta d vdlun iar d.
wfocf bdy hn uksobploionm
Ot os mnpnkmnkt upak akn carrnct tnrb ok i ‸‸( x) n.e. d ar D4
d 2
∘= x
∘
8 3
(an)
= ?
(h) B4
= 3
4 3
Oktnerdtns dx ∘ 48 x wotf akn tnrb carrnct n.e.
3
pracnssnm. D4it
mB4
4 8 48 x dx ar ∘ wotf tfn okmocns 8 =
4 3
=
4 8 3 i ‸( x ) 2 dx ∘ 48 x ⇘ ( i ( x) 2 ) dx ∘ ? x + c iallaw tfrauef ak d ar d kubnrocdl d 8 ar d “bdmn up‒ d hut bust okclumn + c = = 8 x 48 x 3 ? ∘ +c Dllaw sobploionm ar uksobploionm sa dllaw n.e. 8 = 3 Suhstotutns x 2 4 dkm i ( x) 2 ∘6 ta ahtdok d vdlun iar c. Bust fdvn kubnrocdl d kaw.
Mnpnkms ak tfn iorst B bdrg. =
D4
8 5 8 8 5 3 ( i ( x) 2 ) ? x ∘ ? x + ? . Dllaw nquovdlnkt carrnct irdctoaks iar ? , ? ar rncurroke mncobdls
.
.
n.e. :.8, :.5 wotf clndr mats avnr tfn 8 dkm 5. 3
=
= 3
Dllaw n.e. x iar x Laag iar d carrnct nxprnssoak sa ka knnm ta snn i( x) x) 2 … dkm osw oi kncnssdry. 44 = ok Katn tfdt d idorly idorly cabbak nrrar nrrar os ta ahtdok d 2 ∘ ok pdrt (d) lndmoke ta c 2 ? ? pdrt (h)
Zndrsak Nmucdtoak Lobotnm. Xneostnrnm cabpdky kubhnr 658686 wotf ots rneostnrnm aiiocn dt 6: Strdkm, Lakmak, UC8X :XL, ^kotnm Gokemab
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