utatlán matemática 2º sem

7/25/2019 Utatln Matemtica 2 Sem

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4 Dsgfk - Orupk Qtlti`Vaou`hk sajastra - GOAT2

Jltajtgfl


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4kdsgfk - Orupk Qtlti`Vaou`hk sajastra - GOAT

Jltajtgfl

2


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Jltajtgfl 2

Vaou`hk sajastra

Tasarvlhks tkhks iks harafbks. \uahl rgourkslja`ta prkbgdghl il raprkhuffg` tktli k plrfgliha asta jltargli ahufltgvk, pkr fuliqugar jahgk k prkfahgjga`tk, sg` il lutkrgzlfg` hai G`stgtutkOultajlitafk ha Ahuflfg` Tlhgke`gfl, GOAT. Vao` lrtfuik 84 ha il Fk`stgtufg` ^kitgfl haOultajlil qua sa raeara l il lutkrl.

G`stgtutk Oultajlitafk ha Ahuflfg` Tlhgke`gfl, GOAT.As u`l kdrl prkhufghl pkr ai Haplrtlja`tk ha Tahlffg` y Hgsak, plrl ai

G`stgtutk Oultajlitafk ha Ahuflfg` Tlhgke`gfl, GOAT.


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`hgfa................... ..................... ..................... ..................... ..................... ..................... ................... G

Vajl`l


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Vajl`l 4>Juitgpigflfg` ha pkig`kjgks G ................... ..................... .................. 47

^lrl fkja`zlr! Iay ha sgo`ks plrl ai prkhuftk ........................................................... 6> ^rkhuftk ha pkta`fgls ha gouli dlsa ................ ................. ............... 6>

Ai ju`hk ha il jltajtgfl


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Vajl`l 46^rkhuftks `ktldias GG ..................... ..................... ..................... ..................... .. 5?

^lrl fkja`zlr! Ajjy @katbar ......................................................................................... 52

Ai ju`hk ha il jltajtgfl


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Hgvgsg` ha pkig`kjgks..................... ..................... ..................... ..................... ..................... ..... 2

Logighlh ha fifuik ja`tli .................... ..................... ..................... ..................... ...............


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Vajl`l 42Eu`fgk`as ig`alias .................. ..................... ..................... ..................... .............


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Tasuja` ..................... ..................... ..................... ..................... ..................... ..................... ..........


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Vajl`l 66Vujl y rastl fk` `jarks jlyls ............................................ 4


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^lrl fkja`zlr!

Fk`kfa ustah il tkrra g`fig`lhl ha ^gsl;

IlXkrra g`fig`lhl ha ^gslas ai fljpl`lrgk ha il

fltahrli ha il fguhlh ha ^gsl, a` Gtligl. Fkja`z lg`fig`lrsa tl` prk`tk g`gfg su fk`struffg`. Vlda

qu liturl tga`a;

Il praou`tl sa puaha raspk`har l trlvs ha u`l

axprasg` lioadrlgflqua iks fga`tegfks bl` hatar-

jg`lhk plrl flifuilr il liturl ha fuliqugar tkrra.

Vkik bly qua tkjlr ai tgajpk qua tlrhl a` flar u`

kdmatk hasha il plrta js litl blstl ai suaik.

Vg ai kdmatk sa haml flar igdraja`ta, il erjuil k axprasg` lioadrlgfl qua `ks

parjgta flifuilr il liturl as=

b ot

4

4

:

bas il liturl hai ahgegfgk.

oas ai vlikr fk`stl`ta ha il orlvahlh, 7.2 j/s4(lfaiarlfg` fk` qua iks kdmatks

fla` skdra il tgarrl).

tas ai tgajpk qua tlrhl a` flar ai kdmatk.

Lbkrl yl pkhajks flifuilr il liturl. Vg u` kdmatk tlrhl 6.6?saou`hks a` flar

hasha ai pu`tk js litk ha il tkrra ha ^gsl blstl ai suaik, sustgtugjks iksvlikras a` il erjuil, kparljks y kdta`ajks il raspuastl. Ls=

( ) . ( . ) . 11.557.2 6.6?

b4 4

7 2


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Amarfgfgk 6

Kdsarva fk` lta`fg` flhl pkig`kjgk, iuaok asfrdliks a` krha` lieldtgfky hasfa`ha`ta. Bly u`amajpik.

a+ l+ d l + d + a

l+ l6+ l4

l4+ l6d

f6+ f1+ f4

x4y6+x+x8y4

-6r1+s+ r6s

l6d4+ l1d+ l

: ) d+ 4 : 5) h+


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Ial fughlhkslja`ta flhl prkdiajl y iuaok raligfa ils lftgvghlhas. Emasa a` ai amajpik.

L. Ai trg`ouik ha il egourl sgouga`ta ast hgvghghk l su vaz a` ktrks hks trg`ouiks js paquaks

(< y 4). Il jahghl ha flhl ilhk ast raprasa`tlhl pkr u`l vlrgldia.

ld

b

f h

< 4


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Jlrqua fk` u` fbaqua il flsgiil qua jamkr g`hgqua su ra`hgjga`tk.ikorlhk

a`prkfask

`kikorlhk

Haspushaastuhglr...

Filsgefk axprasgkàs lioadrlgfls sao` ai `jark ha trjg`ks.

Gha`tgefk ai orlhk railtgvk ha u` pkig`kjgk.

Asfrgdk axprasgkàs lioadrlgfls a` krha` lieldtgfk, hasfa`ha`ta ylsfa`ha`ta.

Flifuik ai vlikr ùjrgfk ha u`l axprasg` lioadrlgfl.

^rlftgfk ai fifuik ja`tli.

Haslrrkiia ùavls bldgighlhas

Axprasajks a` ekrjl ha pkig`kjgks!

Vldajks qua u`l iatrl a` ioadrl raprasa`tl fuliqugar `jark. ^kr ik tl`tk,il ikògtuh ha iks ilhks y ai ral ha u`ks lzuiamks sa puaha` raprasa`tlr pkr

jahgk ha vlrgldias. ^kr amajpik.

lzuiamk fulhrlhk

ral : l4 l

l

ral : l =d

lzuiamk raftlòuilr

d

l

Qtgigzl`hk il axprasg` ha ral a` flhl flsk, raprasa`tljks jkhaiks ha pkig-`kjgks fk` hgeara`tas lzuiamks. ^kr amajpik.

jkhaik axprasg` pkig`kjgli hai ral

4l4+ 8l =dl4 l4l =d l =d

l =d l =d

Lbkrl ia tkfl l ustah asfrgdgr u`l axprasg` pkig`kjgli qua raprasa`ta ai ralha iks lzuiamks.

jkhaik axprasg` pkig`kjgli hai ral

l4l =d

l =dl =d

l =d d4

d4d4

d4d4

d4

l

d


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^lrl fkja`zlr!

Fjk sa sujl` y rastl` `jarks a`tarks;

Astl sajl`l rafkrhlrajks fjk sa sujl` y rastl` `jarks a`tarks plrl

kparlr pkig`kjgks ha jl`arl rpghl y prafgsl.

A` iks fursks l`targkras lpra`hgjks qua ai fk`mu`tk ha iks `jarks a`tarks

ast ekrjlhk pkr iks `jarks a`tarks pksgtgvks y iks `jarks a`tarks `aoltg-

vks. Vk` amajpik ha astks jarks=

, 1, , 411

Vujl ha `jarks a`tarks

Ils raoils plrl il sujl sa lpigfl` l hks flsks= `jarks fk` gouli sgo`k y

`jarks fk` hgstg`tk sgo`k.

Vujl ha `jarks a`tarks fk` sgo`ks goulias

^lrl sujlr jarks a`tarks fk` sgo`ks goulias, sa sujl` iks vlikras ldskiutks

y sa fk`sarvl ai sgo`k, sg ai sgo`k as pksgtgvk, `k sa asfrgda a` ai rasuitlhk.

Amajpiks=

l.


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Amarfgfgk ) ( 6 5) (1 8 7)l l l l

l l l l

l l l l

l l l

6 5 1 8 7

6 1 8 5 7

2 8


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5) (4y4- 8y+ 6) + (-5y4+ 1y+ 7) ?) (1x6y4- 6x4y+


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F. Taligfa a` su fulhar`k ils sgouga`tas kparlfgk`as fk` pkig`kjgks. ^uaha raskivarik a` sa`tghk

vartgfli y bkrgzk`tli.


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A` astl lftgvghlh ia prkpk`ajks qua, utgigzl`hk su bldgighlh vgsuli, kdsarva hata`ghlja`ta ils -

ourls y raspk`hl ils praou`tls qua sgoua`. Ha astl jl`arl astlr haslrrkiil`hk su flplfghlh ha

lta`fg` l iks hatliias.

) + + + + + + :


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4>

Juitgpigflfg` ha pkig`kjgks G

\u a`fk`trlr astl sajl`l;

Taplsk ha il iay ha sgo`ks y hai prkhuftk ha pkta`fgls

Juitgpigflfg` ha hks k js jk`kjgks y ha jk`kjgk pkr pkig`kjgk

^rkhuftk ha jarks a`tarks y prkhuftk ha hks jk`kjgks

ral lioadrlgfl ha egourls oakjtrgfls y trlhuffg` ha iaòulma fkj` liaòulma lioadrlgfk

Astl sajl`l ikorlr=

Tafkrhlr il iay ha sgo`ks lpigflhl li prkhuftk ha `jarks a`tarks y ai prkfahgjga`tkplrl juitgpigflr pkta`fgls ha il jgsjl dlsa.

Juitgpigflr hks k js jk`kjgks y u` jk`kjgk pkr u` pkig`kjgk. ^rlftgflr ai fifuik ja`tli l trlvs hai prkhuftk ha `jarks a`tarks y ha hks

jk`kjgks.

Flifuilr ai ral lioadrlgfl ha egourls oakjtrgfls.

Xrlhufgr prkdiajls ha iaòulma fkj` l iaòulma lioadrlgfk.

47Jltajtgfl Vajl`l 4>


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^lrl fkja`zlr!

^lrl ajpazlr ùastrk astuhgk astl sajl`l fk`vgaà rafkrhlr fjk juitgpigflr

`jarks a`tarks y pkta`fgls ha il jgsjl dlsa, pkrqua `ks sarvgr plrl juitg-

pigflr pkig`kjgks.

Iay ha sgo`ks plrl ai prkhuftk

Tafuarha=

Ai prkhuftk ha hks `jarks a`tarks fk` ai jgsjk sgo`kas u` `jark

a`tark pksgtgvk.

Ai prkhuftk ha hks `jarks a`tarks fk` sgo`k hgeara`ta as u` `jark

a`tark àoltgvk.

Kdsarva fk` lta`fg` iks prg`fgpgks ha il iay ha sgo`ks y ai amajpik qua ik

lfkjpll.

iay ha sgo`ks amajpik

(+)(+): + (?)(6) : 4) (x+ 7)(x+ 8) : (x)4+ x(7 + 8) + (7 =8) : x4+


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Asfrgdl ai trjg`k qua fkjpiatl il goulihlh. Xgaà u` amajpik plrl flhl flsk.

L. Tasuaivl ils juitgpigflfgkàs ha `jarks a`tarks ik js rpghk qua puahl. Xaòl prasa`ta il iay

ha sgo`ks. Xgaà u` amajpik.

>) 8 x 8 :


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a`

prkfask

`k

ikorlhk

Haspusha

astuhglr...

Tafk`kzfk iks vlikras ha Ajjy @katbar.

Tasuaivk prkhuftks `ktldias pkr sgjpia g`spaffg`= sujl pkr

hgeara`fgl y hks dg`kjgks fk` u` trjg`k fkj` y sgo`ks goulias.

Juitgpigfk y hgvghk `jarks a`tarks3 juitgpigfk pkta`fgls fk` gouli

dlsa.

Tasuaivk u` flsk ha rlzk`ljga`tk iogfk.

Iks prkdiajls ha rlzk`ljga`tk iogfk sgo`gfl` u` ratk qua pkà l pruadl tkhls ùastrls bldgig -

hlhas y hastrazls plrl a`fk`trlr il jamkr skiufg`. Ial ai flsk sgouga`ta, raligfa iks prkfahgjga`tks

qua fk`sghara àfaslrgks y prkpkòl il jamkr skiufg`.

Ai flsk hai plstkr, il fldrl, ai ikdk y il iafbuol

Q` plstkr tgaà qua plslr u` ikdk, u`l fldrl y u`l iafbuol l il ktrl krgiil ha u` rk. Hgspkà ha

u`l paqual dlrfl a` il qua skik flda` i y il fldrl, i y ai ikdk k i y il iafbuol. Ai ikdk sa fkja

l il fldrl sg sa quahl` skiks, il fldrl sa fkja il iafbuol sg sa quahl` skiks. Ha qu jlàrl hada

raligzlr iks vglmas plrl iiavlr li ktrk ilhk hai rk l iks tras;

?2GOAT Qtlti`

Tlzk`ljga`tk iogfk

Tavgsa su lpra`hgzlma


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48

Hgvgsg` ha pkig`kjgks


Taplsk ha il iay ha sgo`ks y ha il hgvgsg` ha pkta`fgls

Hgvgsg` ha u` jk`kjgk a`tra ktrk jk`kjgk, ha u` pkig`kjgk a`tra u`jk`kjgk y ha u` pkig`kjgk a`tra u` dg`kjgk= raoil ha Tueeg`g

^rkhuftk y hgvgsg`

Vafua`fgls iogfls

Astl sajl`l ikorlr=

Taplslr il iay ha sgo`ks y ha il hgvgsg` ha pkta`fgls.

Hgvghgr u` jk`kjgk a`tra ktrk jk`kjgk y u` pkig`kjgk a`tra u` jk`kjgk.

Lpigflr il raoil ha Tueeg`g plrl hgvghgr u` pkig`kjgk a`tra u` dg`kjgk.

^rlftgflr ai fifuik ja`tli l trlvs ha prkhuftks y hgvgsgk`as.

Fkjpiatlr safua`fgls iogfls.

?7Jltajtgfl Vajl`l 48


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^lrl fkja`zlr!

Astl sajl`l fk`vgaà rafkrhlr il hgvgsg` ha `jarks a`tarks y ha pkta`fgls

ha il jgsjl dlsa. ^krqua `ks sarvgr` plrl il hgvgsg` ha pkig`kjgks.

Iay ha sgo`ks ha il hgvgsg`Tafuarha=

Ai fkfga`ta ha hks `jarks a`tarks fk` aijgsjk sgo`kas u` `jark

pksgtgvk.

Ai fkfga`ta ha hks `jarks a`tarks fk` sgo`k hgeara`taas u` `jark

àoltgvk.

Kdsarva il iay ha sgo`ks ha il hgvgsg` y iks amajpiks.

iay ha sgo`ks amajpik+ +: + 5 4 : 6

: + 5 4 : 6

+ : 5 4 : 6

+: 5 4 : 6

Hgvgsg` ha pkta`fgls ha gouli dlsa

^lrl hgvghgr pkta`fgls ha gouli dlsa, sa fkpgl il dlsa y sa rastl` iks axpkà`tas.

Amajpiks

81 86: 816: 84 f? f1: f?1: f4

4? 45: 4?5: 4: ) as gouli l u`k(: ) GOAT Qtlti`


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Amarfgfgk

Ktrk amajpik

Hgvghljks (4x4- 7x-


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Amarfgfgk 6

Lbkrl prlftgqua il hgvgsg` ha hks pkig`kjgks. Ai amarfgfgk > ia sgrva ha amajpik.

>) (x6- 4?) '(x- 6) : > -4?

6 +6 +7 +4?

< +6 +7 >

x4+ 6x + 7,T: >

4) (x4+x- 4>) '(x+ 1) : 6) (4j4- 7j-


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5) (8y6-y+ 8) '(y+ 4) : ?) (z6+ 2) '(z+ 4) :


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Lftgvghlh


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D. Taligfa ils hgvgsgk`as g`hgflhls. Ouasa hai amajpik.

>) (5x8) '(4x) :


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L. A`fua`tra ai prkhuftk k ai elftkr qua elitl. Xga`a u` amajpik.

>) 7 x 2 :

f8) '(-4f) :


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Asta amarfgfgk fk`sgsta a` bliilr il ourl, ai vlikr ujrgfk k iatrl qua fkrraspk`ha sao` il safua`fgl

iogfl. Emasa a` ai amajpik.

>) 6 5 78


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Jlrqua fk` u` fbaqua il flsgiil qua jamkr g`hgqua su ra`hgjga`tk.

ikorlhka`

prkfask`k

ikorlhk

Haspusha

astuhglr...

Taplsk il iay ha sgo`ks y il hgvgsg` ha pkta`fgls.

Hgvghk u` jk`kjgk a`tra ktrk jk`kjgk y u` jk`kjgk a`tra u`

jk`kjgk.

Lpigfk il iay ha Tue`g plrl hgvghgr u` pkig`kjgk a`tra ktrk pkig`kjgk.

^rlftgfk ai fifuik ja`tli l trlvs ha prkhuftks y hgvgsgkàs.

Fkjpiatk safua`fgls iogfls.


Fu`tks fudks bly;

Fk` astl lftgvghlh prata`hajks haslrrkiilr su bldgighlh ha fk`tak y ha kdsar-

vlfg` lta`tl. Il ghal as flifuilr ai tktli ha fudks qua bly a` il ourl sgouga`ta=

Ai rasuitlhk ik pkhajks kdtaàr ls=

A` il egil ha lrrgdl bly 1fudks

A` il egil ha a` jahgk bly ?fudks

A` il egil ha ldlmk bly 2fudks

Taspuastl= A` il eourl bly 1 + ? + 2 : 4> fudks.

Lbkrl ia tkfl l ustah, flifuia il fl`tghlh ha fudks qua bly a` flhl ourl.

Asfrgdl su raspuastl skdra il iàl.


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41

Taplsk= sajl`ls


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\uarghl y quarghk astuhgl`ta=

Va lprkxgjl il tarfarl avliulfg` y hada praplrlrsa lhafulhlja`ta, raplsl`hk iks fk`ta`ghks

ha ils sajl`ls


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Axprasgk`as lioadrlgfls


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Amarfgfgk 4Fkjpiata il tldil asfrgdga`hk sg ai pkig`kjgk sa filsgfl fkjk dg`kjgkk trg`kjgk. Iuaok asfrgdlai orlhk railtgvk raspaftk l il vlrgldia g`hgflhl. Ouasa pkr ai amajpik.

pkig`kjgk filsgflfg` orlhk ha l orlhk ha d

>) 4l6d+ 4l4d + 2d4 trg`kjgk 6 4


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Vujl y rastl ha pkig`kjgks^lrl sujlr k rastlr jk`kjgks, sujljks k rastljks iks fkafga`tas `ujrgfks y fkpgljks il plrtaigtarli.

8j +


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Amarfgfgk 1

Kpara iks jk`kjgks a` sa`tghk vartgfli. Ouasa pkr ai amajpik.

>) 8x

+x


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D. Kpara u` jk`kjgk fk` u` pkig`kjgk a` sa`tghk vartgfli. Bly u` amajpik.

>) 8z -(65x + 4y - 8z):


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Juitgpigflfg` ha pkig`kjgks G


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Amarfgfgk 7

Juitgpigqua u` jk`kjgk pkr u` pkig`kjgk a` sa`tghk bkrgzk`tli. Axprasa ai pkig`kjgk rasuitl`ta a`

krha` hasfa`ha`ta y rahuzfl trjg`ks sajaml`tas. Xga`a u` amajpik.

>) (5x4)(< + 4x-


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Amarfgfgk


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Amarfgfgk ) (x +


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5) (x+ 8)(6x4+ 8x- 4) : ?) (6x+ 4)(8x4- 6x+ ?) :

2) (8l4- 6)(4l4- l+ 5) : 7) (1f6+


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^rkhuftks `ktldias G y GGIks prkhuftks `ktldias sk` juitgpigflfgk`as ha pkig`kjgks qua fujpia` fk` raoils mls y sa puaha`

raskivar pkr sgjpia g`spaffg`.


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b4- c4

Amarfgfgk ) (4l+ 4d)4: (4l)4+ 4(4l =4d) + (4d)4: 8l4+ 2ld + 8d4

) (4b- 6)4: (4b)4-4(4b =6) + (6)4: 8b4-


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Amarfgfgk ) (l+ 1)(l+ 7) :


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Hgvgsg` ha pkig`kjgks


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Amarfgfgk ) (5z6+ 8z4+ 2z) '(4z) : > -2+4 +4 +8 +2 < +4 +8 >

x4+ 4x + 8,T:>

4) (x4+x- 4>) '(x+ 1) : 6) (4j4- 7j-


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8) (4x6+ 5x- 8) '(x+ 8) : 1) (x6+ 4?) '(x+ 6) :

5) (4p4- ?p-


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^rlftgqua il logighlh ha fifuik ja`tli juitgpigfl`hk iks jk`kjgks ik js rpghk qua puahl.

Tafuarha lpigflr ils iayas ha sgo`ks y pkta`fglfg`. Ouasa fk` iks amajpiks.

L. >) (l)(l) :


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Jlrqua fk` u` fbaqua il flsgiil qua jamkr g`hgqua su ra`hgjga`tk.ikorlhk a`prkfask

`kikorlhk

Haspushaastuhgl

r...

Taplsk iks fk`ta`ghks ha il sajl`l


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45

^lras krha`lhks y prkhuftkflrtasgl`k


Aigml su ja`

^lr krha`lhk y prkhuftk flrtasgl`k

Kparlfgkàs fkjdg`lhls

Xrlzlr u`l egourl a` ai pil`k l plrtgr ha il udgflfg` ha plras krha`lhks.

Astl sajl`l ikorlr=

Gha`tgegflr fkjdg`lfgkàs a` lftgvghlhas fktghgl`ls.

Taligzlr plras krha`lhks.

Taprasa`tlr u` prkhuftk flrtasgl`k a` ekrjl aùjarltgvl, tldil ha hkdia a`trlhl y

a` u` pil`k flrtasgl`k.

^rlftgflr ai fifuik ja`tli fk` kparlfgkàs fkjdg`lhls y prkhuftk ha jk`kjgks.

Xrlzlr u`l egourl oakjtrgfl pkr jahgk ha plras krha`lhks a` ai pil`k flrtasgl`k.


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^lrl fkja`zlr!

L trldlmlr!

^rlftgqua ekrjl`hk plramls krha`lhls. Asfrgdl a` il tldil tkhls ils fkjdg`lfgkàs pksgdias ha sldk-

ras ha bailhk (fk`mu`tk L) fk` u` lfkjplljga`tk (fk`mu`tk D). Asfrgdl prgjark ai sldkr hai bailhk

y haspus ai lfkjplljga`tk. Xgaà u` amajpik.

Va puaha` asfkoar 7 ja`s hgstg`tks! Flhl piltk ik pkhrljks filsgegflr fkjk

u` plr krha`lhk. Astl sajl`l lpra`har pkr qu.

jl`ls l`gsgiiks orl`kil

erasl erasl y jl`ls

vlg`giil

fbkfkilta

DL

lrrkz a`slilhl oulfljki

flrà lslhl flrà lslhly lrrkz flrà lslhly a`slilhl flrà lslhly oulfljki

pkiik ergtk pkiik ergtky lrrkz pkiik ergtky a`slilhl pkiik ergtky oulfljki

flihk ha ras flihk ha rasy lrrkz flihk ha rasy a`slilhl flihk ha rasy oulfljki

DL

Aigml su ja`

A` ai rastlurl`ta Ils Haigfgls bly vlrgls kpfgkàs ha lijuarzk l aiaogr. Ual-

jks fui as ai ja` hai hl.

Fu`tls kpfgkàs hgstg`tls ha lijuarzk bly;

Fk` il lyuhl ha il tldil blrajks tkhls ils fkjdg`lfgkàs pksgdias plrl praplrlr

ai ja` hai hl. Iiljljks fk`mu`tk L li fk`mu`tk piltk prg`fgpli y fk`mu`tk D

li fk`mu`tk lfkjplljga`tk. Kdsarva.

Ja` hai hl

Bky ia kerafajks=

flrà lslhl

pkiik ergtk

flihk ha ras

ûaha lfkjpllr su piltk fk`

lrrkz, a`slilhlk oulfljki.

Dua` prkvafbk!


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Li gouli qua u`plr krha`lhk, aiprkhuftk flrtasgl`k`kfujpia fk` ilprkpgahlh fk`ju-tltgvl, L x D !D x L.

4. ^rkhuftk flrtasgl`k

A` il saffg` ^lrl fkja`zlr! ekrjljks u` prkhuftk flrtasgl`k li fkjdg`lr

tkhks iks aiaja`tks hai fk`mu`tk piltk prg`fgpli fk` tkhks iks aiaja`tks hai

fk`mu`tk lfkjplljga`tk.

Ai prkhuftk flrtasgl`k ha hks fk`mu`tks Ly Das ktrk fk`mu`tk L x Dekrjlhkpkr tkhks iks plras krha`lhksqua puaha` ekrjlrsa, fkjdg`l`hk flhl aia-

ja`tk hai fk`mu`tk L, fk` flhl aiaja`tk hai fk`mu`tk D.

Vgjdigflja`ta sa raprasa`tl ls=

L x D, sa iaa= L pkr Dk fk`mu`tk prkhuftk flrtasgl`k ha L pkr D.

4.< Ekrjls ha raprasa`tlr ai prkhuftkflrtasgl`k

Ai prkhuftk flrtasgl`k as u` fk`mu`tk y fkjk tli sa puaha raprasa`tlr a` ekrjl

aùjarltgvl, a` u`l tldil ha hkdia a`trlhl y a` u` pil`k flrtasgl`k. Ualjks.

l. Ekrjl aùjarltgvl

Tafuarha qua aùjarlr as fk`tlr, blfar u`l igstl. ^kr ik tl`tk, raprasa`tlr ai

prkhuftk flrtasgl`k a` ekrjl aùjarltgvl, fk`sgsta a` raligzlr u`l igstl ha

tkhks iks plras krha`lhks pksgdias.

^kr amajpik

Hlhks iks fk`mu`tks L: {


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^lrl vargflr ai`jark ha plraskrha`lhks qua ha-

dajks kdtaàr haiprkhuftk flrtasgl`k,juitgpigfljks a`tras ai `jark haaiaja`tks ha flhlfk`mu`tk.A` ai amajpikL x D : 6 x 6 : 7Hadajks kdtaàr 7plras krha`lhks.

Amarfgfgk ) hk`ha sa uà` iks amas.

Vkdra ai amaxasfrgdgjks iks aiaja`tks hai prgjar fk`mu`tk hai prkhuftk flrta-sgl`k y skdra ai amayiks aiaja`tks hai saou`hk fk`mu`tk. Kdsarva il orfl.

Amarfgfgk 6

Hlhk ai sudfk`mu`tk hai prkhuftk flrtasgl`k U x I: { (l,


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f. Taprasa`tlfg` hai fk`mu`tk prkhuftk flrtasgl`k a` aipil`k flrtasgl`k.

Ualjks u` amajpik ha fjk raprasa`tlr u` prkhuftk flrtasgl`k a` ai pil`k

flrtasgl`k.

Hlhks iks fk`mu`tks F: { l, d, f, } y^: { 4, 8, 5 }

l. Bliiajks ai prkhuftk flrtasgl`k F x ^.

Fkjdg`ljks iks aiaja`tks hai fk`mu`tk Ffk` flhl aiaja`tk hai fk`-

mu`tk ^.

F x ^: {(l, 4), (l, 8), (l, 5), (d, 4), (d, 8), (d, 5), (f, 4), (f, 8), (f, 5) }

d. Taprasa`tajks ai prkhuftk flrtasgl`k F x ^ a` u` pil`k flrtasgl`k.

Hgdumljks ai pil`k flrtasgl`k y udgfljks iks plras krha`lhks hai prk-

huftk flrtasgl`k.

l>

) 6 #1 + 1 :

:


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Hasfudrl il ourl oakjtrgfl qua sa kfuitl a` ai pil`k flrtasgl`k. Vgol astks plsks=


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4?

Tailfgkàs y eu`fgkàs


Kslhl blstl il fujdra!


Uliulr eu`fgkàs

Hglorljl slogtli ha u`l railfg` y u`l eu`fg`

Astl sajl`l ikorlr=

Lpigflr railfgkàs y eu`fgkàs l lspaftks ha il vghl rali.

Gha`tgegflr iks fk`mu`tks hkjg`gk y fkhkjg`gk ha u`l railfg`.

Tailfgk`lr ai prkfahgjga`tk plrl flifuilr ai fk`mu`tk gjloa` ha u`l eu`fg` fk` su

raprasa`tlfg` oregfl.

^rlftgflr il logighlh ha fifuik ja`tli vliul`hk eu`fgkàs.

Hahufgr u`l railfg` k u`l eu`fg` l plrtgr ha su hglorljl slogtli.


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^lrl fkja`zlr!

Kslhl blstl il fujdra!

Q` orupk ha asflilhkras sa bl prkpuastk iiaolr l il fgjl ha astks fultrk vki-

flàs= Ha Loul, Lflta`lòk, Xlfl` y Xlmujuifk.

Vudgr` hks vkiflàs a` mu`gk y iks ktrks hks a` hgfgajdra. Flhl vaz qua

asflia` sudgr` prgjark ai vkif` ha ja`kr liturl.

Il liturl ha flhl vkif` sa juastrl a` il oregfl sgouga`ta.

L trldlmlr!

Taligfa hks fkjdg`lfgkàs ha asflilhl pksgdias, pkr jahgk ha plras krha`lhks. Axpigqua su raspuastl.

Emasa a` ai amajpik.

>)

74 j

Xlmujuifk

844> j


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Amarfgfgk > 4 ids \2.>>

6 ids \>. . . . . .

xids \8(x).>>

Ai prafgk, a` eu`fg` ha ils igdrls ha lzflr, sa puaha raprasa`tlr sgjdigflja`ta=

e(x) : 8x :y

^kr amajpik, sg quarajks bliilr ai prafgk ha 41 id ha lzflr, sustgtugjks xpkr41 a` il eu`fg`e(x) : 8x. Kdsarva=

e(x) : 8x

e(41) : 8(41) : >

Ai vlikr qua bliiljks (>) as ai vlikr qua lhqugara yhaspus ha lpigflr il

eu`fg` li vlikrx.

Vgx : 41 y : >

^kr ik tl`tk, l 41 id ia fkrraspk`ha u` `gfk prafgk, fuyk vlikr as \>.>>.

A` ai amajpik, ai`jark ha igdrls ha

lzflr as il vlrgldiag`hapa`hga`ta. Aiprafgk as il vlrgldiaqua hapa`ha hai`jark ha igdrls halzflr.


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4.< Taprasa`tlfg` oregfl ha u`l eu`fg`

Xljdg` pkhajks raprasa`tlr u`l eu`fg` a` u` hglorljl slogtli, fkjk a` ils

railfgk`as.

Iks plsks l saougr sk` iks sgouga`tas=

Lsgo`ljks u`l sarga ha vlikras li fk`mu`tk hkjg`gk (vlikr ha x), ha

praeara`fgl vlikras farfl`ks l fark, pkr amajpik R : {4, ,

e(

) e (x) :6x :y

e (-4) :6(-4) :-5

e (-) :>

e (

6

5

4) e(x) :x - < :y

e(-4) :

e(-) :

e(

(-4) - < : -6


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Lftgvghlh


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D. Taprasa`ta flhl eu`fg` a` u` hglorljl slogtli. Aigml iks vlikras 4, , < y 4 plrl ai fk`mu`tk

hkjg`gk y lpigqua il eu`fg` l flhl vlikr plrl bliilr ils gjoa`as. Emasa a` ai amajpik.


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Vustgtuyl ja`tlija`ta ai vlikr haxlsgo`lhk plrl bliilr ai vlikr ha su gjloa`. Emasa a` iks amajpiks.

L.

>) e(6) :x+ < :

) :x+ 4 :

4) e(8) :x+ < :

6) e(7) :x+


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L. A` u`l railfg , l flhl aiaja`tk hai hkjg`gk ia fkrraspk`ha u`l k js gjoaàs a` ai fk-

hkjg`gk. A` u`l eu`fg`, l flhl aiaja`tk hai hkjg`gk ia fkrraspk`ha u`l y skik u`lgjloa`

a` ai hkjg`gk. Ha lfuarhk l astls legrjlfgkàs, asfrgdl sg flhl hglorljl slogtli raprasa`tl u`l

eu`fg` k u`l railfg`. Axpigqua su raspuastl. Ai amarfgfgk > ia sgrva ha amajpik.

>)

l

d

f

d

f

h

L D

)

>

pkr flhl bkrl ha astlfgk`ljga`tk. Ai fkstk tktli

hapa`har hai tgajpk qua astlfgkàjks ai flrrk. Kdsarva ai fkstk ha astlfgk-

`ljga`tk plrl


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Amarfgfgk , li juitgpig-flrik pkr fuliqugar`jark, ai rasuitlhkas > (y: >).


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).

Ualjks u` amajpik

Q`l ajprasl ha sarvgfgk ha tlxgs fkdrl 1 quatzlias pkr flhl cgijatrk rafkrrghk

js 1 quatzlias ha tlrgel ml. Kdsarva ai prafgk ha sarvgfgk plrl


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x y

-4 -6

-< - ) (8, )

, )

6) (


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Ial il g`ekrjlfg` y raligfa ils lftgvghlhas.

Q` lutkjvgi sa haspilzl pkr il lutkpgstl ha ^li`-Asfug`til l


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47

Astlhstgfl G


Bgstkrgl ha il astlhstgfl

Astlhstgfl y trjg`ks astlhstgfks


Louhazl vgsuli

Astl sajl`l ikorlr=

Fk`kfar il avkiufg` ha il astlhstgfl l trlvs ha il bgstkrgl.

Haeg`gr qu as il astlhstgfl.

Axpigflr qu as pkdilfg`, juastrl, hltks y vlrgldias.

Krol`gzlr hltks `kjg`lias y krhg`lias a` u`l tldil ha erafua`fgls.

^rlftgflr il logighlh ha fifuik ja`tli fk` kparlfgk`as fkjdg`lhls.

Taskivar amarfgfgks ha louhazl vgsuli.


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^lrl fkja`zlr!

Bgstkrgl ha il Astlhstgfl

Hasha iks fkjga`zks ha il fgvgigzlfg` bl` axgstghk ekrjls sa`fgiils ha astlhs-

tgfl, puas yl sa utgigzldl` raprasa`tlfgkàs oregfls y ktrks sjdkiks a` pgaias,rkfls, pgazls ha jlharl y plrahas ha fuavls, plrl fk`tlr parsk`ls, fksafbls,

l`gjlias u kdmatks.

Il gèkrjlfg` js l`tgoul skdra ai usk ha il astlhstgfl il a`fk`trljks a`

Aogptk, lirahahkr hai lk 6>1> l.F., jufbk l`tas ha fk`strugr ils pgrjghas.

Astl fgvgigzlfg` yl l`ligzldl iks hltks ha il pkdilfg` y iks railfgk`ldl fk` iks

trgdutks.

Xljdg` a` Dldgik`gl,6>>> l.F., usldl` paquals tldigiils ha lrfgiil plrl ra-

fkpgilr hltks skdra il prkhuffg` lorfkil y ils aspafgas va`hghls k fljdglhls

jahgl`ta truaqua.

Ls jgsjk, hamlrk` gèkrjlfg` ha sus raogstrks astlhstgfks a` Orafgl, Tkjl

y Fbg`l.

Wl a` ùastrl arl Ougiiarjk G, ray ha Gòiltarrl, a` 25 a`flro u` fa`sk plrl

raogstrlr `lfgjga`tks y haeu`fgkàs. Asta fa`sk as rafk`kfghk fkjk ai prgjar

fa`sk ha pkdilfg`.

Ai haslrrkiik pia`k ha il astlhstgfl sa g`gfgl a`tra eg`lias hai sgoik RUGG y prg`fg-

pgks hai RUGGG.

Ai trjg`k "astlhstgfl" eua g`trkhufghk pkr Okttergah Lfba`wlii, prkeaskr y afk-

`kjgstl liaj , quga` ik utgigzldl plrl ai lìgsgs ha hltks hai Astlhk. Wl a` ai

sgoik RGR ai trjg`k "astlhstgfl" lhqugrg ai sgo`gegflhk lftuli ha rafkiaftlr y

filsgegflr hltks.

Xkjlhk y lhlptlhk ha bttp=//www.prkeaskraìgàl.fi/jltajltgfl/astlhgstgflBgstkrgl.btj

L trldlmlr!

Fk` il gèkrjlfg` ha il iafturl fkjpiata il iàl ha tgajpk. Asfrgdl ai `kjdra ha il fgvgigzlfg` kparsk`l qua lpigf il astlhstgfl. Xgaà u` amajpik.

6>>> l.F. 25 sgoik RGR6>1> l.F.

aogpfgks


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Amarfgfgk 5 hatarjg`l qua il astlturl jahgl ha iks bkjdras oultajlitafks, ha 4>

lks a` lhail`ta, as ha >> mva`as.

Il pkdilfg` as= Il juastrl as=

Il vlrgldia as= Ai hltk as=

4) Q` orl`mark saiaffgk` 5> pkiiks skdra u` tktli ha 7>>, plrl hatarjg`lr il fl`tghlh ha fk`-

fa`trlhk qua fk`suja flhl pkiik pkr hl. Ai rasuitlhk qua kdtuvk as >.6 igdrls prkjahgk.

l. Il pkdilfg` as= Il juastrl as=


d. Taspk`hl= \u ktrl vlrgldia sa pkhrl astuhglr ha il pkdilfg` ha pkiiks;

4. Xrjg`ks astlhstgfks Fjk `ks fkju`gfljks a` astlhstgfl;

Il astlhstgfl, fkjk tkhls ils fga`fgls, sa axprasl fk` u`l sarga ha trjg`ks qua

ia sk` prkpgks k qua tkjl` u` sgo`gflhk aspafgli. Lpra`hljks ai sgo`gflhk

ha astks trjg`ks.


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Amarfgfgk 4

Taiia`a ai frfuik qua raspk`ha sg il vlrgldia fgtlhl as fuligtltgvl, ful`tgtltgvl, `kjg`li, krhg`li,hgsfratl k fk`tg`ul.

,


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6. Krol`gzlfg` ha hltks

Iks hltks a` astlhstgfl sa krol`gzl` a` u`l tldil ha erafua`fgls.

Erafua`fglas ai `jark ha vafas qua sa rapgta flhl hltk. ^kr ik tl`tk, u`l

tldil ha erafua`fgls as il raprasa`tlfg` ha u` fk`mu`tk ha hltks qua `ks par-

jgta kdsarvlr qua tl` saoughk kfurra liok. Il tldil ha erafua`fgls ast ekrjlhlpkr hks fkiuj`ls, a` u`l asfrgdgjks iks hltks y a` il ktrl ils erafua`fgls.

6.< Krol`gzlfg` ha hltks `kjg`lias

Iks hltks `kjg`lias sa krol`gzl` a` u`l tldil ha erafua`fgls sao` sus fulig-

hlhas k flrlftarstgfls fkjuàs.

^kr amajpik

A` u`l astl ha il fkju`ghlh plrtgfgplrk` parsk`ls (@ : ). Hai tktli

41 arl` `gks, 8> mvaàs, 11 lhuitks y 4> l`fgl`ks. ^khajks raprasa`tlr astlgèkrjlfg` a` il tldil ha erafua`fgls sgouga`ta.

plrtgfgpl tas erafua`fgl

`gks 41

mvaàs 8>

lhuitks 11

l`fgl`ks 4>

tktli (@)

Kdsarva qua a` il itgjl il bajks asfrgtk ai tktli ha iks hltks (@) qua rapra -

sa`tl ai tktli ha il pkdilfg`. Il sujl ha tkhls ils erafua`fgls hada fkg`fghgr

fk` ai tktli (@).

Amarfgfgk 6

Krol`gfa iks hltks a` il tldil ha erafua`fgls.

Va praou`t l > turgstls qu iuolr hasaldl` vgsgtlr. 8> raspk`hgark` Xgcli, 4> Ltgti`, 41 iks Fufbu-

jltlàs y


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6.4 Krol`gzlfg` ha hltks krhg`lias

Iks hltks krhg`lias sa krol`gzl` a` u`l tldil ha erafua`fgls ha lfuarhk l u`

krha` aspaffk. Dga` a` krha` lsfa`ha`ta, as hafgr, ha ja`kr l jlykr vlikr,

k a` krha` hasfa`ha`ta, ha jlykr l ja`kr vlikr.

Ualjks u` amajpik

Va hasal sldar il hgstrgdufg` pkr `gvaias ha > liuj`ks ha u` fkiaogk. Iks

hltks kdta`ghks sk`=


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Amarfgfgk 8


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Lftgvghlh


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Lftgvghlh 4. ^rlftgqua ik lpra`hghk

L. Ial ai sgouga`ta flsk y prasta lta`fg` l il gèkrjlfg` prkpkrfgk`lhl.

Il ajprasl ha loul purl Haigrrgflhasal sldar fu`tks oultajlitafks fk`suja` su prkhuftk a`

tkhk ai pls y qu kpg`g` tgaà` hai loul purl. ^lrl kdtaàr astl gèkrjlfg`, a`fuastlrk` l u`l

juastrl ha 1>>> parsk`ls y ias bgfgark` ils praou`tls sgouga`tas.


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F. Taligfa ils lftgvghlhas qua sa ia pgha`.


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L. Taligfa ils kparlfgk`as fkjdg`lhls ha sujl y juitgpigflfg`. Tafuarha raligzlr prgjark il

juitgpigflfg`.

:

8) < #? + 7 :

1) 5 #5 + :

5) 8 #1 + :

2) 4 #7 + :

) ? #?> + 5 :


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a`prkfask

`kikorlhk

Haspushaastuhglr.

.. Fk`kzfk il avkiufg` ha il astlhstgfl l trlvs ha il bgstkrgl.

Ha`k qu as il astlhstgfl.

Axpigfk qu as pkdilfg`, juastrl, vlrgldias y hltks.

Krol`gzk hltks `kjg`lias y krhg`lias a` u`l tldil ha erafua`fgls.

^rlftgfk il logighlh ha fifuik ja`tli fk` kparlfgkàs fkjdg`lhls.

Tasuaivk amarfgfgks ha louhazl vgsuli.

Kdsarva fk` lta`fg` il ekrjl y ai tljlk ha ils ourls paquals, iuaok hatarjgà fu`tls ourls

ha astls àfasgtlrl plrl ekrjlr il ourl js orl`ha. Ai amajpik > ia sgrva ha oul.

>)


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6>

Astlhstgfl GG


Astlhstgfl a` il fkjputlhkrl

Oregfls astlhstgfls= hglorljl ha dlrrls, pkiok`k ha erafua`fgls y hglorljl ha

saftkras

Kparlfgkàs fkjdg`lhls y pkrfa`tlmas

Lìgsgs a g`tarpratlfg` ha oregfls

Astl sajl`l ikorlr=

Taligzlr oregfls a` u`l bkml ha Axfai.

Aildkrlr hglorljls ha dlrrls, pkiok`ks ha erafua`fgls y hglorljls ha saftkras.

L`ligzlr a g`tarpratlr hltks y bafbks tkjlhks ha il ralighlh jahgl`ta orfls.

^rlftgflr ai fifuik ja`tli fk` kparlfgkàs fkjdg`lhls y pkrfa`tlmas.


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^lrl fkja`zlr!Astlhstgfl a` il fkjputlhkrl

Haslrrkiia su fkjpata`fgl a` taf`kikol

Astl sajl`l lpra`harajks l hgdumlr y fk`strugr hgeara`tas tgpks ha oregfls

plsk l plsk, park sg ustah hgspkà ha u`l fkjputlhkrl ia g`taraslr sldar qua

puaha blfarik hasha ai prkorljl iiljlhk Axfai.

Ai prkorljl Axfai ks parjgta fralr hgeara`tas oregfls ha jlàrl sa`fgiil. Ual-

jks fjk utgigzlr astl barrljga`tl plrl raprasa`tlr oregflja`ta iks olstks

raligzlhks pkr Ear`l`hk a` u` jas.

, >, 4?>.

4. Q`l vaz gòraslhks iks hltks, fkikqua ai pu`tark skdra il plildrl prkhuftks.

6. Dusqua a` il dlrrl ha barrljga`tls il plildrl g`sartlr, haia u` figf y ia lpl-

rafar u` fulhrk fk` ils kpfgkàs ha oregfls qua fk`tgaà ai prkorljl.

8. Asfkml ai tgpk ha oregfl qua hasaa, blol figf y lplrafar il oregfl ha il

gèkrjlfg` qua gòras.

Vg hasal ktrk tgpk ha oregfl, fkikqua ùavlja`ta ai pu`tark skdra il plildrl

prkhuftks y saiaffgkà il kpfg` qua hasaa. ^kòl l pruadl su furgksghlh y

aigml ktrls ekrjls ha oregfls.

L`jasa l uslr astl vligksl barrljga`tl!

4

6

8

parsk`ls a` il fkju`ghlh Il Asparl`zl

A` ai amaxasfrgdgjks ils vlrgldias= lrtasl`ks, astuhgl`tas y flrpg`tarks.A` ai amayhgstrgdugjks ils erafua`fgls ha a` .

Vkdra ai amaxhgdumljks u`l dlrrl plrl flhl vlrgldia, ha liturl prkpkr-fgk`li li vlikr ha il erafua`fgl.

kfuplfg` erafua`fgl

lrtasl`ks 1>

astuhgl`tas 6>

flrpg`tarks 4>

tktli >

Kfuplfg` ha > parsk`ls aìl fkju`ghlh Il Asparl`zl

>

4>

6>

8>

1>

5>

flrpg`tarksastuhgl`taslrtasl`ks

S Xtuik ha il orefl Z

erafua`

fgls

vlrgldiavlrgldiavlrgldia


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Ktrk amajpik.

Ha lfuarhk l iks hltks ha il tldil ha erafua`fgls, raprasa`tajks a` u` hglorl-jl ha dlrrls ai `kjdra y il fl`tghlh ha rdkias pkr aspafga prasa`tas a` ai

Farrk Liux. Tafuarha iks plsks.

Hgdumljks u` pil`k flrtasgl`k y asfrgdgjks ai ttuik a` il plrta supargkr.

A` ai amaxasfrgdgjks ils vlrgldias kjg`lias y a` ai amayils erafua`fgls.

Vkdra ai amax hgdumljks u`l dlrrl plrl flhl vlrgldia, qua lifl`zlr illiturl qua jlrqua su erafua`fgl.

Amarfgfgk 4Bolsa hk`l`ta ha slòra!

Ils trl`seusgkàs ha slòra slivl` vghls. Vkik il hk`lfg` raouilr ha vkiu`tlrgks puaha olrl`tgzlrai sujg`gstrk ha slòra saourl. Il Krol`gzlfg` Ju`hgli ha il Vliuh prasa`t a` ai gèkrja ha 4>>?asta g`fraja`tk a` ils hk`lfgkàs.

Aildkra u` hglorljl ha dlrrls fk` il gèkrjlfg` ha il tldil.

aspafga erafua`fgl

pg`k 11

fgprs 84

aufligptk 4>

plik dil`fk 2tktli

4>

6>

8>

1>

5>

plik dil`fkaufligptkfrgprspg`k

krgoa` ha ils hk`lfgkàs % luja`tk^lsas a` haslrrkiik 41

^lsas a` trl`sgfg` 6>

^lsas haslrrkiilhks 51

>

4>

6>

8>

1>

5>

?>

^lsas

haslrrkiilhks

^lsas a`

trl`sgfg`

^lsas a`

haslrrkiik

^krfa`tlma ha g`fraja`tk ha ilshk`lfgkàs ha slòra


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d. ^kiok`k ha erafua`fgls

Ai pkiok`k ha erafua`fgls as u` orfk qua raprasa`tl il gèkrjlfg` l trlvsha pu`tks y raftls.

Il fk`struffg` ha asta orfk as juy plrafghl li hglorljl ha dlrrls, skik quaa` iuolr ha fkiuj`ls sa trlzl` pu`tks a` ai pil`k flrtasgl`k, plrl iuaok u`griks

pkr jahgk ha iàls raftls.

Ualjks u` amajpik.

Q`k ha iks Kdmatgvks hai Jgia`gk (KHJ), a` fk`fratk ai Kdmatgvk 1, prkpkà ilrahuffg` ha il jkrtlighlh jltar`l. Ai gèkrja hai Vgstajl ha ils @lfgkàsQ`ghls a` Oultajlil, prasa`tlhk a` 4>>2, g`hgfl il avkiufg` ha il jkrtlighlhpkr flhl > >>> `lfghks vgvks. Iks hltks sa prasa`tl` a` il tldil.

lk jkrtlighlh

>> >5

1>

>

4>>

41>

6>>

>> 4>>5

Tahuffg` ha il jkrtlighlh jltar`la` Oultajlil ha >5

Iks Kdmatgvks haHaslrrkiik haiJgia`gk (KHJ) sk`kfbk prkpsgtks hahaslrrkiik bujl`kqua iks


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Amarfgfgk 6

^kòl a` prftgfl il raprasa`tlfg` orfl a` hglorljl ha dlrrls y pkiok`k ha erafua`fgls.

L. Q`l tlzl ha ykour fk`tgaà 1 % ha prkta`ls, 4% ha orlsls y 8% ha lzflr. Asfrgdl iks hltks aìl tldil y hgduma u` hglorljl ha dlrrls fk` il gèkrjlfg`.

ùtrga`tas %

prkta`ls

orlsls

lzflr

^krfa`tlma ha ùtrga`tas qua fk`tgaàu`l tlzl ha ykour

>

4, sao` ai gèkrja hai Vgstajl ha ils @lfgkàs Q`ghls aÒultajlil, prasa`tlhk a` 4>>2.

Aildkra ai pkiok`k ha erafua`fgls.

lk % hasùtrgfg`

61

8>

81

1>

Il orfl `ks g`hgfl qua sa bl rahufghk il hasùtrgfg` gèl`tgi, park ai `gvai l` as litk. ^lrtgfgpaa` iks prkorljls ikflias plrl qua asta prkdiajl haslplrazfl ik l`tas pksgdia.


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f. Hglorljl ha saftkras

Q` hglorljl ha saftkras fk`sgsta a` il raprasa`tlfg` ha hltks a` u`l fgrfu`-

eara`fgl. Astl orfl as juy tgi ful`hk sa fua`tl fk` vlikras ldskiutks y sahasal raprasa`tlr il gèkrjlfg` a` pkrfa`tlmas. Ualjks u` amajpik.

Iks tras ghgkjls js bldilhks a` ai ju`hk sk`=

ghgkjlerafua`fgl

(jgiikàs ha bldil`tas)

fbg`k jl`hlr` 7>>

asplki 65>

gòis 68>

tktli >

Fkjk il gèkrjlfg` a` u` hglorljl ha saftkras sa prasa`tl a` pkrfa`-

tlmas, flifuiljks ai pkrfa`tlma qua ia fkrraspk`ha l flhl ghgkjl.

fbg`k jl`hlr`=7>> x >

>:

7> >>>

>: 15.41%

asplki=65> x >

>:

65 >>>

>: 44.1%

gòis=68> x >

>:

68 >>>

>: 4, quaaqugvlia li >%. lrl raplrtgr iks orlhks ha jlàrl prkpkrfgk`li l flhlpkrfa`tlma, pkhajks lpigflr il erjuil sgouga`ta.

@ ha orlhks : % ha flhl vlrgldia x 65>>%

fbg`k jl`hlr`=15.41% x 65>

>%:

4> 41>

>: 4>4.1 4>6

asplki=44.1% x 65>

>%:

2>

>: 2%:

?51>

>: ?5.1 ?5

^lrl elfgigtlr il jahgfg` a` orlhks, bajks lprkxgjlhk il prgjarl y il tarfarl

fl`tghlh.

Tafuarha quakdtaàjks u`pkrfa`tlma pkr jahgkha u`l raoil ha trassgjpia, ls=

>% 1>> x 441

x:441 x >

1>>: 81%

Lpra`hl u` sjdkik

ùavk, ai sjdkik haaqugvlia`fgl


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Asfrgdgjks iks rasuitlhks a` u`l tldil y fkjprkdljks qua il sujl hails erafua`fgls as gouli li `jark ha hltks, il sujl ha iks pkrfa`tlmas

as >% y il sujl ha iks orlhks 65>.

ghgkjl erafua`fgl % orlhks

fbg`k jl`hlr` 7>> 15.41% 4>6

asplki 65> 44.1% 2 4 >% 65>

A` u` frfuik, jahgjks fk` u` trl`s-pkrtlhkr iks orlhks fkrraspk`hga`tas l

flhl saftkr, ha`trk ha asta asfrgdgjks ai

ghgkjl y su pkrfa`tlma.

Ha`trk ha il oregfl, asfrgdgjks ai hltkhai pkrfa`tlma, `k ha iks orlhks.

Ghgkjls js bldilhks a` ai ju`hk

g`ois4> 41.7% 76

Lsgl, lffk y Krga`ta 5

>>

:

765>

>:76

Lsgl, ^lfegfk y Krga`ta= : : : :

ergfl Vudslblrgl`l= : : : :


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>

4>>6>>

8>>

1>>

5>>

?>>

2>>

7>>

>>

lbkrrkvgvga`hlvastulrgktrl`spkrtaligja`tlfg`

Olstks raligzlhks pkr Ear`l`hk a` u` jas

d. ^kiok`k ha erafua`fglslskfgl pkr jahgk ha pu`tks y raftls iks hltks `kjg`lias fk` sufkrraspk`hga`ta fl`tghlh k erafua`fgl.


>

>

4>>

6>>

8>>

1>>

5>>

?>>

2>>

7>>

>>


f. Hglorljl ha saftkrasraprasa`tl hltks a` u`l fgrfu`eara`fgl. Astl orfl as juy tgi ful`hksa fua`tl fk` vlikras ldskiutks y sa hasal raprasa`tlr il g`ekrjlfg` a` pkrfa`tlmas.


ligja`tlfg`62%

vastulrgk44%

vgvga`hl48%

lbkrrk


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Lftgvghlh 7 a` iks ju`gfgpgks ha Jkrlz`, Fugilpl, Fkd`,Il Q`g` y Bugt ast raogstrlhk a` il tldil. Fk` il g`ekrjlfg` aildkra u` hglorljl ha dlrrls.

hglorljl ha dlrrlshglorljl ha saftkras

pkiok`k ha erafua`fgls

vlrgldiaerafua`fgl

parkhk

hglorljl ha saftkras

hglorljl ha dlrrls

pkiok`k ha erafua`fgls

ils erafua`fgls

ils vlrgldiasai ttuik

ils erafua`fgls

ils vlrgldiasai ttuik

ju`gfgpgk ` hls

Jkrlz`

BugtIl Q`g`Fkd`FugilplJkrlz`


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D. Aildkra u` hglorljl ha saftkrasfk` il gèkrjlfg` ha il tldil skdra iks pkrfa`tlmas ha utrga`tasqua àfasgtl u` lhuitk sl`k. Il gèkrjlfg` yl vgaà a` pkrfa`tlmas, skik hada flifuilr iks orlhksfkrraspk`hga`tas l flhl u`k. Xgaà u` amajpik.

^lrl qua ùastrk fuarpk rafgdl tkhks iks ùtrga`tas afaslrgks hadajks fkjar ligja`tks ha tkhks

iks orupks y a` ils fl`tghlhas lhafulhls.

orlsls41%

^krfa`tlmas ha ùtrga`tas quaàfasgtl u` lhuitk sl`k

ùtrga`tas erafua`fgl (%)

orlsls 41

prkta`ls

orlsls41% x 65>

>%:7> flrdkbghrltks :

prkta`ls : egdrl :

F. Ial il gèkrjlfg` y kdsarva il oregfl plrl raskivar iks sgouga`tas amarfgfgks.

(il `ktl js dlml) blstl < (il `ktl js litl). A`4>, liou`ks plsas ha Ljrgfl Iltg`l y Ai Flrgda kdtuvgark` ils fligegflfgkàs sgouga`tas.

`hgfa ha Haslrrkiik Bujl`k 4>

litk

>.?26 >.??1 >.?1 >.?41 >.578>.517

>.5>8>.151 >.15

>.8>8

Fbgia

Lroa`tg`l

Jxgfk

FkstlTgfl

jahgk

Daigfa

AiVlivlhkr

Bk`hurls

@gflrloul

Oultajlil

dlmk

Blgt

l. Vao` il oregfl, qu pls tgaà ai jamkr GHB;

d. Fui as il hgeara`fgl ha GHBa`tra Fbgia y Oultajlil;

f. Asfrgdl hks lffgkàs plrl luja`tlr ai GHBa` ai ral ha ahuflfg`. Ia lyuhljks fk` u`l=

Fbgia

Jamkrlr il ahuflfg`, sliuh y afk`kjl

GOAT Qtlti`


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4) Il sgouga`ta oregfl prasa`tl iks rasuitlhks ha u` astuhgk a` ai qua sa l`ligzl ai slilrgk ha ils

jumaras, raspaftk li ha iks bkjdras, a` pkrfa`tlma, a` ai saftkr ha il g`hustrgl.

%

86

Mlp`

18

FkralhaiVur

5>

Iuxajdurok

58

Lustrligl

51

AstlhksQ`ghks

5?

Aspll

52

Vugzl

52

Tag`kQ`ghk

?6

Liajl`gl

?8

Diogfl

??

Bkil`hl

?7

Erl`fgl

?7

Orafgl

28

Hg`ljlrfl

27

Vuafgl

Hgeara`fgl ha slilrgk ha ils jumaras raspaftk li ha iks bkjdras, a` pkrfa`tlma

Fk` il g`ekrjlfg` ha il oregfl, raspk`hl ils praou`tls.

l. A` qu pls ils jumaras ol`l` ai ?8% hai slilrgk ha u` bkjdra;

d. Fui as ai pkrfa`tlma hai slilrgk ha u`l jumar fk` raspaftk li

ha u` bkjdra a` Erl`fgl;

f. A` qu pls ai slilrgk ha il jumar sa lfarfl js li hai bkjdra;

h. A` qu pls ils jumaras ol`l` ja`ks hai 1>% qua iks bkjdras;

Lpigqua sus fk`kfgjga`tks skdra pkrfa`tlmas y flifuia.

a. Vg u`l jumar a` Vugzl ol`l > erl`fks, fu`tk ol`l u` bkjdra a` ai jgsjk puastk y fk`il jgsjl fltaokrl prkeasgk`li;

Taspuastl=

e. Q` bkjdra, pkr trjg`k jahgk, ol`l a` ai Tag`k Q`ghk u` suaihk ja`suli ha 4> (igdrlsastarig`ls). Fu`tk ol`lrl sg euasa jumar;

Taspuastl=


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L. Lpigqua il marlrqul ha ils kparlfgk`as li raligzlr iks amarfgfgks. Xga`a u` amajpik.

>) 1 + 7 6 :


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Q` fgfigstl sa ast praplrl`hk plrl il fkjpata`fgl fgfistgfl js gjpkrtl`ta hai lk= Il Uuaitl lOultajlil. Kdsarva fk` lta`fg` il orfl a` ai amaxil hgstl`fgl a` cgijatrks y a` ai amayil liturla` jatrks. Taspk`hl ils praou`tls. Xga`a u` amajpik.

>

>

4>>

6>>

8>>

1>>

5>>

?>>

2>>

7>>

>>

4> 6> 8> 1> 5> ?> 2> 7> > ) Hasha qu liturl g`gfg ai fgfigstl su rafkrrghk;

li ?>, lsfa`hg k hasfa`hg;

1) A` qu cgijatrks tuvk su jlykr hasfa`sk;

5) Fu`tk lsfa`hg hai cgijatrk ?> li 7>;

?) Fu`tks jatrks lsfa`hg hai cgijatrk 6> li 1>;

2) L qu li liturl iiao a` ai cgijatrk 1>;

7) \u hgstl`fgl bl rafkrrghk ful`hk iiaol l u`l liturl ha >> jatrks;

) Fui as il hgeara`fgl ha liturl a`trl il pksgfg` eg`li a g`gfgli;

1> jatrks

Tlzk`ljga`tk iogfk


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Jlrqua fk` u` fbaqua il flsgiil qua jamkr g`hgqua su ra`hgjga`tk.ikorlhk a`prkfask

`kikorlhk

Haspusha

astuhglr...

Taligzk orfls a` u`l bkml ha Axfai.

Aildkrk hglorljls ha dlrrls, pkiok`ks ha erafua`fgls y hglorljls

ha saftkras.

L`ligzk a g`tarpratk hltks y bafbks tkjlhks ha il ralighlh jahgl`ta

orfls.

^rlftgfk ai fifuik ja`tli fk` kparlfgk`as fkjdg`lhls y pkrfa`tlmas.


Ai rdki ha elftkras as u`l ekrjl ha hasfkjpk`ar u` `jark a` sus elftkras

prgjks. Il hasfkjpksgfg` elftkrgli ha u` `jark as ai prkhuftk ha tkhks iks

elftkras prgjks qua lplrafa` li `li ha ils rljls hai rdki.

Ualjks il hasfkjpksgfg` elftkrgli hai `jark 65 y ustah raligfa iks sgouga`-tas. (Vuoara`fgl, asfrgdl li ilhk gzqugarhk ha flhl rljl iks `jarks ja`krasha flhl prkhuftk).

>)

65

6 64 4

5 5 Il hasfkjpksgfg` elftkrgli

ha 65 as 4 x 4 x 6 x 6.

65 : 4 x 4 x 6 x 6


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6) Fui prkdiajl as salilhk pkr ai ja`kr jark ha parsk`ls;

4/>7/> > > >>>

Fui as ai prkjahgk ha jltargli rafgfilhk;

Asfrgdl ai vlikr ha @ :

Fkpga il erjuil= R :

Vuja iks vlikras haxy hgvghl ai rasuitlhk a`tra ai vlikr ha @.

R :

R :

Kdta`ol ai rasuitlhk. R :

Asfrgdl il raspuastl=


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>7.

pls` ha jumaras

hgputlhls

Blgt 8

Oultajlil 7

>

8

2


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Ualjks u` amajpik

Ai Jg`gstargk ha Vliuh kerafa ha ekrjl orltugtl, plrl `gks ha > l 8 lks,

u` sgstajl dsgfk ha vlfu`lfg`.

Il tldil juastrl ai `jark ha vlfu`ls lpigflhl hurl`ta u`l mkr`lhl.

vlfu`ls ` ha vlfu`ls lpigflhlsXudarfuiksgs (DFO) 8?

Vlrljpg` ?2

Hgetargl y tks earg`l (H^X) ?6

Ulrgfail 15

Tktlvgrus 51

Fui as il jahgl`l ha vlfu`ls lpigflhls;

Flifuiljks il pksgfg` ha il jahgl`l.

@ + , 7?,

L qu pask ia fkrraspk`ha il jahgl`l;

^ksgfg` ha il jahgl`l.@ + , 5=86, 5=1>, ?=>4, ?=>1

l. L qu bkrl ia fkrraspk`ha il jahgl`l;

Taspuastl=

d. Fu`tks ajpialhks iiaolrk` l`tas ha il jahgl`l;

Taspuastl=

f. Fu`tks ajpialhks iiaolrk` haspus ha il jahgl`l, park l`tas ha ils ?=>>;

Taspuastl=


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Tasuaivl ja`tlija`ta ils kparlfgkàs y asfrgdl su raspuastl skdra il iàl. Bolik ik js rpghk

qua puahl.

L. Taligfa ils kparlfgkàs fkjdg`lhls, rafuarha il marlrqul ha ils kparlfgkàs. Xgaà u` amajpik.

>) 1 + (4 x 8) :


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L. Lpigqua ik lpra`hghk y rasuaivl iks prkdiajls qua sa pil`tal`.

> jgiikàs ha lhuitks fk` skdrapask y li ja`ks 8>> jgiikàs ha kdasks.A` iks itgjks lks ai skdrapask y il kdasghlh luja`tl` ha jlàrl lilrjl`ta a` iks plsas

a` vls ha haslrrkiik y aspafglija`ta a` ils rals urdl`ls.

A` u` fa`trk fkju`gtlrgk sa hl saougjga`tk l > `gks fk` skdrapask plrl avgtlr qua sal`

lhuitks kdasks. Iks rasuitlhks sk`=

skdrapask a` id `jark ha gks

4 1 67

5 7 8 igtrks ha

loul pkr parsk`l as il fl`tghlh lhafulhl plrl sltgselfar tkhls ils àfasghlhas dsgfls.

Ai fk`sujk ha loul a` hatarjg`lhk hl plrl fg`fk eljgigls fkjpuastls pkr 8 parsk`ls as=

eljgigl fk`sujk ha loul pkr hl a` igtrks

Dltz 5>>Ojaz ?1>

^raz 652

Xufudli 1>>

]urat 82>

l. Fui as ai prkjahgk ha loul fk`sujghk pkr eljgigl;

d. Qtgigfa ai rasuitlhk hai g`fgsk l`targkr plrl flifuilr ai prkjahgk fk`sujghk pkr parsk`l.


a`prkfask

`kikorlhk

Haspushaastuhglr... Axtrlgok gèkrjlfg` ha u`l orfl.

Kdtaòk il jahgl, il jahgl`l y il jkhl l plrtgr ha u`l hgstrgdufg`

ha hltks.

^rlftgfk il logighlh ha fifuik ja`tli fk` kparlfgkàs fkjdg`lhls

y pkrfa`tlmas.

Tasuaivk amarfgfgks hk`ha lpigfk il jahgl jahgl`l y jkhl.

G`tarpratk orfls.

4>>GOAT Qtlti`


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64

Vgstajl ha ùjarlfg` jlyl


Q` rafkrrghk pkr ai ju`hk fk` iks sgstajls ha ùjarlfg`

Vgstajl ha ùjarlfg` jlyl, sjdkiks y raprasa`tlfg` ha fl`tghlhas a`

sgstajl jlyl y hafgjli

Juitgpigflfgkàs y pkta`fglfgkàs

@jarks y erlffgkàs a` Dldgik`gl

Astl sajl`l ikorlr=

Fk`kfar il asfrgturl ha iks `jarks a` hgeara`tas fuiturls.

Ulikrlr y rafk`kfar ai sgstajl ha ujarlfg` jlyl.

Iaar y asfrgdgr fl`tghlhas a` ùjarlfg` jlyl.

Fk`vartgr `jarks hai sgstajl hafgjli l ujarlfg` jlyl y vgfavarsl.

Taskivar fk` logighlh juitgpigflfgkàs y pkta`fgls.

4> > 1>> >>

Iks aogpfgkssa vlil` ha u` sgstajl ha ujarlfg` l dlsa ha markoiegfks3 as hafgr,

sjdkiks k egourls qua raprasa`tldl` fl`tghlhas.

< > >> ,>>> >,>>> >>.>>>

Il ùjarlfg` Fbg`ltljdg` sa dlsldl a` il asfrgturl markoiegfl. Hltl lprkxgjl-

hlja`ta hai lk > l.F.

5 ? 2 7 >< 4 6 8 1

Xaxtk tkjlhk y lhlptlhk ha www.ra`l.ahu.va

L trldlmlr!

Asfrgdl ai sjdkik qua utgigz flhl fgvgigzlfg` plrl raprasa`tlr ai jark hai sgstajl hafgjli qua sag`hgfl. Ouasa pkr ai amajpik.

sgstajl hafgjli fgvgigzlfg` sjdkik

6 Tkjl G G G

Aogptk

sgstajl hafgjli fgvgigzlfg` sjdkik

1> Tkjl

5 Fbg`l

4>4GOAT Qtlti`


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a` 4>

Il jltajtgfl jlyl sa dlsl a` u` sgstajl vgoasgjli. Astk qugara hafgr qua ils

u`ghlhas qua vl` ekrjl`hk ils hgeara`tas fgerls sa tkjl` a` orupks ha vag`ta.

Vldl qua ai `jark 4> tgaà u` sgo`gflhk juy prkeu`hk a` il fuiturl jlyl

y qua ast astrafblja`ta railfgk`lhk fk` il parsk`l bujl`l; ^rasta lta`fg`=

Il sujl ha tkhks iks hahks ha ils jl`ks ha iks oajaiks Mu`Lmpu' y

Gxd'liljca, brkas hai ^kpki _um, as 4>.

Ful`hk u`l jumar y u` bkjdra sa ldrlzl` plrl fralr vghl, il sujl ha

iks hahks ha sus jl`ks as 4>.

Vg juitgpigfljks ùastrls fultrk axtrajghlhas pkr iks fg`fk hahks qua

tgaà flhl u`l, kdtaàjks ai `jark 4>.

,

Ai pu`tk raprasa`tl ai u`k.


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Amarfgfgk ) ) 5 : 5) 2 :


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d. @jarks jlykras ha : + 1 x + 2 x < : 412

Ai vlikr ha u` `jark a` ai sgstajl ha ùjarlfg` jlyl hapa`ha ha su

pksgfg` vartgfli. Astls pksgfgkàs sgoua` ai krha` ha ldlmk blfgl lrrgdl. Iltldil sgouga`ta juastrl iks vlikras pksgfgk`lias blstl il qug`tl pksgfg`.

^rgjarl pksgfg`, ils u`ghlhasha > l l 677.

Xarfarl pksgfg`, ha 8>> l ?777.

Fulrtl pksgfg`, ha 2>>> l >> l 6 . Bloljks il pruadl fk` u`ks amajpiks.

Asfrgdljks ai `jark 41a` ùjarlfg` jlyl.

41 : 4> + 1(u`l vag`ta`l y 1 u`ghlhas).

< vag`ta`l, u` pu`tk a` il saou`hl pksgfg`.

1 u`ghlhas, u`l dlrrl a` il prgjarl pksgfg`.

Asfrgdljks ai `jark 15a` ùjarlfg` jlyl.

15 : 8> + >

8 4>6: 2>>>

6 4>4: 8>>

4 4>

< 4>>: plrl ilùjarlfg`.Ai sgstajl hafgjli,pkta`fgls ha dlsa .

4>1Jltajtgfl Vajl`l 64


217/290

8>> 4

4> ?

< &

8>> 4

4> ?

< :

4

8>> 715

2>>

> 4

4>

> 4

4> ?

, asfrgdgjks ai fkfga`ta ? a`

`ujarlfg` jlyl, a` asa `gvai.

Asfrgdgjks ai rasghuk > a` il

pksgfg` ha ils u`ghlhas.

8>> 4

4> ?

>>

8>> 6

4>

>

>

4>5GOAT Qtlti`


218/290

Amarfgfgk 4

Fk`vgartl iks jarks hai sgstajl hafgjli l ujarlfg` jlyl. Tafuarha il pksgfg` ha flhl fl`tghlh.

Xgaà u` amajpik.

>) 7>>

Hgvghl 7>> a`tra ai vlikr pksgfgk`li hai`gvai jxgjk pksgdia y asfrgdl ai fkfga`ta,

a` ùjarlfg` jlyl, a` asa `gvai.

Hgvghl ai rasghuk > a`tra ai vlikr hai

`gvai gèargkr y asfrgdl ai fkfga`ta, a` ù-

jarlfg` jlyl, a` asa `gvai.

Asfrgdl ai rasghuk > a` ai prgjar `gvai. > ::

Hgvghl 64> a`tra ai vlikr pksgfgk`li hai

jxgjk gvai pksgdia y asfrgdl ai fkfga`ta,


Asfrgdl ai rasghuk > a` ai prgjar `gvai.

4) 581

Hgvghl 581 a`tra ai vlikr pksgfgk`li hai



Hgvghl ai rasghuk 481 a`tra ai vlikr pksg-

fgk`li hai gvai gèargkr y asfrgdl ai fkfga`-

ta, a` ùjarlfg` jlyl, a` asa `gvai.

Asfrgdl ai rasghuk 1 a` ai prgjar `gvai.

6) 765

Hgvghl 765 a`tra ai vlikr pksgfgk`li hai



Hgvghl ai rasghuk 7>>

2>>

>

1

4> >

>

>

64>

765

581

?Jltajtgfl Vajl`l 64


219/290

Amarfgfgk 6

Fk`vgartl iks `jarks jlyls l sgstajl hafgjli. Ouasa pkr ai amajpik.

>)(

1

5

8>> x >

4> x 1 : >

< x 5 : 5

+ ?6>5

:

2

x :

x :

x :

+

1)/

7

:

x :

x :

x :

+

> x 4: 2>>

4> x 2: < x x 1: 4>>>

4> x >: >< x 4: 4

+ 4>>4

4>2GOAT Qtlti`


220/290


221/290

Lftgvghlh


222/290


L. Fk`vgartl iks `jarks hai sgstajl hafgjli l `ujarlfg` jlyl. Xga`a u` amajpik.

>) 1>> > 1>>8>>

>

1

4> >>

>

8) 6521 1) 267?

5) 41 657 ?) 51 678

4


223/290

D. Fk`vgartl iks `jarks jlyls l sgstajl hafgjli. Xga`a u` amajpik.

>) ?

"

7

8>> x ? : 42>>

4> x

< x 7 : 7

+ 6>87

8

4) >

%

:

6)

:

7

5) 4

8

$

?) 7

2

:

2)

82

%

7

7)

>:

$

2

4


224/290

L. Asfrgdl a` il i`al ai rasuitlhk ha il juitgpigflfg`.


225/290

Iks `jarks a` Dldgik`gl

Blfa jgias ha lks, iks bldgtl`tas ha Dldgik`gl haslrrkiilrk` u` sgstajl plrl fk`tlr, fk`sgharlhk

fkjk ai prgjar sgstajl ha ujarlfg` pksgfgk`li. As hafgr, ai vlikr ha u` hogtk plrtgfuilr ast hatar-jg`lhk pkr su pksgfg` a` ai `jark qua sa qugara raprasa`tlr.

Iks sjdkiks qua utgigzlrk` arl` juy plrafghks l iks qua juastrl ai sgouga`ta fulhrk.

< 4 6 8 1 5 ? 2 7 4>

^khajks asfrgdgr erlffgkàs fk` astks sjdkiks, fkjk vajks l fk`tgùlfg`. Ai ratk qua ia prkpk-

àjks as trlhufgr ils erlffgkàs dldgi`gfls li sgstajl hafgjli y rahufgr flhl erlffg` l su j`gjl

axprasg`. Ouasa pkr ai amajpik.

Jlrqua fk` u` fbaqua il flsgiil qua jamkr g`hgqua su ra`hgjga`tk.

ikorlhka`

prkfask`k

ikorlhk

Haspusha

astuhglr...

Fk`kzfk il asfrgturl ha iks `jarks a` hgeara`tas fuiturls.

Ulikrk y rafk`kzfk ai sgstajl ha ùjarlfg` jlyl.

Iak y asfrgdk fl`tghlhas a` ùjarlfg` jlyl.

Fk`vgartk jarks hai sgstajl hafgjli l ujarlfg` jlyl y vgfavarsl.

Tasuaivk fk` logighlh juitgpigflfgkàs y pkta`fgls.

Erlffg`

dldgik`gfl

Erlffg`

hafgjli

5

2: 6

8

4


226/290

66

Vujl y rastlfk` `jarks jlyls


Flia`hlrgk jlyl

Vujl y rastl fk` `jarks jlyls


Lfartgmks jltajtgfks

Astl sajl`l ikorlr=

Fk`kfar il krol`gzlfg` hai tgajpk a` ai flia`hlrgk jlyl.

Vujlr y rastlr fk` axlftgtuh jarks jlyls.

^rlftgflr il logighlh ha fifuik ja`tli fk` kparlfgk`as fkjdg`lhls.

Taskivar lfartgmks jltajtgfks l plrtgr ha u` fk`mu tk ha a`u`fglhks.

4


227/290

^lrl fkja`zlr!

Flia`hlrgk jlyl

fgfiks hurlfg` hurlfg` a` hls

< cg` < hl < hl

< ug`li < jas 4> hls

< tu` < lk 65> hls

< clt` 4> tuàs ?4>> hls< dlct` 8>> tuàs >> hls

< pgct` 2>>> tuàs 4 22> >>> hls

< clildt` >>> tuàs 1? 5>> >>> hls

Vldl ustah qua iks jlyls fralrk` vlrgks flia`hlrgks; Ualjks iks js

fk`kfghks.

Ai flia`hlrgk skilr k Blldha 651 hls, dlslhk a` ai jkvgjga`tk ha rktlfg` ha

il Xgarrl lirahahkr hai Vki. Ai flia`hlrgk slorlhk Xzkicg`ha 45> hls as ai qua

kdsarvljks a` il giustrlfg`.

Il Fua`tl ilrolas ai flia`hlrgk qua utgigzlrk` plrl iiavlr ai raogstrk hai tgajpk

a` ekrjl igàli. A` i astldiafgark` u` hl fark, qua sao` iks fga`tegfks

fkrraspk`ha li


228/290

< 8 868

4>

< 2

8>>1 87

4> 6 848

< 1%

8>> 7

4> 2

< :

4


229/290

Amarfgfgk

4

6

8

tras

fg`fk

hgaz

sustrla`hk y jg`ua`hk

jg`ua`hk y sustrla`hk

`k gjpkrtl ai krha`

:

444GOAT Qtlti`

Lutkfk`trki


234/290


L. Taligfa ils sujls fk` `jarks jlyls. Xga`a u` amajpik.

>) > >+

< > >

4>

75

+ 6) 2

!?%

+

4>

>

4>

>

4>

>

4>

) $

>

?

8

$??

< 8 5

4>

>

4>

) 7 x 4 + 2 :


237/290


a`prkfask

`kikorlhk

Haspusha

astuhglr...

Fk`kzfk il krol`gzlfg` hai tgajpk a` ai flia`hlrgk jlyl.

Vujk y rastk fk` axlftgtuh `jarks jlyls.

^rlftgfk il logighlh ha fifuik ja`tli fk` kparlfgkàs fkjdg`lhls.

Tasuaivk lfartgmks jltajtgfks l plrtgr ha u` fk`mu`tk ha aù`fglhks.

Iks lfartgmks jltajtgfks lftgvl` ùastrks prkfasks ja`tlias plrl hlr u`l fk`fiusg` l plrtgr ha u`

fk`mu`tk ha prajgsls k aù`fglhks.

Lpigqua su rlzk`ljga`tk iogfk y hatarjgà ai `jark qua fujpia fk` ils fk`hgfgkàs ha ils pgstls.^rlftgquajks iks lfartgmks raskivga`hk asta amajpik.

>) l. Ai `jark as ja`kr qua 11 f. Ai `jark as u` jitgpik ha 2

d. Ai `jark as jlykr qua 61 h. As hgvgsgdia a`tra 5

Ai `jark as=

Il prgjarl y saou`hl prajgsls g`hgfl` qua ai `jark ast a`tra 61 y 11.

Il tarfarl y fulrtl, iks jitgpiks a` asta rlòk sk` 8> y 82, park skik 82 as hgvgsgdia a`tra

5, pkr ik tl`tk ai `jark as 82.

> f. Ai `jark as u` jitgpik ha

d. Ai `jark as jlykr qua 2> h. Il sujl ha sus hogtks as y 8>> f. Flhl hogtk as gjplr

d. Ai `jark as hgvgsgdia a`tra 1 h. Hks ha iks hogtks sk` goulias

Ai `jark as=

8) l. Ai `jark as ja`kr qua 5> f. Il sujl ha iks hgogtks as gouli l ?

d. Ai `jark as jlykr qua 4> h. Ai `jark as u` jitgpik ha 8

Ai `jark as=

82

445GOAT Qtlti`

Tlzk`ljga`tk iogfk

Tavgsa su lpra`hgzlma


238/290

68

Taplsk= sajl`ls 45 l 66

Astl sajl`l ikorlr=

Taplslr iks fk`ta`ghks ha il sajl`l 45 l il 66.

Taprasa`tlr ai prkhuftk flrtasgl`k a` ekrjl aùjarltgvl, a` u`l tldil ha hkdia

a`trlhl y a` u` pil`k flrtasgl`k.

Astldiafar ai fk`mu`tk hkjg`gk y fkhkjg`gk a` railfgkàs y eu`fgkàs plrl raprasa`-

tlriks a` u` hglorljl slogtli.

Orlegflr eu`fgkàs igàlias a` ai pil`k flrtasgl`k.

Qtgigzlr il astlhstgfl plrl krol`gzlr hltks, g`tarpratlr gèkrjlfg` a` oregfls ykdtaàr jahghls ha ta`ha`fgl fa`trli.

Fk`vartgr `jarks hai sgstajl hafgjli l ùjarlfg` jlyl y vgfavarsl.

Vujlr y rastlr `jarks jlyls.

44?Jltajtgfl Vajl`l 68


239/290

\uarghl y quarghk astuhgl`ta=

Va lprkxgjl il pruadl `li y hada praplrlrsa lhafulhlja`ta, raplsl`hk iks fk`ta`ghks ha ils

sajl`ls 45 l il 66.

^lrl lprkvafblr asta raplsk ia rafkja`hljks=

Blol u` pil` ha ik qua astuhglr flhl hl y trlta ha fujpigrik. Hahgqua js tgajpk l

iks tajls qua ia rasuita` hgefgias.

Dusqua u` iuolr trl`qugik, giujg`lhk y sgia`fgksk plrl astuhglr.

Ial iks rasjaàs ha flhl sajl`l y asfrgdl ils ghals js gjpkrtl`tas a` su fulhar`k.

Asfufba il filsa rlhgli. Vus jlastrks ikfutkras ia lfkjpllr` a` asta raplsk y ia lyu-

hlr` l raskivar liou`ks amarfgfgks.

Fkjpruada qua blyl raligzlhk dga` iks lutkfk`trkias. Vg tgaà huhls, vuaivl l iaar ilssajl`ls, lb a`fk`trlr axpigflfgkàs y amajpiks.

Fjk sar il pruadl ha avliulfg`;

Il pruadl plrfgli avlil iks jgsjks fk`ta`ghks y ha il jgsjl ekrjl a` qua iks bl trldlmlhk

sajl`l l sajl`l.

A` il pruadl a`fk`trlr=

Q`l sarga ha fifuik ja`tli plrl jahgr su hastrazl y rlpghaz ha fifuik a` u` tgajpkijgta ha tras jgùtks.

Hgeara`tas amarfgfgks qua avlil` ik lpra`hghk a` ils kfbk sajl`ls. Astks amarfgfgks

sar` sajaml`tas l iks qua ustah raskivg a` ils lftgvghlhas hai lutkfk`trki. Va ia pahgr=

raspk`har praou`tls,

raiia`lr ai frfuik ha il kpfg` fkrraftl,

raskivar kparlfgkàs y

raskivar prkdiajls.

Ful`hk rasuaivl amarfgfgks y prkdiajls, hada hamlr asfrgtk ai prkfahgjga`tk.

Juy gjpkrtl`ta=flhl sarga fk`tgaà g`struffgkàs axlftls ha ik qua hada raligzlr a`

flhl lplrtlhk, ls fkjk il vlikrlfg` lsgo`lhl.

Vg ustah sa praplrl fk` tgajpk y hahgflfg`, ai rasuitlhk sar sltgselftkrgk.

442GOAT Qtlti`


240/290

^lras krha`lhks y prkhuftk flrtasgl`k


241/290

4) Fk` iks jgsjks fk`mu`tks L : { l, d, f } y D : { , , } hai amarfgfgk l`targkr. Flifuia aiprkhuftk flrtasgl`kD x L.

( , )

( , )

( , )

ll

( , )

( , )

( , )

( , )

( , )

( , )

Asfrgdl ai fk`mu`tk prkhuftk flrtasgl`k, tkjl`hk tkhls ils plramls qua kdtuvk.

D x L:

D. Hlhks iks fk`mu`tks L : { l, d, f } y D : { 4, 8, 5 }, bliia ai prkhuftk flrtasgl`k L x D a` u`l tldilha hkdia a`trlhl.

4

l (l, 4)

L D

Amarfgfgk 4Hlhks iks fk`mu`tks L : { 4, 8, 5 } y D : { 6, 1, ? }.

GOAT Qtlti`


242/290



243/290

Amarfgfgk 6

Taprasa`ta ils railfgk`as y eu`fgk`as sgouga`tas a` il ekrjl g`hgflhl.

L. Hlhks iks fk`mu`tks L : { prgjk, plr, gjplr, rlfgk`li } y D : { 5,


244/290

F. Fk`sghara ai fk`mu`tkx: { ,


245/290

Eu`fgk`as ig`alias


246/290


247/290

Astlhstgfl G


248/290

Amarfgfgk 1

L. Hatarjg`a pkdilfg`, juastrl, hltk y vlrgldia a` iks a`u`fglhks sgouga`tas. Emasa a` ai amajpik.

>) A` ai fursk ha fga`fgls `lturlias, u` orupk ha astuhgl`tas flifui qua il tajparlturl j`gjl

prkjahgk a` ldrgi ha 4> dads.



4) Ai G`stgtutk @lfgk`li ha Astlhstgfl raogstr a` 4>>5 ai `jark ha bgmks ha > eljgigls oul-

tajlitafls. Hatarjg` qua ai prkjahgk ha bgmks pkr eljgigl as 6.



6) Q` fljpasg`k saiaffgk` ?1 tkjltas hai tktli ha su fksafbl plrl hatarjg`lr ai pask prkja-

hgk ha flhl tkjlta. Ai rasuitlhk eua ha >.4 igdrls.



8) Va flifui qua il ahlh prkjahgk ha iks astuhgl`tas hai orupk Qtlti`, a` ai fa`trk ha krga`-

tlfg` @uavk Bkrgzk`ta, as ha astuhgl`tas.



ai jas ha ldrgi hls

tajparlturl ? F

46?Jltajtgfl Vajl`l 68


249/290

D. Aildkra u`l tldil ha erafua`fgls fk` iks hltks qua sa prasa`tl` a` flhl aù`fglhk y fk`tasta ilspraou`tls.

?6

51 ?> 55 55

52 ?4 ?4 52

pask a` id erafua`fgl51

55

52

?>

?4

?6

tktli (@)

Vaoìl Krol`gzlfg` Ju`hgli ha il Vliuh, ai pask ghali plrl `gks ha lks as ?> id.

Fu`tks `gks ast` pkr hadlmk hai pask ghali;

Fu`tks `gks tgaà` ai pask ghali;

Fu`tks `gks ast` pkr a`fgjl hai pask ghali;

4) Q`l prkjktkrl ha sliuh raogstr il astlturl ha 44 `gks ha 4 lks ha ahlh. Iks hltks a`fa`tjatrks euark`=

26 21 2? 26

22 21 22 27

28 25 25 28

28 21 21 28

22 2? 25 2?

25 21

astlturl a` fj erafua`fgl

26

28

21

25

2?

22

27

tktli (@)

\u astlturl prasa`tl il jlykr erafua`fgl;

Il astlturl ghali a` `gks ha 4 lks as ha 2? fj. Ha lfuarhk l iks hltks, qu puahahahufgr ha il astlturl qua prasa`tl il jlykr erafua`fgl;

462GOAT Qtlti`


250/290

Astlhstgfl GG

>4>>

6>>

8>>

1>>

5>>

?>>

2>>

7>>

>>



d. ^kiok`k ha erafua`fgls lskfgl pkr jahgk ha pu`tks y raftls iks hltks `kjg`lias fk` sufkrraspk`hga`ta fl`tghlh k erafua`fgl.


>

>

4>>

6>>

8>>

1>>

5>>

?>>

2>>

7>>

>>


f. Hglorljl ha saftkrasraprasa`tl hltks a` u`l fgrfu`eara`fgl. Astl orfl as juy tgi ful`hksa fua`tl fk` vlikras ldskiutks y sa hasal raprasa`tlr il g`ekrjlfg` a` pkrfa`tlmas.


ligja`tlfg`62%

vastulrgk44%

vgvga`hl48%

lbkrrk


251/290

Amarfgfgk 5

Aildkra il orfl g`hgflhl fk` iks hltks qua sa prkpkrfgk`l` a` flhl aù`fglhk.

>8. Aildkra u` hglorljl ha dlrrls fk` astl gèkrjlfg`.

flusl ha jkrtlighlh pkrfa`tlma %

àujk`l

8>

1>

5>

Flusls ha jkrtlighlh a` gks ja`krasha < lk, 4>>8

Taeiaxgkà. Ils gèaffgkàs raspgrltkrgls (àujk`l y drk`fkàujk`l) sk` ils prg`fgpliasflusls ha jkrtlighlh a` ùastrks `gks y `gls. Vg u` `gk prasa`tl s`tkjls ha astls aèar-jahlhas, qu hafgsg` hada tkjlr;

Ktrl flusl gjpkrtl`ta as ai s`hrkja hglrragfk, qu bdgtks bgog`gfks puaha` prava`gr astlaèarjahlh;

4) Vao` iks KHJ, plrl arrlhgflr ai bljdra a` iks ja`kras ha 1 lks, bldrl qua rahufgr ai`hgfa ha hasùtrgfg` hai 68% qua bldl a`


252/290

6) Ai kdmatgvk ? ha iks KHJas olrl`tgzlr il sksta`gdgighlh hai jahgk ljdga`ta.A` 4>>5, ai Fk`samk @lfgk`li ha rals ^rktaoghls (FK@L^) prasa`t il sgouga`ta gèkrjlfg`skdra il suparegfga hai tarrgtkrgk oultajlitafk dlmk prktaffg` plrl fk`sarvlr il hgvarsghlh

dgkiogfl. Aildkra u` pkiok`k ha erafua`fgls fk` astks hltks. Ia lyuhljks fk` ai prgjark.

lk suparfga(a` jgiikàs ha baftrals)

4.5

4>>4 6.4

4>>1 6.8

Xarrgtkrgk oultajlitafk dlmk prktaffg`plrl fk`sarvlr il hgvarsghlh dgkiogfl

>

4>>14>>>

Hastg`k hai loul a` ai ju`hk

^lsas

haslrrkiilhks

^lsas a`

trl`sgfg`

^lsas a`

haslrrkiik

lorgfuiturl

51%

orlhks %

lorgfuiturl51 x 65>

>:

46 8>>

>:468

g`hustrgl : :

hkjstgfk : :

48) 2?1 Hgvghl 2?1 a`tra ai vlikr pksgfgk`li hai gvai jxg-

jk pksgdia y asfrgdl ai rasuitlhk a` asa `gvai.

Hgvghl ai rasghuk ?1 a`tra ai vlikr pksgfgk`li hai`gvai g`jahgltk gèargkr y asfrgdl ai rasuitlhk a`

asa `gvai.

Asfrgdl ai rasghuk

4

6

%

48>> 2?1

2>>

?1

6

4> ?1

5>


257/290

6) > 8) 6215

F. Fk`vgartl iks `jarks jlyls a` `jarks hai sgstajl hafgjli. Xga`a u` amajpik.

>) ?

)

8

8>> x ? : 42>>

4> x

< x 8 : 8

+ 6


258/290

Vujl y rastl fk` `jarks jlyls

< " #

4> as u` amajpik.

>) ! 2+

< ! 2 )

4>

) " ?

< $ 2 5

4>

>

4>

) -


266/290

Vajl`l 4) 5( - 4 + 6) : (5 =) - (5 =4) + (5 =6) : 5> -


267/290

4) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

d d d

d d d d d d d

d d d d d

d d d d d

d

6 6 7

6 7 6 6 6 6 7

6 7 6 7 4?

6 6 7 7 4?

4?

4

4 4

6 4 4

6 4 4

6

= = = = = =

- + +

: + + + - + - + -

: + + - - -

: + - + - -

: -

5 5N N

(d- 6)(d4

+ 6d+ 7) : d

6

- 4?

6) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

l d d

l d l d l d d

ld ld l d d

6 8 4 1

6 4 6 6 1 8 4 8 8 1

5 6

4

4 4

4 4

= = = = = =

- + +

: + + + - + - + -

: + + - - -

5 5N N

:

(6l- 8)(4d4+ d+ 1) :

8) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

x x xy y

x x x xy x y x xy y

x x y xy x xy y

x x x y xy xy y

4 4 1

4 1 4 4 4 4 1

4 1 4 8

4 4 1 8

4 4

4 4 4 4

6 4 4 4 4

6 4 4 4 4

= = = = = =

- + +

: + + + - + - + -

: + + - - -

: - + + - -

5 5N N

(x- 4)(x4+ 4xy+ 1y4) : x6- 4x4+ 4x4y + 1xy4- 8xy - y4

D.


268/290

Vajl`l 44L trldlmlr!) (8x + 4)4: (8x)4+ 4(8x = 4) + (4)4:


269/290

Vajl`l 48L trldlmlr!>) -4: x - 4?) '(x - 6)

< > > -4?6 +6 +7 4? < +6 +7 >

x4+ 6x + 7,

T : >


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4) (x4+x- 4>) '(x+ 1) :

< < -4>-1 -1 4> < -8 >

x- 8, T :>

6) (4j4- 7j- -7-6 -6 +7 < -6 >

y- 6, T :>

5) (8y6+y+ 8) '(y+ 4) :

: (8y6+ >y4-y+ 8) '(y+ 4)

8 > -< +8-4 -2 + 8 -2 + > +2-4 -4 +8 -2 < -4 +8 >

z4- 4z+ 8, T :>



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8):

( ) ( )l

d x46

4 4 8

6

4 8 8

62 8

6


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4) (8x)(< + 4>x- 1y) : 8x+ 2>x4- 4>xy :

2>x4+ 8x - 4>xy

6) (4l)(d- ld- 48l4:

-48l4 + 4>ld

8) (d)(l- 65ld+ 8df) : ld- 65ld4+ 8d4f

1) (4j8)(8j4`6-6j`- ) 1x4+ 4xy -8z

8x

4>x6+ 2x4y-


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2) 4l4- l+ 58l4- 6

-5l4+ 6l-


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Amarfgfgk ) (x6- 2) '(x - 4)

< > > -2+4 4 8 2 < +4 +8 >

x4+ 4x+ 8, T: >

-1 -1 +4> < -8 >

x- 8, T: >

6) (4j4- 7j- +5 -8-8 -2 64 - -7 +< +6-6 -6 +7 > -6

< -6 > +< >

x6- 6x4+

7) (-j6

- 5j

4

+ 4j- 6)'

(j-


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Vajl`l 45L trldlmlr!

D

L

jl`ls l`gsgiiks orl`kil

eraslerasl yjl`ls

erasl yl`gsgiiks

erasl yorl`kil

vlg`giilvlg`giil yjl`ls

vlg`giil yl`gsgiiks

vlg`giil yorl`kil

fbkfkiltafbkfkiltay jl`ls

fbkfkilta yl`gsgiiks

fbkfkilta yorl`kil

Amarfgfgk < (l, ) (l, ) (l, )

l

(d, ) (d, ) (d, )

d

F x H : { (l, ), (l, ), (l, ), (d, ), (d, ), (d, ) }

Amarfgfgk 4


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Vajl`l 42L trldlmlr!

51

86

4 pkiiks

Ulrgldia= fl`tghlh ha fk`fa`trlhk Hltk= >.6 igdrls

d. ûaha sar= pask, fkikr ha ils piujls, aspafga...

Amarfgfgk 4Ltgti` 4>Fufbujltlàs 41ûartk ha Vl` Mks

Amarfgfgk 8


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Vajl`l 6>Amarfgfgk

6>

8>

1>

5>

?>

^lsas

haslrrkiilhks

^lsas a`

trl`sgfg`

^lsas a`

haslrrkiik

^krfa`tlma ha g`fraja`tk a` ilshk`lfgkàs ha slòra

Amarfgfgk 6L.

ùtrga`tas %

prkta`ls 1

orlsls 4

lzflr 8

>

61

8>

81

1>

Amarfgfgk 8

pkrfa`tlmas

Ljrgfl Iltg`l=

6>> x >>>:76.4 76

16.6 x 65>>

: 2>>

:

: ?15>>

: ?8.1 ?1

raog` erafua`fgl(a` jgiikàs) % orlhks

Ljrgfl Iltg`l y Ai Flrgda 6>> 41.7% 76

Lsgl, ^lffk y Krga`ta 5


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>) 65

5

4 6 4 6

5

65 : 4 x 4 x 6 x 6

tk`ailhls.

Amarfgfgk 4^ksgfg` : @ + >

4: >

Ja : >

T/ Il jahgl`l a` ai pask ha iks rafgpga`tas hafrajl as > orljks.

6) ^ksgfg` : @ +

>

&

:

4) > 581

8>>481

481 4>

81 8>

1

> 8 2 "

4> > : > !

< & $ >

Amarfgfgk )

4> ) " ?

< $ 2 5

2 4 5

< > 1 1

4) 4> 6 "

< 1 : 1

6) 4> $ < #

< 2 4

8)

4> " ? 1

< & 1 !

1) 8>> 7 5 6

4> 8 "

< " : "

Logighlh ha fifuikja`tliL.4) 426) 4?

8) 841) 82

5) 65 ?) 81 2) 48

7) 87) 18


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Dgdigkorlel

LIJLOQAT, O. y DL]LIHL, M.JltajtgflsG. Igjusl, Jxgfk, >2.

DLIHKT, L.ioadrl. Orupk ^ltrgl Fuiturli, Jxgfk, 4>>2.

DAIIK, G.ioadrl. Xbkjsk`, Jxgfk, 4>>1.

FLDTATL, A. Ai flia`hlrgk jlyl, su krgoa` y ikskel. Igol Jlyl, Fkstl Tgfl, >>.

HA ^L], J. Flia`hlrgk jlyl, ai fljg`k g``gtk hai tgajpk. Orl` mloulr 4.

GOAT. Fgjga`tks 4, Jltajtgfl.GOAT, Oultajlil, 4>>1. Xkjk 4.

GOAT. Jltajtgfl ?, \ugrgou. GOAT, Oultajlil, 4>>?. Xkjks < y 4

GOAT. Jltajtgfl ?, \ugrgou. GOAT, Oultajlil, 4>. Xkjks < y 4.

GOAT. Jltajtgfl 2, Qtlti`. GOAT, Oultajlil, 4>>?. Xkjks < y 4.

CIAA, K.Astlhstgfl. Cljlr, Oultajlil, 4>>2.

JKTA@K, M.ioadrl. JfOrlw Bgii, Jxgfk, 4>>4.

VQIIGUL@, J.ioadrl y Xrgok`kjatrl. ^alrsk`, Jxgfk, 4>>5.

ULTGKV LQXKTAV.Liel fk` ast`hlras ?. @krjl, Dkokt, 4>>5.

ULTGKV LQXKTAV.Liel fk` ast`hlras 2. @krjl, Dkokt, 4>>5.

ULTGKV LQXKTAV.Liel fk` ast`hlras 7. @krjl, Dkokt, 4>>5

ULTGKV LQXKTAV. Jltajtgfl ?. Vl`tgiil`l, Fkikjdgl, 4>>?.

ULTGKV LQXKTAV. Jltajtgfl 2. Vl`tgiil`l, Fkikjdgl, 4>>?.

ULTGKV LQXKTAV. Jltajtgfl 7. Vl`tgiil`l, Fkikjdgl, 4>>?.

^og`ls _ad fk`suitlhls=hasflrtas.f`gfa.jaf.as

a`fgfikpahgl.us.as

bttp=//ahulrhkkfbkl.fkj

bttp=//aifa`trk.u`gl`has.ahu.fk

4??Dgdigkorlel Jltajtgfl


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bttp=//as.wgcgpahgl.kro

bttp=//orupks.ajlogstar.fkj

bttp=//muaokshaikogfl.àt

bttp=//prkeadiko.as/diko

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bttp=//sfbkiilrgs.fkj.jxbttp=//wad.ahuflstur.prg`flst.as

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bttp=//www.tlralsyl.fkj.jx

bttp=//www.u`gfae.kro.ot

bttps=//www.fkhaifkahufl.fi

ravgstlarj.u`gvliia.ahu.fk

www.luilelfgi.fkj

www.fagdli.ahu.uy

www.hgserutlilsjltajltgfls.fkj

www.ahufl.jlhrgh.kro

www.esgfl.usk`.jx

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www.gta.ahuflfgk`.as

www.jltajltgflplrltkhks.fkj

www.rla.as

www.vgtutkr.fkj

4?2GOAT Qtlti`


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Astuhglr

utatlán matemática 2º sem

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