Probabilidades

 

7 7  mg mg mgtf  tfpc pc pc gs gspga pgafb fbl l  ((acst acstc c mg mgl l pb pbsbog1 sb sbog og1 1 mñlb mñlbrg rgs) s). .  Whb W hb  pgrsc pg pgrrschb hbtf tfgh ghg g qugvfbobrghtrglbsmcsafumbmgs(fmbyvugltb)murbhtglcs7mébs lbncrbnlgsmglbsgibhb,ypbrbtrbhspcrtbrsgtcibglprfigrnus qugbpbrgagghgsbrutb,sfhmfegrghafbrgltfpc:³aÿbhtcgspgrbkbstbr gstbpgrschbghlbsgibhb3 Tlbhtgbifghtcmglprcnlgib. MB]CQ? h ?huigrcmgnusgs x ?acstcmgvfbog($/méb) NH?nushcribl NH ?nushcribl ?nushcribl NG?nusgspgafbl ?nusgspgafbl K?kbstcsgibhblmgtrbhspcrtg h924 9 24 NH?7acstcmgpbsbogs?$> NH ?7acstcmgpbsbogs?$> ?7acstcmgpbsbogs?$1 NH?7acstcmgpbsbogs?$1 NH Acic? Mfst Mfstrrrfn fnua fnuafc fch h Hct H ctba tbafc bafch h   G(P)   Ubr(P) Euh. uh.  mg mg mgmg  mghs hsfm fmbm b m  Euh. uh. mgmf g mfst st..   h+2 h +2  2   j Whfecrig Whfec rigmfsarg mfsargtb tb   W(h)   e e((j)9 T( T(P 9j)9 j )9 h   E E((Pj)9 Pj )9 h > > >

^gsclusfñhmglprcnlgib x T( T(x)

  2 >

== ( 2> )9   2=

77 > 2> ( 2> )9   >2

T( T(NH NH) ) 9 T( T(NG) ) 9

2 >

5 ( 2> )9   2=

92

7 2 9 24 > 7 2 9 24 >

K 97 9 7P

G(K) 9   G((7 (7P 7P)97( )97(G(P))97(7)9>7Z$ )97(G ))97(7)9>7Z$9sgibhb ))97(7)9>7Z$9sgibhb 9sgibhbYY G(P)  9  =( =( 2 )+7( 2 )+5( 2 )97Z$ )97Z$9m 9m 9m {bY {b {bY = > = Achalusfñh? Ghlbsgibhbgspgrbkbstbr>7 Ghlbsgibhbgspgrbkbstbr >7 mñlbrgs. GOG^AFAFC =? = ?5?1 5 ?1    pbk  pbk215 pbk215 Ghuhb gsaugl gsauglb b prfibr prfibrfbsg fbsg rgkfst rgkfstrñgl rñgl hÿi hÿigrc grc mgpblbnrbs mgpblbnrbs pcr ifhutcqug ifhutc qug lgébh lgébhlcsgstumfbhtg lcsgstumfbhtgs,ghachtr s,ghachtrâhmcsg âhmcsgqug qug lgébhuh lgébhuh iéhficmg 04 pblbnrbsyuhiâxficmg 216 216.. Nboclbsupcsfafñhmgqug lbvbrfbnlgblgbtcrfbqugmgsarfngglhÿigrcmgpblbnrbslgémbsgstâ uhfecrigightgmfstrfnufmb. b)@bllglbprcnbnflfmbmmgquguhgstumfbhtg,sglgaafchbmcblbzbr, lgbblighcs244pblbnrbs? >

 

7 7  mg mg mgtf  tfpc pc pc gs gspga pgafb fbl l  ((acst acstc c mg mgl l pb pbsbog1 sb sbog og1 1 mñlb mñlbrg rgs) s). .  Whb W hb  pgrsc pg pgrrschb hbtf tfgh ghg g qugvfbobrghtrglbsmcsafumbmgs(fmbyvugltb)murbhtglcs7mébs lbncrbnlgsmglbsgibhb,ypbrbtrbhspcrtbrsgtcibglprfigrnus qugbpbrgagghgsbrutb,sfhmfegrghafbrgltfpc:³aÿbhtcgspgrbkbstbr gstbpgrschbghlbsgibhb3 Tlbhtgbifghtcmglprcnlgib. MB]CQ? h ?huigrcmgnusgs x ?acstcmgvfbog($/méb) NH?nushcribl NH ?nushcribl ?nushcribl NG?nusgspgafbl ?nusgspgafbl K?kbstcsgibhblmgtrbhspcrtg h924 9 24 NH?7acstcmgpbsbogs?$> NH ?7acstcmgpbsbogs?$> ?7acstcmgpbsbogs?$1 NH?7acstcmgpbsbogs?$1 NH Acic? Mfst Mfstrrrfn fnua fnuafc fch h Hct H ctba tbafc bafch h   G(P)   Ubr(P) Euh. uh.  mg mg mgmg  mghs hsfm fmbm b m  Euh. uh. mgmf g mfst st..   h+2 h +2  2   j Whfecrig Whfec rigmfsarg mfsargtb tb   W(h)   e e((j)9 T( T(P 9j)9 j )9 h   E E((Pj)9 Pj )9 h > > >

^gsclusfñhmglprcnlgib x T( T(x)

  2 >

== ( 2> )9   2=

77 > 2> ( 2> )9   >2

T( T(NH NH) ) 9 T( T(NG) ) 9

2 >

5 ( 2> )9   2=

92

7 2 9 24 > 2 7 9 > 24

K 97 9 7P

G(K) 9   G((7 (7P 7P)97( )97(G(P))97(7)9>7Z$ )97(G ))97(7)9>7Z$9sgibhb ))97(7)9>7Z$9sgibhb 9sgibhbYY G(P)  9  =( =( 2 )+7( 2 )+5( 2 )97Z$ )97Z$9m 9m 9m {bY {b {bY = > = Achalusfñh? Ghlbsgibhbgspgrbkbstbr>7 Ghlbsgibhbgspgrbkbstbr >7 mñlbrgs. GOG^AFAFC =? = ?5?1 5 ?1    pbk  pbk215 pbk215 Ghuhb gsaugl gsauglb b prfibr prfibrfbsg fbsg rgkfst rgkfstrñgl rñgl hÿi hÿigrc grc mgpblbnrbs mgpblbnrbs pcr ifhutcqug ifhutc qug lgébh lgébhlcsgstumfbhtg lcsgstumfbhtgs,ghachtr s,ghachtrâhmcsg âhmcsgqug qug lgébhuh lgébhuh iéhficmg 04 pblbnrbsyuhiâxficmg 216 Nboclbsupcsfafñhmgqug 216.. lbvbrfbnlgblgbtcrfbqugmgsarfngglhÿigrcmgpblbnrbslgémbsgstâ uhfecrigightgmfstrfnufmb. b)@bllglbprcnbnflfmbmmgquguhgstumfbhtg,sglgaafchbmcblbzbr, lgbblighcs244pblbnrbs? >

 

n)Mgtgrifhgglhÿigrcmgpblbnrbsqugsggspgrbréblgbuhgstumfbhtgsglgaafchbmcblbzbr. Tlbhtgbifghtcmglprcnlgib

Mbtcs? Ubrfbnlg blgbtcrfb? Ubrfbnlg blgbtcrfb?   P  Hÿigrcmgpblbnrblgfmb Gspbafciugstrbl Gspbafciugstrbl 9  9 ee04????216k h? ?gsglhÿigrcmgpblbnrbs gsglhÿigrcmgpblbnrbs h 976 9 76 @bllbr b)T(P9244)93 9244)93 n)G(P)93 )93 Acic? Mfst Mfstrrrfn fnua fnuafc fch h Hct H ctba tbafc bafch h   G(P)   Ubr(P) Euh. uh.  mg mg mgmg  mghs hsfm fmbm b m  Euh. uh. mgmf g mfst st..   j   h+2 h +2  2 Whfecrig Whfec rigmfsarg mfsargtb tb   W(h)   e(j)9 T(P 9 j )9 h   E(Pj )9 h > > ^gscluafñhmglprcnlgib. b)  244 16 216  abscsebvcrbnlgs 94 ?55 9 9 T 9 76 76 abscpcsfnlgs abscpcsfnlgs >

n) G(P)9

76+2 9 914 14+ +049 049224 224 >

Achalusfñh? Glhuigr Glh uigrcmg cmg mg p pblbnrbsq blbnrbsquggspg uggspgrblg rblggrgs grgs grgs mg mg224 mg224pcrif 224pcrifhut hutc. c. GOG^AFAFC = ?5?= 5 ?=    pbk pbk217 Qgb P uhb Qgb uhb vbr vbrfbn fbnlgblgbt lgblgbtcrf crfb b qugsfkug qugsfkug uhblgy uhfecr uhfecrigscnrg igscnrg {-2,4,2}.Ablaulg

2. Tlbhtgbifghtcmglprcnlgib Mbtcs

P!   e2: 4: 2k @bllbr b)   G(Pj ) pbrbj92,>,... n)   U   U br (Pj ) Acic Mfst Mfstrrrfn fnua fnuafc fch h Hct H ctba tbafc bafch h   G(P) Whfecrig Whfec rigmfsarg mfsargtb tb   W(h)   h+2 > ^gscluafch ^gsclua fchmgl mglprcnlg prcnlgib. ib. h T pj xj   T(P 9 j )9   h2 9 b) G(P))9

  Ubr(P) >

h +2

>

 

Euh. uh.  mg mg mgmg  mghs hsfm fmbm b m  Euh. uh. mgmf g mfst st..   j  2 e(j)9 T(P 9 j )9 h   E(Pj )9 h

j92

2 2 2 (2)j 2 (2)j +2 j j j     G(P )9 )9(( 2) ( )+(4) ( )+(2) ( )9 + 9 1 1 1 1 1 1 j

1

 

U br(Pj )9 G(P>j )

Achalusfñh

f)   U br(Pj )9   1> ff)   U br(Pj )9   1>



 

(G(Pj ))>

= 6



9   6> jgspbr jgsfipbr 49   1>

1 LGV LGV@F @FTG TG^ ^KGCI KGCIÆ]^ Æ]^FAB FAB?? GOG^AFAFC = ?5?7   pbk217

WhbvbrfbnlgPtfghgmfstrfnuafñH@FTG^KGCIÆ]^FAB@(;: =: 7). 7). Ablaulg? b)T(P91): 91): n)lbgspgrbhzbutulfzbhmclbmg…hfafñhyvgrfeéquglbgiplgbhmclb eñriulbmg   G(x): a)lbvbrfbhzbmg   P. TLBH]GBIFGH]CMGLT^CNLGIB Mbtcs? 9 = H 9 ; r 9 7 h

@bllbr 91)93 b)T(P91)93 n)G(P)93 )93 a)U(P)93 )93 Acic

Mf Mfst strf rfnu nuaf afch ch

Hctb Hctbaf afch ch

  G(P)   Ubr(P)

Euh.  mg mg m mgh. 

Euh.  mg m g m mf sstt.

T(P 9 j ) j

@fpgrkgciætrfab   @(H:h:r)

 

rh H

r H

(2

 

r h

)

9

rj

) ( )(   ( ) h

Hh

j

H

^gscluafñhmglprcnlgib b)

171 A1 T(P91)9 91)9 = 1;= 9A1> =1 A;

;

n) G(P))9 9

a) U(P))9 9

7 (2 ;

7(=) >4 9 9> ? 07;2 ; ;

 

7 )9 =

Achalusfñh? =

  

7 9 >0

4?2;07;

  

T(P 9 x)9

r

Hr

P

hP H

) ( )( ( ) h

 

G(P)9> )9>? 07;2 U(P))9 9 4?2;07; GOG^AFAFC = ?5?5   pbk217

 

Ghuhblfhgbmgachtrcl mgablfmbmsg Ghuhblfhgbmgachtrcl mgablfmbmsg rgvf rgvfsbh24brtfaul sbh24brtfaulcs,mgtgr cs,mgtgr-ifhâhmcsgqug`by1 ifhâhm csgqug`by1 qughc aui auiplg plgh h achlbs gspg gspgaf… af…aba abafchg fchgs. s. Qfsg gsackghblbzbrmcsbrtéaulcs,fmghtf…quglcspbrâigtrcsmglblgyy `bllglbgspgrbhzbmglbvbrfbnlgblgbtcrfb   P,qugmgsarfngglhÿigrc ,qugmgsarfngglhÿigrc mgpgfzbsacrrgatbsghtrglbsmcsgsackfmbs. Tlbhtgbifwghtcmglprcnlgib. 2.Mbtcs? Ubrfbnlg blgbtcrfb blgbtcrfbP P- - hÿigrcmgpgfzb hÿigrcmgpgfzbacrrgatb acrrgatbghtrglbgsackf ghtrglbgsackfmb. mb.

H 9 24 h r

9 1 9 >

@bllbr G(P)93 )93 Acic Mf Mfs s ttr r ffn n ua ua ffc c h h

Hc ctt ba ba fc fc h

@fpgrkgciætrfab   @(H:h:r )

  G(P)   Ubr(P)

 

rh H

r H

(2



 r h

)

Euh.  mg mg m mgh.  ( 9 T P j )

9

( )( )   ( ) j

rj

h

Hh j

Euh.  m mg g m mf s stt.

T(P 9 x)9

H

^gscluafñhmglprcnlgib.

)9 G(P)9

>(1) 94 ?5 24

Achalusfñh G(P)94 )94?5 GOG^AFAFC = ?5?;   pbk217 Tbrb Tb rb llghbr llghbr = vbabht babhtgs gs mg ach achtbm tbmcr cr sg prg prgsgh sghtbh tbh 24  pgrs pgrschbs chbs,, ; `cinrgsy 1 iuogrg iuogrgs. s. Qblghsglgaafc Qblghsglgaafchbmcs hbmcs 1 `cinrgsy 2 i iuog uogr. r. Lbs iuogrgsbausbhblgiplgbmcrmgmg iuogrgsbaus bhblgiplgbmcrmgmgsarfifhba sarfifhbafñhsgxubl,pcr fñhsgxubl,pcr lcquglg llgv llgvbhb oufafc oufafc. . Qfgl oug ougzsupch zsupchgquglb gquglb sgl sglgaa gaafcheug fcheugbl bl bzbr,³pugmgmgafrsgquggxfstfñmfsarfi bzbr,³pugmgmgafr sgquggxfstfñmfsarfifhbafñhbl`bagrlbglgaafñh3 fhbafñhbl`bagrlbglgaafñh3 Tlbhtgbifghtcmglprcnlgib Mbtcs

9 ; H 9 24 r 9 = h

@bllbr 7

( )( ) ( ) r

Hr

P

hP H h

   

T(P97)93 97)93

Acic ]bnlbmgmfstrfnuafñh?   G P

M ffst strr fnu fnuaf afc c h h

Hc Hctt b baf afc ch  

@fpgrkgciætrfab   @(H:h:r )

  Ubr P

( )

EuT h(.P  mg mg j mg) h.  9m

( )

rh H

r H

(2    hr )

9

( )( )   ( ) j

rj

h

Hh

j

Euh.  m mg g m mf st st. T(P 9 x)9

H

( )( ) ( ) r

Hr

P

hP H h

^gscluafñhmglprcnlgib.

7 = 1  A7   A7 A2 T(P91)9 91)9 ;   724; 9 ; 7 1 9A1 7; 94 ?7 A24

A24

24

Achalusfñh? Tcrctbhtchcgxfstgmgsarfifh Tcrctbhtchcgxfstgmgsa rfifhbafñ bafñhpcrquglbprcnb hpcrquglbprcnbnflfm nflfmbmgsbltb. bmgsbltb. GOG^AFAFC = ?5?0   pbk217 Glgxbighmgkrbmubafñhmglcsbnckbmcsachstbmg 74 tgibs.Lb ecribmg ecr ibmg gxbifh gxbifhbrgs brgs lbsfkufg lbsfkufght htg? g? pcrscrtgcsg glfkgh5 glfkgh5 tgibsmg tgibsmg lcs qug`byqug ach lcs achtgs tgstbr tbr 1 pbr pbrb b bprcnb bprcnbr. r. Qfgl gst gstumf umfbh bhtgsclc tgsclc `b gstumfbmc14tgibs? b)³Aublgslbprcnbnflfmbmmglcs5tgibssgpbachtgstbracrrgatbightgb13 Mbtcs ! Huigrc vbrfbnlgblgbtcrfb?   _! u igrc mgtgibpbrbbprcnbr ( H) ? 74 Hÿigrctctblmgtgibs( Hÿigrctctblmgtgibs Hÿigrcmgtgibsgstumfbmc ( h) ? 14 Hÿigrcpcrscrtgcmgtgibs( Hÿigrcpcrscrtgcmgtgibs ( r) ? 5 b)Hÿigrcmgtgibspbrbbprcnbr( b)Hÿigrcmgtgibspbrbbprcnbr ( j) ? 1 n)Hÿigrcmgtgibspbrbbprcnbr( n)Hÿigrcmgtgibspbrbbprcnbr ( j) ? e 1: =: 7: 5k @bllbr? Lbprcnbnflfmbm Lbprcnbnf lfmbmmglcs5 mglcs5 tgibssgpbachtgstbr tgibssgpbachtgstbracrrgatb acrrgatbightgb1 ightgb1 Acic M ffst strr fnu fnuaf afc c h h Hc Hctt b baf afc c h   G(P)   Ubr(P) Euh.  mg mg m  mgh.  Euh.  m mg g m mf st st. T(P 9 j ) ( )( ) ( )( )   rh r  r H H h @fpgrkgciætrfab   @(H:h:r ) ( ) (2  ) 9 ( )   T(P 9x)9 Tlbhtgbifghtcmglprcnlgib j

rj

r

Hr

h

Hh

P

hP

j H

T(_9 9 J)9

H h

r j Ahj AH h r AH

b) T(_91)9 91)9

5 1 1   A14 A74 14 5 A74

9

1 1 >>4=   A14 A>4 9 94 ?>62>5 5 A74 ;75;

n) T(_9(1 9(1: =: 7: 5))9

51 1   A14 A74 14 5 A74

+

5= = A14 A74 14 5 A74

5

+

57 7 A14 A74 14 5 A74

+

55 5 A14 A74 14 5 A74

94 ?01755

 

Achalusfñh T(_91)94 91)94?>62>5 T(_94  94?01755 GOG^AFAFC = ?5?6  pbk217

Whbumftcraciprugnblbachtbnflfmbmmguhbgiprgsbytcibaciciugstrb 1aughtbsmguhblfstbmg0aughtbspcracnrbr.Ablaulglbprcnbnflfmbmmgqug glbumftcrghaughtrgpcrlcighcsuhbaughtbvghafmb.sf`by? b)>aughtbsvghafmbsghtrglbs0sglgafchbmbs n)=aughtbsvghafmbs Tlbhtgbifghtcmglprcnlgib.

Mbtcs H 9 0

9 5 r 9 1

h

@bllbr >aughtbsvghafmbsghtrglbs0sglgafchbmbsllbr =aughtbsvghafmbs. ;aughtbsvghafmbs. Acic M ffs s tr tr fn fn u ua a fc fc h h

Hc ctt b ba a fc fc h

@fpgrkgciætrfab

@(H:h:r)

  G(P)  

Ubr(P)

Euh.  m mg g m mgh.  T(P9 9 j )

( )( ) ( ) j

 

rh

r

(2     hr ) H

H

9

h

T(P 8

A51 A>> 2)92   A01

T(P 8

2)92  

T(P 8

>)92  

T(P 8

>)9

n) ;

6 2=

A51 A>> A01 5!   >!  1!1! >!4! 0! 1!7!

( )( ) ( ) r

rj

Hh

j

H

^gscluafñhmglprcnlgib.

Euh.  m mg g m mfs tt..

9 x )9 T(P9

P

Hr

hP

H h

 

T(P 8

2)92 

A=2 A=> 1

A

0

T(P 8

2)92 

T(P 8

2)9

T(P 8

2)92 

T(P 8

2)92 

T(P 8

2)9

  =! =!  >!>! 2!1! 0! 1!7!

9

= ;

9

7 0

= ;

a) A22 A;> A01   ;! 2!  >!7! 2!4! 0! 1!7!

7 0

Achalusfñh 6 T(P 8>)9   2=

T(P 82)9   ;= T(P 82)9   07 GOG^AFAFC = ?5?24  pbk217

Whbgiprgsbrghtbbutcs,blcsqughclgsmbglibhtghfifghtc mgnfmc,pcrlcqugblkuhcseuhafchbhibl.Whmébtfghghmfspchfnlgs 0butcspbrbsgrrghtbmcs,mglcsaublgs1euhafchbnbibl.Gsgmébsg rghtbrch=butcs,ablaulglbprcnbnflfmbmqug? b) Hfhkÿh Hfhkÿhal alfg fgh htgrg tgrgaf afnbu nbuhbu hbutcq tcqu ug euh uhaf afch chg g ibl n) Tcrlc crlc ig ighc hcsuhalf suhalfg ghtgrga tgrgafnbuhbut fnbuhbutcquge cqugeuh uhaf afchgib chgibll a) ]rgs rgsalf alfg ght htgsrga gsrgafnb fnbhbut hbutcsquge csqugeuh uhafc afchghi hghibl bl Tlbhtgbifghtcmglprcnlgib Mbtcs? Ubrfbnlg blgbtcrfb P~huigrcmgpgrschbsblbsqugsglgrghtbrbglbutc H ? huigrctctblmgbutcsH 90 h ? huigrcmgbutcsqugeuhafchbhiblh 91 r ? huigrcmgbutcsqugeuhafchbhnfgh r 9= j ? huigrcmgpgrschbs @bllbr b)T( ( P 94) n)T( ( P  2) a)T( ( P 91)

0

 

Acic M ffst strr fnu fnuaf afc c h h

Hc Hctt b baf afc ch

  G(P)   Ubr(P)

Euh.  mg mg m  mgh.  T(P 9 j ) j

@fpgrkgciætrfab   @(H:h:r ) ^gscluafþhmglprcnlgib. b)

 

rh H

r H

(2    hr )

rj

9 ( )( ( ) )  h

Euh.  m mg g m mf st st.

Hh j

H

r

Hr

T(P 9 x)9 ( )( ( ) ) P

hP H h

1 7  2 T( ( P94)9 94)9 4 0 = 9

2=

=

n) T( ( P  2) 9   T( ( P94)+ 94)+  T( ( P92)9 92)9

2 1 2 1 21 + 9 + 9 94 ?6>07; 2= ; > ; 2=

1 7  1 2 0 1 9 94 ?=>07; (P 9 2)  9 ;

=

a)

2

T( ( P91)9 91)9

2=

+

> ;

9

7 2=

Achalusfñh 2 T( ( P94)9  94)9   2= T( ( P  2)94?6>07; 7 T( ( P91)9 91)9   2=

= LGVG LGVGQMGNG QMGNG^H ^HCW CWLL LLF F VNF VNFHC HCIF IFBL BL?? GOG^AFAFC = ?5?22   pbk215.

WhbvbrfbnlgblgbtcrfbPtfghgmfstrfnuafñhNfh(=,4.>)

Tlbhtgbifghtcmglprcnlgib. Mbtcs? PtfghgmfstrfnuafñhNfh(=,4.>) Nfh(=: 4?>)

Acic M ffst strr fnu fnuaf afc c h h

Hc Hctt b baf afc ch

@fpgrkgciætrfab   @(H:h:r)

  G(P)   Ubr(P)  

rh H

(2    hr ) H r

Euh.  mg m g m mgh.  Euh.  m mg g m mfs tt.. T(P9 9 j ) ) ) ( )( ( )(   T(P9 9 x )9 9 ( ) ( ) j

rj

r

Hr

h

Hh j

P

hP H

H

@bllbr

6

h

 

b)   T(P9>)   a)   T(P 9>)  >) n)   T(P   m)   G(P)  >) ^gscluafþhmglprcnlgib.

 

g)

 

U br (P)

T(P 9   j)9 Ahj   pj  qhj T(P 9  4) 9 A=4 (4?>)4  (4?0)=4 94 ?=465 T(P 9  2) 9 A=2 (4?>)2  (4?0)=2 94 : =465 T(P 9  >) 9 A=> (4?>)>  (4?0)=> 94 : 2715 T(P 9  1) 9 A=1 (4?>)1  (4?0)=1 94 : 4>75 T(P 9  =) 9 A== (4?>)=  (4?0)== 94 : 4425

b) T(P9>)94 9>)94 : 2715

n)  >)9 T(P9>)+ T(P 9>)+ T(P91)+ 91)+ T(P9=)94 9=)94?2715+4?4>75+4?442594?2040

a) 94)+ T(P92)+ 92)+ T(P9>)94 9>)94?=465+4?=465+4?271594?6;>0 T(P  >)9 T(P94)+ m) G(P))9 9 hp 9=  4?>94 ?0

g) U br (P)9 )9 hpq9=  9=  4?>  4?094?5=

Achalusfñh 9>)94 : 2715 T(P9>)94 T(P  >)94?2040  >)94?6;>0 T(P G(P)94 )94?0 U br (P)94 )94?5= GOG^AFAFC = ?5?2>   pbk215 Whbibqufhbllghblbsabobsmg Whbibquf hbllghblbsabobsmg pblf pblfllcsmgeñsecr llcsmgeñsecrc. c. Ghuhbprcpcrafñ Ghuhbprcpcrafñhmgl hmgl 24%lbibqufhbhcllghblbibqufhbpcraciplg 24%lbibqufhbhcllghblbibqufhbpcrac iplgtc. tc. Qgtcibblbzbr>7abobsmg eñsecrcs,Ablau eñsec rcs,Ablaulglbprcnbnf lglbprcnbnflfmbm lfmbmmgqughc mgqughc `bybibsmg `bybibsmgmcsabobsfhac mcsabobsfhaciplgt iplgtbs. bs. 2.Tlbhtgbifghtcmglprcnlgib. Mbtcs.

24

 

Nfh(>7: 4?2)

Trcnbnflfmbmmggxftccerbabsc 94?2 Hÿigrcmggxpgrfightcs 9>7 P9 9 prcnbnflfmbmmgabobsfhaciplgtbs @bllbr Ablaulglbprcnbnf Ablau lglbprcnbnflfmbm lfmbmmg mgqughc`bybibsmgmcsabobsfhaci qughc`bybibsmgmcsabobsfhaciplgtb plgtbs. s. Acic. M ffs s tr tr fn fn u ua a fc fc h h Hc ctt b ba a fc fc h   G(P)   Ubr(P) Euh.  m mg g m mgh.  Euh.  m mg g m mf st st. T(P 9 j ) ( )( ) ( )( ) r @fpgrkgciætrfab   @(H:h:r )   rh (2    hr ) 9   T(P 9 x)9 H H ( ) ( ) j

rj

r

Hr

h

Hh j

P

hP H

H

h

^gscluafñh ^gsclua fñh mglprcnlgib.

T(P 9   j)9 Ahjpj  qhj T r(P    T r(P    T r(P   

9 9 9

 >)9 >7 >)9 >7>  >)9 >7 > 42 > >7 42 2 >7 h j

pj  qhj >7

 >7 2  >7 

2

>7

>

 4?2  4?6

>

 4?6

 ?

2

 4?6

 4?2  4?6

>

+

2

>7

  >7  4  >7  4

2 p  q  +

> p  q  +

 ?

4

4

>

2

>7

 4?2  4?6

>7

p  q

2

+

4  4?2  4?6 4

4

>7

> 94 ?>5706

>7

2 94 ?266=>

>7

4 9; ? 2;64  24>

9 4?>5706+4?266=>+;? 2;64  24> 94 ?71;2 Achalusfñh. T r P ? ?  ( >)94 71;2 GOG^AFAFC = ?5?21   pbk215. Whbghaugstbrgvg Whbghau gstbrgvglbquggl>4%mglbpcnlb lbquggl>4%mglbpcnlbafñhgsebvc afñhgsebvcrbnlg rbnlgbuhpclftfacy buhpclftfacy glrgstcgsmgsebvcrbnlg.Qfsgglfkgh5pgrschbsblbzbr,sgmgsgbsbngr. 2.Tlbhtgbifghtcmglprcnlgib. Mbtcs. Nfh(5: 4?0) Trcnbnflfmbmmgpgrschbmgsebvcrbnlg 94?0 Hÿigrcmggxpgrfightcs 95 @bllbr

22

4

>7



 

T(P95) 95) T(P9=) 9=) Acic?

M ffst strr fnu fnuaf afc c h h

Hc Hctt b baf afc ch

  G(P)   Ubr(P)  

@fpgrkgciætrfab   @(H:h:r )

rh H

r H

(2     hr )

Euh.  mg mg m  mgh.  Euh.  m mg g m mf st st. T(P 9 j ) ( )( ) ( )( ) 9   T(P 9 x)9 ( ) ( ) j

rj

r

Hr

h

Hh j

P

hP H

H

^gsclusfchmglprcnlgib. P9 9 prcnbnflfmbmmg5pgrschbsmgsebvcrbnlgs T(P 9   j)9 Ahjpj  qhj T r(P

 9  5) 9 5 h j

T r(P 9  5) 9

9

55

 

5

 

5

4?0

pj  qhj 5

5   p  q 

5



5

5

4?>

94 ?>5>2=

Trcnbnflfmbmmgpgrschbebvcrbnlg 94?> Hÿigrcmggxpgrfightcs 95 P9 9 prcnbnflfmbmmg=mg5pgrschbsebvcrbnlgs

T(P 9   j)9 Ahjpj  qhj T r(P T r(P

 9  =) 9 5 9  =) 9 5 = h j

9

=

 

 

pj  qhj =

5

=   p  q



4?>=  4?05= 94 ?42715

Achalusfñh T(P95)94 95)94 ?>5>2= T(P9=)94 9=)94 ?42715 GOG^AFAFC = ?5?2=   pbk215.

Whbmgtgrifhbmbrbzbmgpgrrcstfghghaubtrcaba`crrcsghabmb abibmb.Qflbprcnbnfl abibmb .Qflbprcnbnflfmbmmgquguhaba`crrcsgbiba`c fmbmmgquguhaba`crrcsgbiba`cgsmg gsmg4.77, 4.77,sg sg pfmgablaulbr? b) b) Lb Lbpr prcn cnbn bnfl flffmb mbmmg mmgqu quggh gghuh uhbab babib ibmb mb > gxbatbightgsgbh `ginrbs n)  Lb prc prcnb nbnf nfll ffmb mbm m mg qug gh uhb abi abib mb mb  b l ighc ighc s > sgbh `ginrbs 2>

h

 

  G(P)   Ubr(P)

Mfstrfn Mfst rfnuaf uafch ch Hct Hctbaf bafch ch

Nfhcifbl

 

Nfh(h: p)

 

hp 

Eu h h..  m g gm gh gh . . T(P9 9 j )9 h j p (2 p)hj j : j 94: 2: ?????: h 



hpq 

Eu h ha af ñh ñh m m g gm fs fs tt..

j

E ( P  j )9

T

T(xf )

f92

Tlbhtbifghtcmglprcnlgib Mbtcs? P?? Ubrfbnlgblgbtcrfb) n)T(P  > Acic? M ffs s tr tr fn fn u ua a fc fc h h

Hc ctt b ba a fc fc h

  G(P)   Ubr(P)  

@fpgrkgciætrfab   @(H:h:r )

rh H

r H

(2    hr )

Euh.  m mg g m mgh.  T(P 9 j )

9

( )( )   ( ) j

rj

h

Hh j

Euh.  m mg g m mf st st.

T(P 9 x)9

H

j h

j

h

T(P9 9 j )9 A 9 p q j

j 94: 2:???:h

^gscluafñhmglprcnlgib. b) >

>

=

T(P9>)9 9>)9 A= (4?77) (4?=7)

>

=! > > 9 >!(=  >)! (4?77) (4?=7) 94 ?15;7=

n) T(P  >)9 T(P94)+ 94)+ T(P92)+ 92)+ T(P9>)  9>) 94?15;7=+4?>44=0+4?4=2445>94?54641

T(P 9  4) 9 A=4 (4?77)4 (4?=7)=4 94 ?4=2445> T(P 9  2) 9 A=2 (4?77)2 (4?=7)=2 94 ?>44=0

Achalusfñh?

21

( )( ) ( ) r

Hr

P

hP H h

 

b)T(P9>)94 9>)94 ?15;7= n)T(P  >)94?54641 GOG^AFAFC = ?5?25   pbk215. Ghuhbfhstblbafñhiflftbrqugmfspchgmg7rbmbrgs,lbprcnbnflfmbmmgquguhsclcrbmbrmgsaunrbbuhbvfñhmgacinbtggsmg 4,; b)³Auâlgslbprcnbnflfmbm b)³Auâlgslbprcnbnf lfmbmqugsgbhgxbatbight qugsgbhgxbatbightg=rbmbrgslcsqug g=rbmbrgslcsqug mgsaunrghglbvfñh3 n)³Auâlgslbprcnbnflfmbmmgqugpcrlcighcsuhclcsmgsaunrb3 a)³Mgauâhtcsrbmbrgsmgngachstbrlbfhstblbafñhpbrbbsgkurbrsg ghmgtgatb ghmgtg atbrbvfc rbvfchgsblighc hgsblighcsgh sgh uh60%mglbsvgags3 uh60%mglbsvgags3 Tlbhtgbifghtcmglprcnlgib. Mbtcs vbrfbnlgblgbtcrfbP ? hÿigrcmgbvfc hÿigrcmgbvfchgsqugmgsa hgsqugmgsaunrg unrgglrbmbr glrbmbr h 97 p 94: ; @bllbr 9=) b) T(P9=)  2) T(P n)T a) (P 9j ) Acic Mfstrfn Mfst rfnuaf uafch ch Hctbaf Hctbafch ch

Nfhcifbl

 

  G(P)   Ubr(P)

Nfh(h: p)  

hp 

hpq 

Euhafch uhafchmgmgh mgmghsfm sfmbm bm Euha. uha. mg mfst fst.. T(P9 9 j ) E(P  j ) j j 9Ahp (2 p)hj : j 9 j 94: 2:???:h T(xf )

T

f92

Igmfbhtglbmfstrfnuafñhnfhcifbl?  Nfh(h: p) :T(P9 9 j )9 Ahjpj (2 p)hj ^gsclusfñhmglprcnlgib b) 7 T(P9=)9 9=)9 (4?;)= (4?1)2 94 ?15427 =



n) T(P  2)92  T(P < 2)92  ZT(P94)Y92 94)Y92  (A74 (4?;)4 (4?1)74 )94?66;7;

Achalusfñh b)Lbprcnbnflfmbmmgquggxbatbightgsgbh = rbmbrgsgs? 4: 15 n)Lbprcnbnflfmbmmgqugblighcsuhclcmgsaunrbgs? 4: 66 GOG^AFAFC = ?5?2;   pbk215.

GOG^AFAFC = ?5?20   pbk21;.

Qupñhkbsgquglbtbsbmgfhegaafñ Qupñhkb sgquglbtbsbmgfhegaafñhmguhbghegrigmbmacht hmguhbghegrigmbmachtbkfcs bkfcsb b c…afhb `by24 pgrschbsqug pgrschbsqug sgvbauhbrchach sgvbauhbrchachtrb trb gsmgl   >7%. Ghuhb c…afhb lbghegrigmbmyhfhkuhbpgrschbsgachtbkfñ.

2=

 

b) Mgtg Mgtgri rifhg fhglb lb prc rcnb nbnfl nflfm fmbm bm mg qug hfhkuh fhkuhb b pgrsc pgrschb hb sg `unfgrbachtbkfb nfgrb achtbkfbmcbpgsbrmgqughcsg mcbpgsbrmgqughcsg `unfgrbvbauhbmc `unfgrbvbauhbmc.. n) Mggst Mggstg rgsu gsultb ltbmc³M mc³Mg gmuag muag ustg stgm m quglbv uglbvbau bauhbgs hbgsgeg gegatf atfv vb3 Tlbhtgbifghtcmglprcnlgib. Mbtcs.

9 4?>7 q 9 4?;7 h 9 24

p

@bllbr T(P94) 94)

Mgmuagustgmquglbvbauhbgsgegatfvb. Acic Mfstrfn Mfst rfnuaf uafch ch Hctbaf Hctbafch ch   G(P)   Ubr(P)

Nfhcifbl

 

Nfh(h: p)  

hp 

hpq 

Euhafch uhafchmgmgh mgmghsfm sfmbm bm Euha. uha. mg mfst fst.. T(P9 9 j ) E(P  j) j j 9Ahp (2 p)hj : j T T(xf ) j 94: 2:???:h 9 f92

^gscluafñhmglprcnlgib. b) 4  (4?>7)4  (4?;7)244 97 ? 512=  24>  4?47512 T(P94)9 94)9 A24

n)Qf,pcrquglbprcnbnflfmbmmgqugblkufghsgachtbkfggsiuynbob. Achalusfñh. T(P94) 94)  4?47512 Qf,pcrquglbprcnbnflfmbmmgqugblkufghsgachtbkfggsiuynbob. GOG^AFAFC = ?5?26   pbk21;. Lbprcnbnflfmbmmgquguhgstumfbhtgcntghkbgltétulcmgbrquftg tgat atc c gsmg 4.1 4.1. . Ab Abla laulglb ulglb pr prcn cnbn bnflf flfmb mbm m mgqug uh kr krupcmg upcmg sfgtg sfgtg gstumfbhtg gstumf bhtgs s ibtrfaulbm ibtrfaulbmc csgh prfigraursc? prfigraursc? Lcssfgtg…hblfaghlb abrrgrb: 2.  (b) Blighcsmcsbabnghlbabrrgrb: (n) ³Mgauâhtcsbluihcs`bmgachstbruhbprcicafñhpbrb bsgkurbrsgmgqugblighcsuhcaulifhgsuabrrgrb,achuhb prcnbnflfmbmmgl   66%3? Tlbhtgbifghtcmglprcnlgib. Mbtcs _?`` hÿigrcmggstumfbhtgsqug…hblfzghlbabrrgrb ff ach_ Nfh(;:4?1) @bllbr b)lcssfgtg…hblfaghlbabrrgrb: n)Blighcsmcsbabnghlbabrrgrb:

27

 

Acic Mfstrfn Mfst rfnuaf uafch ch Hctbaf Hctbafch ch

Nfhcifbl

 

  G(P)   Ubr(P)

Nfh(h: p)  

hp 

hpq 

Euhafch uhafchmgmgh mgmghsfm sfmbm bm Euha. uha. mg mfst fst.. T(P9 9 j ) E(P  j ) hj j j j h T : 9A p (2 p) T(xf ) j 94: 2:???:h 9 f92

^gscluafñhmglprcnlgib. b) ; ; 9  2(4: 444>20;  2) 9 4: 444>20;

T r(_ 9  ;) 9 A (4: 1); (4: ;);;

n) 2

     _ 

>

 >)9 T r(_94)+ T r(4      _ 94)+  T r(_92)+ 92)+  T r(_9>) 9>)

4 2 > 9   A (4: 1)4 (4: ;); + A (4: 1)2 (4: ;);2 + A (4: 1)> (4: ;);> ; ; ; 9 4: 5=;45

Achalusfñh. T r(_9;)94 9;)94 : 444>20; T r(4  _  >)94 : 5=;45 GOG^AFAFC = ?5?>4   pbk21;.

Ghuhgstumfcigmfcbinfghtblsgmgtgrifhñquglbprgsghafbmg igraurfcghglbkubghvghghbbl>7%mglcspgagsgh>=`crbs.Tbrb ach…ribrglrgsultbmc sgaclcabrch25pgagsghuhtbhqugachbkubachtbifhbmb. sgaclcabrch25pgagsghuhtbhqugachbkubachtbifhbm b. Ablaulg lbprcnbnflfmbmmgquggh>=`crbs? b)scnrgvf b)scnrgvfv vbhgxbatb bhgxbatbigh ightg24 tg24 pgags: n)scnrgvfvbhpcrlcighcs2>pgags a)scnrgvfvbhaubhmciua`c>pgags: m)Ablaulgglhÿircmgpgagsqugsggspgrbscnrgvfvbh g)Ablaulglbvbrfbhzbmglhÿigrcmgscnrgvfvfghtgs Tlbhtgbifghtcmglprcnlgib. Mbtcs? Ubrfbnlgblgbtcrfb?   P    Hÿigrcmgpgagsvfvcsghuhtbhqugachbkub achtbifhbmb. h ? ?gsglhÿigrcmgprugnbs. gsglhÿigrcmgprugnbs.   h 925 gsglhÿigrcmgæxftcs.   j 924 j ? ?gsglhÿigrcmgæxftcs. gslbprcnbnflfmbmmgæxftc.  p 94: >7 p ? ?gslbprcnbnflfmbmmgæxftc. ? gslbprcnbnflfmbmmgerbabsc.  q94  94: ;7 q?gslbprcnbnflfmbmmgerbabsc.

25

 

@bllbr b)T(P924)93 924)93 n)T(P  >)93 a)G(P)93 )93 m)U(P)93 )93 Acic Mfstrfn Mfst rfnuaf uafch ch Hctbaf Hctbafch ch

  G(P)   Ubr(P)

Eu h. h.  m g m m gh gh . . T(P9 9 j )9 h j p (2  p)hj

Eu ha haf ñh ñh m m g gm fs fs tt..

 Nfhcifbl

 

Nfh(h: p)  

hp 

hpq 

j : j 94: 2: ?????: h 

j

E ( P  j )9

T

f92

^gscluafñhmglprcnlgib. b) T(x

9  24) 9 9

25

24 9 4: 44215

(4: >7)24(4: ;7)2524

n) T(P   

>)9 T(P94)+ 94)+  T(P92)+ 92)+  T(P9>) 9>) 25 25 25 9 (4: >7)4(4: ;7)25 + (4: >7)2 (4: ;7)252 + (4: >7)> (4: ;7)25> 4 2 > 9 4: 26;224

 

 

 

a) G(P)9(25)(4 )9(25)(4 : >7)9=

m) U(P)9(25)(4 )9(25)(4: >7)(4: ;7)91?

g) U br (P)9 )9 hpq9=  9=  4?>  4?094?5=

Achalusfñh T(x 924)94 : 44215 T(P  >)94: 26;224 G(P)99= )99= )91? U(P)91 U br (P)94 )94?5= GOG^AFAFC = ?5?>2   pbk21;. Whbacipbhfbpgtrcl Whbacipb hfbpgtrclgrbvbbpgrecrbr grbvbbpgrecrbr >6pczcs,abmbuhc mggllcstfghglb prcnbnflf prcnb nflfmbm mbm mg4.2 mg4.2 mgpr mgprcmua cmuafrp frpgtrþl gtrþlgc gc mgibhgrb mgibhgrbrght rghtbnlg. bnlg. Blbaci Blbacipb pbòfb òfb lg augs ugstb tb 244mñl 244mñlbrg brgs s pb pbrb rb pg pgrec recrbr rbr abm abmb b pcz pczc. c. Wh pc pczcac zcacig igraf rafbl bl gxtrbg gxtrbg pgtrñllgc pcruhvblcrm pgtrñ pcruhvblcrmg g 7ifl 7ifllchgs lchgsmgmñl mgmñlbrgs.Ablaul brgs.Ablaulgg 2. b) Lb kbhbhafbquggspgrb kbhbhafbquggspgrbcn cntghgr tghgr lbacipbòfb lbacipbòfbpcr pcrlcspczcs. lcspczcs. 2;

T(xf )

 

n) Lbmgsvfba Lbmgsvfbafñhgstbhm fñhgstbhmbr br mgvblcrmglbkbhbhafb mgvblcrmglbkbhbhafb.. Tlbhtgbifhghtcmglprcnlgib Mbtcs?

9 J 9 pf 9 a  9 h

>6 ; 4?2 244

@bllbr Auâl l gs lbprcnbnf lbprcnbnfllfmb fmbm m quggl batcr batcr rg rgaft aftg g acr acrg gatb atbigh ightgsu tgsu ³Auâ mfâlckcghlbsgxtbvgz3. ^gscluafñhmglprcnlgib. Acic Hctbaf Hct bafñh ñh Euhafñhmg uhafñhmg mghs ghsfmb fmbm m   e(x)

Euhafñhmg mfstrfnuafñh mfstrfnuafñh   E(P)  Gspg Gspgrb rbhz hzb b   G(x) x

N (h: p)

 

h

j

h

e(x)9 j p (q) j

pj (q)hj

j94

^gscluafñhmglprcnlgib.

T(P

j

P 



T(P

h

E(x)9

  9  ;) 9 ( ) 24 9  ;) 9 (4 1) (4 ;) ; 24 h j h p q j ?

T(P 9  ;) 9

;

j

?

1

(4?1); (4?;)1

; 9 44?>15>? T(P 9  ;) 9

Achalusfþh. Lbprcnbnflfmbmmgqugbprugngh Lbprcnbnflfmbmmgqug bprugngh ghlbs2>prfigrbsprgkuhtb ghlbs2>prfigrbsprgkuhtbsgsmg sgsmg4?>;=?

GOG^AFAFC = ?5?>>   pbk21;.

Whbléhgbbærgb,`bnfgh Whbléhgb bærgb,`bnfghmccnsgrvb mccnsgrvbmcquggl7%mg mcquggl7%mglbspgrschb lbspgrschbsqug squg `baghrgsgrvbaf `baghrgsgrv bafñhhc ñhhc sgprgsghtbh sgprgsghtbh pbrbgl pbrbgl vuglc, vuglc, vghm vghmg g 244 244 nclgtc nclgtcss pbrbuhbvfñhqugtfghg   67bsfghtcs.³Auâlgslbprcnbnflfmbmmgqug, blicightcmglvuglc,`bybuhbsfghtcmfspchfnlgpbrbabmbpbsbogrc3. Tlbhtgbifghtc Mbtcs.

P Ubrfbnlgblgbtcrfbmfsargtb:bsfghtcsmfspchfnlgspbrbabmbpbsbogrc h 9244:hÿigrctctblmgnclgtcs p 97% 967% q967%



20

hp

 

@bllbr T(P  67)93 Acic T(P9 9 j )9 Ahjpj qhj T(P  67)92  T(P 8 67) ^gsclusfñhmglprcnlgib T(P  67)

9  2  T(P 867) 9 2  ZT(P965)+ 965)+ T(P96;)+ 96;)+  T(P960)+ 960)+  T(P966)+ 966)+  T(P9244)Y 9244)Y 244 T j j hj A p q 9 2 h

j965

244

9

2

T

j A244 (4?47)j (4?67)244j

j965

Achalusfñh

244

T

T(P  67)92 

j A244 (4?47)j (4?67)244j

j965

GOG^AFAFC = ?5?>1   pbk21;. Ghuhgxbighsg plbhtgb24prgkuhtbs plbhtgb24prgkuhtbs blbsqugmgnbhrgspchmgrsg blbs qugmgnbhrgspchmgrsg vg vgrmb rmbmbm mbmgrc grc c eblsc. eblsc. uhbluihcbprc uhbluihcbprcnbr nbrb b glgxbigh glgxbigh sf bl ighcs ighcs ; rgspug rgs pugstb stbs s sch sch bag bagrrtbm tbmbs. bs. ³ qug qug prcn prcnbnff bnffmbm mbm mg bp bprrcnb cnbr r tfg tfghg hg uh gstumfbhtgqugrgspchmbblbzbr³yaubhmcuhcsgsbngquggl14% mglbbsfkhbturb. 2. Mbtcs?

9 J 9 p 9 q 9 h

24 ; 4?1 4?;

@bllbr ³Auâ Auâl l gs lbprcnbn lbprcnbnfl flfm fmbm bm qug qug glbatcr glbatcr rg rgaft aftg g acr acrg gatbigh atbightgsu tgsu mfâlckcghlbsgxtbvgz3.

^gscluafñhmglprcnlgib. Acic Hctbaf Hct bafñh ñh Euhafñhmgmg uhafñhmgmghs hsfmb fmbm m   e(x)

Euhafñh mg mfstrfnuafñh   E(P)  Gspg Gspgrb rbhz hzb b   G(x) x

N (h: p)

 

h

e(x)9 j p (q) j



j

h

E(x)9

P  h j

j94

>. ^gscluafñhmglprcnlgib.

26

pj (q)hj

hp

 

h

T(P 9  ;) 9

pj (q)hj

j

T(P

  24 9  ;) 9 (4 1) (4 ;) ; 24 ?

;

?

1

(4?1); (4?;)1

T(P 9  ;) 9

; T(P 9  ;) 9 9 44?>15>?

Achalusfþh. Lbprcnbnflfmb Lbprcnbnfl fmbm m mgqug bprug bprugngh ngh ghlbs 2> prfig prfigrbsprgk rbsprgkuh uhtbsgs tbsgs mg 4?>15>?

7 LGVG LGVGQKG QKGCI CIÆ]^ Æ]^FA FABQ BQV V NFHCI NFHCIFB FBLHG LHGKKB]FUB. GOG^AFAFC = ?5?>=   pbk21;.

Aubhmcsgkrbnbuhacigrafblmgtglgvfsfñh,lbprcnbnfnflmbmmg quguhbatcrrgaftgacrrg quguhbatcrrgaf tgacrrgatbi atbightg ghtg glmfâlckc glmfâlckc mgtutgibgsmg mgtutgib gsmg4.1. 4.1. 2. Mbtcs?

@bllbr Auâl l gs lbprcnbn lbprcnbnfl flfm fmbm bm qug qug glbatcr glbatcr rgaft rgaftg g acrg acrgatbigh atbightgsu tgsu ³Auâ mfâlckcghlbsgxtbvgz3. ^gscluafñhmglprcnlgib. Acic Hctbaf Hct bafñh ñh Euhafñhmgmg uhafñhmgmghs hsfmb fmbm m   e(x) Euhafñh mg mfstrfnuafñh   E(P)  Gs Gspg pgrb rbhz hzb b   G(x) K(p)

e(j)9 TZP 9 j Y9(2

 

NH(r: p)

Achalusfþh.

 

T(P9 9 j )9

j+h 2 h 2

j

p)

 h j  p r

x f94

(2 p)fp sf x 8 4 Fp (r: j+2)   mchmg Fp (x: y )

E(x)9

T







Lbprcnbnflfmb Lbprcnbnfl fmbm m mgqug bprug bprugngh ngh ghlbs 2> prfig prfigrbsprgk rbsprgkuh uhtbsgs tbsgs mg 4?>15>?

GOG^AFAFC =?5?>7   pbk21;.

Ghuhgxbighglprcegscrrgblfzbvbrfbsprgkuhtbsbuhgstumfbhtg. Lbprcnbnflfmbmmgqug bhtg. Lbprcnbnflfmbmmgqug glgstumfbhtgrgspchmbacrrgatbi glgstumfbhtgrgspchmbacrrgatbight ghtg gb b aublqu aublq ufgrprgkuh fgrprgkuhtbgsfkubl tbgs fkubl b4.6. Glprcegscrfhtg Glprcegscrfhtgrrui rruipggl pggl gxbigh gxbigh bpghbsglgstumfbhtgibhf…gstbglmgsachcafifghtcmglbprgkuhtb `ga`b. @bllbr >4

 

 

2

p r p

 

b) Ecr crib ibr r lb lg lgy y mg mfstrfn fstrfnua uafñh fñh mg lbvbr lbvbrfb fbnl nlg g blg blgbt btcr crfb fb qug mgsarfngglhÿigrcmgprgkuhtbsqugrgblfzbglprcegscr. n) @bllbr @bllbrglh glhÿ ÿigrcgs igrcgspgr pgrbmc bmcmgp mgprg rgkuh kuhtbs tbsqug qug`b `brgb rgblfz lfzb bmcgl prcegscr. Acic Hctbafñ Hctb afñh h Euhafñhmgmgh uhafñhmgmghsfmb sfmbm m   e(x) K(p)   e(j)9 TZP9 9 j Y9(2 p)j  h2 h j NH(r: p)   T(P9 9 j )9 j+ p r h2  

Euhafñh uhafñh mg mfstr fnuafñh h   E(P)  Gs Gspg pgrb rbhz hzb b   G(x)  Tx mfstrfnuafñ   2 f E(x)9 p) p sf x 8 4 f94(2 p   r Fp (r: j+2)   mchmg Fp (x: y) p  

^gscluafñhmglprcnlgib. P 2  >  1 j b) T  4.2  4.46  4.402 (4?2) (4?6)j2 e bl bllb  p9%hgkbtfvc r2 Aj2 pr  qjr 2 (4?2)2 (4?6)22 94 ?2 92)9 A22 T(P92)9 2 22 T(P9>)9 9>)9 A>2 (4?2)2 (4?6)>2 94 ?46 22 12 2 1 2 T(P91)9 ?2) (4?6) 94 ?402 P))9 9   rA 9  2 (4 924 ?4 n) G(91)9  

  

 

 

 

  

 

Ub

  

r 9 prfigrb



  



p













4?2

GOG^AFAFC = ?5?>5   pbk210.

Lbprcnbnflfmbmmgqug Lbprcnbnflfmbm mgqug uhtfrbmc uhtfrbmcr`bkbnlbha r`bkbnlbhacghuh cghuh sclcmfspbrc sclcmfspbrc gsfkublb4.>.Bltfrbmcrsglgghtrgkbhabrtua`cs `bstbtbhtchcygrrggltfrc. b)Ecriglblgymgmfstrfnuafñhqugmgsarfngglhÿigrcmgabrtua`cs utflfzbmcs. n)³Aÿbhtc n)³Aÿb htcsabrtua sabrtua`cs `cs sggspgrbqug utflf utflfa aggl tfrbmcr3 tfrbmcr3 Mbtcs T 94 ?> b)QgkÿhlbeñriulbmgTrcnbnflfmbmTr(P9j)9p(2-p) j- 2 , j j92,>  ,  ,  ,h. sgablaulblbsfkufghtglgymgmfstrfnuafñh. T(P92)94 92)94 ?>(2 4?>)4 94 ?> T(P9>)94 9>)94 ?>(2 4?>)94?>(4?0)94?25  

 

T P ? ( 91)94 T P9=)94 9=)94 ?>(2 9 j )94?>(2 T(P9

 

   

? > ? ? ? 4?>)1 94 ?>(4?5=)94 72>)942>0 ?24> j2 4?>) 94 ?>(4?0)j2 94 ?>(4?0)j2

Ghlbsfkufghtgtbnlbsgrgsuig Ghlbsfkufgh tgtbnlbsgrgsuighlcsrgsultbm hlcsrgsultbmcscntgh cscntghfmcs. fmcs. P 2  >  1  =   ???????  j p 4?> 4?25 4?2>0  4?24>    ??? ?????? ??? 4?>(4?0)j- 2 n)Lbgspgrbhzbmglbvbrfbnlgblgbtcrfb"hÿigrcmgabrtua`cs"gstâmbmb pcr? G(P))9 9  p2 9  42?> 97 7 tfrcsgltfrbmcrebllg. Tcrlcstbhtcsggspgrbquggh7 Tcrlcstbhtcsggspgrbquggh GOG^AFAFC = ?5?>;   pbk210.

>2

??????? ???????

 

Ghuh gxbi gxbigh,ghgl gh,ghgl qugsgrgblf qugsgrgblfzbprgk zbprgkuht uhtbssuagsf bssuagsfv vbs,pbrbbpr bs,pbrbbprcnbr cnbr `by qugachstbstbr acrrgat qugachstbstbr acrrgatbigh bightg tg b lbs24 prgku prgkuht htbs. bs. supc supchf hfghmcquggl ghmcquggl blui bluih hc sgpb gl04%mglbsrgsp gl04%mglbsrgspugstb ugstbs. s. 2.  Mbtc Mbtcs? s? P ? prgkuhtbssuagsfvbsqugsgtcibbuhbluhcpbrbqugbprugnggl gxbigh.

9 p 9 q 9 j 9 r

24 4?0 4?> 2>

@bllbr

³Aublgslbprcnbnflf ³Aublgslbprcnbn flfmbmmgqugbpru mbmmgqugbprugngh gngh ghlbs2>prfigrbsprgku ghlbs2>prfigrbsprgkuh htbs. Acic Hctbaf Hct bafñh ñh Euha uhafñhmg fñhmg mghs ghsfmb fmbm m   e(x) K(p)   e(j)9 TZP 9 j Y9(2 p)j

NH(r: p)

 

T(P 9 j

r j

2

 )9  2



Tr (q)jr

Euhafñh mg mfstrfnuaf mfstrfnuafñh ñh   E(P)  Gs Gspg pgrb rbhz hzb b   G(x) x   2 E(x)9 f94(2 p)fp sf x 8 4 p

 T



Fp (r: j+2)   mchmg Fp (x: y )

^gscluafñhmglprcnlgib.

T(P92>) 92>)? T(P

  2  9  2>) 9 62 ?

r j

Tr (q)jr

(4?0)24(4: >)> 22 T(P  9  2>)?94?>15>?

T(P  9  2>)?9

Achalusfþh.

Lbprcnbnflfmb Lbprcnbnfl fmbm m mgqug bprugngh bprugngh ghlbs 2> prf prfig igrbsprgk rbsprgkuh uhtbsgs tbsgs mg 4?>15>? GOG^AFAFC =?5?>0   pbk210.

Whboukbmcrbmgtghfskbhbgl11% Whboukbmcrbmgtgh fskbhbgl11% mglcspbrtfmc mglcspbrtfmcsqugrgb squgrgblfzbh lfzbh. . Gllboukbrâ Gllboukbrâ ghuhtcrhgcifghtrbshcsgbglfifmbmbpcrpgrmgrglpbrtfmc. 2.  Mbtc Mbtcs? s?

9 4?11 q 9 4?5;

p

@bllbr

>>

 

r p

 

b) @bllglbprcnbnflfmbmmgqugsgbglfifhbmbghglsgkuhmcpbrtfmc. n) Qfpbrbkbhbrgltcrhgcsgmgngkbhbr7pbrtfmcsachsgautfvcs.³Auâl gslbprcnbnflfmbmmgquglboukbmcrbpfgrmbpfgrmbghlb…hblmgl tcrhgc.

a) Aubhtcspbrt Aubhtcspbrtfmcsgspg fmcsgspgrbqugllgkug rbqugllgkugb b oukbr oukbrghgl ghgl tcrhg tcrhgc. c. Acic Hctbaf Hct bafñh ñh Euhafñhmg uhafñhmg mghs ghsfmb fmbm m   e(x) K(p)   e(j)9 TZP 9 j Y9(2 p)j  h2 h j NH(r: p)   T(P9 9 j )9 j+ p r h2



Euhafñhmg mfstrfnuafñh Gspgrb rbhz hzb b   G(x)   E(P)  Gspg Tx mfstrfnuafñh   2 f E(x)9 f94(2 p) p sf x 8 4 p   r Fp (r: j+2)   mchmg Fp (x: y ) p



^gscluafñhmglprcnlgib. T(P 9>) T(P 9  >) 9 9 ( (4?5;)>(4?11) T(P 9  >) 9 9 44?>>22

Achalusfþh. @bllglbprcnbnflfmbmmgqugsgbglfifhbmbghglsgkuhmcpbrtfmcgs 4?>>22 Qfpbrbkbhbrgltcrhgcsgmgngkbhbr7pbrtfmcsachsgautfvcs.³Auâl gslbprcnbnflfmbmmgquglboukbmcrbpfgrmbpfgrmbghlb…hblmgl tcrhgcgs 4 ?44;6=5 Aubhtcspbrtfmcsgspgrbqugllgkug Aubhtcspbrtfmcsg spgrbqugllgkugboukbrghgl boukbrghgltcrhgcgs tcrhgcgs 1pbrtfmcs. GOG^AFAFC =?5?>6   pbk210. Whbibrabmgrgergsac Whbibrabmgrgerg sac tfghgfiprgsbs tfghgfiprgsbs,gh ,gh abmbuhbmglbstbpbs,uh abmbuhbmglbstbpbs,uhbmglbs bmglbs …kurbsmglcs =ofhgtgsmgl bpcablfsfs bpcablfsfs yqufghrguhb yqufghrguhbaciplgt aciplgtbkbhbrbuh bkbhbrbuh prgifc. Qfuhaciprbmcr Qfuhacip rbmcr arggqug arggqug`by `byfku fkublblhÿ blblhÿigrc igrc mg…kurb mg…kurbsmgabmbuhc smgabmbuhcsmglcs smglcs pgrschbogs pgrsc hbogs ghlbprcicaf ghlbprcicafñh. ñh. ³Aubhtcsrge ³Aubhtcsrgergsac rgsacs s `bmggspg `bmggspgrbrac rbraciprb iprbrpbrb rpbrb kbhbrglprgifc3. 2. Mbtcs? NH(11?1>: =)

@bllbr Aubhtcsrgergs Aubh tcsrgergsacs acs `bmggspgrbraciprbrpbrbk `bmggspgrbraciprbrpbrbkbhbrglprgifc bhbrglprgifc..

Acic Hctbaf Hct bafñh ñh Euhafñhmg uhafñhmg mghs ghsfmb fmbm m   e(x) K(p)   e(j)9 TZP 9 j Y9(2 p)j  h2 h j NH(r: p)   T(P9 9 j )9 j+ p r h2



>1

Euhafñhmg mfstrfnuafñh Gspgrb rbhz hzb b   G(x)   E(P)  Gspg Tx mfstrfnuafñh   2 f E(x)9 f94(2 p) p sf x 8 4 p   r Fp (r: j+2)   mchmg Fp (x: y ) p



 

^gscluafñhmglprcnlgib. NH(11?1>: =)

11 = G(x) 9 0?11

G(x) 9

Achalusfþh. G(P)90 )90?11: gsmgafr6rgergsacs. GOG^AFAFC = ?5?14   pbk210.

Whpâobrcmgafgrtbgspgafgacigkusbhcsmguhbpcnlbafñhiuy krbh krbhmg mg, , gs gstc tcs s ku kusb sbhc hcs s pu pugm gmgh gh acig acigr, r, b su vgz gz, , mg uh uhb b pl plbh bhtb tb vghghcsb,mgibhgrbqugsfglpâobrcacicuhkusbhcghvghghbmcmgob mgacigrkusbhcsgsgméb,supchfghmcquggl11%mglbpcnlbafñhmg kusbhcsacigmglbplbhtbvghghcsb,`bllbsglhÿigrcigmfcmgkusbhcsacifmcspcruhpâobrcghuhméb. ]bnlbeuhafñhmgmfstrfnuafñh? Mfstrfn Mfst rfnuaf uafch ch Hctbaf Hctbafch ch   G(P)   Ubr(P) Euh uhaf afch chmg mgm mg ghs hsfm fmbm bm j2 T(P 9 j )9 p (2 p) : 2p j 92 : ? ? ? : h  Kgciætrfab   K (p)   p2 p

Euh uha.mg a.mgmf mfst st..



>

2.Mbtcs? Ubrfbnlg blgbtcrfb?   P Hÿigrckusbhcs p ? gslbprcnbnflfmbmmgæxftc. gslbprcnbnflfmbmmgæxftc.   p 94: 11 >.Nusabr? )93 G(P)93 1.Qcluafñh? G(P))9 9  p2 9 4  ?211 91 ? 4141blighcstrgskusbhcspcrméb. GOG^AFAFC = ?5?12   pbk210. Whlgpfmcptgrf Whlgpfmcp tgrfstbsclcgstbfhtgrgs stbsclcgstbfhtgrgsbmcghlcsgogiplbrgs bmcghlcsgogiplbrgs mguhbalbsgibrf 

pcsb qugachstft qugachstftuy uygh gh gl 27% mgtcmb mgtcmbs s lbibrf lbibrfpcsb pcsbs s mglb zchb. zchb. @b @bllglb llglb prcnbnflfmbm prcnbnflf mbm mgquggstb mgquggstbpgrsc pgrschb hb tghkbqug tghkbqugabsb absbr r 0ibr 0ibrfpcsb fpcsbs s mglb mglbsqughc squghc lgfhtgrgsbhbhtgsghachtrbr? 2. Mbtcs? P?

Ibrfpcsbsmglbsalbsgmgsgbmb.

9 4?27 q 9 4?07 j 9 0

p

@bllbr >=

E (P))9 9

P T

qj2p

j92

 

b) Whgogiplbrmglbalbsgmgsgbmb. n) ]rgsgogip rgsgogiplbr lbrgsmglb gsmglb albsgmgsg albsgmgsgbmb bmb.. Acic Mfstrf Mfs trfnua nuafch fch Hctbaf Hctbafch ch Kgciætrfab

 

K (p)

 

Euhaf uhafch chmg mgmg mghs hsfm fmbm bm j 2 T(P9 9 j )9 p (2 p) : j 92: ???: h 

  G(P)   Ubr(P) 2

p

p

2

>

p

j

T(P 9   j)9 p (2 p)

2 ;

T(P 9  0) 9 9 ( (4?27)(4?07) T(P 9  0) 9 9 44?=40;

Achalusfþh.

4?=40;

]rgsgogipl rgsgogiplbrgsmg brgsmg lb albsg albsg mgsgbmb mgsgbmb 4?4175= GOG^AFAFC =?5?1>   pbk210.

Ghuhbebnrfab,glmgpbrtbightcmgachtrclmgablfmbm,rgvfsblcs lctgsmgpfgbsqugghtrbh,mgbaugrmcachglsfk lctgsmgpfgbsqugghtrbh, mgbaugrmcachglsfkufghtg ufghtgarftgrf arftgrfc?sgvbh c?sgvbh gxtrbyghmcpfgzbssuagsfvbightgygllctggsrga`bzbmcsfsgghaughtrb lbprfigrbpfgzbmgegatucsbbhtgsmglbvfkgsfibgxtrbaafch.Qfachcagicsquggl>%mglbspfgzbsschmgegat agicsquggl>%mglbspfg zbsschmgegatucsbs, ucsbs,³aublgslbprcnb ³aublgslbprcnbnflfm nflfmbm bm mgquguhlctgsgbmgegatucsc3 2lctg9>4pfgzbsghtchagshugircmgfhtghtcsbhtg 2lctg9>4pfgzbsghtchag shugircmgfhtghtcsbhtgsmgcntghgruhbpfgzb smgcntghgruhbpfgzb mgegatucsb j926 2 2 Gxftc:pfgzbmgegatucs   74   =6 4 Erbabsc:pfgzbnughb 74 2 20 2 )  4: 2164> (2   74 T(P926)9 926)9   74 GOG^AFAFC = ?5?11 pbk 210.



Ghuhbeânrfabsggxb Ghuhbeâ nrfabsggxbifhbhl ifhbhlbspfgzbs bspfgzbs qugsblghmguhbmg qugsblghmguhbmgtgrif tgrifhbmbiâqufhb. hbmbiâqufhb. Qupchkbsgqugsfghuhb`crbsblghibsmg7pfgzbsmgegatucsbsgsmg4.>ygs lbifsibpbrbtcmbslbpfgzbsebnrfabmbs: 2. Mbtcs.

9 7 p 9 4?> q 9 4?0 h

@bllbr? >7

P

E (P)9 )9

T

qj2p

j92

^gscluafñhmglprcnlgib.

Whgogiplbrmglbalbsgmgsgbmbgs

Euha.mgm uha.mgmffst st..

 

Lbprcnbnflfmb Lbprcnb nflfmbm m mgqugsg tghkbrgab tghkbrgabllfnr fnrbr br lbiâq lbiâquf ufhb hb aub aubhmcsg`b hmcsg`b fhspgaafchbmc>4pfgzbs. Lbprcnbnflf Lbprcnbn flfmbm mbm mgqugsgrgabl mgqugsgrgablfnr fnrbr br lbiâ lbiâquf qufhb hb sfh`b sfh`bngrprcm ngrprcmuaf uafmc mc hfhkuhbpfgzbnughb. Acic Mfstrf Mfs trfnua nuafch fch Hct Hctbaf bafch ch

  G(P)   Ubr(P)

Euhaf uhafch chmg mg mg mghsfm hsfmbm bm

Euh uha. a. mg mg mf mfst st..

j

 

Kgciætrfab

K (p)

 

2

p

T(P9 9 j )9 p (2   p) 2 : j 92: ???: h 

p

2

>

p

E (P)9 )9

P T

qj2p

j92

^gscluafñhmglprcnlgib. j

T(P9 9 j )9 p (2   p)

2

GOG^AFAFC =?5?1=   pbk210.

Qgsbngqugbprcxfibmbightg,gl>4%mglcsusubrfcsmgwfhmcws Qgsbngqugbprcxfibmbightg,gl>4%mg lcsusubrfcsmgwfhmcws hcafgrrbh glprckrbibbmgaubmb glprckrbi bbmgaubmbibh ibhtg. tg. supchkb supchkbsgquggl sgquggl wfhmcw wfhmcws s gstbfhstblbmcghuhb gstbfhstblbmcghuhb aciputbmcr acipu tbmcrb bpÿnl pÿnlfabqug fabquggsutflfzbmb gsutflfzbmb blgbt blgbtcrfbi crfbightg ghtg pcrpgrschbs pcrpgrschbs qugbatÿbh fhmgpghmfghtgightguhbmgctrbs. 2.  Mbtc Mbtcs s q? lcsusubrf lcsusubrfcsmgwfhmcws csmgwfhmcws hcafgrrbhglprckrbibbmgaubm hcafgrrbhglprckrbibbmgaubmbi bibht bhtgg P?Wsubrfcqugafgrbhwfhmcwsbmgaubmbightg

9 4?04 q 9 4?>4

p

@bllbr

³Auâlgslbprcnbnflfmbmmgquggltgragrusubrfcsgblglprfigrcqugafgrrb bmgaubmbightgglwfhmcws3. Acic? Hctbaf Hct bafñh ñh Euha uhafñhmg fñhmg mghs ghsfmb fmbm m

K(p)

 

  e(x)

e(j)9 TZP 9 j Y9(2   p)j

Euhafñh mg mfstrfnuaf mfstrfnuafñh ñh   E(P)  Gs Gspg pgrb rbhz hzb b   G(x)

E(x)9

Tx

f94

 

(2   p)fp sf x 8 4

^gscluafch ^gsclua fch mglprcnlgi mglprcnlgib? b?

T(P 9  1) 9 p (p)>

9 (4?04)   (4?>4) 9 4?41> Achalusfñh

Lbprcnbnflfmbmmgquggltgragrusubrfcsgblglprfigrcqugafgrrbbmgaubmbightgglwfhmcwsgsmg 4?41>? >5

2

p

 

5 LGVG LGVGQTC QTCFQ FQQC QCH. H. GOG^AFAFC = ?5?17   pbk216. QgbVuhbvbrfbnlgblgbtcrfbqugsfkuguhbmfstrfnuafþhmgpcfsschmgigmfb  9>? ablaulg 2. Mbtcs? 9>

@bllbr b)   T(V 9=) n)   T(V =!  25" > T(V 9  =) 9 >= T(V 9  =) 9 9 44?464>

T(V

9  =)  9

T(V )+  T(V91)+  T(V 9=) >4 "> > 2 "> > > "> > 1 "> > = "> + + + + T(V    =)9 =! 1! >! 2! 4! 2 > > = > T(V    =)9 > + > + > + + 1  "> 1  "> " " " 9 4 4?6=;1 T(V 9  =) 9 T(V 8=) T(V

=)92  4?6=;1) T(V 8 =)94?47>; 8

GOG^AFAFC = ?5?15   pbk216. Glprcigmfcmglbsllbibmbsqugrgafnghuhbaghtrbltglge Glprcigmfcmglbsllbibmbsqu grgafnghuhbaghtrbltglgechfabghuhifhutc chfabghuhifhutc gsmg2.7.@bllglbprcnbnflfmbmmgquggh=ifhutcssgrgafnbh. >;



Ub

 

2. Mbtcs?

9 2?7 t 9 = t 9 5 

@bllbr? b) 1llbibmbs. n) ighcsmg1llbibmbs. Acic? Hctbafñh Hctbafñ h Euhafñhmgmgh uhafñhmgmghsfmb sfmbm m   e(x)   T()   g j! j

Euhafñh uhafñh mg mfstr mfstrfnuafñ fnuafñh h   E(P)  Gs Gspg pgrb rbhz hzb b   G(x)   "      p(x: )9 j!

^gscluafchmglprcnlgib. 1llbibmbs.

T(P 9  1) 9 T(P 9  1) 9

  g5  51

1!   5  g  51 5 51

T(P 9  1) 9

5   g   5 >25 T(P 9  1) 9 5   g   5 T(P 9  1) 9  4 ?06>17

Ighcsmg1llbibmbs.

T(P < T(P    T(P    T(P    T(P   

1)9   T(P94)+ 94)+  T(P92)+ 92)+  T(P 9>)  5 4  5    g 5 g  52 g    5  5> 1)9 + + 4! 2! >! 2 5 20 1)9   5 +   5 +   5 g

1)9

g

g

>7

g 5 1) )9 9 4?4526;

Achalusfch?

>0

j



Ub

 

1llbibmbs.gs 4?06>17 Ighcsmg1llbibmbs.gs 4?4526; GOG^AFAFC = ?5?1;  pbk216. Qupchkbqugglhÿigrcmgpbafghtgsqugfhkrgsbhblbsblbmggigrkghafbmg uh`cspftblghlbhca`g uh`cspf tblghlbhca`gmglvfg mglvfgrhgstfgh rhgstfghguhbmfstrf guhbmfstrfnu nuafñh afñh mgTcfsschachig mgTcfsschachigmfb mfb fkublb=.Gvbluglbsprcnbnflfmbmgsmgqug? Mbtcs? s? 2.  Mbtc

P?Hÿigrcmgpbafghtgs P?Hÿigrc mgpbafghtgsqugfhkrgsbh qugfhkrgsbhblbsblb blbsblbmggigrkghafb mggigrkghafbmguh`cspftbl mguh`cspftbl G(P))9 9  9= @bllbr?

(b)Murbhtguhbhca`g`bybgxbatbightg`byb>pgrschbsghlbsblbmg gigrkghafb. (n) Murbhtglb Murbhtglb hca`gbybibsmg hca`gbybibsmg 1pgrschbs. Acic? M fs fs t. t. 

Hc ct t ba ba ffc ch

T( ( )

Tcfssch

  G(P)   Ubr(P)



Eu h ha af c ch h m m g g m g gh hs sff m mb b m m 

T(P9 9 j )9   g



j

 j!

Eu h ha a.  m g m m ffs st t..

: j 94: ?????: h  E (  P))9 9

P T g



j94

^gscluafñh mglprcnlgi ^gscluafñh mglprcnlgib. b. T(P91) 91)

T(P 9  >) 9 T(P 9  >) 9

g= =>

>! 0

g= T(P 9  >) 9 9 44?2=57 T(P 81) T(P 8 T(P 8 T(P 8 T(P 8 T(P 8 T(P 8

1)92  T(P  1) 1)92  ZT(P94)+ 94)+ T(P92)+ 92)+ T(P91)+ 91)+ T(P91)Y 91)Y  = 4  = >  = 2  = 1 g = g = g = g = 1)92  Z + + + Y 4! >! 2! 1! 2 = = 1> 1)92  Z = + = + = + Y g g g 1  g= 1> 6 Y 1)92  Z = + 1  g= g 1)94?7571

Achalusfñh?

Murbhtguhbhca`g`bybgxbatbightg`byb>pgrschbsghlbsblbmggigrkghafbgs 4?2=57 >6

j

 j!

 

2. (b) Murbhtglbhca` Murbhtglbhca`gby gbybibs bibsmg mg 1pgrs 1pgrschbs chbsgs gs 4?7571 .Trcnlgib lgib10. 10. GOG^AFAFC = ?5?10   pbk216.Trcn Ghuh`ctgl,glprcigmfcmgpgmfmcsmgsgrvfa Ghuh`ctgl,glprcigm fcmgpgmfmcsmgsgrvfafcblb`bnftba fcblb`bnftbafñhgs fñhgs fkubl>abmbigmfb`crb.Bllâlbprcnbnflfmbmmgqugghuhb`crbsg rgafnbh? b)  1 p gmfmcs n)  ighcs m  mg 1 1 p p gm gmf m mc cs a)  hc  ig ighcs  mg mg  1 1 p g gm mf mc mcs M ffst st..  Hct Hct b baf afc c h   G(P)   Ubr(P) Eu ha haf ch ch  m mg g m mg gh h ssff mb mb m m Tcfssch

 

T( ( )

 

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Eu ha ha.  m g m m ffss t. t. P Tg     : j 94: ?????: h  E ( ( P))9 9 j! j!

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TlbhtbifghtcmglTrcnlgib Mbtcs? P?gslbvbrfbnlg ? gslbvbrfbnlg ) fhtgrv vblcmg blcmgtfgi tfgipcmg pcmg sgrvfvfcb sgrvfvfcb lb`bnftb lb`bnftbafñh.: afñh.: t 9> t ? fhtgr j ? huigrcmggvghtcsqugpugmghsuagmgrghuhafgrtctfgipc.  ? prcigmfcmgpgmfmcsmgsgrvfvfcblb`bnftbafch: 9> ^gscluafñh? IgmfbhtglbprcnbnflfmbmmgTcfssch   T( ( ) :   T(P 9   j) 9   g j(!t)  : j 9 4: 2: >:???:h >(>) 1 x(>x>) 94 ?26715 b)T(P91)9 91)9   g 1! n)T(P ) >(>) >(>) >(>) 4 4  g x(>x>)  g x(>x>) (>x>)> 94 ?42012+4?4;1>5+ + g + 9 >! 2! 4! 4?2=5794?>1024 a)T(P  1)92  T(P < 1)92  4?>102494 ?;526 GOG^AFAFC = ?5?16   pbk216. t

j

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Whbeânrfab mgkbsgcsbsrgafnfñ244 Whbeânrfab mgkbsgcsbsrgafnfñ244 nctgl nctgllbsvba lbsvbaébs. ébs. LbprcnbnflfLbprcnbnflfmbmmgqugbltrbhspcrtbrlb mbmmgqugbl trbhspcrtbrlbsrgsultguhbnctgllbrctbgsmg srgsultguhbnctgllbrctbgsmg4.41. 4.41. @bllg lbprcnbnflfmb lbprcnbn flfmbmmgquglbeânrfabrgafn mmgquglbeânrfabrgafnbrctbs? brctbs? b)Gxbatbightgmcsnctgllbs n)Iâsmgmcs. a)Tcrlcighcsuhb. TLBH]GBIFGH]CMGLT^CNLGIB Mbtcs Ubrfbnlg blgbtcrfb P  Hÿigrcmgnctgllbsrctbs. h ? ?hÿigrcmgprugnbs hÿigrcmgprugnbs h 9244 p ? ?prcnbnflfmbmmgæxftc prcnbnflfmbmmgæxftc p 94: 1 @bllbr  93 9>)93 T(P9>)93 T(x x8 8>)93 T(P  2)93

14

j

 

Acic? M ffst st.. 

Hct Hct b baf afc ch

  G(P)   Ubr(P)

Eu ha haf ch ch  m mg g m mg gh h ssff mb mb m m 

 g

Eu ha ha.  m g m m ffss t. t. P

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Tcfssch   T(  T(P9 ( P))9 9 ( )    9 j )9 j!   : j 94: ?????: h  E ( ^gscluafchmglprcnlgib. IgmfbhtglbmfstrfnuafñhmgTcfssch T() : T(P 9 j )9 ) 9 g  j! Tcrbprcxfibafñhblbnfhcifblaubhmch gsrglbtfvbightgkrbhmg( h  14) 14) y   p pgqugòc (p  4?47 ),pchfghmc  9 hp(]gcrgibmgbprcxfibafñh 47),pchfghmc  9 hp 9244  4?4191?4 > 9>)9 g 11>! 9   >6 g 1 94 ?>>=4= b) T(P9>)9 n)T(P 8 >)92  T(P  >)92  ZT(P94)+ 94)+T(P92)+ 92)+ T(P9>)Y9 9>)Y9 14 12  6 1 1 1 2  (g 4! + g 2! + > g )94?7;502 4 94)Y92  g 114! 94 ?674> a) T(P  2)92  ZT(P94)Y92 Achalusfch? b)Lbprcnbnflfmbmmgqugrgsultghmcsnctgllbsrctbsgsmg4 b)Lbprcnbnflfmbmmgqugrgsultghmcsnctgllbsrctbsgsmg4 ?>>=4= n)Lbprcnbnflfmbmmgqugrgsultghibsmgmcsnctgllbsrctbsgsmg4 n)Lbprcnbnflfmbmmgqugrgsultghibsmgmcsnctgllbsrctbsgsmg 4 ?7;502 a)Lbprcnbnflfmbmmgqugrgsultghblighcsuhbnctgllbrctbgsmg4 a)Lbprcnbnflfmbmmgqugrgsultghblighcsuhbnctgllbrctbgsmg 4 ?674> GOG^AFAFC = ?5?=4   pbk216. 

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Qgsupchgqugglhÿigrcmgnbatgrfbspcr   ii1 mgbkubghuh gstbhquggs gstbhq uggs uhbvbrfbnlgblgbt uhbvbrfbnlgblgbtcrfbPachMfstrfn crfbPachMfstrfnuafñhmgTcfssch uafñhmgTcfsschmg mg pbrâigtrc  94?7 b) ³A ³Auâ uâlgs lgsl lb bpr prcn cnbn bnfl flfm fmbm bmmg mgqu qug g2 ii1 mgbkubmglgstbhqug hc`byhfhkuhbnbatgrfb3 n) Gh=4 Gh=4 tuncs tuncs mg gh ghsb sby yc sgtcib sgtcibh h iugst ugstrb rbs s mgbkubmg mgbkubmgl l gstbhqug(2   ii1 mg bkub bkub gh abmb abmb tun tunc) c). . ³Ruæ ³Ruæ mfstr mfstrfnu fnua afñh fñh sfku sfkug lb vbrfbnlgV?hÿigrcmgtuncsmgghsbyc,ghtrglcs=4qughcachtfghgh nbatgrfbs3 a) Qfs Qfsbng bngic icsqug squgghuht ghuhtunc`c unc`cynba ynbatg tgrf rfbs, bs,³A ³Auâ uâlgslbp lgslbprc rcnnbnflfmbmmgqug`bybighcsmgtrgs3 ]bnlbeuhafñhmgmfstrfnuafñh? M ffst st..  Hct Hct b baf afc c h   G(P)   Ubr(P) Eu ha haf ch ch  m mg g m mg gh h ssff mb mb m m 

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2.Mbtcs? Ubrfbnlg Ubrfb nlg blgbtcrfb? blgbtcrfb?   P  hÿigrcmgnbatgrfbspcr ii1  94?7 >.Nusabr? T(P94) 94) T(P  >4) 1.Qcluafñh? ( 4 7) (4?7)4 b)T(P94)9 94 ?54571 94)9   g 4! 

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GOG^AFAFC = ?5?=2   pbk216. Whbplbhtbgh Whbplb htbghnctgll nctgllbmcrb bmcrb mgrger mgrgergsacs gsacs tfghguhbiâqu tfghguhbiâqufhb fhb vfgobp vfgobpbrbllg brbllghbr hbr 244 lbsnctgllbs. lbsnctg llbs. Lbiâq Lbiâqufhb ufhb pr prcmuag cmuag uhbkbh uhbkbhbhafb bhafb mg mñlbrgspcrglmfbmg trbnboc:sfhginbrkc,sgmgsacipchg trbnboc:sfhginbrkc ,sgmgsacipchg uhprcig uhprcigmfcmg mfcmg mcsvga mcsvgagsabmb gsabmb 24 mébs? sfV rgprg rgprgsgh sghtb tb glhÿig glhÿigrcmgmgsacipc rcmgmgsacipch hturbs turbsmurbht murbhtggl ggl euhafchb euhafchbifght ifghtc c mglb iâqufh iâq ufhb,sgg b,sggxpr xprgsb gsb pcrt gsglh gsglhÿig ÿigrc rc mgmé mgmébsqu bsqugtrb gtrbnboc nboc lbi lbiâqu âqufhb fhb. . Lb > kbhbafbkghgrbm kbhba fbkghgrbmbpcrlb bpcrlb iâquf iâqufhb hb sggxprgsbpcr sggxprgsbpcrK K 9244t  74V ? Mgtgrifhg lbkbhbafbg lbkbh bafbgspgrb spgrbmb mb ghlcs24 méb mébsmg smg trbnboc. trbnboc. GOG^AFAFC = ?5?=>   pbk216. Ghuhbpcnlbaf Ghuhbpcnlb afñh ñh gl2pcr afgh afghtcmglbpcnlbafñ tcmglbpcnlbafñhsuergmbltc hsuergmbltchfsic hfsic,³ ,³ auâl gs lbprcnbnflf lbprcnbnflfmbmmgqugghtrg244pgrschb mbmmgqugghtrg244pgrschbs. s. 2. Mbtcs? P? pcnlb pcnlbaafñhq fñhqugsuer ugsuerg g mblt mbltchfsi chfsic c h  9  244 p 9 4?42

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@bllbr? b) hfhkuhbpbmgsabmgmbltchfsic. n) mcsbibslcpbmgsabh Acic? M fst fst..  Tcfssch

Hc Hctt b baf afc ch  

T( ( )

 

Euh af af c ch h m mg m  m gh gh ssff mb mb m m

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  gj j!   g2 24

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j

 

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T(P 9  4) 9 9 22  Z

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g

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(P 9  4) 9 9 44?>5== Achaluafñh.

hfhkuhbpbmgsabmgmbltchfsicgs 4 ?15;0 mcsbibslcpbmgsabhgs 4 ?>5==? GOG^AFAFC = ?5?=1   pbk2=4

Ghuh ncsqug ncsqug mgagmr mgagmrc c glhÿigrcmg glhÿigrcmg ârnclg ârnclgs s achplbkbpcr achplbkbpcr `g `gatb atbrgb rgb V tfghg uhbmfstrfnua tfghg uhbmfstrfnuafñh fñh mgTcfss mgTcfsschT(24).Lcs chT(24).Lcs ârnclgs ârnclgs achplbkb achplbkb sg trbtb trbtb ach fhsgatfaf fhsga tfafmbbuhacst mbbuhacstcmg1 cmg1 mñlb mñlbrgspcrârn rgspcrârncl:bmg cl:bmgi iâs mguhacstc…oc,pcr mguhacstc…oc,pcrusc usc mggqufpcytrbhspcrtg,fkublb74mñlbrgs. 2.Mbtcs? V ? Glhÿigrcmgârnclgs achplbkbpcr`gatbrgb. achplbkbpcr`gatbrgb. 

9 24

@bllbr GlvblcrgspgrbmcylbmgsvfbafñhgstbhmbrmglacstcAmgeuifkbr7 `gatârgbsmgncsqug. Acic? M fs fs t t.. 

Tcfssch

Hc ct tb ba a ffc c h   G(P)   Ubr(P)

 

T( ( )

 





Euh a aff ch ch m  mg m m gh gh sf sf mb mb m m 

T(P 9 j )9   g

j

 j!

Euh a. a.  m g m m fs fs t t..

  : j 94: ?????: h  E ( ( P))9 9

P T g



j94

^gscluafchmglprcnlgib Achalusafñh.

G(A)9>44mñlbrgs. )9>44 mñlbrgs.

)9=;?=1mñlbrgs. =1mñlbrgs. (A)9=; GOG^AFAFC =?5?==   pbk2=4?

11

j

 j!

 

Tbrb gl ach Tbrb Tbrbglach achtrcl trcl tr trcl cl mgablfmb mgablfmbm mg mg ablf ablfmb mbm mgmfsac pbrb pbrbacipu pbrb acipu aciputbmc tbmcrbsg tbmc rbsggiplgbuhmfsrbsg gipl giplgbuh gbuh mfspcsftfvcglgatrchfacqugaughtbglh pcsftfvcglgatrchfacqugaughtbglhÿigrc ÿigrc ÿig rc mgnytgsmgegatucscs. mgny mgnytgs tgsmgega mgegatu tucscs. Whbibrabmg Whbibrabmg mfsacsmgaciputbm mfsac smgaciputbmcrbstfgh crbstfghguhprcigmfc guhprcigmfcmg4.2nytgsmgegatucsc mg4.2nytgsmgegatucscspcrmfsac. spcrmfsac. Ablaulgglprcaght Ablau lgglprcaghtbogmgmfsacqug? bogmgmfsacqug? Mbtcs? Mbtcs?? 2.  Mbtcs

P?Hÿigrcsmgmfsacsmgegautucscs. 94 9 4?2 @bllbr?

Hctfghgmgegatc ]fghgblkÿhmgegatc. Acic? Mfst Mfst. . Tcfssch

Hctb H ctbaf afch ch

 

T( ( )

 

  G(P)   Ubr(P)

 

Eu uha hafc hafch h m mg gmg g mghs hsfm fmbm bm  

T T(P 9 j )9   g



j

 j!

Eu uha ha. ha.  m mgmf g mfst st..

  : j 94 9 4: ???????:h ?::h h h  E ( ( P))9 9

P T g



j94

^gscluafch ^gscluafchmgl mglprcnlg prcnlgib. ib. T(P 94)

T(P 9  4) 4) 9 9 T(P 9  4) 4) 9 9

  g4?2 4?24

4! 2

g4?2 T(P 9  4) 4)9 ) 9 4 4?64=0 T(P  2) T(P   

T(P   

2)9 2) 922  ZT(P

T(P   

Achalusafñh.

Hctfghgmgegatcuh 64 ?=0% ]fghgblkÿhmgegatc 6 ?7>% GOG^AFAFC =?5?==    pbk pbk2=4? Glhÿigrcmgbutcicvflgsqugllgkbhbuhgstbafchbifghtc,qugtfghguhb abpbafmbm abpba fmbm mg2>butcsgsuhb mg2>butcs mg2>butcsgsuhb gsuhb vbrfbn vbrfbnlgblgbtc lgblgbtcrfbqugsfkuguhblgymgTcfssc rfbqugsfkuguhblgymgTcfssch, h, achuhprcigmfc=pcr`crb.Qfblfhfafcmglmébglgstbafhb achuhprcigm fc=pcr`crb.Qfblfhfafcmglmébglgstbafhbifght ifghtcgstbvbafc. cgstbvbafc. 1=

j

 j!

 

2.Mbtcs? P ? Hÿigrcmgbutcicvflgsqugllgkbhbuhgstbafchbifghtc

9 = h 9 9 2> 2>  9 9 1> 1> 

@bllbr

Auâlgslbprcnbnflfmbmmgqugllghgmurbtglbprfigrb`crb. Ablaulglbprcn Ablaulglbprcnbnflf bnflfmbmmgqugllgku mbmmgqugllgkugighcsmg14 gighcsmg gigh csmg14 14 vg` vg`faul faulcs cs ghuhturhc ghuh ghuhturh turhc c mg0`crbs. Acic? Mfst Mfst. .

Hctb H ctbaf afch ch

Tcfssch

T( ( )

  G(P)   Ubr(P)

 



Eu uha hafc hafch h m mg gmg g mghs hsfm fmbm bm  

T T(P 9 j )9   g

j

 j!

Eu uha ha. ha.  m mgmf g mfst st..

: j 94 9 4: ???????:h ?::h h  E ( ( P))9 9

P T g



j94

^gscluafch ^gscluafchmgl mglprcnlg prcnlgib ib

T(P 6) 9>6) g1> 1>4 g 1> 1>2 g 1> 1>> g 1>1>4 T(P < 14) 14)9 9 + + + ???????????? + 4! 2! >! >6! T(P < 14) 14)9 944?110

Achalusfñh.

Auâlgslb Auâlgslb âlgslb prcn prcnbn bnfl flfm fmbmmgqugllg bmmgqugllghgmur hg murbt murbtg btg lbprf lb lbprfig prfig igrb rb rb`crb. `crb . 4?4445=27

gsmg gs  mg mg

Ablaulglbprcn Ablaulglbprcnbnflf bnflfmbmmgqugllgku mbmmgqugllgkugighcsmg14 gighcsmg gigh csmg14 14 vg` vg`faul faulcs cs ghuhturhc ghuh ghuhturh turhc c mg0`cr mg0`crbs bs bs gsm gs gsmg mg 4 ?110 GOG^AFAFC =?5?=5  pbk 2=4?

Ghuhgstumfcmg Ghuhgstumfc Ghuhgstumfcmguhfhv mg uhfhvghtbr uhfhvghtbrfcsgmgtgrifhc ghtbrfcsgmgtg fcsgmgtgrifhc rifhc quguhprcig quguhprcigmfc,lbmgibh quguhprcigmfc,lbmgibhmb mfc,lbmgibhm mb pcruhbr pcru hbrté  téau aulc l c gghpbr h pbrtf tfau aulb lbr r  gghuhbncm h  uhbncmgk gkb b  ggrbmg7v rbmg7vga gags gs. .  ³Auâ ³Auâlgslb lgslb prcnbnflfm prcnb prcnbnflfmbm nflfmbm bm quguhmgtgrifhbmcmébgstggstg quguhmgtgrifhbmcmébgstggstgbrtéaulcsgbrg brtéaulcsgbrgqugrfmc. qugrfmc. 2.Mbtcs? P ? Gstumfcmguhfhvghtbrfc. 97 9 7

@bllbr Bibsmg=vgags 17

j

 j!

 

Hfuhbsclbvgz. Hfuhbsclbvgz. Acic?

Mfst Mfst. . Tcfssch

Hctb H ctbaf afch ch T( ( )

  G(P)   Ubr(P)

 



Eu uha hafc hafch h mg mgmg g mghs hsfm fmbm bm  Eu uha ha. ha. m mgmf g mfst st.. P T  g  g  T T(P 9 j )9 j! : j 94 9 4: ???????:h ?::h h  E ( ( P))9 9 j! 

j



j94

^gscluafch ^gscluafchmgl mglprcnlg prcnlgib ib T(P 8=) T(P 8 T(P 8 T(P 8 T(P 8 T(P 8 T(P 8

=)9 =) 922  T(P  =) =)9 =) 922  ZT(P94)+ 94)+ T(P92)+ 92)+ T(P9>)+ 9>)+ T(P91)+ 91)+ T(P9=)Y 9=)Y  7 4  7 2  7 >  7 1  7 = g 7 g 7 g 7 g 7 g 7 =)9 =) 922  Z + + + + Y 4! 2! >! 1! =! g7 74 g 7 72 g 7 7> g 7 71 g 7 7= =)9 =) 922  Z + + + + Y 2 2 > 5 >= =)9 =) 922  Z4?==Y =)9 =) 944?75

T(P 94) T(P 9  4) 4) 9 9

4) 9 9 T(P 9  4)

g7 74

4! 2

g7 T(P 9  4) 4)9 ) 9 4 4?45;=

Achalusfñh. Achalusfñh. Bibsmg=vgagsgs 4?75 Hfuhbsclbvgzgs 4?45;= GOG^AFAFC =?5?=;  pbk 2=4?

Qf`byghprcigmfcuh2%mgzurmcs,³Auâ Qf`byghprcigmfcuh2%mgzurm cs,³Auâlgslbprcnbnf lgslbprcnbnflfmb lfmbmmg mmg tghgrpcrlcighcs=zurmcsghtrg>44pgrschbs3 Qcluafñh ? Qcluafñh MfstrfnuafñhmgTcfssch 

T(P9 9 j )9   g

j

 j!

: j 94 9 4: 2: >:???

Ghgstgabsctghgics? Mg…hficslbvbrfbnlgblgbtcrfb? P ?hÿigr hÿigrcmgzurm cmgzurmcsghtr csghtrguhkrupc guhk gu hkru rupc pc mg >44pgrschbs h9>44 15

j

 

p 94?42  9 hp 9>44  4?429>  =)92  T(P  1) T(P 92  ZT(P94)+ 94)+  T(P92)+ 92)+  T(P9>)+ 9>)+  T(P91)Y 91)Y >

4

>

92  Z g 4!> +   g g > 92    26 1

>2 2!

>

+ g

>> >!

>

+ g

>1 1! Y

t 4?2=>0;5 GOG^AFAFC = ?5?=0   pbk2=4? Ghuhbfhvgstfkbafñhmgigrabmcsgmgtgrifhñquggl>pcrafghtc mglbpcnlbafñhtcibrgkulbrightguhbibrabmgyckcurt.Qggsackfc uhbiugstrbmg144pgrschbs,mgtgrifhglbprcnbnblfmbm.

MB]CQ Gl>%mg14495tcibhuhyckcurtmglbifsibibrab 95  95

EC^IWLBQ t

j

% j ! 

(Mfstrfn (Mfs trfnuaf uaf hmgTcfsch) hmgTcfsch)

^GQCLWAFÑH b)Gxbatbightg7pgrschbstcighyckcurtmggsbpgrschb %t j   %5(2) (5)7   %5 ;;;=   %5 1>= 9 9 9 94 : 252 j! 7! 2>4 7

n)blciâstfghgtrgspgrschbs 91)+Tr (P9>)+Tr 9>)+Tr (P92)+Tr( 92)+Tr( P 94) Tr (P  1)  9  Tr (P91)+Tr  t j  5(2) 1  5(2) >  5(2) %    % % % (5) (5) (5)2 % 5(2) (5)4 9 + + + j! 1! >! 2! 4!   %5(2) >25 % 5(2) (15) % 5(2) (5)2 % 5(2) (5)4 9 + + + 5 4! 2 > 9 4: 406>1+4: 4==5>+4: 42=0; 9 4: 272260

a)blighcstcighafhacpgrschbs Tr (P  7)  9  2  Tr (P9=)+Tr 9=)+Tr (P  1) % 5(2) (5)= 9 2  +4: 272260 =! % 5(2) 7= +4: 272260 9 2  2 9 2  4: 2110+4: 27260 9 4: ;2=>>

1;

 

T(P94)9 94)9

7

  gxp

(7)4

 

4!

 4: 445;1

GOG^AFAFC =?5?=6   pbk2=4? Lbtbsbighsublmgsufafmfcsgsmg= Lbtbsbighsublmgsufafmfc sgsmg= pcruhifllñhmgpgrschbs. pcruhifllñhmgpgrschbs. Ghuhbafumbmmg744444`bnftbh Ghuhbafumbmmg74444 4`bnftbhtgs,`bll tgs,`bllglbprcnbnfl glbprcnbnflfmbmmgqug? fmbmmgqug? b)ghuhigsmbmc,`bybhighcsmg7sufafmfcs: n)³Qgrâscrprghmghtgqugmurbhtguhbòc,blighcsgh>igsgs caurrbhiâsmg=sufafmfcs3

Qcluafñh? b) MfstrfnuafñhmgTcfssch T(P9 9 j )9   g j!   : j 94: 2: >:??? 

j



Ghgstgabsctghgics? Mg…hficslbvbrfbnlgblgbtcrfb?   P ? h uigrcmgsufafmfcsghuhigs u igrcmgsufafmfcsghuhigs Qfpcrifllñhmg`bnftbhtgssgtfghg=sufafmfcspcrigs,gh744444`bnftbhtgs sgrgmuagblbiftbmpcrigs,bsé?    9> T(P )+ 9>)+  T(P91)+ 91)+  T(P9=) 9=) >

9 g t

>4 4!

>

>2 2!

+ g

>

+ g

>> >!

>

+ g

>1 1!

>

>= =!

+ g

4?6=;1=

n) Ghgstgabscmgpghmgicsmgltfgipc,gsachvghfghtgusbr? T(P9 9 j )9 t 9>

t



 g

(t)j

j!

 9> T(P 8=)92  T(P  =) 92 T(P94)+ 94)+T(P92)+ 92)+T(P9>)+ 9>)+ T(P91)+ 91)+T(P9=) 9=) >>

(>>)4 4!

9 2 Z g > >

g

(>>)= Y =!

92    241 1 g t 4: 1;225

> >

+ g

(>>)2 2!

>>

+ g

(>>)> >!

>>

+ g

(>>)1 1!

+

=



Ghtchags,lbprcnbnflfmbmqugmurbhtguhbòc,blighcsgh>igsgscaurrbh iâsmg=sufafmfcsgs iâsmg=sufafmfcs gsmgl1;.22 mgl1;.22%lcaublgsuh %lcaublgsuhvblcr vblcrrglbtfvb rglbtfvbight ightgicmgrbm gicmgrbmc. c. GOG^AFAFC = ?5?74 pbk 2=4?

10

 

Ghgstumfcsmgickrâ…acsscnrgibtrfichfcsqugtfghghblkÿhtfpc mg plbhf…ab plbhf…abafñ afñh h ebifl ebiflffbr,gl hÿigr hÿigrc c P mg `focs `focs pcribtrfic pcribtrfichfcgs hfcgs fkublb>,sblvcafgrtbsmgsvf fkubl b>,sblvcafgrtbsmgsvfbafchg bafchgsmgnfmbsblbzbr. smgnfmbsblbzbr.Qg`b Qg`bacipr aciprcnbmc cnbmc qug,cnfgh? P 9> (V +2) mchmg   V gsuhbvbrfbnlgmgNgrhcullfmg pbrâigtrc  p 9 4?1, y gst gstc c cau caurrg rrg ach ach prc prcnbn nbnfl flfmb fmbm m  p 9   >2 (pugssg auiplgghgl74%mglcsibtrfichfcs),cnfghgs? P 9>+ _  mc mchm hmg g _ sfkuguhbmfstrfnuafñhmgTcfsschmgpbrâigtrc :ygstctbinfæh caurrgachprcnbnflfmbm  p 9   >2 @bllbr? b)glvblcrmg  sbnfghmcqug   G(P)9 > n)lbprcnbnflfmbmmgquguhibtrfichfctghkb2c>`focs.



b) P9>+ 9>+ _ G(P))9 9 G(>+ (>+ _) >9 > 9G(>)+ (>)+ G(_) >9>+   94

n) Mg…hficslbvbrfbnlgblgbtcrfb VqugsfkuguhblgymgNgrhcullf 2:sfgs gxftc: gxftc: p V 9 4:sfgserbabsc:q T(V92)94?1 T(V94)9 q92  92 p 92 4?194?;









Whibtrfichfctghkbuh`foc? (T(V92)+2)9>

 (4?1+2)94?;

Whibtrfichfctghkbmcs`focs? T(P9>)9> 9>)9> (T(V94)+2)9>

 (4?;+2)94?1

92)9> T(P92)9>





(A`bptgr`gbm?)Abpftulc=.24

; LGVW LGVWHF HFE EC^ C^I IG GOG^AFAFC = ?24?2 pbk 272?

WhbvbrfbnlgblgbtcrfbPtfghgmfstrfnuafñhuhfecrigscnrgZ-1,2Y.Ablaulg? b)T(P94): 94): n)T(P 4: :  ssff x x x  > 9 9 ZZb: 2:  sf x 8  n



0