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roop ypxyyJrx Horee uopuftn (6aranuu) xapran3ax ereluesrlT.q Amp HoMoxeA ToAopxoftnoFr
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ro.uopxofinors lrb tar 6aftua.
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xapraJ3ax sJreMenTlTluftn anre6prftn ryftqeenryyasop YpxYYJIx EeMexsA rsr 6afiHa.
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orTopryfiE cJrypb BeKTop rex HopJIsx 6a cyypuap Hb 3a.UJIa)K 6r'rBen Z+ : )r?r flz?z*"13?3 6onno, Ylru: )rrlz,.tr3 roor ? aerroprrx aQr
24 Bxrop, ryya nsglpx vfunrwn
1.
2.
3.
5.
6.
d fT : (rrIrz,gt*yz,z1*.z2),\.? : ()rr, )y,),2)fiL: fr2t Ut: Uz, zL: 22 6afiaar d :T fiafisa.
l"l :tlwA * (*r,yr, zL) 6a B(r2,yz, zz) usrITA Koop.urrnaraapaa eror.qcox 6aftr. Teraeu
ffi : (*, - fittUz - Ut, zz - ,r)
6onno.
7. XoEp qgrufix xoopoulox sa,fir
ITEI:
rorrarEoroop roropxoft nno.
8. Xepvrrauftr ororAcon ) xapruaarrA xyBaa)c uerrafin xoop,utrHarbrr oJrox rousEo rrfir*\rz h*\y2 zr*\zzr:ffi' u: 1+l' z:l11
Xspvrvrirftr raJrJran xyBaax usrraftx KoopAr[rrarlrr onox roun€o rrfit*sz h*Az 4*z.z,:7, U__7, ":T
6ouno. (,\ : 1)
X(uruee 4.1 OreAceu d,l,? aexropyy,uaap ? + 7 + ? aerropur 6afiryyu.
X(uurss 4.2 M{2,4,-2) 6a M2(-2,4,2) uaryln orer,qxee.
xapbqaaraap xyBaalc M qsruftx KoopArrnarbrr oJr.
EoAonr:gt*\xz 2+3(-2)nM:if:ffi:-lh*\yz 4+3.4uM: 1+,\ :-f15:*4*\zz -2+3.2zM: 1+,\ :-113-:t
6ouuo. M(-1,4,L).
MrMz xapvrranfir ,\:3
Bexrop, Tylru Asspx yfrllgJryltq
X(ilnres 4.3 l?l : 1S, l7l : tg, l? + 7.l: rn 6on ;a - 71 :fBoAonr: 2(ldl' + l7l') : ld -ll' + lz} + ?l' tou,beor aIrImIraBaIr
ld -tl':2(L32 + 192) -24:2(169+361) -576:1060 -576:484+ l? =Ti:zz
Xftrriree 4.4 ,M(5,-3,4) qerufin pa*uuyc BeKTopbrH tlurriYrsrrl xocrgycJyAbrr orl.
Bqrdg: Koop,qunatbrg exeec exneursfi Til : (5, -3,4) nertop lYcroe'
tffit: v : tffi:|t/'2
)_: Hrarnyynerq KocuuycyyAbrr oJrox roMb6o 6coop
cos1}: -
:LvDu-r"t - \rt- rt_e
cosB- h:#4p
cosT: l?;i -,rt
6oano.
X(uuree 4.5
uogynufir on.
Bo,qo.rrr:
? : (3, =5,8) AaT' : (-1,1, -4) BeKropyylnir nuftu6gp 6a rurauaphrrr
d +l : (3 + (-1), -5 + 1,8 + (-4)) : (2,-4,4), d -l : (4,-6,12) ryn
l? + 7l : Jz, + t-Ef+ 4z : trq6Ei5:6l" - ?l : J@i ?6)\ 122: \n6 + 36TTaa : vM : t4
Jlacra"rr 6ognoro
4.L. ABC rypBanxuxt AM ruyrryyr ss I BAC oxrlruftx 6raccexrpuc 6onox 6a M uet tss BC rarr Jreep oprurxo. Xepsa lAEl:6, lfri: c 6on ;Z-ili-nr on. lf ll
4.2. d -- (20,30, -60) BeKTopbrlr ypr 6a tlurrYYnertl KoctrHycyy,ublr orl. 7 /U4.3. d : (4,-2, -3) Oa 7 : (0, L,3) rerropyyna.u repren,ulrKyrlnp 7 aerropbrll ypr
Im l?l : 26 6a 0g reuxusrtsft yycrex oHIIer IIb Moxoo 6or- ? BeKTopbrII fkoopgrrrraryyALrr br. t /tt
4.4. A(1,1,1), B(5, 1,2), C(4,13,1) rypBanxlrn opofin [erlYr ororAeB. A erurufirr
6uccexrpnc BC raJrbrr D usresp xyBaarc 6on D usrnftH xoop,urtuarLrr oJI.
4.5. A(0, 0, 1); B(3, 2,1); C(4,6, 5); D(1, 6, 3) 6a d : fr + fr 6ou xooparrnarLru
ToEXJIoTITA .Ilagpx 7 rertopbru [poexqbrr oJr.
4.6. a, B-wfrrx. nMap yrraxA d : (-2,3, B) 6a I : (o,-6,2) BerropyyA xonrrreap 6afix
ss?
4.7. Ad:d,Bl: ? *" ABCD uaparrennorpaMMlru ALIaroHaJryyA 6onAB, Bd,
q,
tt
52 + (-3)2 * 42 :
r
Bexrop, Ty1fr Esgpx yfrwrWt
fr , il Bexropyy.uurr 7,7-eep rrepxufin.
4.8. rrp?? - -2, ry?? : 1 6on up7(3d - Z7)-r on.
4.g. d,t Bexropyyl IIepreHAI{Kynsp 6a ldl : 5, l?l : 12 6on ld + 7h la - 7F"otr.
,
4.10. d,lt BexropyyA sMap 6afigan l? + ?l : l? - 71 ,rturtren 6uengr,usx ss?
4.1L. d,''l BeKropyyAbIH xoopglrgox euuer g: 600, l?l : 5, l7l :8 6on 1? + 71,
l? - 7l-r on.
4.L2. frr,rap Eoxqenr d +l BeKTop d;t Bexropyy,ubrE xoopou.uox onurtrfir ratnan
xyaaa:r ae?
4.13 Xepaee ? BeKTop e reuxnerrsft 600 ollqor lycrex 6a l?l :6 6o.lr np7?-r on.
4.t4" A(-3,6); B(1,2); C(4,-6); D(-2,0) usryyr ororAxse. TE a^CT ".*topyya
4.L7'. ?'- (-3, n,4)'6a b' : (-2,4,p) sexropryA n 6a puftn rMap yrrarr Korrrrlrxeap
4.18. /(3, -1, 2), B(0, -4,2) 6a C(-3, 2, L) ueryya ,usep opofiroft rypaauxrru alrrrr
xa.:ryyrafi 6onoxrrr 6aras. l
, 4.19. A(2,-3,-5), B(-1,3,2) usryyn Hb raparrenorpaMMbrx xo6p opofi bereen rlysufi\, \' ' .uuarouaJryy.unrr orrJroJrqJrhrr rlor E(4, -1,7) 6ou [apanenorpaMMBlll uoroo xo6p
opoftr or.
4.20. A(L,-2,3), B(3, 2,1), C(6,4,4) ueryya raparrenorpaMMbrrr Aapaarrcax rypBan
opoftn rloryyg 6on ryynrfi,uopoa.uex opoft D-r on.
i.,{
r):
--l
^27Befropryacg wa.fisp 6a rexrop Wxaep
CE.UOB V. BEKTOPYY.IIbIH CKAJISP BA BEKTOP YP>KBgP
5.1. BerropyyAErr craJrrp 6a serrop yp)KBep
.+Tqnopxofino.nr 5.1 d 6a b xo6p Berropbrrl xyBbA
. ? : lz'll?lcos(?,7)
routEoroop To.Uopxofiuor,Uox Toor. Te,Urespuftn cKaJIxp YpxBsp r3II3.
d : (rt,yL,zL) 6a 7 : (fi2,A2,22) rscon Koop.qtrEaTaapaa erorAcon 6on
d .l : ntn2*uflz* zrzz..)
6onno.
Cranrp ypxBsp rapaax qauapyyjqraft 6aftsa.J
1. d.T : b' 'd2. d .()7) : (,\7). (?) : )(". 7)g. d .(? + ?) : d 't +d'? ' .r1.,' '
I
4.d.?>0b. 7.d :ldl'6. d .?: l"l . "p??: l7l .
',p??To.qopxofinont 5.2 Rs aa>r d Aa? ,.*topbrlr xyBbA
: dxl:6"11?lsinrp)d
torrarEoroop Tolopxofiror1ox BeKTopbIr ts.4resprftH BeKTop YpxBop reue.
Yyn.u: g: (d,?1 O, d sr d Aa? ,.*ropyyAun xaarrafi,q reprelrlnKyrsp Herx BeKTop
6oroog (d,T,,7rt) nr 6apyyn rypaBT 6aftx .uyprraeep roAopxofrnorAcou qrrrs$Teft. Bexrop
ypxBop Hb .uapaa( varapyyarafi.
1.?x7:-(7x?)2.d x(7+?) : (? * 7) + (? x
")j
3. (k?) * 7: ? x (rT) : k(d " 7)d : (rr,Ar,zL) 6a ? : (rz,y2rzz) BeKTopyyA r(oop.ufiraraapaa orerABeJI BeKTop 1ip*"rp
olx0:
-T) =) =)XJKfi1 Ut 21
fi2 Az 22
At z1
Az 22
fi1 21
fr2 22
n1 nrltfr2 grl
-)1,- V+
28 Bxropyyars cKalrep da nxrop :rrlxslD
6ongo.
Togopxofi.nont 5.3J-
d, b' ,? rypnan BeKTopooc 3oxr{ocou (? , T) . t ypxaspufir xoJrEMor lpx(Bsp rouo.
d : (rtt1t,zt);l : (*r,yz,zz);? : (*r,As,4) BerTopyy.ubrrr xorrrrMor ypxaeprfir
(? * 71 .a:
rourEoroop oJruo.
X(uurge E.l ? : (1, -3,4), 7 : (8, -4,2), ? : (-1,1,4) rexropyy.u ororg)Keo.
fi1 Ut z1
fi2 Uz 22
fi3 Us z3
Tersen op7 *?d-r olr.
BoAo.rrr: 7 + ? : (3+ (_1), _4+ \ ?+ 4): (2, -3,6),rp?7 .d
€coop
:) d\+)Hr;g-,"r(rnr _a':-----i-::'o+c' lb'+Zl Jzz+(_sru _2+9+24: E:.), +62 4+9+36 7
X(ulrreo 5.2 d:3l -2? aat :I +4! aexropyyaHn cKarsp ypxaepufir ou.
Yy*a' l7l : l?l : L,9:X,,r: f{AlBoAour:
d.t: (3? -21).(7'+ s?): t,l.l -2.i.? +L2.l.n -s.? .tr, :a. ly'l2+to .l.l -8.lll' : 3. 1+10. l?1. l?l."os p-8. 1 : 3+10. 1. r.cos
f, -a : 3-8 - -5X(uurss 5.3 ? : (1, -3,4) Oa 7 : (-2,3,1) aerropJryaun xooporrgox ouqrrfir or.EoAorrr: dt - lz,ll?lcos(p, 9:({J) rorrareor aruurnas.
lz'l : tlt, + (-g), *42: \Fgm :18;dT :1 . (-2) + (-3) .B + 4 .L: -2- 9 + 4: -Tl7l : \tW + L, : l4agp- :,,rr4;
d .l (-z) ,/7 . , ,/T ,cos{P : 1"il7j
: m : -ffi I I : arccos(- f-''))t(uruoe 5.4 d : (m,8,4) 6a t : (4,nx,-T) Bexropyy,u ororlxee. m-aftasMap
yrrauA d 6a7 ,.*ropyy,u uepreg,utrKyrrf,p 6afo< ae?
Bo,qonr: Xepss ? . ? : 0 6on a tT oaftsa.
d .t : 4. m*3. m+4. (-Z) - S + Trn :!g e rn : 4.
x(uuraa 5.5 d:67 +37 -21 oa 7:87 -2i +67 BeKropyyAErH Berrop
ypxaspufin uogymrfir on.
Bexropyyxux csrattp.da Bexrgp Wxrryp 29
--, lu -2'-l , 6;*l u 3
l3 -2J+E:14i'-42i-2Lk
x(u1use5.6 xepsa ldl : l?l : r, 1fr1 : [ 6on d:?+g?ou d:3d+tBexTopyy.uLrrr BeKTop ypxnspuftH yprsrr on.
Bo.uorr:
ld* 61l:l("+eT'; *(3?+T')l : l3?x a+sl x?+?x7+s7*?; :ir.o*4. ?, a*8.01 : 8l?ll?l'*tat'l : 8' L' 1'sin[ : s' 1' 1' f,: +
x(umes 5.7 A(2, -1, -2); B(L,Z,L); C(2,3, 0); D(5, 0, 6) usryyr Her xaBrrafi .ueep
opulrxbrr 6atat.
EoAo;1r: Xspsa A, B,C, D usryya Har xaBTrafi aaep opurrrx 6afiaafi AB,AC,AD nex-
TopyyA MoH rrar xasrrafi usap 6afix ryn KoM[nauap cl{creM !'y'cren3. }Ifiua r9.ur39p33p
oafiryynarAcarr rapaJrerom{neJtrbrrr e3nsx'ITH O-tsft ToEIIsIre. Oopeop xender AB,ACTAD
BeKTopyyAbrH xoJrrrMor Ypxmsp rer 6aftx 6onxo.
TE : $ -2,2 + L,l - (-2)) : (*1,3,3)
fr : (2 - 2,3+ 1,0 - (-2)) : (0,4,2)
Tl : (5 - 2,0 + 1, -6 - (-2)) : (3,1, -4)
-1 3 3
0 423 t-4
J J _-J
(AB x AC)- AD : :16*18+0-36+2:0
6ouox ryn A, B,C,D ueryyn Hgr xaBTrafi aegP opurrno.
Jlacra.n doAuoro
b.1. ? : ld - Zl Aa :5d - 67 BerTopyy.ubrrr cxaJrnp lpxnspufir on.
Yyna: l?l :4, l?l :6,(?,]'):T5.2. ? : (3,4,5) 6a t : (4,5, -g) ,u*tlorrotrE xoopos.qox euqrufir ou.
5.3. m,u1msMap yrrarrA d : (m,1,0) 6a 7 : (3,-3,4) BeKTopyy,u repueuarrKyrltp
6aftx sg?
s.4. d:27 +b7+7 oa t :7 +2V -3? BeKropyyAbrr BeKrop ypxraspuftr on.
r l7 i 7l ls -27xb':l 6 3 -2 l:lI B -2 o I l-' 6
+ar,'71 :!@:49.
rI
Bexropyy'w'a<amp 6a rlernop wxasp
5.5. A(2,2,2); B(4,0,3) 6a C(0,L,0) usryynsn opoftrofi rypBanxurr ran6afir on.
5.6. Xepea l?l : 6, l7l : 3 6a d,t Berropyyabrrr xooponAox onrror 600 6on
(2d +t)QA - a?)-r ou.
5.7. Xepsa l?l : 4,lll: 5 6a d,l BeKropyyALrH xoopoxrox ourror 1200 6on
(2d -37)2-ur roropxofiu.i
5.8. A(L1,4,1.);B(3, 4,-2);C(5,2,-1,) ueryya .ussp opofirot rypaanxnu ABC euqrraftr
ronopxoftu
5.9. Xspss ? : (6, -8, -7r 5) aexroprofi ronnr{Heap 7 rexrop su 0z rorDorsrreft xypu
ogqor lycrox 6a l7l :50 6on ryynuft KoopAr.rnaryyALrr on.
5.10. ? : (5,2,5) aerrop"rr 7 : (2, -1,2) aexmp Aaepx rrpoexubrr on.
5.11. Xepsa l?l : 1, l?l :2 6a d,l BeKropyyAbrrr xoopo,,qox o*qor zrn
Aon
l(2d + 7) * (? + z7)12-r ou.
5.12. Xeper l?l : 1, l7l ':
2 6a d,t Berropyy.q'r' xooporr.qox euqor 2ro
Aoo
t(z'+ a7) x (3? - 7y1,-" or,.
5.13. d : (2,-2,1),t : (2r3,6) nexropyyAun xoopox.uox onqruftn crurycbrr orr.
5.14. 'd - (1,-1,3), 7 : (-2,2,L),? - (8,-2,5) 6on (? * 7;.?-" orr.
5.15. d:(2,-3,1),,7: (-3, L,2),t:(L,2,3) 6on (? " 7)
" l,d r. (7 x ?)-r
OJI.
5.16. A(2,-L,1), B(5, 5,4), C(3,2,-L) 6a D(4,1,3) usryya .ussp opofrroft rerpasxpuug3rrsxiTurrftr ou.
5.17. d : (2,1, -1) aexroprofi Konrrrrreap 6ad -d :3 nexuuuftr xaura)c 7 aerroprrOJI.
5.18. Xeper.l?l = 10,171 :2 6ad .l : L2 6otr l? x 7l-r on.
5.19. A(L,2,-1), B(0,1,5), C(-1,2,1) 6a D(2,L,3) ueryya Esr xaema,fi Asep opuux
w!5.20. A(2,3,L), B(4,L, -2), C(6, 3, 7) 6a: D(-5, -4,8) ueryya .qsep opoftroft rerpaeApnn
B opofirooc rarcarr orrAprrftE yprbrr orr.
I
i
h\I - -'---I
COJIOB VI. XABTTAIIH AHAJI}IT}IK TEOMETPIIITH
XffJIEAP BOIIJIOTYYJI
6.1. Xasrrafix anaJrurrrK reoMerprrfix xrn6ap 6ogxoryya
i
L. A(styi, B(nz;g2) xo€p qerufirr xoopox.qox safi
p:1f@z_rr)r+(ur_u)2
2. A, B qerrafir xon6ocox xepvuzfic AM : MB': ,\ xaprqaaEg xyBaax M(*iy)qsrlrftn Koop,ul{Har Irb
n_q*\x2 at*\920+l ; Y: 1+) '
Tyxaftrr roxuonuon.u I : 1 6yroy AB xep.utrufir ra:raalr xyBaax M(*;g) ueruftn ro-
onqr{HaT Hbfiz*frr Uz*Ur"--Tt u: 2 '
3. A(aty), B(xz;yz), C(xs;y3) opofirofi rypaa.rr)KuEr rarr6afi
1
S : -rl@z - r)(as - u) - (rr - r)(az- g/r)1.
TsrEoHqorT(r,3r)xoopaI{HaTHb(p,p)lyfinrurxoop,ruuarrafi,
r : pcos(p, y : psing 6a p: \frT-F, g: arct1f; rout€oroop xon6or.roHo.
Xasrraft Agepx luyraM lrb tarru euqorr KoopAI{IrarLrH cncreMn f (xry) = 0 Teuur{T-
reJreop, ryfirrru roopgrrrrarhrg cr{creMr O(p, g) : 0 terrulrtreJlgep ,uYpcJler.uoxgoc
raAna r: g(t), y : rb?) xsnbepnfin rapaMerpr rarrulruersop erergox.6onxo.
XOY-xoop.UuHaTbrIr cucreunftr O' (a,b)Uor,Usep trapaJureflb 3eox xyBrrpraJrT
I r:x'+a(
Ir:u'+bKoop,urnar1,yArrn rouxlrerlT.uraftr eprlryaex xyBrrpraJrr
{ r:0'cosa-y'sino, - f r':trcosa+usino,{ - 6vrov {
I u:r'sina*y'cosa I r':rsino*3/coso.
4.
5.
6.
r (
.-l
32 Xaarrafu aaarrnrmx rer/uerpfrs xgildap daznoryyz
3.
4.
5.
: j 6.2 Xasrrifi,qaepx uryJrJryE
1. Epeuxufi rermurrerr vt: Ax * By * C: 0
2. Oxqrufir roes$[qrregTTofi rerruurroJr Hb: U : ka +bfiuXeB.ruu .qexb rerrrrrrrreJr Hb: Z* t:,
Hopnna.rrs rgrrurltrgJr Eb: ocoso * gtsin a - p : 0
OrerAces Mr(xtUr) qerufir erer.qcox rrrmnen,q aaftpv rapar( rrrynyyubr rerrur{T-
rEJI IIb:
U At: lc(s - *r)
OrorAcou M(rr;y), Mz(xz;gr2) xo€p qsrnfir raripu rapcau ruynyynbr rorrur.rrreJr
Hb:0-0r _ U-Utfiz-fit Uz-Ut
j-*+
f
6.
7.
8.
9.
10.
U ? kfi * br 6a U : kzr * bz xo6p uryrryyubr xooporrgox onqrnftr:
XoEp uryJryyubr uapaJr"[em 6afix EoxrloJr rts: k1: ft,
Xo€p EynyyEbr rrepnerr.qr[Kyrrf,p 6a^fur EexseJr rrb: k1k2: -1
h-ht$a: L+ krk.,
Orerlcog A(ro;Uo) qeruftr nafipyyaaa rorcronryt onorr ruyrryyu Tarax 6onox 6orooa,
rearespuftr urynyyrJ(yIl'rr 6arq rase. Earqbrr TerrurmeJr rrb: U - Ua: k(s - ro)
6aftra. Yynr k ns 4yprrr rorrMoJr roo.
Barqrra tos,6ouox IIer IIb Ap * BrU * Cr : 0 6a Azx + BzU I C2 :0 receq xo€p
uylyyrrbr orrJroJrqoJr rax ororAson 6aruHrr rerrurrrreJr HL: Afi* Bg*Q*\(Azx*BzU * Cz):0 fiaftsa.
ll. M(as;gro) qereec rcoso * ysino - p = 0 ruy.n5r5rr xlpox gafir:
d,:lrocoso*gssin a-pl
Xepoa uryJlyyu Ax * By + C :0 epeuxuft Torrrrlrrrerroep erorrcen 6on safir
rourEoroop oJrEo.
, lAro * Bao + Cl
I t/Azagz I
\-
Xaariafu aqunurry rcouerpufu xffi*ap *oiaotwa 33
12. Xo6p ruyJryyubr orronucou ugruftr rgrrufiTroJruftr crcreu'rrss 6o.uox 6onuo.
)Kuuree 6.1 A,ABC-ws A$;O) opofi, BE eatpuitn 6a BD uewarlbr rortutrTrorllTA
BE : 2c -',3a * 15 : 0 rex erce' 6ou rypaanx'br raJryyAbrE Terlru*enutr 3ox,o.
BD : 2u +3y - 3:0BogonT: B opoftu KoopAuHarwt BE 6a BD rryJlyyuyyAbrr orroJlqJloJloop oJIIro.
B: [2.-3s*lb:o *[*--Bt)' \2"+Bg-3:o
-- tr:,
AC raauu rerurmrsruir soxuoe. ACLBE y'rtrp AC-nimouqrnftE roeQ$uuuestnfrr onre..3-1 3kaa : i 6yroy kec : krr: -,
oirqrufin xoeesrauraent 6a Her rlsr "" """"n"ui#ep AC raJIbIH tsruurrrsnufir
U - Ue: kec(x - 'o)
rousBoroop oJIIro.
', !:-f,@-n) 6YrcY 2Y*3a-L2:oqsrrfiH KoopluuaTuir BD Me.[I{aH 6a AC TanyyAbllr orToJlqiroJloop.onro'
D trtAC xepuuufir ratnax xyBaarq Uor yqup C opofir C(8;-6) rex orluo' Onoo ryp :
BaJrxrrbr 6yx opofin roopAr{Har Moggr.qcsu y.r[p xo6p uerrafir aaftpcaE ruyJlyyEbr rorurlrr-
- U-Ut fi-fit -----r--^^rSUUfir
-
3 r TOM,beOTOOp OJIHO.
Az-Ut frz-fit ^ , d*;r'. AN: fi=::ffi 6yrov h*\Y-t2:a ,,
:
X(uuree G.2 flapa,nnenorpaMMLru rypBar opofi A(-2;2), B(2;4), C(6; 1) rex oror.ucolr
6on.uopea aex opot 6ouox D qerlrfiH Koop,u[aar, TaJryyALrH Terurllrrer 6a ra^n6afir Irb oJr.
6ogonr: ,Euaronannfin orruolqrrrrr uerufir E rssen rep Eb AC xepvrrtuftr rannau
xyBaarra + E(2;1.5). B 6a E qsryyanftn Kooplr{Harur Monceusep D opofin [email protected]
D(2;-L) Fex oIIHo.
AB:+4:ffi,8c:#:#,AD,#:#,cD,#:# ,
{
Dz Iz*+3s-3:o *[,*:uIar+2y-t2:0 [u:-a
\. Ii
34
6yroy
AB: 2y-0-G:0 AD: 4y*Br-2:0BC :'4y*Zx -22=0 AD : 2y - x*4:0.
TanQofir Eb oJroxlrE TylA BC xspuuufin yprLrr ons€.
IBCI:
/ qsleec BC lrgyayy:c xypralD( saftraap h onapnftr orrEo.
n=ffi- 18 - 6-- 221-- 4. "*rr.
s.raco: h.lBCl:2'.x(uruse 6.3 oY rerxuerufir (0;z) uersep orrrro' rapa)c oz-rcft 4Eo oqqor iycrex
ruynyyuu TsrrurlTroJruftr goxrao
Bolorrr. U: ka*L,U b:7,,t: tg45o:1ryn U:r*7 6onso.
X(nurse 6.4 Xspea ruyrryyu xs A(2;-4), B(b;b) usryyaufir aaftp.r rap.uar 6on ryyrnftenuruiu xoe$$ruuenr & 6a oY-rsfr orrnoJrqcon qerufir roopAr{HaT Lr on.
BoAolr' u : kx* b urynyyg .qoep A, B usrryg opunx yqrp ryc 6ypnfin xoopArrnarHb rgrrrllrrrgrurftr xanrarla.
-{-2k+b5:5&*b
9rs xo6p rgrrrrr[Trerresc ,t : B, b: -10 6omro.
X(uuree 6'5 A(2;L), B(-5;2) ueruir aaftpcax ruyrryyg'r rorrurruerr 3oxlro.BoAorrr. fit : 2; c2 : -5; h : l; yz : 2-aftr U - Ur : a - fiL
u-L x-2 uz-h "-i-AoPrIYYrIoarIfi:
== 6YroY x * 6y - 7 - 0 6orxo.
X(nuroe 6.6 U : -r * 5, U : f,r* 4 uynyyglr xoopou.rox enqrufir on.1
Eo,qorrr. &r : -1, Xr: i,ryn tgg: 3, arctgS: g 6ouuo.
X(unrse 6'7 2x*4y*5:0, r* 2u-3:0 ruyuyxuyygtilH xooporrAox osqrrfir or.EoIour.At:2,Bt-4,Az:L,Bz=2ylupt89:ffittoopt89:
# : -1, ,- arctg (+) . nl^z - ')L)
){(uuree 6.8 l"' --ny f 10 :0 rsrururrsurftr eroJr ,qypco.u rrrrrrxlyn.11_1Eoso.rrr. p: TJ#fr: ;g; C > O ryrr tt: -f ,
34ix+|u-2:0.
X(uurss 6.9 Koop.qrnarr-rr{ sxsec a * 2y - ,fr: 0 urynyyg xypreJD( gaftr ou.
(6 - z1z + (1 - 4)2 :5.
/'
Jlacrarr 6o,4uoro
6.1. KoopAuxarsrx exufir Aaftpcan 6a d : {1;2} uexropT lleptreH,qlffiyrlrp uIyJryyELI
rsrrrrlltrsfi 6rt'r.
6.2. 3x - 4y - !2:0 urynyyru xepquM A3x Terultrrrsnufil 6u'Irx 3yprblr 3yp.
6.4. fr,: {2;-3} nexropT nepregAr{Kyrrf,p Oy tsxxnerufir Graft tanr1YY xsMxIIr,[exITH
6yxnft xepqMegp orrrlox ruyfi]Yrlbr rerlufiTraJluftr 6r'r'
6.4. A(a;3) uerufir Aaftpcan Koop.urHaTbIII Tsgxrlsr1Y'ueoc 3-rafi ToEI1IY ran6aft Oyxuft
rypBairxnbrr orrollx 6aftraa ruyJryyuLr rerulumeu 6n'r'
6.5. lllyayyxyy.uLrr xoopou,qox enuruftr on.
(") 3r -2y*5: 0 6a2r*3U -8:0(6) 2r -3y*8 : 0 6a4r -6U -3 :0(r) 5x-y*5:0 6a 3c *2Y -4:0
6.6. Xoc-urynyyuyyr uapanJlellb 6onoxur 6atan. -(u) 2r*3y -$:06a4r*6A* L:0.(6) U:2t-36aY:2c+5.
6.7. Xoc ruynyyEyyA xapunuaH [eprreEArrKyurp 6onoxrrr 6atan.
(u) 2r*3y * 1:0 6a 6r -4Y* 3:0.(6) x*2Y - 3 - 0 6a 6a -lV* 5:0.
6.g. Opaunar reuxusrfiftr b: L xsp.rrraeep orroJlx, a6cqucc rsnxagrsfir{ gepsilrurnsarefi
2trf euuer lycrgx ruynyylrbr rsrru[trsr 6u'r.o
6.9. A(2;5) qsrrftr gafipcan opArIIaT rslxnsrufir b : 7 xep'ruesp orToJIx rapcafl
uyrlyyubr rsrurumsn 6u'I.
6.10. .ilapail( ruynyyn 6yplrfin xyBbA euqrufin xoeQQnurexr lc, opAtrEar remurerufrr
orroJlx rapa)c b xepvrr,rufiH xgMxtrr.usxlanrftr or'
(") 5r-A*3:0.(b) 2t*3Y-6:0.(.) 5x *3Y *2:0.(d) 3t *2Y : ['(.) s-3:0.
6.11. 2r * 3y * 4:0 ruynyyn orqe3. Mo(2;1) uornfir .ua'frpca'n
i\
(") orcox Iuyuyyntafi rapaJIJIerIb
(b) orcoH uryJryyrr.4 rreprreHAr{Kyrsp ruyflyyrrbr rarurfirrgr 6n.r.
6.L2. Koop,uuuarrrn exufir gafipcar 6a y : 4 - 2r uryuyynra^fi 45o-nftx euuor lYcr3x
' uryuyyuur rarurratrsr 6IFr.
6.18. : A(-1;1) usrnftr safipca* 6a 2x * 3y :6 uryuyynraft 45o-nftu o*qor lTcrex
uryJlyyubr rsrrnumen 6u'r.
erurlr$eJIYYA nr u * 3y : 0, fr : 3, fi - 2y * 3 : 0 6afix rypnaJlxgLr
opofiu esqrly.uufir ou.
6.15. trgraax ruyrlyfryyAtrn xoopoHAox safir on.
(") 3u-49 - 10:0 6a 6r -89*5:0.(b) 24r- 109t39:0 6aL2t-bY- 26:0.
6.16. 2r * 5y - 7 - 0 uynyynraft napanuenr 6ereoA euo rrryrlyynaac &raft rsnuyy aafia
6afix urynyylrbr rgrlumreu 6ll-r.
6.17. 2r*A-2:0,2u*4y*9 : 0 uryryyEyy.qLru xoopoulox ouqrnftu 6rccexrprcyy.uuftx
6.18.
rgruurtren 6ra.r.
llapaur rgrrrrrrrreJrufir xoprraalr xsubspr rurrJlxyTmr, KoopAtrEarbtE 9x9ec IrryJIyyH
xyprsJrx saftr orr.
(r) 8r-Gy*5:0.(b) nrfr+y-t2:0.(.) l2x * 5U : 0.
(d) r*2:0.6.19. Ksanpatrrn xoEp ralr,3s *4V*W=0,3r*4y - L3: Q rnyuyynryl Asep oprrr/Ior
6on xsaaparbrlr ran6afir ol. :,i .
6.20. A(3; -4), B(-1; -3), c(2;L) usryya .usep opofirofi rypranxcnu A opofirooc rarcaE
onapufin yprbrr oJI.
6.21. A(-3;0), B(3; -2), C(L;4) usryyn saep opofitefi rypnanxnn B opofiu Aoroo.q
- ouqrllftu 6uccextpucrfir rgrururren 6r'r.
6.22. A(l;2), B(2;-2), C(6;1) usrlYr .uoep opofttofi rypnarxnn
(u) AB tanun rsrrrrrrrrsJrnftr 6lrr.
(b) CD oxapufiu rsrrunrrslrfir 6uqrx yprhrr lrb orl'
(") CD onaep, BM Me.urraELr xoopou.uox enurufir on'
(d) A opoftn Aoroo[ ra[a6g en11ruin 6uccertpucrftE rerrnqrrea 6u'r.
6.23. Kra.uparnE 5cp3r oplurx xoep opoft A(1;3) 6a c(-1;1) 6ou Horoe xoep opofttt
Koop.qulrarbrr oJIx, TaJryyAbrlr rgrrurltrgfl 6lilr'
6.24. ABC rypBarlxgu A(1;3) opofi, xo6p rvrenran6l TsrIuI{rIorI fi.-2y * 1:0 6a
'y - l: 0 6ou ranyyAbru tsrml[rrsrr 6u'r'
6.25. lypBarlxgbl usr opofi B(2i6) 6a ngr opofirooc TaTcaH enqpufiE TsrIuIITreII
fi*TA+18:0,6uccerrprcuftn rerrrr[TrgJr 7x*g*5:0 6ou raJlyy.qbrH TeilrrllTroJl
6uq.
6.26. fypaauxuhl IIsr opofi B(2; -7) 6aeep opoftnyy'Uaac TaTcaII eugpuftn TorIuIITrsrI
3r * A * 1 1 : 0 6a ueAuasbl TeruII{TreJI s * 2A * 7 : A 6on ranyy.UblH TorIuI{TroJI
6lt'r.
6.27. fypna.nxnbr H3r opon ,4(3; *1), eep opofinyyaaac TarcaII 6uccextpucufix
Tsrrurr*eJr fi-4U*10:0 6a ue.uuaubr rerluutral 6r*10g-59:0 6o.n ranyy.uhrH
tgrurfitren 6n'r.
6.28. lypaanxnbr MeAr.raHyyAbIH TsrIIIrTrsJI 5u + 4y: 0, 3r - A - 0 6a nsr opoftr
Koop.ulluar (-5;2) 6on ta^nyyaLrn Tsuutlrren 6E'r'
6.29. fypaanxrbr xoep opoftB AorooA onurufir 6rccercpucyyaufix rerlrrtrTrsrr ! * 4:0,
Tr * 4y * 5 : 0 6a erreep ouccexrpr4cyyA rapcaH opofiryy.uur xon6ocon rarlhrH
TsrIu[TrsJI 4r * 3y: 0 6ol Horeo xo6p tauurr Tgrrul{Tren 6rl'r'
6.30. n*2A- 11 :0 6a 3r -6y -5:0IIIyrIyyHyy.uHII xoopoH.u YYcex oEUTITgrfiH
A(1;-B) uerraftr aryynx 6afiraa erurufin 6uccexrpucufin rsrururren 6uq.
6.31. 2r -3y - 5 - 0 6a 6r -4a*7:0 rrryrryyEyyAbrE xooporu( IYcsx oEIIrIYAssc
B(2; -t) rlsrrftr aryyncaE ouqsrreft xaMap entlrufin 6uccerrpuclrfin tsrruIrTreJl
6uq.
=):il
OIOI',\_.iit
\ \-"2 /\ s.t
-:fu-12
-8- .v
COJIOB VII. TAIIAPTA BA IITTAMbIH TOTMUTTEJI.XABTTATTH rErItr}ITTOJI
' 7.1. Iagapra 6a uryraubrrr rorrurltreJri
Xasrrafi assp xoEp Yn Mo.qarAextefi f (r,y): 0 TeruurmeJr Hb epeuxr{fi.uoo euap usrrrryraM ro.qopxoftnArrfir geop y3cerr.
Yynrsft rocoorefi orropryfia
F(*,y, z\ : o
Tarrur.rrrgJr epesxuftAoo rMap Hsr ralapra To.uopxoftntor.
Togopxofinoar 7.1 la,qapryy.qesp oprur4x usr dypufiH Koop.qrurar xagra"qar, raaaPryyrlrfir raAHa opuux arb u ugrufir KoopArrHar xaxraAarryfi rsrururrsnrftr lagaprurnTETIIIUTT9JI r3II3.
IrlfiMssc raAapryyrufin rerruurrorr orerAcou 6ou orropryfir sMap Esr qarufir, Top raAaprly.rloop oprulrx gcsxuftr rorroox 6onuo.
Yynu4 Mofl qa*ap Hb ra,apryyr, TITTTT{ft rsrulr,rrrengop cy^nax 6ononqoo oJrox 6afiraa,uoprur{Ho.
Orropryfirr aJrlrBaa ruyraMhrr xo6p ragapryyrufir orrJroJruoJr Mgroep roaopxofiax 6onno"
Tyxafta6an: F(r,U, z) :0, O(r, y, z) :0 raaapryygyyA oror4cor 6o.u re,urseplrftE orrnorr_IIJIoop toAopxofinorgo:r .L ruJaraM rgAsr rrb sArggp rarapryyrraftn aar aJrrrE ,ussp opurrDtqsrrYAIIfiII oJroHJror roMj.::Q-g,pe.op xeq6en.L uyrarrrrrg rlgr ur F(o, urz):0 rsrruurlenraftr
'r, O(r, Urz) :0 tsrurntrrnrta tr xauraua. '
T3TIUI,ITT3N.
oxnes.u F(rril: 0 tsrrurarrenuftr rycrafi.nar aBq y3be. 3ne rsrrufiTrerr"u z xyBbcarq opoJrl
uooryfi Oaftraa yqup rrYHA.uypbru xon6or.(o.rr er.r 6ouuo. Xaprr sxrrfi xo€p xoopgfirrarrruxyBbA uftu 6onoIr,txrW. .[espx TsrrurrTrgrreep Oz rerrxJlorreft uapa.naens 6afiryynar.rrafiqlrnurrAp roAopxofirorAouo.
EneqngH F(t, z) :0, F(U, z) :0 TerrulrrroJryyrsop xapraJr3arr Og, Or reuxJrerrofi uapan-.ue.ur 6afiryynarq Oyxufi ulrrr{H,up rarapryy ro.uopxofinorAoxur rorroox 6onxo.
Anre6pr,n ra,sapryyg)ryA.
ortopryftH arraJrr{Tr{K FeoMerplrfin cy.unax ron syfiu HL reKaprlrg Tsrru ouuorr KoopA}r-
IrarLrrr crlcreM.q aare6prrn rgrrrrETrsJrsgp ro.uopxoftuox rarapryyHyyA rorrz.
(7.1)
;-I
d
A
li(Iit
bIII AJIb H3T TSIIXJI9TT
i
TsgrsspufiIr rsrurllrroJl lrb:
Ar*Bu*Cz*D:0 (7.2)
Axz + Ba, +czz + Dxy* Erz* Fyz*Gr* Hy* Kz* L:0 (7'3)ii.i .----^- \.r---* A p r'1 n ,rrr^Ilwr 'eAer 6orooA ToTTMOJI
xs.nfisprsfi 6afi,uar. Y$ra A,B,C,D, "' roogyy'ubrr xooS$llulleuT r
Toorryy,u 6afina.
Togopxo fttrotr 7.2 (7.2) TerruI{Trerlr'rir ner,qYrsep 33prllftn epenxufi rsrlurr-
rea, (7.3) TerllltrTrerlldr xo6plyraap sepruftn eporxufi ToTIIMTI.SJI r3x Tyc Tyc
"i rrepJrsHe. (7.2) rsruu{Trsrr.q A,B,C-afts. sAax IIer IIb Tereoc enraataft, (7'3) mruur-
rsJr.[ Ar B,C,DrErF roonyyA I$r3II 39psr tsrtsft TsHqsx 6onOxryft' Xspea rrfirvr 6yc
6on reArssp rrb HerAITssp 6a xo€payraap sepruftn rorrulrrrsJl 6aftx 'raAalcYfiA xYplrs'
Aure6pxn rerrur{TreJreep roAopxOfinorAox raAapryyr a.nre6purn ra'4aprl/y r3x H3p-
Jr3A3f. Eul qaauuA 3OBXOH uar, xoepAyraap aepruftx TsrIuETrgJIssp ToAopxofinorAox'
ralapryyEYYAur cy.uanHa'
7.2. xasrrafin TerlllrTreu. Iteregc xaBtrafi xlprsax saft
:
!. Ms(rs,gol zo) qerufir ,qafipcax fr': {A; B;C} BeKTopr rreptrerrArrKynf,p
xa.arrqfis rsrrur[Trerurfi r
A(* -rr) * B(u -so) + C(, -zo) : o 17.4)
tour€orooP 6ra'ure'
2. l-ssprufin rypaan YJI MeAerAsxr3r
Aa*BY*Cz*D:01{.
TorrutrTrsJlrftr xastrafix eponxrfr rerururreu r3H9'
Yynn fr, : {A;B;Cy BeKTop Er xastrafiA treplleHArKyrsp 6a ryyuuftr IlopMzl.rb
Bercrop r9H3.
3. Xas$afin epouxufi .Ierrrrutranraftn ryxafiu TOxlIoJIAIIyyAbIr aBq Y3be:
(r) D : 0 6aftsar Aa * By * cz :0 rgrull,Ilrell IIb KoopII'lHaTbIH exufir 'qaftpcaH
xasrraftr
(b)C:06ouAc-*B.y*D:OrerlrrtrTrerl:sltOzrerxnemsft'
(.)
(d)
i': (.):
' (f)
(s)
A: A 1a.n By * Cz * D :0 rsrruumerl Eb Or rslrxfisrrefi,
B: 0fdn Ar*Cz*^d''= 0 TsrrurrTron as Og relrxrlerrsfi uapanneur xasrraftr
Tyc ryc toAopxoftrro.
C :0,D - 0 6on Ar * BU - 0 rsrurumorl Hb Oz rguxnsrufir
B :0,D : 0 fun Ar * Cz: 0 torrumrelr r5?-ii rlnxngruftr
A : 0, D :0 6on By+:Cz :0 TarrullTreir sr Oz rerxusirfir gafrpcan xaBrr'aftr
Tyc ryc togopxofiluo.
4. KoopauEarrru xaauaftryyArafi napanrerlb xaBTraftnyya:
Ar * D :0 nt Oyz-raii'
By + D: 0 sr Orz-rair
Cz * D :0 tu Ory xastrafiraft EaparrneJlb xastraftr togopxoftnno.
5. Koopnugargu- xa.strafisyyu Hr, xapraJnaH fr :0,U = 0, z : 0 TsrlulrrreJrrsft 6aftua.
6. Koopnunarlrx TsrDffigryy,unftr xapra.n3au a, b, c-xsp.rnaoop orroncon xaeuaftIr rsrurlrr-
(r)
(b)
(.)
rgJI IIb:t,uzr-t:T--rabc
6a^ftsa.YyxufirxasrraftIlxgptlMsspunepxufirrgrAsxTgrIuI[TrgJIr3E9.
7. Koopanxartrr sxosc xaarrafiA 6yynracan repfielr,ql{KyfispbrH yprbrr Psep rgMAerJlse.
XasrrafiE rropManb Eerx BeKTopbrr fi rsx reM.usrrloe. Tgrasn dd : d"ot a+i cos 0+
E"*,y 6onso. Yyra: ar gr^l xr ffi-rrn KoopArrrarLru TouxJlgryytrefr lYcrgceu olrqryyr.
3xe yea xaarraftn terrufitrsnuftr
:rcoso * gtcos 0 + zcosT - p = 0 (7.6)
xen6eprsft 6r,r.rux 6ouOx 6a ryynuftr xa"agafiE srsJl ToTIII[TroJI reIor.
XasrrafiH eponxli TgrrurrrrgJrfifir srsn Aypcs4 rrrrrJlxlYngxuftu rynu rsrmllrrsnuftH
xo6p ranrrr HopMaJrb BeKTopbrH lr,logytnftr D-nfin scpar tsM.qsrrafi aBrr xyBaaHa.
Oepoop xsn6en xastrafis sroJl TorlurrrrgJr Hb
Ar*By+Cz*D :Q*!w4e!z
(7.5)
6onno. Yy",q f,3ryypblll eMIIex rslvlArufir D -niffi. ecpare3p coHroII aBIIa.
(7.6')
8. My(r1,ylrzL) rlersec At * By * Cy * D :0 xaamafi xYprrrox xagafrlr nr:
-
- Ago*Bgt*czt*Do:@ (7.7)
roilr€oroop ToAopxoftnorAox 6a ese xasaftnr xopsB M:' Usr foopAlrllaTlrn exleft xaMT
--^- --^x- A^"r'Dn1 " loo 6aftHa.xaatrairu rrgr Ta.rr4 opIuBoJI copor, xoflp ran.u Eb gpIIIBOJI gepsr TeMAsrTgIl
Xagaftnrrru MoAyJI rrb qer3oc xastrnf, xlpsx iafi 6onox 6a gue rrr
fl,- (i.8)
9. Xo€p xas$a,fi orronqcoH, [apaJrJIeJIb, AaBxaucan 6afix 6onno'
l. A$ * Br! * Ctz * Dt:0lI. A2x * BzU * Czz * Dz:0r3cSrr T9rruuTtrgnrsft omoJlucoH xo6p xarrraftti xoopoBAOX OEItrOr Hb:
cos(p:
iorr,rs0oroop ToAopxofi rorAoso.
3osxon cosP:0 Hoxironu xo6p xaarraft uepuegArKyJrxp 6aftx tyn
AtAz * BtBz * C1C2: g
Er xastraftHyyA rlep[euAlrKyJlsp 6afoc nexqon 6onno'
Hoprraaur BerTqpyyg Hb KorrJrurrap 6afix TOXIrOII.IOJII U xas$aftEyyA llapannelr 6afix
rYrI A,
- Bt
- cr
Az Bz Cz
rrb xaBTrafinyy.u napaJrJreJrb
IIb
6afiua.
6aft-x roxuoJl IoM. XastrafiHyy.u.qaBxucan 6aftx EexqoJI
10. Hsr uynyyEbrr gaftpcan xaatrafiryyAblr 6arq xagrrafi reHe' omonqnrrH IuyJIy-
y*Lrr nr 6arqurr TanxJlgr rgAer. BarU nr xbep xastraftraapaa 6yper rogopxofiror'[oro'
Earq xastrafiH Aypbrlr xasmafir ororAcolr
Ar Bt Ct Dt',,_1-:-:-Az- Bz Cz Dz
t
ArAz* BtBz* CrCz
A?+a?+c? Atr+ BZ+ cZ
Afi*B$*Cfi*Dt:A
':
i
-'T
42 Taaapra 6a rrdrrau:l;e zsfirrrntgJr
AzraBzU*Czz*D2:gxaamafiuyy.qLrr Ter[r]rmgJrufin tycuaMxrafiraap
A1a * Bfl * Ctz * Dt * \(A2n * BzU * Czz* D2) : g (7.e)
6onox y'rup uauafi
rgx( trJropxfiftmrs. Enq ] Ayprflr 6oAsr roo.
X(uruee 7.1 A(2;3;-1) usrnftr Aatpq, B(1; 0; -1), C(-3 ;l; -2) qsryy4uftr Aaftpcau
uIyJIyyHA ueplleEql{Kyurp 6afix xastrafis rsrlul{TroJlrfir 6ffi.
Bogout: BC sertop uanaft xastraftH HopMaJrb BeKTop 6ouxc .raAax y.rup ryynuft rKoop.qrruarlrr oJrx (1) rorrm6ou.[ TaBrrxa,u fi, = {-4;1;-1}
-4. (* * 2) + t. (y -3) - 1 . {z +1) : 0
6onox ryJr xaBTraftn rarururrsfl 4r - y + z - 4: 0 rsx rapua.
X(unres 7.2 Mo(L;3;-1) qgruftr xafipv 2s*U-32*4:0,n-A*52- 1:0xanlraftnyyAa[ fieprrenArlKyJrflp 6afix xaarraftn tsrmfirren 6lr.r.
Boaonr: Oror.qcou xo6p xanmafir uopuaJrb BeKTopyyau[r d,,d-rrp reMgerlrsBoJr
(1) routEo 6coop d: {2;1;-3}, d: {1;*t;5} 6onuo. Manafi xasmafiH xer uer M6
Ms.uorAsx 6afiraa 6ouoxoop ryyNufi HopMaJrb aexrop d-uftr ouoxo.u sopr€. Manafi xasmafi
ororAceu xo6p xantraft.u uepuerqrrKyrrsp ro,ureec d,,fr BerTopyyIrafi napaluenr 6afix €c-
tofi. lltuesc mi, fri-"rn BeKTop ypxBep uanaft xasmaftA reptrelr,qrmyuxp 6a,for rya ryyurfirHopMaJrb BeKTop d-ssp aa.r 6oruo.
n:fr,xfr:{[ :r;l I
2-315
xaamafir TorrutrTrgJl
2(, - 1) - 13(y - 3) - t(z+ 1) :0
6yroy 2x - l3y - 3z *34 : 0 6aftxa.
X(uuree 7.3 Ogt tsxxnsr 6a Ms(-L;4:2) qerufir aafipcar xasrraftn rermurrer 6I,r.r.
BoAour: Ogt reuxneruftr lafipxa reArggc xaamaftH TsrrrrrrrreJr[ftr 6lr.rDrg,u
B :-D: 0 6afix Ectofi. Hfirr,rn Ar * Cz :0 6onno. ffi uelr,rfip gafipua reArsec ryyirufi
KoopAIrIIaryyA eue rorrrrlrmerrufir xaxrax A.(-1) +C .2: 0 6yloy A :2C 6onro. Onassc
2Ca * Cz :0 6yroy 2x * z: 0 rex rrrasari xanrrafin rorrrrrrrrgJr rapua.
X(rurree 7.4 Mo(L;2;3) uerrfir .uafip.r Os rsmffieurfir a - -3, Oz rsflxr:rclrnirr c:2xgprlMggp orrJrox xasrrafiH rerul[rren 6u.r.
',j, lI
-- {z;-13;-B}
la,napra 6a yralutwn rgruEnrrsnt 43
Bo,qo;rr: XasrraftE Koop.ur.ruarrru xo6p renxnsrrfir orrJrox xap'ruuftn xaM)K3o orerAcou
Tyn ryyHrrfi rgrlllrmanrafir xeprrrMT Aypc (2)-oop xaftx flb x-f;noap IoM.
' *u +1:L.-3b2
rorrurrrrelrrgft xagrrair Msuerufir.uafipax Ecrofi ryn * *|*f,: t6afi,ra. Snaeoci
b : -L26orx ruauaft xasuaftu rerrrrr{rrerr { + fi+ i: tbyroy 4x *u - 6z * L2 : 0
6onro.
X(muee 7.5 2n - 2y * z - L8 : 0 xaarrafir A(-1; 2;4), 8(-3;7;0) qer1YAsep
xr3raapJrar,ucag xep.runfir orrrlox ecsxufir luaJrra.
Bogorrr: Orergcon rarrrrrlrrsJrltfir grsn xsn6epr opyyJlx A, B ueryy.uufin xasuafiraac
xasafix xasaftnrbrx onsE.1
22+(-2)2+t2 i rro xas*afiH erorl rsrr'I{*erl 'U" -?o +i, -6: 0 6ouno.
Dnugoc 6a ( 0, dr < 0 6aftraa Oo.noxoop /, B usryya eror.ucolr xasmafts Hor ranq ryxaftu6an
KoopAr.rrrarbru ex Eaftraa r&JrA Hb 6afipna,xc 6a,fira. }Ifirr,rg xasrrafi AB xepurrauftr' ortnoxryfr.
)Efuru#r_!$ Or reux.nsr Aoep 6aftraa M uersec 4x *2y - 4y * 1 : 0 xanrrafi xyp3x
zaft 2 6on M uerufis Kooprlruaryy,ublr oJr.
BoAour: orer.ucos xasrrafiE srsrrurorq ypxur.usxyt, p: -TG+TT6: =t ,"r,2L21
Tarrurrrrsrr ns -!r - ia*i, -;:0 rorr,r. Or rsuxnar asep 6aftpnaxM uarufis xyBb,q
A: z:0 6afix ryJr 3eBxen r-ufir oJrox xepsrrafi. M(r;0;0) uereec orer,ucou xaauafi
xypox sart 2re.qreec 2 : | -?. -1. o * 3 . o - *f 6yroy 2 : | -?" -'^r6afiEa. or,ussc
2r27--132-1^-11-;n- i:*r. -;r-;:2 yeA o - -2, -in- A: -2 ye,u u: i 6onx Oo,uuorun
Hoxqelrr roxropox mretrL;O), Ur(|;0;0) rscsu xo6p uer orrno)K 6a^fina.
Jlacrarr 6o,qnoro
7.L. Koop,uunarbrn exrroec xasrraft.q uepuerrgrrxyrlrp 6yynraaan N(3;5;2) uer rapua.
XasrrafiH tarlunrrsn 6u.r.
7.2. A(3;-1; 2), 8(41-2;-1) Iloryyr ereraoB. A usrsfir aafipv Z-E "u*topr
rleptrer-
-,urrKyrrsp Oaftpna:r xastrafiH Terur]rmeJr 3oxl{o.
7.3. Mo(3; ; -5) qernfir zafip.r d : {3; -1; -2}, 6: {1; 2; -L} aextopyyarat
r
oo:?. (-1) -? r** :4*6: -?o,:'s. (-B) -! z* i .o - o: -?
,
'44 laqapra 6a wJrra,uw rsr l'urroil
trapa,EneJrb'6afix xaamaftH rerrufitrsn 6n.r.
7.4. A(2;t;3), B(2;4;0), C(=3;0;4) usryytrufir Aafipcag xasrrafiH rorIrrr.rrrelr 6r.r.
7.5. l,m-nim nMap yrryyAar 2r *my *22 - 3: 0 3r - y - lz - 9: 0 xanrrafinyy.u
ra$anrenb 6afix ss?
7.6. Me{1;2;3) qer 6a KoopArruarrrx Tenxrrsr Gypufir .uafipcan xarmaftryyrhrH
TSrIUrlTrgJI 30XI40.
7.7. P(3;-2;5), Q(2;3;1) usryy.uuftr aafipcau 6eroeA Oz rsuxnsrrefi uapannenr 6afipna:r
xasrrafix rgrtullrreu 6ra.r.
7.8. KoopArnarbrx xaBTrafinyya 6a P(3; 5;-7)uarufir.uafipv xoop.ulrratLrn .I
TgrxJrgrlTAnftr uxlrn xgptlMagp orrJlox xaarraftraap xf;3raapnarAcalr ffirpauu,uufir
333JIXy!'H On.
7.9 2x * z :0. i + y *32- g : 0 xapmafixyya 6a A(2;L;1), B(1;0;B), C(0;0; 1),
D(-1;5; 1), E(1; a; *3) qeriya orer.uxss. Oarsap uorlyA ororAcog xo6p xaarrafipaap
6aftryynarAcan xo6p raJrcr oHrIrlT'ATsfi :xuruuxeA ,Map dafipnanraft 6afur se?
7.10. Ky6HTT xoEp raacrhru TerrrrrrrreJr x:o2a-2A* z-L:9, 2r-2y* z*5:0 6ou
icy6uN sssuxyyunftr roAopxofia.
7.1L. M(1; -1; 0), N(1; -2; 5) ueryy.uufir nafipv 2r * 39 -,42 * 7 = 0 xa,srrafir
[eprreH.ur{rynspaap ouropox xasrrafiH rsrnrurrer 6u.r.
7.12. l-uira tMap yrrau.{ lr * U - 3z - 8 : 0 6a 7r * 2y - z - 5 :0 xasrraftEyy[ trepuen-
-Aurynrp 6afix ss?
7.13 Jlapaax roxuon,rlnyyAa,A xatsTrafix rerururren 6n.r:
t. M(0;2;0) uernfir Aafipq, Ogr rsnxrsrr reprex,urxynflpaap erropex,
2. P(3;0;0), Q(0;3;0) qsryyarftr .qafipq Oz rsuxuerrafi napa,nrrertrap ouropox,
3. Oa,Oy tsnxnsryy.unftr a : 3, b: -2 xsprrMsep orrlrou i: {2;1;-1} BeKTop
-taft uapaunenr 6atpna>r.
7.14. A(1; 1; 1), B(2;0;0), C(5;6; 1), D(*4;0; 1) usryy.q 6a traparrlrerrb xo6p xaarraft
3r * 4y *22 -10 : 0, 3r * 4y *22 *5 : 0
eror.uceu 6on egnespnftn xapunuan 6aftpmr[JrrLrr ro.uopxoftn
7.15. P(-L,L,-2) usrsac I4I{L;-1;1), Mz(-2;l;3), M3(4;-5;-2) usr1rynuftr aafipcar
xaatraft xyprsJrx gafir on.
7.16. 2a-y -22* 3:0 xasrrafiraac 5 Herx gafi.q 6afiprrax rlerly.reec rorrox ra.qap
-ryyrrfin rgru&rtrgr 6u.r.
h
Illynyys 6a xaarra,fis xapfrlzrgidr dartpnwr
coIIoB vrrr. myJryvH rIrvrAM, ruvJlyyH BA XABTTATaH
xAPrlJruAH EAIaPmTdJI
8.1. ffiyayyu ruyraMr rrrynlnfx 6a xaarrafix xapurr{as 6afiprutrrr
L. Ms(a,b, c) usrufir Aafipcar, 3 : {*;n;p} qilrrlTrorq BeKTopToft uap4nnerib ruyny-
yHbr TsrruuTrgJr Hb:r-a:a-b _r7"
npxsr6sprsft 6afina. Yynuftr ruynJryEbr xrrr6ap rgrrrrrlrreJr reuo.
45
2. (8.1)-mrrur4rreJl .qexb rorrlJty HoorABop 6yprfir t-tsfi ToEqITnx x,y, z'uftr on6ou
fr:a*mt, A:b*nt, z=c*pt (8.2)
6onxo. Yynufir ruynyyurr rapaMerpr rerurrlrroJt, t-r uapaMerp r3.q3r.
3. M1(x;Uti zt), Mz(rz,Uz, zz) receu oror,ucon xo€p qerrfir lafipcax uryJryyutr rsrrrrrrr-
rSJI:fr-fit
(8"1)
(8.3)U-h z-zr:-:-Uz-Ut zz-zrfiz-frt
xsn6sprefi,6aftua.
4. Xo0p xastraftH orrJIoJrqoJI Eb IrryJlyyn 6afu< tyn
( h**\fl*Ctz*D1 :g
Ior, *Bzu*czz*D2:g(8.4)
cucreunftr uynJryrrtr epexxufi TerrrrurraJr reu3.
Illyuyynal epouxufi Tsrrrrr{TreJr (8.4) oreracoH 6aftsan xrn6ap rerru}ITroJlufir nr onx
6onuo. Illyryynu qurrrlynsrrr Bexrop lrb ni = {At,Bt,Ct}, d: {Az,B2,C2} sex-
TopyyAaA rrepuerrArrKyrrp 6afix yqrp sArespufiu BeKTop I?xBopeop roflopxoftnorAoro.
(8.4) crcreunftx aJIb Irsr ruuft.q 6yroy ruynyyrrbr Mo(xo,Ao,zo) qernfir on6ol uryJryylrbr
xrn6ap rgrrrrr4TraJl
6oano.
T-fro U-Uo z-zo
w:w:w
Yyug: S : {m,nrp} rqtrrJrwrertr Berrop
81 Cr
82 Cz, Tl: A1 81
A2 82
CrArlcz o,l' o
5. orropryfix xoEp ruyJr)ryn orrxoJrqcog, uapaJrJreus, coaOrcon 6afix 6ouxo.
Xo€p orronqcoE ruyJrJnfEEr xoopogrox egqer Eb
cos{p:
tn:
6aftEa.
0-a _ y-b _tnn
xoopoEAox oEqor
. sln(P:
rour€oroop rneprulft JrerAorrg.
z-cp ruyJrJrJrE, Aa * Bu + Cz * D
IlAx+By+Czl
: Q 1gena,rafi xo€parn
rourEoroop unepx'rftrrcrAorro. Yyna {mr;nr;pr}, {*r;nzipzl rrb rrrynyygyyAhr' rrrrrryrsrqBexTopyyA roM.
Xo€p uynJrJrrrbr trapaJureJrr 6afix EexueJr Hb
fi,1 tTLl ptt: *r: p,
6aftsa.
flepuerguKyJrrp 6afrx EoxqoJr rrb
tTtttTtz*n1n2*ptpz:0
(8.5)
(8.6)
6.
\ffiawIllynyyx, xanuaftrafi
Am* Bn * Cp: O
EOXqOJr.q uapa.[Jr-e[b
EoxqeJrA rreptreE.qf,xynrp 6aftua.
x(uuree 8.1 Mo(2;1;3) uerrir Aafrpca* ,3: {a; -b; -6} uerroproft raparrnenb ruy-nyyulr rgrlrrumot 6uq.
A -B -Crnnp
trlrtTtz*n1n2*pflzrn?+n?+fr rnT+n?+ p:22
Eo,qoilr. Illynyyurr xrn6ap rorrurrrnsJrsfir canasar rr,ra,na,ft luyJryylrbr rerrulrtreJl Hb
*. a-2:U-L _r-34-5-6
xer6sprsft Gaftna.t'
X(ruuea 8.2 Mo(L;1; 1) uaruftr gafipcari, Or rexxnsrrgfi raparnerrb ruynyyubr rerlrrllT-,:.rsnrfir 6uc.
Bogotr. Illyuyyn Oc reuxrsrrefi uapaunerb yrrpaac ryyuufi tIpIrrITnBrq BeKTopbrH
Oy, Oz tsvxngr ,uospx rtpoerq Eb rgr 6afina. Exg roxuoJlroJlr ruyrtyyubl qrrfilYrerrr
BeKTop ,9-nrr {1;0;0} rexr xeJlx 6onro. Irlfirr,rg uauaft Iuynyyrbr rerlrrl{TreJl Irb
xsn6sprefi 6af,rra. Yyuufir epouxut Aypcasp 6r,rqssr
'ly-1:0 t(z-l:0 I)
6onso.
X(nnros 8.3 A-L;2;3 6a B(2;6i-2) usryylufir .uaftpcag ruynyyglr rorlrrllrronufir
6nvrax, ryyuufi trilrirl !'nsrq Koc[HycyyAbm on.
Bogour. Oror'acon xo€p uerrfir 4aftpcau uyrryyubr rarrlrr{TreJrflfir cauasan uanafr
rrryJryyrrlr rsrurutrgn irrr*L _U -2 : ,73-2+L 6-2 -2-3r*L Y-2 z-3
34-56onno. Ous ruyryyx Hb di r"*roproft uapaJrJreJrb ryrr ryytrlrft vurnyynorq Kocruycyya Hb
AB sexropbru rrurnyyrgrq Kocuuycrsft .raBxqaHa. llfir"r.q
6yroy
32+42+(-5),
::*#Il
Bogonr. XoEp [ryJryyELr xoopoH.uoxb euuor Hb ra.urooplrfin 'rurr1yfi3rrr BeKTopyy-
ALr11 xoopogloxb en[orrsfi AaBxqaxa. Epoxxrft TsurrllrrsJlsep ororgcox lrrynyy{ibr qurnlYfl9rq
BeKTop rlynuftr rogopxofinorq xaBTraftxyytnn HopMaJIb BexropyyAbrlr BeKTof ypxaspreft
rgnqensj I,Iftrrl yxpaac l-p uynyptbr turrriTnorrl BeKTop Eb
*: {l
-J ),
2-p uryuyysBl qtrrrlTnorll BeKTop rlb
,,:{l ;-i
1(-4) + 4-3 + 3. 1a^o/^---:' lEffirtlt-ay, + 32 + 1z
112*2
1 -1 l) : 1,,n,rt2LIJ
,
6aftna. llfrrra,u ororrcen xo€p ruynyyxrr xoopou,u,oxb osqor Hb
112-1
1123 l)
: ,-n,,,,,
:
26
.119: arccos
55 6orno.
)*(unreo 8.5 M(L;3;2) uarxftr aaftpcax x - 2A *22 - 3 = 0 xasuafis reprlen.urlxyrlp
rrryJryynbr rsrurnrren 6u.r.
Bo.qolr. XasmafiH HopMaJrb Bexrop fi : {1; *2;21rrb rrryayyrtaft uapalnenb ryn
ruyJryynbr rrrlrflITfiorq BeKTop 6oaso. Irlfir'aa .i[4 qerrftr aafipcau d vnrrryynsr.r aexroproft
ruynyynLr rsrrullrroJlnfir 6ra.rssn
a-L Y-;3:-:L-2z-2
6aftsa.
X(nurea g.6 * : Y;3 :'!t uyrryylrbr r+2y-Bz-4:}xasrraftraft----.-\^\,l.\,1/z 4 2 5 -
orrorrqcog ugrnftr on.
Bonorrr. gxnas.q uryJryynLr [apaMerpr rgrurrlTrsrrftr 6ra.rnen t : 2 * 4t, y :
3 + 2t, z -- -l * 5t 6orno. Yynuftr oror.ucou xasrrafix rsrrullTroJr.E opfiyynau TaBrrBaJI
2 + 4t+ 2(3 + 2t) -3(-1 + 5r) - 4 : o
6ytoy
-7t +7 :0
\
6ongo. Enqaec t: 1 5aftna.
t : I-u[r iiryJlyyubr rlapaMeTpr r3rrrrumsJlA opflyyJlau TaBIBaJI
fi-2+4.L:6, U:3*2.I:5, z: -1+5.1:4
6onno.,I,lfirraq ruyJIyyH 6a.xasrrafiH orTJIoJIqJIbrH UoI' w M(6;5;4) 6afina.
X(unrss g.Z * : y ;L :' 13 ruyrryyr 6a 2n - 3a - 2z * S: 0 xanrraftn324xoopog,uoxr onrlrufir on.
Bogorrr. Illynyyn xasuaft xcl€prur xooporlAoxb ouqor oJlox roMb6o 6coop
sing -8
\/493
6afiua.
X(rrurgg 8.8 M(*L;2;-3) qsrrfir aaftpcar t ,!3 : Y --^L : u !^3 rrryrryyg.u reP+32flex.qtrKyJrrp xaarlaftH Tgrlu]ITfsrrrrftr 3ox[o.
EoAo.nr. Xearrafi EyrryyrrA rrepregArrryrsp ryJr tyyxrft qurtryYrrerq Betcropr MoE
reprr€H.qrrKynap 6ai-na. ?Ifiua sro roxr{oJr.qonu luyJlyyxbr rrrrrlTrorq BeKTop $ = 1a;S;Z}
xr xasuaftH HopMaJib BeKTop 6onox ryJI xaBTraftx rerurumoJl Hb
4(r + 1) + 3(y -2) +2('z +3) :0 6yroy 4u *3y *22*4:0 oafira.
x(zures g.g * :y;\ :'trz ruyrrypr 2a-u-22-g:0 xaarrafi uaap423opruuxo rs.urlrftr 6arat.
Bo,qoar. OrorAcen uryJrJryE eror.ucou xasrraft ,ueop opruux lrry.Jryyn .[eep opruru(
Mo$;L;-2) usr MoH xasrraft.uoop oprur{no. I,Iftr*a oror.ucou qsrufis Koop.utrEar xasrrafiH
Tgrlur[Tr9nuftr xanrana.
Onoo auo roxrloJrAoJrA xaBTrafi uyuyyx xo6psrr [apalrneJrb 6ouoxarr 6atra:ratr xylnnqserefi.
Xaarrafi ruyJryyu xoep uapanrenr 6aftxbrrr rylr xastra,fix EopMaJIb BeKTop fr,: {2;-t;-2},urynyyrLr qmnlTrarq BeKTop ,9: t4;2;3) nepuelr[ExyJlflp daftna. Oopoep xsn6sn
2'4+ (-1) ' 2+ (-2) '3 : 0
llftr"ra ororAcolr IuyJIyyH ororAcerl xasrrafi [99p oprur{rro.
.3+(-2).4+(-B).21+ (-3)2 a (-z)21/PJT-142
t,
r
Xaarrafi 6a ruynyynrr 6oarroDyyJl
l. M1(lu1,Ur, zt), Mz(xz,Uz, za), Mt(as,Ut, za)rlsrrftr,uafipcau xaarrafin rorrrrrrrreJr 3oxr{o.
JIyFrs qerrfir as M(a,gr,z) repeu @: {rr.,Ur,rrl:Vi,fr:@: {nz,Uz,zz},ffi, : {rr, yr, zsl
" : d, ffi : y- : {r, y,z} rsaen r- - Fi, d - Fi, d - ri BexTopyyl
xoutrnarap yqtrp
(e_ d) .(fr _ d)..(7i _ d): o
xoopltrEaraap 6u'rnst
fr-rt U-Ur Z-Ztfiz-fit Uz-Ut 22-Ztfis-frt Us*Ut Zs-Zt
-0
.r A-fiO U-Uo Z-ZOz. : :
- : ruyJryyu, ryyxufi ra,uua oprlrrx Mr(xrrUr, zt) rlsrufir .raftp-Lrnn
car xaBTraftn rerrurrrgJr 3oxrlo.
tlr.m,nl, @1: {rr -to,Ut-Uo, zt- zs}, M& : {r-rs,y-ui, z-:b} nemopyyr:
KOMIInaIIap y.I[p
fr-to U-Uo z-ZofiL-fro Ut-Uo Zt-Zo
lmrl-0
3. llapanaeur xo6p ruyJryyxhrr Aaftpcau xasrraftH Tsrrurrrrerr 3oxno.
fi-frt _U-Ut:Z-zL fi-rz _y-Uz _Z-Zzl-*:-;-'t:-;:TMt(sryr,zr), Mz(*2,u2,22) {oryy4 xasrraft ,Esop opurrgo. M(*,y,2) au xagrrafin
.Eypbru qer 6on @: {r- frrtu-ut,z- z}, Mfr2: {xz-nttaz-ur,zz- zr},
.f : {1, m,n\ BeKTopyyA xoMrnauap ryn
fi--.01 U-h 27.21
fr.2-fit UZ-Uy 22-Zt
I :m, n
:Q
:t-lDz U-Az Z-zz:_:_lz Tn2 fi2
4.
I'
Orrouucou ? -= " - U - Ut :'z'- zt -
h rfi1 Tty t
aafipcan xasrrafig rgrrrrtrTrgJr 3oxr{o.
urynyyHyylbrr
Illwtyys Ga xaerrafu.xapqwas dafruqs.rr
llyprrn qeruftr us M(r,gt, z) :reaen
fr-frt U-h Z-Zt
h fi,1 Trtl
lz ll,a fTL2
:$
'i
5. Ax* By*tz* D:}xanrraft ry : o;'l : + rryryyrrbr xoopouuoxB
onuruftr on.
Illynyyxu, xaamaftA Earrcarr orqrufrr g(0 < , at) roBsrl xanrraftn rropMailb fr.vfil
ruynyyrbr rrrrrryyrerrrsft yycrecer ollqot { - g ftaftna. }Ifirra y.rpaac ' ,
4
sinrp: *t(; - P):
6ougo.
g --: oon .ff : {I,ffi,n}, fi, -- {A, B,CI BeKropyyA KorrntrEeap y.rrp' 2 \ ', ', ',', \ ', ', 'ABClmr!'Pj.0.6orr-.f.ifrylinp.N.+.Btfi+Cn:0rroxqJryYgrapra::.........
^ fi-fiO A-UO Z-ZO r h ^__^_____e_-6.
--: -: ruyJlyyn Ax*By*Cz*D:0 xasmafiE orroJlqJrbrrlmn
[srnftr on.
Illyryyru rs.rmrrrenrfir trapa,Lrerp xsn6eprsfi 6I.rET xanrrafrn reuutrTroJlg qpJry-
yrx. At * Bm * Cn I 0 6on t : :ffi rex arrgor uIy+yynbr
.j1
t
52 IIIytryW 6a xaarrafu,xapswaq fiartp,'lv.lr
i
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8.1. M(L;-L;-B) uerufir ai,fipcax
r) t2;-3;a) aexroprot ,) +:#:+uynyyrtafi 3) r:3t- 1
U : -2t * 3, z : 5t * 2 ruyJryygTaft uapauuenb myrryyubr rerrutruen 6n.I.
8.2. s-y.3t-2, y:0, z- -f,,*36a r:2t-L, U:0, z-f,,- 3urynyynyyuur
xoopouuoxb en[ruftr on.
8.3. M(-L;l;-2) uereec 4a -5y - z - 3:0 xasrrafiA 6yyuracar reprleEquxyrltpurE
TerlutrTroJrufir 6u.r.
8.4. r*3 :4 z*5Z g =:; rrryrlyyrLr 2a*3y*z-22:0 xaarraftrafi ortnonqcon
ueruftr on.
8.5. A(2; -L;-3), B(5; 2; -T), C(-7;11; 6) ueryy.r aesp opofirofi rypaanx3u A opoftn
raAaa{ euqrrfix 6rccexrpucxftn rerurrugnrftr 6rq.
8.6, ,,!(1; -2;-4), B(3;1;-3), C(5;1;-7) uerITA .qsep opofirofi rypsanxrn C opofrrooc-V--
gcpgr raJr Agep 6yyaracan rrepueguuKyrrsprdE TgrmuTrgJruftr 6[.r.
r*2 u-L z- fi+Y-z:O I8'7' :f : =
: _ 6a n - y - rr -g : 0 I
*"',"yy'u uapaJruenr 6ouoxnr
6aran.3x-u-1:0 I8.8. " I Eyrryyrrrr 2a*Ul z-4:0 xasrrafirafi yycrscsr enqrufir on.
2x*22-2:0 I - t
8,g. +: + : +ruynyyu 2a * a - z -0 xasrra,fiia,i uapanuenr 6a
2 :2= : -T_ rrryJryyE oforAcoB xa,a:raft .4eep oplutrHo rex 6arau.
8.10. A(-L;2;-3) usrufir .uafipcau fr :2, U - z: 1 uyuyyuyyA rlepreuulrKyrltrp
xa,srraftH TgurrrrrreJruftr 6u.r.
8.11. + : # : + ruylryyH 6a A(3;4;0) usrnfir nafipcax xasmaftu reruur?-
-rerrfir 6u.r.
8.12. + : + : + ruylryyrbrr Aafipcau 2r * 3y - z - 4 -- }xasmafiA rlepnen-
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8.13. A(2;t;1) usrufir a + y * 3z *5 : 0 xaamaft A$px upoerqrftr on.
8.14. A(2;3;1) usrufin x:t-7, U:2t-2, z:3t-2 urynyyrr,ueepxnpoexqufir on.
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fr2 , U' -,A- b, -' (e.1)
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rex rrepnox 6a a ) b rex y3oe.u $oxycyyrurrr xooportrox safiH xaracbrr c reB3JI
c2:a2-b2
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nr A1(-a; 0), A2(a;0), B1(0;-b), Bz(O;b) 6oreea exs 4 ugrrftr snurncrfiH opofin u3rlY.u
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82(0,3)
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xapbuaar ryyxufi ercqexTpt[crrrer rsrre. Oxcuextpuclrterufir e Ycr3op tsldAgrnsx 6a
rogopxofinonroop ,=*: Z,
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5
"g
Eo.iorr-. _lg($1I),: Iim(l + 5 )': rirl[(l +,CI=;Tff:- c-:f&\ Bx _ L ,, ,-;Ab\- 3f _ 1t x,-c
6onno.
x(uurss 10.5 I,s w -xr3raapbrr 6o.u.
Bogorrr. 9nq sint' * x 6a11rrrla-(l+')'a+O fr: 1 6onoxbrr aIutrrJIaBaJI
ln(l + sinr) !. ln (1 + sinz)rtrlr -___'-.,...,.....- -
lrlu
--
c+0 era - ] a*0 Sin f
11 11.t.8.1:E6onno.fi
sln 0 : ll[I3(eo' - L)r o+o
3r-
ln (1 + sinc) sinr .1'.'fr3r
n3sln u
limo+0
x(uurae 10.6 "*
*"t'1;1ffi- -xn3raspbrr 60,u.
"-$ ln (lff -xn3ra3Pbrr 6oa'
Eo.qorrr. r .-+ 0 yen In(]" -,r) - (--r), arcsin # * ffi ywrp
.frfiarCiln-7:-_---'"""'-\/ffi: ,,m !ffi: _l 6onno.
ln(l - r) c-o -r
t
7-
62 Esr xyn*aiwea $ywearAm x*sraztp
: : 10.2 Oyuruufix' rac$a.rrrryfi'vauap
1. OpeocreJr x{BraapJyn: /(r) Qyuru fi: a ueruftn opqtrEu tonopxofinorlcou 6aftr.
Xeryeu 0 > a (, < o) 6aftraaa n + a yes /(r) Qyrxnuftr xmraap opruIm 6afisan
ryt'srfir /(r) Qynruufin n I a yeufrn 6apyyn (s1'yx) opoecron xx3raap rex 6a
f (d+0),. (/(".* 0)) rex( EoMAorrose.' Xapsa |rg/(r) : A laftsa$
f(a+0) : /(a - 0) : A 6aima.
2. @ysxqrftr racpanrryfi vanap. Xspar
1) r: o uer neep /(r) gynrrl to,uopxofiror,qcou.
2) [r31/(c) : f (o) (10'1)
6afisar y: f (r) Synxuufir fr: a qor .qeep tacpa.rrrryfi relle.
Ac: n-a, LU: f@-Ar) -/(a) rsx reM.rerrlerl xufiasn 2) xoxunuftr
Iim Au: 0Az+O
(10.2)
xsr6epsep 6tl.rrx 6omro.
llfiMsac U : l@) OyEKU fi : a IIor 1leep TacparrTrYft ro.qrufir ,Uapaa)G Masraap
rogopxoftnx 6ouno. Xepaas
1) o: a [er aeep /(r) Oysxu ro,uopxofiuor.ucou
2) *Eot/(a * Ar) - /(o)l : 0 6aftsan /(r) gynruufir n : o, qar Aeep racpartrvfr
6aina roAar.
3. (r, b) aaacparn 6fx qer Asap f(r) rlyrxq racparrrryfi 6on ryluufir (o, b) 3aBcapr
racpa.rrrrlfi 6aftna rens.
4. @yuxuufin tacpanrErlr qer.
Xspea /(r) gyrr q , : o uer.uegp racpaamyfix Eexqrtrtr xauna:<ryi 6ai"aar fi: a
qerufir /(r) $ynrquftn racpaJrrLrr qor rere. eua rs rapaalrb xexunufrr,uop xaraA
Hgr Hb 6ueusxryft 6afiua roceu IT.
t) /(r) Qynru o: a qer Asep roAopxoftnorscon 6aftx
2) /(a + 0), f (" - 0) opruuu 6atuc
3) /(a + 0) : f (a -0) 6aftx
4 f @+0): f ("- 0): /('a) 6aftx
h
.Efsr xyarcar,refrm AywryftE. x*rrffip 63
5. Tacpanmyfi Syxrrurftn trarrapyyfl. Torcrener roorbr racpairmYft Qynrurfts afirs'
6pufin ulrfin6ep, ypxBep racpanult OyHrrI6afixa. Tacpatrryri QysxqYygrftu uoorlsop
Hb xyBaapb Torosc saraataft Uor Aeop racpantryft 6aftEa. Xepass a : I(u), u : 9(a)
Qy4xqyys racpa,rruyft 6on nuftnMsr Oynru a :. flp@)]-sr * apryMeuraapaa rac-
paqtryfi 6aftxa.
X(raureg LO.? f (r) : xn (n-xarypanr roo) QyEKu (-*, co) saacap aaep racpanrryfi
6*noxurr 6ara"u.
Eogo,nr. OrorAcou qlyxrq c-ufir 61x yrrana roaopxofiuor.qox y.lup 1) Hoxqorr 6uensx
Hb rrlropxrrfi. irlfiua 2) nexunuftr ruanras. (-*, m) sarcapr opIulx Aypbru fi: aqgr aBtI
Ar oopunoJrr orq xapra*r3ax $ynruufix oop.rreflT Ay-rr onSon
Ay: (a+ Ar)* - oi: nan.-tA.+ 1(":-U an-|(Lr)z+ "'+ A$*:zt
: LrLnan-t + '("-- 1) an-2 Ls+ . . . + ac*-11 oaftxa. D'xesc jlgo au : ji1o a, bxan-r +
/ 1. -' ^l^ 1\
;Lt?? - r l::l'--*ir & n-2 Lr +. . " * Arn*'i : rlpo Ac. jimoln an*r +W"o-| L,r *. . . * Ao'--r] :2t
0' (no"-1) : 0 llfir,r.u /(r) : r', Synxq a IIsr .r33p racpanrryft 6afina'
}}(iuree 10.8 U: cusr $ynxu (-*, oo) earcapr racparlTryft Oonoxrrr 5atan.
Bogonr. Oror,qcou SyHKu (--*, x) sarcapr roAopxoft;rorAox yqlrp 2) xoxunufir mal-
t'aBaJI , A \ La :cos(r + Ar) - coso : --zuio ( Ar\ Ar
ionox ryu\"
* ?J sin;- 6onox rYu
jgo^r: jigo [-r.,ot,
**t;+] -'-)ii1o,io (". +) ii3o,io +:-2sinr'0:0 5afina. I{fiMI !!: ensfi OyEKU (-*, m) saacapr racpaarryft dafiua.
)I{rrnrss L0.9 o : ry $ynruufin racparlTlru lrsrrftr on" :.
Boflorrr. cosr, 12 6ynxuyyg (-*, oo) sancaplsop racpanmyfi. Ufir'ag racpanmyi
Syxxqufiu qaHapaap fi * A yeg oror.Econ Qynxu racpaJlTryfi 6a.fiEa. Xaplul s : 0 qer .Ileep
orarAcorr Qyuxr{ to,uopxoftnor.uoxryfr yqrp racapua.
X(uruee 1.0.10 Xspan 0 ( c < 1 6on l@) :24, xspsn I S *( 2 6oa f(r) :3- x
rox ororAcon SyxxU 0 ( c ! 2 sancapT TacpaJlTryft 6onoxgr 6arau.
Boronr. 2r, 3 - s $yrxuyyl rlrlfta61x yrraua racpaurrlfi yqtrp ororncon Qynxu
0 (r ( 1, l<r ( 2saacpyyraf racpalrrryfi6aftxHbrulapxuft. ToaopxofirlToocY3Borr
l(1) : 3*1 : 2. Horoo raJraac J(1+0) : |g1(3-r) : 3-t : 2, l(1-0) : }iIl 2s:2'l:2
6onox tyr /(1 +0): /(1 -0): /(1):2 6oux t:l trer Aeep /(r) rpyxrq racpanrryt
6aftua. I,Ifirvra /(r) $ynxu 0 < n 32 3aBcapr racparrrrYfr 6afira.
Hsr )rystcArfu $ymcwfim xslraap
)E(urues 10.11 U : cosrs QyErU apryMegT s-uiru 61x y.1.rau1 TacpaJrrr'fi 6onoxurr
6arar.
Bo,qonr. a: cosu,, lt : si rDyrrqyyA apryMegTLrnxaa 6Ix yrrau,q racpa.rtryt y.rldp
racpanrdyfi QynrurfiH trarapaap U : cosrs QyHKq r-uirr 6yx ymara TacpallTryft Oaftua.
Jlacra.n 6o,qroro
1+:+1+...+ 1
10.1.m'#a roz jgg-ffi,+5+g+...+3n
1 1 1 . ' " ta*l10.3. #*"(1r+5;+...+ &Ti) 10.4. m'l,"-Tl1o.b.ttuq#10.7.rg4mro.o.B1 ffi
n2 +l10.6.riqffi10.8.,t*ffi
ro.1o. uq6;%11,- -j
10.12. FS ffi (m, n 6vxsn roo)
10.14. ,' "F+ a -'tT
c+0 fi
10.16. li* tio5'o+0 fr
10.18. h* 1 -:ot'c+0 fz
10.20. u*2arcsinrc+o 3n
10.22. ,r* ..'E'-zcosn---7t r - 4x
fr1 -4
10.24. tm (t - l)'c-oo \ f /
10.26. u* (t * 3)'"o+o \ n/
^2a 1
10.28. lim " .-'c+0 Jt
10.30. u* lnr - iu4e fi_e
^olt ^b,10.32. lim" -"
c+0 t
13 -or2 * 11r - 610.11. lim - :-
=s2-3n*2,ffi_310.13. lim'- :^
10.29. Iimo+0
In(r*o)-lno
limcq1
lim------ o-*r-o r_Lo10.15. -tipq(\ffiT - lJrz -t1,+6'
10.17. li- titr'a'o tgSx
10.19. ,r* tgo -=sinoo+0 do
10.21. u% (; - ,) "s*,-,
10.23. r,*f-Fo+o sin, r10.28. u* (r * !)*,+€ \ fi/
10.27,,']Ig(;*)*-'
i:':
r
10.31. lr31(1 * sinr)@'e@
tbywryfia yJrau2otralr 65
CEJIOB Xr. (DyHKUltlrH yJrAMX(JIAJI, TEOMETP BA MEXAHIIK
YTTA, JI}IOOEPEHU}IAJIIIJIAX ITYP OM
11.1. (DynxqnfiE ynaMzrJraJrr reoMetp 6a Me,(aEuK 5rTryy.4ri
(nufiu6ep, yptrBop, Hoor.qBop, AaBxap, ypB5(y1 A&II.[1 IIapaMeTp xatl6gpgop
ororAcorr Qyxxusrn yJraMrrJraJryyA)
.[uQQepeuqqrartrna]( AypMyyA
L. Huft.rr6epufin yrraMxcnarr"
Torcronor roonhr.qu$sepeuqrraJrqJrarra>c Syrxurrn nxfindspuftr yuauxJlaJl EsMer,qsxYl!
ryc 6ypnfiH ynaMxrlarlyyAbrn nrafin6sprsfi rsnuyy 6afina.
(r, + uz* ... a un)' :''tt'L* dz+ .., *u'n
?. Ypxaspufin ynaM)KnaJr.
u(r), a(r) nr nn$$eperquartlrargax Qyrrrllyy,u 6aftr. Terssn
(''o)' : Lt't 'a * u'u'
'Iorrr"ron ypxur.uexyyxnftr yuauxJraJrbrH rsMArlrfiH oMHe raprax 6ouso.
("'o)' : c'a', c- const
3. Hoor.qaopHrr ynaMxcJraJr.
u(r), u(r) nr augQeperquartrrarAa:c Qynruyyn 6afir. a(c) * 0 yen
/u\' 'I..t'u -'u,u'r . (;,) :
T' c- wnst
r9l, : -{''n' u2
4. ,Ilaaxap $ynrqqu YJraMrc[aJr
X rrayxc Aoep roAopxofiuor.ucotty: flv@)l OyEKu 6aftr. z:9(r),y: lQ) Qynruyyn
,quQ$epeuquartularAax 6afi san
f'lp@)l: f'(z)' z'
6yroy
l'lp@)l: f'lp@)1'e @)
-i
-.'.w
-!rt
l
66 iDygr',ryfru yJralrxJrair
X(nmes 11.1 'u : cos2 fr
{ : 2cos r(cos r)' : -Zcos r sin fr, : - sin 2r
5. Ypayy Qynxu*'rr ynaM)KJraJr
Xspsr A:if @) nr auQQepeHrrr.rarqrarlax 6erooA f'(") * 0 6aftxaac ranna n: g(il r'Ecsrr1
ypByy QyHKu Hb opurrru 6afisan ,L: -"-;;;i; ;';.;' ',y +arccos, oy",?fr*" yrraMxlrarrhrr o,r.
3ns $yrrqbru ypByy Hb
1
th-rz
(arccos x)' : -+G. rraparvrerpr xer6epeep ororAcox gyuxqJi ;*rxsraJr
{ *:p(t)
;tol.tl-TI u: d(t)
Tsrrrrr{Tr3Jr3gp ororAcoH rglK y3be.
gG), $(t) Qynxqyys arQQepeuurarrlr{an'.qa:': 6a ,b'Q) *0 6afir yL: ffi"@ynxq*rn ynaM)KnaJrrrrr{ reoMerp 6a Mexax}rr yrra
Mypyft y : f (r)TorIuI{TroJIDop ororAcou 6a ryynufi o0 a6urccrafi qgrt rarcaH ruypier.r::lYix
0o rsuxusrtsfi yycrex ourlon ur o 6on f'(xo) : tga '6aft.rar. Xaprn y : f (r) rrrypyftr
Mo(ro,go) Ilgrt rarcaH uryprsrrruftn rsrurumsn lrb A - Ao: f'(rs)(x - ro).
Illyprenrufin qsrutr nafipcan 6eroeA TIyHlr fleprreHAtrKynf;p ruyrryygbrr yyn uypyfin
EopMaJrb rgx HopJraaer. Hopuanuliu rgrrrrrrrr3ir Hb
a - uo: /-.',;r. - ro)
xsa6epraft.
A : fr(s), y : fz(s) uypyfixyyaLrn orrrrolrrrcou usr Mr,txo,yo)-.q rarcau rrryprsruyygr.uiir
xoopon.uox ouqrufir sAroop uypyfiu xoopougox ourlor rg)K Hepnex 6a ryyxuftr
tgg: l'r@o) - f't@o)1 + /'1(16) f'r(ro)
rontEoroop ro.uopxoft"u"qor. Xspsa Marepr{aJrJrar qer s : s(t) rocsx xyyJrr.xap ruyrxyyu
saMurs xoAonrooll ,usx xypA rrb 3aMaac xyrauaaraap aBcan yJraMxJraJr 6afina.
fi: cosu *L: - sinSr - -\Fcnfr,/ -l --
1
h
a = s'(to)
rb"
r
r7. Jla.rr,q xen6epesp ererAcex, $ynrqr*E y:raMryfl&rr
Apryrvreut r, TIYII33C xaMaapcaH y Qyuxu nt
F(x,v) -- g (1)
i v'=
I
I
" sonncorr 6aftsarr u-I :ergcoE Oynxq rsxs' JIaIA AYpcgspTsrrurrrreJresp xon6oracon 6aftsarr g-Drr raJIE rY[rcaop eI
oror.ucog y eynrueac r-eep aBcan yJraMxnaJrErr oJloxloo (1.) rertrrurrenufia xo€p rap raJrLrr,
U Hb fr-sscxaMaapcag OyHKq 6onoxsrr am(aaptl-33p gnQQepexqrafltl11y1x rapax TsrrutrT-
rsurfiu {*-n*xyBbA 6o.qno.
X(urrree 11.3
g : (sin r)tS', ln g : tgc ' ln sin r, .-ur
1 ' tg* a' lnsinr , ,L 3
-InSInJ+---COS,,
- :
-f
IrA cosz tr sinc A coso s
I lnsinr *1, U:,: (sinr)tgr(l +sec2r.lnsinr)(rt";fs, :6.il, :-' X(uruee 11.4
1
L.U2 -2a:0, 2U{ -2, Y':;
2. 12 +3ry *92 +L :0, 2a*3y *3xy' *2yy' :g, n : -m,Xftrgrse 11.8 az + 2ry2 + 3ya : 6 rr,rypyfin M(1; -1) usri Tarcau rryprerq 6a rop
MaJIbrH TertulrTrsJluftr 6u'r.
.[ana $ynxurrn gu$Qeperutrafltlnar,ua:r'uyprraufir xspbruee'
l,@o): /,(1): -2.r1_r1$r: trulprerrrufin reniuurroll: Ut 1: i@- 1) 6yroy o- 4Y - 5:0rropMarLrs rsrrrrrlTreJl: y + 1: -d1*- 1) 6yroy 4x * U - 3 :0'
+-\-
: , 'yrrarilxl'alt68
11.2 Ynartrxcnalrrn rqGrrrq
rt
:. gl'ti
1. u: aai {,= o,fio-r
y=ri , {:Lg:\fri 1
.'tr^ i' Yr:ry
v:;1 {:-r,a:c {:o
2. U: sinr yt = cosfr
3. U:cosa y'--sino .,,,"'
4. u:tgr y' : * ,
. cos'f ,
5. U:ctgc yt:-,i-l':to-t ' 1'6. y:lod, y' : ilos:: ffi,t
U:lnt U' : a, _fr
7. y = a: U'= a'ln1
u: ea {'= e'.
8. U: arcsinr { : ft,9. l/: arccoco ,.. ,1.: -+v- 'ff';10. y:arctgo 'U':1irr,11. y-atcctgx {:-**
&-o-aL2. y: shc : +_ I : chr
, oa )- o-G13. U: chx': + yt : pha
- shr-14. A:thx:H {:h
cho,1Ic. y : cth,r: ;f,; g/ : -sfi
\'\
:, : l
11.1.
LL.2.i
11p..tr
;$
11.4.
11.5.
11.6. '
t1.7.
11.8.
11.9. .
11.10,
11.11.
LL.72.
11.13.
11.14.
11.15.
11.16.
11.17.
u:n-1"(2+e'+zt@;L I -3U: ;arctgTio1 tirytrlLlaa: 61n{2ni,y:#(\ffi-arcts'F);/^lffi-r\y:2(a_2)\ffi_ln(.ffiJ,
y: arctg("' - r-');_es, g* elr €F
a, -
-.
Y - gsh3z'-'. 2a2u: arcsinfta; (l"l
sm, - cossY-sinr*cosr'v:!*a; .(f+ t)arctg\fru:a;-,
1 z + t6tAaY - A\fr"'2- r,flthx'1 .- shcU: larctg(shr) - mig : (lnr)&;
y:4-'; ,. Lft "
y: (sin.'.6)r*€;( 3t2 +'2l":-ry ;
[ ,: sin(i +2t]( *:arcsin(sin't) I 4 *\,,
11'18' | ,: "'Eo'it*';
t I Vt-^:Tel. i
lr.re. [':F T"r\F,a-
I s, : \/l - ,F7 ' arcsin/i( u :"see2t
11.20. { ' ;
I r:tgtlncost+tgt-t
t^lLK'K
i
[*
',"i#j-=+':;]:-
r,70Ilsslq epsufi.ufu yrrauxilafi 6a- xniffiepe@nan
( CO.IOB XII. JIOOII 9POMEITIAH YJIAM}KJIAJI EA JI}IO(DEPEHUrIAJI
12.1. @yxruufin auQsepeuqrrarr.
Ifee.q epsrr,r6nfin ynaM)KnaJr 6a auQ$eperqtraJr
@yrrrqufix.uuQQepeuquaJr xr ryyxrfi yJraMxJraJrbr apryMeHT*rry anQ$epeuqrraJraap
IpxlTrrceursfi tenuyy.
Toaopxofitorrr 12.1 a : f @) Qyxrurftn ynaMxiraJraac Aaxrs yrraMxnar aBcrrLrr {,. 2-p epean6ufia yxaaa)NcrraJr res,q
rsx r3M.usrnsAgr. Yyrrsft Egrox a,urrnaap 3, 4 6onou,uss.q speM6[tr yaauxuaJrbrr rogopxofinox
6erooA y(n) : (y"-,)'rox reM.rerneger.
Togopxo fr;totr !2.2 Her4yrsep spela6ufin Arr00epeHqrraJraac,ua)rlrn au0Qepexulran
aBcgbrr 2-p speu6ufix au0Qep€Equarr roAsr.
d'a : d(dY)
Mox tyyrvusn dsgt : d,(&y),...,ffA: d,(tr-ry) rsx retvtuernarre. .[ssa epsu6ufts ar6Sepeuunanwr &y:y"(d,x)z, dA: u'o(dr)3,...,ffy - O@(dr)n rsx 6oasor.
Teopenr 12.1 Ponrrufix reopeM. Xepea y : f (r) OyHKu [o, b] xsp.rrM Aoep rac-
panrryfi, (o, b) 3aBcapr auQQepenqrralrAar(aac r&AEa f(o): /(b) 6afiaan f'("):0 6aftx c
usr (a, b) sancpaac f,AarK Hor oJr.qoxo.
Teoperr,r 12.2 Jlarpaxxufix reopeM.'g : f (a) OysxE [o, b] xepvrrM .ueop racparrr-
ryft 6oreer (o, b) assp ,uuQsepenquartrnar.qax 6oa /(b) - f (a) = f'(c)(b - r) 6atu c ser :
eE9 trETepBaJraac f;.qaK *r" orr.qorio. : -:'
Tefinopus ronas€o. U - /(r), Qyrx\ fr :a qerufir aryyncan rMap Hor rrHTepnanq n * 1
yaaa .uuQQepeuqrraJraraax 6afi aan
f(r) : f(a) + ffA- a) + ffa - d2+... + ffa - a)n+ R,(r)
tourOo xl"urnrsi 6aftsa. s :0 qsrrftH opqunA 3aIaJrBaJr
f (,):/(0) + #o, * f"!l) rz+...+ #." + R^(x)
6onox 6a yyuufir Marnopenu ronarEo roxe.
ileex speudfu ylramx<;Irarr 6a zgiWwvvast 77
X(uuree
Eo,qoxr.
)f(rurgg
$oaonr.]Kuurss(Bo,uo.nr.
tz.l y : (2r ' 3)' Qynxuuftr 1, 2,'$-p opsr'46ufrn ynilKllarr6rr on"
U, : Z(2x - 3)22, ,yr' : (6(2r- 3)')' :24(2r - 3), f" : 48
12.2 y: ek'-uitrr ,4g9,u eparvr6nfir ynaMxJlaJrbrr oJI.
y' : kek'r u" : k2ek*r"'a@) - ftn"kt
12.3 arcsin0,5t-ufir ofiponqoo 6o,qoxc on.
f @o+Ar) = /(ro) +/'(ro).Ar -ldr aumrllaBarr uanafi roxlrorlAorlr o0 :
(1)
(2)
0,5,Ar:0,016oreoAarcsin0,51.=arcsin0,5++.0,01:[+0,0]']':0,513{t - (o, b)2 o
6onno.
12.2 Togopxofi 6uruufir tafina:c Jlouura.rrufix Alpsvr
1. Xepea ]g1r(r) :bm|:(r) :0 6a lg,ffiopulffi 6aftsan
wffi:l'sffi6aftua.
2. Xepsa |g1r(r) : hm$(a) : oo 6a Y*W, opruun 6aftaan
E\ffi=rmffi6afisa. (1) 6a (2)-ur Jlouurarruftn gypsM rslre.
X(unree 12.4 I,$ # -nr 6oA.
1.,,,$,ogoar. g@_) .: fi3 - t,^r!(a) : lur Oyxr<uyyr ttb fr + 1 tarraly.rtax YeA rorpYY
TsMWIrox 6a lim ry: [m Y :1im 3zs : B vqrp (Lr-t $'(a) - *- 1 - ;'il323 : 3 yvup (1) rorvrr6o 6coop
-3-'r .t-2 fr
i,gl il:l,11t:3oaftua.;
)Kutlrse12.5 1-coscLY, 2", -brr orl'
Eogonr. p(r) :1 - cosr, ,lt@) :2x2 Syxx\YYA nr r + 0 YeA rerpyy raulynex 6a
lX r"ryO: 1136 Y r" $
*rrospnfin ro.uopxofi 6uru rnepxrftneu 6afina.
Xapun
|, r,,(r) : |glcos n: L, !y5$"(r):4 ryu *Xffi-ur Jlonuramrfix AypMeep olrx(
6onno.
r--
T rI
72 Ilsel4 spsudnfu yJrau)Krralr 6a WWepefrrfiarr
r,e r. 1- cosc sinr r. cos0YflIM.U IiItr --=-- : 11m
-
: lllll --- :c-0 zfr, z+O 4X o*0 4
y.uaa xepsrfleB.
X(lrnrss 12.6 hm iL -Lrr orr.
Eo.iort. p(4'lo*",'rh(r) : er ,ubu + +oo yeA oo-pyy reMrynex 6a
ti+ #: tim I : O 6afina. I4firrra Jlonrrarnfin ayper"r 6coop [* * :0 6aftsa.o+*6 WIA) r+6 At ' r-cr: A0
X(uursg 12.7 lirqrlno -rrr 6b,u.
Eogonr. IIIyyz xr3raapr rrrunxrrxeA (0.m) rocorr xenOspufiu roaopxofi 6uur uuepxuftnsn
6ri,fisa. le:rgsa ororgcox ypxnspuft" Y recolr xsn6apreft 6u.rner r --+ 0 ye.u xyprBep
xyBaapb seroH 3spor xr3raapryft ocox f.rp su.q Jlonuranufts g1pMrfir aurr{rJrax 6onno.1
I,Ifirr,ra limzlnz: limry : U* I- : lim r :0 ii,;-a
X(urrrse 12.8 ri* (1 - .L) -ur 6o.q.c*o\ff e,_l/Eo.uonr. IIIyya xs3raapr rrrr.rJrxBeJr (oo - m) recan rr roAopxofi 6nur
u.uspxufineJr rapua. Xrsraaprrn rau.uruftH noopxu r{nspxr,r1 1 e'-L-r: --:--;T r3x xyBllpra)K u -+ 0 YerfiH xsrrfi e,-L r(ea-l) r r xsn6spnftu
I,Ifiu.u
6onno.
ToAopxoft 6nur urepxrfinsn rapax ryn Jlouuramrftu aypuufir
X(unree 12.9 lim (sinr)t8'-brr orr.?T. ,,*u
Eogonr. IIIyyg xr3raapr rutrrrxBslr (10) xen6spnfiu rogopxofi 6uru urepxnfinsn rapua.
Ens xssraapHr uorapuQrvrrrfificubr Aapaa Jlonnrasuftn ayprraraftr arfirrrJrax 3aMaap xxn6apxax
6o,qox 6or.uor.
Yynufi ryn A:,11(.r, r)tg' rox reMlorrroB.
2Jlorapn$rra QyHKq racpaarryfi y.rrp
6omro. Ena Jlourra,nnftn ayprrrnftr 2
"2
lim (-sinr .cosr) : 6 6onno. lLfi.vrn 1n,4 : 0'tf ',-,
,v lim (sinr)tg'- 1oafixa.rsAreec A: l6yroy ,:oh,rtor)'5'
:
tI
l
InA : ln[ liq(sinr)t8'1 : liu;[in (sinr)t8'1 : liqltsrlnsinr] :,-1
U* Io(.i*r) :7t ctgxfr4-
2
,-l1
7 COSOli%*k-:n+. - - "::"i[--2 sin" 0
r
llest epsudffix ylrael;rxJrafi 6a auiffiepelll\nafi
Canal,rx<. Yyxnfi a4lrnaap d, mo xan69pufiE ro.uopxofr 6urn rugpxuftnnrftr
Jlouura.nufiu gyprr,rrfir amtrrrrarr rafinx 6oauo.
.l1, I
i
73
:*i
dtl
?i
,[apaa;c $yururfiH ?? gpeM6nfis ynaMxsraJrbrr oJr.
t2.5. y:.fra,Jlapaax ymbrr ofiponuoo 6o.u.
12.7. arctg0,97
12.6.U=5-3cos12
t2"8. w 12.9. ln 1,2
I
IIapaaS $ynrqufinLZ.L. Y: cos2 0
L2.3. 9: dtct1x2
L2.t7.lim(zr - r)tsg;:;\" ',-o2
Jlacra.n 6oruoro
2-p eperr,r6ufix yaau2rc.IraJrErr oJr.
l2'2' Y: f"&'L2.4. f (r) : "2a-!, f"(o) :?
12.10. 3,02 pa.uuyctaft ryryfiu.1q16aftr os.
Jlapaax Synruufix x5rnrg,:,Jlarparr:rrfin rourEor 6r.rux Qynruufir yrrur oJr.
12.11. [1, 4] xopturM Asep l@): tfr12.12. [0, 1] xeptluM Aeep f(r): tfrr! +3r
l
Jlouurarrufin .uyprvruftr arrrrrJran,qapaal( xlsraaptrr 6og.
12.18.,'*r-arctga L2.t4.li* " -1,c-0 13 c+0 gggg - ]
i2.15. li*'T - "ott' 12.16. ti* ttq' , 1
r+e gtrx - COSBq a-t Sin4C
12.18. -liq', r(e* - t)c{m
72.t9. rs + 3n2 - 2r *4 oror rfiuyTrrrl{ftr r * 1 sspresp 3a,uaJr.
n.z}. y: \fr $yuxunftH xyBbA fi:Ayer TeftnopLrn ToM66o 6u'r.
Jlapaax Qynrqufirr x5rBb,q MaxnoperEl ToMb€o 6u'r.
l2"2L.y: ,io f 12.22. Y: e'*
,t.
i:
t
@ym<@x eKcrWWM
OKCTPEMTTMbIH UOT
13.1. @yurqufirr ocex, 6yypa:s Bapcap. SrcrpeuyM yrraIi
Toa6pxofinonr 13.1
6rur 6ueau9 r3rrosc
/(rr) < l@z) ff@r) > f(rr))
Touuorron 6uru 6uenx 6aftsan A: f(*) gynxunfir (o,b) sarcapr ocox (6yypax) OyHKu
r3H3,
Xspsa f(x) Syxxu rr (a, b) saacapr anQ0epex{rarrrrrarA.qar 6a x e (a,b) 6yprafis xyBb.q
f'(*) > 0 (l'@) < 0) 6afiaar f(x) Syurq nr (a,b) 3aBcapr ocno (6yypHa). Xxn6ap
ToxuorrgoJr.q y : t@) $yurqsls rogopxofinorgox uyxcufrr rercJror rooubr Mouororr 3aBcpy-
yra,u xyBaax 6onno. Monoron saBcap 6yp nr l'(*) :0 ecssa f'(r) ynopmux usrlylroop
3aarJIarAa,sa.
To.qopxofi.rroar 13.2 Xsprss ro rleiufin rMap nsr 6 opquLr os ltoreoc .f,uraarafi x
6ypuiu xyBbA f (") > /(ro) (.f (") < /(so)) rsurrsrran 6uur 6uenx 6aftsan u6 rrorr,rrtr
y : f (r) QyxrquftB JroKarL MurrrrMyMrrx (rvrarcruy*rux) usr, /(16) yrrglr rroKarrb
@yuxuuftn MrrrrrrMyM 5a uarcnlryMLrrr yrryyALrr aKcrpeMyM yrra raus.
grctpervryr.r opuux saiinuryi Eexrlorr:
Xepass rs IIer nr f(xlSruxqufta axtpeMyMbrr{ u3r 6on y''(rs) : 0 scagr f'(*o) yrr oprujrrro.
/'(ro) : 0 ecBsrl !i''('*o)'yn opurui uo uornftr SyuxuNftx ""*rrirfi qer "r* "rprrirrr.
Csxurrefi usr 6yp.uoopee Oynrcu ercrpeMyMrsfi 6afix an6aryft.
X(uurss 13.1 A : frs QyxxquftH sKcrpeMyMbrr cylnas.
U' :3r2 ynaMxnaJl lrb r : 0 yua rerrsfi TsHulT 6onos'r ouo uor Hb sKcrpeMyMbru qer 6rru
IOM.
X(urrrea 13.2 A: lal @yxxuuiu eKcrpeMyMur conupxoE.
Yaalvrxnanrnrc)0 yeA l*l:* y.rup y':L, x(0 ye.u l*l:-* y.rup
U' : -L 6aftria. r : 0IIor Aeap ynaMxlraJr opurrrx 6afi-xryft 6onosq sgs rlsr ,uesp Synxu
oKcrpeMyMteft (r*,rnnrrr,ryrrarsfi ) 6afina.
9xcrperrayrra 6afix xypenusarefi rexgen:
1. /(c) Qyuxu nr rs coxrrrsft qeruftE sMap ner (rs - 6,no * d) opuunbr rs-ooc 6ycaa
Xspnes Aypbrlr fir,frz €:(a,b)-nfiH xyBb.u r,1'1 fi2 Telrqerrerr,,
.ts*
rae.
I
@unrnu-ng 3K?.TDP-\NM
rrerfyg Aeop.rrooeBeEuuanqrlarAAar 6aftr. Xepsa (ro - 6,16) 6a (*o,ro + d) saacpyy.ua,u
f'(r) ns TsMArso ecparsep oepqIIJIx 6aftral ss flb sKcTpeMyMLIH uor 6aftsa.
Oopeep xeroer (ro - d, rs) sancpbrH o oypnftu xyBbA l'@) > 0 (l'@) < 0) 6a (rs, rs + 5)-
ufts. r 6ypnfir xyBbg /'(r) < 0 (f'(*) > 0) 6on n : no Hb MaKcI{MyMbrE (rraunurtryurm) uer
6afina.
Xepaes f,(r) w r,,€. (as -d,oo+6) 6yp rr t * xo YeA r3Mlr33 xa,Ararlx 6aftaan rs-
sKcrpeMyMLrH rlor 6nru 6afiua.
Xfturree 13.3 , : l*'-1r*'+A*Qynruufin ocox 6yypa:r BaBcap 6onou sKcrpeMyMbrr
OJI.
Eo.qour: @ynxquftr toAopxoftnor.uox ruryx: (-m, oo)
y':fr2-5r*6 12-5r*6:o
TsrIulITrsIIggC 9l(CTpervlyu 6afix CexrrI.sfi qgrufir Ou6On 0r :-2 frz: 3 6onno.
fr (-*,2) (2,3) (3, *){ + +
v
fi (-oo,0) (0, m)
a' +,,
vn
Enasec (-*,2)(3, oo) 3aBcpyy.uafl OyHKq oc.r, (2,3) 3aBcapr QyEKu 6yypna. llfir''r yupaac
x :2 rrb MaKcrrMyMbrH, x :3 Hb MlrHlrMyMhrH usr 6onuo. earaep qgrlY.u asep Qynrurftn
yrrur 6oAson:
umaa : ulr:z :]x - ]i;r, * 6. 2 : 4?g
amin: al,:z --|s. - Zt, * 6 .g : 4;
6onso.
X(uurss 13.4 U : r? @yuxuuftu sKcrpeMyMLrlr qer oJI.
Bolonr: @ynrunfir to,uopxoftnor,uox Myx Hb: (-*, *)o-l ,
Yuarraxuan nr y' : ;* 3 :
# 6onuo. Ysarraxuar Eb rsr 6afix c-ufrn yma onaoxryft,
xapug yrraMxJrarr *" op*r, Oati,:ryn (yuaruxranrru Tacpalrrrrx uer) r:0 6aftna. One xr
excrpeMyM 6afix cextlrrefi uer 6onuo.
76 WwiffiE sKcrwv.w
@ynxq (--,0) 3aBcapr 6yyp.r, (0,oo)eaacapr ocrro. Enaesc x-0 qer Hb MrrEr{MyMLrrr IIsr
6ongo. Oynrurftn exo MrrgDrMyrlr yrra ns /(0) :0 6afina.
2. f(x) QyHKq rrb os csxumeft usr 6oros ryyxufi sMap Her optrun.u rorooc snraaraft II .
spsrr,r6ra{n ynaMxrranrafi 6oroo t f" (ro) > 0 (f" (ro) < 0) 6or rs HL MrrHr{MyMLru (uaxcu.
rr,ryrtrrrn) usr 6onro.
BoAo.nr: TogopxoftJror,qox uyx: (-m, m)
Ynarraxnauaa onx csxurrsft ugrgg on6ou:
a' :3x2 - 8u 3x2 -8s : 0 or': 0 ,, - !-r"*urrefi ueriyr fr
y":6r-8 A"lr:o:-8<0 A"'l 8:8>0,:58y.rrp r : 0. usr Hb MaKcrr\4yMbru rlsr, , : i rler Hb Mr{HrrMyMbrH rlgr 6onro. Oss qer .1
Aoepx Qyuxunftn yrra rrb lfll : -y---- --- r \3/ 27
. 13.2. @ynrqufin xa*rrufirr ux 6a xaurufin 6ara yrra: :
[o, b] sarcapr racpalrrryra f(x) Qynxq rrb yr 3aBcapr xauruftn rx, xaMrr{fts 6ara ymaa
3aBcap aa:r Qynruufin cs)Kurrsfi usryy.u 6a saacprrx yeyypuftE rlerly.urrftu a.ns Hsr rsep
ABIIA.
)*ftruree 13.6 A : xa -212+5 6ynrqr.rftr l*2,213aBcap .ua)G xaMrnftu m 6a xaurufts
6ara yurrr on.
Bo.qorrr: U' : 4r3 - 4a 4x3 - 4x :0 or : 0 fi2: ! r,s: -tgaresp [eryyg ns [-2,2] sancapr xaprsanar,uax 6afina
I,Iftrr,r yupaac oAresp qer1yA Aeepx QyrrqrftH yrryyl 6ouou 3aBcpbrh ysyypufin qeryyA,ueepx a:--
yrrbrr oJrx xaMrr{ftn nx 6a xauruftr Oarrrr Hb courorro.
f(-z) : t3, /(-r) : a, /(o) : s, f (r) : 4, lQ) : L3
OHAgec f (-2) - tQ): 13 ru erergcerr Synxuuftx ercon 3aBcap.,uax xarvrrrfin rx yrra
/(-1) : /(1) : 4 su oror,ucerr QynxqrfiH orcou BaBcap.uax xaMrtrftn 6ara ytra.6ouuo.
X(nuree 13.7 R paanycrat 6orvr6opqorr 6arrcau xaryy ra.qapryyrufis ran6afi xaurrafin
ux 6aftx qrrJrrrx.up on.
BoAonr: Iluuuuapnftn cyyprftH pa4lrychrr r, urrJrrurapuftn engpuftr h, uuuunAprftn
xilKyy raAapryyr ,S rse.
I
I*
ilEr,Fsr:*
ifiVrrcWrtm sxc11pe6y16 77
h:zffi s - 2rr.2.1ffi 0 ( r S 86onno. onsnrssynruufin [0;Rl
saBcap Aax xaMrltftu rx yrra oflox 6otnom ypyy rutrnxlruo.
, ,S' -- 4n(Il
t e:o *ffi:oBR2-2r2:o ,-'-\fr
S(r) Qyrru ur [0;.8] saacapr racpanrryft 3eper $yrxq. Xepvrrauftu ysyypufta uerlY,u .u33p
QyEKq rar yma aBHa.
s(#):znfr.z'W:T
12 Pz -2r2@):+n@
13.2. Y : a2s-a
13.4. y:7.3-l2a*L13.6. Y: h13.8. Y:sinr-z13.10. g:sinc -"++
13.12. U: 12 +2\fr [0,4]
1s.14. y:* [0,4]
13.16. U :Ztgr - tg2r 1O;il
L3.18. U:lnr+a ]0;tl
13.20. u:!+r' l0; l
.tFa, :2trR2
6oruo. One sr yr Qynxunfin xauruftH rrx yrra 6onox 6a R paruyctaft 6ena6opuerr 6amcaxR
ur;urrugplyAeac r : h, "ttryftx panuycrafi urnrrgp xaurnfin u:K rarapryyrafi 6afina.
, JlacraJr 6o.qrroro
@ynruufirr ocex Oyypax BaBcap eKcrpeMyMHB rrer on.
13.1. g--n2-6ct84
13.3. y : rB
13.5. g : cosu * sinr
13.7.U:#13.9. g-e'*e-a
OynruufiH orcou 3aBcap Aar( xaMrris rx 6a xannrnis 6ara yrra on.
13.11. y:a3-6x [-3,4]13.13. U:cosa tO,X]
13.15. Y:sir,2r-r V[;$l13.17. u:na tfi;*113.19. U : cosr* sinr l1i|;l
i{.fu
E
:,:'c
E:!I
t,
78 W orcrperryid
13.21. R paruryctaft.xatrac roiporr 6amcar xaurnftu D( uepuMetprsft Terru ouuor-
-trfir raJryy4brr on. :'
1g.22. E panuycraft .uyryir o reB esqorrefi cerMeEToA 6arrcan xa,urrfrn ux ran6afi-
" -ra,ft Torru esqorruftr on.I
13.2S. Ky61ryaarx arfru6sp ur xa,urrfis 6a,ra Oa,furaap &n roor xoEp rooru ,nraaap!,.-' xsa6eptsft 6u.r.
L3.24. Kaanparyyrur rrfiu6sp'Eb.xaMrtrfis 6arA 6a,braap 36 reces roor xo€p roolrbr
YpxBepT TaBE.
13.25.' Aauni T rypBarryrr*r ueprirerp sb 2p. Yr rypnanxiiur cyyp*frii. t."
tofipyynarr eprlyJreNrA yTcax fiuefuesnJrxiya xarrarrftn ux dafu<rrr ryr.q xarryy
ralr Eb f,Map Qqry as?
13.26. .B pamycra,ft 6err6epuerri6arrcas xaMrrfrs roc ssuuryyareft xorycug enuprftr
Orf;.:: I r: :.. i I
]*,;ii
79iDvwmfu rpafrw<
C3IIOB XIV. OyHKIIUftU rPAOlIKVtfrttt XOTTOP, rY.UrOP.
HYTAPAJITbIH IIOT
1,4.1. @ynrrtrfin npa$nxufin xorrop rY,qrap EyrapaJrrlrr qeri
Toiopxofirrorrc L4.1 (a, b) sarcap .uax $yuxuuftn rpa,QlrK Hb yr 3aBcap Aa)c Aypbrlr
uerr rdrcau ruypr3r.ruftxxse asep (goop) 6aftpuax 6afisan orceH Qyxxuufir (a,b) sancapr
xorrop(rymep) rouo.
Xspsss OyHKu (a,b) saacapr xo€p yAaa Ar{S$epexuuanqrlar,qAar 6a f't (r) > 0(f"(c) < 0)
6ou srs 3aBcapr Syrruufin rpaSnx xomop (ryarep) Oaftna.
Xr.n6ap roxtrorrAorrt A : /(r) 6yrrqufin roaopxoftnor.uox rrayx Qyrr qrfis rpaQur rYrrep
ecBgJr xorrop 6aftx rercJror rooubr 3aBcpyy.ua[ xyBaarAaua. O.urssp 3aBcpyy.uar xyBaa)c
uorITA ur /"(r) : 0 scsen f"(*) Yn opurlrx IIsrIY,q 6afina.
Togopxoir;rratrc 14.2 /(r) QynruufiH rpaQrax f (ao; f (xo)) uerufir aafipan rapa)qaa
xogopooc ry.urop p1y ocBerl ry,qrepeec xouop pyy ruIrrlxnx 6aftsan /(ro; /(ro)) usrufir
HyrapaJrTbIH II9r r3H3.
Hyraparrun qgr 6afix xypsrueerafi Eexuon:
f(x) Qyuxq Hb os ueruftn (ro - d,oo * d) recen sMap ser 6 opqux'u 2 yxaa anSQepeu-
qran.ruarnAar 6orooA (ro - 6,ro) 6a (cs,ro * d) 3aBcpyy.uaA f"(a) Hb rsM,rreo ecpera3p
ooptrrrDr 6afisan (ro;/(ro)) Hb Hyraparrrbrlr u3r 6aftna.
Xftrurss 14.1 U = 3sa - 8r3 + 6n2 + t2 Qyuxurftn rpa$rrrftH xouop, ryarep 6af,-x
3aBcap, HyrapaJITLrn qsrraftr on.
BoAour: Toaopxoftnor,qox Myx Hb (-*;oo) 6aftna-
I6a II epsrra6nftu yJlaMxnanyy.ubr ons6.
u' :L2r3 -2412 *L2r u" :3612 -48r*L2 3612 -48t*L2:0'l
On,qesc q : *;fr2: I rocarr xo€p ,uyraap sperr,r6rftH yJraMxJIaJI Eb tgrtefi TanqIY EyraPJ
aJrrLrH qar 6aftx cgxurrsft qgrlyr onAoHo.
fr (-*,i) ({;r) (1, m)
v" + +
v
1,, : i t2 : 1 usryynufiu xo0p rar.u II epeltdufin ynaMxJlar Irb reMgsreo
oopqrnx 6aftraa y.rrp (1; n*) (1;13) qsrlyr HL HyraparrrLln qer 6ouiro.
N
\\
l
?,
)jl.
I,
l
IIj,
.t:
1
{I
ecp0rsep
80 Oywwtu rpa$w
14.2 @ylirrrtufix acuMrrroT
Tolopxofinoar 14.3 y : f (r) Qynxurfrr xyBb.E Koop.qrurarbrrr exssc M rrerxesraapryftxorl,qoxoA OyErl\ufiu rpa,Qurlrfix M(r; l@D qsreoc sMap rrer rrrynJryn xrprsrD( eaft xxsraapryfi6aracax 61naan yr rryJryyurrr eyrxqufix rpaerv:airs.acuMrrror rere. A6cqucc rerDorsrruepueEA}lryngp acuMtrTorarr 6ocoo actrMrrToT r9H3.
,i
o : a urynyyE Irb p : /(c) Qynxqufin 6ocoo acl{Mrrror 6afi'x rapuaaryft 6oreeA xypenqeereftHexqoJI nr lim /(r) opeocror xs3raapyynLru r.uiDK HOr Hb roncroJrryft 6afrsa.o+*o' 'Tacpanuyfi OyEKq nr 6ocoo acr{Mrrror 6afrxryfi. _^_
Xspsss M qarufin x roopArruar Hb *m ypyy reMlynsx yer y : lcr * b naayy acrrMrroror.uox rapuaaryft 6oreo.u xypenrleergfi Hoxqofl rrr
k_ lim f@)c+*oo f
b - li+ (/(r) - k*),+*m"' '
b*
xr3raapyy.rl opurr{H 6aftx saAan roiu.
x(uuree 14.2 2,,2 + 3s - 5
Boronr: Y -
-GT synxqrfir acr{MrlTorhrr orl'
: l,*yffff:*',n'ffi:*
Engssc x:0 ;x:4 rscsu xo€p 6ocoo acrrMtrTomoft Oaftua.
k: lim f@): 2t2+3r-5-n ,-.r.,--.o-*oo *gH @ : a k: 0 ysup EaJIyy acrrMrrror 6afixryft.
,ItL(f(r) - ka): -lip'4,I" ; u :,o+6 r(a - 4) U : 2 6ocoo actrMtrTorroft 6afina.
ri,It;
i,.i6?
It;r
\tti
$
F
Ovlrrnwfirr roerhux 81
!
'r"!il
X(muss
Bo.qorrr:
14.3 U :2r+ arctSf $yuxqnftr aclrMrlTor orl.
k : _tir,n.. ry :,Ir* '* *:"'rz :,
b : ,If-( f (*) - kx) :,If*(2, + arlcs|, - 2a) : *;!t
,
Snasei
Ir!
i
i{
iirt:'!
i,lH
I
I1{u.;
IjIlIt{rl
I'
-:
,,
i{!I
I
I
I
iia
-,:
-"1:
l
l
U:2*+X y:Z*-tHaJryy acrlMuroryy.qtaft 6aftsa.
14.3 @yxxrtufin rpaQnr 6afiryyra:c cxeM
1. Toaopxofitor.uox Myx onro.
2. Koop.uNxarhrn TeuxnerlY.qrsfi orro.uqox uerYy.u oJrno.
3. Acurrauror oJIHo.
4. Ocox 6yypax 3aBcap, gxcrpeMyMbrn uer oJIHo.
5" Xotrop ry.urap 6afix eascap, tryrapanrlrlr qer oJIHo.
6. Oynxunfiu rpaSrx 6afirYY.ilua
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fit*2az*rs - 6ra=9fitt3rz-fis-7xa:4fit*2x;z-lra:7fit* frz *ca . 3ra : lg
frt*3sz*ra-fin:62*, + Z*z + ax, * 14: lg
b.Lz.rr-2qz*3rs -4xn - -gfiz-fis*Aa:4
5.1.
5.3.
5.5.
5.7.
5.9.
5.11.
5.13.
rt*xztre+4xa:5.:,fit-fi2*ga:10t*fr2-0a+2x4:5..'Zaz - frt * }xt: 4
5.2.
5.4.
5.6.
5.8.
5.10.
x1-2*2-0s:0
s.ra. I 2q + n2 - 7q.- L0
( *, +2rz - ts - fra: -fI -"r.-zxz-2xs*Bxa-4| "r
* 2xz-2as-2aa: -2[ ,, * Axs+2ua- L
( rr.-nz*2rs*2aa+T
J "r * xz - 6xa:7
l rr*rz*xs-3aa:4 ;-I
t ,r - ns-}xa:g( *r+nz*2re-2xa:3II rz-ts*aa:0I ,, + taz+2rr+n4:4
lr*r-2rs*4ra-2( *r+Bxz-ns-n4--1I
) *r+2xz-2rr+2aa- -G| ,r+ Jxz-2*r+at:-7I
I rr+2xz-0s:0I zrr*az- 03laa:!I
I 3*r-2rz*2rs-3xa-2II Srr *rz-ns*2xa:-lI rr, -nz*rs-zxa:4".,( ,,+az-3as:g;';':i
J ,r* 2x2-Bxs-3xa:5 flrr+3rz-3rs-6aa:7[ "; -3rs+ zaa:1
, fit-7*r+Aas:. -193rr - LLxz*2rs: -19
L,
r:i.
bt
h**-I
Bne xaf,lnturr ax;ufi
_-___:___._-__.x1
'95
'iq
5.15.
5.r7.
5.19.
5.2t.
5.23.
5.25.
5.27.
r1*fi2*3r3 *aa:4'2;a-fr2-as*3rn-b5q - fiz* fie,*|oa:9fir*icz*24-laa:7sr*2rz*%-Axa:19
-nz*2as- -4fit * lrs - 2xa: 4
at*fit-aa:|fi1*a2*ra-Sar*r.2*rs-G$z*fry*ta: -2sr*\xz-ra*2xa-2h*2rz-2xs-*ra,- -3fit*\xz-2xs-ol:0ar * }az* 5za - 8
lrt*2sz-Srs -4ra-9fit*az-\xs-6ra:$xz*Zrs*2aa- 3
rt-fr2-9os-10ra:Qx1 - 2r2 * 3r3 - 4xa: Q
fiz - fig* fia: -3at*}xz*rg- fi4:L2a, + 2a, +3rs - p.a : 8
x1- 2a2* 3rs - 4,fi4 - -ofrz-fit*fra:!fit*lrz*os- fit:42xr+3*z+ fit* fra - -5
h*3sz- ns* 6ra - 0
fit*02-rs*Zra:-6a1*5r2-rs*10ua:6
11-2c2* Sss* r,4::Lfit*lr2*613 -Zan-3q*2ss-0n:4q*2sz*as-3xa:d0r*3rz*2rs-Ara:2rt*fr2-2ra:6 :
2r, + \sz * 3os - 7ra: 5
r1*fi2*o3,:32r, + fiz - }e,s: L
h *2az* 6ca : 3
7rr+5rz-3r3-5 ,
ar*fi2-fr3-3rat8rt*gz-fi3-fra:5fit-As*7,a:)Zra*a2-Za1;7' I
11-Zlr2*2ts-ra:52q-1az*4as-2ta- -7
-or * }r,z - 2as I sa- 5
fi1* fr2 - 2xs * \xa -'7
sz*3xs-2x4=6,fir - fiz - 6cs *7aa - -gx1-2r2*3r3 -4xa-4frZ-fr1*ra:-3fir*3rz-\ra: | '
-7a2*3rr * fin: -3'11-2r,2*3r3 -aa:4lr1-3x2*rs-2sn-Aaz*2ss-2at - -$fit-fi2*5r3 -2aa:1
5.16.
5.18.
5.20.
5.22.
5.26.
5.28.
5.29. 5.30.
a:
96 Bne na nrwr a,xutt
-4_,
4 1) l'l), ,,,Il :i)
,rr (: -i:ll.. -T T )(+ -4 2
(z 6 B
'- [r : ],
'"[]ll )
62s[: ] ])
lz -12 o\u.rr. I 10 -le 10 I
l.,, -z4 rr)
F:r
E
r
E-
Bne naa;r:rtm ilKEJr 97
Eo,quoro 7.
1,3,7t uexropyyAaap 7 aextop511 3nrliJrx 6rrr.
7.L.d - (6,12,-1) - L,l: (1,3,0), 7: (2,-1,1),7: (0,-1,2)
7.2. d =t (1, -4, 4), ? - (2,L,-1), ? : (0,3, 2L),7 : (1, -1,1)7.3. 3
=(-9,5,5), 7 - (4, 1, 1), ? : (2,0,-3), ? : ('-1,2,1)
7.4. 1j (-s, -b, b), l : ?2,0, 1), ? - (1, 3, -1), ? : (0,4, 1)
@a = (18,2,7),1- (5,1,0), ? : (2,-L,3),7: (1,0,-1)
7)6. I : (-19, *1,7), 7 : (0, 1,1), ? : (-2,0, 1), ? : (3,1,0)
-7.7. ?: (3,-3,4), 7: (1,0,2),1: (0,1,1), ? : (2,-1,4)
7.8.d - (3,3, -L),1- (3,1,0), ?: (-1,2,L),7: (-1,0,2)
7.9. i : (-1, 7,-4),7 : (-1, 2,L), 7 : (2,0,3), 7 : (1,1,-1)
7.L0. t: (6,5, -14), 7 : (1, L,4),1- (0, -3,2), ? - (2,1, -1)7.LL: d : (6, -1, 7), I : (L,-2,0), ? : (-1, 1,3), ? : (1,0,4)
7.12. d : (5, 15,0), 7 : (1,0,5), ? : (-1, 3,2), ? : (0,-L,1)
7.L3.3 - (2,-1,11), ? - (1,1,0), ?: (0,1, -2),?: (1,0,3)
7.14. i: (11,5, -3), 7 : (1, A,2), A: (-1,0,1), 7 : (2,5, -3)T LW :18, 0,9,*e_=!:ol1], a : fl 1, ol, 17.16. d'- (3, 1,8), 7} : (0, 1,3), ? : (L,2,-1), ? : (2,0,-L)
7.t7. ?: (8, 1, L2), ? : (!,2,-1), ? : (3,0, -2), ?: (-1, 1, 1)
7.18. d: (-9, -8, -3), 7 : (1, 4,L),1: (-3, 2,A), ? : (1, -1,2)7.L9. ? : (-5,9, -13), p} : (0, L,-2),? : (3,-1, 1), ? - (4, 1,0)
7.20.1: (-15,5,6), ?'- (0,5,1), ? - (3,2,-1),7: (-1.,1,0)
7.2L. 3 - (8,9, 4),1 : (1,0, 1), ? = (0, -2, 1), ? : (1,3,0)
7.22.3 = (23,-14,-30), i : (2,1,0), ? - (1,-1,0), 7: (-3,2,5)
7.23. d - (3,1,3), 7 - (2,1,0), ?: (1,0,1), 7 : (4,2,L)
7.24. d : (-1, 7,A),1 : (0,3, 1); ? - (1, -1, 2),7 : (2,-1,0)
7.25. l: (11, -1, 4), ? : (1, -1,2), ? : (3, 2,0), ? - 1---t1, 1;
7.26.1: (-13,2,18), 7: (1, L,4),7: (-3, 0,2),? - (L,2,-1i\7.27.?: (0, -8,9),.}}: (0, -2,I), ?: (3,1,-1), ?: (4,0,1)
7.25. d: (8, -7, -13), 7 : (0, 1,5), ? - (3, -1, 2), ? - (-1,0,1)
7.2g. I : (2,7,5), 7 : (1,0, L), ? : 1i,-2,0), ? - (0,3, 1)
7.30. I : (15,-20, -1), 7 : (0, 2,t), I : (0, 1,-1), 7 : (5,-3,2)
I
1
1
98 Bne xeu;rlrsrg a,zr;frJr
Eo,qnoro 8.
? Oa 7 ""*ropyy,qaap
6afiryynarAcArr d Ca Z| aexropffi upAd-r on.
8.1. ? : (1,0,f); 7 : (-2,3,5); ?1 : ? * 2l; ?z: B? - 7;8.2. d
= (-2,4,1); 7 - (L,-2,7);?r:5d +87; ?r:zd -l,.;
8.3. ?:(L,2,-g); 7-(2,-1,-1); ?r :4d+37; ?r:8?-7;8.4. d = (3,5, +);t: (5,3,7);?t: -2d +7; ?z: ad +21;,$ a: (1,4, -D;t: (1,1,-1); ?, :d *l;?r: B? -21;8.6. ? - (L,-2,r); 7 - (3,-1,0); ?r :4d -21;?r:l -Za;
€1 ?: (3,4,-r); I : (2,-1,r); ?,.:64 - st;?z:t -z-d;g.A. a :(-2,*3,-Z);7: (1,0,5); ?1 :3d +97; ?z:-?-S7;8:g. d: (-1, A,\;t: (3,-2,6); ?, :2d -l;?z:tl -od;8.10. ? = (5,0,-t)' I : (7,2,3); ?1:2d -t; Vr: el - Od;
8.11. ?: (0,3, -4;l - (!,-2,1); ?,:bzi -21;?r:g?+S7;8.12. d -(-2,7, -1); 7 : (-3, 5,2); ?1:2d + 37; ?, = 3d +2T;8.13. ?: (3,7,0); 7- (1,-3,4);?t:4d -21;?r:l -Za;8.14. a: (-1,2,-L);t : (2,-7,t); ?t:6d -2t;?z:7 - e?;8.15. ? : (7,9, -2);t - (5,4,3); ?, : 4d - ?; #r: q,t - d;8.16. ?: (5,0, -D;t: (6,4,S); ?, = b? - 87; ?r:67 - 10?;8.17. d: (8,3,r); ?: (4,1,8); ?, :2d - 7; ?z:Zl -+d;S.18. ? - (3,-1,0); ?: (5,7,10); ?1 :4d -21;?r:l -ZA;8.19. ? - (t,-2,q);t :(7,3,5); ?, :6d -37; ?z:t -Za;8.20. ?: (3,7,0); 7 : (4,6,-1); ?r: B? +21;?r:5d -71;8.2t. d T (2,-L,A);t: (3,-7,-6); ?r :2d - 37; ?r:3d -21;8.22. d: (b, -1, -4;t: (6,0, T);?,,: B? -2t;?r: +T -Ad;8.23. ?: (-9,5,3); 7: (7,1, -2);?r:2d -l;?z:S?+S7;8.24. d - (4,2,g); ? - (0,-1,3); ?r :4t -3?; ?z:ad -3t;8.25. d: (2,-1,6); 7: (-1;3,8); ?1 : E? -21;?z:2d -bt;9.26. d: (5,0, S); ? : (-3, t,7); ?1: 3? - t; ?r: nt - gd;
8.22. d: (-1, 3,9;l - (2, -1,0); ?, : 6? - 2l; ?r:7 - aa;8.28. ? : (4;r,-7)rT : (5,0,-3); ?r:?- g7; ?r: Ol -Zd;8.2g. d - (2,0,-s); 7: (1,-3, 4);?,,:2d -57; Vz:5d -21;8.30. ?:(-1,2,8); 7:(3,7,-1); ?, :4d -37; ?r=97- L2d;
Egle nmnrtrg axuJr
---l99
Bognoro 9.
d AaT ,u*ropyyraap 6afiryyrrarlcau rrapaJreJrorpaMMErE rar6afir on.
9.1. ? - 3l +7,1 : I - 2l,l7l : 1, l?l :, t{;il :[s.z. d - it - s7',l :l +2l,l?]l: |, lal : r, (7;.?l) :;9.g. ? : gl -2t,1 : I +57',1?l : 4, l?ll :f,, tfA) : Tg.4. d : it' - zi,,l : 2l + ?,171 : z, l71 : 3, (tl) : +6tr-: e:: -t-,1
:n--2nl!-.1:?,Wl=P,('tr: 1
\ e.6.?:21 -?,1 :l+3?, l7l :3, l?l :2,({,1):;@ "Wk_iJ?,t :z.(ft):.L_ti.; :7 - a4?: B? +l,l?l : 1, l?l :2, ({,?) : ;9.9. ? :l +47,,1 :2? -l,l?l:7,1?l:2, (f,id):;9.10. ?:s?+zl,l:l-?,171 :10, l?l :t,({,1\:;9.11. ? :41 -l,l :l +2l,l7l : b, l?l :4, (t?):;g.12. d :27*+s?, I :l -21,1?l :6, l?l :7, ({,?):;9.13. ?: s? -l,l :? +2?,1?l : s, l?l :4, ({,1):;$.14..d:2?+g?, l:?-zl,l?l :2, l?l :3,({,1):+9.15. ?e.16. ? :5?+?, l:l -3?,1?l :1, l?l :2,(ild):1
J
s.17. d :71 -z!,1 :v+3?, l7l : f,,wt:2, ({,1) =x9.18. d:67*-?,l:l+l,l7l :3, l?l :4,(ffi):Lq
9.19. ?:1oZ +1,,1 :31 -z!,1?l : 4,lll:1, ({,?):1I Yt - -atlrt -rt:t -r\rrzl 6
' - S.zO. d:61 -l,l :l +2?,1?l :8, l?l :f,, tf,,t):;g.zt. d: 3? + 4l,t : -l +?,1?l : f,, Wl: 2, (t?) :Lg.zz. d :73 +l,l :l- 3?, l7l :3, l?l: ,, j;; :{s.23. d :? +a?, ?: s? -V,l?l :3, l?l :5, (ild):+g.24. d :31+ ?, I : l - 3?, l7l : 7, 1?l:2, (f,?) :;g.25. d:bl -l,l :? +l,l.Pl : b, l?l :3, ({,1) : Ts.26. d:87' -47,,1 :l+3": l7l : 2,l71:3,(f,?):n
't:..:r lryry-r.
100 Eu,e I.aanrbw.a)Kfrrr
s.z\. d:61 -l,l :l+b?, l7l : f,,Wt:4, (y-',V): Tg.28. d:zl+s?, I :l -2?,1?l:z,l?l : L, (flV):;,g.2g. d :2i - 3l,l :51 +?,1?l : 2,lll:3, (ild) :;e.Bo. ?trA +27,1 :2? -?,171 : 4,1?l:8, (il?):T
Eo,uroro 10. B qsrr GexnsrAc"r 7 xy.rsufi A qerrefi xapbqJryncar MoMeHT
6a .ruruyyrrorq KocrEycbrr on.
illl. F B A nlx F B A
lq
4
15)V6
I
8
I10
11 .
t2
13
t4
15
1
2
{3,3,3}
{4,4,4}
{5,5,5}
{6,6,6}
{7,7,7}
{-{ -r, r}{-2,-2,-2]j1-s, -3, -3){-4, -4, -4}{-5, -b;-b}
{3, -3,3}{4, -4,4}{5, -5,5}
{6, -6,6}{7, -7,7}
(3, -1,5)(4,-2,5)
(5, -3,5)(6, -4,5)(7, -5,5)
(8, -6, -5)(9, -7,5)
(10, -4\5)(11, -9,5)(12, -10,5)
(5, -3, 1)
(6, -4,1)(7, -5,1)(8, -6,1)(9;-7,1)
(4,-2,3)
(5, -3,3)(6, -4,3)(7, -5,3)(9, -6,3) -
16
L7
18
19
20
{8, -8,8}{-2,2,-2]'{-3,3, -3}{-4,4,-41-
{-5,5, -5i
(10, -9,1)(11, -9,1)
(12, -10,1)(13, -11,1)(L4,-12,L)
(9,7,3) "'(10, -9,3)(11, -9,3)(12, -10,3)(13, -11,3)(2, -L, -2)(3, 2, -2\(4, -3, -2)(5, -4, -2)(6, -5, -2)(7, =6, -2)(8,7, -2)
(9, -8, -2)(1.0, .9, -2)(L1, -10, -2)
(9, -7,3)(10, -9,3)(L1, -9,3)(12, -10,3)(13, -11,3)(4, -2,3)(5, -3,3)(6, -4,3)(7, -5,3)(9, -5,3)
21,
22
23
24
25
26
27
28
29
30
{3,3, -3}{4,4,-4}{5,5, -5}{6,6, -6}{7,7, -7]'{8,8, -8}
{-2,-2,2}{-3, -3,3}{-4, -4,4}{-5, -5,5}
(0, 1,2)
(1,0,2)
(2, -1,20)(3,-2,2)
(4, -3,2)(5, -4,2)(6, -5,2)(7, -6,2)(8, -7,2)(9, -8,2)
Eqruoro 1lBapranr 11.1 - 11.6 : A;8,C, D qsryyrsg opofirofi Terpao.uptrfiu srenxyyrufir ou.
Bapuanr A B C D
11.1.
LL.2,
11.3.
LL.4.
11.5.
11.6.
{2, -1:1}{1, 2,3}
{2, 1,3}
{1,1, 1}
{1, 2,3}
{0, 1,2}
{5, 5,4}
{0,0,0}
{4, -2,0}{4,4,-21{2, 19,4}
{2,L,7}
{3,2; -1}{1,4,9}
{1,3, -8}{2,0,2I
{0, 15,4}
{2,7,34}
{4, 1,3}
{1,8, 3}
{7,,5,2I
{0,2,21
{2,9,4}
{0,0, 12}
a
:
i
Exe amtrtrg uxfrtr
Bapuanr LL.7 -Ll.l2: ABCD rerpaeAprrfin esenxy"yu V : A,B,C opoiruyyrbilrr rooparr-
HaTyyI ororAcerr 6a D opofi s> OX TeHxIIsr A33p opgl{x 6ol xoopAIIHaTLIr Hb oJI.
101
Bapuaxr A B C V
1L.7.
11.8.
11.9.
11.10.
11.11.
LL.t2.
{5,6,3}
{0, -2, -3}{4, -3, -3}{4,3, -3}{2,16,-7],
{2,0,5}
{3,3, -2}{1,2,6}
t1,2, -lU{2, -1,1}{0,13, 1}
{2,6,321
{4,2,2}
{1,6,0}
{7,4,-t],{0,1, 1}
{2,7,L\
{0, -1,10}
3
t2
48
2
20
N
Bapuanr 11.13 - 11.18 :
XYqHITAITIIH YI{nqngJrrIUII
Buenfin A 6a B usryy.uell F1,F2 xYqEyY,u yfinvruus.
rrryraM Her xaBTraft ,ussp opurux yy ?
Oareep
Bapuanr 11.19 -L1.24: Eue O qsrrfir .uaftpcan rsnxnerrftr rofipon d osqer xyplrafiqaap
eprcJr.uorre. Euenfiu A qsrr ? *v, 6exnarAeng. 3He xyvxrft rraAJrbrr on. ( 3aasap: ganan
Hb xypA 6a xyvnraft cKarsp ypxBsproft tenqyy. lf : (7,F), *ypo nr d er[er xYPA 6a
Bapraanr o t d A
11.19.
11.20.
LL.2L.
LL,22.
11.23.
11.24:'
{0,0,0}
{2,1, -1}{-2, -1,1}{0, -1,1.}
{1,1, 1}
{1,2, 3}
t2,1, -1){-1, 2,0}
{1,3,2}
{L, -2,21
{0,2, 1}
{2,0,3}
{1, -1,2}{0, 1,2}
{1,2,3}
{-1,2, -3}{-1, 1,0}
{0, 1,2}
{2; 1,3}
t2,1,1)
{2,1., 1L}
{1,2, -1}{2,0,2}
{2, 1,3}
D7 ".*topyygbrlr
Bexrop ypxnepmi rauurr. I : ld,dl I
I
{L,-2,2}{1,4, 3}
{0, 1, -1}
{3,0,6}
{2,2,41
{2,4,2I
11.13.
11.14.
11.15.
11.16.
Lt.l7.
11.18.
{2, L, L}
{1, -1,2}{-1,2,3}
{0, 1,3}
{0, 1,1}
{3, 1,2}
{3,3,6}
{1,3,3}
{1,1,
{1,1,2}
{0, 1,2}
{-1, 1,0}
{0, 1,3}
{1, -1,1}
{1,0, 2}
{2,1,1}
{-1, 1,0}
L02 Bue zeenrrl:s a;rc;nnr
Bapuarm tL.25 - LL.27 : ABCD m.tpaMrrrun opofiryy.utrH KoopgrrrrarLrr rvre.qcreep -A
opofrrooc 6yycan errAprftr on.
Bapnanr A B C D
LL.25.
11.26.
LL.27.
{5,0,3}
{2,3, 8}
{0,0,4}
{2, 1,5}
{2,0,0}
{0, -1, -U
{4,0,8}
{0, 3,0}
{2, 1,0}
{6, -2,6}{0,0,6}
{3,2, u
Bapuarr 11.28. d+T ,l +? d+? BeKTopyygaap 6afiryyuarAcarr urpaMrrALru essrxy"prufir r
on. Yyng:d,t r? nu xapunqau rrepreHlrrKyrsp Eer)K BeKTopyyA 6onno.
Bapuanr 11.29. d,l ,? aexropyygssp 6afiryyuargcau TerpaaAplrftx eserxyynufir on. yyng:
? aertop ns ? aa I BeKTopyyraA repuex.rrrKyJrflp 6a d,7 u.*ropyyAlrH xoopouAox
euqer 30, l?l : 1, l7l :2, ?: 3 6onno
Bapuaxt 11.30. Xspen d,t ,? nexropyyn ruL,2,3-p orrarrrrr 6rcceKTprrcly.urftx aaryyrrtrrreceu Eerx BeKTopyyr 6or ? - 7, 7 + ?, ? BeKTopyyAaap 6afi ryynarAcarr rapalrrrerrmr-
ueflrfts s3eJrxtTutrftr on.
Bo.quoro 12.
Bapraut l2.L-L2.3: Iypaanxnrr opofiu roopAlrrrarlry,q erer,qcouoop A opoftrooc rarcau
MeArraE 6a ennpyytuftr xooponuox xypq onurufir or.
Bapraanr A B C
L2.L. C2;3) (5;7) C3;-2)
L2.2. C1;1) (6;5) @ia\L2.3. C3;5) (a;e) Ca;o)
Bapuaur 12.4.L2,6: Iypna^n:rnu opoftx roopArrrrar orcouoep BK lnennalrtsr rerrrrrrrsn 6a <'
yprtlr or. Iypaa^nxnbr reBrliu xoop,unuarbrr oJr.
Bapuanr A B C
L2.4. (5;6) (-2;z) G3;-3)
L2.5. (6;a) C1;o) C2i5)12.6. (a;8) C3;a) (-a;-1)
Bapra,rt L2.7-L2.9: Iypaanxurr opofin Koop.utrrrar ercenoop BK enapufin rsrruurreq BC
TaJr erAep xo6prru xoopouAox xypu errlrnftr on.
E-I
103ErreAaarr.rary ilKw
Bapranr A B C
t2.7. C5;-2) Ca;3) (3;7)
12.8. @;a) C3;1) (a;5)
12.9. C6;o) C5;5) (2;e)
Bapraur iZ.tO-tZ.tZ: Iypaanxxrr opoftn KoopAr.ruar orcoueep AB, AC rarryyrrrlr xoopoulox
onqruftr oy1x, BC ranraft uapaJiJreJrb .[yu.qilK ruyraMbrlr rerrulrrrsJr]rfrn 6lrq.
Bapr{auT A B C
12.10i (2;-5) (1;-3) G;1)
r2.L1. (3;-7) (2; -5) (5i1)
12.L2. (1;-3) (o;-1) (3;3)
Bapuarm L2.L3-L2.15: fyprarxru opofiu usryy.uufix Koop.rtrHar ororlceH 6ox BK Me.ruarl
5a ,4.D ox.upufin orrJlonurlhrrr ueruftr ou.
Bapnanr A B C
12.13. (6;2) (30;-5) (12;0)
12.L4. (7;o) C31;-7) (te;-z)
12.15. (5;a) (29;-3) (11;2)
Bapuaxr 12.16-12.18t A, B uorly.u I urynyyn,ueop oprurx 6orooA A, B usryylnftn opAruar
Ua, Ub.Men C Uerufix Koop,qffiIaT eror4ceu fun AD enApnfiu TerrurrTrsrr, LBAD 6alADl-r
OJI"
Bapuanr Aa Au,L C
t2.L6. 6 -2 2r*A-6:0 (o;6)
L2.17. 4 -4 2s *U - 6:0 (1i2)
12.18. 8 6 2r*U-6:0 (-L;2)
Bapuarr L2.1,g-t2.21: ABC rypBarrxubr opofiu usrlyauftn'roop.q$Ear oror.qcor 6olr, A opoft-
rooc rarcas on,uop 6a ue,ahaubr rgrlurTrsn 6a xoopoggox enqruftr on.
Bapuaur A B C
Lz.tg. G3;1) (2;o) (1;-a)
12.20. (-2rt) (3i2) (2;-6)
t2.2L. Ca;3) (1;2) (oi2)
,l
I
t
104 Bm gaanrm axilJr
Bapuanr 12.22-L2.24: ABC rypBarrxuw. A, B opoin uaryatufiu roopnuuar 6a rypmfi
ouupyyAilftH orrJroJrLrrr qgr M ororAcon 6oa raJryy4brn TerlrluTrgJruir goxuo.
Bapuaxr A B M
L2.22. G3;3) (5;-1) (a;3)
L2.23. C2;1) (6;3) (5;1)
L2.24. Ca;5) (a;t) (3;5)
Bapuarr 72.25-12.27: ABC rypBaJrxxw A, B opoftr usryytrufin roopllruar, Merrra,EyyAEru
orrJroJrbrn E uorrftu roopArruar ororAcorr fut C opofirooc 6yyuracan ouapnftu rgrIIr
Eue Ag.a.;llr.uo a)KEJr
Bapuanr 13.4tr3.6: Poudrur AB 6a BC raryynhrrr rorrurrrsrllTr 6a,D oporir xooptrrr.
Har oror.qcou 6on rar6afir oa:K TaJ$T/rbrE Tsrrslrrrsrufir 3oxrto;
Bapuaur AB BC D
13.4 U *2s - 2:0 2x - U *2:0 (3;a)
13.5. A *2a - 2:0 2a-U-2:0 @;z)
13.6. U *2u * 2 = 0 2a-y*6:0 (2;6)
Bapuanr 13.7-1"3.9: Pgir,r6ux AB, DC rarryyA 6a AC.urarorrarrbrx rerlrrrrrrsllyyx ororAceu
6on opoftnyynurn xoop.rlrruaryyA 6a ran6afir on.
Bapnaxr AB DC AC
t3.7. x*2Y-4:0 u*2Y-L0:0 Y-s-2:013.8. x*29-1:0 x*2y - 7:0 Y-fr*1:013.9. r*2Y -7:A x*2Y-13:0 Y-fi-5:0
Bapuaur 13.10-13.12: Ksqgparsru A opofru Koopglruar 6a BD, AC guaranaJryylLrn Terruur-
rgn ororAcon 6on raJryyAbrx TgrrrrllmgJlfiftr 6lr'r.
Bapnarr A B'D AC
13.10. (-t;8) 4x-5y*3:013.11. (o;6) 4r-5A-L1 :.0 5x*4y -24:013.12. (-2;10) 4r-5y*17:0 5a*4y-30:0
Bapuanr 13.13-13.15 : ABC Dro4nparbrn A opoft 6a Aramsalryf.ram or'ruonunurx K ugruftr
KoopArruar oror,ucon 6on yuacen opoftnyy.uhrr ofix raJryy.rbru TerrrltrTreJlnftr 6rs.
Bapuaxr A K13.13. (2;*1) (-1;-a)L3.t4. (3;-3) (o;-6)
13.15. (1;1) (-2;-2)
Bapuaxr 13.1&13.18: ,[opaex oxqomuftn opofinyynrrs Koop.qunar orer,ucoE 6on exs,uepBoa
ourlorr tparreq Oonoxrrr 6aranx, raa6afir on.
Bapuanr A B C D
13.16. (3;6) (5;2) -1; -3 C5;5)
13.17. (a;a) (6;o) (o;-5) (-a;3)
13.18. (2;8) @;a) (-2;-1) (-o;7)
105
' 106 Eue taaz:rr;ra ilKErr
Bapuart 13.19-13.21: Ksagparllm AB raJrbrE Tarrrrr.rrran 6a Auar.osarryyArrx orrJrorrrlrrbrn
E qsruis roopAuualr eror.qceu 6oa opofin Koop.umrarlyAsrr oJrx, Afiarouanyy.quu Terru]rr-
rertuftr 6rq.
Bapua,ur AB E
13.19. fr+Y-5:0 (a;a)
13.20. r+Y - 4:0 (5;2)
13.21. s+A-6:0 (3;6)
BapllantL3.22.t3.24:ABCD[apaJInenorpa,!lMbIHAD6aABralryyrbIHT9rtuErrgu6a
ArlaForraJrJryELrE orrJronrrrr .E qsr[frs KoopAtrHar ererAcon 6on noroo xo€p rau 6a auaro-HaJryyALrH Tguurlrrgnufir 6llv.
Bapraur AD AB Et3.22. U-a*2:0 59-u-6:0 (o;o)
13.23. Y-fr*5:0 5y-s*5:0 (1;-2)
13.24. Y-fi-1":0 5U-x-17:0 Ct;2)
Bapuani 13.25-13.27: KrarparLrrr scpsr xoBp opoft 6onox A6aC uaryylrftu roopArrrraryyri
erorAceu 6on Eoroo 2 opoftr onx, TaJryyAbrrr rsrrrrfimsnilftr 6n.r.
Bapuaur A C
13.25. (o;1) (2;3)
13.26. (1;-1) (3;1)
L3.27. (-1;3) (1;5)
Bapraant 13.2&13.3a: ABCD a.rrrrr xaxyyr rpanequftx (BCllAD) A, B, C opoftnyyarrn
KoopAlrllaryy,q erorAcox 6ou D opofir KoopArrnar 6a xypq oxqruftr on. ,f
Bapranr A B C
13.28. (1;2) (3;a) (5;a)
13.29. (2;o) @;2) (6;-1)
13.30. (o;a) (z;6) (a;3)
I
Eue aaar:tsu a,x,utr 107
!L
Eo,qaoro 14.
Bapnaur 14.1 - L4.6 : A,B,C, D usryynuftr gafipcan xastra,ftIr rormtrTreJr 3oxtro.
Bapuaxr A B C
L4.1.
L4.2.
L4.3.'
L4.4.
r4.5.
14.6.
(1,4,1)
(0,1,1)
(1,6,2)
(1,2,0)
(0,-1,2)
(0,3,2)
(2,3,-1)
(2,7,L)
(-1,0,2)
(2,5,0)
(2,1,'2)
(1,0,-1)
(0,-1,0)
(1,0,-1)
(1,2,0)
(0,3,2)
(1,4,1)
(1,6,2)
Baprart L4.7 - L4.L2; Oy r3rDsrsr 6a A qgruftr aafipcan xasrraftH TerlrrrreJl 3oxEo.
Bapmnt A Bapnam A
14.7.
14.8.
14.9.
(2,L,6)
(1,3,-3)
(1,4,3)
14.10.
14.11.
14.12.
(-1,0,-3)
(2f5,6)
(3,-8,9)
Bapuasr 14,13- 14.18 : x*y+z-)- 0xamrafiA rleptreEllrKyurp 6a A,B ueryaaufir
aafrpcax xasrraftr rsrurl{Tr3Jl 3oxro.
Bapnaxt A B Bapuanr A B
14.13.
t4.L4.
14.15.
(0,0,-1)
(0,2,3)
(1,0,-2)
(2,2,7)
(-2,0,1)
(-1,0,0)
14.16.
L4.L7.
14.18.
(0,1,1)
(-1,1,2)
(1,2,3)
(2,1,-1)
(2,3,3)
(1,1,0)
, - Bapraant 14.19 - L4.24 : Oz reHxrrorrafi uapanneut 6ereeA A,B uery'gnrfir uaft"pcan
xasrraftrr TguIrETrSJI 3oxso.
Bapuaur A B Bapuarr A B
14.19
L4.20
L4.2L
(-1,0,2)
(1,1,-4)
(5,3,1)
(3,2,5)
(-3,-1,2)
(1,1,2)
L4.22.
L4.23.
L4.24.
(-1,0,5)
(3,2,1)
(-3,-2,0)
(-5,-2,1)
(5,3,-2)
(7,4,L)
Bapranr t4.25- 14.30' + :+: + u"+:t: + rece' coa6ucou
xo€p uryrlyyareft uapatnenr A qerrafir .Uaftpcan xantra,iu rsrrrrtrTrsJl 3oxtro.
108 Bue Eaatrstu axwtr
i
Eo,qnoro 15.
Bapraut A Bapraur A
L4.25.
L4.26.
L4.27.
(3,1,2)
(4,0,3)
(1,0,2)
14.28.
14.29.
14.30"
(0,1,1)
(5,2,2)
(-2,0,1)
Bapranr 15.1- 15.6 : XOY xaarraft da A,B usryy.uufir aafipcar rrrynyyuyyrbrr orrrrolr-
rrnbrg qsr[fiH KooprrruarLrr oJr.
Bapuarr A B Bapnanr A B
15.1.
t5.2.
15.3.
(5,0,5)
(1,2,3)
(-1,3,2)
(-3,4,1)
(-11,9,-3)
(3,-2,7)
15.4.
15.5.
15.6.
(3,1,4)
(-9,7,-2)
(7,-1,6)
(-7,6,-1)
(11,2,3)
(11,-3,9)
Bapua,rt L5.7-L5.L2: ABC rypBanxurr .4 opoftrooc rarcaa Me.uuaabr rermtrTreJr 3oxr{o.
Bapraur A B C
15.7.
15.8.
15.9.
15.10.
15.11.
15.12.
(-2,1,3)
(-1,2,2)
(0,3,1)
(1,4,0)
(2,5,-l)
(3,6,-2)
(2,6,1)
(3,7,0)
(1,4,3)
(1,8,0)
(3,3,1)
(0,3,3)
(0,2,-1)
(1,3,-2)
(-2,0,1)
(3,2,-2)
(-1,5,-1)
C2,1,1)
Bapuaxr 15.13-15.18 : 116a7r2 xoBTrofiEyy.qbrr orrrronuorrg yycex ruyrryyubr trapaMerpr
6a xa,uoxutr TgrrrrtrTroJr 3ox[o.
Bapuanr 711 712
15.13.
15.14.
15,15.
15.16.
15.17.
15.18.
5a-y-9:02r-U*z-5:04r-32*y-33x-z-4:0
2t*5y-92-9:0r*4y-Tzt8:0
n+y-Zz*1:0x-2y*32-6:0
3y-52*7:0fi+A-22*1:0
x-5y*82-l-3:05x*2y-52-2:0
Bapra,at 15.19 - L5.24 i rt 6a tr2 xantrafiryy.urrrr orrJroJrqoJrr lycex ruyJryyuTeft uapa.n-
renr Aqarufir gafrpca,E ruynJryrrbr rgrurrlmsJr soxtro.
Bue r8Enlg'gw :.v.rr-n
! -t
109
Bapuanr A 'r1 712
15.19.
15.20.
15.2L.
15.22.
15.23.
t5.24.
(3,1,0)
(7,-L,2)
(5,0,1)
(-3,4,-3)
(-1,3,-2)
(1,2,-1)
3r*5y-z-5:0'a*a-z*3:02a*3y;z*4:03r*4y-22*7:0
2r *5Y * z:02a*3y-z*4:0
t*2Y*1:02r*U-Zz*8:0r*49*22'3:0
r-22f5:0A+z-2:0
c*3Y*z-1:0
-- (*
D-rfi* sMapyrra'A I n'-u*22 * D:0
rryrryy' 1s.25.-orrsrxnsrufrr, 15.2l.-oy[ 3c+ 2y - z *6:0
rsmrrrsrrfir, L5.27. - Oz ramrxgrsfu orrnox se?
c.-ufra.sMap $Tralr.u { " - U - z- 3 - O,ar - U * Zaz : 0 rrynyyu L5.28.-YOZ xaamafirafr.,(
15.29. - XOZ xanrraftraft, 15.30. - XOY xanrra.firafr traparrneJrb 6afrx se?
Eoruroro 16.
Bapna,ur 16.1- 16.6 : 116a7r2 x&BTr&fiHyygbrrr orrJro'JfrrloJrg yyceiewynyyn ,[eepx A qerfrE
upoerrffi oa.
Bapranr A 711 lf2
16.1.
16.2.
16.3.
16.4.
16.5.
16.6.
(-2,3,1)
(-3,2,2)
C1,1,-3)
(-2,1,-1)
(-1,5,1)
(-3,3,3)
a*3Y*z-1:0A+z-2:0c-22*5:0
r*4A*22-3:02r*A-32*8:0
r*ZY*1:0
3t*W-z*5:0fr+Y-z*3:02r*3y-z*4:0
2r *5Y * z:02r*3y - z*4:03x*4y-22*7:0
Bapraxr 16"7 - 16.12 : A qsreec + : u;2 : '-rL ruyrryyrrA oyynracan uep23-rueEAEKJmSpbrH T9rIUtrTrSJr 3O](EO.
Bapuarr A Bapraur A Bapuaar A
L6.7.
16.8.
(-1,-1,2)
(2,-t,4)
16.9.
.16.10.
(-1,0,1)
(2,-2,1)
16.11.
16.12.
(0,-1,0)
(0,0,3)
Bapraur 16.13 - 16.18 ; A qsreec 7r1 x&BTrofi.u 6yynracaE rreptregrtrryrxprrE cyypuftu
Koop.qffrarbrr oJr.
:=''-
1,10 Erre la8trTw AYrr$
BapNanr A 711 Bapuanr A 711
16.13.
16.14.
16.15.
(3,2,4)
(1,3,0)
(1,1,3)
2x*U*32-6:02y-z-1:0
z-1:0
16.16.
r6.17.
16.18.
(3,0,-2)
(-2,1,2)
(3,2,-3)
2r-y-32*2:03a-z-2:O
2x*U-42*1:0
,\
B"pruur 16.19 :16.24 : Auer 6a * - u -.L : z
Terrrrrl*oJr 3or.,o.Lo',*:' A rl9r oa
. ' -1 = myJIyyEHr 'uaftpca* xasrrafrE
Bapuanr A Baplranr A Baprarr A16.19.
16.20.
(2,3,0)
(1,2,3)
16.21,
16.22.
(-1,5,9)
(-2,6,12)
23
24
(1,-3,3)
(1,4,3)
BaPrarr116.25_16.30:ABCDTeTpaeApIIiropofiroocABCcYypra6yyuraca,uerrr1rrfir
TorrrrrrrrgJrrir 6uq.
Bapranr A B c D
16.25.
16.26.
t6.27.
16.28.
16.29.
16.30.
(1,1,6)
(0,1,7)
(1,0,5)
(0,0,6)
(3,4,7)
(2,3,7)
(L,2,7)
(1,-1,4)
(1,-4,1)
(1,3,8)
(-1,1,8)
(2,2,6)
(3,3,6)
(2,4,8)
(1,-2,3)
(3,5,8)
(1,-3,2)
(-1,2,3)
(5,2,6)
(2,5,3)
(6,1,7)
(3,4,4)
(1,6,2)
(7,0,8)
.-t--
:=f.''r'
a&rF^
111Ez,e zmlrrus axwr
rI BIIE JIAAJIT. Her XyBbcArqlrliH oyHKUlr'IaHlIlrooEPEHUlrAJr T O OJIOJT
[--_
Bonnorj 1. Xrrraapur 6o.q.
',(2-n)2-(2+n)2 1 o,,* (3-n)4-(z-n)4r'r',IHffi r'z'i*@
r'B' rim il -;i: - i:* "i: 1'4' -tim- (t
1:ry)lt,jt- =")'
;+Ary r'uv' ltiB (r - 1)."- (3+ t1..: l.i^ r D\2 r t^ r ,t\i i- - 1\3 - fm +'l\:
r'o',rggW ^'=''i5'Jo (l+2n1.2*n2,^
1.b.mffi r.o.,'gWrz;*ffi 18J*ffir.o.,ri*"ffi 1.1o.JILffi,.rr.,rluffi 1.12.JILffir.ra",g.ffi r.ra.is"ffiruis.ffi 1.16.Jssffirrrjggffi rrai+"ffi,.lro;gffi 1.2o.Jrsffir.zr.)*"ffi 1zz.Mffir.za.js"ffi L24.rimffi1.2b. rim i:".ll:*!:"-il: 1.26. ri* H
1
r.22. Iirnffi 8. lirqffir..2e. rim
(qr' + ?ll + iz" -Lt)z 1.80. rim !" - l[ ; ti i *l;#lU@ ^'vv',,:fr (n-4e + fu+2)s
EoAnoro 2., XrsraaPm 6o.q.
2'1'lim @'- ,'limff-'ff2.8. limn, (d-ffi) 2.a. ]ryn("m-{ffi)
/-2.5. m [ r',, +z,ffi - {m) 2.6. L. Mm2.2.;*ffi 2s.M,fr(rwr-,tffi)
Bne rurru,r. ilKEtr 113
3.28.
4.L. limc+-3
4.4. limn+-2
2n2 +Lffi+g
612-5r*13x-l
2r2-9o*102a-5
2r2-2Lr-LLr-lL
s2+3n-4r*4- 40r + L26
5u2-4r-la-l
6a2+x-l2a*t
5.2. Iimo+-1
5.4. lim
5.10. limc+-
5.12. lim' c+l
5.14. lim01-
o+3
4.6. lim,*j
4.9. lim
4.L2. lim
2r2 -7x *3s-3
gn2 -L3n+13a2 -2r-L
z, 2a *76a2+r-L
rm(n-oo \r*( )"-'3.29.
n _ 2 11*3nI
n+2 ) 3.3o.;g(#')"-*'
Bo.qnoro 4. Xrsraap 6on.
t2n2 +5x - 3
. r*3,3x2+5x-2
a*2612-r-l
;;; 2x -1lim,-* --_1 3r* 1
2s2 +3s *24.g. rim
3tz - 5t-- 2
a+2 t-Z
4.11.rir4#4.14.
''*- 1012-+ 9r - 7
--'-- ,--1 5r * 7
.17. limrry_g4.20. rim
5r2 * 24a - 5
a+5 X-b
4.28. ,* 62-- c - 1-'--' ,--t 3r * t4.26. rim2r2 -- 5t + 2--- ;r* 2t-r4.29. rim
3n2 lt7t - 6-'-- ;* 3a - |
4.2.
4.5.
lima+l
lim,**
4.3. lim
4"7 "
\.---
+.10.rh,Trrylim,*jlim*-\
a+l
limo+1
lim
o+1
limo+1
4.13.
4.16.
, x-j4.15.
,lim
4.18. lim
4.21. lim*+-7
4.24. Iimr--5
4.27. hmc+10
2s-12s2 +l3x*2L
4.19. limr-11
4.22. Iimo3-4
4.30. lim,*-*
;* *r-t2n2 + l5r *7
r*712 +2r *15
-
x.* 5
5; ,2-51r*10r-10
L5s2-2t-l
-
5c*1
o^24.25. lim "'
r*6
Eogrroro 5. X*sraapur 6og.
5.1. lim(*'+3x *2)2
13+2a2-u-213 -3r+2awl .(2r'-0-L)2
i'Iirr*2n2-x-2
o+8
4.28. limc+-6
r-83r2+LTn-6
*-4r+3d-12-r*L
xs -3x +213-fi2-r*!
14-x
rs -3r +2r?-2*+L(r'+ 2a - 3)2
5.6.
5.8.
o+-3
limc+-L
lim:
t sa +2r *l12 -2r+L
i5i2*z-r-L13 -3r-2
13 + 4n2 +3tr.3-2r-L
t n2-t-25s2 +7s *2C+4x2+3s3+5r+6
r;3+3n-2x3 +l2r -32
5.7. 1,gW12+2s-3
5.11. lim
5.13.
5.15. lima+tiifi,2*a-12-L 5.16. limffins-n2-r*1
13 + 4r +16b.rr. l'lld+/l,I:f
(x2 - 2a +l)25.18.,[Trffi
Lt4 Bne gaa,nrtm a.x.Etr
lima+2
5.19.13 -3r-2
12 -2rrz -2x+L
a3-a2-fi+L2x2-fr-L.d3rz-4
13 - 2x':- L
L#+"2n3+L :
as - 4a2 +gm(1 + 0)3 - (r -"s14
x*xa
arctgpx
yTn-2Sarctg2r
cos3r - cosrt
ea -Lsin l5(c + zr)l
e3, -L
5.20.j31,wb.22.ttmffi5.21. lim
a+l
5.23. Iim*a+ti
5.25. fim
5.27. limc-3
5.29. limo+0
5.28. li:qmx2 +2r -3
ra +2r +305.30. lim
a+2i':izst+12+x-L4
y/11 r-t6.18. lim
o+0'--' ;'Jd sin lzr(r + 3)l
5.24. lima2 +2x -3' ffi at - 4r2 -21
5.26. lima+L
xa+3r-412-fi-l
6.7. lim
EoArroro 6. Xrsraaprr 6o,q.
6.1. lim ln L + sinz
c*0 sin 4rL - cos2a
6.3. lim- - tJOcos/r-cos3carcsin 3c
6.5. llm--'-' ;:t ,m _ r,rz
6.2. lim 1 - cos 104
r+0 en-L
6.4. iim 1 - co:3 2-r
c+0 Afr.ol
-166. I'smftfu6.8. 1,36ffio.ro.|1s66ffi
o3, -1c-o 4arctg3o
6.14. lim t-!-gn+t'*o cos t7(c + t)J
6.16. li:qffi
-"' ;;6 sin l2zr(r + 5)l
6eI4r@,,tg3ocos(r + +)6.11.riq JIiJ-1,f4i
6.13.rtuqffi
6.15. limc+0
6.17. limo+0
6.19. limc+0
6.21. lim arc-sin2 r
g-o 8' - t
6.23.riqffi6.2b. Iim tgc- sin?-----' ;;T c ln (1 - a2)
_ ena -l6.27. ]iqililG:T)i
6.29. liq'ffi
6.20. lim 1-'\!@
o+0 ISttzfrcb -L6.22. lim
'-o sin tzr(i + t)l
6.24. lim sin2 t -. tg2a
o+0 tr4
6.26. }itary6.2g. lim .2sin2 a-
a+0 f - COS6O
L-
630.riqtrffi
1ffi2
Ene Aaatttrg unrtr 115
Boguoro 7. Qyxruyyn rs qer rsep racpaJrrryfi re:r 6aran.
7.t. f (x) :5r.2 - L, 7,s: $
7.2. f (x) :3x2 - 3, xs: 4
7.3. /(+) :2n2 - 5, rs: )7.4- f(t):4s2-7,7,e:)7.5. f (r) : -5n2 + 10, 0o : 3
7.6. f (a) : -3r2 * 8, as -- 2
7.7. f(t):2r2 *6,rs:J-l 7.5. f(x) : 4n2 * 4, xs - 2
7.9. f (t) :5n?' * l, xs - 2
7.10. f (t) : -3r2 t 6, rs - 3
Z.Ul /(r) : -2a2 * 9, 16 - 3
7.L2, f{r):4x2 * 1, rs: 1
7.L3./(") :3r2 - 1, rs:17.1a. l@): -4n * 3, rs : I7.15. f (a) : -fu2 t 5, rs : g
7.16. f (r) : -2n2 *7, rs: !7.L7. f (r) : -4r2 * 9, rs : 1
7.L8. f (u):5r2 *3, xs:27.19. f (x) :4rz - L, xs:4,
7.20. f (r) :2r2 - 3, ts: 4
7.2L. f (r) : -3a2 - 4, 7,s: )7.22. l@\ : 5n2 17, rs - L
7.23. f (r):2x2 *9, xs:)7.2a. f (r) : 3r2 t 7, ns : 1
7.25. f (r) : 5n2 - 9, o6 : 1
7.26. f (r) : -2r2 - 9, 16 : 4
7.27. f (x): -3r2 - 1, rs - 3
7.28. f (a): -4n2 + 11, ro:07.29. f (x) : 4n2 * 3, cs - 5
7.30. f (r) :3n2 * 5, rs : I
8.16. /(o): l-2a
8.23. f (r): In(r+1)8.2a. f(x): *
Eognoro 8. @yuruufirr racpaJrrug qerrfir oJrx racpaJrpuifo qerufiH opqurrE
rpaQuKufir g1pce.n.
8.1./(r):ffi8.2.f(t):Y
,,3_fr28.3. /(z):ffi,8.a.f@):#8.5. /(r) : (r * 2arctgl;
8:6. /(r):
8.7. f(r):
lx +21arlHf + z)
#-r)8.8. l(c):#8.9. /(r): +
t-2;
116 Ene aaBilfi"q a)KilJr
8.10.
8.11.
f (r): arctg(r - 1)
r-L
9.2. a. A : In(e' + 1) *
f(*):#
8.13. /(r) :54 - r
8.14./(r):h8.15. /(r): iff
Boguoro 9. Y.nau:rna.rrllr oJr.
9.1. a.a:*""(r- sin 2a - cos2r)
( ^_ 1
^)*-lrtt"'f r:^t*f
8.28. f (r): arctg(o - 2)
tarcsin (, - 1)
lr-11
8.25. f(t):ry8.26, l(r) : ffi8.27. f(a):ffi
, - ^fs128.30. /(r) : ;;tsi _ 2)
lnrb.u:ln--=srn =a
d.y: ln lbrl + \m
e.y = t93(sin(r2'+ 1))
1
9.3. a. u : ftBze" (asin Ba - B cos Bx)
c.Y:r5+5'
18ek+27ea+\tr6(e'f 1)3
, 1, L-Zro.U:6112+2,
d.y:ltffi- 1- -a
. b. y: -;bthzd.12+a2:4
C. ! : gectgx d. u: ale" sin2a
9.b. a.u:ff",m+ f,arcsin T b.u: ,ft - ,,1F- xarcsinttrr
c.y : (lr,r)*
lgnEa.A: t/a * 5arctgr - ,ardOj-nbazFc.a:-+-i-*-afia.fr.
a.Y:W,neaarctpn
L.U-:" lnos1,,
a. y :(arctgr) 2 In arctgr
d. [ ": arcsin(sint)
[ ,: arccos(cost)
.F'
9.6.
9.7.
9.8.
b. s:d,.y*r3+lngr -x2eY:A
b'u : 'sincg
d.U:(1+lncoso)3
b. a:lnlnsln2r
/r"1"*1" T 3y1
ffi'
Bre za6.r11,rr,r Nx4tr . tL7
"T
c.y: o.oF. +. #9.9. a. U,: @3 + 41te"
Ifc.g*i
9.10. a. fo
^2C.U:T9.11. a. y -- rse" sin2r
c. y :cosJsin f)g.12. a. ?! : (r --z)'zW-., (r_b),
C. U :,A,
9.13. a.u:ry-#c. u,: fil"
9.14. o. y: "*l
(a-a)(a-b)c arcsinr:ffia1a'ffi
( ,:arcsin2 td. I t
' b.y=logslgrd. U:arcsin(sittr)
"7*p 1 1b.Y=h , -7arct$r-Wd.y: lnsin @tgb.u: rln(1-t - r) + \m
d.2ylty: a
b.a*2arcsin #-+@m+'nd.y:L+reu
, rt+\mr, 1ln '- ' '- ' - b. U: lnarctg:-r t L*u| ,: t\/t2 +Ld'l ' 1+'ffiIs:t-T
b.y :t lnrcos f
c.y: ln(r* ,vm) -
9.15. a. Y: lnrlog3o
c.y:r-ln(2 +e+2,,1-"a+e"+t1 t' --'ii ' - ;i'.:i;F.l r- -
;.'t, :-:.:!+i' tid. coss.-=g .. .,, :,'
9.16.' @. g,= 2a3 - 5r.2 + 7, -,t"{':':itx''',' .' br. U -: e'atctge' - la 1{1 ga-1c' Y :1'1tgi :
,. -- 1
d. y: ;*u - O"U
ltypyfis o: L a,Oqnccra,ft uerr rarcarr uryprer.r a6[ucc
rsmffict.=eft rMap ogqor yycralq..qs?
sinr 1* sinrv. g - -.....F-rE-coS'fi oos,.rt; * +2A2 +l{g":6 },rypJfir_ M(Li-l) uem TarcaE rrfpmrlr.,,6a no$rrt-arrrin
rsrrufiTrg,fixfir 6u.r. t "'
b.U : (1or)*
i
I '' _-:_l
7
118
'=!FT=E+1e.-%..
Ere nwarru axwr
9.18. a.y :5sin3 fr + a"frtfr ;
c.A:@osa){a+fg.28. a. u : sin 3i - -5c.r-x2-ln4
l. U : logr cos4c * cos8er ,
d. a :8 - x2, u : o2 uapriSonyyx"r"'**po",uox onqrrfir ou.
b.U: (sinc)t8'
, sin, - cos,i.g:
--
slno + cosrd.y:2cos3 i+utn'"i
b. U -- e'arctg* --lm\Md.y:rlfr(3,lnr-2)b.y:ln(a+\m) . :
, x2+4o.a:--n - ctS,
. [ *: arctg3td.<
Iu:7-5tb. yr' lng - y2lnr:0
d.a: (sin2r)1"(5- 0)f
b.a:s@-Zr)+ZI
d.[ ":\fi4[ ,: arcrmzt
:
b.y: (sin5r)2" + (ctsl)" ,,
7l .,1-d.v:ls#+sab.y:(tg3r)3d +(dsl),
9.19. h.y: arc$n,'fi
c' Y : tglr,xg.20. o. y: tge,-s' + rfr
c'Y : ctg6a
g.21. a.U:sin-s(fr)
t'tt :0a - t)?tffi(5r+ 4)2ffi
g.22. a. y:arcsinl +4"or5,fi
c.y:r*arctgy
9.23.. o.y:*.ts] + (ts;)t(l
". l' =;m[ , :2tst
9.24. a.y: (slw)Sr2
c.ye-riury-1:0
rs+c,2- L9.25. o-'tr:" r*1*cos2 (1c.y:cos(r*31) d.l
r:5orcsinSt
s.26. a.y :g1+(sinc)z l r rt : fu+nt
c. U: lognsinSr *cosGzr d. y: arcsin-tE- arccoslf;g.27. a. y :arctg(loge |l - fartft b. y : (cos br) sin 10r * ctglOc
t,'Y'
iu.Er'5
I
"''-I
119Bue gaa:rrsrg axwr
I
I a:9'c. ylnU: 13 d. <
Ir:logetg:2g. a.y:
"*ai,.2a * [e b. y : (sin8r).tg; - (cos8c)"
' I s:larcsinzt \
c.us +xs -\ry:g d. I Ila: nea'
n1 -2s -t- 2- x
9.30. a. u: v'rcosr _ 4 b. a -- '@ i arcsin -;
c.y:.r"rirfr+v66r d.v:# *ryBo.qnoro 10. @yuxuufiE ecex 6yypax 3aBcap ercTpeMyvl.rir qen
10.1. A:2xs -9x2 *t2r -910.3. Y:r2(r-2)'10.5. U:2-3*-xg10.7. A:2x3-3r2-410.9" y:(n-1)'(r-3)'10.1i. A:6a-8c310.13. A:2g.3 + 3c2 - 5
10.15. y : (2a + L)2(2r- 1)'
10.17. U:L2a2-8x3-210.19. ,:z7W- -4
n2(r _.4),10.21. 9:*
16 -tBo2 - 1310.23. Y:T
LA.z. A:3x-s3-S -o-210.4. y-;*6a-e
10.6. y:(n+1)2(r-1)'10.8. A:3s2-2-a3'
13 +3x210.10. A: T 5
10.12. Y: l6r2(t- 1)'
10.14. U:-Br3-L2a2+210.16. y:2xs +9r2 +l2r10.18. y = (2r - t)2(2x- 3)'
r(L2 - x2)10.20. s:
--27(n{ + r,2) rL0.22. Y- ---b
(az'- 41210.24. s: -:-,u-
6n2-d-r010.26 Y:T10.28 U:!6a3-L2a2-4
(o + 1)2(3 'n)210.30 Y:>,16
10.25
L0.27
10.29
a:!6a3 -3612 *24r-9(r-2)2(x-o)'
Y- 16t1+9rJ-3x2-as
v- g
120 Bne gaarrsm a)Kfra
Boguoro 11. Orcen 3aBcap rax QyxxuufiE xaurufis ux 6a xarrarufin 6ara
yTrblr oJI.
11.1. s : n2 ** - 'U [1,4]
11.18. y : if z*1* - a1 l-2,41
tt.zo.r:-#L [-b,1]
L!.22. a : 12 - 2r ** - ,, [2,5]
11.5.i:2tfr-r [0,4] LL.6.y: Lt ilz(*_t)z(r-7) [-1,5]
11.e. y : 1f$+11'1s 4 -z [-3,3] 11.10. y : z*'*W - 5s 12,4)
11.11. a:3-n-# [-1,2] rL.L2.a: lfz*a*-E [-1,6]2(-r2.
11.13. v:# [1,4] 11.1'4. u:n-+'rffi'+a [*1,7]4r
11.15. y : lfZ(n - Z)z(5 - x) [1, 5] 11.16. u: ffi l-4,21
LL.2t. y : lf 2(r - I)z(n - 4) [0.4]
11.23. y:2tF-r*2 [1,5] t1.24. n : flfz(r +2)2(L * x) [-3,4]
11.25., = -* *2r*.. 8 = *5 [-2,1] 11.26. a :8* + 4 f*.a
2 s-2 12 L2--'16
11.27. y: il@+2)2(r -4)+3 l-4,2) 11.28. A : n2 * 4r * *r-S [-1,2]
tL.zs. y: #- 8r - lb l-r,-rl 11.30. y : 1lz(r +r)2(x -2) [-2,5]Boguoro 12r @ynrqnfin xaurufir ux 6a xaurufiu 6ara yrrErE 6o,qrroro
t2.1. D As.aMerprsft ayryft ryaJrnuraac xorrAJrorr orrJroJrhru ran6aft rr xaurrfir rx 6aftxaap
Tgrru ouqorr xonqnorr orrnouroft.uaM Hypyyr orruoopoft. Xoxgnes orrJIoJIhrIr xaM)E(sgr
-{OJI"
1.1, D:0.7u 1.2 D:0.5rvr 1.3 D:O.4rrt 1.4 D:0.6rvr
12.2. Uonx rerrrr,ourlerr Asap xarac lyryfi Tasbcarr roM rutrr xsu6eptsfi. Xepea.qlpcrfiH
trapaMerp nB P 6on qorD(oop xaMrnftH rrx reparr opx 6aftxug TyJIA qomcgbl xeMxeg
.f,Map.6afix sg?
2.5 p:$y 2.6 p:dy 2.7 p:$y1 2.8 p:$y12.3. 3ypar xaEaHA.uoog x.s3raap Eb axrmnar.rufiE Hyruggc rfarr Merp, .usDA xrcraap Eb rrBrt
Merpr 6afta. 3yprldr xaurrafts cafin xapanbrrr ryJrA xaEarraac sMap satuu sorcox ea?
3.9 a:3u b:5u 3.10 a:2rra b:4.5rvr 3.11 a:2rra b:4Ivr 3.12 a:hrl b:3r"r
2
11.8. s: ifi@-zya-x1 -t
121,Ene r&lrrtu a,xutl
12.4. aopronrsfi HrrMrsrr roMop xaBTarr EyraJrx aItrfl xaxyyr rpaueq xeqflfloE'ortnontoft
(AB: BD: CD) onropxoft xqSpqar 6enrrsx 6onlcsg. Tyyrurfi 6amaalraxrftr
xaurnfin llx 6afilra:rrrx TyJIA xsM)Ko3r sax asax se?
4.1$ a:2rra 4.14 a:3.6rra 4.15 a:1.8rvr 4.16 a:2.4u 4.17 a:3u
12.b. Kaprogoop rarnaaryft qr{rrrrrrlp xaftpuar xnfix (6amaarrax rr T 6aftxaap)-ldn ryrl,4
xairmnfin 6ara uateplraJuax opyyna:c rs?
5.18 ?: 30,uu3 5.19 f : 63.ru3 5'20 T: 94 Ar'ra
L2.6, Horrarrs xyygacEbr 6ra.rN,teu tsrcrrfin xgMxeo Q"*'. .[sep 6oucoE .uoopoocoo 4 cM,
-1-
6apyytt
3lTrrsecee ru cu ra.u6aft ynussne. [aac xoMnsxufiH yyAxssc xaurnfix osos'rrofi xyy.qaclrbr
x9Mx33 tMaP 6afix ss?
6.2L a: 3cu b:2co-2 Q :2!6ctur2 6'22 a: Ac.* u : !.*2 Q : l44g*z
6.23 a : 2cl"utll : 3cu2 Q : 294cu2
LZ.T. D.ur{aMerprsfi ayryft ryanr{Hraac rerrr oEqorr omnonroft AaM lrypyy'3Ycex 6onxee'
Ecapryyuenufin MoMeHT (W) xarrarufin ux 6afixrrn ryJIA epren, onnpufir sMpaap aBax
ag? W : fU2 fi Hb orTJIoJrbrE opreH, y lrb orTJroJILrII oE[op
7.24D:Lu7.25D:0.5rr'r7'26D:0'8r"r7'27D:0'7!d- 12.g. D .uuauerprsfi ayryft rya.nunraac rorru eHqorr oruonrofi AaM lrypyy 3Ycox 6ouxee'
,I[au nypyy xsBT3o 6aftpnanlaa xfiIr.q (nsrx ypTa,E Hoorlox aqaaHLI Iraacc torruon)
arraaJrar.Ux(ee. .[[arra EypyyHbr xeluyyH xaurrfin ux (xorofinr xaurufin 6ara) oa,foraap
6TTJI9II6IE OprOH 6a ougpnftr tMpAap aaarc rs? Xoul'yu Eb orTJIoJIrrr 'epreurftr
on.upufiu xy6sep YpxYYrlcsntsft tsHuYY rex aB' . ' .; '
8.28 D=0.8rvr 8.29 D:1.2n4 8'30 D:0'9u-)
t22 Erie Eaanrrll;t a)Kill
Bonrroro 13.
qer (ur.
L7-n213.1.
13.4.
13.7.
13.10.
13.2. u -13.5. u -13.8. y:
\t@t44ss +Jrz -8r -2
13.3. y -16.6. y:
13.9.
13.12. A -13.15. u -13.18. y :
L3.2L. u:
Ilapaarcu QysrrtrfiE xoTrop rygrop 6aix 3aBcap EyraparrrHrr
rs -4xv:i
v':4s-54x2 +9
3x2-4r,2-3
4r +813 -5r
2-3x22s3 +2s2 - zs - |
,ffito -br
2-3x22x3 -3x2 -2r*l
1-3r2s3 -2x2 -3x+2
l-12*+Gr+9
x*4
v-7l:
-
ted a-fr'*-6x+E 2-4r22-n2
@,xz +L6
ffi2rz -l2x2 -gJxz -l13-2r2-3s-1
r*3-n2-4r*13
4r *3
'3x-23r2 -7
13.11. u :
13.14. U -
13.20. u -13.23. U -13.26. s -13.29. y -
t'13.13. y:
13.16.
13.19.
t3.22.
13.25.
13.28.
L4.22.
L4.25.
2x+l2L-x2
u--" 7r*9x2 -Lv:
v-
13.17. trt:-vrffi
4E-g12 +2r -L
2rz -2o2 -2c+213.24. a:13.27. u:13.30. a -
2rs +2r2 - 9r - 3,*2
-q4&U
-8r - 12
t/rz - 4
12 +2rrz -3r+3
Bo,qaoro 14. Oysxqufix
14.1. ,:ryLn2
L4.4. a - -" 3+*14.7. ,:+14.10. ,:(* --t)'
14.13 y:lmo-1
14.16. a: ( -=)'" 'o*1
14.19. ,: 9(" - ll' (r+1)24trt:-
' 3*2x-1212
6ypen ruuxlr(rlJrrss xrfixL4.2. ,-12-s.*1- a-l14.5. y:#14.8. u-a2'-qa+!" *;414.11. A: ;a -" (a-1)2L4.r4. a-9!6i-3n2" rz -2r*LJ14.17. n:#
1 _ 1*3u.za. ,:;
r.2+2a-fL4.23. a--: --
" r!*2t - 3
14.26. ,:ryL4.2g. z:- -1or+10" nz *2a *2
rpa$rrc 6afrr1ryu.
14.3. n:*
14.9. U:T14.12. y:(L*:Y
14.6. y :
i4.15. ,a :
14.18. y :
1.4.21. a:
14.27. a:14.30. 2r =
14.24."- z,4-l
4(r + 1)2
r-12s3+L
(L + x)24
x2+2r-31
*+2r+4I - 1012
8r12+44t
(r +2\23r -2
s2 +L
t/9rz - 813 -zx2 -2r-2
L4.28. n: T
t23xapqy
]IACTAJI EOIIJIOTbIH XAPTIY
-3 0\6 rl-3 3t
: 'l
ijl
5 4\4 5)16
")1622)
b [+1f "\l
) b[; i] )
(: ;) c\::) d, (
(;I) c(;:;) a(
(i:-l T ?lt.t. l-201-b-zl
lroo-b zb BbI1,; o -z 10 ,o)
)
110 (;: )
*il :[i)b) (;
.;) 1la o)4r5
b) - 5 ") 5 r.2L. '2 L.22-
t.26. L44 1.27. -7 1.28. 0 :,
1.32. 1 1.33. 2 1.34. 3'
)26
32
-16
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