More than Just Lines on a Map: Best Practices for U.S Bike Routes
Pat1 ตุลา55 พร้อมเฉลย(2)
1. :j'-
Yad
todounflietfllflet5 (PAT 1) q61nil
i.
nfisr PAT 1 ma1nil
q
1.
55. I
55
rirrauohi'u riluronnvrii'ufivrf uac'lri A, B uas c rflu#urryoaar u :iatqaiolildanoioo
(PAT1 or.o.55)
2. (A n B)-C=(A-C)n (B-C)
4. (AuB)-C=Av(B-C)
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)
2. (-pv-q)=(p=q)
4. [(pnq)=-q]+(p=q)
moil
oou
a
I
v{0'1Til11J 0n?''l
-,
1a
1Ja
oou
?a
U ,J
il0 a L!1*
n. ri'r p, q ttos r rilurJ:vlror.iToufi p = (q n r) frri'rorrruoitufluoi.o
q) (- p
ua"'r r
=[(p = ^
= r11 fieirn'lT ruailriluo?t
u €a
J
I r
?J. fl'llll.tot0nRilfl8JllYl1io0 {x e R x' < 2x + 3} [eJ0 -a n0LtoflA.:01U?1t01.]
R
I
ud'r :x 1:' + 6 : 33-' I firirqrmaroir riluoS.s
rialoei a't:Jdgnoioo 1eRr1 o.n.s5)
1. nnRuaytnn
g{
2. n nn uoi:r fio
q
+. nfioravsfio
g. nfiousirnn
{
4.
ritauotrd I ryrl.ufiora+irururfiru lri' a
trng
B={*. I ll*'
tt
L
={xerll2x+:l<l;
-*-rl, r} fior:rxrfonerilsialild
t )
n. ,iru'luarrfnta":tggr
u. A-grilulro'ir.l
ann
rvirffu 7
riatoeiot:JdqnoTau 1eRr1 0.o.55)
1. n {!n
uas ? an
g. nfiousisnn
2. n nn uEil fio
g
+. n fi'q uns I ficr
I
2. r'- .
2 {oaounfinnran{
s.
(pAT 1) Earnilss.
d'r R rilurryorrirqaura{aailfl'rr (x-2)xz+2 q(x-212x+ro rfio x > 2
ru#r R rilu#u
1.
rsosar?ir{ludolosiolild
2.
{2,3',)
(PAT1 sr.n.5s)
(3.5,5)
3.
(2.5,4)
4.
(4,7)
oou
6.
rirnuo'ld'R ttaurssrro,:*iru:uo?.: urnrilri'I rrflilrsnsa{,iru'rur6l ld'r
a1R R
tllril
n
Tosii
f(x + 5) =
*' - *'+
uor g-'12*- 1) = x+
{
rrns g
ria
rilufltfld'u
Zx arui:.lt1n'iruruo?.: x
q
ehuiurln'iruruo?.: x
fior:nnrioorrilsiotild
n. (f - gxo) < -169
fl. {x etl {s.rxx) + 5=o} rilulqfl'jrt
riotooiot:JdgnoYao 1eRr1 a.n.55)
nn
s. nfiorsiran
1.
2. n aR rroi:r fio
+. nfieruarrfio
R nn Lrn& 1
sl|.u
q
oou
oau
7.
a-
^
fl1fi1.[o[14
l,a
oau
1Ja
I
,-2-cl e
r=j (x,y)eIxl y= zx. -u I rfia t ururrnsot,iruruudru
x'+l
t
'iru:uerur8nfl o.itssl
1.2
8.
da
Dr
J
- R, rvirn-urioloeiahd
2.4
(pAT1 o.n.ss)
3.5
4.7
fiar:a*rriaer:rilsialild
n. cosf+"or$*.orr=1
tt.
eq
,*ff- t*+ = cosec*
, U,J
Y
t!flonoa$
1. n an o*r', un
g. nfioueirnn
{a
LoCIa
(PAT1 a.o.55)
2. n nn ursis fio
q
+. nfiqunsrfio
I
3. Yad
o[odouoolofnffetS (PAT 1) Ea"lnil
?ia
oou
tia
riatoeiatJdqnoiao (PAT1 er.n.55)
1. cos75' : (2 - rE)cosl5"
2. cosl0' + sin40" = cos20"
e. fir R rflu,irurua3.:'loq' uii': *31 cotA
+.
10.
3
sou
e.
55.
cos2A+cos4A
cos2A-cos4A
ri'r R uar B tiluo"'ru?ua3{'[01 ttd'? sin2a+cos2B = 2sin(A-B)cos(A+B)
*.r..* = arcsinl-
rYi
z,,,ccos-2 uii'r cot[4+arcr..*) rvirriurialosiot:Jd
JnJs(2)
(PAT1 sr.fl.ss)
1_
^ 13
.T
_13
g
4.
+'
3._13
16
11. Ltnutanro{?tiriluthu:ra.:r#r.lsr:lfirdarJ:cfi'jlnqosl-olto{l.:nnil
I
,
x- +
2
r,s
y + 6y-7
A
a
= 0 Ltflsly{ftflooilflfl0.:x.:50Ei!ilLEfuqr{
i
13
16
* y'=
25
riurtnnn
a
x+2,13=0 fliln1T10':?.]T
m:rn-uriolosialild (PAT1 n.n.55)
t. i*4y2-}x=O
3.4x2
2. x2+4y"*24y+20=0
+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)
4. (-3,0)
1. (-1,4)
2 (-2,0.5) 3. (0,5)
oou
13. ritauot#n
rilulelsr?ndfififiG
:*:
Lrou
ria
6au
?ro
de(A)+o fiar:rurrian'lrrusiatild
n. (de(A)3 = de(adj(A)
s. d'r a2 = 2A uii': det(A) = 2
q
I
e
, U ,3
datooia L:Jilo.nooo (PAT1 q.n.55)
{
1. nanLIEi?lan
qr{s
3. n fio uoi:r qn
2. nnnuei1fio
+. n fio uac I fio
4. ,
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:anrianrrueiotld
n. 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)
nqnras?Jnn
g. nfiquoitqn
1.
2. nqnueisfio
+. nfiouncsfier
oau
1s.
rirrauo'[r{
d,i
rlflv
t,
rirrrntEraf w rirruru
q
arcsinl
----
Jl
rrnrmaf't ud'rrirrot
1.6
rijurrntnofuu:yuru dt u*v-fr,=0,
n-uurntoof
I
unsl?ntqof
*
i'w=8
rirruru
'
uae
i'w=-2
,-*.rtr{
J,
n-u
II12 +Iv12 rvirn-urialosiat:Jd (PAT1 er.fi.ss)
2.10
3.14
4.18
osu
16. ritauor'lrinec
rlil B raer{rJ c
ria
rilug:.Jarruru6uruto1 ri'r a, b [rns c rflunr1ilur']go.oqirufl:.:?iraJe1iJ
nrrua{roi:-r
Ioufi
--l-.#= r.}*.
tio
A
ua'r sioc rvirn-urialosiat:Jd
(PAT1 fl.n.55)
n J,
t'22"'z
2. _!_
3. Ji
4.
1
sto!
17.
riT
rauold'2, ltau
'
n1?J0{
1.3
22
rilu'l"rurur8tdaudooond'arn-uaernr:
t,, l'' * rrr,
,tr)f!.!'l
li
,rr, ,z )
2.4
='
1io
oau
da
- g.+4 = 0
rvirriudalosiat:Jd lenrr o.n.ss)
3.5
4.6
6. 9t^d
6 {ofloilnoretfllfrer:
(PAT
l)
Ea1nil55.
2g. ritauolrinru#rult'u:f'rfirfi,:rifu:vra'ir.:riaruafiritauol#giot:Ji$tflut#uo:.:
q
x
1
2
3
4
5
v
3
4
b
7
10
, u ,l
?0'l:il11lan?1ila0
[]J1*
n. rirarunr:ra{flx1sJfs.Jfiuf,rfitfl.orifu:!fl,j,1{?iaila
:t. ti"t x = 15 ttti': y =26.4
rialodat:Jdqndao
1. n nn tt03 I
{.U'U
6o v= mx+ c rrii'r m + c Lyiln-ll2.6
(PAT1 er.n.55)
aR
2. n nn uroi :r fio
+. nfiouncsfio
3. nfiotusirnn
t
oou
ria
24. ornnr:aautrrofiosraCIfto.:rinGuu#'otradt :J:rng'jrn;uuusouattinGuufinr:
ua n uo r:J n 6 un c dril uo du;nttti'r fuIA.r :l n fi :vwir.: o fi t z o-r qrr:r r ol:J d
oi
z
0.5
1.0
1.5
2.O
1r.1l{Yl
0.192
0.341
0.433
o.477
ue
e
xe
,u,J
?la1TEU11J 0 0?1 il fl A L:J1{
s v
a
io
o
X
n. nlilnL:Ul,tnilfiU{ 1ilfiO.:UflA:J
3.1'l
r.
OTg'l
U
LtaiX 01 eJ1
#r n ; uuufl a uasl
u
d
q
e
X
n
a:g'l Ut
fi
udX,-
A
o fi 1 fl o
v v
{ Os
ttillAfl a U1 0'l un
o!ev,^{tuX,Lr6[i
[T
U
1.tO
UUtml
n
U -2
or
Sr o {#n 6s urio.: dfi rir eirs a urvir n-u 60 o c ruu un v fr
e
t
ilnr:u1l[1114o{uflauleiflsuuuriosn'ir 54
?ia{il
i
,
t
tOqlcLtUl*1AOUn?lO'l$'lUl[UilAHRA{tYl']fl0.1ff1l*tUU.:LllU
flxrilTsAm-iir a{n1:LrilTr.r1ar
a
flvr[1.[1*
<r
a
--
v
a
60riiufouns 15.9rto.:fin6EJu
a{osttuuflaLutytlnu
0.
1
rialooiat:Jdqnoiao 1enr1 m.n.55)
1. n nn
{g
L[or
fl nn
3. nfiouoiran
{
2. n nn ruoi:r fio
q
+. nfiounrqfio
OAU
v
?JO
7. tyad
55.
7
AEDJ
'[od0unoi9lfl1dst: (PAT 1) Ealnil
1JA
25. 'h"'u6l,iru:u 0,1,2,3 n":luorr-r.i +*+ siotjd (ria.:ar t ,iruru;
a
b
0
3
d
r'
i
UI
t
u
ao
d'or
f irururdru
ttm
At Ltn']A O.:3JO''11{?U[6'J O,
ugi
LO
nsn
v{alTil11
a-n
0 €r?'t rJ 0
0
1
1
c
2
2
3
1,2 ua3 3
o, 1, 2 lLfls 3
,U,J
a t:.lu
n. dha<crd'?b<d
t.
ri"r a
*n
ud'r c <
O
b<dud'rc<d
{, a+b=C+d
€t.
ci"'t
:iatooiat:Jdq:Jka"'qnoiao (PAT1 q.n.55)
1.
3.
n
-
as
: nnLY'IU.1 1
s
2. n -.: nnrfiEJ.s z lio
4, n -.: anrnria
gq
?J0
ql
du
n-{nnL?.lth:3flo
s
I
[-"ffi]
26.
rlrvuotri'u rilura 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
unr.iru':urrurfinso.:ugEr (A-B) u
lvirriurvirto
R
(B-A) rvirriu t t2
uit':dT
u:uam6n:rotrtgr A u B
(PAT1 o.n.sb)
stou
27.
td'
*.[**ro*0./:**r
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)
R
tflluttqriroouta{fliln1r
l;i.
=rn
CIlu
28. rirsa.: r..'(2u,.t*] * *.*f) rvirn-urvirlq
(PAT1 or.n.ss)
gou
8. 9rA.!
8 toflounorsrfllflslr
(PAT
l)
Eo1nil55.
zg. t# e rilu'i'ruruo3.:1o1d'r a uns o uilurirrJ'rnqo?Jo: .or4e-sina0 ltar
mudroi'u ud'r a * o rvirrirrvir'lo (PAT1 o.n.55)
3sin0+4cos0
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
31. d'rrrnnuufitfioogurinnr.:60
"
-. ,,
t
-
c1n, t1 oti:.lur#uor:.:
'
,
a
x+ y +4
'
X,
= 0 uava.:nalflr.iruQo
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* ... *
rirro.:
J3i[+
3s. riruuqld'z
rirra.:
* o]
*
*.. .. #l
ryirn-urvir'lo
on
eArl a.n.ss)
rilu,iru':ur6tdaufiaoon6'orriusrunr: 2lz + 1l = lz + 4l
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:J
q
rvirfi'r sos uai'rzuaurn 5s yrafu:nsotn'ro-::rnrnrflodryirriuryirto (pAT1 er.o.ss)
aou
tl
9. rToaounfinsran{ (pAT t) qlarnil
g7. ri'nauold'R rvr1.u{qsa+irurua3.: trdt:n J R uac g:R
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 =
ql
4. s'(x) = z ihuiurln'ilruruo3.: x
I
s5. 9'"
n rilufi':ri'fi'uloafi
1
9
q
du
flrfifu
a
I
@
;
v
d
t
u
r 0
g ilrirairqoaeJv,ryriryiln!tyi'r
Lo
(PAT1 o.n.55)
aou
38. druuo'lu-p(x)
fl11I0'J lim
x-->2
rilun4urrufiaaood'a.:n-r
tF(lr) -
r(g
rvirriurvirlo
P(x2 + 31= 3xo
* 24x2 +40 [L6t rix;=ir6yor
(PAT1 er.o.55)
qau
39. rirr uold'e1xy
uiiunuurru
loufi
e1o1 = 1 uacsaonii'0.:n":J
3hx + 2h
lim
h+0 P(x + h + 2) + P(h + 2) - P(x + 2)
-
P(2)
=r rirrat
P(12)
rvirriurvirlo (PAT1 q.fl.55)
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:![nuytfr'um:.:Luacu#uq:.i
x+ 2y = 6 rvirn-tifrqr:rtmilrtJ
(PAT1 m.o.55)
oltu
41. ritauold'n = {1,2,3, ..., k} rfia t< riiudrururfituan
rraclfi B={(a,b)eAxAl0<b-a<7} ri'r?0.: r rvirn-rrvir'lofirir1ri',i"rurueilr8nra.:tqo
mtivtlu
B
(PAT1 sl.n.55)
oou
42. ld'x umu,iruruarrno'n abc uas y uvtu,fruauerilua-n coa Iosfi a, b, c e {1,2,3,...,g}
rar a, b, c urondrrniun'oru, fir s rilurror.o * ti.ru, IqEifr x - y firi, urnaor
ruii':
zu
aurns
or
ailr8nturso s ?f{r
er
o ryir n-u
rvirlo
(PAT1 er.n.55)
CIolJ
42. ri'rx,yrtn;ztilu,irurutdarlrnfimoond'0.:n-ux+y+ z=16, y*'=xtBt')uas3v=3(9')
urii'rruanam a.: xyz rrirri'r..r
rvirto
(PATI o.n.55)
oou
10. Yad
tofloilniiletfllcrer: (PAT 1) qla1flil 55.
ilv dt
fi
r
ouqri orirt#au
44.
1
tlJilJ
rin Guun
fr
6i
9
o eru d.:
risi
avnuril uam8nr
11
o.rrilriletlnl: rtareJ:il fi wr utarru:n
ttarvr:J'i1
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
{
u
o
a&
,
,
^da
drururI.roano.tnfllzilyr,lfirJotyrln:JLvl't
q
Lo
(PAT1 o.n.55)
aou
45.
druuo A rnr B rilurnsrnr:nilo
0.30, P(AnB)=0.06 unc P(AuB)-(Ane)=0.:s
P(A - B) rvirn-rrvirlo (PAT1 sr.o.55)
tattuqnr:niE
ruri"lriruar
luutruuilaar:-Jt rnsdrruold' P(E) uvrun']'rrilravrilu
l
d'1 P(B) =
sE!-u 0.14
46.
va
;
'
rirn-r tgs
3A1U?UvlLtglnol':XJ1:lXJnUilrin5?ilt: d'rdoruarodfieireilsoru unr
! I
-,t^r,"vx,#ru:J:sRvrGla.:fid'urvird'r o0 [LdE o.z orardroiu uit'ao':rruuil:il:euto.:riorunsofrrvirnu
U1flAilfl
rvirlo
@
(PAT1 0.n.55)
CI_Et_lJ
47
.
0aa6uA
'Lunr:aa::itrqrrflflaraofror{nl5suoo.:ri'a.: :-Jmng'jr eirtadutarotfiorrotflsrtuuflau
ryirn-:-r
u
a
os orul*u rinu3uuli'0,:ur:nfr +o ou ri'a.:firorfirin6rJu go ou d'rnvul*ufla:Jra{
e
au
t
{
^
unL:u1.rra:u:nru#er:-l:vRmGra.:nr:uil:zuIuuvirn-:-r o.2u1?-r
6s
flrLr1r1.r
dt
v
a
y
ytv
n riiuti.ntTuutiotu:naor'Lt;l"'
6ouilurimrrqr:orurvirn-:.r 1.5 nruuudorsorfnGsud'atfimo.ofirieuifiu.:r:ru
rurn:orurvirn-:.r 12 nvLruu uunvu'ru
4
e.rrm:qrutvirn-r -z rrii'a uru
&
t
r
rflurinGuuli'orfioa.:aartd'octtuudorilu rir
mor'lsi'frflrLLu1* (PAT1 n.r].55)
oau
11. lYad
12 toflouootgtfl'rdgt: (PAT 1) 8a1nil55.
48. tunr:aauisr n fi osr aorSr a oin 6s u#uni
s
ru6n
:*rflifi o fi nrtuon uo.:
nsuldl,t
40
10-14
15-1S
50
30
-24
d
o'ruru (ou1
5-9
o-t
20
20
n'rnsuuurodurarnfimria.:nruuuarudrfiEJuluflJ
{b
lqgd u < b uoe u.i.il. fla{ a un; b lvirn-u
t+9
rda k, a
rn! o rflu'iru':ut6ruurn
1 u#reima.r k + a + O rvirrirrvir'ler
(PAT1 fl.n.55)
ctou
49.
rhrir n= 1,2,g,
qtnthururdsJurn
...
n
rirrauot#
drlrtf
un=r**-{ rrar o" =t-+-#
#ffi=rrr,
(PAT1 0.n.55)
osu
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)
qou
12. rilau PAT
A
-
t
5s n A
n.n.
(A-B) U (A-C)
tBUC)
-(Buct
A
(A-D) U [A-C)
#
rio
(ANB)
_
Ln-cl n Io-c)
C
(Aft
2.
A
I fin
- (B-c)
s)-C = (fr-c) n (0-C)
n t B'n c')
A-(B-t)r'An(B'nc';
"ris
A u tB-t)
IAUB).L
(AUB)-C +
to-t)
Au
+
Ya
1J0
dtrJ
O
efin
ioB
4
'.lfl
nor-r
q,n',1# U""-U",.*
n""1'
fr^uioni filurl,,^qiufr ,,r,rro',Jrrrohruo frranr nl',e.,cit'lr,fro',trr
'1ur3*
'.,
,',1,*,fi q riil ur nt mr *.illuq-:r
6ul-or rjtrnlillv,nl
rr ar nrqdo.r{ r ilr,*,rlru?lr.rfir iriudr raiol*,l,i
^al
lfr,lui ,i*fi, irlu fila{u}'
t?
iou sr or n:'o'ii i r fi Fl {o{et ti 6 r #ori
'uotrl *nnrl cracr trA-1Aloir { r,?n do{nn 6r r,firri fr rtin tr{n: i', dr,.^orid1oiln",ru"riu
",,nol"o*,^.',
'
(P-q)-+(1-P)
ds1
{o
(-P v ^x) + tl -t ar)
L
_/ F'-
't'(P
TF
Ar
FTTF
.
't-cf
J
q
rrrrfr ,lnr^olrili
fi urfl
q
t.
P
F
T
F
I
FTF
/r-t/
Y
T
/
-=F
^
'^-
"iri{orr{r 4tr,,fixirfrr
(p--+n)
q)
,=T
Y
q;nrh;
tl
utl lii d
uil r, rfi
r
6,1
/
"
-t
a,r:-;-?-'
bu"fntlul 4lr,rlutr nt
4,t
/
P=
r, n=
13. $03
L(Pn^%) -+
{s+
- P 1 --+ (P-q, )
t(pnc1l
F ----
TF
-
+ (?-X)
7
-----
-,1,1
TF
F- -, C-B
d-, "" i,,rtr*r/
'fi,.-=-l=7
lri'ilrlnill
rinu6.r
v j
d,
C ou
t r nr'l fr rjr
I
toultltI dlulufli
I
I
Tti.k
TT
fi,i,Jrrruqrl Yv//vz -+
I
I
I
lN
?tr,-umarfi{fi
o,i
.
^
P=t
,%=Y
Ii rilu lfi q
frrrlfr N:-r
rxAr01r1tf,ounr'rxniluflrf,Y"oi
, W=
frrloinr nirruqir.us,r]rrnririoz qrnnr:rlr Na r r,Lir lfiutntmrtrlaqtnila:H1r|uuu
r
ItJ
11
t
unufu V/////A n
.:- TTltlt) =- T
1
Vnn =
IE
1ts
1
2-
v
"Uo-}
1ro +
(P+
cl,)
-
!.
(1+ P )
n)
11n
rr'r
(t+r=r)
:T)
d
nrilfi rljuqilotiirfrua ( F +
F
:
:
t'lr,r,iluXrt*ir*Jr r
T
= (-TV-F) = (FvTl = T
uhtr Ltpn-q,))^? J= ltrnr)-+r] = T*>F
tJrlriuilrnixfi t
=
r
tin{ LtpAX)-+^1i; L(-rnr)-+Tl: F+T: T
L
ril.r*flrilludrl
tluuiludruir Al
3 *
rnrr*r4ou 6rcr
p+
A
/t
(P+r)
q,filr*q'i,16if,r,oifio
i,'liuq't nd;1lii1 [P ))= T , (P )Y)=r frrfl
nnoitfinnr3olfrl4 n riu t'r.lr*l'ur11nfil
rdrlfrtautratrounrJtttn$11 qlti q)1) A (P+r)
nt,.rurdtdrrrair,rilorru,o v uJiuHrilu n hrfi
(t^r I nri11s -+
rrmfir
(p^n)+r
'--)
(p+r)YfP4r)
A ulHuxliuV
:d,fi",,irioudrroido f,',1#',lr,rqrjtfr'u.,^l*?.fiJun1lfro
4
dt
I ,
tllul* lrju [n c
(- PV -%)
p, (XAr) - tprl)
Note:
I
trnx
TF
6-101-l
1 v
X=T, P=F nilrllu (Prq,l= F-+r;
tF
Lfpnl)+-q,l -+ (?*q,)
^4
dd
i,rnld
TF
t(pn-11 -+-Pl'-e P'a q,)
, ,n J
dIil'l)oil
LL6or{nio
F
TF
(^pv-q,)-lpaau) i
G
"
tiAnl it t nn nl6r4rlnl[u uI
I -xvx zT I
I
T/X A = % I -Xnx = f
I
.
F V m = Wh I X--x z *X i
rAW)z I lr-r-r=F
i
lriui{r- TV m = f
I
i
urjnr^enra
XeyE-XvI
r
]'#,Jt,^q,ifidnrnntxflitt',]u,fl.
d'oP
merkrraiudrill?rrn
-Tv@zt
-T4@'-@
-XVX;T
1A la = @
I -t (- p -+ r) J
( =
u ='wrv(-P+r)
=-FV(Pv17
= (-rV r) V
P
= Tvl = O
nluuR
nn,frett
14. 1) l1n LL-- [xeK I x1-. Zltl]
[xeFl x'1-Lx-:l<o] = ixern l(x-:)m+r)-<o] = {xetR [-t-<x<3]
r x
1- -, -f
?tn?sfrlqniroxnlfin'inlt:rl-lrro: ]xIr'+ [= )3-x ] tilo x t w. { - r-< x-<3 , xe RI
q1n ox+b = 3i-'d
i6i A+b: +
It
dt,-{
[1.X1J]1
=
{
J 7
1€ (.l
3{=A
,a ----+ tAt?)tA-3):0 --+ ff "3x=fi,1 .+ X"l
d,rdr* {s l. nnolol
lx[:xt!'33-]l = t
3x7o lnreR
At+gA -L1
--b
fill]lL]
,lve'
daillouotsu 1. x
fiqrtnn A : l1n lZx-t..tl<1
Nq1:6r{r
-1< Ix+'1 -<1
9-l .< 2-x < 1--l
iiun
qrn ll'- x - r
B
x'- x-1 z 1
[il1+
XL-x-1 < x -x <o
x(x- 1) < o
u
x'-x-L 7a
1
2-
u
()(-z)(x+1)70
-16 {ZX< L
71
I
U
- g-<X-(1
tA A=t-r,-.l )-6,-s, ,0,1]
''t
"
[ 6'l
t,
t,fl
-f,-I, -b,-s,-1 r-3,'Lj
A-0 = t-1,"0,1r2J / O
An g = t
-
d
l,
rilit0u010u
2-
< 2n i:dtrlir rn <
ant < a' q:6rdir m 7
^
n
ncrolio
n
firrrfio
lX- 1- )
1) cr <I-2<"1 --+
2<
l*ifrir rltz
(x- +)u+2) )o
nr fl6il1st o 6Nn1tflfr
v <
orotrlllt
0,[l
Z
qrI dir
l
-, 5>Z
1v
bn
--r x>3
xL+L< L/,+10
*'- r-x - 5 <o
[X- 4)r X+ t) < o
-2<x<4 n
11
<x<+1
f
rn)=
X
>3
"/
13,4)
6rrj{0il n or
d*lrria riurnufllidu?urroi.(ir
2_< 5
n r6B
x- L>1
z)
2 <X<3
3<X<3 "
5
5z
lt
3 ). (*)
l, e-)
o
<3
3
rtu
o<a<1
g
flfr (3,+) c (2.5,+)
I
n
L
ldx
1
x<3
> zx+lo
16i ,,<-L u I>+
?>
zX+lO
xL-?-x-B >o
r"fer
1r
x
x'*2
od:rnr: (x- z)" ' '
fi
n" nn
'!
110
J
a-
iiriaiu
+ h(AnB), Y
s X" sJfl
y<
3
x
frr f -'( a, r = E
r'l0iir
{1n g-'tzx-1)= x++ fJsiir $tx+rr)"2-X-1
n) tf-grror = fro)-g(o)
= [t-s )3- t-s)"+
2-(-5)]
- [z(-+) - 11
. ---
(-125 - 7E -10) - (-8-1)
(-lto) - (-q)
-160 +1 = -1E1 /
-111
a
-Y'rftol 11nn1rt!n.,{ X= -5
lu {rx+s) " x1-t'+zx
- 1r1 9 (o) ilnfh,rrnu X. - +
q
l,u 3(x++)=Zx-1
-1/c
., 110 n.
,J6-l
15. cU
)
dxnrt
A0 t111 g'rr oror:rol
t3"f ltxl + 5 = 0
r
g(t(x)l - -5
lv
l,rnnitcrn gcx+r) = L-1 titrrnu |= - 2- 1;1,6-1 llL), Lr-L) -1 ,
(ttn 39)=-s IrA:. qrftx))" -5 )
loi{r ftx: = |
'1
filtna',it qrn f Ix+st " xl- x'+Lx #r,tnu X=1 t:[n fcb),1_1+z=
1r,
r
[n21 T(x) = L tfio I.
u
-5
J
ry
-
'
b
d:rlir gcftxrt- -s otrrlrinuq""i x:6
[oir txel l rg,frcxt+5 =o J / A
dtiJjrdo n, rrfi: fl, f;tu-r irBnuu + I
dil,i s{u nm l'r] uqil
fr
,'. do rr" -2n
4a
qin
2L1
2x-s
LX-+7-1A _ 2(Ii1)-10
)= -:-=X"+
Xt+
1
{tn
l-=
rfixir y
t
e
X,Y)
-'1 ,d,
e 1 bn n
1x
I
X"+
1
Lt'- z I
X'+1 J
lv=
.{
fior[]B
L
1o
-
x'+
1O
_-=-
X'+
1
lfiuir
x,Y e
ri"ilfio
t-l
1
1
I
€l
*
1 0 ?rt dr il Xl+ 1 al oir
16i '+1 = l)?-)s,10 ---+ L=or1,4,9 ----' f,: o,t1,tZ. )13
vrio xL+1: -'l ,-L,-5,-10 ----n x'=-zr-3,-1,-11 ---+ liflrirx'fru'r]uiruluflll
10
cir X=o '--) ) = z- O+1 -- 2-- 1a - -Y ti{lrlr'r Dr = 1o,1,-1, L,-L,t,^3}
'ilwtnulu n1'reJi
1
gir X=!1
----.
):
'
fli )(=12--+ ):
,
f,r
@
n.)
X=
jj--+ t = Z- rc = t-1 .
1+1
rcrsaal [rd:
-,r
otur+
coso
1
drrJ
riuderuu
cas!. + co5 3tr
b57
* 9
-
*
<oos?.u
lnuoernnrrrl
ve
,'. 10 n, gJ6-l
lruuou
,.o
sinacosts- CosAsinB
----*
sin
rrd:
1,. 6ttn
'" (H
f;
cos]{ cosf,
sr-S =
sin I
1
'ilS {
f,od
"0, 3
sin
dtrl{rio n fin
"
L
:inIIcot{-cosf
c sin ti-i51
cos?! c ei6(i-?)
cosl.I
x
3
tanP tan* - sinH sin9f,
1b
B cusry
1bn ;G_
= six ({-e )
t-.
;
firrnerir
r
ilnxrTn a frt
I
cosl +cc,slI = -1 -cos1 = l-(-1)=
-5
7
5
L -
fl.)
R.
++1
cost *cos f + cosr =
/o{
= {_q, _l .,o,x}
1; Dr-Br " -t,L,-L,3)
2_-|0 = 2-5 = -3
1+1
2__ 10 = L_?_ _- o
D
{(
sin
cosuc
{
({
H)
da t,
=
sln (a-B)
_ 3T
zl
o,in
FJN
(i "#)
16. 1)
(2--
c05 15
-
c0:-1:"
cos
oiolfrqr:n*l,lt !os'ts" = Z_fr
G ) cCI515o
sintto'.cos
15"
coeo. sin(lci- e)
rE"t
1so
_ tin 15' .
co9
raislr.
cc,g 150
tan
r
15'
1E"
tane=
t tnR - tang
t + tana trn B
tancA-B),
,'_ g
,3
l::i:_;,'j.,)
(3-Gl'- q-6J3+3
E_ JT
s+il
=
t+E
t + tantE't anld
H
= Z-fi
.'- 1,o n" nn
1-3
3
,1
t-)
Coslcr'+
sin$" =
oos2o"
oiosfieumii sin +ri = co,szd * costd
+a1oIo
rir ros {:floRrilouf rfio O *rrn?u
cc.rsto' > oostc, --+ Ccrslo'- custd < o
ctnntrhlrrXuir
, qJri
tr,or sin +r)' >
-,
O
s
in +o'
r'
@o
c
r,s Ld
( 0< 6 < ttt")
- fos 10'
.l)
y+
1ld "il,
0J6^l
Vo
fld$ A = 0o cot d rarnilulo5 rilrlt*.ferdhir.turil hrralu A 1n1
+)
,'
dild A=B=3d
-
! *i
rin2-R+ LorLb = sin 6o'+ cos od =
=
.'. $0 n. ,J6-l
ry
- zsih(A -Bl crsrA+b) . ?-5in f f d-$l cos (irj+:rjt = Zsindcc,s6rj
a&*it :inzA + cosLh / zsin rA-B) cos tA + Dl &rrair n, B1ar1
drtl{o?r
1;
I
Jrr
------r 5thA
9= ?rccos I
---+
J5
Tordl,.i
1A1 cot
2rcsec
([
X
.
Cos
Jtr
cOS
A
4
=
.l
-----+ >[nD
E
5
=
----+
Jf
zgl
-
tantR_*r_ lann_tanB
r+tan+fins
i+l
I
Ccrs
-4
- qd : --2
I
=_
r
:_
ten tA- zB) =
-
x
tan tA- b) tan g
2t
1+q
9z
-
=o
. I
- tln Itn-s) - B
bans
_
tg
q-1
1
1
Cosqd
sinqd
+
,-qQ)
d:r-liod oro.,
tanE "
?-B
)t
(A -28)
tan rA-o)
J
8
l
2
sinro" Sin(A-lB)
* s in LA -z{b)
L_
rl
fr16-l
"o!
cosrqo"+(A-21)l
= sin Lto+ (A-28)l t I:l::'.l: :l;ffi;:;:lll:
ninl;'c
+z
tanA=
s
+
G
= A-
- zor.ros
cosro" cos r n-
.
tr
t
=
",1
trcs,n
+ Xrcsec x )
B
I
=
,,'. do
x
il
mou 1 x
A= rrcsin
r tor(j) . cr
l3
- lc _ /13tx
-
!
12
13
V{)8)= n
I
17. ut16'roi","uolns'.r+l xt+ 1'= 25
@
d
t
---+
tr,
qfl fr 6-l::rair nat^ldo (+, - 3) 1t61". ( - *,
-v
l=
I
I
+
(u,-,
-+,-3)
(4)-
t-dd
rrJiofl
,l "
LUI
zx) 3(+-
o ,2x
JJ'1-
3.3"
:
3A
tt ou ora
+
3A2+
qLi--x ) = z4T
3t
E1n
I
----+
X= 1f
?'
--+ Xt+ +yt* z+y
b'
+ zcr = 6
orol L x
(l+2x)
dtrl
= 15-(-31-= 1b
-3)
x'*
16i 1u+tY+s1' = I
38r = A
irft1,6
3
-z
Inu urf,unu?trhj rrn u'louosn 6untrils
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
-Jd
-Z
r{ oa
X
lc.rrlutjnal (0, - 3 t --* film:tfr fro r- * (Y+31 =1
-2Ja
I
-fr
al
'
L
T + Y +6Y -Y = 0
2t'L
v - _a --f )(-' t6-)
l-
t -,
fo ,25i+ 6Y-'l = o
Y
iIA:
8t
+
6l
=
A
u
1
,flor"ifdsrifi
L't-*
A
oAL- zz+A *il = o
()A - 1)(A - Et) = o -r
L++
=
3'*
Bl =
tl
I
=)
J
=
+ x, -1
1; r'^.3-',3u
L4+
A
I
)L
d,
l^u11 t-;,2.} c (-t,+)
2?-+
I
10
11nn
x
r io no
tdj A
=
det
irfl: dettrnl=
r*l R-'
chdettnt
)
,et
CI
cf
il,dt n n
n,) qln adjn= dst(A)A-l
take dtl rl,rolirqloi de,ttadinr = deiIdeltnrn-']
dctrftlrtrn.,nrfi - ,_-__-i = tde,ttnll?clettn-'l = (det rRr)'
lJ,
tl.rr
)
delorlfi
lxn: dettrr# o +
q1n
A'=
teke
rn
c rUu
0
n'rnrn
I
-q
-
cTuor ta mr
6
n'f
y<
1rs n. iJfl
?-A
!
det
v
det tL n)
{Y
ril6txl1l tfrl
tor dei R u1r
= Lt de+tA)
1/
,<
l,fi
fl4061
r deI tnr
=
10 1. o,lfli
8
dr,lioil r',ru + x
'fl 1 q {'r 3r xJ fl 0 { il={ bx ru 1 n u n', r
trt/c$
ailn1r10qlnfln0
$x+11 >rb
---+
LX'l1to
--1'
0()-.X
-t
Y
Y
>t -Lx
noruro"toi
e
el o{ or r ""v
i t-l r'k uel
a
I
Atf , 9rr,'b(tct,to),
J
>,'
Lx'
10
i6'l
X,) >tA [r^? 1<X
A
B
y( ... r,rnr'lfinril'
rr
>t
... trlut{un:}l
D
d,.
nllus P:
Nnfrl
tL 6r/
1 '3
(10,
16rA
,
lt
J
I
fito1]
'L
3(lo)+ +(lo) .'10
mih
max
P.3(S)++(o).15
?.9(z't#4(o)-9
10
9r,9) dflfr
"
dlU11"uo1.[
P
(5) o)
(3,0)
t€!
(
I = a(!)++(9,)' r
10 )
vnTil1 P alrnAnno
3k + SY
'u0
F il nr drrAfi
I
*
nqn
rr,o1 I- ) f 3
a1
10 1. hrl
18. l]n
U +V
-
,0
A
Y
q
il
[r'r u+v = w
.
!
!
dot il nr {ot'irlrrotAHnii
v.raj
u.h,l
,JJIJ
ilTlhnl
+ '
=
^,1
,
ur,'H = 8
v -hl
V'
Y'',1
-Z
"
J2
ll,fl: T'W
lhl
=
sQ
lNt
---+ lN.l
d r
J4
V.
-)JJJJ
q';!114m1 X-y=
=l;
I
Tt1}.lil nU
1
lXll)l
€
v
ver
y
8-
L
I
2
{l
JJI:
n = lvltNt
otrlr
-L
,J
CC'S Q
cll
crrs
(lr -arcaiht )
js
. tittGl (- cos tarcctis-r ))
G
= tY l(iu.r(- * /) -__f li I = jz
r 13
_L
J
,
tl fixilei'uor Iin1fi0511n1
fr i 1
IrA:n [fi An0un0
A?-
lhl-= 6
el
'
n"-[rt
JJ
q ?1
m = til,rrilrcc,i(aresin*)
8 = rd,rtfi)eos(Arccosg )
,-.
J
g = ril.rtJT,(E) J5
rdr=
'1
JJ
g+t-Z) "
JZ
rnfi 1qn+r ir',rrr* nll;r
--rLJri
til. t-+
1.{}+
ti l' = *'+
[.lL
lt
=
1ts
dril{oioror + x
*
Noto
Al'csin)t
:
=
t lc cos
X
e
gin ( 0rcsin
cos(1I- 0)
@
11
_+ _
A+c b+c
qrn
( -l-<xs l)
(-l-<x< r)
tFcsinJI-x'
0Fccos
|
- xa
X):X )
: -c059
X) 3 x
sin [n-6) . sine
cos ( arccos
)
(-l<x<t,
d+b+c
I (a+b+cte= a'+b'+ct+ z(eb+bc+ac)
btc + a+c
(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)
1
?'+ bo+cr + zab+ 2ac+ zbc
+ ac+ bcr
a'+b'+zcz+ zab+ *c
=3cx+
ct = lt+bt- ab
-o
jr c'= ?',+ b'- zeb cosc
11nn{no31n+ifr
rur O-O', a- -tb*zabcosd
abtt - 2cosd, ) = o
*.dr
,i,u,
etn
{Jfr
a,
1
nL
6trl ioil
.{r
b
filufr',r-rtsl frruly*ar?
-2cos
nsu
3
cz
t+o
d +-g ----' cos d =
1
= 3tbt sic t 3bc + acz
oab*,+ac
-o
[[1t1,50 l-t
c=
60
slnC= sinod
.E
L
.
R,rnt ir rtndrnrr
1t
fitnol[rn16 0=b=
C=I
3*c
-+-= b+c +8+b+c I + 1=1 dlnr:rilut'rt
1+l=
qJd 1+1 1+1 =31* ?.r1
''*A
z'z 3
,iura:rluneruil A rgc itl" a frruunr lunruunnilr.r = 6d
{'111
oilrrlu sinA = sihB : sind = sin6d = g
zX,
19. .l
q1n 7-gZ+$= 0
lfr
i;
lzrl'=
=
lzn r"
,--+
rz,l'+ r2,r'
3t
Z
2( l)
= 9*r ++
z
(1r)'* ( J;
1
=
_ 3t
32- +U)(+J
)
gtfii
J_Y
Z
2
I la+btl.la-bil. JEu-"
+
4*4 =$
nrrlifrir 6untt o t('* bx * Q= O t{nUanntOrOU " - ! tun flrB ntaror = !
a'a
a
q1n z'-szt4.o lrizuo uln n 'lr1ou -- 7.+ 7, = -!]= 3 , naqorntfl0ll = zlLz
trn
1
oiluur 1*1 - 7t+zz
21
zrz
z,
(lz,lL+ tz,t,
lsi
1
+
i. L t- (n1i1 =
ror +x
drr]{ofl
L
'1
t
o
3n= sin(nn-I.)-cosnr = (sin
11n
)=+
3
=
t
=
nnrf-
- c05n1[
cosnrs
- ZC05nT
=
;h= l)2,.
Notc: Cos(2,nt-O)= cos g Sirulu fr =r,L,r,...
Yv
In
bn= Lco5(znr-{) EcosI = t(1)=g , =r'2''
3
drraiunn n= 1,L,3,
bxru 'ir bn= r
r'r
=
..
lr
!J6i
- 2 CCrS lI _ -21-1) -3
=
b,
33
L
/ 3r
(u.i
/
l br/
L
-jf
i
/-Zcoslnfs/
=
titolrtn 2r
r""loo ?u=
-2CosZT
)
I
z , Io
t
3F
=
1.r$
t:f
olrnnhJfirLb
I
)
-
) = (i)
z3
bn
*(Z )" (:).
3
z
2
2
3
-,rt'
3 I - [3/
fi rr otuu', siu
n
/
darjlo i r t.- 3n
r.,ro
t -z(-r)i
3
r_fu]
( u )=
=
1
-L
1l,
-
d
drillou6'10u 3
li'rr n?urr fir};
{,16i l,, = * (
6o',r',
=)
2
r=
{l
!z
x
r
i(ruiru ullu
oirnourarirrarl r
"
(:l-
tril
r,^tJ;
a, =
)
1,1, er rilu nrflguflordarnrt f rx I = o uunn0 I- 3, X-A x-lr
r
r1l" ui rrjr'" oil 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 +{x -t
tl61
-
<J.Sr
4
)
n
zuu
a
X3- zbx'* bx
-21r,
=
xu- (
^
*
gr+
tr ) xz+ ( 1j* arr + a')
x
-a
fi uufiu.J,tfini {''rna:mr r 11fr
x>
,.. x
7b
, b
. I + !r + ef
L"L
= 3+.a'k+
Y
> z1L = a3 +
16i f tx)
t'1fi
t't1)
2_6
=
=
=
dtrl{oE *o,r
LUz
a'
X=
6+ |+ 6r
a(i+r)
l*r = lj'
r
3
6
x3- ztxt+ lgrrx
-
axl- szx r 156
3- EL+ 156
10'7
--
L*
=
L1L
--+ F=3
L6
3
=Cl
:5
+
6
b2t3) +
(j*r*
'(:)
=
r)
( l1)(13)
,
15b
a
21. n
)
drnr:nfiauLiir Mo-
I-
larat
61,
x
fxrfrlilrinfrrr?rnurlfrorurr*u,'r.lu
qtniuonir x ilosnir M, sadounrtot s
r
X= z5
ncr
x-r
)( = 25
!r
v
A=0.5-o.151 = O.341
)
rrqn
n1?
rr*
s6
dnB
bo c:Jfrir
<
-1 =
fitR=0,3a1 ,Z=1
5-
5+
rr,qr
11 =-2 =l
--z
.S
)
dr
rtrj ir n r ulu
54s
Mo=
X
.'" ?o
n
f,d
nn
IJ
6 n'r: trd:el1*
s
60
x
b
a
G
bo
6o
etY
iul{?110 n. llfl"
1,
@
I
b
3
2
a
7
z
U
1
d
3
C
Z
I
b
x
ulxlllu[8.i
o, 3 tra'.2,
..,')0
= o,1
frrd{oioror I x
nnflgi
-J,
-'t
1
c
L d
J
2
o
1
,i
L
o
1
J
1
fi,lfiu,a, I
:i *
-l'1 ,
0'tr:tln [fl fl ouu no
Jt
6
dt,t
b
1
L
|1ru
2
',l
01
I{
S1
',l
c
Z
4
0
1
J
cl 6l
1]nn:$n
d,
v
br
d
,'
b
ulf;r
c<
d
.'" 1J0 'lt, nn
d. rtlr c<d
a+t = c+d
.'" 'uo n, nn
1
n1
1
[1' ]+?
1
n1 a >
-ri
{1n n[6]bn
1
r'.
r4
qlnnl
rrhr
bu)+?
st_
lli
1.r? 1
d,
2 nr6I+ iuu?'t
drrl{oflmor
",x
n1
t0
?<d
z
41n ntr).tT
1
* b"C =O
' frr 3=3
) frr l=o + [=6=J
1
2
q!flotrilurA"t}
0 uto s tnlhu
o
L
dtadr
nlo d
I
a!,d
1
o, b, c
tnnuX
b<
n.
0.J0^t
v
v
.".lJ0
$.
nn
r
d=3 +
d=o
.la
nrorh
I
-lnrdfiz
lJ
nn
I
22. 1i^ n(AnB) = x
,
frs X -> loo1 1Jo3A fis 5x -? p11n1 = EX
> 2s'/. 1J03 a 6o x I 1oo7" rrol g ''no +x - h(D)= 4x
t-o"l ltos n
11n n((A-B)U(0-A)) = 112Intnl- nrAnBl] + [ n(B) - n(AnB)] =
117
- X) + [ +x - x ) :- 112
= 112- --t X = lt
hthU0). ntA) + n(6) -n(Anb) = 5X+tX- X = 8X = $(1b) =
( 5x
YIX
6lt}{l+
oiorinsuit
"r
A
.
i n+s4zlAs
l
({r;*JJ)*(Jrx+r+
zGxtr
:
1,"'i,
1"
bfri
A=
JTI = 1t
3X+1=25
5
I
x"{*s 1; A'
1*
AuE
trclan
A" {g}
X=E
)1gx+s ---r 'l-A', zxl-(, x r10
Iz)t'-6x+10) +1+L Jr-rx,r = Ls
nin^1L.(t
L-AL+zh -L+ = O
(A+A)(A-3) =O
-r-) A= 3
z = lx'-3x*
=
5
I = xt-3x+5 +
rirriu
1q
1+-3-'-1 ='lo
Jaz'tr
lr; A.
*
ln*G
=
+z.lt3xi1)tt) + Jrnx+1)+(e)+zJtlx+zltq) =
(1)
Jtu+1)+
1ZE
i'- llrr 4 'ct
. { s,4, - 1 J arnurn6alr?nr"r]u u
l = ,1 -)
1,; *ctan{=o -
tanA
I
'
+ " Qr-1
-rJ
X
* tan (Xr))
3
=
r
tans-- 'l
1
rioliir rec'o= 1+tan'o
1rrnl = secz(zA+ts)
=
1
1
- tanxtany
+b anLrzR+0)
tanzA
tan zn + tan
-(
.l-tanzAtan
3* { 12
+Yl
1+
,
12
t+
(f,xfl)
1
=
1
1- (i)t+),}
1+
z
=2 x
z(j )
tanR
- tan'a
=
l-
(rr)'
a
=_
+
23. -a<asinSl
nrruialilr+{oildo
trA
;
n
x'*
A-
C,os'0
Y' <
-:in+B
=
=
.
Bstno+40050
CoS
-+ - 1 < C0Sl0 -< 1 rirlrnfin
Zb
txizx - tt +
x logr-, (x'-t zx
(ir'-x)tlo9r(x1+Lx-1) +
-
1
-x'
Lx
)
(})t, - L) (log, 4tx1zr-u)
q6^r C th,
z)
L o-r
l4xo
oqrrulfruerm y
s
A
"
=
ln
rln:
nsrrarnloidaLnrt 'lX -3Y +
21 .
Bt-z,s)
{7
}t,n. AA0r .
CD
[o hB
! ng.co
*
d.= lAa+Bb+cl
'Ax+BY*Cco
- I rt-r) - 3(3) +21 |
i+1+ q
.
J
(I,.Il ) t
Btxr ry.l
bc =
r
inuh
Rr
J;ffi.
2:
z
) il def 1.on',,virfiri61frrArrrur
axs firu?otn . ri.lld
t,srar dei ",ir., ,,irrr..,* z l:1frflufi n
-5 -L
w,n.
A
h0c
=
J-st
-z s rl
1
-2s+1+ - L +35 ttS
1+, 5
z
-1
a
1
1
1
21
t-21s7'*l5lJl' = J;r
. j,'tr',(ffi)=
Sol z
ldtur,zuninir
O
)
,',.(e't'
n)'-'
.
AB:
" 6-c
(-s -h)'+ (-z+hr+)' = j t-r-hl"+ [r+h++7'
tr+hr'* th.zr' = (N D'+ (h+t)'
urArnr: tkurrr il u'.tr*1ri A ( -s,- z) tr'" B t-t,s)
[6-]
)
'+roh+zs
= '+1$h+rr
rh : - g6 -+ h' -t ) k= -t-.])-t =3
loirn c r -r,l)
A(-5,-1)
A(-5)-2-)
- [t o I -t]
', Zi
Z
B = t r', (rrl', (-9.)' = I n,:r,T
J
rntlnfiufin'lu B = 4+? = to.sx
t
AC=
lv
6
I . -!
Lz
X=
-x-4
B(-Z,s)
{-r,l
fX- 5 = g
(LX - 1 )(Zx r 5) ,
A(X', Y, )
kl
1
4XL+
slr* r.l,,frrt 0i,1, ui,?i,,il u
-h-+l
Sotr)
Iogr 4(x1rx-1) = O
4(XL+ t-x-1) =
:e
*rJ6 c(h,-h-+)
C
=
z---r rrnr*nrh;s lo3 tfr llt-r)
* =X ,
ucrrair lo3 , o"
'1 v
t[o1rt]'+r.x-,r) + 1o9.4) = o
6'urrm,1l,*l-rl lfi
x
1
,,u',
zr
e+b. t
3sip10++cosb -<fi3+4;
Rr sJlndot
ssino + Aooso < 5
z
)x'ogr(xiz)-i) - x log, (x'+ zx-t) zx-x
log,tx'+r-x-1)(ir'-r) r z(lx'- x) :0
1,*-
b=1
' -l?++'<
- s
[ogr,
- sirlo I
(oor'o 1 si nz o ) ( cos'o
(lttcosZo)
5
x'
<a tta: -b*bco:S<b
-r3u
-'lit,^o
L1) - 1a.5
X
-+
I
24. nt
-,i I t nufiurr
",ri,u^
mI 1
'.r1n1flfl061 A (- 3, z
B(a,b)
%11.b I
rnu
X
q
*fr rfl r',[u a
"',
olrffio.t
I qfl f ciorUUlUr r-oruari*rit qat n
l; F(x,z) lrm F otruuitxortr 4lt-3y+14=0
iinunt 4X- 3(t)+1T=o
Y'z
4X. -B -------, l. -L ----+ F(-2,2)
x
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=
4 tX +3 ) llnilhtilccrolnrilAe.rnu 4x-3Y +,14 = o --+ y = 3) - t4
1
(y-zl
e!7
6untens
=
+
nr'luf,amrrrrrluarbfiourqfloinr
rilo )=1
|= -rn qneinfis,-1,,,
()-2)z'+(!J:11 *''1
4t
ilnunt | = 3l:l-1
+
)'-+Y++ L
,fio
3Y-1++1L
(Y-1)()-b)
O
=0
n r y dolntu 4rut.qfrlfl61[6] n ororrr
r
0
1
tr
r
-+
n1 y [rJnnurarnr x lufialnuhlilhuAr vrJs
fi
@
atnrlduorrln
'lY,
[fl 1i[1in]
il= [ ::';"]
JF
,
.
Xnlntnvl
r[fr
J
-l
AF
r1n is1tatcan)= 4
)uunderld
ro d.ts-') ='r tl'
dr*r clet
det
tcl
=
a>o
A(*3,2) F[-2,t) nf,r 8t1,6)
* Lorfr,iorla''ooufru
A'F = l-?-(-3)l = l1l
L 2_- z ) - Lol
iEo Icl
ctl'v
1.,
ror*Blni
Y = 1,6
v.
roolarHo (1,6)
qrnknduonilq6rrio D(a,b)
lrite'' Dfro B(1)6)
')-
) -'1)+t =
1
)=6*x=
l;] lil
FB=
il
det tBt)
=
det cD'r)=
+
ti]
=
=
detr
(1)(i)r(o)(+)--
3
gl
I
EGr
u1 dettn; = o+6+o +1-t = 2o
1^j ds{tctRn = dslcclde,ttRldeltcl= (t)(20)(4)= 3zo*
Z'n
=L+4+6+.. +Zn 2(,1 +L+3+.- {h) = 2[lInr:tl = n'+n
Lz)
h1fr*l)(zpa11 , n(nrt) _
lui bn = f?n = Z tnttnl = Zn'r Zn
-- z
q
h+1
t,
e 4-l=
l:+-+
lim(itlt
; lirn Lb, br " bnJ 6ao6 i=r bn' =tim (i,=r f:r- /t
hJoo
1!flll!l!
11n
ah
-r
=
613'
I
n(n+
2)
,1(1
-
,
n
_1)
n+t/
t
osfl
=
=
Z
nin+1 )tn+z)
2,
h
3-rl
+ nrn+zt i J
!
lirn l./'3 * 3 + 3.5
n_d I 1-3 2-+
-+
i Ig l(i-u.(1 -x)-' (-).',)"(il!.#I*(-X*#)]
a r,
nJil
-a 2
f,*1,
itl
o-o)
=
n(h+l)
g
ti",.(i tn+L) )/
'
.-^ i-r
i Zn=
i=i
_1
n+1
_ 1l
n+zJ
1
+
=
2-.2-5
*
25. '4',t+rur1r46%,q:Afrrurh*ns,iwftiuog
, heu fruAr"f:r u'1,
"i'1":,":ntn
i
1nuinuftlfiu,urfi e [< nr",Aruiunrnrnfio rilufiouriu t t"darr]l"3r-'']ufir )
( 1rn )
:
'r lz t lz,iofiunit"uurnlor z rrrz,tlfirtn I , R.l'- '{]rf
etllqnic'lfr zIa+bi +r= Ia+br++l
z l(a+tl+bi I = lta++l+!il
LBe-r
t-Rcr
!)A-l1
lm
fi
2 .lTa*
b'
1)'*
=i
q1n Sn
=
1ri
=
Sr,
0,,
-Q
q1n
tln 2= e-bi
lt + b'
vl+
atr fu+
16
+ b'
b 6-
r?l=
m(+)'
Ia b,
:
1
fi4il?n
=
Lx
i
=Bgs
l
20
G
ununr
da
hl*l,tlnnn0 5ro', 205+505 ='l1o
tza,+(n-1)A)
t'
=20,
5zo
1,fl
= T (zalrtqJ)= tto
23, +
-O
tqd- qd . Y1-41
=
xt+
q",[fil+t
2n,+eJ, = 41
lii Sss
(3++l'+bt {
aro
frza.uqd)
@-o,
+F-i- ') ,nirxs dor firf,o 1fl1i
cJ+
i.r^il?n rc urid"Ll
t,
uAilln to xl*rrln = S,o = LO
i-20!
t- 305 ------r
#
1,,
= E+bi
lnr
+(ta+t)2+ b') =
4(il+21+r)+4b2
4?2+&+ o + 4b'
a a'+ sbz = 1z
22+ bl . 4
ar
6i1h; z
tod = 3o
----+
J!tta,+s$d) = Xil++
5+(3)) :
d=g
----+
f
rr+
19d = 'rl
-@
t'T'r+ntd'rarA
ol0r=
1+
4 tbt-) = 55(BB) = 4t4o
x
f fin,,d,-,6nflutiniilnrfiuzfix=t dilfildli1 ftu=z u^: fltl=o
+3 ---+ ftxl'9tx) - 9-X+3 6?.1 frxrgixl* f'tr)g(x) . L
i1n 1" tx) . 2- ------+ g'tx I = 2x + C -@t
rrnun"l f rtt,L,+'(1)=o 1u O
fttt g'ut r l'r1) g(1 ) = L
q1n (f.gttxl =
Zx
O
-o
(2)9'(1)
1ri 9'tx)- L/,-1
+ g$tl
------'
=
|
----+
}'rt)=
g(x) = x2-x+D
qrn
1;
1
@ tlTt% x.
g'rn , z+ c
1 .2-1C -*
1
-o
f,ltgcrl. t-+3
2g(1) = 5
3(1) - 2.5
lrnunrt 9(1)=S'lu O : 3c1) " -+D =L'5 '-'-' D'1's
rl"L
oi Jtxl= x--x+2.5
.5
I tn
trl nr rh r
qrn O
l,[Tlt,l
X=1
= (x - L)'- i+t.s
=
(x _ o,s )" +
2.zS
C = -1
nr
r
,'t1J')
a
e cr
or
r{
or o
oi,'ilr, nrr1116cr1a,rin6fro
it
fi
so
?-
Ls
aon
N
10. s,
z.zo)
26. 't"
x1s
Lu
,t,
t1n
[cxt+
t;
P
X'=
t-3
= 3 (r')'+ z+(xz) + +D
= 3(t-a)'+ z+(t-3)+40 = 3t'- lgttzr+ l+t*'12++o
a)
(t r
P ( t)
I crl
= 3t'+ tt - s
= 'o
['F.rtrd.t = r* (stz+rt-srdt = [t'+st' - tt]', = x3+:Xt-sX
l,rn Jprr) Jrt ) = J r,r,lJiu = J (tr+1?.-5)- (r. +1L-1;t)
11 - lcr = J? = J*,
x)L
111
c)
Irt1
y'
=
f 'crl =
!e
Ctotx utnai otnl
"'1
{rn
ahx+ zh
li m
h+
Ptx+h+z) + Prh+z)
cr
o
P
t
- Prx+z) + Pch+zr - Ptzl
li, tJX+L)
h+x+zl
-
X+z)
h-ro
Pcx+z)
+
= 3X+2
lhil:lallr
lor h sjroa,r',ra-,edr.A'rr
h-ro
lirn Bxrz)
I,Jt
- f(x+zt -?ctl
ht,?x+zl
lim
h-+
li, Itr*ht - frxr
h+o
n
L
'hlrrnr+11r
P'rx+z) + P'tzl
Ltn% x
" 0 t" @ '.
Prh+rr
lirn
h+
-?rtt
Itirn,,,',u lim
)
=1
h-ao
h
tr
=t
P'(x+z) r- P'[t )
P'(L) + ?rt2-t = 3(t])+L
---+
"
-o
3x+z
L
LP'r.Lt =
----+
P,(L)
=
O trrilF'(D,1 '" ?'(X+L)+1 = 3X+Z ----+ P'cx+t) .3X11
1a
Ila A = I+z 1 , A-L ". ?' [A) = 3(A-z)+ 1
3A-5
1, PtAt
tcl
= J P',nldn = I(3A-s)dA = 2gL- sR+C
: Piol ,2r*ct)L=E(ol+C =1 ----+ C =
a1n Prol =l
.LL
lAt Prtzt = 2(1L)L* j(1r-)+1 = it1+$) - 60+1 = 216 -5q = 1By
*
41n
=
1
'" X+ZY=U
L1
fn4
Lr : Y = firX+C
r riu L1
16i rYl, ff, = -
: -L
rirul fi
{
-t
(l *1
6-X
-!x+z
z
(0, 1)
urrlo,sio'r L,, riu
ufl;
t,
Y1= )z * --IXt3.
I
1
fn' = L
Lr, )= zx+c
qoiiru ( c', 1)
I = Lt0)+C
C= I
Lr, ). zx+1
i.^.,
Lr, 7=2Xt1
(-l)r, = -,
la
)
rr,m.rx=1,
r! Et/
s, 6
hn.?nfrs2
XL
Lrnr+ nr
6
L1
I= -lx+r
).
2x+
'2
1
-- r= t
,)z.yr=
=l *
E
llho**
a = I to.t l(1 +2. u)
- (O.+)(3.6) = 1,$t
lllttrn--
n.n -rlrrior
ba'l
= j ta-o.t)(?.6)
: (?.6)(2.6); t,.'l!
1,+++ 6,-16
, f,,,X
27. 1ln 0= {ra,bt 6AxAl o< b-1('1 ! riufio B rilul"um$ogFtdudu
t,b e i1,2,t,...,k] 1o u n n Islr.i u fi orrl'u ocb-r-.1
frcr:odt o<b-t-<J ------+ cr < b- a n b-a <-7
t-
( a,br
lSltr
n
der
>1
c,
, "r)
blrr+? 1n 1
?
L
#,0=k
*
Joi
b7
k
n
b-<']+a
rilu{r rJuil nr b froalrr
br,x'ri krutndn1uA trXl
A
r'1,
flnlu ) = k-1 ,
a = k-2
a = k-3 .,
4
d = k-+
d = k-5.,
a = k-6,
. k-y
,
,
d
r
=k
b
=
k-t,
k
.{
k-2, k-t,lr
b " k-:, k-2,k-t,k
:lnl'nrn
b
-- L$
b
=
b
b
= h-b,
k-5,k-*, k-r, k-2, k-t,
b
k-:., k-b,k-5,k-+,
=
a
b
k-r, k-r,
B
sl
k-a, k-2,
h-b, k-5,
k
k-i
b
I
b
k-+,k-r,k-z I d k- r "do doln'- t
't1
g)
I
FI
{
-llu
t=]) fl 0uou
v
I
b,
dnilfrn r14 qdr+oirr
raf Y,
?-z + tk-tt[.1) " ] 1+
"v
I
J,8,1
2-,3,+r5rb,1)8
3, +,5,
ld)
flo}tou
l
iack-
-9
d
h-"1
1+2+3+++5+btT
= k- +, k-a, k-2, k-t, k
= k-8, k-*, k-), k-2., k-t, k
= h-r'
d
l,ri
b
)
fl,,; s
'l .
In
l1n
q1n
=obr = loporlob+c -O
X
--* k= 106 4
++k-c = loL
r,'ilr X= 231 = 2clo + 30 +
. 1oo(L) + 4O(5) +
1
=cbc . looctlob+t -O
0-6' x-) = (100nnibk+c) - (100c ttlb+tr) = 100(a-c) -(a-c) = 19(?-c)
nr q:Iu x-) : a1(a-c) dnrtrnfn oiorrl 2=9 trA: C.1 ( x-c=t l;?Jrndfi)
[u1,6?1 b ,11*rn'uo:rfll6inlxilt t riu 1 rl^,rx',,,ar1orarIm*rrir,r
L6i s = {ltttqb1,q41,qg1,961,111, q81} rnrln'lug = Y(101)+ (u0+30+,..tro) = 665f4
1
Y
dtt
1'l n
11n
x+z
z(X+z)
,
X+2
)=X----+Y=
3Y = 3(qz) ------) 3Y
( x')
=
.l
,x+z
L.
Y'
_-+
1+LV
xr)+z = 16 ttr,filu,rrunr Y= X
1; X+ x'+ lcxt-11 = 16
L
tz
.5L
2-X+2-X, + X^l
.:X2+2X- 3i
=Q
(3x+'tl)(x-3) = o
1{
ln X = 3 ('." x liuirutulfirlrn)
11n
L
--------f
y= 1+L7
nL
)?=
1+2-Z
1- jtx'-rl
,cxt- r) )
z=
]
:3
Y
ldo'loi
= )=
1v
L'L
L5'l
3
L
16i
7
-l
6^l
x
tx{xg
= -(X-1
z
X)2.
(3)(c
1
= 1 tt - t )
Z
-lL
-l
)(+) = 103 x
28. A
:
rfl::r
"(rln 'l {r n r:
c)
: druzu nufrli
dd
n
r",l
r
C
n(nUBUC) = 1t
rrnlqnd n(A)= ?, hlb), Y1(c).10, h(Bnc).
11n
6
n(AnBl: 5 , n(A0c)=3
-,
nr0nc) _ n(Anc) + n (nnBnc)
c) = htn)+ n(B) + nrc) - n(An0)
1b - g+10+10-5- 6- 3 + n(An0nc)
ntAn0nC) = 1b-14 =l
n(AUgu
nr.+ff
rtludrir?nfiros dlmrrBut lnifir,+il
e
-1,
LR0nnuuLL3FI1.s1lnflU 6
=
dr.,fl t,rro',
@
Cr.
X --
= o.tq
++ -
=
6
nrr
rlz
Ir A- B)
=x -N
X
=
PtRr
=
(B)= 0,a6+(0.38-X)
0,3o
1
nUffl 6 C<J x
?i
0.56-x lrndi# F(s)= o,3o
r-t
* 2*
3
0.
o. l+
-
Ptnn sl
l[ x
;
h+uunr'rlu a,b,L ca<b<c)
llmirniuon Me = bo = b ,N A= zx = lS=
)
3
rfir-rir 2=CI.r rL61: c=6r t-.' ol.1+ &r = f,ft*fi-- I ) 6.1.r+. Og = flrl+t) 3)
[5
=
qrn
IX -
Gr
+
oo
1ln dd.fr rruqfiiu
+ Qu = 195
G3
-G, =
Qr+
g*
?1
@ @:
{
1,1
loo'r
,).
or_l
l,,=
'
6S
[cr
S
z1g
=0,L
7-
1.1
_
1'5
Z=
65-i,
x' + NrNz
-----r
Gr
- G t -- Ltl
6i =
,L:
-.
11x[o'tqrn
O
-
= W _ tlt x
,= lt
b5-Xr
-|
'l.3X.,.OS
4U+ 30
=
fr)
-b
4550,
+ir=50
Zoo0+ 3O-Y,2 ----n
_L
x
X
x
- t5
-r5
61
85
fi,rJfr n,l,rufit"urfouldl'r]u x
1z
-2+
l,r'-
,.
lL
7= x-x
1.s1
tJll
rouJd c1 fr:t[l+u
x
'
rrn @' er =
5=O,lXt
$,JX1 =
t4otc sol+ r :o )t
c.
)
B5
X.6s
2Qc =
Q3
121+25,+ 2,5$
r otrfior dsl roulai n: rruuEn ilu drmortiru , - 2
'1
0+O:
I ilru = 65;
Na+N2
=
l
o,t^1,t
N4
=
X.
uru
0.2
(g4-6,gtz+ (60-6s)2+(gl-65)1
*
[[h%Fi1111
=
135 -O
33
s = b5- I,
q1n X:,,
Q,r+&. =
tl
uA u t* rA
1
6Ir
135
x4
a
t1 u
Gr-G1
G1
.
r,=x,,
J
fl
fl(x-il'
NJ
-_
----+
1[2
{81
5J|
29. aonl narr'(u
n:rrlt1d
i= Zfx
n14
)ti
5-7
-?
+0
10-1+
a1
5o
s-11
1"1
36
-
LL
Lo
(I)t+0) + (12)tso) + ( 1't ) (30) + (LDtLo)
+0-t scl + bo + 2_O
ZCr
?-t+
h
l,i ( An
b"l-
.n=
l-
1t
1bb1
u
=
15t1 i
m*l
1+
=
,rll
I
on
a+b+k= 1+14+1). LKX
nt-n-t
nn'
h2.
L
l't+n-
nLn-
1
11,
t511
ll ,
....
1
t
(fl)
(fr Xfi X+;)
9,
b
,
1
n2- nn
-
3
h+
rl"
nt+n-
= 183 -- 13+!
1+
1+
1+
I
t+-!-- n' - nt+n-1
4d6 -
[ln
Lr+60+s1+44
,
dotr
)
nt+h -
n'-rt-
1
r
rrnh n 1 l'l = 2 ,9,$,
= ht+h-1
L
h +h - thXL = A
(l"t+ari( I'l-16) = Q
@
11n
5t2 + rb
+ C
=
1^t.:6 y
41b
r(8a+b ) = 41to - L
aa+b = 416-t
7
<$
tlt0
t {1 n
n1[vl
2,b e Ir,L,... 1t trr'r]uiru:urfiu
416-C e t* dta
'7
,dr
c'10
i [fl0 n
tnr
r
[nrl", +1b-1 =413=51
C=3
,,
,e+ b = 51
) = sg-b
I
#rrlri te+b e/ t+ frtu
J,
1
rln x tiiuiru**1fir" drhi sq-h e 1+
fllrfisn b=3 1; r, 5q:s3. 56-v
-t
x
cilrir* 2'-'1 , b= e, C =3
?+b+c = 'r+3r3 = 10
1;
,do,
8
x
dt- -.
d
"[c tilur']%tLi]01lt
fri'lu