Post on 27-Feb-2018
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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
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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 axprasgk`as lioadrlgfls sao` ai `jark ha trjg`ks.
Gha`tgefk ai orlhk railtgvk ha u` pkig`kjgk.
Asfrgdk axprasgk`as lioadrlgfls a` krha` lieldtgfk, hasfa`ha`ta ylsfa`ha`ta.
Flifuik ai vlikr `ujrgfk ha u`l axprasg` lioadrlgfl.
^rlftgfk ai fifuik ja`tli.
Haslrrkiia `uavls bldgighlhas
Axprasajks a` ekrjl ha pkig`kjgks!
Vldajks qua u`l iatrl a` ioadrl raprasa`tl fuliqugar `jark. ^kr ik tl`tk,il ik`ogtuh 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 raftl`ouilr
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 ia`oulma fkj` lia`oulma 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 ia`oulma fkj` l ia`oulma lioadrlgfk.
47Jltajtgfl Vajl`l 4>
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^lrl fkja`zlr!
^lrl ajpazlr `uastrk astuhgk astl sajl`l fk`vga`a 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 `aoltgvk.
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. Xga`a u` amajpik plrl flhl flsk.
L. Tasuaivl ils juitgpigflfgk`as ha `jarks a`tarks ik js rpghk qua puahl. Xa`ol prasa`ta il iay
ha sgo`ks. Xga`a u` amajpik.
>) 8 x 8 :
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Jlrqua fk` u` fbaqua il flsgiil qua jamkr g`hgqua su ra`hgjga`tk.ikorlhk
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 pk`a l pruadl tkhls `uastrls bldgig -
hlhas y hastrazls plrl a`fk`trlr il jamkr skiufg`. Ial ai flsk sgouga`ta, raligfa iks prkfahgjga`tks
qua fk`sghara `afaslrgks y prkpk`ol il jamkr skiufg`.
Ai flsk hai plstkr, il fldrl, ai ikdk y il iafbuol
Q` plstkr tga`a qua plslr u` ikdk, u`l fldrl y u`l iafbuol l il ktrl krgiil ha u` rk. Hgspk`a 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 jl`arl 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
\u a`fk`trlr astl sajl`l;
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`vga`a 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
`aoltgvk.
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 axpk`a`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 hgvgsgk`as.
Fkjpiatk safua`fgls iogfls.
Haslrrkiia `uavls bldgighlhas
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 kdta`ar 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 i`al.
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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
\u a`fk`trlr astl sajl`l;
Aigml su ja`
^lr krha`lhk y prkhuftk flrtasgl`k
Kparlfgk`as 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`lfgk`as a` lftgvghlhas fktghgl`ls.
Taligzlr plras krha`lhks.
Taprasa`tlr u` prkhuftk flrtasgl`k a` ekrjl a`ujarltgvl, tldil ha hkdia a`trlhl y
a` u` pil`k flrtasgl`k.
^rlftgflr ai fifuik ja`tli fk` kparlfgk`as 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`lfgk`as 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. Xga`a 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
flr`a lslhl flr`a lslhly lrrkz flr`a lslhly a`slilhl flr`a 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 kpfgk`as ha lijuarzk l aiaogr. Ual-
jks fui as ai ja` hai hl.
Fu`tls kpfgk`as hgstg`tls ha lijuarzk bly;
Fk` il lyuhl ha il tldil blrajks tkhls ils fkjdg`lfgk`as 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=
flr`a lslhl
pkiik ergtk
flihk ha ras
^uaha 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
a`ujarltgvl, a` u`l tldil ha hkdia a`trlhl y a` u` pil`k flrtasgl`k. Ualjks.
l. Ekrjl a`ujarltgvl
Tafuarha qua a`ujarlr as fk`tlr, blfar u`l igstl. ^kr ik tl`tk, raprasa`tlr ai
prkhuftk flrtasgl`k a` ekrjl a`ujarltgvl, 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 kdta`ar haiprkhuftk flrtasgl`k,juitgpigfljks a`tras ai `jark haaiaja`tks ha flhlfk`mu`tk.A` ai amajpikL x D : 6 x 6 : 7Hadajks kdta`ar 7plras krha`lhks.
Amarfgfgk ) hk`ha sa u`a` 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?
Tailfgk`as y eu`fgk`as
\u a`fk`trlr astl sajl`l;
Kslhl blstl il fujdra!
Tailfgk`as y eu`fgk`as
Uliulr eu`fgk`as
Hglorljl slogtli ha u`l railfg` y u`l eu`fg`
Astl sajl`l ikorlr=
Lpigflr railfgk`as y eu`fgk`as 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`fgk`as.
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-
fl`as= Ha Loul, Lflta`l`ok, Xlfl` y Xlmujuifk.
Vudgr` hks vkifl`as 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`lfgk`as 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 gjoa`as 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 legrjlfgk`as, 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 astlfgk`ajks 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
\u a`fk`trlr astl sajl`l;
Bgstkrgl ha il astlhstgfl
Astlhstgfl y trjg`ks astlhstgfks
Kparlfgk`as fkjdg`lhls
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`tlfgk`as 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 g`ekrjlfg` 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` g`ekrjlfg` ha sus raogstrks astlhstgfks a` Orafgl, Tkjl
y Fbg`l.
Wl a` `uastrl arl Ougiiarjk G, ray ha G`oiltarrl, a` 25 a`flro u` fa`sk plrl
raogstrlr `lfgjga`tks y haeu`fgk`as. 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 l`igsgs 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.prkeaskra`ig`al.fi/jltajltgfl/astlhgstgflBgstkrgl.btj
L trldlmlr!
Fk` il g`ekrjlfg` ha il iafturl fkjpiata il i`al ha tgajpk. Asfrgdl ai `kjdra ha il fgvgigzlfg` kparsk`l qua lpigf il astlhstgfl. Xga`a 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=
Il vlrgldia as= Ai hltk 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 fkju`as.
^kr amajpik
A` u`l astl ha il fkju`ghlh plrtgfgplrk` parsk`ls (@ : ). Hai tktli
41 arl` `gks, 8> mva`as, 11 lhuitks y 4> l`fgl`ks. ^khajks raprasa`tlr astlg`ekrjlfg` a` il tldil ha erafua`fgls sgouga`ta.
plrtgfgpl tas erafua`fgl
`gks 41
mva`as 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-
jltl`as 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 g`ekrjlfg` prkpkrfgk`lhl.
Il ajprasl ha loul purl Haigrrgflhasal sldar fu`tks oultajlitafks fk`suja` su prkhuftk a`
tkhk ai pls y qu kpg`g` tga`a` hai loul purl. ^lrl kdta`ar astl g`ekrjlfg`, 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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Jlrqua fk` u` fbaqua il flsgiil qua jamkr g`hgqua su ra`hgjga`tk.ikorlhk
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` kparlfgk`as fkjdg`lhls.
Tasuaivk amarfgfgks ha louhazl vgsuli.
Kdsarva fk` lta`fg` il ekrjl y ai tljlk ha ils ourls paquals, iuaok hatarjg`a fu`tls ourls
ha astls `afasgtlrl plrl ekrjlr il ourl js orl`ha. Ai amajpik > ia sgrva ha oul.
>)
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6>
Astlhstgfl GG
\u a`fk`trlr astl sajl`l;
Astlhstgfl a` il fkjputlhkrl
Oregfls astlhstgfls= hglorljl ha dlrrls, pkiok`k ha erafua`fgls y hglorljl ha
saftkras
Kparlfgk`as fkjdg`lhls y pkrfa`tlmas
L`igsgs 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` kparlfgk`as 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 hgspk`a 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 jl`arl 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 g`oraslhks 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 kpfgk`as ha oregfls qua fk`tga`a ai prkorljl.
8. Asfkml ai tgpk ha oregfl qua hasaa, blol figf y lplrafar il oregfl ha il
g`ekrjlfg` qua g`oras.
Vg hasal ktrk tgpk ha oregfl, fkikqua `uavlja`ta ai pu`tark skdra il plildrl
prkhuftks y saiaffgk`a il kpfg` qua hasaa. ^k`ol 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 a`il 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 sl`ora!
Ils trl`seusgk`as ha sl`ora slivl` vghls. Vkik il hk`lfg` raouilr ha vkiu`tlrgks puaha olrl`tgzlrai sujg`gstrk ha sl`ora saourl. Il Krol`gzlfg` Ju`hgli ha il Vliuh prasa`t a` ai g`ekrja ha 4>>?asta g`fraja`tk a` ils hk`lfgk`as.
Aildkra u` hglorljl ha dlrrls fk` il g`ekrjlfg` 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`lfgk`as % 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`lfgk`as ha sl`ora
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d. ^kiok`k ha erafua`fgls
Ai pkiok`k ha erafua`fgls as u` orfk qua raprasa`tl il g`ekrjlfg` 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 i`als raftls.
Ualjks u` amajpik.
Q`k ha iks Kdmatgvks hai Jgia`gk (KHJ), a` fk`fratk ai Kdmatgvk 1, prkpk`a ilrahuffg` ha il jkrtlighlh jltar`l. Ai g`ekrja hai Vgstajl ha ils @lfgk`asQ`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
^k`ol a` prftgfl il raprasa`tlfg` orfl a` hglorljl ha dlrrls y pkiok`k ha erafua`fgls.
L. Q`l tlzl ha ykour fk`tga`a 1 % ha prkta`ls, 4% ha orlsls y 8% ha lzflr. Asfrgdl iks hltks a`il tldil y hgduma u` hglorljl ha dlrrls fk` il g`ekrjlfg`.
`utrga`tas %
prkta`ls
orlsls
lzflr
^krfa`tlma ha `utrga`tas qua fk`tga`au`l tlzl ha ykour
>
4, sao` ai g`ekrja hai Vgstajl ha ils @lfgk`as Q`ghls a`Oultajlil, prasa`tlhk a` 4>>2.
Aildkra ai pkiok`k ha erafua`fgls.
lk % has`utrgfg`
61
8>
81
1>
Il orfl `ks g`hgfl qua sa bl rahufghk il has`utrgfg` g`el`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 g`ekrjlfg` a` pkrfa`tlmas. Ualjks u` amajpik.
Iks tras ghgkjls js bldilhks a` ai ju`hk sk`=
ghgkjlerafua`fgl
(jgiik`as ha bldil`tas)
fbg`k jl`hlr` 7>>
asplki 65>
g`ois 68>
tktli >
Fkjk il g`ekrjlfg` 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%
g`ois=68> x >
>:
68 >>>
>: 4, quaaqugvlia li >%. lrl raplrtgr iks orlhks ha jl`arl 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 quakdta`ajks u`pkrfa`tlma pkr jahgkha u`l raoil ha trassgjpia, ls=
>% 1>> x 441
x:441 x >
1>>: 81%
Lpra`hl u` sjdkik
`uavk, 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.
Olstks raligzlhks pkr Ear`l`hk a` u` jas
>
>
4>>
6>>
8>>
1>>
5>>
?>>
2>>
7>>
>>
lbkrrkvgvga`hlvastulrgktrl`spkrtaligja`tlfg`
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.
Olstks raligzlhks pkr Ear`l`hk a` u` jas
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 g`ekrjlfg` ha il tldil skdra iks pkrfa`tlmas ha utrga`tasqua `afasgtl u` lhuitk sl`k. Il g`ekrjlfg` yl vga`a a` pkrfa`tlmas, skik hada flifuilr iks orlhksfkrraspk`hga`tas l flhl u`k. Xga`a u` amajpik.
^lrl qua `uastrk fuarpk rafgdl tkhks iks `utrga`tas afaslrgks hadajks fkjar ligja`tks ha tkhks
iks orupks y a` ils fl`tghlhas lhafulhls.
orlsls41%
^krfa`tlmas ha `utrga`tas qua`afasgtl u` lhuitk sl`k
`utrga`tas erafua`fgl (%)
orlsls 41
prkta`ls
orlsls41% x 65>
>%:7> flrdkbghrltks :
prkta`ls : egdrl :
F. Ial il g`ekrjlfg` 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 fligegflfgk`as 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 tga`a ai jamkr GHB;
d. Fui as il hgeara`fgl ha GHBa`tra Fbgia y Oultajlil;
f. Asfrgdl hks lffgk`as 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.
Haslrrkiia `uavls bldgighlhas
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 kparlfgk`as y asfrgdl su raspuastl skdra il i`al. Bolik ik js rpghk
qua puahl.
L. Taligfa ils kparlfgk`as fkjdg`lhls, rafuarha il marlrqul ha ils kparlfgk`as. Xga`a u` amajpik.
>) 1 + (4 x 8) :
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L. Lpigqua ik lpra`hghk y rasuaivl iks prkdiajls qua sa pil`tal`.
> jgiik`as ha lhuitks fk` skdrapask y li ja`ks 8>> jgiik`as ha kdasks.A` iks itgjks lks ai skdrapask y il kdasghlh luja`tl` ha jl`arl 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 `afasghlhas 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.
Jlrqua fk` u` fbaqua il flsgiil qua jamkr g`hgqua su ra`hgjga`tk.ikorlhk
a`prkfask
`kikorlhk
Haspushaastuhglr... Axtrlgok g`ekrjlfg` ha u`l orfl.
Kdta`ok il jahgl, il jahgl`l y il jkhl l plrtgr ha u`l hgstrgdufg`
ha hltks.
^rlftgfk il logighlh ha fifuik ja`tli fk` kparlfgk`as 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 `ujarlfg` jlyl
\u a`fk`trlr astl sajl`l;
Q` rafkrrghk pkr ai ju`hk fk` iks sgstajls ha `ujarlfg`
Vgstajl ha `ujarlfg` jlyl, sjdkiks y raprasa`tlfg` ha fl`tghlhas a`
sgstajl jlyl y hafgjli
Juitgpigflfgk`as y pkta`fglfgk`as
@jarks y erlffgk`as 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` `ujarlfg` jlyl.
Fk`vartgr `jarks hai sgstajl hafgjli l ujarlfg` jlyl y vgfavarsl.
Taskivar fk` logighlh juitgpigflfgk`as 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 `ujarlfg` 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> tga`a 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 `uastrls fultrk axtrajghlhas pkr iks fg`fk hahks qua
tga`a flhl u`l, kdta`ajks 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 `ujarlfg` jlyl hapa`ha ha su
pksgfg` vartgfli. Astls pksgfgk`as 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` `ujarlfg` 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` `ujarlfg` jlyl.
15 : 8> + >
8 4>6: 2>>>
6 4>4: 8>>
4 4>
< 4>>: plrl il`ujarlfg`.Ai sgstajl hafgjli,pkta`fgls ha dlsa .
4>1Jltajtgfl Vajl`l 64
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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`
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Amarfgfgk 4
Fk`vgartl iks jarks hai sgstajl hafgjli l ujarlfg` jlyl. Tafuarha il pksgfg` ha flhl fl`tghlh.
Xga`a u` amajpik.
>) 7>>
Hgvghl 7>> a`tra ai vlikr pksgfgk`li hai`gvai jxgjk pksgdia y asfrgdl ai fkfga`ta,
a` `ujarlfg` jlyl, a` asa `gvai.
Hgvghl ai rasghuk > a`tra ai vlikr hai
`gvai g`eargkr y asfrgdl ai fkfga`ta, a` `u-
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,
a` `ujarlfg` jlyl, a` asa `gvai.
Asfrgdl ai rasghuk > a` ai prgjar `gvai.
4) 581
Hgvghl 581 a`tra ai vlikr pksgfgk`li hai
jxgjk gvai pksgdia y asfrgdl ai fkfga`ta,
a` `ujarlfg` jlyl, a` asa `gvai.
Hgvghl ai rasghuk 481 a`tra ai vlikr pksg-
fgk`li hai gvai g`eargkr y asfrgdl ai fkfga`-
ta, a` `ujarlfg` jlyl, a` asa `gvai.
Asfrgdl ai rasghuk 1 a` ai prgjar `gvai.
6) 765
Hgvghl 765 a`tra ai vlikr pksgfgk`li hai
jxgjk gvai pksgdia y asfrgdl ai fkfga`ta,
a` `ujarlfg` jlyl, a` asa `gvai.
Hgvghl ai rasghuk 7>>
2>>
>
1
4> >
>
>
64>
765
581
?Jltajtgfl Vajl`l 64
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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`
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Lftgvghlh
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Lftgvghlh 4. ^rlftgqua ik lpra`hghk
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
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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
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L. Asfrgdl a` il i`al ai rasuitlhk ha il juitgpigflfg`.
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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 erlffgk`as fk` astks sjdkiks, fkjk vajks l fk`tg`ulfg`. Ai ratk qua ia prkpk-
`ajks as trlhufgr ils erlffgk`as 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 `ujarlfg` jlyl.
Iak y asfrgdk fl`tghlhas a` `ujarlfg` jlyl.
Fk`vgartk jarks hai sgstajl hafgjli l ujarlfg` jlyl y vgfavarsl.
Tasuaivk fk` logighlh juitgpigflfgk`as y pkta`fgls.
Erlffg`
dldgik`gfl
Erlffg`
hafgjli
5
2: 6
8
4
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66
Vujl y rastlfk` `jarks jlyls
\u a`fk`trlr astl sajl`l;
Flia`hlrgk jlyl
Vujl y rastl fk` `jarks jlyls
Kparlfgk`as fkjdg`lhls
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
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^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> tu`as ?4>> hls< dlct` 8>> tu`as >> hls
< pgct` 2>>> tu`as 4 22> >>> hls
< clildt` >>> tu`as 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 ig`ali. A` i astldiafgark` u` hl fark, qua sao` iks fga`tegfks
fkrraspk`ha li
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< 8 868
4>
< 2
8>>1 87
4> 6 848
< 1%
8>> 7
4> 2
< :
4
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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
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Lftgvghlh 4. ^rlftgqua ik lpra`hghk
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 :
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Jlrqua fk` u` fbaqua il flsgiil qua jamkr g`hgqua su ra`hgjga`tk.ikorlhk
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` kparlfgk`as fkjdg`lhls.
Tasuaivk lfartgmks jltajtgfks l plrtgr ha u` fk`mu`tk ha a`u`fglhks.
Iks lfartgmks jltajtgfks lftgvl` `uastrks prkfasks ja`tlias plrl hlr u`l fk`fiusg` l plrtgr ha u`
fk`mu`tk ha prajgsls k a`u`fglhks.
Lpigqua su rlzk`ljga`tk iogfk y hatarjg`a ai `jark qua fujpia fk` ils fk`hgfgk`as 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 rl`ok 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
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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 a`ujarltgvl, 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` railfgk`as y eu`fgk`as plrl raprasa`-
tlriks a` u` hglorljl slogtli.
Orlegflr eu`fgk`as ig`alias a` ai pil`k flrtasgl`k.
Qtgigzlr il astlhstgfl plrl krol`gzlr hltks, g`tarpratlr g`ekrjlfg` a` oregfls ykdta`ar jahghls ha ta`ha`fgl fa`trli.
Fk`vartgr `jarks hai sgstajl hafgjli l `ujarlfg` jlyl y vgfavarsl.
Vujlr y rastlr `jarks jlyls.
44?Jltajtgfl Vajl`l 68
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\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 rasja`as 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 tga`a huhls, vuaivl l iaar ilssajl`ls, lb a`fk`trlr axpigflfgk`as 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 jg`utks.
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 kparlfgk`as y
raskivar prkdiajls.
Ful`hk rasuaivl amarfgfgks y prkdiajls, hada hamlr asfrgtk ai prkfahgjga`tk.
Juy gjpkrtl`ta=flhl sarga fk`tga`a g`struffgk`as 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`
7/25/2019 Utatln Matemtica 2 Sem
240/290
^lras krha`lhks y prkhuftk flrtasgl`k
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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`
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Tailfgk`as y eu`fgk`as
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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,
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F. Fk`sghara ai fk`mu`tkx: { ,
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Eu`fgk`as ig`alias
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Astlhstgfl G
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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.
Il pkdilfg` as= Il juastrl as=
Il vlrgldia as= Ai hltk as=
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.
Il pkdilfg` as= Il juastrl as=
Il vlrgldia as= Ai hltk as=
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.
Il pkdilfg` as= Il juastrl as=
Il vlrgldia as= Ai hltk as=
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.
Il pkdilfg` as= Il juastrl as=
Il vlrgldia as= Ai hltk as=
ai jas ha ldrgi hls
tajparlturl ? F
46?Jltajtgfl Vajl`l 68
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D. Aildkra u`l tldil ha erafua`fgls fk` iks hltks qua sa prasa`tl` a` flhl a`u`fglhk y fk`tasta ilspraou`tls.
?6
51 ?> 55 55
52 ?4 ?4 52
pask a` id erafua`fgl51
55
52
?>
?4
?6
tktli (@)
Vao`il 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 tga`a` 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`
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Astlhstgfl GG
>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`fgls lskfgl pkr jahgk ha pu`tks y raftls iks hltks `kjg`lias fk` sufkrraspk`hga`ta fl`tghlh k erafua`fgl.
Olstks raligzlhks pkr Ear`l`hk a` u` jas
>
>
4>>
6>>
8>>
1>>
5>>
?>>
2>>
7>>
>>
lbkrrkvgvga`hlvastulrgktrl`spkrtaligja`tlfg`
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.
Olstks raligzlhks pkr Ear`l`hk a` u` jas
ligja`tlfg`62%
vastulrgk44%
vgvga`hl48%
lbkrrk
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Amarfgfgk 5
Aildkra il orfl g`hgflhl fk` iks hltks qua sa prkpkrfgk`l` a` flhl a`u`fglhk.
>8. Aildkra u` hglorljl ha dlrrls fk` astl g`ekrjlfg`.
flusl ha jkrtlighlh pkrfa`tlma %
`aujk`l
8>
1>
5>
Flusls ha jkrtlighlh a` gks ja`krasha < lk, 4>>8
Taeiaxgk`a. Ils g`eaffgk`as raspgrltkrgls (`aujk`l y drk`fk`aujk`l) sk` ils prg`fgpliasflusls ha jkrtlighlh a` `uastrks `gks y `gls. Vg u` `gk prasa`tl s`tkjls ha astls a`ear-jahlhas, qu hafgsg` hada tkjlr;
Ktrl flusl gjpkrtl`ta as ai s`hrkja hglrragfk, qu bdgtks bgog`gfks puaha` prava`gr astla`earjahlh;
4) Vao` iks KHJ, plrl arrlhgflr ai bljdra a` iks ja`kras ha 1 lks, bldrl qua rahufgr ai`hgfa ha has`utrgfg` hai 68% qua bldl a`
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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 g`ekrjlfg`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` jgiik`as 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 g`eargkr y asfrgdl ai rasuitlhk a`
asa `gvai.
Asfrgdl ai rasghuk
4
6
%
48>> 2?1
2>>
?1
6
4> ?1
5>
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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
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Vujl y rastl fk` `jarks jlyls
< " #
4> as u` amajpik.
>) ! 2+
< ! 2 )
4>
) " ?
< $ 2 5
4>
>
4>
) -
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Vajl`l 4) 5( - 4 + 6) : (5 =) - (5 =4) + (5 =6) : 5> -
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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.
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Vajl`l 44L trldlmlr!) (8x + 4)4: (8x)4+ 4(8x = 4) + (4)4:
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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 :>
Haslrrkiia `uavls bldgighlhas
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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. ^uaha sar= pask, fkikr ha ils piujls, aspafga...
Amarfgfgk 4Ltgti` 4>Fufbujltl`as 41^uartk 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`lfgk`as ha sl`ora
Amarfgfgk 6L.
`utrga`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` jgiik`as) % orlhks
Ljrgfl Iltg`l y Ai Flrgda 6>> 41.7% 76
Lsgl, ^lffk y Krga`ta 5
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Haslrrkiia `uavls bldgighlhas
>) 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.`at
bttp=//prkeadiko.as/diko
bttp=//rlzk`ljga`tkjltajltgfktktli.dikospkt.fkj
bttp=//sfbkiilrgs.fkj.jxbttp=//wad.ahuflstur.prg`flst.as
bttp=//www.llljltajltgfls.fkj
bttp=//www.lpigflfgk`as.g`ek
bttp=//www.fk`avyt.kro.jx
bttp=//www.aujah.`at/fursafk /igdrargl
bttp=//www.oakily.fkj
bttp=//www.k`u.kro.ot
bttp=//www.saftkrjltajltgfl.fi
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
www.okdgar`khafl`lrgls.kro
www.gta.ahuflfgk`.as
www.jltajltgflplrltkhks.fkj
www.rla.as
www.vgtutkr.fkj
4?2GOAT Qtlti`
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Astuhglr