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t
i-
I
I
I
MIIHXUISX y)(ltAII TE}ffiOJIOryTftH ID( C}?[YYJI}
}IATEMCYPTXVJIb
IIIYTIIC-ufitt 35 xrurrfin.-.
oEA 3OpI{yJIaB
.MT1Ol ltr_
XIrI33JtrffiI CEMHHAPbIH
fAPbIH ABIIAIA
- - lgugrop: tlp@eceop Ia.Iknncypeu
,[en trPo$eccop f.flPrafi
YMXUT
2OO5 on

3. -
: faPtry*
crar@
T. 1. It{arprq ........o.................or...............a............."""c"'e "tottt 1
L.2. TOXOpXOfuOISI ............o.................o.r.t............. tttt S
1.3. MaryAur pam .............rooorooorooooo.............oo.oo......t"e""tttt 8
A, Eacrafi bomo* ..o.r..........o...o................o..............""'""t"" I
Cexen[I)
V,L. Ill,,.ntau reru*rreflrrEn cgg,e*. I{panreprrrrr AypeM .....,...12
212. Ypqry Mamprr...o.o...................................oo............o.t;ttttt lq
2.3 " ypBii/ **ir*IF* apnaap'cgcreMgi1. 6opox "o " o
" " " " "" " "" 14
- ,--)gd;" 6o4rroro .....................................................o....""'15
Cexeri lt /'' Yf . IUyranran reruffrxunrrH cf,snen{ffi 6ogox safifiyyuax apra ..17
3,2. Marpmpm xXrBufu yrItIII donrroro ...c..o...o...........or...o........19
,fl fiaCtat, fiOpalfOfO ......e ....o.....o...oe .e ........o.ro..o.......oo......c..ttt' 2L
CegeU'I/
- l(1. Bernop, *ilseep Aaep xxrifo( EyraMarr rfirEJrYYA'
BelInopbrE Koopprrrar .......... o......... o... o.... o....... c.... . o n...... " n22
/' -r lpcrarr 6oBrroro ..........o...o....................................r'......."'25
Cenen V I
V.L, Berrnoplyplu cKaJrflp 6a nerrop yprc3p .,.. o......... .
".......... i27
,- ." flpcrali 6omo* .....o..............i....r.....o........t......oo..... "tt"tttt29
CexenVI i )
6.2. )Grmaf Aaepx Eyuy5m .o............o.o....o...............................32
,r-- ) ,[acra.tr Oolruoto ...o...............................o.............o...o..ttttttt35
Ceaen {1Li
f3,. IqXepna 6a m5rra*rmr ranIIIIrraII ...oo.....................o............38
7.2. >fanrfnam OefUUffferf. I{afeeC XAn''tfafi ,ryprreJD( 3ilil ..............39
i- .a flgcrarr 6op*irom...........................e...o....o................."ttttttt43
Cexen VIII
8Y Ulysrlx6 rr{iray, ruyJr}IE 6a xannrafu xapf,Jqeg 6afiplmir ...45
g'x*'fi.
or.mc ..oo..........o....o...........................o.........d.'...o...........53
g.2. fhep6o.lr ........o.......................o....o.......oo............oo."""" 55
9.3. napaOor ..................o.............r.....o.....................t"""t "ttt 57
[acran 6op*iroro ..................o.......o..o......o........................t 58
Caaeu X
10.2. OJrryft Ttacpatrrrrlfi rraEap .........o.........o............o.......... 62
[art*rt 6on*1roro...........o.....................q.....o.3................ttttt(i4
r-

4. r r: -q-!Y-- !'- ' -..^ - - -- ,.
Ce4ea 11
11.1. Oy@ ynanffirraJr, nsoMgrp 6a ruexam,ryyn,
.nfQ0epelrrqraJrut,aEr AYpIuYYX .r'........................................ 65
LL.2. Vrranmnarrrur TadJrrilt ....o....o............................................ (i€l
,[acial 6o4rorrc.......o......G.................oo.................o..o.......69
Cetea 12
12.1. Armo@,qpQ$epenrpan.
,[egX epeM6trfiE yJrarffiJraJr 6a pOOepeErEaIr .........!......,., 7O
12.2. Togopxofi 6nffi raf.rrax Jlomralrlfr AypoM ......... r.... ..... 7 I
,[acrall 6oprrorrr o.....................................o........................73
Ceaea 13
13.1. Oym@ ogox 6,XDar( gaBcalr. SrcfpenqM !,fnB ........o...... 74
L3.2. Oym@ xaMrlrrr rD( 6a xanurtdn 6ana yma .....................76
flacra.lr 6ollrroro ...............r....o.......................................... 77
Csnen 14
14.1. OJrrt@ rpa$ffi xornop, ryFep EyrapaJrrirn rfor ....,. 79
L4.2. OJrIm@ acEltfiuor ......................................o................6]O
14. 3. qnlK@ rpa$rm 6afryyaax cxeM ...................... o............8 1
,[gcral 6on*1roro..................c................................!..........83
CeAer 15
"15.1.
I(onrurrerc roo, c)Seep Agep xffiKIfirXIrfyA.........o............€14
,[acrarr 6og*lroro ..................................o...................o........6]€]
I EI{E IA^{IIT .....r...........r.....................................o.......r..........89
II EI'IE IA^dItT......................o................................................111
,IACUIJI EOIfIIOIbIH )(APIIy ..........o.................rooioooooooooo......123

5. Maroru. Toropxofrnotu
colloB I. I{ATPUU, MATP}Iq II9OPX ylln.UnyY.u, To,IIoPXOIf[Orq,
T OIIOp XO finOr q Vfit11 rIAHApyy.U, MATrIIIIbIH p AHr
1.1. Marpuu
I
Tonopxofirrour 1.1 rn Mop n 6arana 6yxufi ra6nuq xsu6apsep 6a;fipnyynarreafi m'n
Toor rn x ?z xoM)rcoscr Marpurr regA A**n rox rgM.qqrJle,qsr.
i_Amxn -
AtL An . ... &tn
AZt A2Z Azn
Aml Arn2 Amn
Matpuqaq 6fiqceu roonyy.qbrt' Marprrqbrg sJreMerrr r9lr3. Marpuurrx
6araua.q 6afipna:r sneraextfiftr o6i rgx r9ldl3rJle.q3r.
Togopxoft-rorrr 1.2 Xspsn m t n 6on rerru ouqerr Marplru, xapun
KBa/Ipar Marprlq 6yroy n'epeuoufir l,rarpuq r3He.
?-p Mep ,-p
m'= n 6oll,
an at2
azt azz
Ant An2 Ann
Kaaapar MaTpI{IIaJI atb a22t ... t ann sJIeMeHTIY,r TOJI,Ul{arollanb YTCr3H3.
Tonopxofiaotr 1.3 A rsailpar MaTprrrrhIII xyBbl i * j 6aftxa.q on! = 0 6afisan .A
MaTplrubrr ArlaroEaJrb MaTp[rI r9H9.
Togopxofinorrr 1.4 a]t: a22: ... : ann: d,6afu..q[arolraJlb MarplrqLlr cKaJrsp
MaTpuq reHg.
Togopxofiuorrr 1.5 d : t 6afix cxansp Marprqurr Eer)K MarpErI rosl E-esp
T9M.q3r[3.U3r.
E_
10 0
0 1 ...'0
00 1
To4opxofinonr 1.6 Xspsn KBaIIpar Marplrqbrrr roJl.ulrarorranrfts.qae,u (.uoo.r) raiu
oaftpnax 61x enerraeHTiyA ror 6on AooA (neen) rypBarDKrrr Marprrq reHe.
aln
a2n
q:'.

6. r-E
a
r, E5.
2 Mawry. Tonopxofrtror'
m,xL
X(uursaa6sa:
[:
traarpuqur
ltl6a1larrerr
rooA rypBaJrxfirE Marprrq 6aftna.
Marpuq, xapr{H 1x n xelaxoeqr Marprlqbrr Mepou
Marpuqirager.rn xn xsMxeecr / 6a B Marpuuurr iri fyp:airrr xyBbA aii,: b;7 6aftoan
TeErIyY MaTpUrryyA r3He.
Togopxofirroar 1.7 E1x euer,,reETly,u ns rgrrefr TsHrIfy Marprrqbrr rer MarpErI
rsea O-sep reMAgrrroAor. X
)Kuureen6gn: o:(o o) o---o o) )
Tolopxofino4r 1.8 Jlapaa:r Marpr{qur rpaueq xen6eprfiE MaTprrq rsns.
An
0
A_
0 0 an ..: arn
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Yy.qAax#0,(t=17)6a,isa.. . i,,,: .., ,^, : . , ,
Marpuu Aeopx yfirr.qrr1ryg
1. Ma.rpIqJrygEr ueyp..{; Xepeu A 6a B Eb rn x rz xeMxsecr Marptrg,6on ++B
'. Q,i -- au *bii,(i:ffi,i :fi')
eneMenTlTlrofi C Marprrrl 6a,ftra. ','
,l':
2. Marpuqur rooroop ipxryyaox: .
Xepea . 6oarr roo 6oa A rrrarpuqtr .-aap tpxryyncerr yp)KBep
atz
azz
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X. a*t A'."o;2 ::. | a*n

7. Marpw. Totopxofrnors
6aftxa.
A 6a B Marpr{uyynbrn rnraBprrr A - B - A+ (-B) rex ro,qopxofiux 6orno.
Jlgsp ro.qopxoftncor xo€p yftu.umrfix'xyar.u Aapaax qauapyy[ xyvunrsfi.
Xepea;A, BrC xs rn x n xgMxsgcr Marpnqyy[, )r, ]z 6ogtrT roonyy,u 6ou:
't
'2.(A+B)+c:A+(B+c)
3. ()t . ,z) .A : )r . (fz'A)
4..'(A+B):),'A+l'B
X(uuree 1.1 -r oJr.
-- -
BoAo*.
^(-: I l).'7
-1 0
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(-r.4+T s.4+(-1) 4.4+o (t 11 16
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r.n+(-b) 5.4+L -7.4+8) a 2L -20)
3. MarpuqJ4flur ypxlynex: Ansmnepreft k6arasarafi , B nukrraeprsfi n6aranarafi
Marpuu 6afir.
& _
cii: a;1b1i * qzW +... + ai*b*i:
Eorurt
(i,:ffi,i:1,r)
erreMenriy,rsfi 6afix c uaul^'fri_fr"T
arprur
rorro'
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3 4 8l
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1.1+2.2+7.9
3.1+4.2+8.9
68 20 10 -Z
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8. r
MW,ry.fo49@_
EporrrfiUoe Marplrqyygbr yp)r(yyrlax yfrngruftn xyBb,q A'B + B'.4 6afinar.
Togopxoinorr 1.9 Xspaa A,B - ff.A6aftsan A,B MaTprluyy.ublr coJrrsx
(rorunryrarua) uarpuq4yr rsre.
x(rruesu6gu: A: ( :,r) ,r: ( _; :^) 6o,r A.B - B.A:(
; _:
)
MarpmyirAurr ypxryriex yrrdoufin xyurn Aapaa)c qarapyyr xtqtrETefi.
1.A.(B-C):(A.B).C
2. A.@+C)-A.B+A.C
3. A'E-fi'A:A
4. (1. A) . B: .. (A.B) .- 6osur roo
4. Marpuqur xepslyflex: A*rnMarpr{uHr xepBIYJrcgE Marpnu nx?71, xoMxsscr
A" rr,rarpuq 6atna.
AT:
At AZt Anl
AtZ AZZ dn2
A1n A2n Anm
X(uursa 1.3 / -
Bo,qarr. AT : 6aftEa.
MarpuUsrr xepBITJIex yftnnnnfis xyBb.II Aapaax qa,EapyyA xyqtrETetr.
L (Ar)r : A
2. (A+ B)r - [t + Br
3' @qr : BrAr
4. (,A)r : )*{ ,- 6osut too
Xspaa KBaApar Marprru A-uirs.xyBB.u A: .{ftaftsafl rsrurxetarsfi Marplrq reue.
't.
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Marpw. Toropxofrtrots b
1.2 To.qopxofiJrorq
Togopxofiaoar 1.10 A xaaupar Marprru&A To.qopxoft gypMeep roor xapraJl3yynx,
ryynrfiriyr Marpuqbrr ToAopxofinorq rsw, detA rex rsM.q3rJlsAor.
detA:
Att AtZ Atn
AZt AZZ A2n
AnL An2 dnn
n x n xeMxsecr roAopxoft[orqrfir n ep3M6uftu ro,uopxofiJrorq r9s9.
Xo€payraap epsrra6uftn roAopxoftJlor.rrftr
at atz
azt azz
: a!!'a2z - &tz' azt
-1 3l
l-r 6on.
7 4l
BoAorrr. -1 3
74
lypaaryraap spsrvr6ufru ro.uopxofiJrorqufrr.uapaax Capprocuftu (rypnanxurr) nypusap6orox
nr xrn6ap 6ainar.
rous€oroop 6o,qno.
X(unres 1.4
:-|.4-3.7:-25
-'
Att AtZ Atg
AZt AZZ Ay3
Att ASZ Ag3
Dns .uyprrauftr cxeusep xapyynBfl.
: A1yA22AsS * aDABAg * A13o,ZtAtz - a1g622o,s - a126210gg - AgA2gAs2
I
I
l. .:ll l ,Dll:,(H:,/H q
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10. r . .-.5 E--
6 Marpw. To4opxofr$ors.
)f(uruee 1.5
-1 3 2l
o 7 al-"or,.
1 -3 ol
,i
Eo.qdnr.k
t
-1 32
0 7 4 I
: -1.7.0+3.4.L+2.0.(-3)-2.7.r-3.0.0-(-1).4.(-B) : -74
1'30
A, B xs,a/par MarprrrlyyAbrH xJrBbA
t. det(AB) : detA' detB
2. d'et(A)r : d,etA 6afina. .;
Munop 6a a.nre6prfiu rytueeJrr
Togopxofiuonr 1.11 zt-p epsrvrOnfin ro,uopxoftnorqrfiH i -p uop j -p 6aranur xacaxaA
rapa)c (n- 1)-p speu6ufin roaopxoliuor.rutr a;3 sJreM€HTe.u xapraJr3ax MrrIIop rseA Mij r3)K
Tgtvl.q9rJI3Ir9.
(-L)i+i.M.. -nir a;i eJreMeHTeA xapran3ax anrre6pufis rlfiqeanr rooA ,4ri rsx rsMIerrrsAer.
137
X(unres 1.6. -4 5 8 I togopxoftnorqrfin,46 -Lrr oJr.
0 9-6
13 13Bo.qorn. Azs: (-t;z+t - Mzs: (-t;'+a.
0 9l l0I
Kaanpar MaTpxrlr.rrr ro.qopxoiuor.r Eb rITEr[fi anrr Eor Mepufix (6arauu) sne.
M€ET dypnfir xapraJr3ac arrredpuir ryfiuearrreep rrb ypxciyncex yp)KBepyT'gufir
EeMceE uufin6eprefi reuqyy.
112 3
)&(uurgs 1.7 A -.i
12311
I rtratpraqun ro.uopxofitorquftr anre6prftn 'l-*
0 -42 0l
L 6 47
ryftUeett arrrslfia,g 6oa.
BoAorrr. To,uopxoftnor.rrfir xaurnftn ouou 0 aryyrcaa Mop roMyy barauaap Hb 3aAJrax
6ogox rr xxu6ap 6afitar ryu III rraepuftr conro€.
detA : 0. /er + (-4) . Atz * 2. A$+ 0. AB4 - -4. (-l)'*'Mrz * 2. (-1)3+3M$ :
-4.
L23 113
1 3 1l+2.1f 2 ll:4.8*2.l4:60
L47 167

11. ---T
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Marpru.Tozop:afi{otu ., ,, . . ..
To,qopxofinor.rnft n qaBaPYYJr
l. Xspaa KBa,upar Marprrubru Aypbrrr xo6p ruep (6arana) r:rl{Jr 6aftsan loAopxoftnoFr rarrgft
ToHIIIy 6aftxa.
2. Xegsa rBaApar Marplrqbrr aJrb rrer rr,ropuftx (6araxurn) 61x suerraeHTIYA rsrteft TsHuYY
6on rogopxoftuonr Hb r3r 6aftua.
'|
B. Xsper B rraarpuu A xuaapar Marplrrlbrrr.[ypbru xo6p rr,ropufiu (6aranurx) 6afiprff coJlrrxoA
rapcan Marprrq 6on rogopxoftnorl ss d,etB : -d,etA 6afixa'
4. Xspen B rr4arpuu A xaatpar MarprrqrilH sMap uer rraepnftn (6araurrE) 6yx sJIeMeH-
^ ryylrafir rereoc xuraarafr Ic rooroop lpxqyJlsxo.u rapcan Marptru 6ou to.uopxofiuorq Hr
d,etB : k.d,etA6afina.
b. Kaanpar Marpr{quu alrb nsr uopuftn (6aranrrn) sneuentyyuuftr rereec snraaraft roo-
roop ypxyyJrx Horee uopuftn (6aranuu) xapran3ax ereluesrlT.q Amp HoMoxeA ToAopxoftnoFr
eepvnor.uexryt.
6. Xepau xBaApar Marprrlbrn xo6p uopufin (6araxrru) 6yx oJIeMeIrrIY[ rpouopqroHanr 6on
ro.uopxofinors lrb tar 6aftua.
7. Ksalpar Marpr.r[Hs arb nsr uopufin (6araxux) enelurexryyuuftr oop ruropufin (6aranrrn)
xapraJ3ax sJreMenTlTluftn anre6prftn ryftqeenryyasop YpxYYJIx EeMexsA rsr 6afiHa.
Men rypaaJrxr{H xen$sprsft Marprrubrx rogopxoftnortl Eb roJI Al{arouauufis gJIeMeH-
ryy,uraftx ypxBopreft rsnuyy . Xapun xBa.rpar Marpl{qbru xyBbA xilKyy.qtraroEaJrrrftu alr
Egr raJrLrE grreMeHTITA 6yrn rer 6on roAopxofiuor.r lrb xaxyy.IIuaroHaJIb A99px rooHyyrHlr
ypxaspufir (-f)arP-oop YpxcYYncsxtsft TeIrqYY 6aft.uar.
10-2
X(llrusg 1.8 2 1 0 | ro.uopxofinor.rllftn rrauapbrr aruur;ran 6oA.
-34 5
Bogorr. 5-p.ranapLrr x3perJrsu ToAopxoftror.rrfir rypBaJlxl{n xen6epr EI{JIxIYrIbe.
1 0 -2 I I 1 0 0 10 0
2 1 0 l:l 2 L 4l: 2 L 0 l:t'1'(-17):-LT
-3 4 5 I l-S 4 -1 -3 4 -L7
1. I6aranrr 6yx steueuruftr 2-oop ypxlTnx, &aap 6aranrr xapraJnax sJIeMeIrr Asep HsMHe.
2. Exruft xyBtrpraJrr xufir.ucen ro.uopxoftnor.rrftH II 6aranu 6yx eueueuruftr (-4)-eop yp
xIyJIx, &aap 6araurr xapraJr3ax gJIeMeET.usep rrsMng.
l

12. r'--
8 Marpw.Togopxofi$ors
1.3 Marpuqbra parrr
Togopxofrrronr 1.12 A uatpuurrx Terosc snraarafi Mr{HopyyrHn xanarrfin ux opaM-
6nftr rvrarPuqnrx pasr rss4, rangA rgx r9M.q3rJro.u9r.
Ilapaa:c fru,qnuftr Marpuu Asepx snrrfix xyBrrpranr rsno.
1.. Matpdutru aJIb nar uoputn (6aranu) 6f* srreMerrryyrrftr reregc snraaraft eJreMerrreop
YpxYYaex
2. MarpuqgrE .uypLrH xo€p uopufin(6araxrr) Oaftprrr colrrrx
3.MatpuurrxueruopufiH6yx9neMeHTIYI[firrgregcf,JIraaTaftrooroopYpxYy[x(oop
uopnftn (6ararar) xapraJr3a)c eneMeHT Aeop rroMsx
4. Marpuutilr xepBlyrrox
Tpaueu xen6epuftn Marpl{qhrg paHr rsrgoc ,Jrraarafi rvropuftn roorofimo rsnuyy 6afi.qar.
Marprauaa snrnfin xyBl{praJrr xuftxa.u paHr rrb oopqner.qexryft . I4irr4.q Marpr.rrrbrr snrrftu
xyBl[praJrraap rpaueu xsuOspt ruuJDrryTnx( panrufir onx 6onAor.
X(uursg 1.9
10 4 -1
2111 2
it4b65
2-15-6
MaTpr{rlbrE parrfBrr on.
1 0 4 -1 lt 0 4 -1
Eo.rour.
01 3 41 1013 4
0 0 0 0l 1000 0
0 -1 -3 -41 0 0 0 0
Tersgc rmaarafi Murrop 2 + rangA:2
Yyu.u:
. l. I rrreprftr (-2),(-11), (-2) -oop ypxyyrx xaprarr3arr II ,[I,IV Mep Aeop HeMHe
2. Oxruft yfiu,qnrfir xlrftceu MarprrULrH IV rr,repufir II rraep [e3p HsMne.
10 4 -1
2 L 11 2
11 4 56 5
2-15-6
h

13. :9Mal,rrui.Totopxofrtrors
Jtracral Gogaoro
Marpuqry,u .qsopx lfinqruir ryfiqerrs.
1.1. Tsnqyy MatpmlyyArrr orl.
.i
1::: )[ii)
il, -
l
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14. 10 Marpw. Tcraopxofurcrs
(4
1.11.A:f 8 lurr:(z 4 s) 6oa a)ArA qBrB dA+Br-ron.
l-,0/ " -/ '
L.l2.A:(:i)'"":(],1)6ola)(AB)Tb)BTAr:foJI.
Ilapaax lp)rBop Marpf,ql1rAr.qtil xJrBbI A uarpuqaur .xeMx(eor rogopxofii.
11& a) (;: ::),
b
[; i 1),(: i)
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6onox ruaJrra.
a) (Ar)r:tr b).(A+B), - lr qgr e @B)r:Br.{ d) (6,A)r:6Ar
(,'
1.15. xspea n:| 6 B Ioon B:AAT rerurxsMrefiMarplru6onoxurxapyyn.
l., ')Tolopxofrrror.r.StrE qatrapErr arrrf,rJran .qapaax regqerrerrrfir 6ararr.
1.16.
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26. r
Bexrop, ryyr Eespx vfrwrwn
coJIoB rv. BEKTOP, TO.{rOOp /IOOP XI/IraX mvrAMAH yIaJrJIJrYvA
4.L. Berrop, To,qrsep Aeep xufrx ruyraMar Ifrr.qryyg.
Berropsrx KoopIrrxar
,:
To.uopxofi.rronr 4.1 gurneur xepuuufir Berrop reus. Xepss.A exnsn, B rorcro,n
6ou aeriroprrr,4l rex reM.usrngAar. Berropain yprbrr rypruft Morynb rox Hepnex 6oroo.u
lfrl,l?l rex Mer roMrerrrene.
Bexropyynbrr EeMsx, xacax, Tooroop lp)Klynex yfiuanyyaufir aemop,uespx ruyraMau yftngan
rex Hoprrene. .[ypuu d Aa b' xo6p BeKTop aBaag rgArsgprftr HoMgx 6a xacax yftuanufir 'F
aap'aax 3ypraap y3lynbe. 3ypar 1.
22
bb
Berropur rooroop 1p)KyyJrgx,qos.uapaax ayprrarftr 6apuurnana. a- Aypbrrr BeKTop,
V, e E 6o.uur roo 6aftr. ?: ,7 BeKTopHr
1. l"l : lllial,
2. Xapar, > 0 6on ?-raft rxr4Jr.rrruemefi, ) ( 0 6o.n ?-raft ecpor qr,rrrsuteft 6aftxaap
roAopxofiuAor.
Berrop nssp xnfix ruyraMar yfinun1ryguftn xyara Aapartx qarapyyA xy.rrnreft 6afira.
JJ
f. ?+ b' : b'+d
2.(d+7)*?:?+(7+?)
3. 7+ li :d
4.d+(:d):T'
5.,(?+7) :)?+)7, )eE
7. .1(.2?) : (.1,2)?
8. 1.?:7
Xasrraft .uospx Aypbru rypBau, orropryfi .uax.uypbrH AopBoH BeKTop uyraMaH xauaapanraft
6afiua.
Togopxofinoar 4.2 Xaerrafi lloapx ruyraMarr xaMaapiuryft ayprrn xo6p nerropbr
xanrrafin cyypb BeKTop rox Hspnex 6oroo.u cyypuap 3aAnau 6[.rseu
I

27. Bexrop;ryw,asspx'yftwrryyp ., , ,. ,. , 23
6orno. Yyug: lr,lz toor ? BeI(TopbrH a$(run xoop,qnrrar rsx3.
Togopxofinonr 4.3 Ormpryfi Aax ruyraMall xaMaapanryfi AypbIH rypBag BeKTopHr
orTopryfiE cJrypb BeKTop rex HopJIsx 6a cyypuap Hb 3a.UJIa)K 6r'rBen Z+ : )r?r f
lz?z*"13?3 6onno, Ylru: )rrlz,.tr3 roor ? aerroprrx aQr![lur'koop,qrrrar rel{3.
'.
Berroprur rexxJrer lleepx upoeKrrbm opVTE : lTfilcosg roMb6oroop togopxofinxo.
On,u g nr ? TsHxJIor E ru*ropbru xoopou,qox eHuor.
Ilpoexuuru xyBbl Aapaa)c rrauapyyA xy'ruxtefi.
rp?(? + ?) : rry?d + uP7? (4'1)
np7)?:I.up??, V.eR (4.2)
i', j' , k us roopar{Harbru Oa,Oy,Oz rsrxusryyareft xapran3au }Ixlrrr el;IrJrerr3[, Herr(
i
yprraft r.*"opyy,u 6afir. ,Ilypuru ? aexrophrr KoopAIrHarLIIr cyypuap 3aArIaH 6u'rsar
d:x.T +y.i +z.l
6onuo..
fi,U,2 xosseuuueuryyAnfir ? aexropn, {7, i,l} cyypb,qaxb Koop.urnar rsx 6a 3Ar33p
Hb KoopAlrrrarbrn rsnxnerlyit Aeepx,upoeKuyyA tora. Te.qrsspnftr 7} nertoprrn Tsrur eHqerr
KOOpAUEaTJ(y.q r3H3.
To,qopxofitonr 4.4 ? aexrop fitUt z roopArnarraft rsssn d : (r,y, z) rex 6rqrs.
o' BeKTopbrH xyBbA ypr Hb nsrrgfi ToHuIy, d -tafi.IuKrrJr .rfirrsrrsff Bel(Topbrr o BeKTo-
prrrg Hgr)K BeKTop 6yroy opr reat ?o.rex'TeMlernoxe. Opr nexropbru xyBb.II
--+o d:ra
rourEo xy'ruureft.
o BeKTop Hb KoopArruarbrrr r3rrxJraryTt OarOyrOz -rsft xaprau3aH a,B,7 enuor lYcroAsr
;
6on vurryy[ertl KocruycyyALrr oJlox roMb6o Hr
r'naz
cos.o:
6i
cosfl:
,1i
cosT:
lf;i
ffIJ:r#"- rJffiff:vv,
'usepx ,"vraMa* vfiuanYYr lrb r,Ar,3puftu xoop'ur
d : (rt,Ur,zr) Oa 7 : (nz,Uz,zz),d,1 € J?3 ororacos 6afir
I

28. 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 :tlw
A * (*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 rr
fir*rz h*y2 zr*zzr:ffi' u: 1+l' z:l11
Xspvrvrirftr raJrJran xyBaax usrraftx KoopAr[rrarlrr onox roun€o rr
fit*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:-l
h*yz 4+3.4
uM: 1+,
:-f15:*
4*zz -2+3.2zM: 1+,
:-113-:t
6ouuo. M(-1,4,L).
MrMz xapvrranfir ,:3

29. Bexrop, Tylru Asspx yfrllgJryltq
X(ilnres 4.3 l?l : 1S, l7l : tg, l? + 7.l: rn 6on ;a - 71 :f
BoAonr: 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 : trq6Ei5:6
l" - ?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 t
ss BC rarr Jreep oprurxo. Xepsa lAEl:6, lfri: c 6on ;Z-ili-nr on.
lf ll4.2. 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
I
m l?l : 26 6a 0g reuxusrtsft yycrex oHIIer IIb Moxoo 6or- ? BeKTopbrII f
koopgrrrraryyALrr 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,
t
t
52 + (-3)2 * 42 :

30. 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.,
{

31. 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' 'd
2. d .()7) : (,7). (?) : )(". 7)
g. d .(? + ?) : d 't +d'? ' .r1.,' '
I
4.d.?>0
b. 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)
=) =)XJK
fi1 Ut 21
fi2 Az 22
At z1
Az 22
fi1 21
fr2 22
n1 nrlt
fr2 grl
-)1,- V+

32. 28 Bxropyyars cKalrep da nxrop :rrlxslD
6ongo.
Togopxofi.nont 5.3
J-
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+(_s
ru _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{Al
BoAour:
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 - -5
X(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: -T
l7l : 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.

33. Bexropyyxux csrattp.da Bexrgp Wxrryp 29
--, lu -2
'-l , 6
;*l u 3
l3 -2
J+
E:14i'-42i-2Lk
x(u1use5.6 xepsa ldl : l?l : r, 1fr1 : [ 6on d:?+g?ou d:3d+t
BexTopyy.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 42
3 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,(?,]'):T
5.2. ? : (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:l
I B -2 o I l-' 6
+ar,'71 :!@:49.

34. 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 rerpasxpuu
g3rrsxiTurrftr ou.
5.17. d : (2,1, -1) aexroprofi Konrrrrreap 6ad -d :3 nexuuuftr xaura)c 7 aerroprr
OJI.
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

35. 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*92
0+l ; Y: 1+) '
Tyxaftrr roxuonuon.u I : 1 6yroy AB xep.utrufir ra:raalr xyBaax M(*;g) ueruftn ro-
onqr{HaT Hb
fiz*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'+b
Koop,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 (

36. .-
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$[qrregTTofi rerruurroJr Hb: U : ka +b
fiuXeB.ruu .qexb rerrrrrrrreJr Hb:
Z* t:,
Hopnna.rrs rgrrurltrgJr Eb: ocoso * gtsin a - p : 0
OrerAces Mr(xtUr) qerufir erer.qcox rrrmnen,q aaftpv rapar( rrrynyyubr rerrur{T-
rEJI IIb:
U At: lc(s - *r)
OrorAcou M(rr;y), Mz(xz;gr2) xo€p qsrnfir raripu rapcau ruynyynbr rorrur.rrreJr
Hb:
0-0r _ U-Ut
fiz-fit Uz-Ut
j-*+
f
6.
7.
8.
9.
10.
U ? kfi * br 6a U : kzr * bz xo6p uryrryyubr xooporrgox onqrnftr:
XoEp uryJryyubr uapaJr"[em 6afix EoxrloJr rts: k1: ft,
Xo€p EynyyEbr rrepnerr.qr[Kyrrf,p 6a^fur EexseJr rrb: k1k2: -1
h-ht$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

37. -
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$;O) opofi, BE eatpuitn 6a BD uewarlbr rortutrTrorllTA
BE : 2c -',3a * 15 : 0
rex erce' 6ou rypaanx'br raJryyAbrE Terlru*enutr 3ox,o.
BD : 2u +3y - 3:0
BogonT: B opoftu KoopAuHarwt BE 6a BD rryJlyyuyyAbrr orroJlqJloJloop oJIIro.
B: [2.-3s*lb:o *[*--Bt)'
2"+Bg-3:o
--
tr:,
AC raauu rerurmrsruir soxuoe. ACLBE y'rtrp AC-nimouqrnftE roeQ$uuuestnfrr onre.
.3-1 3
kaa :
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:o
qsrrfiH 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 K@p.urrHarbrr
D(2;-L) Fex oIIHo.
AB:+4:ffi,8c:#:#,AD,#:#,cD,#:# ,
{
Dz Iz*+3s-3:o *[,*:u
Iar+2y-t2:0 [u:-a

38. . I
i
34
6yroy
AB: 2y-0-G:0 AD: 4y*Br-2:0
BC :'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 ryyrnft
enuruiu xoe$$ruuenr & 6a oY-rsfr orrnoJrqcon qerufir roopAr{HaT Lr on.
BoAolr' u : kx* b urynyyg .qoep A, B usrryg opunx yqrp ryc 6ypnfin xoopArrnar
Hb rgrrrllrrrgrurftr xanrarla.
-{-2k+b
5:5&*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-AoPrIYYrIoarI
fi:
==
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 ,
34
ix+|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.

39. /'
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
2tr
f 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

40. (") 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.

41. 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

42. -8-
.v
COJIOB VII. TAIIAPTA BA IITTAMbIH TOTMUTTEJI.
XABTTATTH rErItr}ITTOJI
' 7.1. Iagapra 6a uryraubrrr rorrurltreJr
i
Xasrrafi assp xoEp Yn Mo.qarAextefi f (r,y): 0 TeruurmeJr Hb epeuxr{fi.uoo euap usr
rrryraM 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, raaaP
ryyrlrfir raAHa opuux arb u ugrufir KoopArrHar xaxraAarryfi rsrururrsnrftr lagaprurn
TETIIIUTT9JI 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,u
oprur{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 opurrDt
qsrrYAIIfiII 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 xoopgfirrarrru
xyBbA uftu 6onoIr,txrW. .[espx TsrrurrTrgrreep Oz rerrxJlorreft uapa.naens 6afiryynar.rrafi
qlrnurrAp 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
(
I
i
t
bIII AJIb H3T TSIIXJI9TT

43. i
TsgrsspufiIr rsrurllrroJl lrb:
Ar*Bu*Cz*D:0 (7.2)
Axz + Ba, +czz + Dxy* Erz* Fyz*Gr* Hy* Kz* L:0 (7'3)
i
i.
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'

44. (.)
(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')

45. 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:0
lI. A2x * BzU * Czz * Dz:0
r3cSrr 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

46. ':
i
-'T
42 Taaapra 6a rrdrrau:l;e zsfirrrntgJr
AzraBzU*Czz*D2:g
xaamafiuyy.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 r
Koop.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- 1:0
xanlraftnyyAa[ 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 ryyurfir
HopMaJrb BeKTop d-ssp aa.r 6oruo.
n:fr,xfr:{[ :r;l I
2-3
15
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:2
xgprlMggp orrJrox xasrrafiH rerul[rren 6u.r.
',j, lI
-- {z;-13;-B}

47. 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. Snaeoc
i
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,
2L21Tarrurrrrsrr 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: -?

48. ,
'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

49. 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"
np
xsr6sprsft 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

50. Yyug: S : {m,nrp} rqtrrJrwrertr Berrop
81 Cr
82 Cz
, Tl:
A1 81
A2 82
CrArl
cz 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-c
p ruyJrJrJrE, Aa * Bu + Cz * D
I
lAx+By+Czl
: Q 1gena,rafi xo€parn
rourEoroop unepx'rftrrcrAorro. Yyna {mr;nr;pr}, {*r;nzipzl rrb rrrynyygyyAhr' rrrrrryrsrq
BexTopyyA roM.
Xo€p uynJrJrrrbr trapaJureJrr 6afix EexueJr Hb
fi,1 tTLl pt
t: *r: p,
6aftsa.
flepuerguKyJrrp 6afrx EoxqoJr rrb
tTtttTtz*n1n2*ptpz:0
(8.5)
(8.6)
6.
ffiaw
Illynyyx, 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*pflz
rn?+n?+fr rnT+n?+ p:22

51. Eo,qoilr. Illynyyurr xrn6ap rorrurrrnsJrsfir canasar rr,ra,na,ft luyJryylrbr rerrulrtreJl Hb
*. a-2:U-L _r-3
4-5-6
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
'l
y-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 irr
r*L _U -2 : ,73-
2+L 6-2 -2-3
r*L Y-2 z-3
34-5
6onno. 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

52. 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. 1
a^o/^---:
' lEffirtlt-ay, + 32 + 1z
11
2*2
1 -1 l) : 1,,n,rt
2LIJ
,
6aftna. llfrrra,u ororrcen xo€p ruynyyxrr xoopou,u,oxb osqor Hb
11
2-1
11
23 l)
: ,-n,,,,,
:
26
.11
9: 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-2
z-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

53. 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 xanrraftn
324
xoopog,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
+32
flex.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 uaap
423
opruuxo 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,

54. 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-Zt
fiz-fit Uz-Ut 22-Zt
fis-frt Us*Ut Zs-Zt
-0
.r A-fiO U-Uo Z-ZO
z. : :
-
: 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-Zo
fiL-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-Zz
l-*:-;-'t:-;:T
Mt(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
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lz Tn2 fi2
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Orrouucou ? -=
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-
h rfi1 Tty t
aafipcan xasrrafig rgrrrrtrTrgJr 3oxr{o.
urynyyHyylbrr

55. 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 ', ', ',', ', ', '
ABC
lmr!'
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 orroJlqJrbrr
lmn
[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

56. 52 IIIytryW 6a xaarrafu,xapswaq fiartp,'lv.lr
i
Jlacra.rr 6o,qrroro
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*5
Z 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 I
8.8. " 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-
-AIlr(yJIrp xastra,ftr rgrlulrrrgJlnftr 6n.I.
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.
]-

57. Xr/;paqaulpap?*r64sr{WyWqx, , , , , ,,, 53
csIIgB Ix. XoEPJIyTAAP gPgMsllriH vtypyIAHvvJI
9.1. 9.rarruuc
To,ilopxoirronr 9.1 @oryc r3x r€pnerAex 6gxnsracsr xo€p q3r x!?Tenx rafiryyxrrn
nrafinosp rorrMorr roo 6afix xasrrafiE 6y:r usrufiE onoEnorufir gJrJrErrc rsHe.
fr2 , U'
-,A- b, -' (e.1)
(9.1) tsrurrtrsnuftr snntaucrftH xru6ap rorrurrrroJl 6ytoy raEoEoK TsrmETrgJr reH3.
Yyng a6abrrb erorAcerr oeper roouyyA 6eroe.q te4rsepr.rfir ennuucrftE xarac roElGrrorIYA
rex rrepnox 6a a ) b rex y3oe.u $oxycyyrurrr xooportrox safiH xaracbrr c reB3JI
c2:a2-b2
6afina. eus ye.u (9.1) snnrucr.rfir OoxycyyA rs Fr(-c;0), &(c;0) 6onno.
(9.1) rsrururrggefi srnrucnfiE a6cuucc 6oaon op,IIuIIar rellxnorlY.Ursfi orrnon[ox qsrIY.u
nr A1(-a; 0), A2(a;0), B1(0;-b), Bz(O;b) 6oreea exs 4 ugrrftr snurncrfiH opofin u3rlY.u
r3rr9.
X(uures 9.1
Eo,qoar:
L ennuucuftr tylceux $oxycyyaEr oJr.
82(0,3)
-r/i ,fr
rrU Fz fr
81(0,-3)
a2.:!6 -, a:4rb, - g -* b :8, c2.:a2 -*:L6-B:7 + c:*.r/7.
To4opxofi.rronr 9.2 Onnraucrfin $oxycyyabrrl xoopox,{ox safir rr( TgID(JI3rr xapbqyyJrcaH
xapbuaar ryyxufi ercqexTpt[crrrer rsrre. Oxcuextpuclrterufir e Ycr3op tsldAgrnsx 6a
rogopxofinonroop ,=*: Z,
o> c ryrr *Map rr srrrrcsts srcqespuctrTer e < L 6aftna.
Xepsa M(a;y) qsr sJITII{rc Aeap oprllmc,4ypbIII [3r MoE rt, t2 Eb TITtseoc Ft, Fz Qorycyyl
xypsx saftuyyt 6ou
Tt: o,* e'r
T2: a - €'fi
u2.a2
-+-:16'9
Ar(
(e.2)
e.urespufir M qerraftn $orycux pa,urryc reua.

58. 54
,afi : *; xo6p urynyyuLrr (9.1) arrnrucufis ,supeKTpuc reHe. enuuucnftE .ulrpeK-
Teopena 9.1 Xepea r rrb gJrJrrrucufiu Aypbrrr qorsoc aJrb rrer Qoryc xl?gx aa,ft, d ur
iruos qerebc erre Qorycr xapraJnax AlrpeKTptrc xlpex saft 6on -: e Oafiua..CT
X(u'i,ree 9.2 9r2 + 25y2 : 225 Ontrurc orcex 6ou
1. Xarac rsruffierly,urftr ou
Boao.nr: Orces rorllrr{Trer ufrn 2Tarlrrr Z2*rxyaaa6an
*.*:
1 =}
1. a:5, b: 3
2. a2 - E : b2 * c= 4, fi(- ;o),&(a;o)
B. s: g:! roM.-
a 5 a:--- " i
' )&(unrge 9.3 +.* - 1 snnrnc.Eeop opurro( M{Z;f) rr"rnn goxycxu pa.4}ryc
nssp 6afix ruyryyrrbr rerrurrrrsJruftr 6rq. :
EoIoar: Onox tuynyygLl TerrlrmeJiuirr y = aa * b rse.
Mr qgruftr.uaftpa:r ryu
+
: 2a*b(i) 6onno. Orarncufir Qorycyya"r" oo"'6. a2-C : b2
) s - 2, fi(-2;0),F2(2;0) 6onox ryr 2 roxr{orrlolr 6afisa.
1. Fl Qorycrrr onox uryrryyn aafipax ryn 0 : -2a* b yyruftr (ii) reBerl (i) 6a (ii)-ssc
_-5-5b : *, a :
i rex rapxa. Oepoop xsn6sr 5a +l2g* 10 = 0 ruyrlyyr 6aftna.
oLz
2. FzQorycrrr orceu ruyJlyyE raftpv 6onox ryn MeE r - 2: 0 urynyys 6a,ftra.
X(umee 9.4
#. {: tennnrcraftu 6apyyn Soxycaac prlrlgtrc xypex ga,ft ur 14 xerx<
oalx uSrr{}Ir oJr.
Eo.qonr: o : 10, b'= 6 6yroy rD( TerDsref trO ryn., 6apyyx $orycaac L4 norx, 6afix
uei roop,uurrarbrr'3IyH xarac'xdauafia 6aftra repxly qsrrftr M(ro;yo) ree. 'Onnnucnftr
Qoxycrr oa6on c : *8 6afina. (ro + 8)''+ Uzs - L42 (a) rae. Mer M qer anntrrrc Aeep
op*m*y,fr*+#:1(b)rse.
(o) 6a (b)-ssc M(-5;3y'5) , M(-5;-3/3) 6omro.
42
,{ :1 ennuucrafin orrnonqorrbr,X(urues 9.5 3r * 10y - 25 :0 ruynyyn Aa
frl
qgruftror. ' i" ::" ; ':

59. ___-_
:i
' i ' '$":-t '
Xolpwreurp spsufifrE Mwyfuiyyr 55
EoAonr: Onox qsr uryJryyn 6a saurucrftr aaftpax ryJr Aapail( cucreMxfir 6oAoxog
..':
xanranttafi
:8
cucre*igc r:l y:; roxrapua.
Tofipor
Xspsr srnrucufix xyBb,q a: b 6on sunuuc Eb rofipor 6onuo. Tofipqrfiu xru6ap
rsrrurrmelr Hb (* - o)' + (y - g)2 : R2 xsu6eprsfi 6afisa. Ena C(o;d) qe" ,ueep extefi
Rpaarycraft rofipor Oaftna. o:0 , g:0 yen fiz +g2: R2 TsrrulrrrsJr rapa)c 6a eue nr
r(oopAuuarbru sx .uosp roBTefi rofipruftn terruutrgJl IoM.
)*(rruse 9.6 n -2A - f : 0 uyuyynrafi M(3;1) usm Iuyprenllox R: fi pa,,urryc
dyxnft roftpruftn rgrrulrrrsrlraftr 6u.I.
Bo.qonr. Tofiprrafin rsrruumorr ur (r - a)2 + (y - P)' :5 xsu6sptsfi 6ai-x Ecrofi.
M qer roftprnftu uar ryn (3 - o)'+ (1 - g)' :5. (*) Men rofiprnfiE to.seoc orcor ruyryyn
xypsx gafi nr panuycrafi re*uyy 6aftx tyu fr :la-20 -tl
| ffl snaasc c :6 *24 scsen
a:20 - 4 6aftra. Oue 2-nftr (*,)-mft clrcreM 6onrox 6oAaon
o,. d,: 4 , 0: -l
b. a : 2, fr :3 rsx orr.uox 6a (r - 4)'+ (gr+ 1)' : 5, (r -2)' + @ -3)' : 5 ror'r.
9.2 fuuep6on
To.uopxofitoar 9.3 @oryc rox rropJrslgax 6exuerAcsrr xo0p uer xyproiD( 3a,ftnyy,lrur
.,rnraBap rorrMoJr roo 6afi"x xasrrafir 6yx usrufiH oJlonnorufir runep6on rsus.
#-fi:, (e.3)
(9.3) mrrunrrannir snlrnucufin xrrr6ap rgrrrrr{TraJr 6yroy KasoEoK TorrurtrrgJr rgH3.
Yynn a, b sepsr rorrMoJr roorryyA 6ereog a-r 6o,qur xarac reuxJrgr, br xyypr,rar xarac
TorrxJrar rsns. (9.3) ruuep6onufir Qorycyyabrrr xooporrox gafin xarac ns c 6ou
3:a2+b2
6afixa. (9.3) ruuepOonrftn Qorycyy,u nr F1(-c;0), Fr(c;0) 6afina.
b_
a: *1r uryJlyygyyA ur ruuep6ouufts acl[Mrrroryyg 6oauo.
a
( S*+10y -25:0
| ,!'*t:'
--
I.

60. rT .-ry-r
56 Xafipsyrmp spsufitu r4ypyWyz
fiz y2
X(unres 9.7 '
^ -
v-
- 1 rmepEonuftr aypcsnx Qoxycyyabrr oJr.
Eo.qour.
a2:9 + a:3,rb2:4 + b:2, &: a2 *b2:g*4: l_3
-r c: ,8.
Xspen a: b6on agun xarrJ(yr ruuep6on reue.
Togopxofirro.nr 9.4 fuuep6onnfiu Soxycyyabrg xooporrrox safir 6ogur rsuxlrerr
xapbllyyJlcau xapbllaar ryprr.rft grcqerrrprrcurer rsrre. ercuexrpucurerrafir € ycreop
Tsr,r.rlornex 6a roAopxofinorroop ,:'fi: :, a < c ryrr sMap ,r runep6or." ,*"uar-
ptrctrTer e > 1 6afina. Xspes M(*;y) ir" riuep6on .qeep oprrrrc( Aypbru uer MeE 11, 12 Hb
ryyEeeq Ft, Fz oo*ycry: xypox
irTr:iT: "r>U yeA (
t12:-a*e'x
^ ( rt:-o'-e'no<U yefl (
.aftsa. [ "':
a- €'r
116a 12 -afir M qgrrfir Qorycrrn paluyc roue.
.a
" +; xo6p uyuyyrLlr (9.3) ruuep6o.rurfin ,,qrrpexTprc rens. .Ilupexrprcufin xyrra
Aapaax reopeM xyuuurefi .
Teopeu 9.2 Xepes r rrb rrruep6onufin nypurr rtrereac aJrb Hor Qoryc lrypex sa,fi, d us
Morr rlereec oue Soxycr xapralr3ax.uupexTprrc xlpex safi 6ou
fi: r 6aftna.
X(uruss 9.8 l6a2 - 9U' :144 ruuep6on orcos 6ou
1. Xarac rerrxJrorlTruftr or.
2. @orycyy.ubrr orr.
3. OrcuenTptrcrrrerrfir on.
Bogoar: Orcos rarrurrrrerr ufrn 2Tarrbrr L44rrxyaaa6au * -*: 1 =+
916
L. a:3, b: 4
2. & - a2 : b2 * c: b, Fr(-b;o), r'z(5;o)

61. .'#'-.-Fft'!
57
X&tpzyraap epstutfi@.uwYrtEwa , .
c5
3'6:;:Eo*'
x2 t :1 ennrncrftr opofiuyy'a AoapX(unrea 9.9 Xepsa ruuep6ounu Qoxycyyn fOO
+
Oa
opurrx 6a gupespuc rrb BHs eJrJrrr[cuftn Qoxycrm aafip,uar 6or ryymft TsrrrrlrTr3rlufir 6r'r'
Eogour: t -+ :1 xsn6apxfis rsrrurrrenreft ruuep6on 6afix nr uuspxrfi. O,uoo
a,Lr rfnox OctJn. Sirnlrucuftu Or Tegxner .U33px xo€p opoft ns -L0,L0 Tyr rlmep6onarn
xyBb.[c:10, + ]-02 : a2*b2 (**'!) MoE sJllluucllfiaQoxyc ur c3: a2-bz 6yroy
c3 : tr00 - 64 : 36 co : *6. knep6ouux .ul{perrptrc slre qsrrfir .uafipax lyu I : 6
oopoop xeu6sn o:69 + u2: 6c =+ a2 :6'10 a: ",ma'
02 lf
(**)-ooc b: tR:+ orlox Terrrrl{.Trorl
* - ;O
: O 6onro. :
X(ruree g.1o ! - * :1 runepSonuu 6apyyn $oxyca ac 4.5 Horx aafi,q oprurx
64 36
runep6o;rarn rlsrlrfir on.
6o,qour. 3 : a2 *fr :64+36 : 100 + c -_L0 6a 6apyyn $oxyc xr F2(10;0),
Orox qgruftr M(ro;gs) rsasn ruuep6onarr usr r)'n ffi - *.= t 6o5o. Horoo rallaac
aB + @o- 10)2 : (4.b)2 cyyrufiu xoEp rsrururrelreoc (ro'l) , (to,-|) *"rn, oIIAoEo.
9.3 flaPa6on ., . ,
To.aopxofinon'r 9.5 Ooryc rsx HopnerAsx oexJler.ucoH qer 6onox,uEpeKTplrc x3M33x
orcor Iuynyygaac trxI{JI safirafi oprurlx Usryy.Ulrftn ononnorufir uapa6On roxg'
oorycaac AupeKTpEc xi?ex aaftr p rssg uapafonEnl uapaMetp r3E3-. xspea
"
: -e,
*yrryy* uapa6onn' ,ulrpeKrp ac, F (;tt) uapa,6onnu $oxyc oon uapa6orrbllr TorruuTrsr Hb
Y2 * 2Pr
xsJlosptsfi 6a[na. Yyuufir uapa6onxu xru6ap TerIuETrsJI r3H3.
Tlnopxofironr 9.6 flapa6onrrs Aypbrr usreec $oxyc xl?ox gair AllperTpllc xYpex
aaft.U xapsqyyncax xapbUaar uapa6o.nux ercqexTp[cllTeT rex 6a rMarT e : t Oafiua'
xepsa M uer napabon A33p opHr{x AypEilII q3r, r sb TIYHs3C $oxyc xYpsx eafi 6on
(e.4)
(e.5),:*+PZ
6aftna. Yynuftr M rtsrufiH Qoxycarn paArryc rolre.
X(uures g.11 ]r(-7;0) Qoxycrafi gnpexrprrchrE Terllrlrrrerl ss u-7: 0 6afix uapadourrn
TgrrrrrrrrgJrufir raPra.
Eo.qour:

62. XapryEaap epeu6frE uwyWya 59
g.5.
*-+: 1 gnruucrfiu :yyx Qoxycaac 2.5 Harx gafts opIurx srrtrrcrftd qsrufir ou.
9.6.:!n -5U - 3:0 6a2a -39 - L0:0 uox 3r -2y * 5:0lrynyyflyy.utm lulpreceu
(V
rofipruftn rgrrrrltrrgJrnfir 6n'r.
g.7. A(1;0) qerufir safipcau 2s * U *2 : 0 6a 2x * U -18 : 0 luyrlyyryyllr luIprecolr
roftprrftu rsrlrrutrgJllrfir 6n'r.
9.8. Tofipruftn roB sb 2r * U: 0 IuynyYII Asep 6ai1y- 4x - 3U *10 : 0 6a 4r - 3y -30 : 0
urynyyHyy.qbrr ruiprgcsE 6afix roftprufin TerrurrTrsJlufrr 6r.I.
_* 9.9. (r + 3)2 + (g + !), :25 6a (, - 2)' + (y + 4)' :9 rofipryynull orrrlorlqorlLlr{ ugrrhfir
' raftpca,r rer Hb xoopAlrrrarDrrr gx A99p oprutrx tofiprtlu rerlutrTreJrrfrs 6r'r.
g.10. A(L;-2) rreresc n2 +y2 *x- 3gr- 3:0 rofiporr rrrlprocon rulprer.rufru Jprbrr orl.
g.11. frauep6ontlu sKcueuTprrcr{Ter ,:l Qoxyc nr F(5;0) 6a,utrpeKTprlcrftu rerururrsu
+
5r - 16 :0 6on ruuep6ourru TgrlurlTrsnnfir 6u'r.
9.12 5x-6y-16:06a LSr -l}y-48:0 ruynyyryyn ruuep6onnr rrrYprsx 6onryyunfi
T3rrrrtrTrgJlufir 6lr'r.
fi2 y2
9.13. 2x-y* 1:0 myryyr 6" ? -?: l ruuep6orbrg orTnonqorblr qsruftr ou.
94
x2 y2
9.14
t - ft
: 1 ruuep6onbrg 3IyE $orycaac 7 Hsrx aaft4 oprrrln( rlrnep6onnn usruftr on.
9.15. .Ilapaa:c rr,rypyftuyy.ubrr,qypcerl.
n
L.y:it/rz-9
2.Y:4!Gq
t.0: -t trr y
4.y:?rm
42 P;-
t4v
9.10.
fo +
h: l srnruc 6ay':24a uapabonrrr orrJrorrqorlHs uerrftr or.
!- 9.17. U:fi2 *L,r:y2 -Gy*7 napa6onyy.ubru orrnoirqoJlbrn qerrftr on.
9.18. U2 : 64a napa6onooc 4a * 3y - L4: 0 uynyyn xy?ox safi sr xaurrfis 6ara 6a,fo<
M qsrufir or.
9.19. x2 :16y uapa,6onurr ruiprece s.2a * 4y *7: 0 urynyyEu ueptreguuxynxp 6a,for
ruyJryyuLr rorlul{Trgnllfir 6lrr.
a2 ' Y2 L oa ?; - f* :l
^avpvixvvaur
9.20. Hsr roopnrruar cl{creM,q optuu*
A +
3
:
orrJroJluoJlxn usrfifir on.
1,-5-/
u
+
+-,
c5
*-'
)-r?
- v-
I
6to
r tu'/
.Ytr
6r,

63. 60 Hsr xyBbca;.wfirs ibywryfra x*lr.aap
c9.u3B X. OyHKIII,rraH Xg3rAAP, TACPArrrrYri I{AHAP
10.1. @ynxuufirr xs3raap
To.q[pxofirorrr 10.1 Xepass xrrrnoeu .r 6ara 6afix 6ouox ( > 0 Too aBa)ra,u ??, >
N({) raLsu 6pc .uyraapyyra.u l*o - al < € reuqamerr 6uur ypraux 6ueuer.uax 6afix ,n[(f)
AyraapoJrlox oaftsan a roor fiLrfi2r... tfint...AapaaJUrErE xs3raap rexrrspngsn j$r" :
o r3x TeMAerJrSIIg.
llypnr 6ara ( ) 0 roo aBum,[ l* - ol < 6 tsnusmen 6nurrfir xanra>c 6p< o-ufir xyBb,u
lf(") - /l < € 6uenerAsx 6afixaap d(() roo onAox 6or A roor /(r) r[ynxqnfir n -> a
yeuftn xf,3raap rex rrepJrar[
]41 l@): A rgx Hopne.uor.
Xspss
lr]t"(r) :0 6on a(r)-r Qyxxqufir 6aracra:r 6aparga"rryfi 6ara
XSMTKETA9](IYE r9A3r.
Baracrax 6apar,uauryfi 6ara a(r), B(r) rscsu xo6p xsuxrrAexl1rrrrft xyar4
t. a(r
}i* 5
: , 6aftsan s.ursepufir eKBlrBaJrerm (ax .rarryy) 6ara 6afixc 6ouox
xeM)Kr{r.uexiTrryyA reeA o - B rsx roMAerneAsr. BoAroro 6ogo:14oo r -r 0 ye4 sinr -:
*.. fi, tgr - r, arcsin fr - fit arctgr - r,
ln(l * r) " r 6ouoxrrr M,,*IIer 6aftx auru*afi.
Xspea
]q1/("), ]r4Xo(o)
xn3raapyy,q oprurn 6afisan
1) #*t/(r) + g(s)l : I,3' f (a) + |g1a(r)
2) ]g1t/(u)
.g(r)l : ]'g1 f (s) .
|,1g0(r)
B) ll*#: ffi , (!glo@)*oyer ) qa*apvv,u.xv'.ururefi6afisax.
Mou
u* rio, : 1
o+0 fr
(1-p rafixaMrurmr xxsraap)
1
J* (r * *)' : Iga(l + a)a : e : 2,71828"'
(2-p raftxaMrutrrr xxsraap) 6aftna.
Xrsraaprru 6o,uuoro 6ogoxon Aapaalo Tsuqniy.qlrfir xspernex rrb aruurrafi.
I
,r*In(1+c) -, li*o'-1 :lna rr*(1+r)--1 -*o+0 fr c+0 fi o+0 fi
X(unres 10.1 mffi-x.E3raap'r 6o,q.
I

64. i ..-- ----:."-
BoAoar. IIIyya xs3raapr rurnxBaJr 0f0 xsn6epuftH Toxopxofirlri unepxuftusu rapu
6aftsa. To.uopxoftrvfir rairla)rrrx TyJrA xyprBsp, xyBaapr.rftr^lpxuraaxyYE 6onrox 3a,uJlat.
*##: I,rl ffi: l's ffi : Is, #: |
: o
. 13 *2r2 *3n * 4
X(n,'iee l0.2 l*ffi-xn3raaprrr 6oa.
Ess trr 5'xen6gpufin ro,uopxofiryft unspxnftuen 6aftra. llftrvr repuufiu xreraaprrf 6ogo:rgoo
oo
xaurnftn a)rMan saport (r3-{) xYplasp xyBaapb ryc 6yprarir xyBaalar'
s":* ** * a
Eosoar. )!Xffi:i,n+;+ *r+ r,
X(nurse 10.3 r;yrffi-xx3raapblr 6oa.
Eogonr. Xyptrep 6a xyraapuftr xocuot ,,,G4* 2 -oop YpxBsn:
ri,q$- q:4: FqI Yrq+2) : 4. 4 :16
r-o1/r+4-2 {a+4+? ,:o
X(uurss 10.4 ,rgg(##)' -xrsraapurr 6o.u. Oxs Hr 1o xsr6spufin to.uopxofiryft
6afiua.
3z-1 5s
5
"g
Eo.iorr-. _lg($1I),: Iim(l + 5 )': rirl[(l +,CI=;Tff:- c-:f& Bx _ L ,, ,-;Ab- 3f _ 1t x,-c
6onno.
x(uurss 10.5 I,s w -xr3raapbrr 6o.u.
Bogorrr. 9nq sint' * x 6a11rrrla-(l+')'a+O fr
: 1 6onoxbrr aIutrrJIaBaJI
ln(l + sinr) !. ln (1 + sinz)
rtrlr -___'-.,...,.....-
-
lrlu
--
c+0 era - ] a*0 Sin f
11 1
1.t.8.1:E6onno.
fi
sln 0
: ll[I
3(eo' - L)r o+o
3r-
ln (1 + sinc) sinr .1'.'
fr3r
n3sln u
lim
o+0
x(uurae 10.6
"*
*"t'1;1ffi-
-xn3raspbrr 60,u.
"-$ ln (lff -xn3ra3Pbrr 6oa'
Eo.qorrr. r .-+ 0 yen In(]" -,r) - (--r), arcsin
#
*
ffi
ywrp
.frfi
arCiln-7:-_---'"""'-/ffi:
,,m !ffi: _l 6onno.
ln(l - r) c-o -r

65. 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: 0
Az+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

66. .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-2Lr +. . " * 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.

67. 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*l
10.3.
#*"(1r+5;+...+ &Ti) 10.4.
m'l,"-Tl
1o.b.ttuq#
10.7.rg4m
ro.o.B1
ffi
n2 +l
10.6.riqffi
10.8.,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 - i
u4e fi_e
^olt ^b,
10.32. lim" -"
c+0 t
13 -or2 * 11r - 6
10.11. lim - :- =
s2-3n*2
,ffi_310.13. lim'- :^
10.29. Iim
o+0
In(r*o)-lno
lim
cq1
lim------ o-*r-o r_Lo
10.15.
-tipq(ffiT - lJrz-t1,+6'
10.17. li- titr'
a'o tgSx
10.19. ,r*
tgo -=sino
o+0 do
10.21. u% (; - ,)
"s*,-,
10.23. r,*f-Fo+o sin, r
10.28. u* (r * !)*,+€
fi/
10.27,,']Ig(;*)*-'
i:'
:
r
10.31.
lr31(1
* sinr)@'e@

68. 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.4r
i
(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

69. -.'.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'Ecsrr
1
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-T
I 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)

70. rb"
r
r
7. 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
Ir
A 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: t
rulprerrrufin reniuurroll: Ut 1: i@- 1) 6yroy o- 4Y - 5:0
rropMarLrs rsrrrrrlTreJl: y + 1: -d1*- 1) 6yroy 4x * U - 3 :0'
+--

71. : , 'yrrarilxl'alt68
11.2 Ynartrxcnalrrn rqGrrrq
rt
:. gl
't
i
1. u: aai {,= o,fio-r
y=ri , {:L
g: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-a
L2. y: shc : +_ I : chr
, oa )- o-G
13. U: chx':
+ yt : pha
- shr-
14. A:thx:H {:h
cho,1
Ic. y : cth,r:
;f,; g/ : -sfi

72. '
:, : 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 tirytrlLla
a: 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'
-'. 2a2
u: arcsinfta; (l"l
sm, - coss
Y-sinr*cosr'
v:!*a; .
(f+ t)arctgfr
u:a;-,
1 z + t6tAa
Y - Afr"'2- r,flthx'
1 .- shc
U: larctg(shr) - mi
g : (lnr)&;
y:4-'; ,. Lft "
y: (sin.'.6)r*€;
( 3t2 +'2
l":-ry ;
[ ,: sin(i +2t]
( *:arcsin(sin't) I 4 *,,11'18'
| ,: "'Eo'it*';
t I Vt-^:Tel. i
lr.re. [':F T"rF,a-
I s, : /l - ,F7 ' arcsin/i
( u :"see2t
11.20. { ' ;
I r:tgtlncost+tgt-t
t^l
LK'K
i
[*
',"i#j-=+':;]:-

73. r,70 Ilsslq 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 ar6
Sepeuunanwr &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.

74. 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'sffi
6aftua.
2. Xepsa
|g1r(r)
: hm$(a) : oo 6a Y*W, opruun 6aftaan
Effi=rmffi
6afisa. (1) 6a (2)-ur Jlouurarruftn gypsM rslre.
X(unree 12.4 I,$ # -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 (L
r-t $'(a) - *-
1 - ;'il323
: 3 yvup (1) rorvrr6o 6coop
-3-'r .t-2
fr
i,gl il:l,11t:3oaftua.
;
)Kutlrse12.5 1-cosc
LY, 2", -brr orl'
Eogonr. p(r) :1 - cosr, ,lt@) :2x2 SyxxYYA 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--

75. T r
I
72 Ilsel4 spsudnfu yJrau)Krralr 6a WWepefrrfiarr
r,e r. 1- cosc sinr r. cos0
YflIM.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,
;-aX(urrrse 12.8 ri* (1 - .L) -ur 6o.q.
c*off e,_l/
Eo.uonr. IIIyya xs3raapr rrrr.rJrxBeJr (oo - m) recan rr roAopxofi 6nur
u.uspxufineJr rapua. Xrsraaprrn rau.uruftH noopxu r{nspxr,r
1 1 e'-L-r: --:--;T r3x xyBllpra)K u -+ 0 YerfiH xsrr
fi 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.
2
Jlorapn$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'
:
t
I
l
InA : ln[ liq(sinr)t8'1 : liu;[in (sinr)t8'1 : liqltsrlnsinr] :
,-1
U* Io(.i*r) :7t ctgx
fr4-
2
,-l
1
7 COSO
li%*k-:
n+. - -
"::"i[--
2 sin" 0

76. r
llest epsudffix ylrael;rxJrafi 6a auiffiepelllnafi
Canal,rx<. Yyxnfi a4lrnaap d, mo xan69pufiE ro.uopxofr 6urn rugpxuftnnrftr
Jlouura.nufiu gyprr,rrfir amtrrrrarr rafinx 6oauo.
.l
1, 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 $ynrqufin
LZ.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@): tfr
12.12. [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