pat1 ตุลา55 พร้อมเฉลย(2)
TRANSCRIPT
:j'- i.Yadtodounflietfllflet5 (PAT 1) q61nil 55. I
nfisr PAT 1 ma1nil 55q
1. rirrauohi'u riluronnvrii'ufivrf uac'lri A, B uas c rflu#urryoaar u :iatqaiolildanoioo
(PAT1 or.o.55)
1. A-(B uC):(A-B)u (A-C)
3. A-(B-C)=4n(B'nC')
oOU ria
rirrauo'lrip rar q rflu:Jfiflef :l:n^rofludalooiotldritu#ofiiuo{ (pAT1 q.n.s5)
1. (p=q)=(q=p),3. [(pn-q)3-p]=(p=q)
1. nnRuaytnng{
g. nfiousirnn{
n. ,iru'luarrfnta":tggr ann rvirffu 7
u. A-grilulro'ir.l
riatoeiot:JdqnoTau 1eRr1 0.o.55)
1. n {!n uas ? an
g. nfiousisnn
4. ritauotrd I ryrl.ufiora+irururfiru lri' a ={xerll2x+:l<l;
trng B={*. I ll*' -*-rl, r} fior:rxrfonerilsialildL tt t )
a I -,
U ,Jv{0'1Til11J 0n?''l il0 a L!1*
n. ri'r p, q ttos r rilurJ:vlror.iToufi p = (q n r) frri'rorrruoitufluoi.o
ua"'r r =[(p = q) ^
(- p = r11 fieirn'lT ruailriluo?tu €a I r J -a?J. fl'llll.tot0nRilfl8JllYl1io0 {x e R
I x' < 2x + 3} [eJ0 R n0LtoflA.:01U?1t01.]
ud'r :x 1:' + 6 : 33-' I firirqrmaroir riluoS.s
rialoei a't:Jdgnoioo 1eRr1 o.n.s5)
2. (A n B)-C=(A-C)n (B-C)
4. (AuB)-C=Av(B-C)
2. (-pv-q)=(p=q)4. [(pnq)=-q]+(p=q)
2. n nn uoi:r fioq
+. nfioravsfio
2. n nn uEil fiog
+. n fi'q uns I ficr
moil 1a
oou 1Ja
Ioou ?a
r'- . 2 {oaounfinnran{ (pAT 1) Earnilss.
s. d'r R rilurryorrirqaura{aailfl'rr (x-2)xz+2 q(x-212x+ro rfio x > 2
ru#r R rilu#u rsosar?ir{ludolosiolild (PAT1 sr.n.5s)
1. {2,3',) 2. (3.5,5) 3. (2.5,4) 4. (4,7)
oou ria
6. rirnuo'ld'R ttaurssrro,:*iru:uo?.: urnrilri'I rrflilrsnsa{,iru'rur6l ld'r rrns g rilufltfld'u
a1R R tllril n Tosii f(x + 5) = *' - *'+ Zx arui:.lt1n'iruruo?.: xq
uor g-'12*- 1) = x+ { ehuiurln'iruruo?.: x
fior:nnrioorrilsiotild
n. (f - gxo) < -169
fl. {x etl {s.rxx) + 5=o} rilulqfl'jrt
riotooiot:JdgnoYao 1eRr1 a.n.55)
1. R nn Lrn& 1 nn 2. n aR rroi:r fiosl|.u
s. nfiorsiran +. nfieruarrfioq
oou da
^ a- I7. fl1fi1.[o[14 r=j (x,y)eIxlt
'iru:uerur8nfl o.itssl Dr -1.2 2.4
8. fiar:a*rriaer:rilsialild
n. cosf+"or$*.orr=1
tt. ,*ff- t*+ = cosec*
eq , U,J Y{a LoCIa t!flonoa$ (PAT1 a.o.55)
1. n an o*r', un
g. nfioueirnn
,-2-cl ey= zx. -u I rfia t ururrnsot,iruruudru
x'+l J
R, rvirn-urioloeiahd (pAT1 o.n.ss)
3.5 4.7
2. n nn ursis fioq
+. nfiqunsrfio
oau l,a
Ioau 1Ja
Yado[odouoolofnffetS (PAT 1) Ea"lnil 55. 3
e. riatoeiatJdqnoiao (PAT1 er.n.55)
1. cos75' : (2 - rE)cosl5"
2. cosl0' + sin40" = cos20"
e. fir R rflu,irurua3.:'loq uii': *31 - cos2A+cos4A' cotA cos2A-cos4A
+. ri'r R uar B tiluo"'ru?ua3{'[01 ttd'? sin2a+cos2B = 2sin(A-B)cos(A+B)
sou ?ia
10. rYi *.r..* = arcsinl- z,,,ccos-2 uii'r cot[4+arcr..*) rvirriurialosiot:JdJnJs(2)(PAT1 sr.fl.ss)
1_ _13 ^ 13 3._13 4. 13g .T 16 +'
16
oou tia
11. Ltnutanro{?tiriluthu:ra.:r#r.lsr:lfirdarJ:cfi'jlnqosl-olto{l.:nnil i * y'= 25 riurtnnnI, 2 r,s A a a
x- + y + 6y-7 = 0 Ltflsly{ftflooilflfl0.:x.:50Ei!ilLEfuqr{ x+2,13=0 fliln1T10':?.]T
m:rn-uriolosialild (PAT1 n.n.55)
t. i*4y2-}x=O 2. x2+4y"*24y+20=0
3.4x2 +y2+6y-7=A 4.4x2*y'-32x+48=0
eo! fa
12. ritauold'n urrursodlnauta{drJnr: 3(1+2x) *9(2-x):z++ ud'rtqfl R rflu#utrqrot
ri:rloEiatrJd (PAT1 sr.o.s5)
1. (-1,4) 2 (-2,0.5) 3. (0,5) 4. (-3,0)
oou ria
13. ritauot#n rilulelsr?ndfififiG :*: Lrou de(A)+o fiar:rurrian'lrrusiatild
n. (de(A)3 = de(adj(A)
s. d'r a2 = 2A uii': det(A) = 2
e q , U ,3 Idatooia L:Jilo.nooo (PAT1 q.n.55){1. nanLIEi?lan 2. nnnuei1fioqr{s
3. n fio uoi:r qn +. n fio uac I fio
6au ?ro
, 4 doaounfinqran{ (PAT 1) Ealnilss.
14. ritauolri'P = 3x * +y rilufl,orifuno:.J:ratri-Ioufiasilnr:{odrn-q eird
2x+3y26
2x-y<10
0<y(x
fior:anrianrrueiotldn. p firirrurnqorvirriu zo
:r. firne (a, b) idfi1td'e frriroirqo uii'rqo (a, b) flaornd'0.:niraunr: x - y = 3
rialeoia'lrJdgndoo (PATI o.n.ss)
1. nqnras?Jnn 2. nqnueisfiog. nfiquoitqn +. nfiouncsfier
oau ria
1s. rirrauo'[r{ d,i rlflv t, rijurrntnofuu:yuru dt u*v-fr,=0, i'w=8 uae i'w=-2
rirrrntEraf w rirruru arcsinl n-uurntoof I unsl?ntqof * rirruru ,-*.rtr{ n-uq ---- Jl ' J,rrnrmaf't ud'rrirrot II12 +Iv12 rvirn-urialosiat:Jd (PAT1 er.fi.ss)
1.6 2.10 3.14 4.18
osu tio
16. ritauor'lrinec rilug:.Jarruru6uruto1 ri'r a, b [rns c rflunr1ilur']go.oqirufl:.:?iraJe1iJ A
rlil B raer{rJ c nrrua{roi:-r Ioufi --l-.#= r.}*. ua'r sioc rvirn-urialosiat:Jd
(PAT1 fl.n.55)
n J, 2. _!_ 3. Ji 4. 1t'22"'z
sto! 1io
17. riT rauold'2, ltau 22 rilu'l"rurur8tdaudooond'arn-uaernr: =' - g.+4 = 0
' t ' ,tr)f!.!'l rvirriudalosiat:Jd lenrr o.n.ss)n1?J0{ li ,, l' * rrr, ,rr, ,z )1.3 2.4 3.5 4.6
oau da
lYad{oaounsrinnrcari (PAT 1) qatnil 55. 5
18. rirrauold' u, = rm(n,r-i)- *r* rhrair n=1,2,3,..
rLar bn = o.o, [2n,. - ]) ehrail n -- l, 2, 3, . .\ 3l
,.rn!?Rra{aun:ru fr .(t)'. [fr)' . . (#)'+... rvir#uriolosiat:Jd
(PAT1 et.n.55)
1.2 z.-? 2.2 4.-255
nou :ia
19. ritauotd't1x) = x'- 26x2 + r*-#rdo o riludruruoir d'r d1 , a2,a. tflu,iruruoi't
au,iru':u6utn-uuuuairo-r.,rr:Trnfio ttactfllurirqo]JlJo{flflJR.l: f(x) = 6 rr#rri'rto.: r1go,u,J
rvirnurialoda'L:Jfl (PAT1 o.n.55)
1. 211 2. 1A7 3. 101 4. 85
oou ria
20. ritauold't(x1 rflunrlurrudra-.:aot Iou;i (o) = 1 uos f(x + 1)= f(x- 1) + x + 1
shraf,i,irurua3.o x'lo 1 ui',r.t ] r1*y ax tvirn-uriatosi at:Jd (PAT1 ot.o.55)
-2
1. 3 2. 2 ^ 2 |3' i 4' ioou :ia
It+x-2x21 , e ,, , u21. rirra.: ri- [3------r rvirnu:l-olosio'L:Jd (pAT1 o.n.ss)
x+l+ {; +3 -21. -12 2. o 3. 12 +. rarrirhilei
oou fa
22. lunr:maqanrsr"roa{anilfarun-u nrurirasuilufirarironrsl.ronudtduuui'r a uasu#rontsirgsq{q
6nonra*durusi'ru o IoErfi 1 *l=* rvirri'rriatqrsia'l:Jd (PAr1 4.n.55){ab2
1.1 2.1 3. I 4. I" g 6 "' 18 " 12
oou ria
9t^d6 {ofloilnoretfllfrer: (PAT l) Ea1nil55.
2g. ritauolrinru#rult'u:f'rfirfi,:rifu:vra'ir.:riaruafiritauol#giot:Ji$tflut#uo:.:q
, u ,l?\0'l:il11lan?1ila0 []J1*
n. rirarunr:ra{flx1sJfs.Jfiuf,rfitfl.orifu:!fl,j,1{?iaila 6o v= mx+ c rrii'r m + c Lyiln-ll2.6
:t. ti"t x = 15 ttti': y =26.4
rialodat:Jdqndao (PAT1 er.n.55)
1. n nn tt03 I aR 2. n nn uroi :r fio{.U'U
3. nfiotusirnn +. nfiouncsfiot
oou ria
24. ornnr:aautrrofiosraCIfto.:rinGuu#'otradt :J:rng'jrn;uuusouattinGuufinr:
ua n uo r:J n 6 un c dril uo du;nttti'r fuIA.r :l n fi :vwir.: o fi t z o-r qrr:r r oi ol:J d
z 0.5 1.0 1.5 2.O
xe1r.1l{Yl 0.192 0.341 0.433 o.477
,u,J?\la1TEU11J 0 0?1 il fl A L:J1{
s v a io o X ue e t , in. nlilnL:Ul,tnilfiU{ 1ilfiO.:UflA:J tOqlcLtUl*1AOUn?lO'l$'lUl[UilAHRA{tYl']fl0.1ff1l*tUU.:LllU
udX,-3.1'l OTg'l U LtaiX 01 eJ1 a:g'l Ut A { Os ttillAfl a U1 0'l un [T U 1.tO UUtml n U -2
r. #r n ; uuufl a uasl n fi o fi 1 fl o Sr o {#n 6s urio.: dfi rir eirs or urvir n-u 60 o c ruu un v frau d q e X v v e t a <r a -- - v ailnr:u1l[1114o{uflauleiflsuuuriosn'ir 54 flvr[1.[1* 60riiufouns 15.9rto.:fin6EJuo!ev,^{tuX,-?ia{il Lr6[i flxrilTsAm-iir a{n1:LrilTr.r1ar a{osttuuflaLutytlnu 0. 1
rialooiat:Jdqnoiao 1enr1 m.n.55)
1. n nn L[or fl nn{g
3. nfiouoiran{
2. n nn ruoi:r fioq
+. nfiounrqfiov
OAU ?JO
x 1 2 3 4 5
v 3 4 b 7 10
tyad'[od0unoi9lfl1dst: (PAT 1) Ealnil 55. 7
25. 'h"'u6l,iru:u 0,1,2,3 n":luorr-r.i +*+ siotjd (ria.:ar t ,iruru;
r' i t u aoLO UI ttm At Ltn']A O.:3JO''11{?U[6'J O,
ugi nsn a-n d'or f irururdru o,
,U,Jv{alTil11 0 €r?'t rJ 0 a t:.lu
n. dha<crd'?b<dt. ri"r a *n ud'r c < O
€t. ci"'t b<dud'rc<d{, a+b=C+d
1,2 ua3 3
1, 2 lLfls 3
:iatooiat:Jdq:Jka"'qnoiao (PAT1 q.n.55)
asn - \: nnLY'IU.1 1 ?J0s
dun-{nnL?.lth:3flos
1.
3.
IAEDJ 1JA
nnl,r#e.nl-nrf'rra;1rf,n unr g rflu#rlsqta.] u
dt zoo/o ra.:aurfinturser A riJufimfrnlurqm e
2s% ta.rffxn8nturqo e uiluaurfinlursfl R
unr.iru':urrurfinso.:ugEr (A-B) u (B-A) rvirriu t t2 uit':dT u:uam6n:rotrtgr A u B
lvirriurvirto (PAT1 o.n.sb)
stou
27. td' R tflluttqriroouta{fliln1r l;i. *.[**ro*0./:**r =rn
ura;td' g tilutqor{rfl aura{ffiJfl1t zx2 - 5x * 11 * 2,,1 p a* * 5 = 25
i{ail?Rllo.rdlrfinvru.:fi eJq'[urser A u s uyirfiuryirto eATl o.€r.55)
28. rirsa.: r..'(2u,.t*] * *.*f) rvirn-urvirlq (PAT1 or.n.ss)
2. n -.: nnrfiEJ.s z lioql
4, n -.: anrnriagq
[-"ffi]rlrvuotri'u rilura26.
CIlu
gou
a b 2
3 0
c d 1
2 0 1 3
9rA.!8 toflounorsrfllflslr (PAT l) Eo1nil55.
zg. t# e rilu'i'ruruo3.:1o1d'r a uns o uilurirrJ'rnqo?Jo\: .or4e-sina0 ltar 3sin0+4cos0
mudroi'u ud'r a * o rvirrirrvir'lo (PAT1 o.n.55)
oou
30. rirrauo'l#R ttvtuttflrlrflalta.]flaln1: *2loga1*2 +2x-I)+xlogr(x2+2x-l):2x-x2
uartd' B={r2 lxea} ,{au?nga\:6rur8nyr".:uruoturssr B ryirn-!,r1'r1, (ror, o.n.55)
oau
- A d t A t - X,31. d'rrrnnuufitfioogurinnr.:60 c1n, t1 oti:.lur#uor:.: x+ y +4 = 0 uava.:nalflr.iruQo" -. ,, ,
' a
' ^,
t
AGs, -2) Lrasoo B(-2, 5) uii'rdufigoq:Jarruuadur ABC rvirn-rrvir'lo lenrt o.fl.55)
32. vrr:rTuorg:Jradrfinoaaoaqjfi n1-a, 2) frunuaurl$:flu1rarlnil x uavIiln-a F a{rur#uor:r
I dtfieunr:rilu +x - 3y + 14 = o drl'nmIua'rd'qr-qn-:rr#l.{er:r r- fino B(a, b) Ioufi a " o
u#'lzuagruta{r?nLma{ Ar. FB rvirri'urvirlq (PAT1 a.o.55)
33. rirrauqld'n, B rai c rilurilsrindfifiG :": Ioufi aete +o dt4=[] ? ll[: -r o.]
rLnr det(BrCB-')=-o ud'r o.t(Qtec) uvirn-lrvir'lo (PAT1 el.fi.ss)
9-0u
34. thfAiU n = 1,2,g, ... fiff4Ufrt#an = 2 + 4 +6 + ... + 2n tldg bn = ?r + dz + ds* ... * on
rirro.: J3i[+ * o] * *.. .. #l ryirn-urvir'lo eArl a.n.ss)
3s. riruuqld'z rilu,iru':ur6tdaufiaoon6'orriusrunr: 2lz + 1l = lz + 4l
rirra.: 121 rvirn-urrirlo lrdo z rmu#{qn (conjugate) ro.: z) (PAT1 o.o.55)
oou
30. fireiro-uuo:rofiflrofid.ifrzun:r?n 1o nofrr:nryirn-u 20s u0rrl0:rrn6n to yrar[o-ol:Jq
rvirfi'r sos uai'rzuaurn 5s yrafu:nsotn'ro-::rnrnrflodryirriuryirto (pAT1 er.o.ss)
aou
tl
rToaounfinsran{ (pAT t) qlarnil s5. 9'"
g7. ri'nauold'R rvr1.u{qsa+irurua3.: trdt:n J R uac g:R I n rilufi':ri'fi'uloafi
1. (fgxx) = 2x + s ahraiu4n'iluruait x
z. fitrifu f uff! g fior4n-urfr'r1no'uoiuihrirrln,iru:uo?t x
s. firf i'u t firira.oao#ln-r:f'rvirn-u 2 fi x = 1ql 9
4. s'(x) = z ihuiurln'ilruruo3.: x
q du a I ; @ v d t u r 0flrfifu g ilrirairqoaeJv,ryriryiln!tyi'r Lo (PAT1 o.n.55)
38. druuo'lu-p(x) rilun4urrufiaaood'a.:n-r P(x2 + 31= 3xo * 24x2 +40 [L6t rix;=ir6yor
limx-->2
tF(lr) - r(g rvirriurvirlo (PAT1 er.o.55)
39. rirrlimh+0
fl11I0'J
uold'e1xy uiiunuurru loufi e1o1 =
3hx + 2h
aou
1 uacsaonii'0.:n":J
=r rirrat P(12) rvirriurvirlo (PAT1 q.fl.55)
qau
oou
P(x + h + 2) + P(h + 2) - P(x + 2) - P(2)
CIllu
40. lri't-rilur#uo:tfihiruoo(0, 1) rua;ote.:arnn-:it#uo:.o x+2y=odufito.:u3t':rx
luoraoiud'(quadrant)frtd.:#aa.r:o:.tsi'rrunuxttn:
Yadtofloilniiletfllcrer: (PAT 1) qla1flil 55. 11
44. rin Guun 6i ilv dt fi r o eru d.: risi avnuril uam8nr o.rrilriletlnl: rtareJ:il fi wr utarru:n9
ouqri orirt#au 1 tlJilJ ttarvr:J'i1
fr a nu rilumrur8nsa.oser:ritrnr:
fr to ou riJuarur8nrarrr:rfiwr
fr to oru rfluarur8nra.:rer:ilousri
fi o eru uilusrur6nro.:rer:ilousr?runctilrsJfivn
ft s eru riluarrfrnro.:tru:ruitrnr:iunrtru:rufiwr
uarfr 3 ou tiiumrurBnta.:rr:ruit1n1:tLnEril:ruouo3
d'0,:nr:udanr'irr?lu 1 01* ra.:usiarrru:uloufirrsincnurlt"'a.:tflumrur8ntfiu"rteJ:ilrdurrvirrfu{o ^da u , a& , q
drururI.roano.tnfllzilyr,lfirJotyrln:JLvl't Lo (PAT1 o.n.55)
45.
aou
druuo A rnr B rilurnsrnr:nilo l luutruuilaar:-Jt rnsdrruold' P(E) uvrun']'rrilravrilu
tattuqnr:niE d'1 P(B) = 0.30, P(AnB)=0.06 unc P(AuB)-(Ane)=0.:sruri"lriruar P(A - B) rvirn-rrvirlo (PAT1 sr.o.55)
sE!-u 0.14
; ' v a rirn-r tgs d'rdoruarodfieireilsoru unrU1flAilfl 3A1U?UvlLtglnol':XJ1:lXJnUilrin5?ilt: ! I @
-,t^r,"vx,-#ru:J:sRvrGla.:fid'urvird'r o0 [LdE o.z orardroiu uit'ao':rruuil:il:euto.:riorunsofrrvirnu
rvirlo (PAT1 0.n.55)
CI_Et_lJ
0aa6uA47 . 'Lunr:aa::itrqrrflflaraofror{nl5suoo.:ri'a.: :-Jmng'jr eirtadutarotfiorrotflsrtuuflau
ryirn-:-r os orul*u rinu3uuli'0,:ur:nfr +o ou ri'a.:firorfirin6rJu go ou d'rnvul*ufla:Jra{u a e au t
^ { dt v a y ytv
unL:u1.rra\:u:nru#er:-l:vRmGra.:nr:uil:zuIuuvirn-:-r o.2u1?-r n riiuti.ntTuutiotu:naor'Lt;l"'
6s flrLr1r1.r 6ouilurimrrqr:orurvirn-:.r 1.5 nruuudorsorfnGsud'atfimo.ofirieuifiu.:r:ru
rurn:orurvirn-:.r 12 nvLruu uunvu'ru r rflurinGuuli'orfioa.:aartd'octtuudorilu rir4
e.rrm:qrutvirn-r -z rrii'a uru t mor'lsi'frflrLLu1* (PAT1 n.r].55)&
46.
oau
lYad12 toflouootgtfl'rdgt: (PAT 1) 8a1nil55.
48. tunr:aauisr n fi osr aorSr a oin 6s u#uni s ru6n :*rflifi o fi nrtuon uo.: o-t dnsuldl,t o'ruru (ou1
5-910-14
15-1S
20 -24
40
50
30
20
50. a'1 a,b, ce{1,2,3,...,9} tta:aaend'arrYu lsf xa)+(7xb)+c=416
uf,'l a + b + c rvirn"uryir'lq (PATI q.n.55)
n'rnsuuurodurarnfimria.:nruuuarudrfiEJuluflJ t+9 rda k, a rn! o rflu'iru':ut6ruurn{b
lqgd u < b uoe u.i.il. fla{ a un; b lvirn-u 1 u#reima.r k + a + O rvirrirrvir'ler
(PAT1 fl.n.55)
ctou
49. rhrir n= 1,2,g, ... rirrauot# un=r**-{ rrar o" =t-+-#qtnthururdsJurn n drlrtf #ffi=rrr, (PAT1 0.n.55)
osu
qou
rilau PAT t n.n. 5s n A
(A-B) U (A-C)
Ln-cl n Io-c)
n t B'n c')
{o L (-P't'(P
\\\
\
FTF/r-t/
T Y / 6,1 /\\ Y "4,t -t a,r:-;-?-'bu"fntlul 4lr,rlutr nt P= r, n=q;nrh; tl utl lii d uil r, rfi r
A - tBUC)
(ANB) _ C
IAUB).L
A -(Buct # (A-D) U [A-C)
rio I fin
(Aft s)-C = (fr-c) n (0-C)2.
A-(B-t)r'An(B'nc';"ris
(AUB)-C + A u to-t)Ya1J0 4
'.lfl
efin
+
dtrJ ioB nor-r
+
F
A - (B-c)
A u tB-t)
O fr^uioni filurl,,^qiufr ,,r,rro',Jrrrohruo frranr nl',e.,cit'lr,fro',trr q,n',1# U""-U",.* '.,
n""1' '1ur3*
rr ar nrqdo.r{ r ilr,*,rl\ru?lr.rfir iriudr raiol*,l,i ^al
6ul-or rjtrnlillv,nl ,',1,*,fi q riil ur nt mr *.illuq-:r
",,nol"o*,^.', cracr trA-1Aloir i r,?n do{nn 6r r,fir\ri fr rtin tr{n: i', dr,.^orid1oiln",ru"riu lfr,lui ,i*fi, irlu fila{u}''uotrl *nnrl iou sr or n:'o'ii { r fi Fl {o{et ti 6 r #o\ri fr
' t?
ds1 (P-q)-+(1-P)F_/ '\-\
TFAr '^\-FTTF. 't-cf J /
"iri{orr{r 4tr,,fixirfrr P -=F ,\=T
q rrrr\fr ,lnr^olrili fi urfl q
v ^x)
^ q)
Tt. I
tl -t ar)
(p--+n)
F
$03 L(Pn^%) -+ - P 1 --+ (P-q, )
F ----TF
F- -, C-Bd-, "" i,,rtr*r/'fi,.-=-l=7
{s+ t(pnc1l - -,1,1 + (?-X)7-----\
TF
rinu6.r lri'ilrlnill ?tr,-umarfi{fi P=t ,%=Yt r nr'l fr rjr .
^ o,i Ii rilu lfi q
TTI
I
v j d, C ou ItoultltI dlulufli
Tti.kII
I
IE 1
1ts 2-
v"Uo-}
1ro +
.:- TTltlt) =- T
Vnn = F
(P+ cl,) - (1+ P ) !. i,rnld
TF(^pv-q,)-lpaau) i trnx
TF
t(pn-11 -+-Pl'-e \P'a q,) :tF
Lfpnl)+-q,l -+ (?*q,) :TF
, ,n JdIil'l)oil 6-101-l 3 *^4
fi,i,Jrrruqrl Yv//vz -+ lN rxAr01r1tf,ounr'rxniluflrf,Y"oi frrrlfr N:-r , W=frrloinr nirruqir.us,r]rrnririoz qrnnr:rlr Na r r,Lir lfiutntmrtrlaqtnila:H1r|uuu\r 11
-
tItJ unufu V/////A n 1
G dd " 1 v I I , dt dtiAnl it t nn nl6r4rlnl[u uI rr'r tllul* lrju [n c
LL6or{nio nrilfi rljuqilotiirfrua ( F + F
X=T, P=F nilrllu (Prq,l= F-+r; T
(- PV -%) = (-TV-F) = (FvTl = T
uhtr Ltpn-q,))^? J= ltrnr)-+r] = T*>F
(t+r=r):T)
t'lr,r,iluXrt*ir*Jr r
tJrlriuilrnixfi t
= r L ril.r*flrilludrl
tin{ LtpAX)-+^1i; L(-rnr)-+Tl: F+T: T tlu\uiludruir Al
n) 11n p, (XAr) - tprl) /t (P+r) i,'liuq't nd;1lii1 [P )\)= T , (P )Y)=r frrfl
rnrr*r4ou 6rcr A q,filr*q'i,16if,r,oifio nnoitfinnr3olfrl4 n riu t'r.lr*l'ur11nfil
Note: p+ (t^r I nri11s -+ rdrlfrtautratrounrJtttn$11 qlti q)1) A (P+r)
rrmfir (p^n)+r nt,.rurdtdrrrair,rilorru,o v uJiuHrilu n hrfi
A ulHuxliuV \'--) (p+r)YfP4r)
:d,fi",,irioudrroido f,',1#',lr,rqrjtfr'u.,^l*?.fiJun1lfro r ]'#,Jt,^q,ifidnrnntxflitt',]u,fl.4
lriui{r- TV m = f I -xvx zT d'oP
I I merkrraiudrill?rrn -Tv@ztT/X A = % I -Xnx = f II . i -T4@'-@F V m = Wh I X--x z *X i
rAW)z I lr-r-r=F i -XVX;T
urjnr^enra ( = I -t (- p -+ r) J 1A la = @
XeyE-XvI u ='wrv(-P+r) =-FV(Pv17= (-rV r) V P
= Tvl = O nluuR nn,frett
1) l1n LL-- [xeK I x1-. Zltl] = [xeFl x'1-Lx-:l<o] = ixern l(x-:)m+r)-<o]- -, -f J 7 { r x 1-?tn?sfrlqniroxnlfin'inlt:rl-lrro: ]xIr'+ [= )3-x ] tilo x t w. { - r-< x-<3
q1n ox+b = 3i-'\d 3{=A
= {xetR [-t-<x<3]
, xe RI
i6i A+b: + --bIt
dt,-{
[1.X1J]1 1€ (.l fill]lL]
Y 110
''t "[ 6'l
t,t,fl
,lve'daillouotsu 1. x
fiqrtnn A : l1n lZx-t..tl<1
-1< Ix+'1 -<19-l .< 2-x < 1--l
-16 {ZX< L- g-<X-(1
tA A=t-r,-.l )-6,-s, ,0,1]
a- < 2n i:dtrlir rn <
ant < a' q:6rdir m 7
^ x'*2iiriaiu od:rnr: (x- z)" ' '
1) cr <I-2<"1 --+ 2<
l*ifrir rltz > zx+lo
xL-?-x-B >o
(x- +)u+2) )o16i ,,<-L u I>+ n
fi gr"fer nr fl6il1st o 6Nn1tflfr o
Iflfr (3,+) c (2.5,+)
An g = t -f,-I, -b,-s,-1 r-3,'LjA-0 = t-1,"0,1r2J / O
- l, Jd rilit0u010u 2-
x
+ h(AnB),y<
1r s X" sJfl
ldx
rtu
n" nn'!
2_< 5
-, 5>Z
n
n
ncrolio ?> 1
firrrfio o<a<1
zX+lO v <lX- 1- ) orotrlllt 0,[l
z)
5<35z). (*)
L3
lt\3
n r6B
L>1
dir
Z
x-qrI
x<3
l2 <X<3
--r x>3
xL+L< L/,+10*'- r-x - 5 <o[X- 4)r X+ t) < o
1vbn
l, e-)
-2<x<4 n X
11 <x<+1 "/
>3
3<X<3 " 13,4)
6rrj{0il n or 3
(-lto) - (-q)
-160 +1 = -1E1
f rn)= a
-Y'rftol 11nn1rt!n.,{ X= -5lu {rx+s) " x1-t'+zx
x
d*lrria riurnufllidu?urroi.(ir frr f -'( a, r = E r'l0iir
{1n g-'tzx-1)= x++ fJsiir $tx+rr)"2-X-1n) tf-grror = fro)-g(o)
= [t-s )3- t-s)"+ 2-(-5)] - [z(-+) -(-125 - 7E -10) - (-8-1)
11. - - 1r1 9 (o) ilnfh,rrnu X. - +
\-- q
l,u 3(x++)=Zx-1
-1/c., 110 n. ,J6-l
At+gA -L1 ,a ----+ tAt?)tA-3):0 --+ ff "3x=fi,1 .+ X"l
lx[:xt!'33-]l = t
Nq1:6r{r B
iiun [il1+
qrn ll'- x - r I
x'- x-1 z 1 u
x'-x-L 7a u()(-z)(x+1)70 U
d,rdr* {s l. nnolol
71
XL-x-1 < - 1
2-x -x <ox(x- 1) < o
3x7o lnreR
/ -111
cU ) A0 t111 g'rr oror:rol dxnrt
t3"f ltxl + 5 = 0rg(t(x)l - -5
l,rnnitcrn gcx+r) = L\-1 titrrnu |=loi{r ftx: = | (ttn 39)=-sfiltna',it qrn f Ix+st " xl- x'+Lx #r,tnu1r, r '
[n21 T(x) = L tfio I. b
d:rlir gcftxrt- -s otrrlrinuq""i x:64a[oir txel l rg,frcxt+5 =o J / A
dtiJjrdo n, rrfi: fl, f;tu-r irBnuu + I
ulv- 2- 1;1,6-1 llL), Lr-L) -1 ,
IrA:. qrftx))" -5 )J
-
'1 ryX=1 t:[n fcb),1_1+z=
fr dil,i s{u nm l'r] uqil
,'. do rr" -2n
1O_-=-X'+ 1
-5
2L12x-s LX-+7-1A _ 2(Ii1)-10-:-=-
qin
{tn
)=
l-=
X"+ 1 Xt+ 1
lv=X,Y) e 1x I
-'1 ,d, .{\1 bn n fior[]B L -
Lt'- z I\
X'+1 J1o t-l
x'+ 1
X"+ 1
lfiuir x,Y e I
rfixir y
te ri"ilfio * € l
'ilwtnulu n1'reJi 1 1 0 ?rt dr il Xl+ 1 al oir
16i \'+1 = l)?-)s,10 ---+ \L=or1,4,9 ----' f,: o,t1,tZ. )13vrio xL+1: -'l ,-L,-5,-10 ----n x'=-zr-3,-1,-11 ---+ liflrirx'fru'r]uiruluflll
@
cir X=o '--) ) = z- 10 -- 2-- 1a - -YO+1
gir X=!1 ----. ): 2_-|0 = 2-5 = -3' 1+1
fli )(=12--+ ): 2__ 10 = L_?_ _- o, ++1
f,r X= jj--+ \ = Z- rc = t-1 . 1t 1+1
n.) cost *cos f + cosr = ;cosl +cc,slI = -1 -cos1 = l-(-1)= 35 -5 7 L -
L
tanP tan* - sinH sin9f,1b B cusry ;G_
firrnerir rcrsaal [rd: -,r <oos?.u lnuoernnrrrl/o{otur+ cas!. + co5 3tr * 9 lruuoub57
ti{lrlr'r Dr = 1o,1,-1, L,-L,t,^3}R. = {_q, _l .,o,x}
1; Dr-Br " \-t,L,-L,3)ilnxrTn a frtdrrJ riuderuu t-. x "
I
ve,'. 10 n, gJ6-l
,.o
sinacosts- CosAsinB = sln (a-B)fl.)
1bnr coso = six ({-e ) :inIIcot{-cosf
----* cos]{ cosf,
sr-\S = 1
'ilS sin {f,od "0, 3
{(
sin f; '" (H_ 3T\
zlsin ({ H)
cosl.I c sin ti-i51 -cos?! c ei6(i-?) *
cosuc {
o,in (i "#)
FJNda t,sin ID
dtrl{rio n fin rrd: 1,. 6ttn
1)
9=
Tordl,.i
c05 15
c0:-1:"
cos 15"
(2-- G ) cCI515o
- sintto'- rE"t.cos 1so
_ tin 15' . tan 15'
oiolfrqr:n*l,lt !os'ts" = Z_fr raislr.cc,g 150
coeo. sin(lci- e)
r tane= Hco9 1E"
l::i:_;,'j.,)tancA-B),
,'_ g,3=
t tnR - tangt + tana trn B
E_ JT (3-Gl'- q-6J3+3t + tantE't anld
Coslcr'+ sin$" = oos2o"
oiosfieumii sin +ri = co,szd * costd +a1oIo
tr,or sin +r)' > O
------r 5thA =
---+ Cos B =
t + E s+il3
----+ cOS A =
-----+ >[nD =
1-3
t-)
.l)
+)
1; A=
Xrcsec x )
* s in LA -z{b)
i+lI
I
- -4 2- : --qdJ
8
=_
:_2t9z l3
- lc _ /13\tx\ 13! - V{)\8)= n12
d:r-liod oro.,1 x
ctnntrhlrrXuir rir ros {:floRrilouf rfio O *rrn?u ( 0< 6 < ttt"), qJri cc.rsto' > oostc, --+ Ccrslo'- custd < o
-, s in +o' r' c r,s Ld - fos 10'@ofld$ A = 0o cot d rarnilulo5 rilrlt*.ferdhir.turil hrralu A 1n1
dild A=B=3d- rin2-R+ LorLb = sin 6o'+ cos od = ! *i = ry- zsih(A -Bl crsrA+b) . ?-5in f f d-$l cos (irj+:rjt = Zsindcc,s6rj
a&*it :inzA + cosLh / zsin rA-B) cos tA + Dl &rrair n, B1ar1
drtl{o?r mou 1 x x+il2
4
.l trt
G
I
Jtr
E",1 5
ten tA- zB) =
rl+z
1A1 cot ([ +
. L_tantR_*r_ lann_tanB
r+tan+fins
rrcsin IJrr
?rccos IJ5
2rcsec X . trcs,n Jf - zor.ros
"o! = A - ?-B
Cosqd = osinqd . I
- tln Itn-s) - B I
y+,' 1ld "il, 0J6^l
Vo.'. $0 n. ,J6-l
r tor(j) . cr
,,'. do s fr16-l
= Z-fi.'- 1,o n" nn
,1
tanA= l
tanE "
= cosrqo"+(A-21)l \ I:l::'.l: :l;ffi;:;:lll:sin Lto+ (A-28)l tcosro" cos r n- zgl - sinro" Sin(A-lB)ninl;'c ) t
Ccrs (A -28)
tan rA-o) - bans
-
r + tan tA- b) tan g
1+q_ tg
q-11
,-qQ)
@ ut16'roi","uolns'.r+l xt+ 1'= 25Y d t t -,
fo ,25i+ 6Y-'l = o ---+tr,
qfl fr 6-l::rair nat^ldo (+, - 3)
-vl= -2JaI
I
I
-fr +
-+,-3)(u,-,
(4)- 3
t-d- -z
a\l' Inu urf,unu?trhj rrn u'louosn 6untrilsY +6Y -Y = 02t'L
--f )(-' t6-) = 15-(-31-= 1b ----+ X= 1f
-3)
LiIA: T +
v - _al-
1t61". ( - *,
rrnlniilridoqtruunuion qJfr F Ix,-:1 .L[01 f oi3r.ru
rXuorl I+2Is=o --+x= -zJZ \6i F<-z|r,-a)qrnn:rr{ lsi c = Lfi , 3 = {, ---+ b'" :'- t2 = 15- 17= 4
lc.rrlutjnal (0, - 3 t --* film:tfr fro r- *-Jd -Zr{ oa X
=1
16i x'* tY+s1'1u+
?' b'= I --+ Xt+ +yt* z+y + zcr = 6
d rrJiofl orol L x,l
"LUI
E1n
38r = A(l+2x)3t
o ,2xJJ'1-
3.3" +
: 3A + 6l
qLi--x ) = z4T
3(+- zx) - L++
8t = L4+3'*
= 2?-+
?-A! v {Y
ril6txl1l tfrl1/
u1r fl4061 l,fi
3A2+ Bl = L't-* A
oAL- zz+A *il = o
()A - 1)(A - Et) = o -r A =
1; r'^.3-',3u + x,d,
l\^u11 t-;,2.} c (-t,+)
tL n) = Lt de+tA) =
,<
10 1. o,lfli
) rnrUu n'rnrncf
-q - 6
n 0 I cTuor ta mr n'f
y<1rs n. iJfl
mih
max
nqna1
rr,o1 I- ) f 3 10 1. hrl
irft1,6
I
=)JI-1
10
tl
)L
I 11nn
dtrl tt ou ora uI
A
1 x
,flor"\ifdsrifi r io no tdj A = det r*l R-' irfl: dettrnl= chdettnt
n,) qln adjn= dst(A)A-l
take dtl rl,rolirqloi de,ttadinr = deiIdeltnrn-']
dctrftlrtrn.,nrfi - = tde,ttnll?clettn-'l,_-__-i
,et CI
il,dt
= (det rRr)'
c
n
lJ, ) q1n A'=
teke dettl.rr delorlfi
lxn: dettrr# o + tor dei R
det
8
dr,lioil r',ru + x
t\rt/c$ailn1r10qlnfln0
'fl 1 q {'r 3r xJ fl 0 { il={ bx ru 1 n u n', r rr e el o{ or r ""v i t-l r'k uel a I
noruro"toi Atf , 9rr,'b(tct,to),$x+11 >rb ---+
Y
LX'l1to --1' Y
0()-.X -t X,)
>t -LxJ
>,' Lx' 10
>tA [r^? 1<X
i6'lNnfrl nllus P: 3k + SY
AtL 6r\1 '3 / I = a(!)++(9,)' r
B (10, 10 ) P 3(lo)+ +(lo) .'10
(5) o) P.3(S)++(o).15D (3,0) ?.9(z't#4(o)-9
y( ... r,rnr'lfinril'
\ >t ... trlut{un:}ld,. t€!
vnTil1 P alrnAnno
16rA ( 9r,9) dflfr
, lt "
JdlU11"uo1.[ fito1] 'L
I
10 'u0
F il nr drrAfiI
*
(Y+31
r deI tnr
Al]n U +V - Y\ ,0\v.raj[r'r u+v = w
! ! . '1 'dot il nr {ot'irlrrotAHnii n"-[rtJJ
u.h,l + \' \^,1 = v\ -hl
,JJIJ
ilTlhnl ur,'H = 8 , V' Y'',1 "A?-g+t-Z) " lNt
JZlhl-= 6 ---+ lN.l
el J4 d r , J
tl fixilei'uor Iin1fi0511n1 I-)JJJJq';!114m1 X-y= lXll)l
dril{oioror + x
* Noto : Al'csin)t =
0Fccos X e
gin ( 0rcsin
cos(1I- 0)
q rnfi 1qn+r ir',rrr* nll;r q ?1 fr i 1
il m = til,rrilrcc,i(aresin*)8 = rd,rtfi)eos(Arccosg )
J ,-. J5g = ril.rtJT,(E) rdr=sQ ver y {l
IrA:n [fi An0un0JJI:
V. n = lvltNt crrs (lr -arcaiht )js. tittGl (- cos tarcctis-r ))
G= tY l(iu.r(- * ) -__f li I = jzr 13 /
--rLJritil. t-+ ti l' = *'+ [.lL lt = 1ts
8- LI
2-Z ll,fl:
=l;Tt1}.lil nU
1
CC'S Q
€ otrlr
JJ
T\'W =
v
J2lhl
_L
-L
,Jcll 1.{}+
tFcsinJI-x't lc cos
X):X ) cos
: -c059 )
( -l-<xs l)(-l-<x< r)
( arccos X) 3
sin [n-6) .x (-l<x<t,sine
@ qrn 11_+ _A+c b+c
btc + a+c
d+b+c1
1t
-+-=3*c b+c
=31* ?.r1
11nn{no31n\+ifr jr c'= ?',+ b'- zeb cosc
rur O-O', a- -tb*zabcosdabtt - 2cosd, ) = o
*.dr,i,u, etn a, b filufr',r-rtsl frruly*ar? t + o
{Jfr 1 -2cos d +-g ----' cos d = 1
nL6trl ioil nsu 3 ..{r
R,rnt ir rtndrnrr
qJd ''*A1+1 1+1
(a+c)tb+c) t+b+ C
Ltt+btc) + cl ta+b+t) = 3(i+c.)(b+c).z(E+b+c)'+ c (atb+c) - 3(ab+ ac+ bctcz)
?'+ bo+cr + zab+ 2ac+ zbc + ac+ bcr cz = 3tbt sic t 3bc + acz
a'+b'+zcz+ zab+ \*\c =3cx+ oab*\,+a\cct = lt+bt- ab -o
I (a+b+cte= a'+b'+ct+ z(eb+bc+ac)
slnC= sinod
fitnol[rn16 0=b= C = I
+ 1=1 dlnr:rilut'rt
-o
[[1t1,50 l-t
c=
8+b+c+ 1+l= Iz'z 3
.EL
60
,iura:rluneruil A rgc itl" a frruunr lunru\unnilr.r = 6d{'111
oilrrlu sinA = sihB : sind = sin6d = gzX,
| - xa
32- +U)(+J.lq1n 7-gZ+$= 0 ,--+ Z
lfr lzn r" = lzrl'= (1r)'* (
i; rz,l'+ r2,r' = 4*4trn nrrlifrir 6untt o t('* bx *
q1n z'-szt4.o lrizuo
oiluur 1*1 - 7t+zz =21 z, zrz t
lsi (lz,lL+ tz,t, \i. L
3n= sin(nn-I.)-cosnr
Notc: Cos(2,nt-O)=
bn= Lco5(znr-{) =
'ir bn= r drraiunn n=
gtfii2
la+btl.la-bil. JEu-"
- ! tun flrB ntaror = !a a'- a
-!]= 3 , naqorntfl0ll = zlLz =1
3t _ 3t J_Y
2( l)
= 9*r - +++
O t{nUanntOrOU "
'lr1ou -- 7.+ 7, =
(n1i1 = L
o
z
)J;1
Z
I
11n
1- 2r=
{l- l, d !z
drillou6'10u 3 x
I z , I o fi rr otuu', siu li'rr n?urr fir}; 6o',r', i(ruiru ullu
t olrnnhJfirLb I ) {,16i l,, = * ( oirnourarirrarl r )
drr]{ofl ror'1
Sirulu fr =r,L,r,...
= t(1)=g , r'r =r'2''
r.,ro darjlo i r
1.r$ - z2
- <J.Sr 4uunn0 I- 3,
+x
- c05n1[ = - ZC05nT ;h= l)2,.
" (:l-
tril
n
) = (i)z3)" (:).
=)
YvIn
bxru
!J6i
titolrtn 2r )
r""loo ?u= 3F
tl61 1,1,r
r r,^tJ; a, = a
er rilu nrflguflordarnrt f rx I = o X-A x-lr)
r1l" ui rrjr'" n oil zuu r { tx l .q rloi Irrr = (I- + )(t-r)(x- ar)
1A { rx r = x}- z6,,t'+ bx - zru = (x - +)(x- a)tx- ?r)
x'- zsx'+bx -ztt = (x'- r, - + n 1'l(I-art
xt^ t-txr+b x - zrg = x3- lx'- ll'* & - arx'+ x'rx
X3- zbx'* bx -21r, = xu- ( ^ * gr+ tr ) xz+ ( 1j* arr +
fi uufiu.J,tfini {''rna:mr r 11fr
-t
-a
x> 7b . I + ! + efr 6+ |+ 6r
a(i+r)= lj' --+ F=3
3
,.. xL"L, b = 3+.a'k+ a'
Y
> z1L = a3 + X= 6
2_6 =
LUz
l*rr
- L1L
10'7
L63
=Cl
:5
a+{x
a') x
l1)(13) ,
+ b2t3) +
6 (j*r* r)
'(:) = (
16i f tx) =
t'1fi =
t't1) =
dtrl{oE *o,r
x3- ztxt+ lgrrx
axl- szx r 156
3- EL+ 156 --
L*
= (sin nnrf- cosnrs
)=+1
=$Q=
uln n
3+
t-
/ 3nt.-
\ bn
*(Z\3
23
-jf = t:f-z(-r)i -,rt'3 I - [3/
33L/ -2CosZT \ i( u )= \
/-Zcoslnf- t\s/\
lr =
b,L
/ 3r\(u.i =
3
r_fu] =l br/
- 2 CCrS lI _ -21-1)
cos g
EcosI3
1,L,3, ..
1-L-3
t
15b
^el,-((9 Lqnilrirrar+n\'fi {cn,"riunru,orirfir6ol 1orrfi
fiilfr {txl = axt+ bx + c rror {tor = 1 ---r
q1n {tx+t)= ftx-r)+xt1
fco:=1
1'- 0ttl)+ btol+c ---+ c-- 1
Id a(xt1)zt btx+ll+\ - a(x-1)'+btx-r)"t\tx+trItxtri- tx-rit + bitxtl) -LX-1)] ' x+1
2txztzx+1 - xt+zx -1) + btxtl^xt1) ' X+1
4tX+ I-b = x+1 ----+ ,fiorXr,ril,7n-i4lrrlnsrtr'tdi s?=1r zb'-t
i; J'{rnd* " J' (lr'* 1x+t) dx = [frr'* *xix]', = (*rtt*+r)-(t;-L-z+L
6rrJ{odo'rsil 1X
li rn lr+ x- zx'\ =xrrt J-Xtt - 2
ne+cJ-xtg+L) -:-r = lirn ()t-1)(2x+{)(ll+3+2).l{ur{urrarharr xrl+ (Jxt}-zltJ-xt:+z)
loi htEl= o ----+ PcEt, nrEl
lln., lt-xl I t+zxl
--+ l=1,+
+1-z).i*
.lt-xl .I-1 faoxz.l. ll+zxl s t*2X ,da X rr -Llir l(1-xlt1+zxll
=r_i1t I-x* _ Z
nt5 )
x,1+ Jl* - z
3a = 15fi+5C -O10'1 = E\nn+ 15C -O10 = *51"fl + 15c -O
-+ C= 01
b=lL
frz=3
, ,t J6tr.lrrou eiotr 3X
lrun',r1'l0flantfrtdosnnh,ioaifiu flcs) = 6x6 = 3r
frrrr,fr:..dssrrfix{snniorril l*} = I 1frit E = lt+,+ ),(ttL),tb,a)ieb z
x+1t (x +3 - + )
dxnuritlrl ) = rflxtL
dzunit,\nfi Z)= mZx+ nc
ZxY = m 7x'+ cZx
[l'n%nl h,5r 1,t\= 15, 2y = 3o, ZX2=55, Zr) = 1o1
lirn (2X+tttfx+l +Z) " (ltt)+1)(It+l +z) " ())(tf ) = 17X-rtt
dtd{oanot + *
(""1*l=1*1-l+1 =lr'' r'4-7-1-1''7-L'3llb lL
['5
Z
b 6-)
rir $ xl
*t @ -O : 1o.l -10 -- ssrq - +5m
1'1 = 1otflrn = 1.'J
ttnunl fl-l " 1,f \m 0 '. bo = L5'5 + 5c --+ 9L= *'5
loinalnradilrJdo y = 1,'lX.+ o.qdtIuual ffl+[= 1,]*0.9 = L.b ,',"]o nqn
frt x" 15' Y' 1,'1(15)+ 0'n : ?'5'5t0'q'1-b'+,Y Y ,
eirru,rnnfi{ n 
n ) fxrfrlilrinfrrr?rnurlfrorurr*u,'r.lu x qtniuonir x ilosnir M, sadounrtot s
drnr:nfiauLiir Mo- X=r !r v dr
ncr rtrj ir n r ulu n1? rrqn rr,qr dnB
--z 11 =-2 =l.Slarat nn-'t
z5
x-r)
Mo= X
.'" ?o nI- )( = 25
A=0.5-o.15161, )
= O.341
fitR=0,3a1 ,Z=1
rr* s6 < bo c:Jfrir
-1 = 54- 60
s5- G
IJn'r: trd:el1*
s
xb = o,1
bo
f,d 6
a ..,')0 lJ nnI6o5+
etYiul{?110 n. llfl" 1, frrd{oioror I x
fi,lfiu,a, I
nnflgi
@-J,ulxlllu[8.io, 3 tra'.2,
-l'1 , :i *0'tr:tln [fl fl ouu no
dt,t
|1ru 2 ',l
01 I { S1 ',l tnnuX
Ia!,d0 uto s tnlhu
dtadrnlo d q!flotrilurA"t}
r .lad=3 + nrorh I
d=o -lnrdfiz
o, b, c
')
frr 3=3 * b"C =O
frr l=o + [=6=J
cl 6l1]nn:$n z
st_41n ntr).tT\ 1
-ri{1n n[6]bn 1
r'.r4qlnnl 2 nr6I+
?<d rrhr
a > b ulf;r
b < d. rtlra+t = c+d
,' t0 n.
.'" 1J0 'lt,v
.'" 'uo n,
v.".lJ0 $.
d,bu)+? 1
[1' ]+? 1
lli 1.r? 1
d,iuu?'t
",x
br d
c< d
c<d
0.J0^t
nn
nn
nn
vn1
n1
n1
drrl{oflmor1
x b 7
I 3 z U
C d 1
Z a 1 J
,i I b L
1 3 2 o
c L d 1
2 o 1 J
Jt6 1 b L
1 L o
c Z 4 1
2 0 1 J
1i^ n(AnB) = x, t-o"l ltos n frs X -> loo1 1Jo3A fis 5x -?> 2s'/. 1J03 a 6o x I 1oo7" rrol g ''no +x -
11n n((A-B)U(0-A)) = 112-
Intnl- nrAnBl] + [ n(B) - n(AnB)]( 5x - X) + [ +x - x ) :- 112
YIX = 112- --t X = lt
p11n1 = EX
h(D)= 4x
= 117
6lt}{l+ hthU0). ntA) + n(6) -n(Anb) = 5X+tX- X = 8X = $(1b) = 1ZE *
oiorinsuit i n+s4zlAs ="r l ln*G
A . Jtu+1)+ (1) +z.lt3xi1)tt) + Jrnx+1)+(e)+zJtlx+zltq) = 1q
JTI = 1t({r;*JJ)*(Jrx+r+zGxtr : 1+-3-'-1 ='lo
5 3X+1=25 X=E A"
lr;1,"'i,
I x"{*s 1; A' =
Iz)t'-6x+10) +1+LL-AL+zh -L+ = O
(A+A)(A-3) =O
z = lx'-3x* 5
I = xt-3x+5 +
)1gx+s ---r 'l-A', zxl-(,
{g}
x r10
+ \" Qr-1
-rJ
I
31
'l
+b
* tan (Xr)) =
r - tanxtany
anLrzR+0)
tan zn + tan tanzA =
z tanR =
1 - tan'a
z(j )-(
,
.l-tanzAtan3* { 12+Yl1- (i)t+),}
12(f,xfl)=2
a
=_+
1+
1
1+
t+ x
Jaz'tr
A.
1"bfri A=
i'- llrr 4 'ct
rirriu AuE . { s,4, - 1 J arnurn6alr?nr"r]u u X
1* trclan l = ,1 -) tanA '1,; *ctan{=o - tans--
rioliir rec'o= 1+tan'o
1rrnl = secz(zA+ts) = 1
1
Jr-rx,r = Ls
nin^1L.(t-r-) A= 3
l- (rr)'
nrruialilr+{oildo
{7[6-] CD
[o hB
! ng.co
trA ;
nA-
C,os'0
q6^r C th, kl oqrrulfruerm y =
*rJ6 c(h,-h-+)
Bstno+40050 ' -l?++'< 3sip10++cosb -<fi3+4;
- 5 s ssino + Aooso \< 5
-a<asinSl <a tta: -b*bco:S<b
= (oor'o 1 si nz o ) ( cos'o - sirlo I
= (lttcosZo)
. CoS Zb -+ - 1 < C0Sl0 -< 1 rirlrnfin b = 1
Rr sJlndot1
xe+b. t
x' [ogr, txizx - tt + x logr-, (x'-t zx - 1 )
)x'\ogr(xiz)-i) - x log, (x'+ zx-t),,u',
6'urrm,1l,*l-rl lfi slr* r.l,,frrt 0i,1, ui,?i,,il u
Lx -x'zzx-x
'1 vL o-r
l4xo
log,tx'+r-x-1)(ir'-r) r z(lx'- x)
(ir'-x)tlo9r(x1+Lx-1) + z) = s
1,*- zr t[o1rt]'+r.x-,r) + 1o9.4) = o
(})t, - L) (log, 4tx1zr-u) : e
-x-4
AC=lvln
:0
A(X', Y, ) -t AB:\. Btxr ry.l
Sotr)
C {-r,l )
B(-Z,s)
urArnr: tkurrr il u'.tr*1ri A ( -s,- z) tr'\" B t-t,s)
nsrrarnloidaLnrt 'lX -3Y + 21 . O
(-s -h)'+ (-z+hr+)' = j t-r-hl"+ [r+h++7'tr+hr'* th\.zr' = (N D'+ (h+t)'\'+roh+zs = \'+1$h+rrrh : - g6 -+ h' -t ) k= -t-.])-t =3
loirn c r -r,l)
Sol z ) \il def 1.on',,virfiri61frrArrrur
ldtur,zuninir axs firu?otn . ri.lld 1
t,srar dei ",ir., ,,irrr..,* z l:1frflufi n
rln: bc = r inuh Rr " 6-c
-5 -L 1
,',.(e't' * d.= lAa+Bb+cln)'-' J;ffi.Bt-z,s) 'Ax+BY*Cco
A(-5)-2-) - I rt-r) - 3(3) +21 | 21
-
i+1+ q J-st
1
1z
-1a
-z s rl -'lit,^o-r3u. J t-21s7'*l5lJl'. j,'tr',(ffi)=
= J;r-2s+1+ - L +35 ttS -+ I
L1) - 1a.5 X2\:z
x'* Y' \<
-:in+B
z---r rrnr*nrh;s lo3 tfr llt-r)* =X , ucrrair lo3 , o"
Iogr 4(x1rx-1) = O
4(XL+ t-x-1) = 1
4XL+ fX- 5 = g(LX - 1 )(Zx r 5) , 6
X = I . -!LzA - [ o I -t]" t ', Zi Z )
B = t r', (rrl', (-9.)' J = I n,:r,T\rntlnfiufin'lu B = 4+? = to.sx
(I,.Il ) t
-h-+l
A(-5,-1)
}t,n. AA0r . 1+, 5
w,n. A h0c =
Y'z
x
e!76untens (y-zl =
ilnunt | = 3l:l-1+
()-2)z'+(!J:11 *''1\4tL
)'-+Y++ - 3Y-1++1L')-
) -'1)+t = O
(Y-1)()-b) = 0
Y = 1,6
iEo Icl n r y dolntu 4rut.qfrlfl61[6] n ororr\rr tr r 0 1 -+n1 y [rJnnurarnr x lufialnuhlilhuAr vrJs
v. ctl'v 'lY, - . ,
fi atnrlduorrln [fl 1i[1in] X nlntnvl
4 tX +3 ) llnilhtilccrolnrilAe.rnu 4x-3Y +,14 = o --+ y = 3) - t4
nr'luf,amrrrrrluarbfiourqfloinr +
nt ",ri,u^ -,i I t nufiurr mI 1 %11.b I rnu X q *fr rfl r',[u a "', olrffio.t
'.r1n1flfl061 A (- 3, z I qfl f cior\UUlUr r-oruari*rit qat n
l; F(x,z) lrm F otruuitxortr 4lt-3y+14=0
iinunt 4X- 3(t)+1T=o
4X. -B -------, l. -L ----+ F(-2,2)
6arnr:nurlualfi0 ty - hl'= 4c (x * b) I xrrror:rnr
Irnuritqnuofri A(-l,z) ---+ h. -1, k=2
rr'truro^rt C = r:tr:urt{flh0olfi!q6-lI = -L-(-3tr= 1
rilo )=1 |= -rn qneinfis,-1,,,
,fio )=6*x= 1 roolarHo (1,6)
qrnknduonilq6rrio D(a,b) 1., a>o
lrite'' Dfro B(1)6)
ror*Blni A(*3,2) F[-2,t) nf,r 8t1,6)
A'F = l-?-(-3)l = l1l * Lorfr,iorla''ooufruL 2_- z ) - LolJFil= [ ::';"] = ti]
r[fr -lAF
J
FB= l;] lil = (1)(i)r(o)(+)-- 3
@ r1n is1tatcan)= 4 )uunderlddr\*r clet ro d.\ts-') ='r tl'
il det tBt) =
det cD'r)=
detr glI
EGrdet tcl = +
u1 dettn; = o+6+o +1-t = 2o
1^j ds{tctRn = dslcclde,ttRldeltcl= (t)(20)(4)= 3zo*
Z'n11n ah
lui bn
\; lirn
=L+4+6+.. +Zn
= f?n = Z tnttnlt, e h+1 -rl:+-+ 4-l=Lb, br " bnJ
2(,1 +L+3+.- {h) = 2[lInr:tl = n'+nLz) i Zn= n(h+l)i=i Z
=
osfl
=
= Zn'r Zn
lim(itlt6ao6 \ i=r bn'
ti",.(i ' ).-^ \ i-r h tn+L) /
lirn l./'3 * 3
n_d I\ 1-3 2-+
h1fr*l)(zpa11 , n(nrt)q --
z=tim (i f:r- thJoo \ ,=r 1!flll!l! /
_ nin+1 )tn+z)2,
!+ -+3.5
g
+ 3-rlnrn+zt i J
613'
tI
n(n+ 2 )
_1)n+t/
,1(1- ,\ n
f,*1,
*
B(a,b)
= i Ig l(i-u.(1 -x)-' (\-\).'\,)\"(il!.#I*(-X*#)]a r,
-a 2 nJil_1
n+1_ 1l
n+zJ
1 = 2-.2-5+
= itl o-o)
'4',t+rur1r46%,q:Afrrurh*ns,i\wftiuog , heu fruAr"f:r :
u'1, "i'1":,":ntn ( 1rn )
1nuinuftlfiu,urfi e [< nr",Aruiunrnrnfio rilufiouriu t t"darr]l"3r-'']ufir i )
!)A-l1 'r lz t lz,iofiunit"uurnlor z rrrz,tlfirtn I , R.l'- '{]rf 6i1h; z = E+bi
etllqnic'lfr zIa+bi +r\= Ia+br++lz l(a+tl+bi I = lta++l+!il
t-Rcr lm LBe-r lnr
2 .lTa* 1)'* b' = i cJ+ +F-i- ') ,nirxs dor
+(ta+t)2+ b') = (3++l'+bt {fi
4(il+21+r)+4b24?2+&+ o + 4b'
a a'+ sbz = 1z
22+ bl . 4
t,
uAilln to xl*rrln = S,o = LO\
i-20! t- 305 ------r#
i.r^il?n rc urid"Ll
q",[fil+t 1 fi4il?n 20
xt+ vl+ lt + b'atr fu+ 16 + b'
e-bi
m(+)'Ia\ b, ununr
G = Lx
tlnb 6-\
1fl1ifirf,o
2=r?l=
:
ar
q1n Sn =
1ri Sr, =
1,, 0,, aro
-Q tza,+(n-1)A)
frza.uqd) =20,
2n,+eJ, = 41 -Otqd- qd . Y1-41 ----+
J!tta,+s$d) = Xil++
5zo = T (zalrtqJ)= tto
23, + 19d = 'rl -@----+ d=g t'T'r+ntd'rarA ol0r= 1+
f rr+ 4 tbt-) = 55(BB) = 4t4o x
qrn @ tlTt% x. 1
1; g'rn , z+ c1 .2-1C
t'1,fl
tod = 3o
5+(3)) :
-* C = -1
@-o,lii Sss =
q1n f fin,,d,-,6nflutiniilnrfiuzfix=t dilfildli1 ftu=z u^: fltl=oq1n (f.gttxl = Zx +3 ---+ ftxl'9tx) - 9-X+3 - O
6?.1 frxrgixl* f'tr)g(x) . L -oi1n 1" tx) . 2- ------+ g'tx I = 2x + C -@trrnun"l f rtt,L,+'(1)=o 1u O
fttt g'ut r l'r1) g(1 ) = L
(2)9'(1) + g$tl = | ----+ }'rt)= 1
1ri 9'tx)- L/,-1 ------' g(x) = x2-x+D -oqrn O l,[Tlt,l X=1 f,ltgcrl. t-+3
2g(1) = 53(1) - 2.5
lrnunrt 9(1)=S'lu O : 3c1) " \-\+D =L'5 '-'-' D'1'srl"L\oi Jtxl= x--x+2.5
= (x - L)'- i+t.s= (x _ o,s )" + 2.zS
=Bgs i
l dahl*l,tlnnn0 5ro', 205+505 ='l1o
.5I tn n r trl nr r rh a r e cr or r{ or o it fi so aon 10. s, z.zo)
,'t1J')
oi,'ilr, nrr1116cr1a,rin6fro ?- Ls N
(stz+rt-srdt = [t'+st'
't"Lu
t1n
t;
111
Irt1
[cxt+ a )
P (t r
P ( t)I crl
y' !e
Ctotx utnai otnl"'1
= ['F.rtrd.t = r*'o c)
,t, x1s X'= t-3= 3 (r')'+ z+(xz) + +D
= 3(t-a)'+ z+(t-3)+40 = 3t'- lgttzr+ l+t*'12++o
= 3t'+ tt - s
l,rn Jprr) Jrt ) = J r,r,lJiux)L
- tt]', = x3+:Xt-sX
= J (tr+1?.-5)- (r. +1L-1;t) = = J? = J*,
f 'crl = li,h+o
ahx+ zh
Ptx+h+z) + Prh+z) - f(x+zt -?ctlht,?x+zl
P t h+x+zl - Prx+z) + Pch+zr - Ptzl
li, tJX+L)h-ro
X+z) - Pcx+z) Prh+rr -?rtt+ lirnh+ tr
lor h sjroa,r',ra-,edr.A'rr
Itirn,,,',u limh-ao
Itr*ht - frxr
{rn li mh+ cr
limh-+ o
)lirn Bxrz)h-ro
= 3X+2I,Jtlhil:lallr n
'hlrrnr+11r
=1
Ltn%
41n
1aIla
1,tcl
a1n
lAt
=tP'rx+z) + P'tzl
x " 0 t" @ '. P'(L) + ?rt2-t = 3(t])+L ---+
O trrilF'(D,1 '" ?'(X+L)+1 = 3X+Z ----+
A = I+z 1 \, A-L ". ?' [A) = 3(A-z)+ 1 =
PtAt = J P',nldn = I(3A-s)dA = 2gL-
Prol =l : Piol ,2r*ct)L=E(ol+C =1
Prtzt = 2(1L)L* j(1r-)+1 = it1+$) - 60+1.LL
h
P'(x+z) r- P'[t ) " 3x+z
-!x+z
Lr, 7=2Xt1
-oLP'r.Lt = L ----+ P,(L) =
P'cx+t) .3X11
3A-5
sR+C
----+ C = 1
= 216 -5q = 1By *
L1
Lr
'" X+ZY=U fn4
: Y = firX+Cufl; r riu L1
16i rYl, ff, = - 1
(-l)r, = -,fn' = L
{: --L
rirul fi-t
(l\ *1
(0, 1)
6-Xz )
urrlo,sio'r L,, riu t, ba'l ). '\2I
Y1= )z * --IXt3. 2x+ 1
i.^., -- r= trr,m.rx=1, ,)z.yr=
E
a = I to.t l(1 +2. u)XL
- (O.+)(3.6) = 1,$t
lllttrn-- = j ta-o.t)(?.6)
: (?.6)(2.6); t,.'l!
1,+++ 6,-16 , f,,,X
r! Et\s, 6 /
la Lr, )= zx+cLrnr+ nr qoiiru ( c', 1)
I = Lt0)+C
C= I
Lr, ). zx+1
L1
6
I= -lx+r
11 - lcr
L
hn.?nfrs2 = l * llho**
n.n -rlrrior
=k= k-t, k
= k-2, k-t,lr
" k-:, k-2,k-t,k= k- +, k-a, k-2, k-t, k= k-8, k-*, k-), k-2., k-t, k
= h-b, k-5,k-*, k-r, k-2, k-t, k
c, , "r)blrr+? 1n 1 L
? r'1,
flnlu )a
a
4d
d
a
d
I
FI{
k-a, k-2, k-i Ik-+,k-r,k-z I d k-
iack-I
)
b
b
d
a
B
d
1ln 0= {ra,bt 6AxAl
t,b e i1,2,t,...,k]frcr:odt o<b-t-<J
o< b-1('1 ! riufio
1o u der n n Islr.i u fi orrl'u
------+ cr < b- a n
, ,x+z( x')
1+LV.l
B rilul"um$ogFtdudu ( a,br
ocb-r-.1
b-a <-7
t-lSltr n
br,x'ri krutndn1uA trXl
l
>1 n b-<']+a#,0=k
= k-1 ,
= k-2 ,
= k-3 .,
= k-+ ,
= k-5.,= k-6,. k-y r
k-:., k-b,k-5,k-+,k-r, k-r, h-b, k-5,
3, +,5, b, J,8,12-,3,+r5rb,1)8
s dnilfrn r14 qdr+oirr
?-z + tk-tt[.1) " ] 1+ raf Y, ++k-c = loL
=obr = loporlob+c -O=cbc . looctlob+t -O
b 7 k rilu{r rJuil nr b froalrr A* Joi
l,ri fl,\,;'l .In
= h-r'
= h-"1
-9
Isl
b
b
b
b
b
b
b
b
b
x+z z(X+z) X+2
--* k= 106 4
r,'ilr X= 231 = 2clo + 30 + 1
. 1oo(L) + 4O(5) + 1
.{:lnl'nrn1+2+3+++5+btT
ld)
-- L$ flo}tou"v
r "do doln'- t't1
-llug) t=]) fl 0uouv
)?= 1+2-Z
1- jtx'-rl
l1n X
q1n Y
0-6'nr q:Iu
dtt[u1,6?1 b
L6i s
x-) = (100nnibk+c) - (100c ttlb+tr) = 100(a-c) -(a-c) = 19(?-c)
x-) : a1(a-c) dnrtrnfn oior\rl 2=9 trA: C.1 ( x-c=t l;?Jrndfi)
,11*rn'uo:\rfll6inlxilt t riu 1 rl^,rx',,,ar1orarIm*rrir,r
= {ltttqb1,q41,qg1,961,111, q81} rnrln'lug = Y(101)+ (u0+30+,..tro) = 665f4
1'l n --------f Y'_-+ y=
)=X----+Y=3Y = 3(qz) ------) 3Y =
L.Y= X nL
Lx
1+L7
L'L= )= 3
L
= -(X-1z
11n
11n
1;xr)+z = 16 ttr,filu,rrunr
X+ x'+ lcxt-11 = 16L
tz2-X+2-X, + X^l .5L.:X2+2X- 3i = Q
(3x+'tl)(x-3) = o
X = 3 ('." x liuirutulfirlrn)
z = ,cxt- r) )
Y
7
1{ln-l
6^l tx{xg
1
= 1 tt - t )Z
)(+) = 103
-lL-l
x
ldo'loi ] : 3
1vL5'l
16i
X)2. (3)(c
@ @: r,=x,,{ J a tl
'1,1 fl t1 u uA u t* rA or_l [cr1
loo'r ,). s = b5- I,
*
[[h%Fi1111
q1n X:,, =N4 x' + NrNz I ilru = 65;
Na+N2
6i = t4otc sol+ r :o )t fr)4U+ 30
c)
C : druzu nufrli
-b 4550, Zoo0+ 3O-Y,2 ----n l,r'- 85
n(nUBUC) = 1t
n(AnBl: 5 , n(A0c)=3nr0nc) _ n(Anc) + n (nnBnc)
A : "(rln 'l {r n r:dd
rfl::r n r",l r
rrnlqnd n(A)= ?, hlb), Y1(c).10, h(Bnc). 6
11n n(AUgu c) = htn)+ n(B) + nrc) - n(An0)
1b - g+10+10-5- 6- 3 + n(An0nc)ntAn0nC) = 1b-14 =l
-,
0.56-x lrndi# F(s)= o,3o
r-t (B)= 0,a6+(0.38-X)0,3o = Cr. ++ - X
dr.,fl t,rro', = o.tqX -- o. l+
@ h+uunr'rlu a,b,L ca<b<c)
rfir-rir 2=CI.r rL61: c=6r t-.'
qrn IX - Gr + oo + Qu = 195 ----+
1ln dd.fr rruqfiiu G3 -G, = Gr-G1
; zx) A=,N
= flrl+t) =
?1
Qr+ G1 135
g* -_ fl(x-il' . (g4-6,gtz+ (60-6s)2+(gl-65)1NJ
l\,,= z1g
6S 1.1 Z=
_ 65-i,o,t^1,t
S7-
x4
1'5 l
+ir=50
,L:
X. = B5 c. ,.\) = lL
rouJdx
nr.+ff rtludrir?nfiros dlmrrBut lnifir,+il 3 * 2* 1 = 6 nrr
-1, e rlz C<JLR0nnuuLL3FI1.s1lnflU 6 nUffl 6 ?i x
Ir A- B) = PtRr - Ptnn sl=x -N= 0. l[ x
llmirniuon Me = bo = b
ol.1+ &r = f,ft*fi-- I ) 6.1.r+. Og
Q,r+&. = 135 -O
=0,L 5=O,lXt
X.6s ,\= lt
$,JX1 = b5-Xr -| 'l.3X.,.OS
= lS= [53
3)0+O: 2Qc = 1[2
Q3 ' {81
rrn @' er = 5J|
_ tlt x
= 0.2 -----r Gr - G t -- Ltl - O
121+25,+ 2,5$ = W33
6Ir uru r otrfior dsl roulai n: rruuEn ilu drmortiru , - 2 fi,rJfr n,l,rufit"urfouldl'r]u x'1 -.11x[o'tqrn 7=
_L
-2+
X
x-x
x - t51z
x -r5
61
1.s1 tJll c1 fr:t[l+u
i= Zfx (I)t+0) + (12)tso) + ( 1't ) (30) + (LDtLo)
+0-t scl + bo + 2_O
Lr+60+s1+441+ = 183 -- 13+!1+ 1+
a+b+k= 1+14+1). LKX
nt-n-th2.
h+ 3 =b
m*l1+
4-d6 -
l,i (
1t[ln u
t+-!--h
An\-
l-
b"l-
I - nt+n-1 I
, on
Ll't+n- 1
nLn- t
,rllnn'n' rl"
nt+n- 1
.n=n2- n- 1
1bb1 =
15t1 i
Lh +h -(l"t+ari(
@ 11n 5t2 + rb
<$tlt0 t {1 n
n1[vl
,drc'10 i [fl0 n
tnr r
,do, rln x tiiuiru**1fir" drhi
fllrfisn b=3 1; r, 5q:s3.
cilrir* 2'-'1 , b= e, C =3
1; ?+b+c = 'r+3r3 = 10
r(8a+bC = 41b
= 41to - L
aa+b = 416-t7
2,b e Ir,L,... 1t trr'r]uiru:urfiu416-C e t* dta
'7
C=3 [nrl", +1b-1
,,,e+ b = 51
) = sg-bI
#rrlri te+b e t+ frtu/
J, dt- -. d"[c tilur']%tLi]01lt
n
(fr Xfi X+;)
9, 11, ll , .... ,t511thXL = A
I'l-16) = Q
(fl)nt+h - 1
n'-rt- r
1^t.:6 y
=413=511
sq-h e 1+8
56-v-t
x
dotr rrnh n 1
= ht+h-1
fri'lu
l'l = 2 ,9,$,)
+
)
x
n:rrlt1d aonl narr'(u )ti n14
5-710-1+
\s-11
ZCr - ?-t+
-?
a1
1"1
LL
+0
5o
36
Lo