CK-12: Álgebra Edición-Española v1 s1

7/30/2019 CK-12: lgebra Edicin-Espaola v1 s1

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2/955

ikajzg H # Abhdho At|gfig

Tgx R`goet rf r`a Gqr`fztDihde `rr|>&&{{{/de7:/fzk&tgxr`goet

(Of thko ho zawqhzab
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3/955

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Rf gddatt g dqtrfch~gjia }azthfo fm r`ht jffe) gt {aii gt fràzhorazgdrh}a dforaor) }hthr {{{/de7:/fzk

DE#7: Mfqobgrhfo ht g ofo#|zfr fzkgoh~grhfo {hr` g chtthfo rfzabqda rà dftr fm raprjffe cgrazhgit mfz rà E#7: cgzear jfr`ho rà Q/T/ gob {fzib{hba/ Qthok go f|ao#dforaor) {aj#jgtabdfiigjfzgrh}a cfbai razcab rà MiapJffe) DE#7: horaobt rf|hfoaaz rà kaoazgrhfo gob bhtrzhjqrhfo fm `hk`#wqgihrx abqdgrhfogidforaor r`gr {hii taz}a jfr` gt dfza rapr gt {aii gt |zf}hba gogbg|rh}a ao}hzfocaor mfz iagzohok) |f{azab r`zfqk` rà MiapJffe\igrmfzc/

Df|xzhk`r :&&{{{/de7:/fzk&tgxr`goet (|igdab ho g }hthjia ifdgrhfo hogbbhrhfo rf rà mfiif{hok razct/

Apda|r gt fr az{hta ofrab) gii DE#7: Dforaor (hodiqbhokDE#7: Dqzzhdqiqc Cgrazhgi ht cgba g}ghigjia rf Qtaztho gddfzbgoda {hr` rà Dzagrh}a Dfccfot Grrzhjqrhfo&Ofo#Dfccazdhgi&T`gza Gihea 4/< Qo|fzrab (DD JX#OD#TG Ihdaota(`rr|>&&dzagrh}adfccfot/fzk&ihdaotat&jx#od#tg&4/&&{{{/de7:/fzk&razct/

\zhorab> Nqix 77) :


4/955

h} {{{/de7:/fzk

Dforaort

7 Adqgdhfoat x Mqodhfoat 77/7 Ap|zathfoat ]gzhgjiat / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / :7/: Fzbao ba igt F|azgdhfoat Gikajzghdgt / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / 7

G 3 i { G 3 i { G 3 i{

Ofrg> Qo anac|if ba ap|zatho }gzhgjia at :i % :{5 qo anac|if ba qog adqgdho at \ 3 :i % :{/ Ig bhmazaodhg|zhodh|gi aorza adqgdhfoat x ap|zathfoat at ig |zataodhg ba qo thkof hkqgi wqa (f thkof ba hkqgibgb (3/

Ao ai anac|if gorazhfz) of `gx mfzcgt ct thc|iat |gzg igt ap|zathfoat ba |azcarzf x zag/ Aiigt tfo) thoacjgzkf) ap|zathfoat |azmadrgcaora kaoazgiat |gzg ai |azcarzf x ai zag ba qo zadrokqif/ Aiigt bgzo zatqirgbftdfzzadrft |gzg dqgiatwqhazg }gifzat oqczhdft ba ig ifokhrqb x ai god`f ba qo zadrokqif |gzrhdqigz/ Gt) rq rhaoat)taodhiigcaora) wqa tqtrhrqhz ift }gifzat at|addft ba ig ifokhrqb x god`f ba qo zadrokqif zagi baorzf ba oqatrzgtadqgdhfoat ba |azcarzf x zag/ Atra |zfdabhchaorf at g caoqbf dfofdhbf dfcf tqtrhrqxaobf (fz zaac|ig~gobf}gifzat ao qog ap|zatho/ Ao atra dg|rqif qrhih~gzacft ai |zfdatf ba tqtrhrqdho |gzg a}giqgz ap|zatthfoat dqgobfta bht|fokg ba }gifzat |gzg igt }gzhgjiat ho}fiqdzbgbgt/

D`g|raz 7/ Adqgdhfoat x Mqodhfoat
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10/955

2 {{{/de7:/fzk

A}giqgdho ba Ap|zathfoat Gikajzghdgt

Dqgobf ta rhaoa qog ap|zatho gikajzghdg) qof ba ift |zfdabhchaorft ct dfcqoat wqa raobzacft wqa zagih~gz dfoaiig at a}giqgzig |gzg giko }gifz ocazhdf ba ig }gzhgjia dforaohbg ao ig ap|zatho/ Ai thkqhaora anac|if hiqtrzgatra |zfdatf/

Anac|if :Tag p 3 7:/ Aodqaorzg ai }gifz ba :p ;/\gzg aodfozgz ig tfiqdho) tqtrhrqhcft 7: ao p baorzf ba ig ap|zatho bgbg/ Dgbg }a~ wqa rq aodqaorzat qog p)aorfodat bajat zaac|ig~gzig |fz 7:/ Ofrg> Ao atrg arg|g dfifdgcft bhd`f }gifz aorza |gzoratht>

:p ; 3 :(7: ;3 :2;3 7;

Ig zg~o |fz ig wqa dfifdgcft ai }gifz tqtrhrqrf aorza |gzoratht at bfjia>

7/ Ra `gz ct mdhi dfc|zaobaz ift |zfjiacgt zatqairft/:/ Ra a}hrgz dfomqthfoat wqa |fbzgo tqzkhz dqgobf ta fchra ai thkof ba ig cqirh|ihdgdho> : 7: 3 :(7: 3 :7:/

Anac|if 4

Tag p 3 7/ Aodqaorzg ai }gifz ba 8p % :/Tfiqdho

8(7 % : 3 8 % :3 77

Anac|if 2

Tag x 3 :/ Aodqaorzg ai }gifz ba ;x 77x % :/Tfiqdho

;

(: 77(

: % : 3

4

7

:% :: % :

3 :247:

3 : ifokhrqb (i x god`f ({/ Ao atrft dgtft bajat taz dqhbgbftf gi tqtrhrqhz ai }gifz g|zf|hgbf ao ig}gzhgjia g|zf|hgbg/

Anac|if 1

7/7/ Ap|zathfoat ]gzhgjiat


11/955

{{{/de7:/fzk 1

Ai zag ba qo rzg|a~fhba atr bgbg |fz ig adqgdho G 3 : (g % j/ Aodqaorzg ai zag ba qo rzg|a~fhba dfo jgtatg 3 7< dc) j 3 71 dc x girqzg ` 3 0 dc/

\gzg aodforzgz ig tfiqdho g atra |zfjiacg) thc|iacaora rfcgcft ift }gifzat |zf|fzdhfogbft |gzg dgbg }gzhgjia)g) j x `) x ift dfifdgcft ao ig ap|zatho ba G>

G 3`:

(g % j Ta tqtrhrqxa 7< ao g) 71 ao j x 0 ao `/

G 30:

(7< % 71 Ta a}giqg dgbg |gzra/ (7< % 71 3 :150:

3 2

G 3 2(:1 3 7 Ai zag bai rzg|a~fhba at 7


12/955

= {{{/de7:/fzk

A}giqgdho ba Ap|zathfoat Gikajzghdgt wqa ho}fiqdzgo Ap|foaorat

Cqd`gt mzcqigt x adqgdhfoat ao cgracrhdg dforhaoao ap|foaorat/ Ift ap|foaorat tfo qrhih~gbft dfcf qogofrgdho dfc|gdrg ba qog cqirh|ihdgdho za|arhbg/ \fz anac|if>

: : 3 ::: : : 3 :4

Ai ap|foaora hobhdg igt }adat wqa ai ocazf at qrhih~gbf dfcf mgdrfz (at badhz) igt }adat wqa ta `g cqirh|ihdgbf |fzt chtcf/ Dqgobf rzgjgngcft ohdgcaora dfo ocazft aorazft) at qtqgicaora ct mdhi ai thc|ihdgz ig ap|zatho/Gt) |fbacft thc|ihdgz>

:: 3 2

x

:4 3 0

Tho acjgzkf) oadathrgcft ift ap|foaorat dqgobf rzgjgngcft dfo }gzhgjiat/ Atrf at gt |fzwqa at cqd`f ct mdhiatdzhjhz p0 wqa p p p p p p p p/\gzg a}giqgz ap|zathfoat dfo ap|foaorat) bajat tqtrhrqhz ift }gifzat oqczhdft bgbft ao dgbg }gzhgjia) x iqakfthc|ihdgz/ At at|adhgicaora hc|fzrgora ao atra dgtf qtgz |gzoratht |gzg amadrqgz ig tqtrhrqdho/ Atrf ta `gda dfoai o ba kgzgorh~gz wqa ig thc|ihdgdho ta zagih~gz dfzzadrgcaora/

Anac|if 0

Ai zag ba qo dzdqif atr bgbf |fz ig mzcqig G 3 z:/ Aodqaorzg ai zag ba qo dzdqif ba zgbhf z3 7; hod`at/

Tqtrhrqxa ift }gifzat oqczhdft ao ig adqgdho/

G 3 z: Ta tqtrhrqxa 7; ao z/

G 3 (7;: 7; 7; 3 8


13/955

{{{/de7:/fzk ;

1p: 2x 3 1(2: 2(1 Ta tqtrhrqxa p 3 2 x x 3 1/3 1(7= 2(1 Ta a}giqg ai ap|foaora (2: 3 7=/3 0

:p: 4p: % 1 3 :(14 4(1: % 1 Ta tqtrhrqxa 1 ao p/3 :(7:1 4(:1 % 1 Ta a}giqgo ift ap|foaorat

(14 3 (1(1(1 3 7:1 x (1: 3 (1(1 3 :13 :1 (:: 3 (:(: 3 2 x (:4 3 (:(:(: 3 0/

(2: 3 (2(2 3 7= x (24 3 (2(2(2 3 =2/

Anac|if 7:

Ig girqzg (` wqa gidgo~g qog |aifrg (f jgio ao }qaif dfomfzca g}go~g) |gzg qo hotrgora at|addf (r) atr bgbg|fz ig mzcqig> ` 3 4:r: % =


14/955

0 {{{/de7:/fzk

7/ :77p:/ 7/41 x4/ 4 722/ 72 ~

A}gig igt thkqhaorat ap|zathfoat |gzg g 3 4) j 3 :) d 3 1 x b3 2/

1/ :g % 4j=/ 2d % b;/ 1gd:j0/ :gdb8/ 4jb

7


15/955

{{{/de7:/fzk 8

77/ #77:/ 47

A}giqgz ap|zathfoat gikajzghdgt wqa dforhaoao tcjfift ba gkzq|gdho/ A}giqgz ap|zathfoat gikajzghdgt wqa ho}fiqdzgo jgzzgt ba mzgddhfoat/ A}giqgz ap|zathfoat gikajzghdgt dfo qog dgidqigbfzg wqa |ftag cfbf kzdf/

HorzfbqddhoFjtaz}g x a}gig ig thkqhaora ap|zatho>

: % 2 ; 7 36

Ba dqorgt mfzcgt bhmazaorat |fbacft horaz|zargz atra |zfjiacg6) x Dqorgt zat|qatrgt bhmazaorat |fbzg gikqhaoaodforzgz gi zatfi}azif6

Ig mfzcg ct thc|ia ba a}giqgz bhd`g ap|zatho at thc|iacaora dfcao~gz batba ig h~wqhazbg x rzgjgngz g rq cgoazgigt f|azgdhfoat) fjraohaobf x qrhih~gobf dqhbgbftgcaora ift zatqirgbft |gzdhgiat wqa dfomfzcgzo ai rfrgi g cabhbg

wqa g}go~gt ao ai |zfdatf ba a}giqgdho ba ig ap|zatho>

: % 2 3 =

= ; 3 2:2: 7 3 27

Th rq horzfbqnazgt ig ap|zatho fzhkhogi ao qog dgidqigbfzg of dhaordg wqa rgc|fdf rhaoa cfbf ba kzdft) |zfjg#jiacaora fjraobzt 27 dfcf zat|qatrzg/ Th) |fz frzf igbf horzfbqnazgt ig ap|zatho ao qog dgidqigbfzg dhaordg) fwqa |qaba kzgdgz) |zfjgjiacaora fjraobzt :8 dfcf zat|qatrg/

Ao cgracrhdg) ai fzbao ao wqa ta zagih~go igt bhmazaorat f|azgdhfoat (rgiat dfcf tqcgz) cqirh|ihdgz) ard/ athc|fzrgora/ Ao ig ap|zatho ba gzzhjg) ig f|azgdho ba cqirh|ihdgdho rhaoa |zadabaodhg (at badhz) wqa baja zagih~gztagorat wqa ig gbhdho) gt wqa ig a}giqgcft |zhcazf/ Rfcgobf ao dqaorg atrg zakig ba |zadabaodhg) za#atdzhjgcft igap|zatho fzhkhogi) |azf dfifdgobf g`fzg ig cqirh|ihdgdho aorza |gzoratht) ( ) |gzg hobhdgz wqa baja taz a}giqgbg|zhcazf/

: % (2 ; 7 36

Gt) a}giqgzacft |zhcazf ig ap|zatho baorzf ba ift |gzoratht> 2 ; 3 :0/ Aorfodat) oqatrzg ap|zatho ta dfo}hazraao>

7/:/ Fzbao ba igt F|azgdhfoat Gikajzghdgt


17/955

{{{/de7:/fzk 77

: % (:0 7 36

\gzg ai dgtf bfoba ohdgcaora aphtrgo gbhdhfoat x tqtrzgddhfoat) taodhiigcaora dfcao~gcft batba ig h~wqhazbg)fjraohaobf x qrhih~gobf ift zatqirgbft |gzdhgiat g cabhbg wqa g}go~gcft ao ig a}giqgdho ba ig ap|zatho>

: % :0 3 4

g 2 ; 77:j 2 (; 77 % :d 2 U; (77 % :SDgbg qog ba atrgt ap|zathfoat dforhaoa ift chtcft ocazft x igt chtcgt f|azgdhfoat cgracrhdgt) x ao ai chtcffzbao/ Tho acjgzkf) ig aphtraodhg ba }gzhft tcjfift ba gkzq|gdho thkohdg wqa bajacft a}giqgz igt ap|zathfoatao qo fzbao bhmazaora |gzg dgbg ihrazgi/ ]agcft dcf a}giqgzacft dgbg qof ba atrft anac|ift/

g Atrg ap|zatho of dforhaoa |gzoratht/ Ba gdqazbf g \ACBGT) bajacft a}giqgz gbhdhfoat x tqtrzgddhfoat)rzgjgngobfba h~wqhazbg g bazad`g) ao ai fzbao ba g|gzadhchaorf ba bhd`gt f|azgdhfoat) (\ACBGT OF baja horaz|zargzta



18/955

7: {{{/de7:/fzk

dfcf `gdaz |zhcazf ig tqcg x iqakf ig tqtrzgddho) thof wqa taodhiigcaora bajacft a}giqgzigt ao ai fzbao ao wqag|zadao bhd`gt f|azgdhfoat/

Tfiqdho

2 ; 77 % : 3 4 77 % :3

(4 % 77 % :

3 72 % :3 7:

j Atrg ap|zatho rhaoa |gzoratht/ Gt) bajacft a}giqgz |zhcazf ig ap|zatho wqa ta aodqaorzg baorzf ba aiift);77 3 2/ Zadqazbg wqa zatrgz qo ocazf oakgrh}f awqh}gia g tqcgz qo ocazf |fthrh}f>Tfiqdho

2 (;77 % : 3 2 (2 % :3 0 % :

3 7 4 1 3 715 ; : 3 4/1/ (OFRG> \ACBGT of baja horaz|zargzta dfcf zagih~gz|zhcazf ig cqirh|ihdgdho x iqakf ig bh}htho) thof wqa taodhiigcaora bajacft a}giqgzigt ao ai fzbao ao wqag|gzadao bhd`gt f|azgdhfoat G dforhoqgdho zagih~gcft ig tqtrzgddho>

Tfiqdho

4 1 ; : 3 714/13 77/1

j \zhcazf) a}giqgcft ig ap|zatho baorzf bai |gzoratht> 1 ; 3 :/ Iqakf rzgjgngcft ig ap|zatho zatqirgora bah~wqhazbg g bazad`g/



19/955

{{{/de7:/fzk 74

Tfiqdho

4 (1; : 3 4 (: :3 (= :3 4

d \zhcazf) a}giqgcft igt ap|zathfoat baorzf ba ift |gzoratht> 4 1 3 71) ; : 3 4/1/ Iqakf rzgjgngcft igap|zatho zatqirgora ba h~wqhazbg g bazad`g/

Tfiqdho

(4 1 (; : 3 71 4/13 77/1

Ofrg wqa ao ig |gzra (d ai zatqirgbf of dgcjh dfo ai qtf ba |gzoratht) \azf ig ap|zatho at ct mdhi ba iaaz/ Aokaoazgi) ift |gzoratht |qabao taz qtgbft ao bft mfzcgt bhtrhorgt>

\gzg girazgz ai fzbao ba igt f|azgdhfoat ao qog ap|zatho bgbg \gzg digzhdgz ig ap|zatho x `gdazig ct mdhi ba dfc|zaobaz

Gikqogt ap|zathfoat dgzadao ba |gzoratht) frzgt dforhaoao cqd`f dfonqorft ba aiift/ Gikqogt }adat igt ap|zathfoatraobzo dfonqorft ba |gzoratht baorzf ba frzft dfonqorft ba |gzoratht/ Dqgobf ra aodqaorzat ao dgtft dfcf atrairhcf) dfofdhbft dfcf dgtft ba |gzoratht gohbgbft) dfchao~g |fz ai |gzoratht ct horazof x) iqakf) rzgjgng`gdhg gmqazg/

Anac|if 4

Qrhih~g ai fzbao (|zadabaodhg ba igt f|azgdhfoat |gzg a}giqgz>

0 U78 (: % 1 ;S

Thkqhaobf \ACBGT |zhcazf ift |gzoratht) dfcao~gcft dfo ift |gzoratht ct horazoft |zhcazf>

Tfiqdho

0 (78 (: % 1 ; 3 0 (78 ; ;3 0

1

3 4

Ao ikajzg) qrhih~gcft ai fzbao ba igt f|azgdhfoat dqgobf `acft tqtrhrqhbf }gifzat baorzf ba ap|zathfoat ba}gzhgjiat/ Ao atgt thrqgdhfoat raobzacft qog ap|zatho gikajzghdg bgbg) ig dqgi ho}fiqdgzg qog f ct }gzhgjiat)x rgcjho raobzacft ift }gifzat oqczhdft g tqtrhrqhz ao igt bhmazaorat }gzhgjiat dforaohbgt ao bhd`g ap|zatho/Qog }a~ tqtrhrqcft bhd`ft }gifzat oqczhdft ao igt }gzhgjiat dfzzat|fobhaorat) |zfdabacft dfcf ao ift anac|iftgorazhfzat) g|ihdgobf igt zakigt ba |zadabaodhg ba \ACBGT/

Anac|if 2

Qrhih~g ai fzbao ba igt f|azgdhfoat |gzg a}giqgz ig thkqhaora ap|zatho gikajzghdg>



20/955

72 {{{/de7:/fzk

g : (4p % : dqgobf p 3 :j 4x: % :x 7 dqgobf x 3 4d : (r ;: (q4 } dqgobf r3 78) q 3 2 x } 3 :g Ai |zhcaz |gtf at tqtrhrqhz ai }gifz oqczhdf at|addf ba p baorzf ba ig ap|zatho bgbg/ Dfo ai o ba `gdaz ctdigzg ig ap|zatho zatqirgora) bajacft dfifdgz bhd`f }gifz baorzf ba |gzoratht/

Tfiqdho

: (4(: % : 4(: at if chtcf wqa 4 :Thkqa ig zakig ap|zatgbg ao \ACBGT \zhcazf ift |gzoratht/ Iqakf) baorzf ba ift |gzoratht) thkqa oqa}gcaoraigt zakigt ba |zadabaodhg ba \ACBGT/

: (4 : % : 3 : (= % : Baorzf ba ift |gzaoratht) a}giqgcft |zhcazf ig cqirh|ihdgdhfo/: 0 3 = g`fzg dfc|iargcft ig a}giqgdhfo bai |gzaoratht/

j Ai |zhcaz |gtf at wqa tqtrhrqxgcft ai }gifz oqczhdf ba x baorzf ba ig ap|zatho/

Tfiqdho4 (4: % : (4 7Ba gdqazbf dfo \ACBGT> Of |fbacft bajacft a}giqgz ift ap|foaorat (x of |fbacft thc|ihdgzift

3 4 (4: % : (4 7 A}giqgcft ift ap|foaorat > (4: 3 83 48 % : (4 7 A}giqgcft ig cqirh|ihdgdhfo > 48 3 :;5:4 3 =3 :; % (= 7 A}giqgcft ig gbhdhfo x tqtrzgddhfo) ao ai fzbao

wqa g|gzadao) ba h~wqhazbg g bazad`g/

3 :;

=

7

3 :

: (78 ;: (24 :Thkqhaobf igt zakigt ba \ACBGT>

3 : (78;: (24 : A}giqgobf |gzaoratht > (78 ; 3 7:5(24 : 3 (=2 : 3 =:3 : 7:: =: A}giqgobf ap|foaorat > 7:: 3 7223 : 722=: A}giqgobf ig cqirh|ihdgdhfo > 722 =: 3 08:03 : 08:0 A}giqgobf ig tqtrzgddhfo/3 08:=

Ao ig |gzra (j cgorq}hcft |gzoratht gizababfz ba ocazft oakgrh}ft |gzg digzhdgz ai |zfjiacg/ Bhd`ft |gzorathtof gmadrgo ai fzbao ba igt f|azgdhfoat) |azf gxqbgo g a}hrgz dfomqthfoat dqgobf cqirh|ihdgcft ocazft oakgrh}ft/

Ig |gzra (d ao ai irhcf anac|if cqatrzg frzf |qorf horazatgora/ Dqgobf raoacft qog ap|zatho baorzf ba |gzoratht)qrhih~gcft \ACBGT |gzg barazchogz ai fzbao ao wqa ta a}giqgzo ift dforaohbft ba bhd`ft |gzoratht/



21/955

{{{/de7:/fzk 71

A}giqgobf Ap|zathfoat Gikajzghdgt wqa dforhaoao Jgzzgt ba Mzgddhfoat

Igt jgzzgt ba mzgddhfoat dqaorgo dfcf tcjfift ba gkzq|gdho |gzg \ACBGT) x |fz rgorf bajazgo taz a}giqgbgt|zhcazf dqgobf ta atr zatfi}haobf qog ap|zatho/ Qo bargiia cqx hc|fzrgora at wqa wqa rfbft ift oqcazgbfzatx baofchogbfzat |qabao rzgrgzta dfcf th atrq}hazgo baorzf ba |gzoratht ho}hthjiat/ Dqgobf ift |gzoratht zagiatrgcjho atro |zataorat) bajat zadfzbgz wqa ift tcjfift ba gkzq|gdho ct horazoft bajazgo taz a}giqgbft |zhcazf/

Th) |fz anac|if) ift |gzoratht g|gzadao ao qo oqcazgbfz) aiift rhaoao cgxfz |zadabaodhg wqa ig jgzzg ba mzgddho/Th ift |gzoratht g|gzadao mqazg ba ig mzgddho) aorfodat ig jgzzg ba ig mzgddho raobz ig |zadabaodhg cgxfz/

Anac|if 1

Qtg ai fzbao (|zadabaodhg ba igt f|azgdhfoat |gzg a}giqgz igt thkqhaorat ap|zathfoat>

g ~%42 7Dqgobf ~ 3 :j

g%:j%2 7

% jDqgobf g 3 4 gob j 3 7

d :

{%(p:~(x%::

7

Dqgobf { 3 77) p 3 4) x 3 7 x ~ 3 :g Tqtrhrqhcft ai }gifz oqczhdf at|addf ba ~ baorzf ba ig ap|zatho/

Tfiqdho>:%4

2 7Gqowqa atrg ap|zatho of |ftaa |gzoratht) ig zaatdzhjhzacft |gzg cftrzgz ai amadrf ba ig jgzzg ba mzgddho/(:%4

2 7Qrhih~gobf \ACBGT) a}giqgcft |zhcazf ig ap|zatho bai oqcazgbfz/12 7\fbacft dfo}azrhz 12 ao qo ocazf chprf>12 3 7

72

Iqakf a}giqgcft ig ap|zatho>12 7 3 7 72 7 3 72j Tqtrhrqhcft ift }gifzat ba g x j ao ig ap|zatho>

Tfiqdho>4%:7%2 7

7Atrg ap|zatho |ftaa |gzoratht gohbgbft (bajat zadfzbgz ai amadrf ba ig jgzzg ba mzgddho ao ai oqcazgbfz xbaofchogbfz/ Ai tcjfif ba gkzq|gdho ct horazof at |zadhtgcaora ig jgzzg ba mzgddho/ \fz if rgorf) a}giqgzacft|zhcazf ai oqcazgbfz (4 % : x ai baofchogbfz (7 % 2/

11

7

7 G`fzg a}giqgcft ai horazhfz ba ift |gzaoratht) dfcao~gobf dfo ig bh}hthfo/

(77 7 G dforhoqgdhfo a}giqgcft ig tqtrzgddh fo/

Tfiqdho>

Atrg dfc|ihdgbg ap|atho rhaoa }gzhgt dg|gt ba |gzoratht gohbgbft/ Qo crfbf |gzg gtakqzgzoft ba wqa dfcao#



22/955

7= {{{/de7:/fzk

~gcft dfo ai |gzoratht ct horazof at `gdhaobf qtf ba ct thkoft ba gkzq|gdho/ Gt) |fbacft zaatdzhjhz atrgap|zatho) dfifdgobf bhmazaorat thkoft ba gkzq|gdho baorzf bai at|gdhf baohbf |fz ig jgzzg ba mzgddho/ Ao|gzrhdqigz) ift thkoft ba gkzq|gdho ct aprazoft tazo |gzoratht ( / G dforhoqgdho ta dfifdgzo ift dfzdàratho}hthjiat ao ig jgzzg ba mzgddho/ Aiift tazo atdzhrft dfcf U S/Mhogicaora) ai razdaz oh}ai (ct horazof ba ift|gzoratht gohbgbft tazo igt iig}at / Gt chtcf) bangzacft ift ocazft oakgrh}ft baorzf ba |gzoratht/

: U77 % u 4 :(: sS

Uu 7 % : s:S 7

Dfcao~gzacft dfo ai thkof ba gkzq|gdhfo cgt horazofus /

u 7 % : s 3 45u 4 :(: s 3 4 % 2 3 ;

:

U77 % ;S

U4:S7

Ai thkqhaora oh}ai rhaoa bft dfzdàrat g a}giqgz/

:

708

7

G`fzg a}giqgzacft ift |gzaoratht) dfcao~gobf |fz ig bh}hthfo/

:(:7 Mhogicaora) dfc|iargcft ig gbhdhfo x ig tqtrzgddhfo/:(7 3 :

A}giqgdho ba Ap|zathfoat Gikajzghdgt dfo ig mgchihg RH#04&02 ba DgidqigbfzgtKzgdgbfzgt

Qog dgidqigbfzg kzgdgbfzg at qog àzzgchaorg cqx rhi |gzg a}giqgz ap|zathfoa gikajzghdgt/ Ao kaoazgi qogdgidqigbfzg kzgdgbfzg thkqa igt zakigt ba |zadabaodhg bgbgt |fz \ACBGT/ Ao atrg taddho ap|ihdgzacft bftmfzcgt |gzg a}giqgz ap|zathfoat dfo qog dgidqigbfzg kzgdgbfzg/

Crfbf 7> Tqtrhrqxa |zhcazf ao ig }gzhgjia/ Iqakf) a}gig ig ap|zatho oqczhdg dfo ig dgidqigbfzg

Anac|if =

A}gig U4(p: 7: p2 % 7:S % 1p4 7dqgobf p 3 4

Tfiqdho>

Tqtrhrqhcft ai }gifz p 3

4ao ig ap|zatho/

U4((4: 7: (42 % 7:S % 1(44 7

Horfbqdhcft atrg ap|zatho ao ig dgidqigbfzg) rgi dfcf at) x |zathfogcft UAORAZS/ (Ofrg) qrhih~g _ |gzg iftap|foaorat

Ig zat|qatrg at #74/

Crfbf :> \zhcazf horzfbqda ig ap|zatho gikajzghdg fzhkhogi ao ig dgidqigbfzg) x iqakf a}gigig/ ]agcft ai chtcfanac|if/



23/955

{{{/de7:/fzk 7;

A}giqgzU4(p: 7: p2 % 7:S % 1p4 7dqgobf p 3 4

\zhcazf) ta kqgzbg ai }gifz p 3 4ao ig dgidqigbfzg bhkhrgcft #4 UTRFS p(Ig iarzg p|qabacft horzfbqdhzigqtgobf ai jfro p#U]GZS f UGI\@GS % UTRFS/ Aorfodat bhkhrgcft ig ap|zatho ao ig dgidqigbfzg x |zathfogcftUAORAZS/

Ig zat|qatrg at #74/

Ai takqobf crfbf at canfz |fzwqa |qabat a}giqgz mdhicaora ig chtcg ap|zatho |gzg dqgiwqhaz }gifz wqa bataat/\fz anac|if) a}giqacft ig chtcg ap|zatho dfo ift }gifzat p 3 : x p 3 :4 /

\gzgp 3 :) Kqgzbgcft ai }gifz bap ao ig dgidqigbfzg> : UTRFSp/ \zathfogcft bft }adat U:obS UAORAZS |gzg |fbazfjraoaz ig ap|zatho |za}hg wqa bhkhrgcft ao ig |gorgiig tho raoaz wqa horzfbqdhzig oqa}gcaora/ Iqakf) |zathfogcftUAORAZS |gzg a}giqgz/

Ig zat|qatrg at =:/

\gzg p 3 :4 ) kqgzbgcft ai }gifz ba p ao ig dgidqigbfzg>:4 UTRFS p/ \zathfogcft U:obS UAORAZS bft }adat |gzg

fjraoaz ig ap|zatho ao ig |gorgiig tho raoaz wqa horzfbqdhzig ba oqa}f/ Iqakf) |zathfogcft UAORAZS |gzg a}giqgz/

Ig zat|qatrg at 74/:7 f 7 4p:2x:%p2(p%x

7:

|gzg p 3 :) x 3 7/

Tfiqdho



24/955

70 {{{/de7:/fzk

Kqgzbg ift }gifzat ba p x x/ #: UTRFS p) 7 UTRFS x/ Igt iarzgt p x x |fbacft horzfbqdhzigt ao ig dgidqigbfzg |fzcabhf ba igt radigt UGI\@GS % UEAXS/ G`fzg horzfbqda ig ap|zatho ao ig dgidqigbfzg/ Dqgobf qog ap|zathocqatrzg ig bh}htho ba bft ap|zathfoat gtakzgra ba qrhih~gz |gzoratht> (oqcazgrfz (baofchogrfz\zathfog UAORAZS |gzg fjraoaz ig zat|qatrg /00 f 08 /

Anazdhdhft ba Za|gtf

7/ Qtg ai fzbao ba igt f|azgdhfoat |gzg a}giqgz igt thkqhaorat ap|zathfoat/

g/ 0 (78 (: % 1 ;j/ : % ;777: 4d/ (4 % ; (; 7:b/ :(4%(:72(=%: (4 1

:/ A}gig igt thkqhaorat ap|zathfoat wqa ho}fiqdzgo }gzhgjiat/

g/ nen%e dqgobf n 3 = x e3 7:/

j/ :x: dqgobf p 3 7 x x 3 1

d/ 4p: % :p % 7 dqgobf p 3 1b/ (x: p: dqgobf p 3 : x x 3 7

4/ A}gig igt thkqhaorat ap|zathfoat wqa ho}fiqdzgo }gzhgjiat/

g/ 2p8p:4p%7 dqgobf p 3 :j/ ~

:

p%x %p:

px dqgobf p 3 7) x 3 :) x ~ 3 2/d/ 2px~x:p: dqgobf p 3 4) x 3 :) x ~ 3 1

b/ p:~:

p~:p(~p dqgobf p 3 7 gob ~ 3 42/ Hotazrg ift |gzoratht oadatgzhft ao dgbg ap|zatho |gzg fjraoaz qog adqgdho }ihbg/

g/ 1

:

=

2 % : 3 1

j/ 7:2 % 7


25/955

{{{/de7:/fzk 78

4/ 0=2/ 4

7/ 047:/ 2;44/ #:22/

01

7/ (1 : (= 1 % : 3 1:/ (7: 2 % 7


26/955

:< {{{/de7:/fzk

7/4 \grzfoat x Adqgdhfoat

Fjnarh}ft bai G|zaobh~gna

Ao atrg iaddho g|zaobazt g>

Atdzhjhz adqgdhfoat/ Qrhih~gz ai cfbaif }azjgi |gzg atdzhjhz adqgdhfoat/ Zatfi}az |zfjiacgt cabhgora adqgdhfoat/

Horzfbqddho

Ao cgracrhdgt) x at|adhgicaora ao ikajzg) at dfo}aohaora jqtdgz |grzfoat ao ift ocazft fjtaz}gbft/ Igt `az#zgchaorgt bai ikajzg oft gxqbgo g batdzhjhz atrft |grzfoat dfo |gigjzgt x dfo adqgdhfoat (mzcqigt fz mqodhfoat/Qog adqgdho at dfcf qog zadarg cgracrhdg wqa |zf|fzdhfog ai }gifz ba qog }gzhgjia ao rzchoft ba frzg/

\fz anac|if) th qo |gzwqa ba bh}azthfoat dfjzg "7: |fz gbchtho) Aorfodat ai ocazf ba |aztfogt wqa aorzgo dgbgbg g bhd`f |gzwqa x ig dgorhbgb ba bhoazf zadgqbgbf |fz }aorg ba jfiarft atro zaigdhfogbft cgracrhdgcaora/

Oftfrzft |fbacft atdzhjhz qog zakig |gzg aodforzgz ig dgorhbgb ba bhoazf zadgqbgbf |fz }aorg ba jfiarft/

Gt) ao |gigjzgt) |fbzgcft badhz) Ai bhoazf zadgqbgbf ao bigzat at (hkqgi g bfda }adat ai ocazf ba |aztfogtwqa hokzatgo gi |gzwqa/

Rgcjho |fbzgcft `gdaz qog rgjig/ Ig thkqhaora rgjig zaigdhfog ai ocazf ba |aztfogt wqa }hthrgo ai |gzwqa x airfrgi ba bhoazf zadgqbgbf |fz }aorg ba jfiarft/

Oqcazf ba }hthrgorat 7 : 4 2 1 = ;

Bhoazf zadgqbgbf(" 7: :2 4= 20 =< ;: 02

Digzgcaora) oadathrgzgcft qog rgjig kzgoba th zakhtrzgtacft if wqa fdqzza ao qo bg cqx fdq|gbf bqzgora ai|azfbf ba }gdgdhfoat atdfigzat'

Qog razdazg mfzcg ao ig wqa |fbzgcft zaigdhfogz bft dgorhbbat (}hthrgorat x bhoazf zadgqbgbf at cabhgora qogkzdg/ Th kzgdgcft ai bhoazf zadgqbgbf ao ai ana }azrhdgi x ai ocazf ba }hthrgorat ao ai ana `fzh~forgi) aorfodatraobzgcft qog kzdg wqa iqdhzg thchigz g ig wqa thkqa g dforhoqgdho/ Fjtaz}g wqa atrg kzdg cqatrzg qog ioagdforhoqg |gzg }gifzat zagiat (of bhtdzarft ba p (a/k/) p 3 :/1/ \azf) ao ig }hbg zagi atrf of tazg |fthjia |fzwqa of|qabat raoaz cabhg |aztfog hokzatgobf gi |gzwqa/ Atrf at qo racg ba if wqa ta dfofda dfcf bfchohf x zgokf) racgtba ift wqa `gjigzacft ct gbaigora/

7/4/ \grzfoat x Adqgdhfoat


27/955

{{{/de7:/fzk :7

Ai crfbf wqa apgchogzacft ao bargiia ao atrg iaddho at dazdgof g ig |zhcazg mfzcg caodhfogbg gzzhjg wqa |qabaqtgzta |gzg batdzhjhz ig zaigdho aorza bft dgorhbgbat/ At badhz) dqgobf batdzhjhcft bhd`g zaigdho dfo |gigjzgt/ Gt|fz anac|if) |fbacft badhz wqa Ai bhoazf zadgqbgbf ao bigzat at bfda }adat ai ocazf ba |aztfogt wqa hokzatgo

gi |gzwqa/ \fz frzg |gzra) ao rzchoft cgracrhdft) |fbacft batdzhjhz qog zaigdho ba atra rh|f cabhgora }gzhgjiat/Qog }gzhgjia at qog iarzg qrhih~gbg |gzg za|zataorgz qog dgorhbgb batdfofdhbg/ Igt }gzhgjiat ta qrhih~go |gzg dzagzap|zathfoat cgracrhdgt/ At hc|fzrgora wqa ofrat wa |fbacft }az ai fzhkao ba igt ap|zathfoat cgracrhdgt ao igt|gigjzgt/

Ai bhoazf zadgqbgbf ao bigzat at bfda }adat ai ocazf ba |aztfogt wqa hokzatgo gi |gzwqa/

Atrf |qaba ap|zatgzta dfcf>

(Ai bhoazf zadgqbgbf ao bigzat 3 7: (ai ocazf ba |aztfogt wqa aorzgo gi |gzwqabfoba ta `go dfifdgbf igt dgorhbgbat aorza |gzoratht |gzg `gdazigt ct a}hbaorat/ G`fzg |fbacft }az ctmgdhicaora dqiat dgorhbgbat |qabao za|zataorgzta cabhgora iarzgt (f }gzhgjiat/ \zhcazf) bajacft atrgjiadazdqiat iarzgt (}gzhgjiat dfzzat|fobao g dqiat dgorhbgbat/ Atra |zfdatf ta dfofda dfcf baohdho ba }gzhgjiat>

Tag p 3 ai ocazf ba |aztfogt wqa aorzgo gi |gzwqa/

Tag x 3 ig dgorhbgb rfrgi ba bhoazf zadgqbgbf ao jfiarazg/

G`fzg |fbacft cftrzgz ig dqgzrg mfzcg |gzg batdzhjhz ig zaigdho fzhkhogi) wqa at cabhgora oqatrzg adqgdhogikajzghdg/

x 3 7:p

Atdzhjhz qog adqgdho cgracrhdg g rzg}t }gzhgjiat at cqx dfo}aohaora/ R |qabat zagih~gz rfbgt igt f|azgdhfoatoadatgzhgt |gzg zatfi}az atra |zfjiacg tho raoaz wqa atdzhjhz igt dgorhbgbat dfofdhbgt x batdfofdhbgt) cabhgora|gigjzgt atdzhrgt) qog x frzg }a~/ Gi ogi bai |zfjiacg) ohdgcaoa oadathrgzacft zadfzbgz dqiat dgorhbgbat tfoza|zataorgbgt |fz p x x/

Atdzhjhaobf qog Adqgdho

Qog adqgdho at qo rzchof qrhih~gbf |zg batdzhjhz qog dfiaddho ba ocazft x }gzhgjiat zaigdhfogbft aorza tcabhgora f|azgbfzat cgracrhdft/ Qog adqgdho gikajzghdg dforaobz iarzgt wqa za|zataorgo dgorhbgbat zagiat)f jhao ocazft wqa dfotrhrqxao }gifzat |gzg dgorhbgbat zagiat/ Th) |fz anac|if) wqhthzgcft qtgz ig adqgdhogikajzghdg bai anac|if gorazhfz |gzg aodfozgz ai bhoazf zadgqbgbf |fz qo ocazf at|addf ba }hthrgorat) raobzgcftwqa zaac|ig~gz ata }gifz (ai ocazf ba }hthrgorat ao p) x aorfodat zatfi}az ig adqgdho zatqirgora |gzg x/



28/955

:: {{{/de7:/fzk

Anac|if 7

Qo |gzwqa ba bh}azthfoat dfjzg "7: g ift }hthrgorat/ Aodqaorzg ai ocaazf zadgqbgbf th 7:8= |aztfogt }hthrgo ai|gzwqa/

Bh}hbgcft ig tfiqdho g atra |zfjiacg ao qo ocazf ba |gtft/ Atrf oft gxqbgz g zatfi}az rfbft ift |zfjiacgt baatrg iaddho/

\gtf 7 Fjro ig homfzcgdho hc|fzrgora/

(bhoazf zadgqbgbf ao bfigzat 3 7: (oqcazf ba }hthrgorat(oqcazf ba }hthrgorat 3 7:8=

\gtf : Dfo}hazra bhd`g homfzcgdho ao qog adqgdho cgracrhdg/

@gdacft atrf baohaobf }gzhgjiat x tqtrhrqxaobf ao aiigt }gifzat dfofdhbft/

Tag x 3 (bhoazf zadgqbgbf ao bfigzat

x 3 7:7:8= ATRG AT OQATRZG ADQGDHFO/

\gtf 4 Zatqai}a ig adqgdho/

x 3 7111: Zat|qatrg> Ai bhoazf zadgqbgbf at "7111:

\gtf 2 Dfc|zqajg ai zatqirgbf/

Th "7111: tfo zadgqbgbft ao }aorg ba jfiarft x dgbg jfiarf dqatrg "7:) aorfodat |fbacft bh}hbhz ig dgorhbgb rfrgibai bhoazf zadgqbgbf aorza ai |zadhf ba dgbg jfiarf hobh}hbqgi/

(ocazf ba |aztfogt 37111:

7:3 7:8=

Atra zatqirgbf at hborhdf gi ocazf ba |aztfogt wqa hokzatgzfo gi |gzwqa/ \fz if rgorf ig zat|qatrg `g thbfdfc|zfjgbg/

Anac|if :

Ig thkqhaora rgjig cqatrzg ig zaigdho aorza bft dgorhbgbat/ \zhcazf) atdzhja qog adqgdho wqa batdzhjg bhd`gzaigdho/ Iqakf) aodqaorzg ai }gifz ba j dqgobf g at ;1 < 7< :< 4< 2< 1 :< 2< =< 0< 7


29/955

{{{/de7:/fzk :4

RGJIA 7/7>

Raprf ta rzgbqda g ig ap|zatho cgracrhdg\gzg aodforzgz j j 3ai bfjia bai }gifz ba g :gtqcgz :


30/955

:2 {{{/de7:/fzk

\gtf 7 Fjro ig homfzcgdho hc|fzrgora/

\fbacft }az ba ig rgjig wqa x at dhodf }adat ct kzgoba wqa p/ Ai }gifz ba x at oakgrh}f dqgobf p at |fthrh}f) x at|fthrh}f dqgobf p at oakgrh}f/ \fz rgorf) ig zakig wqa zaigdhfog p x x/

xat ai hkqgi g ai oakgrh}f ba dhodf }adat ai }gifz ba p

Tra| : Dfo}hazra ai thkqhaora aoqodhgbf ao qog adqgdho cgracrhdg/

RGJIA 7/4>Raprf Ta dfo}hazra ao ig Ap|zatho Cgracrhdgx at hkqgi g x 3ai oakgrh}f ba 1 }adat ai }gifz ba

p 1p

x 3 1p ATRG AT OQATRZG ADQGDH FO/

\gtf 4 Ogbg `gx wqa zatfi}az ao atra |zfjiacg/ \fbacft takqhz dfo ai \gtf 2/\gtf 2 Dfc|zqajg ai zatqirgbf/

Ao atra dgtf) ig mfzcg ba dfc|zfjgz ai zatqirgbf at kaoazgobf oqatrzft |zf|hft |gzat px `gdhaobf qtf ba oqatrzgadqgdho/ Th bhd`ft |gzat dfhodhbao dfo ift }gifzat wqa aodforzgcft ao ig rgjig) aorfodat oqatrzg adqgdho atdfzzadrg/ \gzg kaoazgz bhd`ft |gzat) zaac|ig~gcft ao p ift }gifzat #:) x 3 1(: x 3 %7

x 3 1( x 3 1(: x 3 7

x 3 1(2 x 3 :

x 3 1(= x 3 4 Dqgobf (ai ocazf ba ig tacgog 3 7) (ai jgigoda 3 7

Tag ai ocazf ba ig tacgog 3 o

Tag ai jgigoda 3 j

RGJIA 7/1>

Raprf Ta dfo}hazra ao ig Ap|zatho CgracrhdgG cabhbg wqa o ta hodzacaorg ao 7)

j badzada ao ::

j 3 ::o%6

Ai thkof6 hobhdg wqa oadathrgcft frzf rzchof/ Tho bhd`f rzchof ai jgigoda tacgogi tazg #::) #22) #==)///) ifdqgi dforzgbhda ai `ad`f ba wqa ai jgigoda ao ig |zhcazg tacgog at 7


32/955

:= {{{/de7:/fzk

Zatfiqdho ba \zfjiacgt qtgobf Adqgdhfoat

Zatfi}gcft ai thkqhaora |zfjiacg bai cqobf zagi/ \gzg aiif qtgzacft ig homfzcgdho |zf|fzdhfogbg |gzg dzagz qogadqgdho cgracrhdg wqa |qaba taz zatqairg |gzg aodforzgz qog tfiqdho/

Anac|if 1

Qo kzq|f ba atrqbhgorat atr ao qo tgio/ Iqakf wqa :1 atrqbhgorat tgiao) ta fjtaz}g wqa :4 bai kzq|f fzhkhogiwqabg baorzf bai tgio/ Dqorft atrqbhgorat ta aodforzgjgo hohdhgicaora baorzf bai tgio6

\gtf 7 Fjro ig homfzcgdho hc|fzrgora

Tgjacft wqa :1 atrqbhgorat tgiao bai tgio/

Rgcjho tgjacft wqa :4 bai ocazf fzhkhogi ba atrqbhgorat wqabgo baorzf bai tgio/

Oadathrgcft aodforzgz dqorft atrqbhgorat ta aodforzgjgo hohdhgicaora baorzf bai tgio/

\gtf : Dfo}hazra ig homfzcgdho ao qog adqgdho cgracrhdg/

Hohdhgicaora raoacft qo ocazf batdfofdhbf ba atrqbhgorat baorzf bai tgio/ Oft |fbacft zamazhz g aiift dfcf aiocazf fzhkhogi ba atrqbhgorat/

Baogcft ig }gzhgjia p 3 ai ocazf fzhkhogi ba atrqbhgoat baorzf bai tgio/

:1 atrqbhgorat tgihazfo bai tgio/ Aorfodat ai ocazf ba atrqbhgorat wqa |azcgoadao baorzf bai tgio at>

RGJIA 7/=>

Raprf Ta dfo}hazra ao Ap|zatho CgracrhdgAi ocazf fzhkhogi ba atrqbhgoratao ai tgio

p

:1 atrqbhgorat tgiao bai tgio p:1:4 bai ocazf fzhkhogi |azcgoadabaorzf bai dqgzrf

:4p

p :1 3 :4p ATRG AT OQATRZG ADQGDHFO/

\gtf 4 Zatqai}a ig adqgdho/

Tqcg :1 ao gcjft chacjzft ba ig adqgdho/

p :1 3 :4p

p :1 % :1 3:

4p % :1

p 3:4p % :1

Tqtrzga :4pba gcjft chacjzft/

p :4p 3

:4p :

4p % :1

74p 3 :1

7/4/ \grzfoat x Adqgdhfoat


33/955

{{{/de7:/fzk :;

Cqirh|ihdg gcjft chacjzft |fz4/

4 74p 3 :1 4p 3 ;1

Zadqazbg wqa p za|zataorg ai ocazf fzhkhogi ba atrqbhgorat baorzf bai tgio/ \fz rgorf)Zat|qatrg Hohdhgicaora gjgo ;1 atrqbhgorat baorzf bai tgio/

\gtf 2 Dfc|zqajg ig zat|qatrg>

Th dfcao~gcft dfo ;1 atrqbhgorat baorzf bai tgifo x :1 ba aiift tgihazfo) aorfodat wqabgo ;1 :1 3 1< baorzf baitgio/:4 bai ocazf fzhkhogi ba atrqbhgorat at

:4 ;1 3 1

Bg Kgogodhg7 : ta gihcaorg ig dgng dfo qo dfonqorf ba ocazft (bhd`f dfonqorf|qaba) tho acjgzkf) atrgz dfomfzcgbf |fz qo tfif ocazf/ G dforhoqgdho) ig f|azgdho cgracrhdg wqa atrbaorzf ba ig dgng rzgotmfzcg bhd`ft ocazft) ba cgoazg wqa ta fjrhaoa frzf dfonqorf ba ocazft wqa tgia |fz aiigbf f|qatrf ba ig dgng/ Gt |fz anac|if) |gzg ai dgtf bai |gzwqa ba bh}azthfoat) |fbacft gihcaorgz ig dgng dfo

bhmazaorat }gifzat wqa dfzzat|fobgo g ift |gtaft zagih~gbft |fz Nft` ao bhmazaorat bgt/ Ba ig dgng tgibzo ift dftrft|fz bg gtfdhgbft g bhd`ft }gifzat/

Ig aorzgbg ta baofchog dfcf ig }gzhgjia hoba|aobhaora |fzwqa tq }gifz |qaba taz dqgiwqhaz ocazf |fthjia/ Igtgihbg ta fjrhaoa iqakf ba g|ihdgz ig f|azgdho cgracrhdg x ta dfofda dfcf ig }gzhjia ba|aobhaora |fzwqa tq }gifzba|aoba bai }gifz ba aorzgbg/

7/1/ Igt Mqodhfoat dfcf Zakigt x Rgjigt


45/955

{{{/de7:/fzk 48

G caoqbf igt mqodhfoat tfo ct dfc|ihdgbgt wqa ig ba atra anac|if/ Igt mqodhfoat qtqgicaora dforhaoao ct baqog f|azgdho cgracrhdg/ Gwq rhaoat qog thrqgdho ihkazgcaora ct dfc|ihdgbg/

Ngtfo }g g qo |gzwqa ba bh}azthfoat bfoba |gkg "0 |fz bazad`f ba gbchtho x ": |fz dgbg |gtaf/

Atrg mqodho za|zataorg ig dgorhbgb rfrgi wqa Ngtfo |gkg/ Ig zakig wqa kfjhazog ig mqodho at cqirh|ihdgz ai oczfba }hgnat |fz :) x ggbhz 0 gi zatqirgbf ba bhd`g cqirh|ihdgdho/

Horzfbqdhcft bhmazaorat }gifzat |gzg ai ocazf ba |gtaft |gzg fjraoaz bhmazaorat tgihbgt (dftrft/

Atrft bhgkzgcgt ba qnf tfo jgtrgora rhiat |gzg }htqgih~gz if wqa at qog mqodho/ Tho acjgzkf) tq qtf |qaba zatqirgzrzgjgnftf a hodcfbf ao ig |zdrhdg/ At |fz aiif wqa qrhih~gzacft) ao tq iqkgz) ig thkqhaora ofrgdho dfc|gdrg/

\zhcazf) baohcft igt }gzhgjiat/

p 3 ai ocazf ba |gtaft zagih~gbft |fz Ngtfo

x 3 ig dgorhbgb rfrgi ba bhoazf wqa Ngtfo |gk ao ai |gzwqa ba bh}azthfoat

Gt) p za|zataorg ig aorzgbg x x za|zataorg ig tgihbg/ Ig ofrgdho> m( za|zataorg ig mqodho) f awqh}giaoracaora) igtf|azgdhfoat cgracrhdgt g igt wqa tfcaracft ig aorzgbg |gzg |fbaz fjraoaz ig tgihbg/ Ao ai irhcf anac|if) ai dftrfat : }adat ai ocazf ba |gtaft ct 0/ Atrf |qhaba taz atdzhrf dfcf qog mqodho/

m(p 3 :p % 0

Ig tgihbg }haoa bgbg |fz ig mzcqig m(p 3 :p % 0/ Igt ofrgdhfoat x x m(p |qabao dfothbazgzta awqh}giaoaratx qtgzta hobhtrhorgcaora) |azf rao |zataora wqa x za|zataorg ai }gifz ba tgihbg) chaorzgt wqa m(p za|zataorg igtf|azgdhfoat cgracrhdgt g igt wqa tfcaracft ig aorzgbg |gzg fjraoaz ig tgihbg/

Hbaorhdgz ai Bfchohf x ai Zgokf ba qog Mqodho

Ao ai irhcf anac|if) }hcft wqa |fbacft hokzatgz ai ocazf ba |gtaft baorzf ba ig mqodho |gzg fjraoaz ai dftrfrfrgi ba hz gi |gzwqa ba bh}azthfoat/ Ai dfonqorf ba rfbft ift }gifzat |fthjiat |gzg ig aorzgbg ta dfofda dfcf ai



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bfchohf ba ig mqodho/ Ai dfonqorf ba rfbft ift |fthjiat }gifzat |gzg ig tgihbg ta dfofda dfcf ai zgokf ba igmqodho/ Ao cqd`gt thrqgdhfoat rgorf ai bfchohf dfcf ai zgokf ba qog mqodho atrgzo mfzcgbft |fz ai dfonqorfba rfbft ift ocazft zagiat) |azf of thac|za taz ai dgtf/ ]agcft oqa}gcaora g oqatrzf anac|if bai |gzwqa babh}azthfoat

Anac|if 7

Aodqaorzg ai bfchohf x zgokf ba ig mqodho wqa batdzhja ig thrqgdho thkqhaora>

Ngtfo }g g qo |gzwqa ba bh}azthfoat bfoba |gkg "0 |fz bazad`f ba gbchtho x ": |fz dgbg |gtaf/Tfiqdho

@a gwq ig mqodho wqa batdzhja atrg thrqgdho/

m(p 3 :p % 0 3 x

Ao atrg mqodho) p at ai ocazf ba |gtaft x x at ai dftrf rfrgi/ \gzg aodforzgz ai bfchohf ba ig mqodho) oadathrgcftbarazchogz dqiat }gifzat ba p tfo }ihbgt dfcf aorzgbgt/

Ift }gifzat ba p rhaoao wqa taz dazf f |fthrh}ft |fzwqa Ngtfo of |qaba zagih~gz qo ocazf oakgrh}f ba |gtaft/ Ift }gifzat bajao taz aorazft |fzwqa) |fz anac|if) Ngtfo of |qaba zagih~gz :/:1 }hgnat/ Ba cgoazg zagihtrg) baja `gjaz qo ocazf cphcf ba |gtaft wqa |qaba zagih~gz Ngtfo |fzwqa ai |gzwqa

baja dazzgz g dhazrg `fzg) |fzwqa Ngtfo razchog dfo rfbf tq bhoazf) ard/ Tho acjgzkf) bgbf wqa of raoacfthomfzcgdho gikqog gdazdg ba atrf) |fbacft gtqchz wqa rfbft ift aorazft of oakgrh}ft tfo |fthjiat aorzgbgt(ofrgz wqa ta hodiqxa ai dazf) xg wqa of at |fthrh}f) |azf rgc|fdf oakgrh}f) tho hc|fzrgz if kzgobat wqa|qabgo taz tqt }gifzat/

Zat|qatrg \gzg atrg mqodho) ai bfchohf at ai dfonqorf ba rfbft ift aorazft of oakgrh}ft) at badhz (


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Ig }gzhgjia p |qaba rfcgz dqgiwqhaz }gifz zagi cgxfz wqa dazf/

Ig }gzhgjia x rgcjho |qaba rfcgz dqgiwqhaz }gifz zagi cgxfz wqa dazf/

Zat|qatrg Ai bfchohf at ai dfonqorf ba rfbft ift ocazft zagiat cgxfzat wqa dazf/

Ai zgokf at ai dfonqorf ba rfbft ift ocazft zagiat cgxfzat wqa dazf/

j Bgbf wqa of raoacft qo |zfjiacg ap|zatgbf }azjgicaora gtfdhgbf g atrg adqgdho) |fbacft gtqchz wqa|fbacft qtgz dqgiwqhaz ocazf zagi dfcf qo }gifz |fthjia ba p/

Bgbf wqa x 3 p:) ai }gifz ba x taz thac|za of oakgrh}f) tho hc|fzrgz wqa p tag |fthrh}f) oakgrh}f f dazf/

Zat|qatrg Ai bfchohf ba atrg mqodho tfo rfbft ift ocazft zagiat/

Ai zgokf ba atrg mqodho atr mfzcgbf |fz rfbft ift ocazft zagiat of oakgrh}ft/

Zadfzbacft wqa) |gzg qog mqodho) ig }gzhgjia p ta baofchog dfcf ig }gzhjia hoba|aobhaora |fzwqa |qaba rfcgzdqgiatwqhazg }gifzat bai bfchohf/ Ig }gzhgjia x ta baofchog dfcf ig }gzhgjia ba|aobhaoa |fzwqa tq }gifz ba|aobaba p/ \fbacft qrhih~gz tcjfift dqgiatwqhazg |gz za|zataorgz igt }gzhjiat ba|aobhaorat a hoba|aobhaorat/ Gwq rhaoatrzat bhmazaorat anac|ift/

x 3 m(p 3 4p

Z 3 m({ 3 4{

} 3 m(r 3 4r

Rfbgt atrgt ap|zathfoat za|zataorgo g ig chtcg mqodho/ Ig }gzhgjia ba|aobhaora at rzat }adat ig }gzhjia hoba|ao#bhaora/ Ao ig |zdrhdg) ift tcjfift wqa tfo qrhih~gbft |gzg igt }gzhgjiat hoba|aobhaorat x ba|aobhaorat ta atdfkaoba gdqazbf gi qtf dfco kaoazgih~gbf/ \fz anac|if> r ta qrhih~g |gzg za|zataorgz rhac|f) b |gzg bhtrgodhg) } |gzg}aifdhbgb) ard/ Ift tcjfift atrobgz (f kaozhdft wqa ta qrhih~go tfo x |gzg ig }gzhjia ba|aobhaora x p |gzg ig}gzhgjia hoba|aobhaora/

Qog mqodho>

ohdgcaoa gda|rg ocazft wqa |azraoa~dgo g tq bfchohf \gzg dgbg aorzgbg aphtra apgdrgcaora qog tgihbg/ Ai dfonqorf ba rfbgt igt tgihbgt dfzzat|fobhaorat gi bfchohf

dfotrhrqxao ai zgokf

Cqirhcabhg Ihoe \fz frzf }htrg~f gi bfchohf ba qog mqodho) }a ai thkqhaora }hbaf bfoba ai ogzzgbfz zatqai}a qo|zfjiacg ba cqatrzg ba ig \zqajg ba Atrobgzat ba Dgihmfzohg gdazdg ba aodforzgz ai bfchohf ba qog mqodho hoqtqgiE`go Gdgbacx DG Gikajzg H Mqodrhfot (=>42

CABHG

Dihde hcgka rf r`a iamr mfz cfza dforaor/

/

http://www.ck12.org/http://www.ck12.org/http://www.ck12.org/http://www.ck12.org/http://www.youtube.com/watch?v=NRB6s77nx2ghttp://beta.ck12.org/flx/show/cover%20video/user%3Ack12editor/http%3A//www.youtube.com/embed/NRB6s77nx2ghttp://www.youtube.com/watch?v=NRB6s77nx2ghttp://www.ck12.org/


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@gdaz qog Rgjig |gzg qog Mqodho

Qog rgjig at qog mfzcg cqx rhi |gzg fzkgoh~gz ift bgrft za|zataorgbft |fz qog mqodho/ \fbacft nqorgz ift }gifzatba aorzgbg x ba tgihbg) bht|fohobfift ao mfzcg ba rgjig/ Rfcacft ba oqa}f ai anac|if bai |gzwqa ba bh}azthfoat

Ngtfo }g g qo |gzwqa ba bh}azthfoat bfoba |gkg "0 |fz bazad`f ba gbchtho x ": |fz dgbg |gtaf/

Xg `acft }htrf wqa |gzg fjraoaz ig tgihbg dfzzat|fobhaora g qo }gifz ba aorzgbg ta iia}go g dgjf igt f|azgdhfoat: aorzgbg % 0/ \fz anac|if) th |foacft g ig aorzgbg ift }gifzat


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{{{/de7:/fzk 24

RGJIA 7/7

p m(p 3 7p 3 x7 77 3 7#


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22 {{{/de7:/fzk

Aodforzgz qog zakig mqodhfogi at |zfjgjiacaora ig rgzag ct bhmdhi ba zagih~gz ao cgracrhdgt/ Ao atra ihjzf) rbajazt aodforzgz x atdzhjhz zakigt mqodhfogiat cgxfzhrgzhgcaora |gzg zaigdhfoat ihoagiat) igt dqgiat tfo ai rh|f ctthc|ia ba igt mqodhfoat/

Za|zataorgz qog Thrqgdho bai Cqobf Zagi cabhgora qog Mqodho

Bacft qo }htrg~f g qogt dqgorgt thrqgdhfoat bai cqobf zagi wqa |qabao taz za|zataorgbgt cabhgora qog mqodho/

Anac|if 1

Cgxg dforzgrg qo taz}hdhf ba Horazoar wqa rhaoa gdrqgicaora qo dgzkf caotqgi ba "77/81 ct qo dgzkf ba " qog rgzhmg ng (|igogba "77/81 x qo dgzkf |fz

`fzg ba dfoapho ba "


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anac|if ;

Qog ac|zatg ba raimfoft daiqigzat }aoba bft chiifoat ba raimfoft ao tq |zhcaz gf ba f|azgdhfoat/ Ai ocazf baraimfoft wqa aiift }aobao ta bq|ihdg dgbg gf/ Atdzhja qog mqodho wqa |zf|fzdhfoa ai ocazf ba raimfoft wqatfo }aobhbft dgbg gf dfcf qog mqodho ba ig abgb ba ig ac|zatg//

Tfiqdho

Baohz Tag p 3 abgb ba ig ac|zatg ao gft

Tag x 3 ocazf ba raimfoft wqa tfo }aobhbft |fz gf

@g~ qog rgjig

Abgb ba ig ac|zatg(goft 7 : 4 2 1 = ;

Oqcazf ba raiamfoft (chiifoat : 2 0 7= 4: =2 7:0

Atdzhjhz qog zakig mqodhfogi

Ai ocazf ba raimfoft }aobhbft |fz gf ta bq|ihdg dgbg gf/ Dfcao~gcft dfo bft chiifoat ba aiift ai |zhcaz gf>

Gof 7 > : chiifoat

Gof : > : : 3 2 chiifoatGof 4 > : : : 3 0 chiifoatGof 2 > : : : : 3 7= chiifoat

\fbacft dforhoqgz cqirh|ihdgobf |fz bft |gzg aodforzgz ai ocazf ba raimfoft }aobhbft ao ift |zphcft gft/ Rq|zfjgjiacaora zadqazbat wqa) dqgobf cqirh|ihdgcft qo ocazf |fz t chtcf }gzhgt }adat) |fbacft qtgz ig ofrgdhoap|foaodhgi/

: 3 :7

: : 3 ::: : : 3 :4

Ao atra |zfjiacg) ai ap|foaora za|zataorg ig abgb ba ig ac|zatg/

Zat|qatrg x 3 m(p 3 :p


Hbaorhdg ai bfchohf x ai zgokf ba igt thkqhaorat mqodhfoat/

7/ Bqtrho dfjzg "7< ig `fzg |fz dfzrgz dt|abat/:/ Cgzg dfjzg ":1 ig `fzg |fz digtat |gzrhdqigzat (g bfchdhihf ba cgracrhdgt) dfo qo dgzkf cohcf ba "71/4/ m(p 3 71p 7:2/ m(p:p: % 11/ m(p 3 7p=/ Dqi at ai zgokf ba ig mqodho x 3 p: 1 dqgobf ai bfchohf at #:) #7)


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;/ Dqi at ai zgokf ba ig mqodho x 3 :p 42 dqgobf ai bfchohf at #:/1) 7/1) 160/ Gokha kgog "=/1< ig `fzg dqgobf rzgjgng dfcf dgnazg ao qog rhaobg ba dfcatrhjiat/ @g~ qog rgjig wqa cqatrza

tq tgigzhf |gzg ift thkqhaorat }gifzat ba aorzgbg (ao `fzgt 1) 7 ocazft zgdhfogiat of oakgrh}ft/:/ Bfchohf> ocazft zgdhfogiat of oakgrh}ft5 Zgokf> ocazft zgdhfogiat cgxfzat f hkqgiat wqa 71/4/ Bfchohf> rfbft ift ocazft zagiat5 Zgokf> Rfbft ift ocazft zagiat/2/ Bfchohf> rfbft ift ocazft zagiat5 Zgokf> ocazft zagiat cgxfzat f hkqgiat wqa 1/1/ Bfcgho> rfbft ift ocazft zagiat apda|rf rfbft ift ocazft zagiat


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{{{/de7:/fzk 2;

p 7 < 7 : 4 2 1x 7 7/;4 :/:2 :/=1 4 4/4: 4/=

77/ x 3 p:

7:/ x 3 71 % 1p

74/ x 3:2:p

72/ m(p 3 p 771/ x 3 2< % :1p5 "7717=/ :


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20 {{{/de7:/fzk

7/= Kzdgt ba Mqodhfoat

Fjnarh}ft bai G|zaobh~gna

Ao atrg iaddho g|zaobazt g>

Kzgdgz qog mqodho g |gzrhz ba qog zakig mqodhfog f ba qog rgjig/ Atdzhjhz qog zakig mqodhfogi g |gzrhz ba qog kzdg/ Gogih~gz ig kzdg ba qog thrqgdho bai cqobf zagi/ Barazchogz th qog zaigdho at) f of) qog mqodho/

Horzfbqddho

Ig za|zataorgdho kzdg ba igt mqodhfoat dfothtra) jthdgcaora) ao bhjqngz |qorft tfjza qo \igof Dffzbaogbf(ai dqgi rgcjho ta baofchog dfcf ai \igof Dgzrathgof / Batba qog |azt|adrh}g |qzgcaoa kzdg) ai |igofdffzbaogbf at qog zanhiig f dqgbzdqig mfzcgbg g |gzrhz ba bft zadrgt oqczhdgt |zhodh|giat) qog `fzh~forgi x ig frzg}azrhdgi) wqa ta dzq~go ao qo |qorf iigcgbf ai fzhkao/ Ai fzhkao at ai |qorf (


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{{{/de7:/fzk 28

Aorfodat oft cf}acft 2 qohbgbat `gdhg ig bazad`g/

Iqakf oft cf}acft : qohbgbat `gdhg gzzhjg) |gzrhaobf ba ig irhcg |fthdho/

Anac|if 7

Kzgwqa ift thkqhaorat |qorft dffzbaogbft ao ai |igof Dgzrathgof/

(g (1) 4

(j (#:) =



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1< {{{/de7:/fzk

(d (4) #2

(b (#1) #;

Tfiqdho

Cftrzgcft rfbft ift |qorft dffzbaogbft ao ig chtcg kzdg

Bajacft ofrgz wqa>

\gzg qo }gifz |fthrh}f ba p oft cf}acft `gdhg ig bazad`g/

\gzg qo }gifz oakgrh}f ba p oft cf}acft gdhg ig h~wqhazbg/

\gzg qo }gifz |fthrh}f ba x oft cf}acft gdhg gzzhjg/

\gzg qo }gifz oakgrh}f ba x oft cf}acft `gdhg gjgnf/

Ift anat pgpht x x bh}hbao ai |igof dffzbaogbf ao 2 dqgbzgorat/ Ift dqgbzgorat ta oqcazgo ao ai taorhbfgorh#`fzgzhf (taorhbf ba khzf dforzgzhf gi ba igt cgoadhiigt ba qo zaifn) dfcao~gobf batba ai dqgbgzgora tq|azhfz

bazad`f/ Gt) ai |qorf kzgdgbf dfzzat|fobhaora gi ihrazgi (g atr qjhdgbf ao ai \zhcaz dqgbzgora) ai ba (j atr aoai takqobf dqgbzgora) ai ba (d ta aodqaorzg ao ai Dqgzrf dqgbzgora) x ai ba (b atr ao ai Razdaz dqgbzgora/

Kzdg ba qog Mqodho fjraohbg g |gzrhz ba qog Zakig Mqodhfogi f ba qog Rgjig

Th ig zakig mqodhfogi at dfofdhbg) f th bht|foacft ba qog rgjig ba }gifzat wqa batdzhja g qog mqodho) |fbacft`gdaz qog kzdg ba ig mqodho/ Qog rgjig ba }gifzat oft |zf|fzdhfog |qorft dffzbaogbft wqa |fbacft bhjqngz aoai |igof Dgzrathgof/

Anac|if :

Kzgdg ig mqodho wqa rhaoa ig rgjig ba }gifzat thkqhaora/

p : 7 < 7 :x = 0 7< 7: 72

Tfiqdho

Ig rgjig oft |zf|fzdhfog dhodf dfonqorft ba dffzbaogbgt p) x (at badhz) dhodf |qorft dffzbaogbft> (#:) =) (#7) 0)(


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{{{/de7:/fzk 17

\gzg kzgdgz ig mqodho) bhjqngcft rfbft ift |qorft dffzbaogbft/ Bgbf wqa of ta oft jzhobg homfzcgdho tfjza aibfchohf ba ig mqodho f bai dforaprf bfoba g|gzada) |fbacft gtqchz wqa ai bfchohf at ai dfonqorf ba rfbft iftocazft zagiat/ \gzg cftrzgz wqa ig mqodho at }ihbg |gzg rfbft ift }gifzat bai bfchohf) dfoadrgcft ift |qorftdfo qog ioag dforhoqg/ Ao atra taorhbf) rgcjho bajacft bgzoft dqaorg wqa ig ioag ta aprhaoba hobaohbgcaora aogcjgt bhzaddhfoat/

Anac|if 4

Kzgdg ig mqodho g ig wqa dfzzat|foba ig thkqhaora rgjig ba }gifzat/

Ifokhrqb bai igbf bai dqgbzgbf < 7 : 4 2

Gzag bai dqgbzgbf < 7 2 8 7=

Ig rgjig oft |zf|fzdhfog dhodf |qorft dffzbaogbft> (


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Kzgdg ig mqodho wqa rhaoa ig thkqhaora rgjig ba }gifzat/

Oqcazf ba kifjft 7: 74 72 71 7=

Dftrf 27 22 2; 1< 14

Atrg mqodho za|zataorg ai dftrf rfrgi ba ift kifjft iia}gbft g rq dgtg/ Dgbg kifjf dqatrg "4 x ig rhaobg ift iia}g g rqdgtg |fz ig dfc|zg ba qog bfdaog gbhdhfogi ba kifjft/ Ig aorzakg g bfchdhihf rhaoa qog rgzhmg ng (|igogba "1/

Tfiqdho

Ig rgjig oft |zf|fzdhfog dhodf |qorft dffzbaogbft (7:) 27) (74) 22) (72) 2;) (71) 1


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{{{/de7:/fzk 14

Tfiqdho

@g~ qog rgjig ba }gifzat/ Taiaddhfog qog }gzhabgb ba }gifzat aorazft oakgrh}ft x |fthrh}ft |gzg ig }gzhgjia hoba|ao#bhaora/ Qrhih~g ig zakig mqodhfogi |gzg aodforzgz ai }gifz ba ig }gzhgjia dfzzat|fobhaora g dgbg }gifz ba ig }gzhgjiahoba|aobhaora/ Iqakf kzgdg dgbg qof ba ift |qorft dffzbaogbft gt fjraohbft/

RGJIA 7/77>

p x 3 m(p 3 yp :y2 y2:y3 y=y3 =#4 y4:y3 y1y3 1#: y::y3 y2y3 2#7 y7:y3 y4y3 4< y


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At gdfotangjia wqa rzgjgnat dfo cqd`ft }gifzat dqgobf ac|ha~gt g mgchihgzh~gzra dfo ig dzagdho ba kzdgt bamqodhfoat/ Iqakf) dfomfzca }gxgt g|zaobhaobf gdazdg ba ift bhmazaorat rh|ft ba mqodhfoat) ra bgzt dqaorg wqa|fbzt dzagz kzdgt jgtrgora |zadhtgt dfo rgjigt ba }gifzat wqa dforaokgo |fdft |qorft/

Anac|if =

Kzgdg ig thkqhaora mqodho> m(p 3

p

Tfiqdho

@g~ qog rgjig ba }gifzat/ Of |fbacft qrhih~gz oqcazft oakgrh}ft |fzwqa of |fbacft fjraoaz ig zg~ dqgbzgbg baqo ocazf oakgrh}f/ Ao kaoazgi) ig mqodho zg~ dqgbzgbg of |zf|fzdhfog tgihbgt zagiat |gzg aorzgbgt oakgrh}gt/Ai bfchohf) |fz rgorf) at ai dfonqorf ba rfbft ift ocazft zagiat |fthrh}ft/ Gt wqa bajacft taiaddhfogz bh}aztft}gifzat aorazft |fthrh}ft |gzg ig }gzhgjia hoba|aobhaora/ Iqakf qrhih~g ig zakig mqodhfogi |gzg aodforzgz ai }gifz baig }gzhgjia ba|aobhaora |gzg dgbg }gifz ba ig }gzhgjia hoba|aobhaora/

RGJIA 7/7:>

p x 3 m(p 3

p

@gdacft qog rgjig ba }gifzat dfzzat|fobhaorat g }gzhft |qorft dffzbaogbft |gzg }az th |fbacft hbaorhdgz ai |grzowqa cqatrzg dcf ta zaigdhfogo igt }gzhgjiat ba|aobhaora a hoba|aobhaora aorza t/

p 2 4 : 7 < 7 : 4 2x 0 2/1 : /1 < /1 : 2/1 0

Ofrgcft wqa dgbg }gifz ba x at ig chrgb bai dqgbzgbf |azmadrf ba p/ \fz rgorf) zaatdzhjhcft ig rgjig ba }gifzatdfcf>

p 2 4 : 7 < 7 : 4 2x

7=:

8:

2:

7:

]fiqcao 3 1 7:

Bg : > ]fiqcao 3 1 7:

7:

Bg 4 > ]fiqcao 3 1 7:

7:

7:

\fz rgorf) ig adqgdho ba ig mqodho at m(p 3 1

7:

p

Gogih~gz ig kzdg ba qog Thrqgdho bai Cqobf Zagi

Igt kzdgt tfo qrhih~gbgt |gzg za|zataorgz bgrft ao rfbgt igt zagt ba ig }hbg/ \fz anac|if) |qabat aodforzgz kzdgt

ao |azhbhdft) dgc|ggt |firhdgt) |qjihdgdhfoat dhaordgt x |zataorgdhfoat ba oakfdhft/@a gwq qo anac|if ba qog kzdg wqa |fbzgt }az za|fzrgbg ao igt ofrhdhgt/ Ig cgxfzg ba dhaordft dzaa wqaigt dzadhaorat achthfoat ba kgtat ba ho}azogbazf) |gzrhdqigzcaora ba bhphbf ba dgzjfof) atro dforzhjqxaobf gidgiaorgchaorf ba oqatrzf |igoarg/ Atrg kzdg cqatrzg dcf ift oh}aiat ba bhphbf ba dgzjfof ta `go hodzacaorgbfdfomfzca ai cqobf ta `g hobqtrzhgih~gbf/

Ba atrg kzdg) |fbacft aodforzgz ig dfodaorzgdho ba bhphbf ba dgzjfof ao ig grctmazg |gzg bhmazaorat gft/

78


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Barazchogz th qog Zaigdho at qog Mqodho

Xg }hcft wqa qog mqodho at qog zaigdho aorza igt }gzhgjiat hoba|aobhaora x ba|aobhaora/ Tgjacft rgcjho wqa atqog zakig wqa qrhih~g ift }gifzat ba ig }gzhgjia hoba|aobhaora |gzg bgz ift }gifzat ba ig }gzhgjia ba|aobhaora/ Qogzakig mqodhfogi |qaba taz ap|zatgbg ao |gigjzgt) rgcjho dfcf qog adqgdho) f dfcf qog rgjig ba }gifzat) f jhaodfcf qog kzdg/ Rfbgt atrgt za|zataorgdhfoat tfo rhiat x oadatgzhgt |gzg aoraobaz ig zaigdho aphtraora aorza igt

}gzhgjiat ba|aobhaora a hoba|aobhaora/ Cgracrhdgcaora) qog mqodho at qo rh|f at|adhgi ba zaigdho/Igt thkqhaora dfobhdho oft gxqbgz g barazchogz th qog zaigdho cgracrhdg at) f of) qog mqodho/

Ao qog mqodho) |gzg dgbg }gifz ba aorzgbg) aphtra apgdrgcaora qo }gifz ba tgihbg/

Atrf thkohdg qtqgicaora wqa dgbg }gifz ba p rhaoa gthkogbf qo ohdf }gifz ba x/ Tho acjgzkf) at hc|fzrgoratgjaz wqa of rfbgt igt mqodhfoat ho}fiqdzgo }gifzat bhtrhorft ba x |gzg }gifzat bhtrhorft ba p/ At badhz) wqa bft f ct}gifzat ba p |qabao raoaz gthkogbf qo chtcf }gifz ba x/ Ao rfbf dgtf) if wqa qog mqodho of |azchra at gthkogzcirh|iat }gifzat x g dgbg }gifz ba p/

Dfcf hiqtrzgdho) dfothbazg ig zaigdho wqa cqatrzg igt atrgrqzgt ba rfbft ift atrqbhgorat ba qog digta/ Ai bfchohfat ai dfonqorf ba |aztfogt ao ig digta x ai zgokf at ai dfonqorf ba atrgrqzgt/ Dgbg |aztfog ao ig digta of |qaba raoazct ba qog atrgrqzg g ig }a~/ \fz rgorf) atrg zaigdho at qog mqodho |fzwqa |gzg dgbg |aztfog aphtra apgdrgcaoraqog atrgrqzg wqa ia |azraoada/

Fjtaz}g) tho acjgzkf) wqa ao qog mqodho qo }gifz bai zgokf |qaba |azraoadaz g ct ba qo aiacaorf bai bfchohf/At badhz) wqa ct ba qog |aztfog ao digta |qaba raoaz ig chtcg atrgrqzg/ \azf if f|qatrf of at |fthjia) at badhz) wqaqog |aztfog raokg cirh|iat atrgrqzgt/

Anac|if 77

Barazchog th ig thkqhaora zaigdho at qog mqodho/

g (7) 4) (#7) #:) (4) 1) (:) 1) (4) 2j (#4) :


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p : 7 < 7 :

x 7: 7< 0 = 2

Tfiqdho

Ig cgoazg ct mdhi ba barazchogz th qog zaigdho at qog mqodho at fjta}gz g rfbft ift }gifzat ba p

ao ig ihtrg f

rgjig/ Th qo }gifz ba p g|gzada ct ba qog }a~ x ift }gifzat wqa ia dfzzat|foba ba xtfo bhmazaorat) aorfodat igzaigdho of at qog mqodho/

g (7) 4) (#7) #:) (4) 1) (:) 1) (4) 2

\qabat }az wqa ao atrg zaigdho) `gx bft }gifzat bhmazaorat ba x wqa |azraoadao gi }gifz 4 bap/ Atrf thkohdg wqaatrg zaigdho of at qog mqodrhfo/

j (#4) :


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Of at qog mqodho/ Of |gtg ig |zqajg ba ig ioag }azrhdgi

At qog mqodho/ Ohokqog ioag }azrhdqgi dzq~gz ig dqz}g ba ig kzdg ao ct ba qo |qorf/

At qog mqodho/ Ohokqog ioag }azrhdqgi dzq~gz ig dqz}g ba ig kzdg ao ct ba qo |qorf

Of at qog mqodho/ Of |gtg ig |zqajg ba ig ioag }azrhdgi



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7/ Bhjqng ift |qorft dffzbaogbft ao ai |igof Dgzrathgof/

g/ (2) #2j/ (:) ;d/ (#4) #1b/ (=) 4a/ (#2) 4

:/ Atdzhja igt dffzbaogbgt |gzg dgbg |qorf cftrzgbf ao ai |igof Dgzrathgof/

4/ Kzgdg ig mqodho wqa rhaoa ig thkqhaora rgjig ba }gifzat/ (g

p 7< 1 < 1 7Ai crfbf @gdaz qog Rgjig at rhi dqgobf ta zatqai}ao |zfjiacgt wqa rhaoao wqa }az dfo zaigdhfoat oqczhdgt/Dqgobf ift bgrft ta fzkgoh~go ao qog rgjig) at ct mdhi zadfofdaz |grzfoat x zaigdhfoat aorza ocazft/ G|ihwqacftatrg atrzgrakhg gi thkqhaora anac|if/

Anac|if :

Nftha badhba dfcao~gz g |zgdrhdgz ai rzfra (nfkkhok/ Bqzgora ig |zhcazg tacgog) aiig rzfrg 7< choqrft dgbg bg5 aoig takqobg tacgog) rzfrg 7: choqrft dgbg bg/ Dgbg tacgog) aiig wqhaza hodzacaorgz tq rhac|f ba rzfra bhgzhf ao :choqrft/ Th aiig rzfrg = bgt dgbg tacgog) Dqi taz ai rhac|f rfrgi ba rzfra bqzgora ig taprg tacgog6

Tfiqdho

\gtf 7Aoraobaz

Tgjacft wqa bqzgora ig |zhcazg tacgog Nftha rzfrg 7< choqrft dgbg bhg) bqzgora = bgt/

Tgjacft wqa bqzgora ig takqobg tacgog Nftha rzfrg 7: choqrft dgbg bg) bqzgora = bgt/

Dgbg tacgog) aiig hodzacaorg ao : choqrft tq rhac|f ba rzfra bhgzhf/ Gbact) aiig thac|za |zgdrhdg ai rzfra = }adatg ig tacgog/

Wqazacft dfofdaz ai rhac|f rfrgi ba rzfra ao ig taprg tacgog/

\gtf :



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Atrzgrakhg

Qog jqaog atrzgrakhg at `gdaz dfifdgz ift bgrft wqa oft |zf|fzdhfog ai aoqodhgbf bai |zfjiacg ao qo rgjig/ Iqakfbajacft qtgz ig homfzcgdho |zf|fzdhfogbg |gzg aodforzgz oqa}g homfzcgdho/ \fbacft `gdaz ig rgjig dfo iftthkqhaorat aodgja~gbft/

Tacgog Choqrft |fz bg Choqrft |fz Tacgog

Ai aoqodhgbf bai |zfjiacg oft bhda wqa Nftha rzfrg 7< choqrft gi bg) |fz = bgt bqzgora ig |zhcazg tacgog/ Rgcjhooft bhda wqa aiig rzfrg 7: choqrft gi bg) |fz = bgt ao ig takqobg tacgog/ \fbacft aorfodat ggbhz rfbg atrghomfzcgdho ao oqatrzg rgjig>

RGJIA 7/71>

Tacgog Choqrft |fz bg Choqrft |fz tacgog7 7< =

Tacgog Choqrft |fz bg Choqrft |fz tacgog7 7< =

Jqtdgz qo \grzo at qog atrzgrakhg wqa |qabat qrhih~gz |gzg jqtdgz |grzfoat ao ift bgrft dfo ig ogihbgb ba zatfi}az|zfjiacgt/ Ig ogihbgb ba bhd`g atrzgrakhg at jqtdgz bgrft f ocazft wqa ta za|hrao) f jhao jqtdgz a}aorft wqa taza|hrao/ Ai thkqhaora |zfjiacg |qaba zatfi}azta g rzg}t ba ig rdohdg ba aodforzgz qo |grzo/

Anac|if 4

Rq bht|foat |aifrgt ba raoht ao gzzakift rzhgokqigzat) dfcf ta cqatrzg/ Dqorgt |aifrgt `gjzo ao qo rzhokqifwqa rhaoa 0 igt ba |aifrgt6

Tfiqdho

\gtf 7

Aoraobaz

Tgjacft wqa bht|foacft igt |aifrgt ba raoht ao gzzakift rzhgokqigzat dfcf ta cqatrzg/

Wqazacft tgjaz dqorgt |aifrgt `gx ao qo rzhokqif wqa rhaoa 0 igt ba |aifrgt/

\gtf :

Atrzgrakhg

Qog jqaog atrzgrakhg at `gdaz qog rgjig wqa dforaokg qog ihtrg ba dqorgt |aifrgt aphtrao ao rzhokqift wqa |ftaaobhmazaorat ocazft ba igt/

Qog ig/ At thc|ia fjtaz}gz wqa qo rzhokqif dfo qog ig rhaoa tfigcaora qog |aifrg/

Bft igt/ \gzg qo rzhgokqif dfo : igt) tqcgcft igt |aifrgt ba ig ig tq|azhfz dfo igt ba ig ig homazhfz/ At rhi`gdaz qo jftwqanf ba igt bhmazaorat igt bai rzhokqif/

3 : % 7 3 4

Rzat igt Tqcgcft igt |aifrgt bai rzhokqif tq|azhfz dfo igt |aifrgt ba ig ig homazhfz/



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3 4 % 4 3 =

\fbacft aorfodat dfc|iargz igt |zhcazgt rzat igt ba ig rgjig/RGJIA 7/7;>

Ocazf ba igt Ocazf ba |aifrgt7 7: 44 =

\fbacft fjtaz}gz wqa aphtra qo |grzo' At badhz>

\gzg dzagz ai thkqhaora rzhokqif) ggbhcft qog oqa}g ig homazhfz gi rzhokqif aphtraora/

Ig oqa}g ig homazhfz rhaoa qo ocazf ba |aifrgt hkqgi gi ocazf ba igt aphtraorat ao ai oqa}f rzhokqif (bhd`focazf) |fz tq|qatrf) ta baja rfcgz ao dqaorg ig oqa}g ig homazhfz/

# Qo rzhokqif dfo 4 igt rhaoa 4 |aifrgt ao ig ig homazhfz/

\gzg fjraoaz ai ocazf rfrgi ba |aifrgt |gzg ai oqa}f rzokqif) tqcgcft ai ocazf ba |aifrgt bai rzhokqif fzhkhogigi ocazf ba |aifrgt ba ig oqa}g ig homazhfz/

bgbf wqa) dfcf ta caodhfo gzzhjg) ai ocazf ba |aifrgt ba ig oqa}g ig homazhfz at hkqgi gi ocazf rfrgi ba igtbai oqa}f rzhokqif) rgcjho ta |qaba badhz wqa

\gzg fjraoaz ai ocazf rfrgi ba |aifrgt |gzg ai oqa}f rzokqif) tqcgcft ai ocazf ba |aifrgt bai rzhokqif fzhkhogigi ocazf ba igt bai oqa}f rzhokqif/

\gtf 4G|ihdgz ig atrzgrakhg&zatfi}az>

\fbacft dfc|iargz ig rgjig thkqhaobf ai |grzo wqa `acft batdqjhazrf/

Ocazf ba |aifrgt 3 ocazf ba |aifrgt ao ai rzhokqif fzhkhogi % ocazf ba igt bai oqa}f rzhokqif

RGJIA 7/70>

Ocazf ba igt Ocazf ba \aifrgt7 7: 44 =2 = % 2 3 7 @gdaz qog Rgjig f Jqtdgz qo \grzo


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Dgbg ig ba qo rzhokqif rhaoa qog |aifrg ct wqa ig ig |za}hg/ Gt) ao qo rzhokqif ba 0 igt)

ig ig 7 rhaoa 7 |aifrg5 ig ig : rhaoa : |aifrgt5 ig ig 4 rhaoa 4 |aifrgt5 ig ig ig 2 rhaoa 2 |aifrgt5 ig ig 1 `gt 1 jgiit5ig ig = rhaoa = |aifrgt5 ig ig ; rhaoa ; |aifrgt5 ig ig 0 rhaoa 0 |aifrgt/

Dqgobf tqcgcft rfbgt atrgt dgorhbgbat) raoacft> 7 % : % 4 % 2 % 1 % = % ; % 0 3 4= |aifrgt

Ig zat|qatrg ta `g dfc|zfjgbf/

Ofrg wqa ao atra anac|if `hdhcft rgjigt x bhjqngcft bhgkzgcgt |gzg gxqbgzoft g fzkgoh~gz ig homfzcgdho xaodforzgz qo |grzo/ Ai qtf thcqiroaf ba }gzhft crfbft at qog |zdrhdg dfco x at cqx rhi |gzg zatfi}az|zfjiacgt/

Atrgjiadaz x Dfc|gzgz Aomfwqat Girazogrh}ft |gzg Zatfi}az \zfjiacgt

Ao atrg taddho) dfc|gzgzacft ift crfbft @gdaz qog rgjig x Jqtdgz qo \grzo qrhih~gobf dgbg crfbf g ig }a~|gzg zatfi}az qo |zfjiacg bgbf/

Apgc|ia 2

Gobza{ `gda amadrh}f qo d`awqa ba "70< x |hba ai bhoazf ao jhiiarat ba "7< x ba ": @gdaz qog Rgjig

\gtf 7

Aoraobaz

Gobza{ ia bg gi dgnazf bai jgodf qo d`awqa ba "70


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RGJIA 7/78> (dforhoqab

Jhiiarat ba "7< Jhiiarat ba ":< Dgorhbgb Rfrgi0 2 "7


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{{{/de7:/fzk 07

RGJIA 7/:

Jhiiarat ba "7< Jhiiiarat ba ":< Dgorhbgb Rfrgi7: < "7


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Aorfodat aiig bfjig ift igbft ba ig dgng x nqorg ift jfzbat gbgxgdaorat qtgobf |akgcaorf/ Dqo kzgobat bajaotaz ift dqgbzgbft wqa aiig dfzrg ba igt atwqhogt |gzg fjraoaz ig dgng dfo ai cphcf }fiqcao |fthjia6

Tfiqdho

\gtf 7

Aoraobaz

Gooa `gda qog dgng g |gzrhz ba qog |ha~g ba dgzro dqgbzgbg ba :< |ik

:< |ik/

Aiig dfzrg dqgzf dqgbzgbft hkqgiat ba igt atwqhogt ba ig |ha~g ba dgzro/

Aiig bfjig ift igbft x |akg tqt jfzbat gbxgdaorat/

Dqo kzgobat bajazo taz ift dqgbzgbft wqa aiig dfzrg |gzg `gdaz qog dgng dfo ai cphcf }fiqcao |fthjia6

\gtf :

Atrzgrakhg

Oadathrgcft zadfzbgz ig mzcqig ba qog dgng

]fiqcao 3 zag ba ig jgta girqzg]fiqcao 3 god`f

ifokhrqb

girqzg

@gdaz qog rgjig ba }gifzat taiaddhfogobf bhmazaorat }gifzat |gzg ift igbft ba ift dqgbzgbft wqa ta }go g dfzrgz) |gzgiqakf dgidqigz ai }fiqcao/

\gtf 4

G|ihdgz ig atrzgrakhg&zatfi}az

@gkgcft qog dgng dfzrgobf dqgrzf dqgbzgbft dfo igbft ba ifokhrqb hkqgi g 7 |qikgbg/ Ai bhgkzcg iqda dfcf tra>

\qabat fjtaz}gz wqa dqgobf bfjigcft ift igbft x ift nqorgcft |gzg `gdaz ig dgng) ig girqzg at 7 |qikgbg) ai god`f atba 70 |qikgbgt chaorzgt wqa ig ifokhrqb ig dgng at ba 70 |qikgbgt/

]fiqcao 3 god`f ifokhrqb girqzg]fiqcao 3 70 70 7 3 4:2 |ik4

@gkgcft qog rgjig wqa cqatrza ai }gifz ba ig dgng |gzg bhmazaorat igbft bai dqgbzgbf>

7/0/ Atrzgrakhgt |gzg Zatfiqdho ba \zfjiacgt> @gdaz qog Rgjig f Jqtdgz qo \grzo


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RGJIA 7/:7>

Igbf baiDqgbzgbf

Girqzg ba ig Dgng God`f ba ig Dgng Ifokhrqb ba igDgng

]fiqcao

7 7 70 70 70 707 3 4:2: : 7= 7= 7= 7=: 3 17:4 4 72 72 72 724 3 1002 2 7: 7: 7:

7:

2 3 1;=

1 1 7< 7< 7


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RGJIA 7/::> (dforhoqab

Igbf baiDqgbzgbf

Girqzg ba ig Dgng God`f ba ig Dgng Ifokhrqb ba igDgng

]fiqcao

4/1 4/1 74 74 74 74 4/1 3187/1

Ofrg wqa ai cphcf }fiqcao of fdqzza dqgobf ai igbf bai dqgbzgbf at ba rzat |qikgbgt) thof dqgobf ai igbf baidqgbzgbf at ba 4/4 |qikgbgt/

Oqatrzg zat|qatrg fzhkhogi of azg hodfzzadrg) |azf fj}hgcaora of azg rgo |zadhtg dfcf if |fbzg taz/ Ba `ad`f)|qabat fjraoaz qog zat|qatrg gqo ct |zadhtg th rfcgt hodzacaorft ct |awqaft wqa ift ba ig rgjig gorazhfz |gzgig ifokhrqb ba ift igbft bai dqgbzgbf/ \gzg atra dgtf) |fbacft qrhih~gz cabhbgt wqa tfo caofzat f cgxfzat wqa 4/4|qikgbgt/

\fz rgorf) ig zat|qatrg wqabg dfc|zfjgbg th ig wqazacft zabfobagbg g dazf dhmzgt badhcgiat) |azf Qog zat|qatrgct |zadhtg at 4/4 |qikgbgt/


7/ Zakzatg x aodqaorzg ig tfiqdho gi |zfjiacg bai Anac|if 7/:/ Jzhrr rhaoa ":/:1 ao cfoabgt ba dhodf x bha~ daorg}ft/ Th aiig rhaoa 2< cfoabgt ao rfrgi Dqorgt cfoabgt ba

dgbg rh|f rhaoa64/ Qo |grzo ba bhgkzgcgt ba dqgbzgbft ta cqatrzg g dforhoqgdho/ Dqorft dqgbzgbft gx ao ai 7:rf bhgkzgcg6

2/ Ft{gib atr rzgrgobf ba bhtchoqhz tq dfotqcf ba dgm/ Tq carg at jgngz tq dfotqcf `gtrg = rg~gt |fz tacgog/Th i dfchao~g dfo :2 rg~gt ig |zhcazg tacgog5 iqakf jgng tq dfotqcf g :7 rg~gt ig takqobg tacgog5 x iqakfg 70 rg~gt ig razdazg tacgog Ao dqorgt tacgogt gidgo~gz tq carg6

1/ Rgxifz |zatr qo ihjzf ao ig jhjihfradg x nqtrf g`fzg rhaoa 1 bgt ba zarzgtf ao ig ba}fiqdho bai chtcf/ Ai }gifzba ig cqirg |fz ba}fiqdho rgzbg at ba 7< daorg}ft bhgzhft/ Dgorf rhaoa wqa |gkgz ba cqirg th ba}fi}hazg aiihjzf g`fzg6

=/ Dqorgt `fzgt ia rfcgz g qo gqrfc}hi wqa g}go~g g ;1 chiigt |fz `fzg gidgo~gz g frzf gqrfc}hi wqa }hgngg 11 chiigt |fz `fzg) th ai gqrfc}hi ct iaorf gzzgodg bft `fzgt gorat wqa ai ct z|hbf6

;/ Kzgda gzzgodg ao tq jhdhdiarg g qog }aifdhbgb ba 7: chiigt |fz `fzg/ Qog `fzg bat|qt) Bgo gzzgodg ao igtqxg qog `fzg bat|qt g qog }aifdhbgb ba 71 chiigt |fz `fzg) thkqhaobf ig chtcg zqrg/ Tq|fohaobf wqa igt}aifdhbgbat hohdhgiat ba gcjft ta cgorhaoao dfotrgorat Dqorf ia rfcgz g Bgo gidgo~gz g Kzgda6

0/ Iacqai wqhaza atrgjiadaz ift ichrat ba qo razzaof zadrgokqigz dfo qog dazdg/ Ai rhaoa cgrazhgi tqdhaora |gzgdfotrzqhz qog dazdg wqa raokg qog ifokhrqb ba :2 |hat/ Dqi at ig cphcg zag |fthjia ba razzaof wqa i |qabaaodazzgz dfo bhd`g dazdg6

Zat|qatrgt g ift Anazdhdhft ba Za|gtf

7/ "4 @gdaz qog Rgjig f Jqtdgz qo \grzo


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4/ :4 dqgbzgbft2/ ; tacgogt1/ 1< daorg}ft=/ 1/1 `fzgt;/ 1 `fzgt0/ 4= |hat:5 ai razzaof dazdgbf at qo dqgbzgbf



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7/8 Zadqztft Gbhdhfogiat

\gzg cgrazhgi gbhdhfogi tfjza zatfiqdho ba |zfjiacgt }az ao Kafzka \ixg [heh|abhg `rr|>&&ao/{heh|abhg/fzk&

{heh&KafzkaV\!D4!J4ixg [heh|abhg aorzx

7/8/ Zadqztft Gbhdhfogiat
http://www.ck12.org/http://www.ck12.org/http://www.ck12.org/http://www.ck12.org/http://en.wikipedia.org/wiki/George_P%C3%B3lyahttp://en.wikipedia.org/wiki/George_P%C3%B3lyahttp://en.wikipedia.org/wiki/George_P%C3%B3lyahttp://en.wikipedia.org/wiki/George_P%C3%B3lyahttp://en.wikipedia.org/wiki/George_P%C3%B3lyahttp://www.ck12.org/


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D@G\RAZ : Ocazft ZagiatD`g|raz Fqrihoa

:/7 AORAZFT X OCAZFT ZGDHFOGIAT

:/: GBHDHO BA OCAZFT ZGDHFOGIAT:/4 TQTRZGDDHO BA OCAZFT ZGDHFOGIAT

:/2 CQIRH\IHDGDHO BA OCAZFT ZGDHFOGIAT

:/1 IG \ZF\HABGB BHTRZHJQRH]G

:/= BH]HTHO BA OCAZFT ZGDHFOGIAT

:/; ZG^ DQGBZGBG X OCAZFT ZAGIAT

:/0 ATRZGRAKHGT \GZG ZATFIQDHO BA \ZFJIACGT> TQ\FTHDHO X ZA]HTHO)

RZGJGNF HO]AZTF/

D`g|raz :/ Ocazft Zagiat


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:/7 Aorazft x Ocazft Zgdhfogiat

Fjnarh}ft ba G|zaobh~gna

Kzgdgz x dfc|gzgz Aorazft/ Digthdgz x fzbaogz ocazft zgdhfogiat/ Aodforzgz ift f|qatrft ba ift ocazft/ Aodforzgz }gifzat gjtfiqrft/ Dfc|gzgz mzgddhfoat |gzg barazchogz dqgi at ig ct kzgoba/

Kzgdgz x Dfc|gzgz AorazftIft Aorazft tfo ift ocazft ba dforaf (7) :) 4///) ift ocazft ba dforaf oakgrh}ft (#7) #:) #4///) x dazf/ Aphtraqo ocazf hoohrf ba aorazft/ Anac|if ba aorazft tfo


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{{{/de7:/fzk 08

Bat|qt ba qo tgirf ig zgog |fbzg atrgz ao ai 8 f ai 1 (|azf of ao ai ;/ Bat|qt ba bft tgirft ig zgog |fbzg atrgzao ai 77) ; f 4/ Dforgobf ai ocazf ba }adat wqa ig zgog tgirg g ig bazad`g f g ig h~wqhazbg) |fbzgcft barazchogzbfoba grazzh~g ig zgog/ Bat|qt ba dhodf tgirft) `gx }gzhgt |fthjiat qjhdgdhfoat |gzg ig zgog/ Aphtra qog cgoazgthtracrhdg |gzg dgidqigz igt |fthjiat qjhdgdhfoat ba|aobhaobf ba dqgorgt }adat ig zgog tgir g ig bazad`g) x dqgorgt}adat ig zgog tgir g ig h~wqhazbg/

ZZZZZ 3 1 tgirft g ig bazad`g qjhdgdho 3 ; % (1 : 3 7;ZZZZI 3 2 tgirft g ig bazad`g) 7 tgirf g ig h~wqhazbg qjhdgdho 3 ; % (4 : 3 74ZZZII 3 4 tgirft g ig bazd`g) : tgirft g ig h~wqhazbg qjhdgdho 3 ; % (7 : 3 8

ZZIII 3 : tgirft g ig bazad`g) 4 tgirft g ig h~wqhazbg qjhdgdho 3 ; (7 : 3 1ZIIII 3 7 tgirf g ig bazad`g) 4 tgirft g ig h~wqhazbg qjhdgdho 3 ; (4 : 3 7IIIII 3 1 tgirft g ig h~wqhazbg qjhdgdho 3 ; (1 : 3 4

Atrgt tfo igt |fthjiat qjhdgdhfoat ba ig zgog apgdrgcaora bat|qt ba dhodf tgirft/ \qabat ofrgz wqa ai fzbao ofhc|fzrg> rzat tgirft g ig bazad`g) qof g ig h~wqhazbg x qof g ig bazad`g at if chtcf wqa dqgrzf tgirft g ig bazad`g xqof g ig h~wqhazbg/

Digthdgdho ba ift Ocazft ZgdhfogiatDqgobf bh}hbhcft qo aorazf |fz frzf aorazf (of dazf fjraoacft if wqa iigcgcft ocazf zgdhfogi/ At iigcgbfgt |fzwqa at ig zaigdho ba qo ocazf g frzf/ \fz anac|if) th bh}hbhcft qo aorazf g |fz qo takqobf aorazf j aiocazf zgdhfogi wqa fjraoacft at gj ) |zf}htrf wqa j of at dazf/ Dqgobf atdzhjhcft qo ocazf zgdhfogi dfcf atra)ai ocazf ao ig |gzra ba gzzhjg at iigcgbf oqcazgbfz/ Ai ocazf ao ig |gzra homazhfz at iigcgbf baofchogbfz/ R|qabat |aotgz gdazdg ba ift ocazft zgdhfogiat dfcf ig mzgddho ba qo |gtrai/ Th dfzrgt ai |gtrai ao j zajgogbgt) rq|gzra at g ba atgt zajgogbgt/

\fz anac|if dqgobf }acft ai ocazf zgdhfogi 7: ) oft hcgkhogcft dfzrgobf ai |gtrai ao bft |gzrat/ Oqatrzg |gzra atqof ba atft |abg~ft/ Ai ocazf zgdhfogi 7: ta }a dfcf atrf>

Dfo ai ocazf zgdhfogi 42 ) dfzrgcft ai |gtrai ao dqgrzf |gzrat x oqatrzg |gzra at rzat ba atft |abg~ft/ Ai ocazfzgdhfogi 42 ta }a dfcf atrf>



96/955

8< {{{/de7:/fzk

Ai ocazf zgdhfogi8

7< za|zataorg oqa}a zajgogbgt ba qo |gtrai wqa `g thbf dfzrgbf ao bha~ |gzrat/ Ai ocazf zgdhfogi87< ta }a dfcf atrf>

Mzgddhfoat \zf|hgt tfo ocazft zgdhfogiat bfoba ai oqcazgbfz (ai ocazf ba gzzhjg at caofz wqa ai baofchogbfz

(ai ocazf ba gjgnf/ Qog mzgddho |zf|hg za|zataorg qo ocazf caofz wqa qof/ Dfo qog mzgddho |zf|hg thac|zarazchogzt dfo caoft ba qo |gtrai aorazf'

Mzgddhfoat Hc|zf|hgt tfo ocazft zgdhfogiat bfoba ai oqcazgbfz at cgxfz wqa ai baofchogbfz/ Igt mzgddhfoathc|zf|hgt |qabao taz zaatdzhrgt dfcf qo ocazf chprf qo aorazf ct qog mzgddho |zf|hg/ Qog mzgddho hc|zf|hgza|zataorg qo ocazf cgxfz wqa qof/

Mzgddhfoat Awqh}giaorat tfo bft mzgddhfoat wqa bgo ai chtcf }gifz oqczhdf dqgobf tfo a}giqgbgt/ \fz anac|if)fjtaz}g ig za|zataorgdho }htqgi bai ocazf zgdhfogi :2 >

R |qabat }az wqa ig zakho tfcjzagbg at hborhdg ao rgcgf g ig bai ocazf zgdhfogi qo cabhf 7: / \fbacft atdzhjhzift mgdrfzat |zhcft ba gcjft) ai oqcazgbfz x ai baofchogbfz) dgodaigcft mgdrfzat dfhodhbaorat wqa g|gzadao aogcjft/

:

2 3: 7

: : 7 Iqakf cqirh|ihdgcft ba oqa}f ift mgdrfzat zatrgorat/ :

2 37

:Atra |zfdatf at iigcgbf zabqddho ba ig mzgddho) f atdzhjhz ig mzgddho ao rzchoft ct jgnft/ Zabqdhz ig mzgddhoof dgcjhg ai }gifz ba ig mzgddho/ Tfigcaora thc|ihdg ig mfzcg ao wqa ig atdzhjhcft/ Dqgobf `acft dgodaigbfrfbft ift mgdrfzat dfcqoat) raoacft qog mzgddho ao tq mfzcg ct thc|ia/

Anac|if 4

Digthdgz x thc|ihdgz ift thkqhaorat ocazft zgdhfogiat

g

4;

j

84

:/7/ Aorazft x Ocazft Zgdhfogiat


97/955

{{{/de7:/fzk 87

d

1 1 7: )42 )

:4

]gcft g bhjqngz qog za|zataorgdho ba dgbg mzgddho/

\fbacft fjtaz}gz wqa ai ocazf ct kzgoba at 42 x ai ct |awqaf at7: >

Tfiqdho7: 9

:4 9

42

Dfo mzgddhfoat thc|iat) at cqx mdhi fzbaogzigt/ \haotg ao ai anac|if gorazhfz/ Tgjacft wqa qo cabhf at cgxfzwqa qo dqgzrf) x tgjacft wqa bft razdhft at cgxfz wqa qo cabhf/ Dfo mzgddhfoat ct dfc|iangt) oadathrgcft qogcanfz cgoazg |gzg dfc|gzgz/

Anac|if 1

Dqi at cgxfz) 4; f 28 6

Ao fzbao ba barazchogz atrf) oadathrgcft aodforzgz qog cgoazg ba zaatdzhjhz oqa}gcaora igt mzgddhfoat gt |fba#cft dfc|gzgzigt canfz/ Tgjacft wqa |fbacft atdzhjhz mzgddhfoat awqh}giaorat |gzg gcjgt/ Th `gdacft iftbaofchogbfzat hkqgiat ao oqatrzgt mzgddhfoat awqh}giaorat) aorfodat |fbacft dfc|gzgzigt bhzadrgcaora/ Atrgcftjqtdgobf ai cirh|if dfco ct jgnf ba dgbg baofchogbfz/ G atrf ta ia iigcg aodforzgz ai cohcf dfcobaofchogbfz (cdb/

Ai cohcf dfco cirh|if ba ; x 8 at =4/ Oqatrzg mzgddho atrgz za|zataorgbg |fz qog mfzcg bh}hbhbg ao =4taddhfoat/ Atrg }a~ qtgzacft qo zadrokqif dfzrgbf ao 8 |fz ; 3 =4 |abg~ft>

; bh}hba g =4 oqa}a }adat aorfodat>



98/955

8: {{{/de7:/fzk

4;

3

88

4;

3

:;=4

Ofrg wqa cqirh|ihdgz |fz 88 at if chtcf wqa cqirh|ihdgz |fz 7/ \fz rgorf):;=4 at qog mzgddho awqh}giaora g

4; / Gwq ta

cqatrzg if gorazhfz>

8 bh}hba g =4 thara }adat aorfodat>

28

3

;;

28

3

:0=4

:0=4 at qog mzgddho awqh}giaora g

28 / Gwq rhaoat dfcf ta fjtaz}g/

Atdzhjhaobf ig mzgddho tfjza qo dfco baofchogbfz ba =4) |qabat mdhicaora dfc|gzgzigt/ Gwq rfcgcft :0 baift dqgbzft tfcjzagbft ba =4 (ba oqatrzg hcgkao ba 28 ba gzzhjg x fzkgoh~gbgt ba cgoazg wqa tag mdhi dfc|gzgzigdfo oqatrzg za|zataorgdho ba 4; / Ofrg wqa `gx qo |awqaf dqgbzf wqa tfjzg/

Tfiqdho

Xg wqa :0=4 at cgxfz wqa:;=4 )

28 at cgxfz wqa

4; /

Zadqazbg

\gzg dfc|gzgz ocazft zgdhfogiat bajat zaatdzhjhzift dfo qo dfco baofchogbfz/



99/955

{{{/de7:/fzk 84

Aodforzgz ift F|qatrft ba ift Ocazft

Dgbg ocazf rhaoa qo f|qatrf/ Ao ig zadrg oqczhdg) qo ocazf x tq f|qatrf tfo f|qatrft aorza t/ Ao frzgt|gigjzgt) atro g ig chtcg bhtrgodhg g |gzrhz ba dazf) |azf atro ao igbft f|qatrft ba ig zadrg oqczhdg/

\fz baohdho) ai f|qatrf g dazf at dazf/

Anac|if =

Aodforzgz ai }gifz ba ift f|qatrft ba dgbg qog ba igt thkqhaorat ap|zathfoat/

g 4 % (

4

j 1 % (1d (77/1 % (77/1b 4; %

4;

Dgbg qog ba igt |gzangt ba ocazft ao ift anac|ift gorazhfzat tfo f|qatrft/ Ai f|qatrf ba 4 at (#4) ai f|qatrf ba1 at (#1) ai f|qatrf ba (#77/1 at 77/1 x ai f|qatrf ba 4; at

4; /

Tfiqdho

ai }gifz ba dgbg tqcg ao atra |zfjiacg at g 78/=

j 28d p

b px:

a (p 4Xg wqa tgjacft wqa ift ocazft f|qatrft atro ao igbft f|qatrft g |gzrhz bai dazf) |fbacft thc|iacaora cqirh#|ihdgz dgbg ap|zatho |fz #7/ Atrf dgcjhg ai thkof bai ocazf g tq f|qatrf/

g TfiqdhoAi f|qatrf ba 78/= at #78/=/

j Tfiqdho

Ai f|qatrf ba 28 at 28 /d Tfiqdho

Ai f|qatrf ba p at p/b Tfiqdho

Ai f|qatrf ba px: at px:/



100/955

82 {{{/de7:/fzk

a Tfiqdho

Ai f|qatrf ba (p 4 at (p 4 3 4p/Ofrg> Dfo ai irhcf anac|if rq bajat cqirh|ihdgz ig ap|zatho dfc|iarg |fz #7/ Qo azzfz dfco ao atra anac|ifat gtqchz wqa ai f|qatrf ba (p 4 at (p % 4/ OF DFCARGT ATRA AZZFZ'

Aodforzgz ]gifzat Gjtfiqrft

Dqgobf `gjigcft gdazdg ba }gifz gjtfiqrf) atrgcft `gjigobf gdazdg ba bhtrgodhgt ao ig zadrg oqczhdg/ \fz anac|if)ai ocazf ; atr ; qohbgbat giangbgt ba dazf/ Ai ocazf #; atr rgcjho ; qohbgbat giangbgt bai dazf/ Ai }gifzgjtfiqrf ba qo ocazf at ig bhtrgodhg g ig wqa atr g |gzrhz ba dazf) aorfodat ai }gifz gjtfiqrf ba ; x ai }gifz gjtfiqrfba #; at |gzg gcjft ;/

Oftfrzft iaacft ig ap|zatho ypy gt ai }gifz gjtfiqrf ba p/ f Oftfrzft atdzhjhcft ai }gifz gjtfiqrf ba #; gty;y

Rzgrgz ap|zathfoat ba }gifz gjtfiqrf dfcf |gzoratht/ Th aphtra qog f|azgdho gbaorzf ba ift tcjfift bai}gifz gjtfiqrf |zhcazf bajat a}giqgz ig f|azgdho/

Ai }gifz gjtfiqrf ba qo ocazf f ba qog ap|zatho at thac|za |fthrh}g f dazf/ Of |qaba taz oakgrh}f/ Dfoai }gifz gjtfiqrf) atrgcft horazatgbft ohdgcaora ao wq rgo ianft atr qo ocazf g |gzrhz ba dazf) x of igbhzaddho/

Anac|if 0

A}giqgz igt thkqhaorat ap|zathfoat ba }gifz gjtfiqrf/

g y1 % 2yj 4 y2 8yd y1 77yb

y;

::yZadqazbg wqa bajat rzgrgz rfbgt igt ap|zathfoat baorzf bai tcjfif bai }gifz gjtfiqrf dfcf th atrq}hazgo baorzf ba

qo |gzoratht) x a}giqgzigt |zhcazf/

Tfiqdho

g y1 % 2y3 y8y3 8

j 4 y2 8y3 4y1y3 4 13 :d

y1

77

y3

y7=

y3 7=b y;::y3 y71y3 (713 71

Zatqcao ba ig Iaddho

Aorazft (f Ocazft aorazft tfo ift ocazft ba dforaf (7) :) 4///) ift ocazft ba dforaf oakgrh}f (#7) #:)



101/955

{{{/de7:/fzk 81

#4///) x dazf/ Qo ocazf zgdhfogi at ig zaigdho ba qo aorazf zat|adrf g frzf) dfcf gj f

41 / Ai ocazf ao ig |gzra ba gzzhjg

at iigcgbf oqcazgbfz x ai ocazf ao ig |gzra homazhfz (ai dqgi of |qaba taz dazf at iigcgbf baofchogbfz/ Mzgddhfoat \zf|hgt tfo ocazft zgdhfogiat ao igt wqa ai oqcazgbfz at caofz wqa ai baofchogbfz/ Mzgddhfoat Hc|zf|hgt tfo ocazft zgdhfogiat ao igt wqa ai oqcazgbfz at cgxfz wqa ai baofchogbfz/ Mzgddhfoat Awqh}giaorat tfo bft mzgddhfoat wqa |zf|fzdhfogo ai chtcf }gifz oqczhdf dqgobf tfo a}giqgbgt/ \gzg zabqdhz qog mzgddho (atdzhjhzig ao tq mfzcg ct thc|ia Atdzhjhz rfbft ift mgdrfzat |zhcft bai oqcaz#

gbfz x baofchogbfz) dgodaigz mgdrfzat dfcqoat) iqakf atdzhjhz ba oqa}f ig mzgddho ao tq mfzcg thc|ihdgbg/ \gzg dfc|gzgz bft mzgddhfoat at dfo}aohaora atdzhjhzigt dfo qo dfco baofchogbfz> ai chtcf aorazf ao ig|gzra homazhfz (baofchogbfz ba dgbg mzgddho/

Ai }gifz gjtfiqrf ba qo ocazf at ig bhtrgodhg ao ig dqgi atr ao ig zadrg oqczhdg g |gzrhz ba dazf/ Ai }gifzgjtfiqrf ba qo ocazf taz thac|za |fthrh}f f dazf/

Bft ocazft tfo f|qatrft th atro g ig chtcg bhtrgodhg g |gzrhz ba dazf ao ig zadrg oqczhdg x ao igbftf|qatrft gi dazf/ Ai f|qatrf ba qog ap|zatho |qaba taz aodforzgbf cqirh|ihdgobf ig ap|zatho rfrgi |fz #7/


7/ Igt cgzdgt ao ig zadrg oqczhdg za|zataorgo aorazft at|gdhgbft qohmfzcacaora/ aodqaorzg ift }gifzat bag) j) d) bx a/

:/ Barazchogz wqa mzgddho bai rfbf za|zataorg dgbg zakho tfcjzagbg/

g/

j/

d/4/ Dfifdgz ift thkqhaorat dfonqorft ba ocazft zgdhfogiat ao fzbao) ba caofz g cgxfz/

g/ 7: )74 )

72

j/ 777: )7:77 )

74772

34 742

34

7:

Aorfodat qog tqcg awqh}giaora |gzg oqatrzf |zfjiacg at =7:

% :7:

% 47:

3 (= % : % 47:

3 777:

Tfiqdho

\abzf |qaba dforgz dfo foda bfdag}ft bai dftrf bai }hgna (":):


109/955

{{{/de7:/fzk 7


110/955

7


111/955

{{{/de7:/fzk 7

Ai zjfi ba mgdrfzat |gzg 8< ta fjtaz}g gt>

8< 3 8 7

7:= 3 8 728 3 4 4

72 3 ; :7:= 3 4 4 ; :



112/955

7

\zhcf Mgdrfzat ao 8< Mgdrfzat ao 7:= Rfcgcft: 7 7 7

4 : : :1 7 < 7; < 7 7

Aorfodat oadathrgcft> qo :) bft 4t) qo 1 x qo ;/ Ao frzgt |gigjzgt> : 4 4 1 ; 3 =4 gd jd 3 gjd

Ai ocazf qof at g }adat iigcgbf baofchogbfz ho}hthjia) dfcf dgbg ocazf aorazf |qaba taz |aotgbf dfcfqo ocazf zgdhfogi dqxf baofchogbfz at qof/

Ig bhmazaodhg aorza bft }gifzat at ai dgcjhf ao atg dgorhbgb/


7/ tqtrzga ift thkqhaorat ocazft zgdhfogiat/ Gtakzgra wqa rq zat|qatrg atr ao ig mfzcg ct thc|ia/

g/ 17: 870j/ :4 72d/ 42 74b/ 7177 8;a/ :74 777m/

;

:; 8

48k/ =77 4::`/ 74=2 ;2

Dgorhbgb ba @:F: |qzg 3


123/955

{{{/de7:/fzk 77;

\fbacft apgchogz atra |zfjiacg qtgobf ocazft zgdhfogiat/ Cgoraoacft qo rfrgi gdqcqigbf ba wq mzgddho ba igjgzzg wqabg/ Zadqazbg) dqgobf iaacft qog mzgddho takqhbg |fz ba ao ai |zfjiacg) thkohdg wqa oftfrzft bajacftcqirh|ihdgz |fz atg mzgddho/

Dfcao~gcft dfo qog jgzzg dfc|iarg ba d`fdfigra Ai rfrgi dfo ai wqa dfcao~gcf

UQ%


124/955

770 {{{/de7:/fzk


7/ Cqirh|ihdgz igt thkqhaorat ap|zathfoat |fz caoft qof/

g/ :1j/ #7


125/955

{{{/de7:/fzk 778

:/1 Ig \zf|habgb Bhtrzhjqrh}g


G|ihdgz ig |zf|habgb bhtrzhjqrh}g/ Hbaorhdgz |gzrat ba qog ap|zatho/ Zatfi}az |zfjiacgt bai cqobf zagi qtgobf ig |zf|habgb bhtrzhjqrh}g/

Horzfbqddho

Gi ogi bai gf atdfigz) qo |zfmatfz ba atdqaig aiacaorgi `gda qog |awqag jfitg ba zakgif |gzg dgbg qof ba tqtatrqbhgorat/ Dgbg jfitg dforhaoa qog mfrfkzgmg ba ig digta) bft |awqagt jfithrgt dfo fjtawqhft x dhodf bqidat/ Ai|zfmatfz bhtrzhjqhz igt jfitgt aorza tqt :0 atrqbhgorat/ Dqorft gzrdqift ba dgbg rh|f oadathrg ai |zfmatfz6

G|ihdgz ig \zf|habgb Bhtrzhjqrh}g

Dqgobf raoacft qo |zfjiacg dfcf ai cftrzgbf ao ig horzfbqddho) Ig \zf|habgb Bhtrzhjqrh}g |qaba gxqbgzoft gzatfi}azif/ \gzg ac|a~gz) |fbacft atdzhjhz qog ap|zatho |gzg ift dforaohbft ba dgbg jfitg/

Dforaohbft 3 (mfrfkzgmg % : jfithrgt ba fjtawqhft % 1 bqidat

Dforaohbft 3 (| % :m% 1d \fbzgcft qtgz qog iarzg |gzg atdzhjhz igt }gzhgjiat ba qog ap|zatho/

Tgjacft wqa ai |zfmatfz rhaoa :0 atrqbhgorat) ba dqgiwqhaz cgoazg |fbacft atdzhjhz ig thkqhaora ap|zatho |gzg aiocazf ba gzrdqift wqa ai |zfmatfz oadathrgz/

gzrdqift 3 :0 (| % :m% 1d :0 }adat ai dforaohbf hobh}hbqgi ba dgbg jfitg/

Kaoazgicaora fchrhcft dqgiwqhaz thkof ba cqirh|ihdgdho wqa of tago atrzhdrgcaora oadatgzhft/

gzrdqift 3 :0(| % :m% 1d

g Ig |zf|habgb Bhtrzhjqrh}g ba ig Cqirh|ihdgdho thkohdg wqa dqgobf aodforzgcft baorzf ba |gzoratht qorzchof cqirh|ihdgobf frzft rzchoft) ai rzchof ba gmqazg ta cqirh|ihdg dfo dgbg qof ba ift rzchoft baorzf bai|gzoratht/



126/955

7:< {{{/de7:/fzk

3 :0(| % :m% 1d 3 :0(| % :0(:m % :0(1d 3 :0| % 1=m% 72


127/955

{{{/de7:/fzk 7:7

:;

4x: 77 3 =x:::;

b :p;

4x: 77px

3 :p; (4x

: % :p;

77px

3 =p

:x;

::p;px

Tfiqdho:p;

4x: 77px

3 =px

4::;x

Hbaorhdgz Ap|zathfoat wqa Ho}fiqdzgo ig \zf|habgb Bhtrzhjqrh}g

Ig \zf|habgb Bhtrzhjqrh}g g|gzada dfo mzadqaodhg ao ap|zathfoat) x cqd`gt }adat of rzga |gzoratht dfcf tc#jfift ba gkzq|gchaorf/ Ao ig iaddho 7/:) fjtaz}gcft dfcf ig jgzzg ba ig mzgddho gdrg dfcf qo tcjfif bagkzq|gchaorf/ Ai thkqhaora anac|if ho}fiqdzg ai qtf ba ig \zf|habgb Bhtrzhjqrh}g dfo mzgddhfoat/

Anac|if 2

Thc|ihdgz igt thkqhaorat ap|zathfoat/

g :p%02

j 8x:4d ~%=:Gqowqa atrgt ap|zathfoat of atro atdzhrgt ao qog mfzcg wqa ofzcgicaora gtfdhgcft dfo ig \zf|habgb Bhtrzhjqrh}g)ai `ad`f wqa ai oqcazgbfz ba igt mzgddhfoat bajhazg taz rzgrgbf dfcf th atrq}hazg ao |gzoratht `gda atrf qo |zfjiacgwqa ig \zf|habgb Bhtrzhjqrh}g |qaba gxqbgzoft g zatfi}az/

g :p%02 |qaba zaatdzhjhzta dfcf72 (:p % 0/

\fbacft aorfodat |zfdabaz g bhtrzhjqhz ai 72 /

72(:p % 0 3

:p2 %

02 3

:p: : %

2:

2

Tfiqdho:p%0

2 3p: % :

j 8x:4 |qaba taz zaatdzhrf dfcf74 (8x :/

\fbacft aorfodat |zfdabaz g bhtrzhjqhz ai 74 /

74

(8x : 3 8x4

:4

34 4p

4 :

4

Tfiqdho8x:

4 3 4x :4d ~%=: |qaba taz zaatdzhrf dfcf

7= (~ % =/

\fbacft iqakf |zfdabaz g bhtrzhjqhz 7: /

7:

(~ % = 3~

:%

=:



128/955

7:: {{{/de7:/fzk

Tfiqdho~%=

4 3~: % 4

Zatfi}az \zfjiacgt bai Cqobf Zagi Qtgobf ig \zf|habgb Bhtrzhjqrh}g

Ig \zf|habgb Bhtrzhjqrh}g at qog ba igt |zf|habgbat cgracrhdgt ct dfcqoat }htrgt ao ig }hbg bhgzhg/ Wqa gfzgao ift oakfdhft x ao ig Kafcarzg/ Dgbg }a~ wqa ta rhaoao bft f ct kzq|ft ba fjnarft) Ig \zf|habgb Bhtrzhjqrh}g|qaba gxqbgzoft g bat|angz qog hodfkohrg/

Anac|if 1

Qo ehftdf fdrfkfogi `g ba taz dfotrzqhbf dfcf ta cqatrzg ao ig hiqtrzgdho/ Ai zakigcaorf ba dfotrzqddho zawqhazatf|fzrat ba gdazf ba dhodf |hat ba ifokhrqb |gzg taz gkzakgbft g if igzkf ba ig ig jgta x tf|fzrat ba gdazf ba dqgrzf

|hat ba ifokhrqb |gzg taz gkzakgbft g ig ioag bai rad`f bai ehftdf/ Wq ifokhrqb ba gdazf taz zawqazhbg |gzgdfc|iargz ai |zfxadrf6

Dgbg igbf zawqazhz bft ifokhrqbat) qog ba dhodf x qog ba dqgrzf |hat za|adrh}gcaora/ @gx fd`f igbft) gth wqa gwqatr oqatrzg adqgdho/

gdazf zawqazhbf 3 0(2 % 1 |hat/

\fbacft qtgz ig |zf|habgb bhtrzhjqrh}g |gzg aodforzgz ig dgorhbgb rfrgi ba gdazf>

Gdazf zawqazhbf 3 02 % 01 3 4: % 2< |hat/

Tfiqdho

Qo rfrgi ba ;: |hat ba gdazf tfo zawqazhbft |gzg ai |zfxadrf/

:/1/ Ig \zf|habgb Bhtrzhjqrh}g


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Anac|if =

G dgbg atrqbhgora ia taz bgbf qo awqh|f ba acazkaodhg x tq|az}h}aodhg |gzg qo }hgna gi jftwqa/ Ai awqh|f dforhaoaqog ihorazog) qo awqh|f ba |zhcazft gqphihft) x zgdhfoat ba dfchbg |gzg acazkaodhg/ Igt ihorazogt dqatrgo "7: dgbgqog) ift ehr ba |zhcazft gqphihft ; dgbg qof) x igt zgdhfoat ba dfchbg dqatrgo ": |fz bg/ @gx "1

7 "7: ihorazog/7 "; awqh|f ba |zhcazft gqphihft/

p ": zgdhfoat bhgzhgt/

Ai ocazf ba awqh|ft 3 7;) aorfodat ai dftrf rfrgi at awqh}giaora g ig thkqhaora adqgdho>

Dftrf rfrgi 3 7;(7: % ; % :p

\fbacft qtgz ig \zf|habgb Bhtrzhjqrh}g ao atrg ap|zatho/

7;(7: % ; % :p 3 :


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7:2 {{{/de7:/fzk


7/ Qtgz ig \zf|habgb Bhtrzhjqrh}g |gzg thc|ihdgz igt thkqhaorat ap|zathfoat/

g/ (p % 2 :(p % 1j/ 7: (2~ % =d/ (2 % 1

(1 % :

b/ (p % : % ;a/ x(p % ;m/ 74p(4x %~

:/ Qtgz ig \zf|habgb Bhtrzhjqrh}g |gzg zacf}az ift |gzoratht ba igt thkqhaorat ap|zathfoat/

g/ 7: (p x 2j/


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{{{/de7:/fzk 7:1

7/ :p % 4:/ 4p % 24/ 77p: % =2/ x: %

74

1/ :4 :~=/ ;4 :|

:/ Ai atrgora dforhaoa 41 ihjzft ba |fatg x 11 of}aigt/

7/ Gcgz |qaba `gdaz 74 kgiiargt (: f~ tfjzgo/:/ Dgbg kgiiarg rhaoa 2 tachiigt ba cgdgbgchg (0 tfjzgo/



132/955

7:= {{{/de7:/fzk

:/= Bh}htho ba Ocazft Zgdhfogiat


Aodforzg

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CK-12: Álgebra Edición-Española v1 s1