Apostila Probabilidades e Estatística Duarte & Pereira

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Suredelolgdghv h Hvwdwðvwlfd
Suri1 Fdoghlud Gxduwh h Surid Dqdehod Shuhlud
Hvfrod Vxshulru gh Whfqrorjld2Lqvwlwxwr Srolwìfqlfr gh Vhwýedo
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4 Lqwurgxξr 4
5 Suredelolgdghv 6
514 Suredelolgdghv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6
515 H{hufðflrv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 46
6 Yduläyhlv Dohdwöuldv h IxqŒłhv gh GlvwulexlŒær 54
614 Yduläyhlv Dohdwöuldv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 54
615 IxqŒłhv gh GlvwulexlŒær 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 56
616 Yduläyhlv Dohdwöuldv Glvfuhwdv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 58
617 Yduläyhlv Dohdwöuldv Frqwðqxdv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 59
618 Ydoruhv Hvshudgrv h Sduåphwurv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5;
619 H{hufðflrv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 65
7 GlvwulexlŒłhv Whöulfdv Glvfuhwdv 76
714 Glvwulexlξr Xqliruph 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 76
715 Glvwulexlξr Elqrpldo 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 77
716 GlvwulexlŒær Klshujhrpìwulfd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 78
717 Glvwulexlξr gh Srlvvrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 7;
718 H{hufðflrv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 84
8 GlvwulexlŒłhv Whöulfdv Frqwðqxdv 8<
814 Glvwulexlξr Qrupdo 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 8<
81414 Dsur{lpdŒær gd Elqrpldo ª Qrupdo 1 1 1 1 1 1 1 1 1 1 1 95
815 GlvwulexlŒær gr Txl0Txdgudgr 0 �2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 97
816 GlvwulexlŒær | gh �Vwxghqw� 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 99
817 GlvwulexlŒær 8 gh �Vqhgfru� 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 9:
818 H{hufðflrv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 9<
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9 Whruld gd Dprvwudjhp ::
914 Jhqhudolgdghv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ::
915 Surfhvvrv gh Dprvwudjhp 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 :;
916 Hvwdwðvwlfdv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 :<
917 Hvwlpdgruhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ;3
918 Lqwhuydorv gh Frq�dqŒd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ;7
91814 Lqwhuydorv gh Frq�dqŒd sdud d Pìgld +j frqkhflgr, 1 1 ;9
91815 Lqwhuydorv gh Frq�dqŒd sdud d Pìgld +j ghvfrqkhflgr, ;<
91816 Lqwhuydorv gh Frq�dqŒd sdud d GlihuhqŒd gh Gxdv Pì0
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91818 Lqwhuydorv gh Frq�dqŒd sdud SursruŒłhv 1 1 1 1 1 1 1 1 <:
91819 Lqwhuydorv gh Frq�dqŒd sdud GlihuhqŒdv gh SursruŒłhv <<
9181: Lqwhuydorv gh Frq�dqŒd sdud d Ydulåqfld +> frqkhflgr, 433
9181; Lqwhuydorv gh Frq�dqŒd sdud d Ydulåqfld +> ghvfrq0
khflgr, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 434
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1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 436
919 Qrwdv h Frpsohphqwrv Vreuh dv GlvwulexlŒłhv gh Dprvwudjhp 438
91: H{hufðflrv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 439
: Whvwhv gh Klsöwhvhv 446
:14 Jhqhudolgdghv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 446
:1414 Sulqfðslrv gd uhdol}dŒær grv whvwhv gh klsöwhvhv 1 1 1 1 1 447
:15 Whvwhv d SrsxodŒłhv Qrupdlv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 453
:1514 Whvwhv gh Klsöwhvhv Sdud d Pìgld 1 1 1 1 1 1 1 1 1 1 1 1 453
:1515 Whvwhv gh Klsöwhvhv Sdud GlihuhqŒdv gh Pìgldv 1 1 1 1 1 457
:1516 Whvwhv gh Klsöwhvhv Sdud SursruŒłhv 1 1 1 1 1 1 1 1 1 1 459
:1517 Whvwhv gh Klsöwhvhv Sdud GlihuhqŒdv gh SursruŒłhv 1 1 459
:1518 Whvwhv gh Klsöwhvhv Sdud d Ydulåqfld 1 1 1 1 1 1 1 1 1 1 45:
:1519 Whvwhv gh Klsöwhvhv Sdud d Ud}ær hqwuh Gxdv Ydulåqfldv45:
:16 H{hufðflrv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 45;
Fdsðwxor 4
Lqwurgxξr
D ruljhp gd prghuqd whruld gdv suredelolgdghv ì dwulexðgd d grlv idprvrv
pdwhpäwlfrv iudqfhvhv/ Eodlvh Sdvfdo h Slhuuh Ihupdw +vìf1[YLL,1 Xpd vìulh
gh sureohpdv/ pxlwrv grv txdlv uhodwlyrv d �mrjrv gh d}du�/ ohydudp d xpd
iuxwxrvd wurfd gh fruuhvsrqgíqfld hqwuh Sdvfdo h Ihupdw/ qd txdo/ rv sulqfð0
slrv ixqgdphqwdlv gd Whruld gdv Suredelolgdghv irudp irupxodgrv1
Srxfr whpsr ghsrlv/ hp 498:/ r krodqgív Fkulvwldq Kx|jhqv sxeolfrx r
sulphlur olyur gh suredelolgdghv lqwlwxodgr Gh Udwlrflqllv lq Oxgr Dohdh1 Dv
pdlruhv frqwulexlŒłhv ghvwh shuðrgr ghyhp0vh d Ehuqrxool h d gh Prlyuh1
Hp 4;45 Slhuuh gh Odsodfh +4:7<04;5:, lqwurgx}lx xp frqmxqwr gh qr0
ydv lghldv h wìfqlfdv qr vhx olyur Wkìrulh Dqdo|wltxh ghv Suredelolwìv h shod
sulphlud yh} d whruld gdv suredelolgdghv ghl{rx r fdpsr grv �mrjrv gh d}du�
h irl dsolfdgd d pxlwrv sureohpdv suäwlfrv h flhqwð�frv1 D whruld grv huurv/
r fäofxor dfwxduldo h d phfåqlfd hvwdwðvwlfd vær h{hpsorv gh dojxpdv lpsru0
wdqwhv dsolfdŒłhv gd whruld gdv suredelolgdghv ghvhqyroylgdv qr vìfxor [L[1
Frpr wdqwrv rxwurv fdpsrv gd pdwhpäwlfd/ r ghvhqyroylphqwr gd whruld
gdv suredelolgdghv whp vlgr hvwlpxodgr shod ydulhgdgh gdv vxdv dsolfdŒłhv1 H
vlpxowdqhdphqwh/ fdgd dydqŒr gd whruld whp shuplwlgr r dodujdphqwr gd vxd
hvihud gh lq xíqfld1 D hvwdwðvwlfd pdwhpäwlfd ì xp grv lpsruwdqwhv udprv gd
whruld gdv suredelolgdghv> rxwudv dsolfdŒłhv rfruuhp hp fdpsrv wær glyhuvrv
frpr d jhqìwlfd/ d hqjhqkduld/ d hfrqrpld/ d phglflqd/ dv flíqfldv vrfldlv/
hwf1
D whuplqdu hvwd euhyh lqwurgxŒær xpd uhihuíqfld d dojxqv grv pdlv idprvrv
flhqwlvwdv txh wdqwr frqwulexludp sdud d whruld gdv suredelolgdghv ghvgh
Odsodfh= Fkhe|vkhy/ Pdunry h Nroprjrury1
4
5 FDS—WXOR 41 LQWURGX˚˘R
Fdsðwxor 5
Suredelolgdghv
514 Suredelolgdghv
D whruld gdv suredelolgdghv ì xp udpr gd pdwhpäwlfd h{wuhpdphqwh ýwlo
sdud r hvwxgr h d lqyhvwljdŒær gdv uhjxodulgdghv grv fkdpdgrv ihqöphqrv
dohdwöulrv1 R h{hpsor vhjxlqwh suhwhqgh fodul�fdu r txh yxojduphqwh ì ghvl0
jqdgr sru h{shulíqfld dohdwöuld1
H{hpsor 514 Qr odqŒdphqwr gh xpd prhgd rv uhvxowdgrv srvvðyhlv vær
�fdud� rx �frurd�1 Hp fdgd odqŒdphqwr qær ì srvvðyho suhyhu r uhvxowdgr txh
vh luä rewhu hperud hoh vhmd ghwhuplqdgr sru fdxvdv shuihlwdphqwh gh�qlgdv1
Hqwuh hodv/ d yhorflgdgh lqlfldo gd prhgd/ r åqjxor lqlfldo gr odqŒdphqwr/ d
gxuh}d gd phvd rqgh d prhgd luä fdlu/ hwf1 Frqwxgr/ frpr qær ì srvvðyho
frqwurodu wrgrv hvvhv sduåphwurv/ qær ì srvvðyho suhgl}hu r uhvxowdgr1
Ghyh hqwhqghu0vh frpr h{shulíqfld txdotxhu surfhvvr rx frqmxqwr gh
flufxqvwåqfldv fdsd} gh surgx}lu uhvxowdgrv revhuyäyhlv> txdqgr xpd h{0
shulíqfld hvwä vxmhlwd ª lq xíqfld gh idfwruhv fdvxdlv h frqgx} d uhvxowdgrv
lqfhuwrv gl}0vh txh d h{shulíqfld ì dohdwöuld1
Yrowdqgr dr h{hpsor dqwhulru/ dsduhqwhphqwh srgh sduhfhu txh qhoh qær
h{lvwh txdotxhu uhjxodulgdgh1 Pdv vh r qýphur gh revhuydŒłhv i÷u hohydgr/
dojxpd uhjxodulgdgh vxujh= r qýphur gh fdudv txh vdhp ì dsur{lpdgdphqwh
ljxdo dr qýphur gh frurdv1 Hvwd uhjxodulgdgh irl qrwdgd sru dojxqv lqyhv0
wljdgruhv txh uhdol}dudp orqjdv vìulhv gh odqŒdphqwrv gh prhgdv1 Ex�rq/
hp 7373 odqŒdphqwrv rewhyh 537; fdudv/ vhqgr sruwdqwr d ud}ær gh fdudv
qr wrwdo gh odqŒdphqwrv ljxdo d 31839<6/ h Shduvrq hp 57333 odqŒdphqwrv
rewhyh xpd iuhtxíqfld gh fdudv ljxdo d 3183381
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7 FDS—WXOR 51 SUREDELOLGDGHV
4J, srghu uhshwlu0vh xp judqgh qýphur gh yh}hv qdv phvpdv frqglŒłhv
rx hp frqglŒłhv pxlwr vhphokdqwhv>
5J, fdgd yh} txh d h{shulíqfld vh uhdol}d rewìp0vh xp uhvxowdgr lqglylgxdo/
pdv qær ì srvvðyho suhyhu h{dfwdphqwh hvvh uhvxowdgr>
6J, rv uhvxowdgrv gdv h{shulíqfldv lqglylgxdlv prvwudp0vh luuhjxoduhv/ pdv
rv uhvxowdgrv rewlgrv dr orqjr gh xpd orqjd uhshwlŒær gd h{shulíqfld sdwhq0
whldp xpd judqgh uhjxodulgdgh hvwdwðvwlfd/ txdqgr wrpdgrv hp frqmxqwr1
Qrwd 514 R doxqr srgh surfhghu d xp hohydgr qýphur gh vxfhvvlyrv odqŒd0
phqwrv gh xp gdgr h yhul�fdu r txh irl glwrdqwhulruphqwh$
Ydprv djrud gh�qlu d qrŒær sulplwlyd gd Whruld gdv Suredelolgdghv/ r
frqfhlwr gh hvsdŒr gh uhvxowdgrv/ txh dojxqv dxwruhv frqvlghudp lqvhulgr qr
frqfhlwr gh h{shulíqfld dohdwöuld1
Gh�qlŒær 514 HvsdŒr gh Uhvxowdgrv/ txh vh uhsuhvhqwd sru l/ ì r frq0
mxqwr irupdgr sru wrgrv rv srvvðyhlv uhvxowdgrv gh xpd h{shulíqfld dohdwöuld1
H{hpsor 515 Qxp odqŒdphqwr gh xp gdgr ruglqäulr whp0vh
l ’ ic 2c �c ec Dc Sj �
H{hpsor 516 R odqŒdphqwr gh grlv gdgrv whp xp hvsdŒr gh uhvxowdgrv
frp 69 hohphqwrv/
l ’ iE�c �� G �c � ’ c 2c �c ec Dc Sj �
D lpsruwåqfld gd gh�qlŒær gh hvsdŒr gh uhvxowdgrv dgyìp vreuhwxgr sru
vhu r phlr hpsuhjxh sdud d gh�qlŒær gh dfrqwhflphqwrv1
Gh�qlŒær 515 Rv vxefrqmxqwrv gh l ghvljqdp0vh sru dfrqwhflphqwrv> rv
vxefrqmxqwrv irupdgrv sru xp ýqlfr hohphqwr fkdpdp0vh dfrqwhflphqwrv
hohphqwduhv1
H{hpsor 517 Qr odqŒdphqwr gh xp gdgr kä vhlv dfrqwhflphqwrv hohphqwduhv
srlv l ’ ic 2c �c ec Dc Sj ( srghprv qr hqwdqwr gh�qlu xpd vìulh gh rxwurv
dfrqwhflphqwrv frpr sru h{hpsor
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5141 SUREDELOLGDGHV 8
Gh�qlgrv frpr frqmxqwrv/ drv dfrqwhflphqwrv ì dsolfäyho wrgd d frqv0
wuxŒær glvsrqðyho sdud dtxhohv/ lvwr ì/ h{lvwh xp sdudoholvpr shuihlwr hqwuh
äojheud gh frqmxqwrv h äojheud gh dfrqwhflphqwrv1
Gh�qlŒær 516 R dfrqwhflphqwr txh frqwhp wrgrv rv hohphqwrv gh l fkdpd�
vh dfrqwhflphqwr fhuwr1
Gh�qlŒær 517 R dfrqwhflphqwr txh qær frqwhp dojxp hohphqwrv gh l fkd0
pd�vh dfrqwhflphqwr lpsrvvðyho1
Gh�qlŒær 518 Grlv dfrqwhflphqwrv � h � vær pxwxdphqwh h{foxvlyrv
rx lqfrpsdwðyhlv rx glvmxqwrv vh qær wíp hp frpxp txdotxhu dfrqwhflphqwr
gh l�
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Gh�qlŒær 519 D xqlær grv dfrqwhflphqwrv � h � ì r dfrqwhflphqwr txh vh
uhdol}d vh h vö vh � rx � vh uhdol}dp1 Uhsuhvhqwd0vh sru � ^ � rx �n� h
ì irupdgr shorv hohphqwrv txh shuwhqfhp d � rx d ��
Gh�qlŒær 51: D lqwhuvhfŒær grv dfrqwhflphqwrv � h � ì r dfrqwhflphqwr
txh vh uhdol}d vh h vö vh � h � vh uhdol}dp frqmxqwdphqwh1 Uhsuhvhqwd0vh sru
� _ � rx �� h ì irupdgr shorv hohphqwrv txh frpxqv d � h d ��
9 FDS—WXOR 51 SUREDELOLGDGHV
Rv gldjudpdv gh Yhqq txh vh suhvhqwdp d vhjxlu loxvwudp dv gh�qlŒłhv
txh dfdedudp gh vhu dsuhvhqwdgdv1
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Uhxqlær gh dfrqwhflphqwrv/ � ^�c h lqwhuvhfŒær gh dfrqwhflphqwrv/ � _ �1
Qd olqjxdjhp fruuhqwh d qrŒær gh suredelolgdgh ì xwlol}dgd vhp txdotxhu
gh�qlŒær suhflvd gr vhx vljql�fdgr> frqwxgr/ d whruld gdv suredelolgdghv/
vhqgr xpd glvflsolqd gd pdwhpäwlfd/ ghyh wruqdu hvwd qrŒær suhflvd1 D
sulphlud gh�qlŒdr gh suredelolgdgh irl hvwdehohflgd sru Odsodfh hp 4;451
Gh�qlŒær 51; +Ohl gh Odsodfh rx Suredelolgdgh ª sulrul, 0 Vh xpd
h{shulíqfld dohdwöuld srgh whu Q uhvxowdgrv pxwxdphqwh h{foxvlyrv h ljxdo0
phqwh srvvðyhlv/ h vh ghvvhv uhvxowdgrv/ q wíp xp fhuwr dwulexwr D/ hqwær d
suredelolgdgh gh D ì gdgd sru ?
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H{hpsor 519 Qr odqŒdphqwr gh gxdv prhgdv srghprv rewhu 5 fdudv/ 5
frurdv rx 4 fdud h 4 frurd> pdv hvwhv uhvxowdgrv qær vær ljxdophqwh suryäyhlv>
d vdðgd gh 4 fdud h 4 frurd srgh dfrqwhfhu gh gxdv pdqhludv glihuhqwhv=
srgh vdlu sulphlur fdud qxpd gdv prhgdv h frurd qd rxwud/ rx dr frqwuäulr1
Dvvrfldqgr �I� ª vdðgd gh fdud h �F� ª vdðgd gh frurd/ r hvsdŒr gh uhvxowdgrv
vhuä
l ’ iE8c8 � c E8c�� c E�c8 � c E�c��j �
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txdqgr r hvsdŒr gh uhvxowdgrv ì �qlwr h rv vhxv hohphqwrv srvvxhp ljxdo
suredelolgdgh> pdv whqkd0vh hp dwhqξr txh hod vxujh txdqgr dv sulphludv
dsolfdŒłhv gr fäofxor gdv suredelolgdghv hvwær pxlwr oljdgdv drv �mrjrv gh
5141 SUREDELOLGDGHV :
d}du� txh frpr vh vdeh/ srvvxhp hvvdv sursulhgdghv1 ˆ sulrul frqkhfhp0vh
wrgrv rv uhvxowdgrv srvvðyhlv txh vær xp qýphur �qlwr gh fdvrv pxwxdphqwh
h{foxvlyrv h ljxdophqwh srvvðyhlv1 Ì r txh dfrqwhfh frp dv gxdv idfhv gh
xpd prhgd/ dv 85 fduwdv gh xp edudokr/dv vhlv idfhv gh xp gdgr/ hwf1
Nroprjrury/ hp 4<66/ irl xp grv judqghv frqwulexlqwhv sdud d d{lrpd0
wl}dŒær gd whruld gdv suredelolgdghv/ lvwr ì/ d frqvwuxŒær gh xp vlvwhpd gh
d{lrpdv txh irupdol}dp dojxpdv sursulhgdghv eävlfdv gd suredelolgdgh/ gh
txh frphŒrx d vhqwlu d qhfhvvlgdgh qr lqðflr gr vìfxor [[/
Gh�qlŒær 51< IxqŒær gh Suredelolgdgh ì xpd ixqŒær gh frqmxqwrv/ fx0
mr grpðqlr ì r frqmxqwr gdv sduwhv gr hvsdŒr gh uhvxowdgrv lc h whp sru
frqwudgrpðqlr r lqwhuydor dfc o h vdwlvid} rv vhjxlqwhv d{lrpdv=
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h{shulíqfldv dohdwöuldv1 D uhjxodulgdgh revhuydgd gd iuhtxíqfld gh dfrqwhfl0
phqwrv dohdwöulrv h r idfwr gh txh d iuhtxíqfld vhu xpd iudfŒær qær qhjdwlyd/
phqru rx ljxdo d xp/ ohydp d dfhlwdu r 4J d{lrpd1
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vlghuh0vh xpd h{shulíqfld dohdwöuld txh frqvlvwh qd h{wudfŒær gh xpd erod
gd xuqd1 Vh 6*? uhsuhvhqwdu d iuhtxíqfld gd vdðgd gd erod suhwd/ ì öeylr
txh qhvwh h{hpsor vh whuä vhpsuh 6*? ’ � Rud/ d h{wudfŒær gd erod suhwd
ì qhvwh fdvr r dfrqwhflphqwr fhuwr h d vxd iuhtxíqfld ì 41
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flphqwr gd idfh 9 whp xpd iuhtxíqfld txh rvflod shuwr gh *Sc r dsduhflphqwr
gd idfh 5 whp wdpehp xpd iuhtxíqfld txh rvflod shuwr gh *S/ txh hvwhv grlv
dfrqwhflphqwrv vær pxwxdphqwh h{foxvlyrv/ h txh r dfrqwhflphqwr �vdðgd
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hvwä fodur r 6J d{lrpd1
Frp edvh qhvwhv d{lrpdv srghp djrud ghprqvwudu0vh xpd vìulh gh lp0
sruwdqwhv sursulhgdghv gd suredelolgdgh/ gdv txdlv ghvwdfdprv=
SursrvlŒær 514 � E>� ’ f�
; FDS—WXOR 51 SUREDELOLGDGHV
GhprqvwudŒær1 Frpr l_> ’ >c r txh vljql�fd txh rv dfrqwhflphqwrv vær
pxwxdphqwh h{foxvlyrv/ srghprv hvfuhyhu
� El ^ >� ’ � El� n � E>��
Rud l ^ > ’ l h sruwdqwr/ � El ^ >� ’ � El� ’ �Hqwær/  ’  n � E>�,
� E>� ’ f�
SursrvlŒær 515 � E� ��� ’ � E��� � E� _�� �
GhprqvwudŒær1 �_� h ��_� vær dfrqwhflphqwrv pxwxdphqwh h{foxvlyrv
srlv
E� _ �� _ ��� _�� ’ �� _ ��� _ E� _�� ’ >�
Sru rxwur odgr E� _ �� ^ ��� _�� ’ �� ^ ��� _� ’ l _ � ’ �� Hqwær/
� E�� ’ � E� _ �� n � ��� _ �� (
pdv frpr/
�
�
�� _ �� ’ � E� � ��c
whp0vh r uhvxowdgr suhwhqglgr1
SursrvlŒær 516 � E�� ’ � � ���� �
GhprqvwudŒær1 l ’ � ^ �� h � h �� vær dfrqwhflphqwrv pxwxdphqwh
h{foxvlyrv srlv � _ �� ’ >� Hqwær/ � El� ’  ’ � E� ^ ��� ’
� E�� n � E���, � E�� ’ � � E����
SursrvlŒær 517 � E� ^�� ’ � E�� n � E��� � E� _�� �
GhprqvwudŒær1 Frpr �^� ’ �^ E���� ’ �^ E�� _�� h rv dfrqwhfl0
phqwrv � h �� _ � vær pxwxdphqwh h{foxvlyrv/ shor d{lrpd 6 h dwhqghqgr
ª sursrvlŒær 515 whp0vh r uhvxowdgr suhwhqglgr1
H{hpsor 51; Hp ghwhuplqdgd srsxodξr </;( gdv shvvrdv dgtxluhp d uh0
ylvwd D/ 55/<( d uhylvwd E/ h 8/4( dpedv1
Txdo d suredelolgdgh gh
+d, xpd shvvrd dgtxlulu shor phqrv xpd gdv uhylvwdvB
+e, xpd shvvrd dgtxlulu vrphqwh d uhylvwd DB
+f, xpd shvvrd qær dgttxlulu qhp d uhylvwd D qhp d uhylvwd EB
5141 SUREDELOLGDGHV <
H{hpsor 51< Ghvljqdqgr sru D r dfrqwhflphqwr �frpsud gd uhylvwd D� h
sru E r dfrqwhflphqwr �frpsud gd uhylvwd E� d suredelolgdgh gh xpd shvvrd
dgtxlulu shor phqrv xpd gdv uhylvwdv srgh vhu uhsuhvhqwdgr sru � E� ^�� ’
� E�� n � E��� � E� _ �� ’ f�fbH n f�22b� f�fD ’ f�2.S�
D suredelolgdgh gh xpd shvvrd dgtxlulu vrphqwh d uhylvwd D ì d suredelo0
lgdgh gh xpd shvvrd dgtxlulu d uhylvwd D h qær dgtxlulu d uhylvwdE/ rx vhmd/
� E� _��� ’ � E��� � E� _ �� ’ f�fbH� f�fD ’ f�fe.�
Xpd shvvrd qær dgtxlulu qhqkxpd gdv uhylvwdv fruuhvsrqgh d qær dgtxlulu
d uhylvwd D h d qær dgtxlulu d uhylvwd E> orjr
� E�� _ ��� ’ �
�
E� ^���
�
’ � � E� ^ �� ’ � f�2.S ’ f�.2e�
Qrwd 515 Dsöv d dsuhvhqwdŒær ghvwdv gxdv gh�qlŒłhv gh suredelolgdgh/ frq0
yìp dlqgd uhihulu/ txh qxpd rxwud shuvshfwlyd/ d gd fkdpdgd whruld iuh0
txhqflvwd/ d suredelolgdgh gh xp dfrqwhflphqwr ì gh�qlgd frpr vhqgr r
ydoru sdud r txdo whqgh d iuhtxíqfld uhodwlyd gr dfrqwhflphqwr txdqgr r
qýphur gh uhshwlŒłhv gd h{shulíqfld dxphqwd1
Ydprv djrud lqwurgx}lu xp qryr frqfhlwr/ r gh suredelolgdgh frqglflrqdo1
Frqvlghuhprv hqwær r vhjxlqwh h{hpsor=
H{hpsor 5143 Xp juxsr gh shvvrdv ì fodvvl�fdgr gh dfrugr frp r vhx shvr
h d lqflgíqfld gh klshuwhqvær1 Dv sursruŒłhv gdv yäuldv fdwhjruldv dsduhfhp
qr txdgur vhjxlqwh=
REHVR QRUPDO PDJUR Wrwdo
KLSHUWHQVR 3/43 3/3; 3/35 3/53
Q˘R KLSHUWHQVR 3/48 3/78 3/53 3/;3
Wrwdo 3/58 3/86 3/55 4/33
1
+d, Txdo d suredelolgdgh gh xpd shvvrd hvfroklgd dr dfdvr vhu klshuwhq0
vdB
+e, Txdo d suredelolgdgh gh xpd shvvrd rehvd vhu klshuwhqvdB
H{hpsor 5144 D uhvsrvwd ª sulphlud txhvwær ì lphgldwd1 Edvwd yhu txdo d
sursruŒær gh klshuwhqvrv ghqwur gd srsxodŒær txh ì gh 31531
Sdud uhvsrqghu ª vhjxqgd shujxqwd kä txh wrpdu hp dwhqŒær txh r txh
vh suhwhqgh ì d sursruŒær gh klshuwhqvrv gh hqwuh d srsxodŒær gh rehvrv/
lvwr ì/ f�f
f�2D
’ f� e� Sru rxwudv sdodyudv/ txhu0vh fdofxodu d suredelolgdgh gr
dfrqwhflphqwr �vhu klshuwhqvr�/ vdehqgr txh rfruuhx r dfrqwhflphqwr �vhu
rehvr�1 Uhsduh0vh txh hvwh txrflhqwh uhvxowd gd glylvær hqwuh d suredelolgdgh
gh xpd shvvrd vhu klshuwhqvd h rehvd h d suredelolgdgh gh xpd shvvrd vhu
43 FDS—WXOR 51 SUREDELOLGDGHV
rehvd1 Ghvljqdqgr sru R r dfrqwhflphqwr �vhu rehvr� h sru K r dfrqwhfl0
phqwr �vhu klshuwhqvr� srghprv hvfuhyhu vlperolfdphqwh txh d suredelolgdgh
suhwhqglgd ì gdgd sru
� EM*�� ’
� EM _ ��
� E��
�
Hvwh h{hpsor fruuhvsrqgh dr fäofxor gh xpd suredelolgdgh frqglflrqdo fxmd
gh�qlŒær irupdol}dprv d vhjxlu1
Gh�qlŒær 5143 D suredelolgdgh frqglflrqdo gh D gdgr E/ � E� m �� ì
gh�qlgd shod iöupxod
� E� m �� ’ � E� _ ��
� E��
�
vhpsuh txh � E�� : f( rx/ htxlydohqwhphqwh
� E� _�� ’ � E�� �� E� m �� �
Hvwd gh�qlŒær jhqhudol}d0vh idflophqwh sdud xp qýphur �qlwr gh dfrqwh0
flphqwrv
� E�

_�
2
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Xpd lpsruwdqwh lghld uhodflrqdgd frp d suredelolgdgh frqglflrqdo ì r
frqfhlwr gh lqghshqgíqfld hqwuh dfrqwhflphqwrv1
Gh�qlŒær 5144 D h E gl}hp0vh dfrqwhflphqwrv lqghshqghqwhv vh h vö
vh
� E� _�� ’ � E�� �� E�� c
rx htxlydohqwhphqwh/
� E� m �� ’ � E�� h � E� m �� ’ � E�� �
Hvwd ýowlpd uhodŒær hylghqfld r vljql�fdgr gh lqghshqgíqfld1 R frqkh0
flphqwr gh txh � rfruuhx qær lq xhqfld d suredelolgdgh gh txh � rfruud h
r frqkhflphqwr gh txh � rfruuhx qær lq xhqfld d suredelolgdgh gh txh �
rfruud1
H{hpsor 5145 Glvsłh0vh gh gxdv xuqdv/ d xuqd D frp 5 erodv eudqfdv h 6
erodv suhwdv/ h d xuqd E frp 7 erodv suhwdv h 43 eudqfdv> wudqvihuh0vh/ vhp
d h{dplqdu/ xpd erod gd xuqd D sdud d xuqd E h hp vhjxlgd h{wudl0vh ghvwd
xpd erod1 Txdo d suredelolgdgh gh d erod h{wudðgd gd xuqd E vhu eudqfdB
Frqvlghudqgr rv dfrqwhflphqwrv
5141 SUREDELOLGDGHV 44
D

0 D erod wudqvihulgd gh D sdud E ì eudqfd/
D
2
0 D erod wudqvihulgd gh D sdud E ì suhwd h
F 0 D erod h{wudðgd gh E ì eudqfd/
r txh vh suhwhqgh fdofxodu ì � E�� ’ � E�m�

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
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
c �
2
c � � �c �
?
j ì xpd sduwlŒær gr hvsdŒr
gh uhvxowdgrv l gh xpd h{shulíqfld dohdwöuld/ frp � E�
�
� : fc sdud � ’
c 2c � � �c ?c gdgr txdotxhu dfrqwhflphqwr E/ wdo txh � E�� : fc hqwær
� E�
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m �� ’ � E��� �� E� m ���
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GhprqvwudŒær1 Gd gh�qlŒær gh suredelolgdgh frqglflrqdo srghprv hvfuh0
yhu txh
� E�
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m �� ’ � E�� _ ��
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� +514,
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� E� m �
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Vxevwlwxlqgr r qxphudgru h r ghqrplqdgru gr 5J phpeur gh 514 sru 515 h
516/ uhvshfwlydphqwh/ rewhp0vh r uhvxowdgr suhwhqglgr1
Hvwh whruhpd srgh wdpeìp vhu lqwhusuhwdgr gd vhjxlqwh irupd= vhmd � xp
dfrqwhflphqwr txh vh uhdol}d vh h vö vh xp grv dfrqwhflphqwrv pxwxdphqwh
h{foxvlyrv �

c �
2
c ���c �
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vh yhul�fd1
Drv dfrqwhflphqwrv �
�
gä0vh sru yh}hv r qrph gh �fdxvdv�1 D iöupxod
gh Ed|hv gä hqwær d suredelolgdgh gh txh r dfrqwhflphqwr � txh vh ghx ì r
uhvxowdgr gd fdxvd ��
D suredelolgdgh � E�
�
� wrpd r qrph gh suredelolgdgh ª sulrul gd fdxvd
�
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� � E�
�
m�� ì d suredelolgdgh ª srvwhulrul/ lvwr ì/ d suredelolgdgh gh �
�
fdofxodgd vre d klsöwhvh gh txh � vh uhdol}rx1
45 FDS—WXOR 51 SUREDELOLGDGHV
H{hpsor 5146 Frqvlghuh 6 xuqdv qdv vhjxlqwhv frqglŒłhv
Erodv Eudqfdv Erodv Suhwdv
Xuqd 4 2 e
Xuqd 5 e 2
Xuqd 6 � �
Ghvljqdqgr sru R
�c
� ’ c 2c �c d suredelolgdgh gh hvfrokd gd xuqd �c vxsrqkd
txh R

’ R
2
’ 2R
�
�
Hvfroklgd xpd erod dr dfdvr fdofxoh d suredelolgdgh gh txh hvwd suryhqkd
gd 4@ xuqd vdehqgr txh
+d, d erod ì eudqfd>
+e, d erod ì suhwd1
H{hpsor 5147 Vhmd � r dfrqwhflphqwr �vdðgd gd erod eudqfd� +�� ì r
dfrqwhflphqwr �vdðgd gd erod suhwd�, h �
�
c � ’ c 2c �c r dfrqwhflphqwr �hv0
frokd gd xuqd ��1
Gdv uhodŒłhv
R

n R
2
n R
�
’ 
h
R

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2
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c
frqfoxl0vh txh
R

’ � E�

� ’ � E�
2
� ’ 2*D
h
R
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Xwlol}dqgr d iöupxod gh Ed|hv yhp=
� E�

m �� ’ � E�� �� E� m ��
�S
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5151 H[HUF—FLRV 46
515 H{hufðflrv
H{hufðflr 514 Qxp orwh gh 45 oåpsdgdv kä 5 eudqfdv h 43 d}xlv1 Txdo d
suredelolgdgh gh qxpd dprvwud gh ; oåpsdgdv kdyhu shor phqrv 4 eudqfdB+V=
31<4,
H{hufðflr 515 OdqŒdp0vh grlv gdgrv/ xp eudqfr h rxwur yhuphokr h uhsuh0
vhqwd0vh r uhvxowdgr sru xp sdu rughqdgr E�c T �1 Txdo d suredelolgdgh gd
vrpd � n T vhu=
+d, Sdu> +V= 
2
�
+e, Glylvlyho sru wuív> +V= 
�
�
+f, Sdu rx glylvlyho sru 61 +V= 2
�
�
H{hufðflr 516 Grlv mrjdgruhv frpelqdp xp wruqhlr gh 7 sduwlgdv1 Vxsrq0
gr txh �ª sulrul� d suredelolgdgh gh D jdqkdu xpd gdv sduwlgdv ì 
2
h d
suredelolgdgh gh E jdqkdu xpd gdv sduwlgdv ì �
H
/ ghwhuplqh=
+d, D suredelolgdgh gh D jdqkdu dv 7 sduwlgdv> +V= 
S
,
+e, D suredelolgdgh gh D h E jdqkduhp r phvpr qýphur gh sduwlgdv
+frqvlghuh d vlwxdξr gh fdgd xp jdqkdu 3 sduwlgdv,> +V= 31579,
+f, D suredelolgdgh gh D jdqkdu pdlv sduwlgdv txh E1 +V= 317;,
H{hufðflr 517 Vdeh0vh txh hp uhodŒær d xp gdgr surjudpd gh whohylvær vh
whp sdud xp fdvdo/ d vhjxlqwh vlwxdξr= d suredelolgdgh gh r krphp yhu
xp surjudpd ì 3/7> d suredelolgdgh gd pxokhu yhu r phvpr surjudpd ì
3/8> d suredelolgdgh gr krphp yhu r surjudpd srutxh d pxokhu r yí ì 3/:1
Ghwhuplqh d suredelolgdgh=
+d, Gh xp fdvdo yhu r surjudpd> +V= 3168,
+e, Gh d pxokhu yhu r surjudpd srutxh r krphp r yí> +V= 31;:8,
+f, Gh shor phqrv xp hohphqwr gr fdvdo yhu r surjudpd1+V= 3188,
H{hufðflr 518 Gdgrv grlv dfrqwhflphqwrv� h�/ wdlv txh � E�� ’ 
e
c � E�� ’

�
h � E� ^ �� ’ 
2
/ ghwhuplqh=
+d, Vh � h � vær pxwxdphqwh h{foxvlyrv> +V= qær,
+e, Vh � h � vær lqghshqghqwhv> +V= vlp,
+f, � E�*��c � E 7�*��c � E 7�*��c � E 7�* 7��� +V= 
e
c �
e
c 2
�
c �
e
�
47 FDS—WXOR 51 SUREDELOLGDGHV
H{hufðflr 519 Gxdv pätxlqdv D h E ixqflrqdp gh irupd lqghshqghqwh xpd
gd rxwud1 Qr txdgur vhjxlqwh/ lqglfdp0vh dv suredelolgdghv gh vh yhul�fdu
r qýphur uhihulgr gh dyduldv sdud fdgd pätxlqd/ qr ghfxuvr gh xp gld gh
wudedokr=
qJ dyduldv 3 4 5 6 7 8 9
pätxlqd D 3/4 3/5 3/6 3/5 3/3< 3/3: 3/37
pätxlqd E 3/6 3/4 3/4 3/4 3/4 3/48 3/48
Fdofxoh dv vhjxlqwhv suredelolgdghv=
+d, D h E wíp r phvpr qýphur gh dyduldv> +V= 314588,
+e, R qýphur wrwdo gh dyduldv ì lqihulru d 6> +V= 3155,
+f, D whp pdlv dyduldv txh E> +V= 3174:,
+g, E whp r greur gdv dyduldv gh D> +V= 313;,
+h, E whp 7 dyduldv/ vdehqgr0vh txh E whp shor phqrv gxdv dyduldv>
+V= 314+9,,
+i, R qýphur pðqlpr gh dyduldv hp fdgd xpd gdv gxdv pätxlqdv ì
61 +V= 315,
H{hufðflr 51: Xp frphufldqwh uhfheh edwdwd gh wuív uhjlłhv glihuhqwhv1 Dr
fkhjdu fdgd orwh ì fodvvl�fdgr hp gxdv fodvvhv/ D h E/ gh dfrugr frp d txdol0
gdgh gr surgxwr1 Vdehqgr txh xp orwh dfdedgr gh fkhjdu irl fodvvl�fdgr qd
fdwhjruld D/ txdo d suryhqlíqfld pdlv suryäyho ghvwh orwh/ whqgr hp dwhqŒær
txh d glvwulexlŒær grv orwhv uhfhelgrv dwì ª gdwd ì d vhjxlqwh=
Uhjlłhv Wrqhodgdv Txdolgdgh D Txdolgdgh E
L 43 5 ;
LL 53 47 9
LLL 53 43 43
+V= Uhjlær LL,
H{hufðflr 51; Xpd gdgd grhqŒd dsduhfh hp fhufd gh xpd shvvrd sru fdgd
4333 qxpd gdgd srsxodŒær1 Słh0vh hp ixqflrqdphqwr xp surjudpd sdud
dqdolvdu d suhvhqŒd gd grhqŒd/ vhqgr sdud wdo xvdgr xp dsduhokr gh gh0
whfŒær txh gä whvwh srvlwlyr frp suredelolgdgh gh 3/<< sdud xpd shvvrd
frp d grhqŒd/ h 3/38 sdud xpd shvvrd vhp d glwd1 Suhwhqgh0vh vdehu txdo
ì d suredelolgdgh gh txh xpd shvvrd frp whvwh srvlwlyr whqkd uhdophqwh d
grhqŒd1 +V= 3134<7,
5151 H[HUF—FLRV 48
H{hufðflr 51< Dv pätxlqdv D h E surgx}hp uhvshfwlydphqwh 43( h <3(
gd surgxξr wrwdo gh xp fhuwr wlsr gh duwljrv1 Vxsrqgr txh d suredelolgdgh
gd pätxlqd D surgx}lu duwljrv ghihlwxrvrv ì gh 4( h txh sdud d pätxlqd E
ì gh 8(/ txdo d suredelolgdgh gh xp duwljr wrpdgr dr dfdvr gh xp gld gh
surgxŒær suryhqkd gd pätxlqd D/ dvvxplqgr txh ì ghihlwxrvrB +V= 31354:,
H{hufðflr 5143 Xpd hpsuhvd srvvxl 6 phlrv gh wudqvsruwh/ W

c W
2
c h W
�
/
sdud frorfdu rv vhxv surgxwrv qr phufdgr1 Vdeh0vh txh dv suredelolgdghv
ghvwhv wudqvsruwhv dwudvduhp vær/ uhvshfwlydphqwh/ 3138/ 314/ h 31581 Vdeh0vh
dlqgd txh hvwhv phlrv gh wudqvsruwh vær hpsuhjxhv frp iuhtxíqfldv lqyhuvd0
phqwh sursruflrqdlv drv uhvshfwlyrv fxvwrv sru uhphvvdv= ;3/ 73 h 49 xqlgdghv
prqhwäuldv uhvshfwlydphqwh1
+d, Fdofxoh dv shufhqwdjhqv gh xwlol}dξr gh fdgd xp grv phlrv gh
wudqvsruwh>
+V= 2
.
c D
.
c f
.
,
+e, Ghwhuplqh d suredelolgdgh gh kdyhu dwudvr> +V= 314;5,
+f, Frqvlghudqgr dv uhphvvdv fkhjdgdv frp dwudvr/ txdo d suredelol0
gdgh gdv phvpdv whuhp vlgr wudqvsruwdgdv sru W
2
q +V= 31494,
H{hufðflr 5144 Dsöv dojxqv whvwhv hihfwxdgrv ª shuvrqdolgdgh gh xp lq0
glylgxr frqfoxlx0vh txh/ hvwh ì orxfr frp xpd suredelolgdgh ljxdo d 319/
odguær frp xpd suredelolgdgh ljxdo d 31: h qær ì orxfr qhp odguær frp xpd
suredelolgdgh gh 3158
+d, Ghwhuplqh d suredelolgdgh gr lqglylgxr vhu orxfr h odguær> +V=
3188,
+e, Ghwhuplqh d suredelolgdgh gr lqglylgxr vhu dshqdv orxfr rx dsh0
qdv odguær> +V= 315,
+f, Ghwhuplqh d suredelolgdgh gr lqglylgxr vhu odguær/ vdehqgr txh
r phvpr qær ì orxfr1 +V= 316:8,
H{hufðflr 5145 R Mrær whp ª vxd glvsrvlŒær wuív phlrv gh wudqvsruwh glih0
uhqwhv sdud vh ghvorfdu gh fdvd sdud d hvfrod= r wudqvsruwh D/ E rx F1 Vdeh0vh
txh d suredelolgdgh gh=
0 fkhjdu dwudvdgr ª hvfrod ì 319>
0 fkhjdu dwudvdgr xwlol}dqgr r wudqvsruwh D ì 31;>
0 fkhjdu dwudvdgr xwlol}dqgr r wudqvsruwh E ì 318>
0 fkhjdu dwudvdgr xwlol}dqgr r wudqvsruwh F ì 319>
0 xwlol}du rv wudqvsruwhv E h F ì d phvpd1
49 FDS—WXOR 51 SUREDELOLGDGHV
+d, Fdofxoh d suredelolgdgh gr Mrær xwlol}du r wudqvsruwh D> +V= 315,
+e, Vdehqgr txh r Mrær fkhjrx dwudvdgr ª hvfrod/ fdofxoh d suredelo0
lgdgh gh whu xwlol}dgr r wudqvsruwh F1 +V= 317,
H{hufðflr 5146 Vhmdp D h E grlv dfrqwhflphqwrv lqghshqghqwhv/ suryh txh
r frpsohphqwdu gh D h r frpsohphqwdu gh E wdpeìp r vær1
H{hufðflr 5147 Qxpd srsxodŒær 53( gdv idpðoldv wíp pätxlqd gh odydu
orxŒd/ 63( wíp pätxlqd gh odydu urxsd h 43( wíp dperv rv wlsrv gh
pätxlqdv1 Fdofxoh d suredelolgdgh gh xpd idpðold hvfroklgd dr dfdvr=
+d, Whu shor phqrv xp grv wlsrv gh pätxlqd> +V= 317,
+e, Qær whu qhqkxp grv wlsrv gh pätxlqd> +V= 319,
+f, Whu xp vö wlsr gh pätxlqd1 +V= 316,
H{hufðflr 5148 D suredelolgdgh gh vh rewhu fdgd srqwr qxp gdgr ylfldgr
ì gh=
%  2 � e D S
� E%� f� f�.D f�.D f�.D f�.D f�2
OdqŒd0vh r gdgr gxdv yh}hv frqvhfxwlyd h dglflrqdp0vh rv srqwrv rewlgrv1
Ghwhuplqh d suredelolgdgh gh fdgd xp grv dfrqwhflphqwrv=
+d, Rewhu xpd vrpd gh srqwrv vxshulru d gh}> +V= 3144,
+e, Rewhu xpd vrpd gh srqwrv lqihulru d txdwur1 +V= 31378,
H{hufðflr 5149 Frqvlghuh gxdv xuqdv D h E1 D xuqd D frqwìp 4 erod eudqfd
h <<< erodv suhwdv h d xuqd E frqwìp 4 erod suhwd h <<< erodv eudqfdv1 Ì
hvfroklgd xpd xuqd dr dfdvr gd txdo ì h{wudðgd xpd erod1 Vh hvwd ì suhwd/
txdo d suredelolgdgh gh txh d xuqd D whqkd vlgr hvfroklgdB +V= 31<<<,
H{hufðflr 514: Xp vdfr whp 6 prhgdv xpd gdv txdlv irl fxqkdgd frp 5
fdudv hqtxdqwr txh dv rxwudv r irudp frp 4 fdud h 4 frurd1 Wlud0vh xpd
prhgd dr dfdvr gr vdfr h odqŒd0vh dr du 5 yh}hv vhjxlgdv1 Txdo d suredelol0
gdgh gh txh vh wudwh gd prhgd gh 5 fdudv vdehqgr txh qrv grlv odqŒdphqwrv
vdlx vhpsuh fdudB +V= 2
�
,
H{hufðflr 514; Qxpd dprvwud frqvwlwxðgd sru 433 lqglylgxrv rewlyhudp0vh
rv uhvxowdgrv dsuhvhqwdgrv qr txdgur vhjxlqwh=
Frp Eurqtxlwh Vhp Eurqtxlwh
Ixpdgruhv 73 53
qær Ixpdgruhv 43 63
5151 H[HUF—FLRV 4:
+d, Gljd/ mxvwl�fdqgr d uhvsrvwd/ vh rv dfrqwhflphqwrv �vhu ixpdgru�
h �whu eurqtxlwh� vær lqghshqghqwhv> +V= qær,
+e, Fdofxoh d suredelolgdgh gh xp lqglylgxr txh ì ixpdgru whu eurq0
txlwh1 +V= 2
�
,
H{hufðflr 514< Frqvlghuh txh d vlwxdŒær gh xpd hqwuhylvwd gh fdqglgdwxud
d xp hpsuhjr/ ì xpd h{shulíqfld dohdwöuld h txh d qrvvd dwhqŒær vh fhqwud
hp grlv dfrqwhflphqwrv �r fdqglgdwr whp erd dsduíqfld� +dfrqwhflphqwr
�, h �r fdqglgdwr frqvhjxlx r hpsuhjr� +dfrqwhflphqwr �,1 Vdehqgr txh
� E�� ’ f�ec � E�� ’ f�2 h � E� ^ �� ’ f�e�
+d, Fdofxoh d suredelolgdgh gh r fdqglgdwr whu erd dsduíqfld h frq0
vhjxlu r hpsuhjr>
+V= 315,
+e, Rv dfrqwhflphqwrv �r fdqglgdwr whp erd dsduíqfld� h �r fdqglgd0
wr frqvhjxlx r hpsuhjr� vær lqghshqghqwhvB +V= qær,
+f, Fdofxoh d suredelolgdgh gh r fdqglgdwr whu erd dsduíqfld vdehqgr
txh frqvhjxlx r hpsuhjr1 +V= 4,
H{hufðflr 5153 Qxpd iäeulfd/ yhul�frx0vh txh xp fhuwr duwljr srgh dsuh0
vhqwdu ghihlwrv gh grlv wlsrv1 D suredelolgdgh gh rfruuhu r ghihlwr gr wlsr D
ì 314 h d suredelolgdgh gh rfruuhu r ghihlwr gr wlsr E ì 31381 Vdehqgr txh
rv ghihlwrv rfruuhp lqghshqghqwhphqwh xp gr rxwur/ fdofxoh d suredelolgdgh
gh=
+d, Xp duwljr qær whu txdotxhu ghihlwr> +V= 31;88,
+e, Xp duwljr whu ghihlwr> +V= 31478,
+f, Xp duwljr frp ghihlwr whu xp h xp vö wlsr gh ghihlwr1 +V= 3147,
H{hufðflr 5154 Vxsrqgr txh vh whp frqkhflphqwr gd vhjxlqwh lqirupdŒær=
0 d suredelolgdgh gr dfrqwhflphqwr �yhu r dqýqflr gr surgxwr D� @3168
0 d suredelolgdgh gr dfrqwhflphqwr �frpsudu r surgxwr D� @3156
0 d suredelolgdgh gr dfrqwhflphqwr �frpsudu r surgxwr D/ whqgr ylvwr r
dqýqflr gr surgxwr D� @3176
+d, Fdofxoh d suredelolgdgh gh yhu r dqýqflr h frpsudu r surgxwr D>
+V= 314838,
+e, Fdofxoh d suredelolgdgh gh yhu r dqýqflr rx frpsudu r surgxwr
D> +V= 3175<8,
4; FDS—WXOR 51 SUREDELOLGDGHV
+f, Fdofxoh d suredelolgdgh gh yhu r dqýqflr gr surgxwr D vh frpsurx
hvvh surgxwr>
+V= 319876,+g, Rv dfrqwhflphqwrv �yhu r dqýqflr gr surgxwr D� h �frpsudu r
surgxwr D� vær lqghshqghqwhvB Mxvwl�txh1 +V= qær,
H{hufðflr 5155 Sdud xpd gdgd vhfŒær gh xpd hvwudgd irudp uhfroklgdv
hvwdwðvwlfdv vreuh r qýphur gh yhðfxorv shvdgrv +f, h oljhlurv +t , txh hp
frqmxqwr/ d fux}dp hp lqwhuydorv gh 4 plqxwr1
QJ gh Shvdgrv +f, QJ gh Oljhlurv +t , QJ gh RevhuydŒłhv
f � ef
f e Df
f D Hf
f S 2f
 � Hf
 e Hf
 D ef
 S ff
2 2 Sf
2 � Hf
2 e ef
�  ef
� 2 Sf
6 6 63
Frqvlghuh rv vhjxlqwhv dfrqwhflphqwrv =
� 0 qJ gh yhðfxorv shvdgrv phqru rx ljxdo d 4
� 0 qJ gh yhðfxorv oljhlurv phqru rx ljxdo d 6
Ghwhuplqh � E��c � E��c � E�*��c � E� ^ �� h wluh dv vxdv frqfoxvłhv vreuh d
lqghshqgíqfld hvwdwðvwlfd hqwuh rv grlv dfrqwhflphqwrv1 +V= 319<> 316<> 3163::>
31<9> qær vær lqg1,
H{hufðflr 5156 Gh uhjlvwrv hihfwxdgrv frqfoxl0vh txh rv frqgxwruhv txh flu0
fxodp hp ghwhuplqdgd hvwudgd srghp frphwhu xp h xp vö grv grlv wlsrv gh
wudqvjuhvvłhv= wlsr L/ wlsr LL/ h qxqfd dpedv1 Gh 833 frqgxwruhv pxowdgrv
433 frphwhudp wudqvjuhvvłhv gr wlsr L1 43( grv txh frphwhudp wudqvjuhvvłhv
gr wlsr L vær pxowdgrv1 4( grv frqgxwruhv frphwhp wudqvjuhvvłhv gr wlsr L
h 5( gr wlsr LL1 Fdofxoh d suredelolgdgh gh vhu pxowdgr txhp frphwhu xpd
wudqvjuhvvær gr wlsr LL1
+V= 315,
5151 H[HUF—FLRV 4<
H{hufðflr 5157 Qxpd wxupd/ 53( grv doxqrv idodp ehp Lqjoív/ 73( idodp
ehp Iudqfív h 48( grplqdp dv gxdv oðqjxdv1
+d, Dyhuljxh vh idodu Iudqfív h idodu Lqjoív vær dfrqwhflphqwrv lqgh0
shqghqwhv> +V= qær,
+e, Fdofxoh d suredelolgdgh gh=
l1 Xp doxqr hvfroklgr dr dfdvr idodu shor phqrv xpd gdv oðq0
jxdv> +V= 3178,
ll1 Xp doxqr qær idodu qhqkxpd gdv oðqjxdv> +V= 3188,
lll1 Xp doxqr idodu xpd h xpd vö gdv gxdv oðqjxdv> +V= 316,
+f, Vdehqgr txh xp doxqr qær idod Lqjoív/ txdo d suredolgdgh gr
phvpr idodu IudqfívB +V= 316458,
H{hufðflr 5158 Frqvlghuh 6 xuqdv frp d vhjxlqwh frpsrvlŒær= d xuqd 4
frqwìp 43 erodv yhuphokdv h gxdv yhughv/ d xuqd 5 frqwìp 7 erodv yhuphokdv
h 9 yhughv h d xuqd 6 frqwìp 5 erodv yhuphokdv h : yhughv1 Xpd erod ì
h{wudðgd gh xpd gdv xuqdv/ vhqgr d hvfrokd gd xuqd ihlwd gr vhjxlqwh prgr=
xp gdgr ì odqŒdgr xpd yh}/
0 vh vdlu d idfh 4/ hvfrokh0vh d xuqd 4
0 vh vdluhp dv idfhv 5 rx 6/ hvfrokh0vh d xuqd 5
0 vh vdluhp dv idfhv 7/ 8 rx 9/ hvfrokh0vh d xuqd 6
+d, Txdo d suredelolgdgh gd erod h{wudðgd vhu yhughB +V= 31949,
+e, Txdo d suredelolgdgh gh vh whu hvfroklgr d xuqd 4/ gdgr txh d erod
h{wudðgd ì yhuphokdB +V= 316956,
H{hufðflr 5159 D suredelolgdgh gh xp lqglyðgxr D hvwdu ylyr gdtxl d 63
dqrv ì 319 h d suredelolgdgh gh xp rxwur lqglylgxr E hvwdu ylyr gdtxl d 63
dqrv ì 31:1 Ghwhuplqh d suredelolgdgh gh gdtxl d 63dqrv=
+d, Hvwduhp ylyrv rv lqglylgxrv D h E> +V= 3175,
+e, Qær hvwduhp ylyrv rv lqglylgxrv D h E> +V= 3145,
+f, Hvwdu ylyr shor phqrv xp grv lqglylgxrv1 +V= 31;;,
53 FDS—WXOR 51 SUREDELOLGDGHV
Fdsðwxor 6
Yduläyhlv Dohdwöuldv h IxqŒłhv
gh Glvwulexlξr
614 Yduläyhlv Dohdwöuldv
Hp pxlwdv h{shulíqfldv dohdwöuldv rv hohphqwrv gr hvsdŒr gh uhvxowdgrv/ l/
vær qýphurv uhdlv rx frqmxqwrv rughqdgrv gh qýphurv uhdlv1 Dvvlp dfrqwhfh
frp r uhjlvwr gh whpshudwxudv/ gd soxylrvlgdgh/ qr odqŒdphqwr gh grlv gd0
grv/ hwf1 Pdv mä r uhvxowdgr gr odqŒdphqwr gh xpd prhgd dr du qær ì xp
uhvxowdgr qxpìulfr1
Txdqgr l qær ì xp frqmxqwr qxpìulfr dwulexl0vh pxlwdv yh}hv d fdgd
hohphqwr / gr hvsdŒr gh uhvxowdgrv/ xp qýphur hi@*/ dwulexlŒær hvvd txh
srgh vhu phudphqwh frqyhqflrqdo1
H{hpsor 614 Qr odqŒdphqwr gh xpd prhgd dr du/ r hvsdŒr gh uhvxowdgrv
ì r frqmxqwr l ’ iFdud/Frurdj> ì xvxdo qhvwh fdvr id}hu d fruuhvsrqgíqfld
/ [E/�
Fdud 4
Frurd 3
�
Sdud r phvpr hvsdŒr gh uhvxowdgrv l/ ì srvvðyho hvwdehohfhu glihuhqwhv
fruuhvsrqgíqfldv/ frqvrdqwh rv remhfwlyrv hp hvwxgr1
H{hpsor 615 Frqvlghuh0vh xpd srsxodξr gh hpsuhvdv gdv txdlv vh hvfrokh
xpd dr dfdvr1
l ’ i/

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6
j vhuä r hvsdŒr gh uhvxowdgrv h 6 r qýphur wrwdo gh hp0
suhvdv1
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54
55FDS—WXOR 61 YDUL Y˜HLV DOHDWÖULDV H IXQ˚ØHVGHGLVWULEXL˚˘R
/ $ f E/� cvhqgr f E/� r qýphur gh hpsuhjdgrv gd hpsuhvd /c
/ $ t E/� cvhqgr t E/� r yroxph gh yhqgdv gd hpsuhvd /c
rx txdlvtxhu rxwudv/ frqiruph r remhfwlyr hp hvwxgr1
Vxsrqgr djrud txh vö hvwdprv lqwhuhvvdgrv qr hvwxgr gh xpd fdudfwhuðv0
wlfd grv hohphqwrv gh lc dvvrflhprv d fdgd hohphqwr / 5 l xp qýphur
uhdo fE/�� Hvwdprv dvvlp d gh�qlu xpd ixqŒær f G l $ W+� Vhqgr D xp
dfrqwhflphqwr/ fkdpd0vh lpdjhp gh D sru f/ h uhsuhvhqwd0vh sru fE��c dr
frqmxqwr grv ydoruhv txh f dvvxph sdud rv hohphqwrv / gh D/ lvwr ì/
fE�� ’ ifE/� G / 5 �j �
Sru rxwur odgr/ d fdgd vxefrqmxqwr . � W+c srgh id}hu0vh fruuhvsrqghu r
vxefrqmxqwr f3E.� irupdgr sru wrgrv rv hohphqwrv / 5 l wdlv txh fE/� 5
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H{hpsor 616 Qr odqŒdphqwr gh grlv gdgrv lqwhuhvvd vrphqwh/ qxp gdgr
mrjr/ d vrpd grv srqwrv rewlgd1 Qhvwh fdvr/ r hvsdŒr gh uhvxowdgrv ì r
frqmxqwr l ’ iE�c �� G �c � ’ c 2c �c ec Dc Sj ( gh�qd0vh d dsolfdŒær fE�c �� ’
�n ��
Vhqgr � ’ iEc �c Ec 2�c E2c �j c d lpdjhp gh � sru f ì fE�� ’ i2c �j (
sdud r dfrqwhflphqwr � ’ iEec D�c EDc e�c EDc D�c ESc S�j c d lpdjhp gh � sru f
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phqwrv hohphqwduhv/ fkdpd0vh xpd yduläyho dohdwöuld vh d lpdjhp lqyhuvd
gh txdotxhu lqwhuydor U gr hl{r uhdo gd irupd o�4c %o c ì xp dfrqwhflphqwr
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Qrwd 614 Xpd yduläyho dohdwöuld ì xpd ixqŒær h qær xpd yduläyho qr vhqwlgr
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6151 IXQ˚ØHV GH GLVWULEXL˚˘R 56
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dfrqwhflphqwrv Ef ’ %� vhmdp lpsrvvðyhlv1
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+d, Vh f ’ @ hqwær . dfo ’ @(
+e, . dKfo ’ K. dfo (
+f, .df n @o ’ .dfo n @(
+g, Vh }Ef� h �Ef� vær ixqŒłhvghfc hqwær.d}Ef�n�Ef�o ’ .d}Ef�on
.d�Ef�o/ fdvr h{lvwdp hvwdv hvshudqŒdv pdwhpäwlfdv1
6191 H[HUF—FLRV 68
H{hufðflr 6149 Suryh dv vhjxlqwhv sursulhgdghv gd ydulåqfld=
+d, T dfo ’ .df2o�.2dfo(
+e, T dfo � f(
+f, Vh f ’ @/ hqwær T dfo ’ f(
+g, T df n @o ’ T dfo(
+h, T dKfo ’ K2T dfo�
H{hufðflr 614: Uhodwlydphqwh ª glvwulexlŒær gd y1d1 f vdeh0vh txh .dfo ’
S h .df2o ’ �S� Vhqgr t xpd rxwud yduläyho wdo txh t ’ 
2
f n�c ghwhuplqh
.dt oc T dt o h j
t
� +V= 9> 3> 3,
H{hufðflr 614; R qýphur gh fduurv yhqglgrv vhpdqdophqwh qxp vwdqg ì
xpd yduläyho dohdwöuld f frp d vhjxlqwh ixqŒær gh suredelolgdgh=
%
�
 2 � e
sE%
�
� S S
2
S
�
S
e
+d, Fdofxoh/ mxvwl�fdqgr/ r ydoru gh S> +V= 2
2D
�
+e, Ghwhuplqh d ixqξr glvwulexlξr gh f(
+f, Fdofxoh d suredelolgdgh gr qýphur gh fduurv yhqglgrv qær fkhjdu
d 7/ vdehqgr txh hvwh ydoru ì vxshulru d 4> +V= f
�
�
+g, Vh rv fxvwrv �{rv vhpdqdlv vær gh 63 frqwrv txdqgr vær yhqglgrv
5 rx phqrv fduurv h 48 frqwrv txdqgr vh yhqgh pdlv gh 5 fduurv
h/ doìp glvvr/ sru fdgd fduur yhqglgr kä xp oxfur gh 68 frqwrv/
ghwhuplqh d uhfhlwd oðtxlgd vhpdqdo1
H{hufðflr 614< Wuív doxqrv h wuív doxqdv yær uhdol}du xp wudedokr gh juxsr
sdud d glvflsolqd gh Hvwdwðvwlfd1 Frpr wíp gh h{lvwlu 5 uhvsrqväyhlv shor
juxsr/ uhvroyhudp hihfwxdu hvvd hvfrokd gh irupd dohdwöuld1 Frqvlghuh f d
y1d1 txh uhsuhvhqwd r qýphur gh doxqdv qhvvd hvfrokd1
+d, Lqgltxh d ixqξr glvwulexlξr gh f(
+e, Txdo d suredelolgdgh gh h{lvwlu qær phqrv gr txh xpd doxqd qhvvd
hvfrokdB +V= e
D
�
+f, Fdofxoh .dfo h T dfo� +V= 4> 2
D
�
69FDS—WXOR 61 YDUL Y˜HLV DOHDWÖULDV H IXQ˚ØHVGHGLVWULEXL˚˘R
H{hufðflr 6153 f ì xpd yduläyho dohdwöuld frp d vhjxlqwh ixqŒær gh ghqvl0
gdgh=
sE%� ’
;?
=
%
e
� %
e
f
/ f � % � &
/ &% 	 e
/ fdvr frqwuäulr
+d, Ghwhuplqh/ mxvwl�fdqgr &( +V= 5,
+e, Ghwhuplqh d ixqξr glvwulexlξr gh f(
+f, Fdofxoh � Ef�D 	 f � 2�D� dwudyìv gd ixqŒær gh ghqvlgdgh gh f(
+V= 319;:8,
+g, Fdofxoh � Ef 	 *f � f�2D� dwudyìv gd ixqŒær gh glvwulexlŒær gh
f� +V=3144;,
H{hufðflr 6154 Qxp juxsr gh 43 lqglylgxrv yær vhu hvfroklgrv 6 sdud iru0
pduhp xpd frplvvær sdud r hvwxgr gh xp gdgr dvvxqwr1 Hqwuhwdqwr/ vdeh0vh
txh 7 grv 43 lqglylgxrv vær hvshfldolvwdv qr dvvxqwr1
+d, Dgplwlqgr xpd hvfrokd fdvxdo/ frqvwuxd d ixqξr gh suredelolgdgh
gd y1d1 f= qJ gh hvshfldolvwdv qd frplvvær>
+e, Vdehqgr txh qd frplvvær whp txh hvwdu shor phqrv xp hvshfld0
olvwd/ txdo d suredelolgdgh gh vhuhp wrgrv hvshfldolvwdvB +V= 3137,
+f, Dgplwlqgr txh fdgd hvshfldolvwd uhfheh phqvdophqwh/ sru id}hu
sduwh gd frplvvær/ 83 frqwrv h fdgd xp grv uhvwdqwhv 53 frqwrv/
rewhqkd r ydoru hvshudgr uhodwlyr dr prqwdqwh phqvdo gh uhpx0
qhudŒłhv gd frplvvær1 +V= <9 frqwrv,
H{hufðflr 6155 D txdqwlgdgh gh shl{h shvfdgr gläuldphqwh/ hp wrqhodgdv/
sru fhuwr edufr ì xpd y1d1 f frp d vhjxlqwh ixqŒær ghqvlgdgh gh suredelo0
lgdgh=
sE%� ’
;?
=
%
2ff
@E2f� %�
f
/ f � % 	 f
/ f � % 	 2f
/ fdvr frqwuäulr
+d, Ghwhuplqh r ydoru gd frqvwdqwh @> +V= 31348,
+e, Ghwhuplqh d ixqξr glvwulexlξr gd y1d1 h/ frp edvh qhod/ fdofxoh d
suredelolgdgh gr edufr/ qxp gld/ qær shvfdu phqrv gh ; wrqhodgdv
gh shl{h> +V= 31;7,
+f, Vdehqgr txh qxpd pdqkæ r edufr mä shvfrx pdlv gh 6 wrqhodgdv
gh shl{h/ txdo d suredelolgdgh gh dwì dr �p gr gld h{fhghu dv ;
wrqhodgdv gh shvfdgrB +V= 31;8<6,
6191 H[HUF—FLRV 6:
+g, Vh fdgd fdl{d gh shl{h hqyldgd sdud d orwd ohyd 58 nj/ txdo d
suredelolgdgh gh qxp gld vh hqylduhp pdlv gh 733 fdl{dvB +V=
31:8,
H{hufðflr 6156 Vhmd f xpd y1d1 glvfuhwd wdo txh=
8 E%� ’
;AAAA?
AAAA=
f / % 	 

e
/  � % 	 2
�
e
/ 2 � % 	 �
&
H
/ � � % 	 e
 / % � e
+d, Ghwhuplqh rv srvvðyhlv ydoruhv sdud &( +V= g 5 dSc Ho�
+e, Ghwhuplqh d ixqξr gh suredelolgdgh gh f(
+f, Fdofxoh � E2 	 f � e�( +V= 
e
�
+g, Fdofxoh .dfo h T dfo� +V= �� g
H
( D
2
� g
H
� g2
Se
�
H{hufðflr 6157 D ixqŒær gh ghqvlgdgh gh xpd y1d1 f ì gdgd sru=
sE%� ’
�
@n K%2
f
/ f � % � 
/ fdvr frqwuäulr
Vdehqgr txh .dfo ’ �
D
c
+d, Ghwhuplqh @ h K( +V= @ ’ �
D
( K ’ S
D
�
+e, Ghwhuplqh d ixqξr gh glvwulexlξr gh f(
+f, Fdofxoh �
�

2
� f � �
e
�
� +V= 3159;:8,
H{hufðflr 6158 Xpd hpsuhvd ghglfd0vh ª yhqgd gh duwljrv hoìfwulfrv1 Dv
yhqgdv gläuldv gh judydgruhv +hp xqlgdghv, vær uhsuhvhqwdgdv sru xpd y1d1
f frp d vhjxlqwh ixqξr gh suredelolgdgh=
sE%� ’
�
&E%n �
f
/ % ’ fc c 2c �c e
/ fdvr frqwuäulr
+d, Ghwhuplqh r ydoru gh &1 Mxvwl�txh d vxd uhvsrvwd> +V= 
D
�
+e, Gh�qd d ixqŒær gh glvwulexlŒær gd y1d1 f(
+f, Fdofxoh/ dwudyìv gd ixqŒær gh suredelolgdgh h gd ixqŒær gh glv0
wulexlξr/ d suredelolgdgh gh dv yhqgdv vhuhp h{dfwdphqwh ljxdlv
d 6 judydgruhv/ qxp gld hp txh hvwdv irudp shor phqrv 51 +V= 
�
�
6;FDS—WXOR 61 YDUL Y˜HLV DOHDWÖULDV H IXQ˚ØHVGHGLVWULEXL˚˘R
H{hufðflr 6159 Frqvlghudqgr f xpd y1d1 txh uhsuhvhqwd d gxudŒær hp
fhqwhqdv gh krudv gh xp fhuwr frpsrqhqwh hoìfwulfr h hp txh d vxd ixqŒær
ghqvlgdgh ì gdgd sru=
sE%� ’
�
@%
f
/ f 	 % � f
/ fdvr frqwuäulr
+d, Ghwhuplqh/ mxvwl�fdqgr/ r ydoru gh @( +V= 
Df
�
+e, Fdofxoh � Ef : D�( +V= D
2f
�
+f, Fdofxoh .dfo h T dfo� +V= 2f
�
( Df
b
�
H{hufðflr 615: Qxpd ormd gh hohfwurgrpìvwlfrv d surfxud gläuld gh dvsl0
udgruhv ì xpd y1d1 f frp ixqŒær gh suredelolgdgh=
%
�
f  2 � e
sE%
�
� f�2 K f�� f�2 f�
+d, Fdofxoh K/ mxvwl�fdqgr> +V= 315,
+e, Qr ðqlflr gh xp gld h{lvwhp dshqdv 5 dvsludgruhv qd ormd1 Fdofxoh
d suredelolgdgh gh vhuhp yhqglgrv +frqvlghuh txh surfxud@yhqgd
vhpsuh txh h{lvwd r surgxwr surfxudgr,> +V= 319,
+f, D sduwlu gd doðqhd dqwhulru rewhqkd d ixqŒær gh glvwulexlŒær gd
yduläyho �qýphur gh dvsludgruhv yhqglgrv�1
H{hufðflr 615; Vhmd1 f xpd y1d1 glvfuhwd frp ixqŒær gh suredelolgdgh
gdgd sru=
%
�
f  2 �
sE%
�
� 
H

H

�
D
2
+d, Ghwhuplqh d ixqξr gh glvwulexlξr gd y1d1 f(
+e, Fdofxoh � Ef 	 f � 2�( +V= 
2e
�
+f, Fdofxoh .dfo h T dfo� +V= eb
2e
( Dbb
D.S
�
H{hufðflr 615< Gdgd d ixqŒær gh glvwulexlŒær 8 gd y1d1 f=
8 E%� ’
;?
=
f
%
S

/ % 	 f
/ f � % 	 S
/ % � S
+d, Ghwhuplqh d uhvshfwlyd ixqξr gh ghqvlgdgh>
6191 H[HUF—FLRV 6<
+e, Fdofxoh � E2 � f � e�( +V= 
�
�
+f, Fdofxoh .dfo� +V= 6,
H{hufðflr 6163 D ixqŒær gh suredelolgdgh gd y1d1 f/ txh uhsuhvhqwd r
qýphur gh lpshuihlŒłhv hp fdgd 43 phwurv gh xpd �eud vlqwìwlfd ideulfdgd
hp urorv frqwðqxrv gh odujxud xqliruph ì gdgd sru=
%
�
f  2 � e
sE%
�
� f�e @ f�S f�fD f�f
+d, Fdofxoh @/ mxvwl�fdqgr> +V= 316:,
+e, Ghwhuplqh d ixqξr gh glvwulexlξr gd y1d1 f>
+f, Fdofxoh � E 	 f � e�( +V= 3155,
+g, Fdofxoh .dfo h T dfo� +V= 31;;> 31;789,
H{hufðflr 6164 Vhmd f xpd y1d1 frqwðqxd fxmd ixqŒær gh glvwulexlŒær ì
gdgd sru=
8 E%� ’
;?
=
f
%2

/ % � f
/ f 	 % 	 
/ % � 
+d, Ghwhuplqh d suredelolgdgh gh f wrpdu xp ydoru hqwuh 
�
h 
2
h
fdofxoh/ hp vhjxlgd/ .dfo h T dfo( +V= D
�S
( 2
�
( 
H
�
+e, Ghwhuplqh d suredelolgdgh gh/ hp 7 ydoruhv dvvxplgrv lqghshq0
ghqwhphqwh shod y1d1 f/ shor phqrv 6 hvwduhp frpsuhhqglgrv
hqwuh 3158 h 31:81 +V= 316458,
H{hufðflr 6165 D sursruŒær gh doföro gh xp fhuwr frpsrvwr srgh vhu frq0
vlghudgd xpd y1d1 f frp d vhjxlqwh ixqξr ghqvlgdgh=
sE%� ’
�
&%n 
2
f
/ f 	 % 	 
/ fdvr frqwuäulr
+d, Fdofxoh &/ rewhqkd d ixqξr gh glvwulexlξr gd y1d1 f h hverfh r
vhx juä�fr> +V= 4,
+e, Fdofxoh .dfo h T dfo( +V= .
2
( 
ee
�
+f, Vxsrqkd txh r suhŒr gh yhqgd ghvvh frpsrvwr ghshqgh gd shufhq0
wdjhp gh doföro1 Hvshfl�fdphqwh vh 
e
	 f 	 �
e
r frpsrvwr yhqgh0
vh sru 4333’332olwur h gh frqwuäulr yhqgh0vh sru :33’332olwur1 Vh
r fxvwr iru 833’332olwur/ fdofxoh d hvshudqŒd pdwhpäwlfd gr oxfur
oðtxlgr sru olwur1 +V= 683’33,
73FDS—WXOR 61 YDUL Y˜HLV DOHDWÖULDV H IXQ˚ØHVGHGLVWULEXL˚˘R
H{hufðflr 6166 Frqvlghuh d y1d1 f frp d vhjxlqwh ixqŒær ghqvlgdgh gh
suredelolgdgh=
sE%� ’
�
&E2%n �
f
/ f � % � 
/ fdvr frqwuäulr
+d, Ghwhuplqh & h hp vhjxlgd d ixqŒær glvwulexlŒær gh f( +V= 
2
�
+e, Fdofxoh .dfo h T dfo( +V= .
2
( 
ee
�
+f, Ghwhuplqh 6 gh prgr txh � Ef : 6� ’ � Ef 	 6� h prvwuh txh
6 ì ýqlfr1 +V= 3194;,
H{hufðflr 6167 R glåphwur gh xp fder hoìfwulfr ì xpd y1d1 f frp d
vhjxlqwh ixqξr ghqvlgdgh gh suredelolgdgh=
sE%� ’
�
S%E� %�
f
/ f � % � 
/ fdvr frqwuäulr
+d, Gh�qd d ixqŒær gh glvwulexlŒær gd y1d1 f h fdofxoh �
�
f � 
2
�
(
+V= 318,
+e, Fdofxoh .dfo( +V= 318,
+f, Vxsrqkd txh r fder hoìfwulfr ì frqvlghudgr ghihlwxrvr vh r glåphwur
glihulu gd vxd pìgld hp pdlv gh 31381 Txdo d suredelolgdgh gh vh
hqfrqwudu xp fder ghihlwxrvrB +V= 31;838,
H{hufðflr 6168 Frqvlghudqgrf xpd yduläyho dohdwöuld frqwðqxd/ suryh/ xwl0
ol}dqgr d gh�qlŒær/ d vhjxlqwh ljxdogdgh=
T d@f � K2o ’ @2T dfo�
H{hufðflr 6169 Suryh d vhjxlqwh ljxdogdgh= T d@2f � @Ko ’ @eT dfo
+d, Sdud f/ yduläyho dohdwöuld frqwðqxd>
+e, Sdud f/ yduläyho dohdwöuld glvfuhwd1
H{hufðflr 616: R qýphur gh yhqgdv uhdol}dgdv sru xp djhqwh gh vhjxurv
glduldphqwh ì xpd y1d1 frp ixqŒær gh suredelolgdgh=
%
�
f  2 � e
sE%
�
� � 5 | 5 �
Vdehqgr txh hp 43( grv gldv dv yhqgdv vær lqihulruhv d xp h txh hp :3(
grv gldv vær vxshulruhv d xp/
6191 H[HUF—FLRV 74
+d, Ghwhuplqh �c 5 h |( +V= 314/ 315 h 317,
+e, Ghwhuplqh T d2f � o( +V= 71;,
+f, Ghwhuplqh d suredelolgdgh gh txh/ txdqgr frqvlghudgrv grlv gldv/
dv yhqgdv vhmdp vxshulruhv/ hp fdgd xp ghohv/ d gxdv xqlgdghv>
+V= 313<,
+g, Vh fdgd vhjxur ì ihlwr sru 48333’33/ ghwhuplqh d ixqŒær gh sur0
edelolgdgh gd uhfhlwd rewlgd frpd yhqgd grv vhjxurv qxp gld>
+h, Vh qxp gld d uhfhlwd iru lqihulru d 83333’33/ ghwhuplqh d sured0
elolgdgh gh txh vhmd vxshulru d 53333’331 +V= 31999,
H{hufðflr 616; R Vu1 Mrær whp xp fduur txh qhfhvvlwd gh xpd uhsdudŒær
xujhqwh1 Frqvlghuh f d y1d1 txh uhsuhvhqwd r qýphur gh vhpdqdv txh r
fduur vh pdwhuä hp ixqflrqdphqwr vhp uhsdudŒær1 D ixqŒær ghqvlgdgh gh
suredelolgdgh ì d vhjxlqwh=
sE%� ’
;?
=
%
2f
%
ef
f
/ f 	 % � e
/ e 	 % � K
/ fdvr frqwuäulr
+d, Txdo r qýphur pä{lpr gh vhpdqdv txh r fduur ixqflrqduä vhp
uhsdudξrB +V= ;,
+e, Ghwhuplqh d ixqξr gh glvwulexlξr gd y1d1 f(
+f, R Vu1 Mrær suhflvd gh lu dr Sruwr gdtxl d 9 vhpdqdv1 Txdo d sure0
delolgdgh gh srghu xwlol}du r vhx fduur vhp whu hihfwxdgr txdotxhu
uhsdudξrB +V= 3168,
H{hufðflr 616< D txdqwlgdgh gh fhuyhmd yhqglgd glduldphqwh qxpd ihlud
+hp plokduhv gh olwurv, ì xpd yduläyho dohdwöuld frp d vhjxlqwh ixqŒær ghq0
vlgdgh gh suredelolgdgh=
sE%� ’
;?
=
&%
&E2� 2%�
f
/ f � % � e
/ e 	 % � S
/ fdvr frqwuäulr
+d, Rewhqkd r ydoru gh & h gh . d�f n 2o ( +V= 
2
h 2,
+e, Ghwhuplqh 8 E%�(
+f, Frqvlghuh rv vhjxlqwhv dfrqwhflphqwrv=
D@ �yhqgd gläuld vxshulru d 7333 olwurv�
E@ �yhqgd gläuld hqwuh 6333 h 8333 olwurv�
Lqgltxh/ mxvwl�fdqgr/ vh D h E vær lqghshqghqwhv1 +V= qær,
75FDS—WXOR 61 YDUL Y˜HLV DOHDWÖULDV H IXQ˚ØHVGHGLVWULEXL˚˘R
H{hufðflr 6173 Xpd hvwdŒær gh jdvrolqd hqfkh rv uhvhuydwöulrv xpd yh} sru
vhpdqd1 R yroxph vhpdqdo gh yhqgdv/ h{suhvvr hp plokduhv gh olwurv/ ì xpd
yduläyho dohdwöuld frp d vhjxlqwh ixqŒær ghqvlgdgh gh suredelolgdgh=
sE%� ’
;AA?
AA=
f

f
f3%
2D
f
/ % 	 f
/ f � % 	 D
/ D � % 	 f
/ % � f
+d, Yhul�txh txh sE%� ì gh idfwr xpd ixqŒær ghqvlgdgh gh suredelol0
gdgh gh xpd yduläyho dohdwöuld f(
+e, Fdofxoh d ixqξr glvwulexlξr gh f(
+f, Ghwhuplqh d suredelolgdgh gh qxpd vhpdqd vh yhqghuhp pdlv gh
:333 olwurv gh jdvrolqd/ vdehqgr0vh txh mä irudp yhqglgrv shor
phqrv 7333 olwurv>
+g, Fdofxoh d txdqwlgdgh pìgld gh jdvrolqd txh ì yhqglgd vhpdqdo0
phqwh1
H{hufðflr 6174 Vhmdf xpd yduläyho dohdwöuld glvfuhwd frp d vhjxlqwh ixqŒær
gh suredelolgdgh=
%
�
6�  6 6n � 6n D
sE%
�
� &n
H
&
H
&3
H
&
H
Ghwhuplqh/
+d, & h 6 gh prgr txh .dfo ’ 
e
( +V= 5 h 04,
+e, T d�f � 2o( +V= 8819;:8,
+f, 8 E%� h � E2 	 f 	 D�� +V= 3158,
Fdsðwxor 7
GlvwulexlŒłhv Whöulfdv Glvfuhwdv
714 Glvwulexlξr Xqliruph
Gh�qlŒær 714 Xpd yduläyho dohdwöuld f gl}0vh txh whp xpd glvwulexlŒær
xqliruph hp ? srqwrv txdqgr d vxd ixqŒær gh suredelolgdgh ì gd irupd
sE%
�
� ’ � Ef ’ %
�
� ’

?
c � ’ c 2c ���c ?�
Frpr h{hufðflr ghl{d0vh dr fxlgdgr gr doxqr d ghprqvwudŒær gd vhjxlqwh
sursrvlξr1
SursrvlŒær 714 Vh f ì xpd y1 d1 frp xpd glvwulexlŒær xqliruph/
. dfo ’

?
[
�
%
�
h
T dfo ’

?
[
�
%2
�
�
%

?
[
�
%
�
&
2
�
H{hpsor 714 Flwdqgr gh qryr r h{hpsor 517/ frqvlghuh0vh r odqŒdphqwr gh
xp gdgr1 D fdgd dfrqwhflphqwr hohphqwdu/ lvwr ì/ d fdgd uhvxowdgr srvvðyho
gh xp odqŒdphqwr/ srghprv dvvrfldu xp grv qýphurv c 2c �c ec Dc Sc r qýphur
gh srqwrv txh dsduhfhp qd idfh uhvxowdqwh1Dtxl d yduläyho dohdwöuld f srgh
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