Libro Quiriguá Matemática 1er.sem.15

7/26/2019 Libro Quirigu Matemtica 1er.sem.15

1/350


2/350


3/350

0 Dsilg - Brupg Puiribu]riokr skokstrk - IBKT

O`tkotil`2


4/350

O@^KOÎL@ 2]riokr skokstrk

Tkskrv`ngs tgngs cgs nkrklhgs. Pukn` riburgsòkftk prghidin` c` rkprgnullif tgt`c g p`rli`cnk kstk o`tkri`c knul`tivg, pgr lu`cquikr oknig g prglknioikftg, sif c` ùtgriz`lif nkc IfstitutgBu`tko`ctklg nk Knul`lif T`nigafil`, IBKT. \kbf `rtlucg >4 nk c` Lgfstitulif ]gctil` nkBu`tko`c` quk sk rkkrk c` ùtgr`.

Ifstitutg Bu`tko`ctklg nk Knul`lif T`nigafil`, IBKT.Ks uf` gdr` prgnulin` pgr kc Nkp`rtòkftg nk Tkn`llif y Niskg, p`r` kc

Ifstitutg Bu`tko`ctklg nk Knul`lif T`nigafil`, IBKT.

00 vkfin` 08->;, Liun`n Fukv`, zgf` 4Liun`n nk Bu`tko`c`.

]DX5 4>04 1111A`x5 4>04 12=>

Lgrrkg kckltrfilg5 ibkrMibkr.knu.bt]bif` wkd5 www.ibkr.knu.bt

Knilif 4=0>Ioprksg kf IBKT t`cckrks brlgs

Lnibg5 000=2=>4=0

I\DF 92899944940>


5/350

fnilk................... ..................... ..................... ..................... ..................... ..................... ................... I

Dikfvkfin` y dikfvkfing!..................... ..................... ..................... ..................... ................ 0

\ko`f` 0Lgfeuftgs ......................................................................................................................... 00

]`r` lgokfz`r! Bkgrb L`ftgr Lrk`ngr nk c` ^kgr` nk Lgfeuftgs......................... 0

Kc oufng nk c` o`tkotil`

0. Lgfeuftgs .................... ....................... ..................... ..................... ..................... ....................... 0;4. Lc`sks nk lgfeuftgs ..................... ..................... ......................... ..................... .................... 01

4.0 Lgfeuftg fitg.................. ..................... ........................ ..................... ..................... ....... 01

4.4 Lgfeuftg ufit`rig ................... ..................... ......................... ..................... .................... 01

4.< Lgfeuftg v`lg ................... ..................... ......................... ..................... .................... 02

4.> Lgfeuftg iffitg................... ..................... ........................ ..................... ..................... ... 02

4.; Lgfeuftg ufivkrsg [ .................. ..................... ........................ ..................... ................. 08

Tksuokf ........................................................................................................................................ 08

@utglgftrgc..................... ..................... ..................... ..................... ..................... ..................... .... 09

@bicin`n nk lclucg okft`c.................... ..................... ..................... ..................... ................ 44

T`zgfòikftg cbilg..................... ..................... ..................... ..................... ..................... ...... 4>


0. Tkc`lif nk pkrtkfkfli` .................... ..................... ..................... ..................... ............. >; 0.0 Tkc`lif nk fg pkrtkfkfli` ..................... ..................... ..................... ................... >1

4. Tkc`lif nk lgftkflif ..................... ..................... ..................... ..................... ............. >2

4.0 Tkc`lif nk fg lgftkflif .................. ..................... ..................... ..................... .. >8

Tksuokf ........................................................................................................................................ >9

@utglgftrgc.................... ..................... ..................... ..................... ..................... ..................... ..... ;=

@bicin`n nk lclucg okft`c................... ..................... ..................... ..................... ................. ;>

T`zgfòikftg cbilg.................... ..................... ..................... ..................... ..................... ....... ;;

\ko`f` >Kc lgfeuftg F nk cgs fokrgs f`tur`cks............................ ;2

]`r` lgokfz`r!Log f`likrgf cgs fokrgs7.................. ..................... ..................... . ;9

Ckfbuèk o`tkotilg.................. ..................... ..................... ..................... ..................... ....... 1=


0. Cgs fokrgs f`tur`cks (F) ..................... ..................... ..................... ..................... ............. 10

0.0 L`r`ltkrstil s nkc lgfeuftg nk cgs fokrgs f`tur`cks .................... ............. 10

0.4 Tkprkskft`lif nkc lgfeuftg nk fokrgs f`tur`cks sgdrk c`

skoirrklt` fuoril` .................... ..................... ..................... ..................... ................. 14

0.< Lgop`r`lif nk fokrgs f`tur`cks ................... ..................... ..................... ........ 1 Grnkfkogs skriks nk fokrgs f`tur`cks................................... ..................... .... 1>

Tksuokf................... ..................... ..................... ..................... ..................... ..................... ............ 1;

@utglgftrgc.................... ..................... ..................... ..................... ..................... ..................... ..... 11

@bicin`n nk lclucg okft`c................... ..................... ..................... ..................... ................. 2=

T`zgfòikftg cbilg.................... ..................... ..................... ..................... ..................... ....... 20

\ko`f` ;Gpkr`ligfks y prgpikn`nks kf kc lgfeuftg nk

cgs fokrgs f`tur`cks (F) ............................................................ 2 Nivisif nk fokrgs f`tur`cks ................... ..................... ..................... ................... 28

4. ]rgpikn`nks nk c`s gpkr`ligfks nk cgs fokrgs f`tur`cks (F) ..................... ..... 29

4.0 ]rgpikn`nks nk c` suo` ..................... ..................... ..................... ..................... ......... 29

`. ]rgpikn`n lgfout`tiv .................. ..................... ..................... ..................... ......... 29

d. ]rgpikn`n sgli`tiv` .................. ...................... ..................... ..................... ............ 29

l. Kckokftg fkutrg .................... ..................... ..................... ..................... .................... 29 Kf rksuokf... .................... ..................... ......................... ..................... ..................... .............. 8=

4.4 ]rgpikn`nks nk c` rkst` .................. ...................... ..................... ..................... ............ 8=

`. Kckokftg fkutrg .................... ..................... ..................... ..................... .................... 8=

4.< ]rgpikn`nks nk c` ouctipcil`lif .................. ..................... ..................... ............... 80

`. ]rgpikn`n lgfout`tiv .................. ..................... ..................... ..................... ......... 80

d. ]rgpikn`n sgli`tiv` ..................... ..................... ..................... ..................... ......... 80

l. Kckokftg fkutrg .................... ..................... ..................... ..................... .................... 80

n. Nistridutiv` nk c` ouctipcil`lif rkspkltg c` suo` y ` c` rkst` ......... 84

Kf rksuokf... ..................... ..................... ......................... ..................... ..................... .............. 84 4.> ]rgpikn`nks nk c` nivisif .................... ..................... ..................... ..................... ..... 8>

`. Kckokftg fkutrg .................... ..................... ..................... ..................... .................... 8>

Tksuokf ........................................................................................................................................ 8;

@utglgftrgc..................... ..................... ..................... ..................... ..................... ..................... .... 82

@bicin`n nk lclucg okft`c.................... ..................... ..................... ..................... ................ 94

T`zgfòikftg cbilg..................... ..................... ..................... ..................... ..................... ...... 9;-0908)

Lrk`ngr nk c` tkgr`nk lgfeuftgs.

0 Lgfeuftg ifaifitg Lgf ifaifin`n nk kckokftgs

Luftgs kckokftgs tikfk kc lgfeuftg Knk c`s kstrkcc`s nkc roòkftg7 Ksiopgsidck cckb`r ` s`dkr luft`s kstrkcc`s h`y pgrquk fg sk puknkf lgft`rtgn`s. Ks uf lgfeuftg iffitg.

[f lgfeuftg ifafitg ks `qukc quk fg pgnkogs cckb`r ` s`dkr luftgs kck-okftgs tikfk.\u l`rnif`cin`n ks ifnktkroif`n`.

Gtrgs kekopcgs5 kc lgfeuftg Fnk fokrgs f`tur`cks, kc lgfeuftg Onk octi-pcgs nk uf fokrg, kc lgfeuftg Nnk fokrgs nklio`cks, ktl.

Kekrlilig Lgfeuftg ifafitg5 fg pgnkogs cckb`r ` s`dkr luftgs kckokftgs tikfk. \u l`rnif`cin`n ksifnktkroif`n`.

4.; Kc lgfeuftg ufivkrsg ks kc lgfeuftg o`ygr nkc quk sk gdtikfkf gtrgs lgfeuftgs.

08IBKT Puiribu

Tksuokf


31/350

@ltivin`n 0.Nkoukstrk cg prkfningNkoukstrk cg `prkfningNNsu

@. Iftkftk fg lgfsuct`r su cidrg, lgoprukdk qu `prkfni.

0) Kslrid` lgf sus p`c`dr`s qu ink` tikfk nk lgfeuftg.

4) Kslrid` lgf sus p`c`dr`s qu ks uf kckokftg.

D. Tk`cilk kstk kekrlilig p`r` vkriail`r su lgoprkfsif nk c`s lc`sks nk lgfeuftgs. Inkftiaiquk si cgs

lgfeuftgs sgf v`lgs, ufit`rigs, aifitgs g ifaifitgs. ^ikfk uf kekopcg.=) Kc lgfeuftg@nk c`s cktr`s nkc `ca`dktg.

0) Kc lgfeuftg Onk cgs oksks lgf 48 g 49 n`s.

4) Kc lgfeuftg]nk cgs fokrgs p`rks.

) Kc lgfeuftg Gnk cgs gegs nk uf` pkrsgf`.

;) Kc lgfeuftgKnk cgs fokrgs kftrk > y ;.

L. Kslrid` kf c` lgcuof` nkrklh` kc lgfeuftg [fivkrsg nkc quk sk gdtuvg l`n` lgfeuftg nk c`lgcuof` izquikrn`. ^ikfk uf kekopcg.

Kc lgfeuftg@ks fitg.

Lgfeuftgs Lgfeuftg ufivkrsg ([)

Kc lgfeuftg Kagro`ng pgr c`pilkrg,s`l`puft`s, dgrr`ngr.

Kc lgfeuftgKsk gdtuvg nkc lgfeuftg

[ nk

Kc lgfeuftg ] agro`ng pgr cgs pc`fkt`s\`turfg, [r`fg, O`rtk.

Kc lgfeuftg ]sk gdtuvg nkc lgfeuftg

[nk

Kc lgfeuftg N agro`ng pgr cgs nkp`rt`-okftgs nk Ksluiftc`, ]ktf, Euti`p`.

Kc lgfeuftg Nsk gdtuvg nkc lgfeuftg

[nk

Kc lgfeuftg ] agro`ng pgr cgs fokrgsp`rks 4, >, 1, 8.

Kc lgfeuftg ]sk gdtuvg nkc lgfeuftg

[nk

09O`tkotil` \ko`f` 0

@utglgftrgc


32/350

@ltivin`n 4. ]r`ltiquk cg `prkfning

@. Kslrid` nkftrg nkc lu`nrg c` l`rnif`cin`n nk l`n` lgfeuftg nkslritg. \i sk tr`t` nk uf lgfeuftgifaifitg, kslrid` c` cktr` i. ^ikfk uf kekopcg.

=) Kc lgfeuftg Qnk c`s cktr`s vgl`cks nk c` p`c`dr` o`ripgs`.

0) Kc lgfeuftg Nnk cgs nkngs nk c` o`fg.

4) Kc lgfeuftg Pagro`ng pgr sus lgop`krgs y lgop`kr`s nkc brupg Puiribu.

) Kc lgfeuftg Ink cgs fokrgs iop`rks.

;) Kc lgfeuftg Lnk cgs br`fgs nk `rkf` nkc nksikrtg.

D. Kslrid` sgdrk c` cfk` luc ks kc lgfeuftg [nk cgs lgfeuftgs n`ngs. ^ikfk uf kekopcg.=) Luc ks kc lgfeuftg [ nkc lgfeuftg]agro`ng pgr cgs fokrgs p`rks 4, >, 1 7

Kc lgfeuftg [nkc lgfeuftg]ks kc

0) Luc ks kc lgfeuftg [nkc lgfeuftg Aagro`ng pgr c`s arut`s p`p`y`, pi` y pkr`7

Kc lgfeuftg [nkc lgfeuftg Aks .

4) Luc ks kc lgfeuftg [ nkc lgfeuftg O agro`ng pgr cgs oksks kfkrg, o`rzg ynilikodrk7

Kc lgfeuftg [nkc lgfeuftgOks .


33/350

4) Kc lgfeuftg I agro`ng pgr trks hkrròikft`s nkc `briluctgr.

Lgfeuftg Iagro`ng pgr5 , ,

) Kc lgfeuftg Fagro`ng pgr lu`trg fokrgs okfgrks quk 0=.

Lgfeuftg Fagro`ng pgr5 , ,

,

;) Kc lgfeuftg Aagro`ng pgr trks lc`sks nk acgrks.

Lgfeuftg Aagro`ng pgr5 , ,

D. Ck` kc kfufli`ng, `f`cilk y rkspgfn` ` c`s prkbuft`s. îkfk uf kekopcg.

=) \i lgfsinkrògs kc lgfeuftgInk c`s liar`s iop`rks kf kc fokrg 4>85

Luftgs kckokftgs tikfk7

]gr qu7

Kftgflks, qu tipg nk lgfeuftg ks I7

0) \i lgfsinkrògs kc lgfeuftg ]nk prksinkftks nk uf` f`lif5


]gr qu7

Kftgflks, qu tipg nk lgfeuftg ks ]7

4) \i lgfsinkrògs kc lgfeuftg Fnk fokrgs f`tur`cks o`ygrks quk ;5


]gr qu7

Kftgflks, qu tipg nk lgfeuftg ks F7

Lkrg kckokftgs

v`lg

]grquk tgn`s c`s liar`s nkc fokrg sgf p`rks



34/350

Kslrid` kc rksuct`ng nk c`s ouctipcil`ligfks cg os rping quk pukn`. Hb`cg lgf cpiz, `s si skkquivgl`, puknk dgrr`r kc rksuct`ng k iftkft`rcg nk fukvg. Fg uticilk c` l`cluc`ngr`.

0) 0 x < ?4) 4 x 2 ?

) 0 x 0= ?

;) 4 x ; ?

1) 0 x 1 ?

2) 0 x 0 ?

8) 4 x 8 ?

9) 4 x < ?

0=) 0 x 9 ?

00) 4 x > ?04) 0 x 4 ?

0) 0 x ; ?

0;) 4 x 9 ?

01) 4 x 0 ?

02) 0 x > ?

08) 4 x 0= ?

09) 0 x 8 ?

4=) 0 x 2 ?

40) 0 x 9 ?44) 4 x 2 ?

4) 0 x > ?

4;) 4 x < ?

41) 0 x 4 ?

42) 4 x 8 ?

48) 0 x 1 ?

49) 4 x 0 ?

? >

0 x ; ? ;

0 x 1 ? 1

0 x 2 ? 2

0 x 8 ? 8

0 x 9 ? 9

0 x 0= ? 0=

^`dc` nkc 4

4 x 0 ? 4

4 x 4 ? >

4 x < ? 1

4 x > ? 8

4 x ; ? 0=

4 x 1 ? 04

4 x 2 ? 0>

4 x 8 ? 01

4 x 9 ? 08

4 x 0= ? 4=

44IBKT Puiribu

@bicin`n nk lclucg okft`c


35/350

Tksgculif nk prgdcko`s Tksgcvkr prgdcko`s o`tkotilgs fgs `yun` ` nks`rrgcc`r kc r`zgfòikftg cbilg. Kst` lgo-

pktkfli` fg sk cgbr` nk c` fglhk ` c` o``f`, pgr ksg sko`f` ` sko`f`, ck prgpgfkogsuf` skrik nk prgdcko`s p`r` quk v`y` ùokft`fng sus h`dicin`nks. Lgokflkogs!

]`r` rksgcvkr prgdcko`s, ck rklgokfnògs5

0. Ckkr lgf `tkflif y `skbur`rsk nk quk lgoprkfnk kc prgdcko`.

4. Tkackxigf`r5

Pu ok prkbuft`f7Pu n`tgs tkfbg7 Pu n`tgs ok a`ct`f7

. Gpkr`r

Tk`ciz`r gpkr`ligfks fklks`ri`s

;. Kslridir c` rkspukst`

@hgr` ck tgl` ` ustkn rksgcvkr cgs prgdcko`s. Hb`cg kf su lu`nkrfg y `pciquk cgs p`sgs `ftkrigrks.

0) O`ri`f` lgopr uf quift`c nk ariegc ` >;= quktz`cks y ngs quift`cks nk `zl`r ` 4;=quktz`cks l`n` ufg. Luftg p`b pgr c` lgopr`7 Luftg ck sgdr si cckv`d` 0===quktz`cks7

4) Luft`s dgcs`s fklksitògs p`r` kop`l`r 004; bu`l`tks, si kf l`n` dgcs` pgfkogs0; `bu`l`tks7

> hkltrk`s y uf` argftkr` nk> jicoktrgs.

`. Luftgs `gs nk ifnkpkfnkfli` luopci c` Liun`n nkc Q`til`fg kf kc `g 4===7

d. Kf qu `g c` Liun`n nkc Q`til`fg luopcir 0== `gs lgog Kst`ng ifnkpkfnikftk7

l. \i uf` hkltrk` kquiv`ck ` 0=,=== o4, luc ks c` kxtkfsif tkrritgri`c nk c` Liun`nnkc Q`til`fg7

n. \i uf jicoktrg oink 0,=== o, luftgs oktrgs oink c` argftkr` nk c` Liun`n nkcQ`til`fg7

4IBKT Puiribu


37/350

4

Tkprkskft`lif nklgfeuftgs

4;O`tkotil` \ko`f` 4


38/350

Pu kflgftr`r kst` sko`f`7

Cgs cgbrgs quk lgfskbuir kst` sko`f`sgf5 Lgfglkr y pr`ltil`r kc sibfg (cc`vks).

Nkslridir k inkftil`r c`s agro`s nk rkprkskft`r cgslgfeuftgs.

Tkprkskft`r lgfeuftgs kf agro` nkslriptiv`, brl` ykfuokr`tiv`.

Nks`rrgcc`r c` h`dicin`n nk lclucg okft`c pr`ltil`fngc`s t`dc`s nkc ouctipcil`r nkc 0 `c >.

Nks`rrgcc`r kc r`zgfòikftg o`tkotilg rksgcvikfngprgdcko`s.

Nks`rrgcc`r c` gdskrv`lif y c` `tkflif kflgftr`fngtgngs cgs lu`nr`ngs quk agro`f uf` bur`.

@bicin`n nk lclucg okft`c ^ dc s nk ouctipcil`r nkc 0 c >

Ckfbuèk o`tkotilg Kc sibfg (cc vks)

Kc oufng nk c` o`tkotil` Tkprkskft`lif nk lgfeuftgs

T`zgfòikftg cbilg ]rgdcko`s o`tkotilgs lgf

gpkr`ligfks dsil`s

]`r` lgokfz`r! Digbrà nk Eghf Qkff

41IBKT Puiribu


39/350

Eghf Qkff

@ tr`d`e`r!

0. go`fng kf lukft` c` ckltur`, luc auk kc `pgrtk nk Eghf Qkff ` c` o`tkotil` y c`cbil`7

4. ]`r` qu sirvkf cgs ni`br`o`s nk Qkff7


40/350

C`s cc`vkssgf sibfgs nk `brup`lif quk sirvkf p`r` kflkrr`r cgs kckokftgs nk uf lgfeuftg.

C` cc`vk nk pkrtur ( ) c` kslridiogs ftks nkc priokr kckokftg y c` cc`vk nk likrrk ( ), cukbg nkc ctiogkckokftg.

C`s cc`vks sk tr`z`f nk `rrid` h`li` `d`eg. Aeksk kf c`s klh`s.

Kstk sibfg sirvk p`r`5

Iftkftk ustkn kc tr`zg, niduek c`s cc`vks sgdrk l`n` cfk`.

@.

gok su c`pilkrg y rkp`sk l`n` cc`vk. \ib` c` nirkllif quk ck ifnil`f c`s klh`s.

Cc`vk nk `pkrtur`5 Cc`vk nk likrrk5

48IBKT Puiribu

Ckfbu`ek o`tkotilg


41/350

0. Tkprkskft`lif nk lgfeuftgs

Log rkprkskft`r` uf` o`fz`f`7 H`y niakrkftks agro`s nk h`lkrcg5 puknk ccò`rc` pgr su fgodrk5 o`fz`f`

nkslridirc`5 arut` rkngfn` hufnin` pgr cgs kxtrkogs, nk lsl`r` nkcb`n` ycis`, nk lgcgr rgeg, vkrnk g ò`riccg, lgf s`dgr ling g nuclk .

g nidue`rc`5

Lgog c` o`fz`f`, cgs lgfeuftgs tòdif sk puknkf rkprkskft`r nk niakrkftksagro`s. Qkògs5

0.0 Agro` nkslriptiv` [f lòifg os lgrtg

C` agro` nkslriptiv` ks uf` o`fkr` `drkvi`n` nk rkprkskft`r lgfeuftgs pgr-quk sgcg fgodrògs uf` prgpikn`n lgof ` tgngs cgs kckokftgs.

]`r` rkprkskft`r lgfeuftgs kf agro` nkslriptiv`5

Kslridiogs c` cktr` quk rkprkskft` kc fgodrk nkc lgfeuftg, skbuing nk ufsibfg ibu`c.

Nidueògs c` cc`vk nk `pkrtur`, nkslridiogs uf` l`r`ltkrstil` nk cgskckokftgs y lkrrògs c` cc`vk.

Kekopcg5 ]? { ]`sks nkc oufng}

\k ckk 5 ]ks ibu`c `c lgfeuftg nk cgs p`sks nkc oufng.

Kekrlilig 0Kxprksk kf agro` nkslriptiv` kc lgfeuftg n`ng. îkfk uf kekopcg

=) Kc lgfeuftg ^nk cgs fokrgs 4, >, 1, 8, 0=, 04, 0>, 01, 08, 4=.

^? { }

0) Kc lgfeuftg Fnk cgs fokrgs =, 0, 4, , ;, 1, 2, 8, 9.

F? {

Fokrgs p`rks okfgrks quk 44




42/350

Kekrlilig 4Gdskrvk cgs lgfeuftgs Ly ], kxprks`ngs kf agro` nkslriptiv`, y cukbg rkprkskftk kc lgfeuftg ]kf agro` brl`. ^ikfk uf kekopcg.

=) L? {Liar`s nkc fokrg >

; ;

1 2

F


43/350

@ufquk c` ` skrkpitk ngs vklks kf c`p`c`dr` o`p`, sgcgsk kslridk uf` vkz.

0.< Agro` kfuokr`tiv` [f cist`ng nk kckokftgs

Kfuokr`r ks lgft`r, h`lkr cist`s. C` rkprkskft`lif kfuokr`tiv` fgodr` ufg` ufg tgngs cgs kckokftgs nkc lgfeuftg. \ukck uticiz`rsk p`r` lgfeuftgs lgfpglgs kckokftgs.

]`r` rkprkskft`r lgfeuftgs kf agro` kfuokr`tiv`5

Fgodrògs kc lgfeuftg lgf uf` cktr` o`ysluc` y lgftifu`lif kslri-diogs kc sibfg ibu`c.

Nidueògs c` cc`vk nk `pkrtur`, kslridiogs cgs kckokftgs skp`rfngcgslgf uf` lgo` y lkrrògs c` cc`vk.

Lu`fng uf kckokftg sk rkpitk, sgcg sk kslridk uf` vkz.

Kekopcgs5

Kc lgfeuftg Qnk c`s cktr`s vgl`cks.

Q?{`, k, i, g, u }

\k ckk5 Q ks ibu`c `c lgfeuftg agro`ng pgr cgs kckokftgs `, k, i, g, u.

Kc lgfeuftg Cnk c`s cktr`s nk c` p`c`dr` o`p`, kf agro` kfuokr`tiv`, skkxprks`5

C? {o, `, p}

\k ckk5 C ks ibu`c `c lgfeuftg agro`ng pgr cgs kckokftgs o, `, p.

Kekrlilig @hgr` ck tgl` ustkn. Tkprkskftk cgs sibuikftks lgfeuftgs kf c` agro` kfuokr`tiv` kspkli`c. ikfkuf kekopcg.

=) Kc lgfeuftg ]nk cgs fokrgs p`rks nkc 8 `c >;1.

] ?{ }

0) Kc lgfeuftg Cnk c`s cktr`s nkc `dklkn`rig nksnk c` e h`st` c` w.

C ?{

4) Kc lgfeuftg cnk cgs fokrgs iop`rks nkc 0 `c 999.

c?{

) Kc lgfeuftg Hnk cgs fokrgs f`tur`cks nkc = `c ;==.

H?{

8, 0=, 04,... >;1


45/350

0. [f lgfeuftg puknk rkprkskft`rsk kf trks agro`s5 nkslriptiv`, brail`y kfuokr`tiv .

0.< Agro` kfuokr`tiv`

C` agro` nk rkprkskft`lif kfuokr`tiv` lgfsistk kf kslridir tgngs cgs kckokftgs nkc lgf-euftg, sif rkpktir fifbufg,kflkrr`ngs kftrk cc`vks { } y skp`r`ngs pgr lgo`s. Kekopcg5

Q?{`, k, i, g, u }

`. Agro` kfuokr`tiv` kspkli`c

C` agro` kfuokr`tiv` kspkli`c sk uticiz` p`r` lgfeuftgs ouy fuokrgsgs luygs kckokftgstikfkf uf grnkf kst`dckling. ]`r` rkprkskft`rcg5

\k kslridkf cgs trks priokrgs kckokftgs nkc lgfeuftg.

Cukbg sk lgcgl`f puftgs suspkfsivgs p`r` rkprkskft`r cgs kckokftgs quk fg sk kslridkfy sk `fgt` kc ctiog kckokftg nkc lgfeuftg. Kekopcg5

C ?{`, d, l, z }

0.0 Agro` nkslriptiv`

C` agro` nk rkprkskft`lif nkslriptiv` lgfsistk kf kslridir nkftrg nk cc`vks uf` gr`lif quknkslridk c` l`r`ltkrstil` lgof nk tgngs cgs kckokftgs nkc lgfeuftg. Kekopcg5

Q ?{Cktr` vgl`c }

0.4 Agro` brail` g Ni`br`o` nk Qkff

C` rkprkskft`lif brail` lgfsistk kf nidue`r uf ni`br`o` nk Qkff y, nkftrg nk c, kslridircgs kckokftgs nkc lgfeuftg, lgcglfngcks uf puftg ` c` izquikrn`. \i uf kckokftg kst rk-

pkting, scg sk kslridk uf` vkz. Kc fgodrk nkc lgfeuftg sk kslridk aukr` nk c` aibur` kf gtrglrlucg pkqukg. Kekopcg5

Q

k

i

u g


46/350

@ltivin`n 0Nkoukstrk cg prkfning.Nkoukstrk cg `prkfningNNsu@. Lgopcktk kc ni`br`o` kslridikfng c`s agro`s nk rkprkskft`lif nk uf lgfeuftg.

Cgs lgfeuftgs

sk puknkf rkprkskft`r kf agro`5

D. Inkftiaiquk c`s agro`s nk rkprkskft`lif nk cgs lgfeuftgs. Tkcckfk kc lu`nrg quk lgrrkspgfnk `c` rkspukst` lgrrklt`.

=) Luc ks c` agro` nk rkprkskft`r uf lgfeuftg kf c` quk sk h`lkuf` cist` nk sus kckokftgs7

0) Kf qu agro` kst rkprkskft`ng kc lgfeuftg@? {Fokrgs o`ygrks quk 2 }7

4) Kf qu agro` kst rkprkskft`ng kc lgfeuftg T? {`, o, n }7


47/350

@ltivin`n 4]r`ltiquk cg `prkfning.uNNsu

@. Tkprkskftk cgs lgfeuftgs kf agro` nkslriptiv`. ^ikfk uf kekopcg.

=) Kc lgfeuftg L agro`ng pgr Bu`tko`c`, Kc \`cv`ngr, Hgfnur`s, Fil`r`bu`, Lgst` Til` y]`f`o.

L? { }

0) Kc lgfeuftg Kagro`ng pgr cufks, o`rtks, oirlgcks, eukvks, vikrfks, sd`ng y ngoifbg.

K? {

4) Kc lgfeuftg Aagro`ng pgr cgs fokrgs iop`rks 0,


48/350

@ltivin`n < Nks`rrgcck fukv`s h`dicin`nks.u

@. Kslrid` log sk ckkf cgs lgfeuftgs. ^ikfk uf kekopcg.

=) @ ? {8, 9, 0=,2; }

0) D? {04, 0>, 01,981 }

4) L? {Fokrgs o`ygrks quk 08}

D. Kslrid` kf agro` kfuokr`tiv` kspkli`c cgs sibuikftks lgfeuftgs.

0) Kc lgfeuftg Fagro`ng pgr cgs fokrgs nkc 08 `c ;2>.

F? {

4) Kc lgfeuftg Oagro`ng pgr cgs oksks nkc `g.

O? {

, 01, 08, 4=

L. Tkprkskftk cgs lgfeuftgs kfagro` kfuokr`tiv`.

=) Kc lgfeuftg Fagro`ng pgr cgs fokrgs p`rks kftrk 9 y 40.

F? { }

0) Kc lgfeuftg Oagro`ng pgr cgs oksks nkc `g lgf


49/350

@ltivin`n 0Kslrid` kc rksuct`ng nk c`s ouctipcil`ligfks cg os rping quk pukn`. Fg uticilk l`cluc`ngr`.Kslrid` lgf cpiz. @s, si sk kquivgl`, puknk dgrr`r. ^ikfk uf kekopcg.

]`r` b`f`r bicin`n okft`c y r`pinkz, h`y quk pr`ltil`r lgfst`ftkokftk. Tkp`sk c`s t`dc`s nkc < y nkc >y lkrlirksk nk quk c`s s`dk nk okogri`. \i ustkn y` c`s ngoif`, rk`cilk c` `ltivin`n 0 nirklt`okftk.

Nk okogri`!

^`dc nkc

< x 9 ? 42

< x 0= ?

> x 0 ? >

> x 4 ? 8

> x < ? 04

> x > ? 01

> x ; ? 4=

> x 1 ? 4>

> x 2 ? 48

> x 8 ? x 9 ? x 0= ? >=

0=) 4 x 2 ?00) < x 1 ?

04) < x ; ?

0) 0 x 9 ?

0;) < x 1 ?

01) 0 x ; ?

02) < x > ?

08) > x ; ?

09) 0 x < ?

=) < x 4 ?0) 0 x > ?

4) 4 x ; ?

) > x 4 ?

;) 4 x < ?

1) > x 8 ?

2) 4 x 0 ?

8) 4 x 9 ?

9) 0 x 8 ?

4=) 4 x 4 ?40) < x 0 ?

44) > x 1 ?

4 x > ?

4>) > x < ?

4;) < x 9 ?

41) 4 x 8 ?

42) < x 2 ?

48) 0 x 9 ?

49) > x 2 ?

1


50/350

Tksgculif nk prgdcko`s@. Tklukrnk quk p`r` lgoprkfnkr kc prgdcko` y kflgftr`r c` rkspukst` nkdk ckkrcg lgf

`tkflif.

0) \i l`n` p`p`y` lukst` 9 quktz`cks, luftg lgst`rf < p`p`y`s7 \i p`b` lgf uf diccktk nk P;=.==, luftg vukctg rklidk7

4) O`r` vkfnk lhulhitgs cgs aifks nk sko`f`. Kc sd`ng vkfni 04 y kc ngoifbg 0;. \il`n` ufg v`ck 4 quktz`cks, qu l`ftin`n nk nifkrg rkufi kf tgt`c7

quktz`cks.`. Luftg tuvg quk p`b`r kf tgt`c7

d. \i p`b lgf uf diccktk nk 0== quktz`cks, luftg ck nkvgcvikrgf7

>) \i uf` l`e` nk 4> eubgs kf c`t` ok lukst`f 91 quktz`cks,

`. luft`s l`e`s pukng lgopr`r lgf 218 quktz`cks7

d. Luc ks kc v`cgr ifnivinu`c nk l`n` c`t` nk eubg7

;) O`rt` b`st 0;= quktz`cks c` v`r` lu`nr`n`, luftg b`f` kf kc fkbglig7

8) Kc cufks, Tglg t`rn < hgr`s lgf 44 oifutgs p`r` rklgrrkr 444 jicoktrgs kf su l`rrg.\i kc o`rtks t`rn 4 hgr`s lgf ;; oifutgs, qu tikopg `hgrr7

9) 42 kstuni`ftks nkc brupg Puiribu agro`rgf uf` didcigtkl`. L`n` kstuni`ftk rkufi 8 cidrgs, luftgs cidrgs h`y kf c` didcigtkl`7


51/350

D. Eukbg cbilg

Luftgs lu`nr`ngs h`y7

Kstk eukbg ck `yun`r ` okegr`r su `tkflif pgrquk tkfnr quk lgflkftr`rsk kf c`s agro`snk c`s aibur`s. Cgs ifvkftgrks y lrk`ngrks nks`rrgcc`f uf` br`f l`p`lin`n nk lgflkftr`lifquk cks pkroitk skftir su ink` nk agro` iftkfs`.

Kekrlitk su okftk rk`ciz`fng kstk kekrlilig p`r` lgfvkrtirsk kf uf br`f lrk`ngr!

Fg tgng ks cg quk p`rklk. Gdskrvk lgf `tkflif y kflukftrk tgngs cgs lu`nr`ngs pgsidckskf c` sibuikftk aibur`. O`rquk l`n` lu`nr`ng lgf uf lgcgr niakrkftk.

Luftgs lu`nr`ngs kflgftr7


52/350

Fgt`s5Kslrid` `qu sus ifquiktunks, nksludrioikftgs g nun`s p`r` lgop`rtir kf kc lrlucg nk kstunig.

O`rquk lgf uf lhkquk c` l`sicc` quk okegr ifniquk su rkfnioikftg. cgbr`ngkf

prglksgfg

cgbr`ng

Nkspusnkkstuni`r...

Inkftilg y pukng nidue`r kc sibfg nk cc`vks.

Lgoprkfng c`s niakrkftks agro`s nk rkprkskft`r lgfeuftgs.

]ukng rkprkskft`r cgs lgfeuftgs kf niakrkftks agro`s.

]r`ltilg c`s t`dc`s nk ouctipcil`r nkc 0 `c >, rk`ciz`fng cgs kekrliligsprgpukstgs.

Lgfgzlg y uticizg lgrrklt`okftk c`s t`dc`s nk ouctipcil`r nkc 0 `c >.

Tksukcvg cgs prgdcko`s pc`ftk`ngs.

Nks`rrgccg `tkflif y gdskrv`lif kflgftr`fng pgr cg okfgs 00lu`nr`ngs.

Tkvisk su `prkfniz`ek

>=IBKT Puiribu


53/350

0O`tkotil` \ko`f`


54/350

Cgs cgbrgs quk lgfskbuir kst` sko`f`sgf5 Lgfglkr y pr`ltil`r cgs sodgcgs o`tkotilgs5

(pkrtkfklk), (fg pkrtkfklk), (lgftkfing)y (fg lgftkfing).

Nkslridir k inkftil`r c`s rkc`ligfks nk lgfeuftgs5pkrtkfkfli` y lgftkflif.

@sgli`r c`s rkc`ligfks nk pkrtkfkfli` y lgftkflif `situ`ligfks lgtini`f`s.

Lgfstruir kekopcgs nk rkc`ligfks nk pkrtkfkfli` ylgftkflif.

@biciz`r kc lclucg okft`c pr`ltil`fng c`s t`dc`s nkouctipcil`r nkc 0 `c 1.

Nks`rrgcc`r kc r`zgfòikftg cbilg rksgcvikfngprgdcko`s o`tkotilgs.


@bicin`n nk lclucg okft`c ^ dc s nk ouctipcil`r nkc 0 c 1

Kc oufng nk c` o`tkotil` Tkc lif nk lgfeuftgs5

]krtkfkfli`

Lgftkflif

]rgdcko`s o`tkotilgs lgf

gpkr`ligfks dsil`sT`zgfòikftg cbilg

]`r` lgokfz`r! C`s `vks ]hicip Rhitkcn

Ckfbuèk o`tkotilg Cgs sibfgs (pkrtkfklk), (fg pkrtkfklk),(lgftkfing) y (fg lgftkfing)

>4IBKT Puiribu


55/350

C`s `vks

@ tr`dè`r!

0. Kslrid` ngs l`r`ltkrstil`s nk c`s `vks quk sk okfligf`f kf c` ckltur`.

4. Lgopcktk c`s gr`ligfks lgf c` `yun` nk c` ifagro`lif nk c` t`dc`. îkfk uf kekopcg.

C` cklhuz` ks uf vk quk pkrtkfklk ` c` lc`sk...

Kc p`vg rk`c ks uf vk quk pkrtkfklk ` c` lc`sk...

Kc pkcl`fg ks uf vk quk pkrtkfklk ` c` lc`sk...

C`s vks sgf uf brupg nk fio`cks quk sk h` n`pt`ng ouy dikf cgs lòdigs

nkc kftgrfg. H`f lgfquist`ng kc ìrk, c` tikrr`, kc `bu` y sk kflukftr`f kf tgngkc pc`fkt`, iflcusg kf c`s hkc`n`s tikrr`s nk c` @ftrtin`.

H`y uf`s 9,=== kspkliks nk `vks nistift`s, lgf tò`gs quk v`f nksnk kc

nioifutg lgcidr h`st` cgs br`fnks `vkstrulks. @ pks`r nk sus niakrkfli`s, tgn`s

c`s `vks sk rkprgnulkf ` tr`vs nk hukvgs. C` lr` kf nks`rrgccg kst prgtkbin`

kf kc iftkrigr nkc hukvg nk lsl`r` nur` y sukck skr iflud`n` pgr cgs p`nrks

kf uf fing. C` iflud`lif nur` kftrk 0= y 04 n`s kf `vks pkquk`s y puknk

prgcgfb`rsk h`st` 8> n`s kf kc l`sg nk `vks br`fnks, lgog kc `cd`trgs. ^gn`s

c`s `vks tikfkf ngs p`t`s, uf p`r nk `c`s y kc lukrpg ludikrtg lgf pcuo`s.

Qk` kf c` t`dc` `cbuf`s lc`sks nk `vks y c`s kspkliks quk pkrtkfklkf ` kcc`s.

Lc`sk @cbuf`s kspkliks

tkrrlgc s p`vg rk`c, b ccg, a isf, lhgopipk, `vkstruz

z`flun`s lisfk, brucc`, acòkflg, b`rz`

o`rif`s b`vigt`, pkcl`fg, pifbifg

r`p`lks cklhuz`, dhg, zgpicgtk, buic`

pe`rgs lgcidr, bgcgfnrif`, lkfzgftck, oircg, quktz c

r`p`lks

^kxtg `n`pt`ng nk C` kflilcgpkni` nk cgs `fio`cks. ]hicip Rhitaikcn.

>@. Lgfsinkrk cgs lgfeuftgs5

Q? { `, k, i, g, u}:K? { d, o, t}:A? { l, n, a, b} yB? { f, x, y}

Kst`dckzl` kf l`n` iflisg c` rkc`lif nk fg lgftkflif kslridikfng kc sodgcg . îkfk ufkekopcg.

=) K Q 0) A Q 4) B Q

D. Lgfsinkrk cgs lgfeuftgs5

]? { fokrgs p`rks okfgrks quk 08}:H? { 0=, 0> }:I? { 0, , 01} yJ? { 0, < , ; } Kslrid` kf kc ksp`lig nk l`n` iflisg (lgftkfing) g (fg lgftkfing), skbf lgrrkspgfn`.

îkfk uf kekopcg.

=) H ] ) H I 2) J I

4) E ] ;) H E 8) J E

>8IBKT Puiribu


61/350

Tkc`ligfks kftrk lgfeuftgs5

0. ]krtkfkfli`

C` pkrtkfkfli` ks c` rkc`lif kftrk uf kckokftg lgf ufg g os lgfeuftgs nk cgs lu`cksagro` p`rtk. ]`r` kst`dcklkr kst` rkc`lif sk uticiz` kc sodgcg quk sk ckk5 pkrtkfklk `.

Kekopcg5

\i kc lgfeuftg @? { `, k, i, g, u }, pgnkogs nklir quk k @

0.0 Fg pkrtkfkfli`

C` fg pkrtkfkfli` ks c` rkc`lif quk sk kst`dcklk kftrk uf kckokftg y ufg g os lgfeuftgs

nk cgs quk fgagro` p`rtk. \k ifnil` lgf kc sodgcg quk sk ckk5 "fg pkrtkfklk `..." Kekopcg5

\i kc lgfeuftg D? { o, g, r, ` },nkliogs quk kc kckokftg d D

C`s rkc`ligfks nk pkrtkfkfli` y fg pkrtkfkfli` sgcg sk kst`dcklkf kftrk kckokftgsy lgfeuftgs.

4. Lgftkflif

C` lgftkflif ks c` rkc`lif quk sk kst`dcklk kftrk ngs lgfeuftgs lu`fng tgngs cgs kck-okftgs nk uf lgfeuftg okfgr kstf iflcuings kf uf lgfeuftg o`ygr. C` lgftkflif sk

rkprkskft` lgf kc sodgcg y sk ckk kst lgftkfing kf.

Kekopcg5

N`ngs cgs lgfeuftgs5 @? { l, `, a, k }y D? { `, k } nkliogs quk D @

Kf agro` brl` kst` rkc`lif sk rkprkskft`5`

D

@l

a k

4.0 Fg lgftkflif

C` fg lgftkflif ks c` rkc`lif quk sk kst`dcklk kftrk ngs lgfeuftgs lu`fng `cbufgs gtgngs cgs kckokftgs nk ufg fg kstf iflcuings kf kc gtrg. C` rkc`lif nk fg lgftkflifsksiodgciz` y sk ckk5 fg kst lgftkfing kf.

Kekopcg5

N`ngs cgs lgfeuftgs5 Q? { 4, < } y R? { 4, ;, 2 } nkliogs quk Q R

C`s rkc`ligfks nk lgftkflif y fg lgftkflif sgcg sk kst`dcklkf kftrk lgfeuftgs.

>9O`tkotil` \ko`f` p`t`s } Q \

, ;, 9 } y ]? {Fokrgs f`tur`cks p`rks } P ]

>) R? {0=, 0;, 4= } y O? {Octipcgs nk ; } R U

;) N ? {]ktf, Puilh, \gfsgf`tk }y

J ?{Nkp`rt`okftgs nk Bu`tko`c` } N J

1) D? {=, ;, 8 } y ]? {Fokrgs nbitgs p`rks } D ]2) T? { `, k, n } yC? {k, s, p, ` n } T C

8) B ? {o`fz`f`, pi`, pkr` } y A? {arut`s } B A

9) I? {fgrtk, kstk, gkstk } y H? {]uftgs l`rnif`cks } I H

0=) \ ? {0


65/350

) K? {=, 4, >, 1, 8, 0=, 04 }D? {f, r } A? {4, 1, 0=, 04 }D @ A K

@ltivin`n } L? {0, ) ] 8) ]

0) ] ;) ] 9) ]

4) ] 1) ] 0=) ]


66/350

Y okogriz c`s t`dc`s nkc 0 `c >7 L`n` sko`f` tikfk c` gpgrtufin`n nk pr`ltil`r y `biciz`r kc lclucgokft`c. \u ksaukrzg y lgfst`fli` ck b`r`ftiz`rf dukfgs rksuct`ngs. Ck prgpgfkogs quk pr`ltiquk c`st`dc`s nkc ; y nkc 1 O`fgs ` c` gdr`!

Okogrlkc`s!

@ltivin`n 0Lgoprukdk si `prkfni c`s t`dc`s. Kslrid` kc rksuct`ng nk c`s ouctipcil`ligfks, prglurk fg lgpi`r.

=) ; x 0 ?

0) > x 4 ?

4) ; x 8 ?

?

>) ; x ; ?

;) ; x 2 ?

1) > x > ?

2) ; x 9 ?

8) > x 1 ?

9) ; x 4 ?

0=) > x 0 ?

00) > x 8 ?

04) > x < ?

0) > x 9 ?

0;) 1 x < ?

01) 1 x ; ?

02) 1 x 8 ?

08) 1 x 9 ?

09) 1 x 0= ?

4=) ; x 0= ?

40) < x 9 ?

44) 1 x > ?

4) ; x 1 ?

4;) > x 2 ?

41) < x 1 ?

42) 1 x 1 ?

48) < x 8 ?

49) 1 x 2 ?

^`dc nkc ;

; x 0 ? ;

; x 4 ? 0=

; x < ? 0;

; x > ? 4=

; x ; ? 4;

; x 1 ? ;

; x 0= ? ;=

^`dc` nkc 1

1 x 0 ? 1

1 x 4 ? 04

1 x < ? 08

1 x > ? 4>

1 x ; ? 8

1 x 9 ? ;>

1 x 0= ? 1=

;

;>IBKT Puiribu



67/350

Tksgculif nk prgdcko`s\ko`f` ` sko`f` ck prgpgfkogs kc rktg nk rksgcvkr prgdcko`s p`r` nks`rrgcc`r kc pkfsòikftgcbilg. ^r`dèk kf su lu`nkrfg lgf grnkf y ciopikz`. Nkek kslrit`s tgn`s c`s gpkr`ligfks qukrk`cilk p`r` quk su grikft`ngr vgcuft`rig pukn` rkvis`rc`s.

^gok kf lukft` quk p`r` rksgcvkr uf prgdcko` nkdk5

0. Ckkr lgf `tkflif y `skbur`rsk nk quk lgoprkfnk kcprgdcko`.

4. Tkackxigf`r.

. Gpkr`r.

;. Kslridir c` rkspukst`.

0) Luftgs n`s h`y kf 1 sko`f`s7

4) \i uf` nglkf` ks kc ngdck nk okni` nglkf , luft`s okni`s nglkf`s h`y kf 1 nglkf`s7

) @ykr auiogs `c okrl`ng y lgoprògs 9 nglkf`s nk ciogfks, > nglkf`s nk o`fn`-rif`s y < nglkf`s nk f`r`fe`s. Luft`s arut`s lgoprògs kf tgt`c7

;) O`rb`rit` dgrn` ; acgrks ni`ri`s. \i su bipic tikfk >; acgrks,

`. kf luftgs n`s tkroif`r 0 bipic7

d. kf luftgs n`s tkroif`r 1 bipicks7

1) [f lgokrli`ftk nispgfk nk uf tkrrkfg nk 4,;== o4p`r` lgfstruir uf` tikfn` nkùtgskrvilig, lgf rk` nk kst`ligfòikftg. \i c` tikfn` glup 0,24= o4,

`. qu rk` qukn nispgfidck p`r` kc kst`ligfòikftg7

d. luftgs l`rrgs puknkf p`rquk`r si c` fgro`tiv` gdcib` l`cluc`r 0= o4 pgrl`rrg7

2) Ngs gdrkrgs tr`dè`rgf ,;== quktz`cks kftrk cgs ngs. \i ufg nkkccgs b`f`d` 9= quktz`cks ni`rigs, luc kr` kc egrf`c ni`rig nkc gtrg7

8) @fnrk` tikfk < `gs y su hkro`fg tikfk kc ngdck nk kn`n. \i kc p`p nk @fnrk` tikfk4> `gs os quk su hieg, qu kn`n tikfk kc p`p7

;;O`tkotil` \ko`f`

Kc lgfeuftg F nk cgsfokrgs f`tur`cks

;2O`tkotil` \ko`f` >


70/350

Cgs cgbrgs quk lgfskbuir kst` sko`f`sgf5 ]r`ltil`r kc tr`zg nk cgs sodgcgs o`tkotilgs5

6 (o`ygr quk), 3 (okfgr quk), ? (ibu c `).

Nkfir y rkprkskft`r sgdrk uf` skoirrklt` fuoril`

kc lgfeuftg nk cgs fokrgs f`tur`cks. Lgop`r`r y grnkf`r fokrgs f`tur`cks.

Tklgrn`r y pr`ltil`r c`s t`dc`s nkc ouctipcil`r nkc 0 `c 8.

Nks`rrgcc`r kc r`zgfòikftg cbilg rksgcvikfng prg-dcko`s o`tkotilgs.


^ dc`s nk ouctipcil`r nkc 2 y kc 8@bicin`n nk lclucg okft`c

\odgcgs 6 (o`ygr quk),

3 (okfgr quk), ? (ibu`c `)Ckfbuèk o`tkotilg

Kc oufng nk c` o`tkotil` Cgs fokrgs f`tur`cks (F)

]rgdcko`s o`tkotilgs lgf c`s

t`dc`s nkc 0 `c 8T`zgfòikftg cbilg

Log f`likrgf cgs fokrgs7

C` o`r`viccgs` histgri` nk cgs fokrgs]`r` lgokfz`r!

;8IBKT Puiribu


71/350

Log f`likrgf cgs fokrgs7

]`r` lgokfz`r!

@ tr`dè`r!

0. \kbf c` ckltur`, luc auk c` priokr` agro` nk lgft`r nkc skr huo`fg7

4. Kc lgftkg nk gdektgs sk nig log uf` nistr`llif, lgog uf` fklksin`n g pgrl`su`cin`n7

H`lk oulhgs, oulhsiogs `gs, c`s pkrsgf`s viv`f kf pkqukgs brupgs,

kf lukv`s ngfnk sk kslgfn`f nk cgs `fio`cks pkcibrgsgs y sk prgtkb`f nkc

o`c tikopg. L`z`d`f `fio`cks p`r` `ciokft`rsk y p`r` s`dkr luftgs h`d`f

`tr`p`ng, o`rl`d`fuf p`cg lgf sk`cks.

]`s`rgf oulhgs `gs p`r` lòdi`r c` agro` nk vin`5 c`s pkrsgf`s sk grb`fi-

z`rgf kf tridus, sk nivinikrgf kc tr`dèg kftrk sus oikodrgs, ktl. Cgs p`stgrks,

pgr kekopcg, sk kfl`rb`rgf nk bu`rn`r cgs rkd`gs, rklgbkr c` c`f` nk c`s

gvke`s y su cklhk.

Y kstgs p`stgrks, log h`l`f p`r` lgft`r c`s gvke`s7 ]rgd`dckokftk, skbf

s`c` l`n` `fio`c` p`st`r `c lòpg, kc p`stgr okt` uf` piknr`kf su ogrr`c.

Lgop`r`fng l`ftin`nks, c` huo`fin`n lgfstruy kc lgflkptg nk fokrg. ] r`

cgs prioitivgs, kc hklhg nk lgft`r nkd` nk kst`r ouy rkc`ligf`ng lgf piknr`s,

p`cgs, o`rl`s, nkngs, ktl. Kc lgflkptg nk fokrg surbi lgog lgfsklukfli`

nk c` fklksin`nnk lgft`r gdektgs.

^kxtg `n`pt`ng nk C` o`r`viccgs` histgri` nk cgs fokrgs.

Ouskg virtu`c nk c` likfli`. Lgfskeg \upkrigr nk Ifvkstib`ligfks Likftail`s (L\IL), Ksp``.

;9O`tkotil` \ko`f` >


72/350

\ib` pr`ltil`fng kc ckfbu`ek o`tkotilg! Kst` sko`f` uticiz`rkogs cgs sodgcgs5

6(O`ygr quk)

3(Okfgr quk)

?(Ibu`c quk) g (ibu`c `)

gok su c`pilkrg y rkp`sk l`n` sibfg. \ib` c` nirkllif quk ck ifnil`f c`s klh`s.

@. 6 (O`ygr quk)

6 6 6 6 6 6 6 6 6 6 6

6 6 6 6 6 6 6 6 6 6 66 6 6 6 6 6 6 6 6 6 6

D. 3 (Okfgr quk)

3 3 3 3 3 3 3 3 3 3 3

3 3 3 3 3 3 3 3 3 3 3

3 3 3 3 3 3 3 3 3 3 3

L. ? (Ibu`c `)

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

? ? ? ? ? ? ? ? ? ? ?

1=IBKT Puiribu

Ckfbu`ek o`tkotilg


73/350

Kekrlilig 0@. Kslrid` uf` l`r`ltkrstil` nkc lgfeuftg nk fokrgs f`tur`cks.

D. Kslrid` kc fokrg quk `ftklknk y kc quk sulknk ` cgs sibuikftks fokrgs f`tur`cks. ^ikfkuf kekopcg.

, 08, , 0=;, , 0429,

0. Cgs fokrgs f`tur`cks (F) [f` hkrr`oikft` p`r` lgft`r

[stkn lgfglk ouy dikf cgs fokrgs f`tur`cks pgrquk cgs uticiz` sikoprk nksnkquk `prkfni ` lgft`r y cgs sibuk uticiz`fng kf sus lukft`s ni`ri`s, vkrn`n7

Kc lgfeuftg nk cgs fokrgs f`tur`cks kst agro`ng pgr cgs fokrgs quksk uticiz`f p`r` lgft`r. Kstk lgfeuftg sk inkftial` lgf c` cktr` o`ysluc`

F.

Kf kc priflipig, kc lgfeuftg nk cgs fokrgs f`tur`cks kst`d` agro`ng pgr cgsfokrgs nkc 0 kf `nkc`ftk. Cukbg, kf kc sibcg QII n.L., cgs hifnks ifvkft`rgfkc sodgcg quk rkprkskft` kc v`lg, c` f`n`, ksk sodgcg ks kc fokrg lkrg.

@ p`rtir nk kftgflks kc lgfeuftg nk cgs fokrgs f`tur`cks kst agro`ng pgrcgs fokrgs nkc = h`st` kc iffitg5

F? =, 0, 4, , ;, 1, 2, 8, 9, 0=, 00,

0.0 L`r`ltkrstil`s nkc lgfeuftg nk cgsfokrgs f`tur`cks

Kc lgfeuftg nk fokrgs f`tur`cks tikfk c`s sibuikftks l`r`ltkrsitl`s5

\u priokr kckokftg ks kc fokrg lkrg. Ks uf lgfeuftg ifaifitg. Fg pgnkogs lgfglkr kc ctiog fokrg.

F? =, 0, 4,



74/350

0.4 Tkprkskft`lif nkc lgfeuftg nk fokrgsf`tur`cks sgdrk c` skoirrklt` fuoril`

^gngs tikfkf su cub`r

@c ibu`c quk uf` agtgbrà` ks c` rkprkskft`lif nk uf` pkrsgf`, cgs fokrgsf`tur`cks sk puknkf rkprkskft`r pgr oknig nk puftgs sgdrk uf` skoirrklt`.

Lgfgzlògs c` agro` nk rkprkskft`rcgs5

Nidueògs uf` skoirrklt`.

O`rlògs uf puftg quk ifnil` kc ifilig nk c` skoirrklt` kf kc c`ngizquikrng, y h`li` c` nkrklh` o`rlògs puftgs kquinist`ftks0.

\k kslridk lkrg (=) nkdèg nkc priokr puftg y ` su nkrklh` cgs fokrgslgfsklutivgs.

= 0 4 < > ; 1...

\koirrklt`5 l`n` uf`nk c`s p`rtks kf qukqukn` nivinin` uf`rklt`. îkfk uf puftgnk gribkf y, pgr gtr`p`rtk sk kxtikfnkh`li` kc iffitg.

C` puft` nkklh` ifnil` quk kclgfeuftg nk fokrgsf`tur`cks ks iffitg.

Kekrlilig 4Cgl`cilk y kslrid`, dèg kc puftg lgrrkspgfnikftk, cgs fokrgs nk l`n` lgfeuftg. @cbufgs f-okrgs y` kstf inkftil`ngs.

0) @? =, , 2, 00

) N? =, ;, 9, 0 hiegs. Kf c` t`dc` tkfkogs kc fgodrk y c` kn`n

nk l`n` ufg.

Fgodrk Kn`n

L`rcgs 02

Koici` 0;

Tkf 0;

Tgs`rig 04

@c lgop`r`r c` kn`n nk `cbuf`s p`rke`s nk hkro`fgs tkfkogs5

L`rcgs ks o`ygr quk Koici` pgrquk 02 ks o`ygr quk 0;.

O`tkotilòkftk cg kxprksògs5 02 60;\k ckk5 02 ks o`ygr quk 0;

Tgs`rig ks okfgr quk Tkf pgrquk 04 ks okfgr quk 0;.

O`tkotilòkftk cg kxprksògs5 04 30;\k ckk5 04 ks okfgr quk 0;

Koici` tikfk c` oiso` kn`n quk Tkf pgrquk 0; ks ibu`c ` 0;.

O`tkotilòkftk cg kxprksògs5 0; ?0;\k ckk5 0; ks ibu`c ` 0;

Kekrlilig 8 o).

]krg, qu p`s`r` si c` `cpifist` hilikr` su rklgrring `c rkvs7

Qkògs5

]`r` grnkf`r uf` skrik nk o`ygr ` okfgr (grnkf nkslkfnkftk)5

H`lkogs c` cist` nk c` `ctur` nk c`s ogft``s, cukbg inkftiailògs cògft`` os `ct` y c` sk`cògs5

8,4=0 8,8>8 8,01< 8,;81

Kslridiogs c` o`ygr `ctur` (8,8>8) p`r` ifili`r c` skrik. Cukbg kslridiogs` c` nkrklh` y kf grnkf nk v`cgr, c`s `ctur`s okfgrks5

8,8>8 8,;81 8,4=0 8,018 o

O`f`scu 8,01< o

J`flhkfeufb` 8,;81 o

1>IBKT Puiribu


77/350

0. Cgs fokrgs f`tur`cks agro`f kc lgfeuftg fuorilg quk sirvk p`r` lgft`r.

0.0 Kc lgfeuftg nk cgs fokrgs f`tur`cks sk inkftiail` lgf c` cktr` o`ysluc` F.

Kc lgfeuftg nk cgs fokrgs f`tur`cks ks uf lgfeuftg ifaifitg,pgrquk fg sk puknklgfglkr kc ctiog fokrg. Kf agro` kfuokr`tiv` sk rkprkskft` `s5

F? =, 0, 4, , ;, 1, 2, 8, 9, 0=, 00,...

0.4 ^ odif sk puknk rkprkskft`r sgdrk uf` skoirrklt`. Kc puftg nk gribkf nk c` skoirrklt` lg-rrkspgfnk `c lkrg. C` puft` nk klh` `c f`c nk c` rklt` ifnil` quk kc lgfeuftg ks iffitg.

0. ; 1 2 8 9 0= 00

Kekrlilig >@. Gdskrvk l`n` skrik y cukbg kslrid` sgdrk c` cfk` si kst grnkf`n` kf agro` `slkfnkftk g

nkslkfnkftk.

0) 2>99 2;08 2;41

4) ;12; ;;99 ;;8;

4188

D. Gdskrvk l`n` skrik, inkftiaiquk kc fokrg okfgr y cukbg grnfkc` nk agro` `slkfnkftk.

0) 02; 4 098 >00 10 000 , , , , ,

4) 2>0 ;01 28; 8=1 ;;1 , , , ,

L. Gdskrvk l`n` skrik, inkftiaiquk kc fokrg o`ygr y cukbg grnfkc` nk agro` nkslkfnkftk.

0) 4< 02 ;1 12 04> >0 , , , , ,4) 00 0; 8 0= 0> , , , ,

0 08; 0=4 4

Tksuokf


78/350

@ltivin`n 0 Nkoukstrk cg `prkfning af` su `prkfniz`ek.@. Cgs fokrgs f`tur`cks agro`f uf lgfeuftg fuorilg lgf likrt`s l`r`ltkrstil`s. Kslrid` ngs

l`r`ltkrstil`s.

0)

4)

D. Nkf` kc lgfeuftg nk fokrgs f`tur`cks lgf sus p`c`dr`s.

L. ^gng fokrg f`tur`c, okfgs kc lkrg, tikfkf uf `ftklksgr y uf sulksgr. Lgopcktk c` t`dc`kslridikfng kc `ftklksgr y kc sulksgr nk cgs fokrgs n`ngs. ^ikfk uf kekopcg.

=) 0==

0) 2;8

4) 4==


79/350

A. Lgop`rk l`n` p`rke` nk fokrgs y kslrid` sgdrk c` cfk` kc sodgcg 6 , 3 ? , skbf lgrrkspgfn`^ikfk uf kekopcg.

0) 8 ;

4) 01 40

< ;0

>) 21 1>

;) 0; 02

1) >< >0

2) 8; 4

12O`tkotil` \ko`f` >


80/350

@ltivin`n 4 ]ikfsk y `pciquk cg quk `prkfni

0) Grnkfk cgs sibuikftks nkp`rtòkftgs nk Bu`tko`c` nk o`ygr` okfgrskbf kc fokrg nkh`dit`ftks. îkfk uf kekopcg.

Nkp`rtòkftgs L`ftin`n

nk h`dit`ftks]ktf ;0

Ksluiftc` 1


81/350

]`s Kxtkfsiftkrritgri`c kf jo4

Bu`tko`c` 0=8,889

Dkcilk 44,911Kc \`cv`ngr 40,=>0

Hgfnur`s 004,>94

Fil`r`bu` 04=,=

Lgst` Til` ;0,0==

]`fò 2;,;02

Fgodrk \`c`rigIro` 4,

Cgrkf` 0,8>9

Egrbk 0,99=

>) Grnkfk kc fgodrk nk cgs kopck`ngs nk `lukrng ` su s`c`rig okfsu`c. Nkc s`c`rig os `ctg(o`ygr)`c s`c`rig os dèg (okfgr). îkfk uf kekopcg.

]`sks Kxtkfsif kf jo4

Kc \`cv`ngr 40,=>0

^go`ng nk5 www.wijipkni`.grb



82/350

C` fil` agro` nk ngoif`r c`s t`dc`s nk ouctipcil`r ks pr`ltilfngc`s.

Tkp`sk y okogrilk c`s t`dc`s nkc 2 y kc 8.Lu`fng y` c`s ngoifk, rk`cilk c` `ltivin`n 0.

@ltivin`n 0Nkspus nk kstuni`r c`s t`dc`s, kslrid` cgs rksuct`ngs nk l`n` ouctipcil`lif. Gdskrvk quk `ufquksk l`odik kc grnkf nk cgs fokrgs kc rksuct`ng sibuk sikfng ibu`c. Hb`cg cg os rping qukpukn`.

0) 2 x 0 ?

4) 8 x 4 ?

) 2 x > ?

;) 2 x 2 ?

1) 2 x < ?

2) 8 x 2 ?

8) 2 x 8 ?

9) 2 x 9 ?

0=) 8 x 0= ?

00) 8 x 2 ?

04) 8 x 8 ?

0) > x 2 ?

0;) ; x 2 ?

01) 1 x 2 ?

02) 8 x 8 ?

08) 8 x 9 ?

09) 9 x 2 ?

4=) 0= x 2 ?

^`dc nkc 2

2 x 0 ? 2

2 x 4 ? 0>

2 x < ? 40

2 x > ? 48

2 x ; ? 4

2 x 2 ? >9

2 x 8 ? ;1

2 x 9 ? 1

8 x > ? =

8 x 1 ? >8

8 x 2 ? ;1

8 x 8 ? 1>

8 x 9 ? 24

8 x 0= ? 8=

40) 0 x 2 ?

44) 8 x 0 ?

4) 8 x < ?

4;) > x 8 ?

41) 8 x ; ?

42) 1 x 8 ?

48) 8 x 2 ?

49) 2 x > ?


83/350

Tksgculif nk prgdcko`s@. Tksukcv` cgs prgdcko`s kf su lu`nkrfg. Nkek kslrit`s tgn`s c` gpkr`ligfks quk rk`cilk.

0) Ngf Dktg tikfk ; lòigfks y l`n` ufg tikfk 8 cc`ft`s.

`. Luft`s cc`ft`s tikfkf tgngs sus lòigfks euftgs7d. \i l`n` cc`ft` lukst` P.;==, luftg nifkrg fklksit` p`r` pgfkr cc`ft`s fukv`s ` uf

lòif7

4) O`ritz` tikfk < b`ccif`s. [f` b`ccif` pgfk uf hukvg ni`rig `prgxio`nòkftk pgr 4>=n`s.

`. Luc skr c` prgnullif tgt`c nk sus b`ccif`s kf 4>= n`s7d. \i lgopr` 4 b`ccif`s os, luc skr su prgnullif tgt`c nk hukvgs kf kc oisog

tikopg7) \k prkp`r`rgf dgcs`s nk nuclks p`r` cgs figs y c`s fi`s nk c` lgoufin`n. L`n` dgcs`tikfk 8 l`ròkcgs, < p`ckt`s y 1 lhilcks.

`. Luft`s p`ckt`s sk kop`l`rgf kf 0= dgcs`s7d. Luftgs l`ròkcgs h`y kf 2 dgcs`s7l. Luftgs lhilcks h`y kf 8 dgcs`s7

;) Kf kc okrl`ng h`y 8; pukstgs nk vkft`. L`n` pukstg p`b` 8 quktz`cks pgr c` kxtr`l-lif nk d`sur`.

`. Luftg p`b`f kf tgt`c tgngs cgs pukstgs7d. \i sk `brkb`f 0; pukstgs os, luc skr kc tgt`c quk nkdkf p`b`r7

1) Kc l`ckfn`rig brlgc` nk cgs o`y`s sk nivin` kf 08 oksks nk 4= n`s l`n` ufg, os ufpkrgng nk ; n`s. Luftgs n`s tkf` kf tgt`c kc l`ckfn`rig `brlgc` nk cgs o`y`s7

2) \k rkp`rtk uf` lgckllif nk nislgs kftrk trks b`f`ngrks nk uf prkoig. @ l`n` b`f`ngrsk ck kftrkb`f 0; nislgs y sgdr`f 4. Luftgs nislgs h`d` kf c` lgckllif7

8) @cili` lgopr uf kquipg nk sgfing kf 8 lugt`s nk 4=kslritgrigs pgr br`ng, luftgs kslritgrigs fklksit` kf tgt`c kc oufilipig7


T`zgfòikftg cbilg


84/350



prglksgfg

cgbr`ng

Nkspusnkkstuni`r...

Inkftilg y tr`zg cgs sibfgs 6, 3, ?.

Nkfg lgf ois p`c`dr`s kc lgfeuftg F nk fokrgs f`tur`cks.

Tkprkskftg sgdrk uf` skoirrklt` fuoril` kc lgfeuftg F nkfokrgs f`tur`cks.

Lgop`rg y grnkfg fokrgs f`tur`cks.

Nks`rrgccg oi `bicin`n nk lclucg okft`c pr`ltil`fng c`s t`dc`snkc 0 `c 8.

Tksukcvg prgdcko`s o`tkotilgs kf cgs quk `pcilg cgs lgfglioikf-tgs `nquirings.


24IBKT Puiribu


85/350

;

Gpkr`ligfks yprgpikn`nks kf kc

lgfeuftg nk cgs fokrgsf`tur`cks (F)

2IBKT Puiribu


87/350

Kc d`lg

@ tr`dè`r!

0. ]`r` qu sirvk kc d`lg7

4. Gdskrvk qu l`ftin`n sk rkprkskft` kf kc d`lg nk `dèg. Cukbg, rkcckfk kc lu`nrgnk c` gplif quk ifnil` c` l`ftin`n lgrrklt`. \i tikfk nun`, lgop`rk lgf kc d`lg

nk c` icustr`lif nk c` ckltur`.

Kc d`lg ks lgfsinkr`ng lgog kc os `ftibug ifstruokftg nk lclucg, `n`pt`ng

y `prkli`ng kf nivkrs`s luctur`s. \k uticiz` p`r` rk`ciz`r suo`s, rkst`s y ouc-

tipcil`ligfks skflicc`s. Bkfkr`cokftk, kc d`lg kst agro`ng pgr uf lu`nrg

nk o`nkr` `tr`vks`ng pgr nikz v`ricc`s p`r`ckc`s, lgf nikz dgc`s ovicks l`n`

uf`. Kf c`s v`ricc`s nkc d`lg pgnkogs rkprkskft`r5 ufin`nks ([), nklkf`s (N),

lkftkf`s (L), ufin`nks nk oicc`r ([O), ktl. Kst` nispgsilif pkroitk l`cluc`r

lgf a`licin`n y r`pinkz.

C` p`c`dr` lclucgvikfk nkc c`tf l`clucus quk sibfiail` piknr`. Kf cgs priok-

rgs d`lgs, c`s dgc`s ovicks kr`f nk piknr`.

Kf c` `ltu`cin`n, kc usg nk d`lg ks lgof kf Lhif` y kf E`pf. Cgs usu`rigs

kxpkrtgs sgf l`p`lks nk rk`ciz`r gpkr`ligfks os rpinòkftk quk lgf uf`

l`cluc`ngr` kckltrfil` ognkrf`.@n`pt`ng nk Oilrgsgat Kfl`rt` 4==2

4,;12

21;,>

2;O`tkotil` \ko`f` ;

]`r` lgokfz`r!


88/350

gok su c`pilkrg y rkp`sk l`n` sibfg. \ib` c` nirkllif quk ck ifnil`f c`s klh`s.

@. Kc sibfg

? (ibu`c) kst agro`ng pgr ngs cfk`s hgrizgft`cks p`r`ckc`s quk sk tr`z`f nk

izquikrn` ` nkrklh`5

? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ?Niduek kc sibfg ibu`c kf l`n` cfk`.

Kc sibfg ? sibfil`5

D. Kc sibfg (fg ibu`c) kst agro`ng pgr ngs cfk`s hgrizgft`cks p`r`ckc`s y t`lh`ng lgf uf`cfk` iflcif`n`.

Niduek kc sibfg fg ibu`c kf l`n` cfk`.

Kc sibfg sibfil`5

Hgy rklgrn`rkogs cgs sibfgs5 ?(ibu`c) y (fg ibu`c).

Kc sibfg ? (ibu`c) kst`dcklk c` rkc`lif nk ibu`cn`n kftrk ngs fokrgs, ngs gpkr`ligfks, ktl.

Kc sibfg (fg ibu`c) kst`dcklk c` rkc`lif nk nksibu`cn`n kftrk ngs fokrgs g ngs gpkr`ligfks.Gdskrvk cgs kekopcgs5

4 + < ? ; 0 + < ;

21IBKT Puiribu

Ckfbu`ek o`tkotilg


89/350

0. Gpkr`ligfks kf kc lgfeuftg

nk cgs fokrgs f`tur`cks (F)C`s gpkr`ligfks suo`, rkst`, ouctipcil`lif y nivisif, sgf vike`s lgfglin`s. C`shkogs pr`ltil`ng nur`ftk tgn` c` prio`ri` y pgnkogs rksgcvkrc`s lgf a`licin`n.Kst` sko`f` c`s rklgrn`rkogs.

0.0 \uo` nk fokrgs f`tur`cks Tkufif nk l`ftin`nks

C` suo` g `nilif ks c` gpkr`lif quk rkfk v`ri`s l`ftin`nks kf uf` sgc`.

Kf ckfbuèk o`tkotilg sk kxprks`5` + d ? l

C`s l`ftin`nks quk sk rkfkf sk ccò`f suo`fngsy c` l`ftin`n quk rksuct`ks c` suo` g tgt`c.

`+ d

l` + d ? l

suo`fngs

suo` g tgt`c

Kekopcg5

>+ + < ? 2

0.4 Tkst` nk fokrgs f`tur`cks Niakrkfli` kftrk ngs l`ftin`nks

C` rkst` g sustr`llif ks c` gpkr`lif ifvkrs` c` suo`. \k uticiz` p`r` h`cc`r c`niakrkfli` kftrk ngs l`ftin`nks. Kf ckfbuèk o`tkotilg sk kxprks`5

` d ? l

Kc oifukfngks kc fokrg o`ygr, c quk sk ck rkst` gtr` l`ftin`n. Kc sustr`kfng

ks kc fokrg okfgr, ks c` l`ftin`n rkst`n`.Kc rksuct`ng ks c` niakrkfli`.

`d

l` d ? l

oifukfng

niakrkfli`

sustr`kfng

Kekopcg5

84

1

8 4 ? 1

22O`tkotil` \ko`f` ;



90/350

@c ouctipcil`fng y`c ouctipcil`ngr skcks lgfglk t`odiflgog a`ltgrks.

suo` rkst` ouctipcil`lif nivisif; + 1 ? 00 1 ? 8 x < ? 4> 8 ?

9 + 8 ? 02 8 ? 9 x ; ? >; 9 ?

< + 2 ? 0= 2 ? 2 x > ? 48 2 ?

Kekrlilig 0Tksukcv` okft`cokftk y kslrid` kc rksuct`ng. ^ikfk uf kekopcg kf l`n` lgcuof`.

00 4>;

4> < ? 8

28IBKT Puiribu


91/350

4. ]rgpikn`nks nk c`s gpkr`ligfksnk cgs fokrgs f`tur`cks (F)

[f` prgpikn`n ks uf` l`r`ltkrstil`, uf` lu`cin`n kskfli`c. C`s prgpikn`nks

o`tkotil`s fgs pkroitkf `biciz`r c`s gpkr`ligfks y s`dkr qu rksuct`ngs pg-nkogs kspkr`r.

4.0 ]rgpikn`nks nk c` suo`

`. ]rgpikn`n lgfout`tiv` Lòdi`fng kc grnkf

C` prgpikn`n lgfout`tiv` kxprks` quk kc grnkf nk cgs suo`fngs fg `ctkr`

kc rksuct`ng. Kf ckfbuèk o`tkotilg sk kxprks`5

` + d ? d + `

Kekopcg5 08 + 04 ? 04 + 08

Lgfout`tiv`vikfk nk c` p`c`dr`lgfout`r quksibfil` iftkrlòdi`rg lòdi`r uf` lgs`pgr gtr`.

@sgli`tiv`vikfk nk c` p`c`dr``sgli`r quk sibfil`

euft`r g `sgli`r lgs`sg ink`s.

l. Kckokftg fkutrg

Kc kckokftg fkutrg ks `qukc quk `c suo`rcg ` lu`cquikr fokrg f`tur`c, n`lgog rksuct`ng kc oisog fokrg. Kc kckokftg fkutrg nk c` suo` ks kc = .Kf kc ckfbuèk o`tkotilg sk kxprks`5

` + =? `

Kekopcg5 0> + =? 0>

d. ]rgpikn`n `sgli`tiv` Agro`fng brupgs

C` prgpikn`n sgli`tiv` ro` quk fg iopgrt` log sk brupkf cgs suo`f-ngs p`r` gpkr`rcgs, kc rksuct`ng ks kc oisog. Kf ckfbuèk o`tkotilg skkxprks`5

(` + d) + l ? ` + (d + l)

Kekopcg5 (9 + 8) + 4 ? 9 + (8 + 4)

02 + 4 ? 9 + 0=

09 ? 09



92/350

D. Lgopcktk c` gpkr`lif y lgoprukdk quk sk luopck c` prgpikn`n `sgli`tiv` nk c` suo`.^ikfk uf kekopcg.

=) (2 + >) + 4 ? 2 + (> + 4) 0) (; + 9) + 1 ? ; + (9 + 1)

00 + 4 ? 2 + 1 + ? +

0< ? 0< ?

L. @pciquk c` prgpikn`n nkc kckokftg fkutrg p`r` lgopckt`r c`s suo`s. ^ikfk uf kekopcg.

=) 019 + ? 019 4) + = ? 4==

0) 4> + ? 4>


93/350

4.< ]rgpikn`nks nk c` ouctipcil`lif

`. ]rgpikn`n lgfout`tiv` Kc grnkf fg lòdi` kc rksuct`ng

C` prgpikn`n lgfout`tiv` nk c` ouctipcil`lif nilk quk kc grnkf nk cgs a`ltg-rks fg `ctkr` kc prgnultg. Kf ckfbuèk o`tkotilg sk kxprks`5

` x d ? d x `

Kekopcg5 ; x 8 ? 8 x ;

>= ? >=

d. ]rgpikn`n `sgli`tiv`

H`likfng brupgsC` prgpikn`n `sgli`tiv` nk c` ouctipcil`lif nilk quk c` o`fkr` kf quk sk`brup`f cgs a`ltgrks fg `ctkr` kc prgnultg. Kf ckfbuèk o`tkotilg skkxprks`5

(` x d) x l ? ` x (d x l)

Kekopcg5 (> x 4) x ; ? > x (4 x ;)

8 x ; ? > x 0=

>= ? >=

C` prgpikn`n `sgli`tiv` nk c` ouctipcil`lif fgs pkroitk `brup`r cgs a`ltgrksnk c` agro` os lgfvkfikftk p`r` a`licit`r c` gpkr`lif.

Gtrg kekopcg5 ; x (< x 4) ? (; x


94/350

Kf rksuokf...

n. Nistridutiv` nk c` ouctipcil`lif rkspkltg `c` suo` y ` c` rkst`

C` prgpikn`n nistridutiv` kxprks` quk p`r` ouctipcil`r uf fokrg f`tur`cpgr uf` suo` g rkst` ifnil`n`, sk ouctipcil` nilhg fokrg pgr l`n` ufg

nk cgs v`cgrks y sk suo`f g sk rkst`f cgs rksuct`ngs.Kf ckfbu`ek o`tkotilg sk kxprks`5

]rgpikn`n nistridutiv` rkspkltg ` c` suo`5 ` x (d + l) ? (` x d) + (` x l)

]rgpikn`n nistridutiv` rkspkltg ` c` rkst`5 ` x (d l) ? (` x d) (` x l)

Gdskrvk cgs sibuikftks kekopcgs5

0) ]rgpikn`n nistridutiv` nk c` ouctipcil`lif rkspkltg ` c` suo`5

2 x (< + 4) ? (2 x

x x ;)

Nistridutiv` rkspkltg` c` rkst`

` x (d l) ? ( x d) (` x l) > x (; x ;) (> x


95/350

Kekrlilig ? x x 2 x 0 ?

4) (4 x


96/350

4.> ]rgpikn`nks nk c` nivisif

`. Kckokftg fkutrg

C` nivisif sgcg luopck lgf c` prgpikn`n nkc kckokftg fkutrg sikoprk

quk kc 0 glupk c` pgsilif nkc nivisgr. Kf ckfbuèk o`tkotilg sk kxprks`5 ` 0? `

Qkògs uf kekopcg5 2 0? 2

Kekrlilig >@. Kslrid` uf kekopcg nk l`n` uf` nk c`s prgpikn`nks nk c` ouctipcil`lif quk kstuniògs. Ck

`yunògs lgf c` prgpikn`n lgfout`tiv`.

Gpkr`lif N n l

=) 4> < ? 4> < 8

0) 4) ;=< 0 ?

0) 4=> ? 4=> IBKT Puiribu


97/350

0. Gpkr`ligfks kf kc lgfeuftg nk cgs fokrgs f`tur`cks

Gpkr`ligfks y sus p`rtks

0.0 \uo` (rkufif nk l`ftin`nks)

C` suo` ks c` gpkr`lif quk rkfk v`ri`s l`ftin`nks kf uf`sgc`.

0.4 Tkst` (niakrkfli` kftrk ngs l`ftin`nks)

C` rkst` ks c` gpkr`lif ifvkrs` ` c` suo`. \k uticiz` p`r` h`cc`rc` niakrkfli` kftrk ngs l`ftin`nks.

0.< Ouctipcil`lif (prgnultg nk l`ftin`nks)

C` ouctipcil`lif ks c` gpkr`lif quk lgfsistk kf ùokft`r uffokrg (ouctipcil`fng) t`ft`s vklks lgog ifnil` kc ouctipci-l`ngr, p`r` gdtkfkr uf rksuct`ng g prgnultg.

@c ouctipcil`fng y `c ouctipcil`ngr tòdif sk cks ccòà`ltgrks.

0.> Nivisif (rkp`rtg kf p`rtks ibu`cks)

C` nivisif ks c` gpkr`lif ifvkrs` ` c` ouctipcil`lif. Lgfsistkkf rkp`rtir uf` l`ftin`n kf p`rtks ibu`cks.

`+ d

l

suo`fngs

suo` g tgt`c

` d

l niakrkfli`

oifukfng

sustr`kfng

`x d

l prgnultg

ouctipcil`fng

ouctipcil`ngr

nivisgr

l

n N

lglikftk

nivinkfng

4. ]rgpikn`nks nk c`s gpkr`ligfks nk cgs fokrgsf`tur`cks

4.0 ]rgpikn`nks nk c` suo`

C` suo` luopck c`s prgpikn`nks5 lgfout`tiv`, `sgli`tiv` y kckokftg fkutrg.

`. Lgfout`tiv`

Kc grnkf nk cgs suo`fngs fg `ctkr` kc rksuct`ng.

d. @sgli`tiv`

Fg iopgrt` log sk `brupkf cgs suo`fngs p`r` gpkr`r-cgs, kc rksuct`ng ks kc oisog.

` + d ? d + `

(` + d) + l ? ` + (d + l)

8;O`tkotil` \ko`f` ;

Tksuokf


98/350

l. Kckokftg fkutrg

Kc kckokftg fkutrg nk c` suo` ks kc =. Lu`cquikr fokrgsuo`ng ` lkrg, n` lgog rksuct`ng kc oisog fokrg.

4.4 ]rgpikn`nks nk c` rkst`

`. Kckokftg fkutrg

C` rkst` sgcg luopck lgf c` prgpikn`n nkc kckokftg fkutrgsikoprk quk kc = glupk c` pgsilif nkc sustr`kfng.

` + = ? `

` = ? `

` 0 ? `

4.> ]rgpikn`nks nk c` nivisif

`. Kckokftg fkutrg

C` nivisif luopck lgf c` prgpikn`n nkc kckokftg fkutrgsikoprk quk kc 0 glupk c` pgsilif nkc nivisgr.

4.< ]rgpikn`nks nk c` ouctipcil`lif

C` ouctipcil`lif luopck c`s prgpikn`nks5 lgfout`tiv`, `sgli`tiv`, kckokftg fkutrgy nistridutiv` lgf rkspkltg ` c` suo` y ` c` rkst`.

`. Lgfout`tiv`

Kc grnkf nk cgs a`ltgrks fg `ctkr` kc prgnultg.

d. @sgli`tiv`

C` agro` kf quk sk `brup`f cgs a`ltgrks fg `ctkr` kcprgnultg.

l. Kckokftg fkutrg

Kc kckokftg fkutrg nk c` ouctipcil`lif ks c` ufin`n, kc 0.

Ouctipcil`r lu`cquikr fokrg f`tur`c pgr 0, n` lgog

rksuct`ng kc oisog fokrg f`tur`c.

` x d ? d x `

` x (d x l) ? (` x d) x l

` x 0 ? `

`x(d + l) ? (` x d) + (` x l)

`x(d l) ? (` x d) (` x l)

n. Nistridutiv` lgf rkspkltg ` c` suo` y ` c` rkst`

]`r` ouctipcil`r uf fokrg f`tur`c pgr uf` suo` g rkst`ifnil`n`, sk ouctipcil` nilhg fokrg pgr l`n` ufg nk cgsv`cgrks y sk suo`f g sk rkst`f cgs rksuct`ngs.

Nistridutiv` rkspkltg ` c` suo`5

Nistridutiv` rkspkltg ` c` rkst`5

81IBKT Puiribu


99/350

@ltivin`n 0 Nkoukstrk cg prkfning af` su `prkfnizèk.@. Kslrid` c`s p`rtks nk l`n` gpkr`lif. îkfk uf kekopcg.

\uo`5

D. Ck` l`n` lgflkptg y kslrid` uf` ar`sk u gtr`s p`c`dr`s quk sibfiqukf cg oisog. îkfk ufkekopcg.

ouctipcil`r5suo`r5

rkst`r5

nivinir5

L. Kslrid` kc fgodrk nk c` prgpikn`n quk sk `pcil` kf l`n` gpkr`lif. îkfk uf kekopcg.

=) 0; + = ? 0;

0) 0> + 4 ? 4 + 0>

4) (> + 1) + 8 ? > + (1 + 8)

x 0 ? ;>

>) (4 x 0 1> 8 9 =8 < = 1

< > 1 20 4 < >4 4 < 8 > ; 4

Tkst`5

Ouctipcil`lif5

@c ouctipcil`fng y ouctipcil`ngr tòdif

sk cks ccò`5

ùokft`r uf fokrg, suo` `drkvi`n`

suo`fngs

Nivisif5

4 =02 < > = < > =

Kckokftg fkutrg nk c` suo`


@utglgftrgc


100/350

N. Lgopcktk c` t`dc` kslridikfng kc fgodrk nk l`n` prgpikn`n nk c` suo` y uf kekopcg.

@ltivin`n 4 ]r`ltiquk cg `prkfning

@. Tksukcv` okft`cokftk y kslrid` kc rksuct`ng. îkfk uf kekopcg kf l`n` lgcuof`.

42 ;0 ? 0= 4 ?

0> + ; ? 4> > ? 9 x ; ? 4= > ?

01 + > ? 09 2 ? 0> x 0 ? 0> 4 ?

4; + 0 ? >= 1 ? 1 x ; ? + 9 ? 11 2 ? 0== x 8 ? 2; < ?

80 + 8 ? 2; 04 ? 4== x ; ? 0;= ; ?

D. Lgopcktk l`n` gpkr`lif y lgoprukdk quk sk luopc` c` prgpikn`n lgfout`tiv` nk c` suo`.îkfk uf kekopcg.

=) < + 2 ? 2 + < + 2 ? +

0= ? 0= ?

0) ; + ? 2 + >) 02 + 1 ? +

? ?

4) 04 + ? < + ;) ;= + ? 4; +

? ?

]rgpikn`nks nk c` suo` nk cgsfokrgs f`tur`cks5

\k kxprks` kf ckfbuèko`tkotilg5

Kekopcg5

` + d ? d + `

(` + d) + l ? ` + (d + l)

` + = ? `

88IBKT Puiribu


101/350

L. Lgopcktk l`n` gpkr`lif y lgoprukdk quk sk luopc` c` prgpikn`n lgfout`tiv` nk c` ouctipci-l`lif. ^ikfk uf kekopcg.

=) < x 9 ? 9 x < 4) ; x 8 ? x

42 ? 42 ?

0) x ? 8 x 1 + ; ?

0) ; + 1 + 0 + < ?

4) 04 + 04 + < + 8 ?

+ 04 + 8 ?

K. @pciquk c` prgpikn`n `sgli`tiv` nk c` ouctipcil`lif p`r` rksgcvkr cgs kekrliligs. Buksk pgr kkekopcg.

=) (2 x 4 ? >4 ?

0) ; x (4 x

? ?

? ?

A. @pciquk c` prgpikn`n nkc kckokftg fkutrg nk c` suo` p`r` lgopckt`r cgs kekrliligs. ^ikfk ufkekopcg.

=) >4 + ? >4 4) + ;40 ? ;40

0) 982 + ? 982

=

0



102/350

H. Lgoprukdk quk sk luopck c` prgpikn`n nistridutiv` nk c` ouctipcil`lif lgf rkspkltg ` c` suo`.^ikfk uf kekopcg.

=) < x (4 + ;) ? (< x 4) + (< x ;) 4) (; + + 01 ? 1= \ga` suo (4< + 2) + (40 + 9) ?


103/350

; + 1) + (0; + ;)

`. Kf qu prgpikn`n g prgpikn`nks sk pgy p`r` kckbir ksk l`oifg7

d. Luc ks kc rksuct`ng7

;) Kf uf` lc`sk lgf 4> ouekrks y 4< hgodrks, l`n` ufg tikfk < lu`nkrfgs. \i qukrkogs`vkribu`r luftgs lu`nkrfgs tikfkf kf tgt`c, cg pgnkogs pc`ftk`r nk ngs agro`s5

< x (4> + 4) + (< x 4 + 4) + (< x 4


104/350

Kf c`s sko`f`s `ftkrigrks hkogs tr`dè`ng c`s t`dc`s nkc 0 `c 8, y` c`s `prkfni7 C`s t`dc`s sgf uf`hkrròikft` iopgrt`ftk y nkdkogs s`dkrc`s nk okogri`. \i h`lkogs l`n` sko`f` kc ksaukrzg nktr`dè`rc`s, okegr`rkogs fukstr` h`dicin`n nk lclucg okft`c. Kst` sko`f` ck prgpgfkogs kstuni`r c`s

t`dc`s nkc 9 y kc 0=.

@ltivin`n 0Tksukcv` c`s ouctipcil`ligfks. ^r`tk nk fg lgfsuct`r c`s t`dc`s. Ks ouy iopgrt`ftk quk c`sokogrilk.

0) 8 x 9 ?

4) 2 x 9 ?

) < x 9 ?

;) 2 x 0= ?

1) > x 9 ?

2) < x 0= ?

8) 4 x 9 ?

9) 9 x 9 ?

0=) 0= x 4 ?

00) 9 x 0= ?

04) 0= x ; ?

0) 2 x 4 ?

0;) ; x < ?

01) < x > ?

02) 4 x ; ?

08) < x 1 ?

09) 9 x 2 ?

4=) < x 8 ?

^`dc nkc 9

9 x 0 ? 9

9 x 4 ? 08

9 x < ? 42

9 x > ? ;

9 x 1 ? ;>

9 x 2 ? 1=

0= x ; ? ;=

0= x 1 ? 1=

0= x 2 ? 2=

0= x 8 ? 8=

0= x 9 ? 9=

0= x 0= ? 0==

40) > x 9 ?

44) 8 x 4 ?

4) 2 x > ?

4;) 9 x ; ?

41) < x 1 ?

42) 1 x 2 ?

48) 8 x 0= ?

49) 1 x 9 ?

?

94IBKT Puiribu



105/350

Tksgculif nk prgdcko`s

@. Tksukcv` cgs prgdcko`s kf su lu`nkrfg. Nkek kslrit`s tgn`s c` gpkr`ligfks quk rk`cilk.0) Tkufiogs kf c` lgcklt` < diccktks nk ; quktz`cks. Luftg

nifkrg rkufiogs kf tgt`c7

4) Cgrkf` lgopr 0> lu`nkrfgs kop`st`ngs ` 9 quktz`cks l`n` ufg. `. Luftg p`b7

d. \i p`b lgf 4 diccktks nk 0== quktz`cks, luftg vukctg rklidk7

91 rdgcks kf p`rtksibu`cks.

`. Luftgs rdgcks sk pc`ft`rgf kf l`n` skllif7

d. \i l`n` rdgc lgst 2 quktz`cks, luftg sk ifvirti kf c` sikodr`7

8) Eu`f f`li kf 098


106/350



prglksgfg

cgbr`ng

Nkspusnkkstuni`r... Tklukrng y rk`cizg gpkr`ligfks nk suo`, rkst`, ouctipcil`lif y

nivisif nk fokrgs f`tur`cks.

Tklgfgzlg k inkftilg c`s p`rtks nk c`s gpkr`ligfks.

Lgoprkfng y `pcilg `cbuf`s prgpikn`nks nk c`s gpkr`ligfks lgffokrgs f`tur`cks.

Tksukcvg lgrrklt`okftk cgs kekrliligs nk c`s t`dc`s nk ouctipcil`r.

Tksukcvg lgf xitg cgs prgdcko`s o`tkotilgs.


9>IBKT Puiribu


107/350

1

Octipcgs y nivisgrks

9;O`tkotil` \ko`f` 1


108/350

Cgs cgbrgs quk lgfskbuir kst` sko`f`sgf5 Nkaifir cgs lgflkptgs nk octipcg y nivisgr nk uf

fokrg f`tur`c.

Inkftiail`r octipcgs y nivisgrks nk uf fokrg f`tur`c.

Agro`r lgfeuftgs lgf cgs priokrgs octipcgs y lgftgngs cgs nivisgrks nk uf fokrg f`tur`c.

Inkftiail`r c`s prgpikn`nks nk cgs octipcgs y nkcgs nivisgrks.

@pcil`r kc lgflkptg nk octipcg ` situ`ligfks lgtini`f`s.

Nks`rrgcc`r c` `bicin`n nk lclucg okft`c lgf c` prltil`nk c`s t`dc`s nk ouctipcil`r y nivinir.

Nks`rrgcc`r su r`zgfòikftg cbilg rksgcvikfng

prgdcko`s o`tkotilgs.


@bicin`n nk lclucg okft`c ^ dc`s nk ouctipcil`r nkc 0 c 0=

Kflgftr`r nivisgr

]`r` lgokfz`r! Digbrà` nk ]itbgr`s

T`zgfòikftg cbilg ]rgdcko`s o`tkotilgs lgf

gpkr`ligfks dsil`s nk cgs

fokrgs f`tur`cks

Kc oufng nk c` o`tkotil` Octipcgs y nivisgrks

91IBKT Puiribu


109/350

]itbgr`s

]itbgr`s f`li kf c` isc` nk \`ogs, c` `ltu`c Brkli`. Auk aicsgag y

o`tkotilg. H`li` kc `g ;


110/350

0. Octipcgs nk uf fokrg f`tur`c

Cgs octipcgs nk uf fokrg f`tur`c sgf cgs fokrgs quk sk gdtikfkf `couctipcil`r nilhg fokrg pgr gtrgs fokrgs f`tur`cks5 =, 0, 4, 2 x < ?402 x > ?482 x ; ?42 x 2 ?>92 x 8 ?;12 x 9 ?1, 40, 48, 4, >9, ;1, 1, 40, 48, 4, >9, ;1, 1. ]`r` h`lkrcg, ouctipcilògs kc pgr l`n` ufg nk cgs fokrgs f`tur`cks5 =, 0, 4, .

= ? =

0 ? ? 04

< x

> x

= ? =

0 ? >

4 ? 8

< ? 04

> ? 01

O(, 8, 04, 01 }

Bkfkr`cokftk l`clucògs cgs priokrgs octipcgs nk uf fokrg, pkrg pgnr`-ogs l`cluc`r tgngs cgs octipcgs nk uf fokrg7

Fg, pgrquk lgog y` kstuniògs, kc lgfeuftg nk cgs fokrgs f`tur`cks ksiffitg, pgr cg t`ftg, kc lgfeuftg nk octipcgs tòdif ks ifafitg.

98IBKT Puiribu



111/350

Kekrlilig 0

@. Gdtkfb` cgs priokrgs liflg octipcgs nk cgs fokrgs n`ngs. ]`r` h`lkrcg ouctipciquk kcfokrg quk kst kf kc lkftrg pgr l`n` ufg nk cgs fokrgs5 =, 0, 4, . Cukbg rkprksftkcgskf agro` nk lgfeuftg. Ck `yun`ogs lgf kc ifilig.

0)

1 x

= ?0 ?4 ?< ?> ?

=

=O(1) ? { , , , , }

?

O( ) ? { , , , , }

4)

4 X

= ?

0 ?

4 ?

< ?

> ?

=

=O( ) ? { , , , , }

>)

9 x

= ?0 ?4 ?< ?> ?

O( ) ? { , , , , }

]`r` `vkribu`r si uf fokrg ks octipcg nk gtrg, niviniogs kc fokrg o`ygrkftrk kc fokrg okfgr.

\i c` nivisif ks kx`lt`, kc fokrg ks octipcg.

\i c` nivisif ks ifkx`lt`, kc fokrg fgks octipcg.

Kekopcg5

Ks 0; octipcg nk < y nk 17 ]`r` s`dkrcg, niviniogs 0; kftrk < y 15

;< 0;

0;

=

41 0;

04

ks octipcg nk 47 ? < x

0) 4; ks octipcg nk ; pgrquk 4; ? ; x

4) 0> ks octipcg nk 2 pgrquk 0> ? 2 x

) 0== ks octipcg nk 0= pgrquk 0== ? 0= x

;) 1; ks octipcg nk ; pgrquk 1; ? ; x

1) 01 ks octipcg nk > pgrquk 01 ? > x

N. Ck` c` prkbuft` y rkcckfk kc lu`nrg nk c` rkspukst` lgrrklt`.

Ks pgsidck kslridir tgngs cgs octipcgs nk uf fokrg7

Fg, ks iopgsidck pgrquk sgf ifaifitgs.

\, d`st` lgf ouctipcil`r kc fokrg pgr tgngs cgs fokrgs f`tur`cks.

Fg, pgrquk skr` ouy l`fs`ng ouctipcil`r uf fokrg pgr tgngs cgs fokrgs f`tur`cks.

8

4 0>

; 4;

< 02

1


113/350

0.4 ]rgpikn`nks nk cgs octipcgs nk fokrgsf`tur`cks

C`s rkbc`s nk cgs octipcgs

Cgs octipcgs nk cgs fokrgs f`tur`cks luopckf lgf `cbuf`s prgpikn`nks.

Kc lkrg ks kc priokr octipcg nk tgng fokrg f`tur`c.

Kekopcgs5

=? < x =

= ? 0; x 4< x =

^gng fokrg, nistiftg nk lkrg, ks octipcg nk s oisog y nk c` ufin`n (0).

Kekopcgs5


114/350

Kekrlilig 4@. Tkspgfn` ` c`s prkbuft`s y eustiaiquk su rkspukst` `pgyfngsk kf c`s prgpikn`nks nk cgs

octipcgs. ^ikfk uf kekopcg.

=) Luc ks kc priokr octipcg nk >;17

0) Kc fokrg 0>;> ks octipcg nk s oisog7

4) Kc fokrg ;89 ks octipcg nkc fokrg 07

? 9 x 4 x ks octipcg nk 9, 4 y ? 9 x 1

;> ? 4 x 42

;>? < x 08 ]gr cg t`ftg,0) >= ? > x ; x 4

Kc fokrg >= ks octipcg nk >, ; y 4. Lgoprkdkcg5

>= ? x

>= ? x

>= ? x ]gr cg t`ftg,

L. Gpkrk p`r` lgoprgd`r quk c` suo` nk ngs octipcgs nk uf fokrg ks t`odif octipcg nk

ksk fokrg.04 y


115/350

4. Nivisgrks nk uf fokrg f`tur`c

Kf c` sko`f` liflg rklgrnògs quk5 uf nivisgr ks kc fokrg quk nivink `gtrg uf fokrg kx`ltg nk vklks.

04 4 ? 1

N n ? l

]`r` nktkroif`r kc lgfeuftg nk nivisgrks kopkzògs ` nivinir sulksivòkftkkc fokrg kftrk 0, 4, >=

0

< > > >=

>

0 > >=

Cgs fokrgs 0, 4, y > agro`f kc lgfeuftg nk cgs nivisgrks nk >. Cg kslridiogs`s5

N(>) ? {0, 4, > }

Kc lgfeuftg nk nivisgrks sk fgodr` lgf c` cktr` No`ysluc`, cukbg kftrk p`-rftksis sk lgcgl` kc fokrg nkc lu`c kfuokrògs cgs nivisgrks.

@ niakrkfli` nkc lgfeuftg nk cgs octipcgs quk sgf iffitgs, cgs nivisgrks s

puknkf nktkroif`rsk, pgr cg t`ftg agro`f uf brupg afitg.

4.0 Log s`dkogs si uf fokrg ks nivisgrnk gtrg7

]`r` `vkribu`r si uf fokrg ks nivisgr nk gtrg, niviniogs kc fokrg o`ygrkftrk kc fokrg okfgr.

\i c` nivisif ks kx`lt`, kc fokrg okfgr ks nivisgr.

\i c` nivisif ks ifkx`lt`, kc fokrg okfgr fg ks nivisgr.Lgoprgdkogs si < y 9 sgf nivisgrks nk 4>. Niviniogs5

8< 4> 4>

=

49 4> 08

1C` nivisif ks kx`lt`.]gr cg t`ftg < ks nivisgrnk 4>

C` nivisif ks ifkx`lt`.]gr cg t`ftg 9 fg ks nivisgr nk 4>

0=5

0> ? 0>

0> ? 2

0> ? 4

0> ? 0

N(0>) ? { , , , }

4) Nivisgrks nkc fokrg 0=5

0= ? 0=

0= ? ;

0= ? 4

0= ? 0

N(0=) ? { , , , }

D. Nktkroifk si cgs fokrgs o`rl`ngs kf fkbrit` sgf nivisgrks nk 01.

0) 01 8 ? lgf rksinug

C` nivisif ks

8 ks nk 01

4) 01 , tkfnr` 4 brupgs nk >c`t`s

8 >? 4,tkfnr` > brupgs nk 4c`t`s

8 8? 0,tkfnr` 8 brupgs nk 0c`t`

0=>IBKT Puiribu


117/350

Kekrlilig >@. Kc lgfeuftg nk nivisgrks nk uf fokrg ks aifitg. Gdtkfb` tgngs cgs nivisgrks nk cgs fokrgs

n`ngs. Y` tikfk `cbufgs nivisgrks kslritgs.

0) N(0;) ? {0, , ;, }

4) N(04) ? { , ,


118/350

0. Octipcgs nk uf fokrg f`tur`c

Cgs octipcgs nk uf fokrg f`tur`c sgf cgs fokrgs quk sk gdtikfkf `c ouctipcil`r nilhgfokrg pgr gtrgs fokrgs f`tur`cks5 =, 0, 4,


119/350

@ltivin`n 0 Nkoukstrk cg prkfning.a@. Kslrid` lgf sus p`c`dr`s uf` nkaifilif nk octipcg.

D. Kslrid` lgf sus p`c`dr`s uf` nkaifilif nk nivisgr.

@ltivin`n 4 ]r`ltiquk cg `prkfning@. Kslrid` cgs liflg priokrgs octipcgs nk ;, 4, 9 y ?

=O( ) ? { , , , , }

; x

= ?0 ?4 ?< ?> ?

=

=O( ) ? { , , , , }

< x

= ?0 ?4 ?< ?> ?

O( ) ? { , , , , }

9 x

= ?0 ?4 ?< ?> ?

O( ) ? { , , , , }

D. Kslrid` kc lgfeuftg nk tgngs cgs nivisgrks nk cgs fokrgs ifnil`ngs. ^ikfk uf kekopcg.

=) N(>) ? { } 4) N(1) ? {

0) N(;) ? {

=) 4)

0)


120/350

L. Kslrid` nkftrg nk l`n` lu`nrg vsi c` kxprksif ks vkrn`nkr` y a, si c` kxprksif ks a`cs`. Eustiaiquksu rkspukst`. ^ikfk uf kekopcg.

=) 8 ks octipcg nk < a ]gr qu7

0) 0; ks octipcg nk ; ]gr qu7

4) 4> ks octipcg nk 8 ]gr qu7

1

=

0

;

=

0==

ifaifitg

0=

4;


121/350

@ltivin`n < Nks`rrgcck fukv`s h`dicin`nks

@. Ck` c` ifagro`lif y rkspgfn` c`s prkbuft`s kxpcil`fng kc pgrqu nk su rkspukst`.

0) Kf c` tikfn` nk o`ygrkg, `cbufgs prgnultgs sgf kop`l`ngs kf brupgs.

Cgs ygburks sk vkfnkf kf p`quktks nk >.

`. ]ukng lgopr`r 1 ygburks7

d. ]ukng lgopr`r 8 ygburks7

Cgs hukvgs sgcg sk vkfnkf kf p`quktks nk 1.

l. Pu l`ftin`n nk hukvgs pukng lgopr`r7 N `c okfgs ngs kekopcgs.

4) \i uf lèkrg ùtgotilg sgcg n` nifkrg kf octipcgs nk 0==, pukng s`l`r nkc lèkrg 0>4

quktz`cks7

D. Tk`cilk c`s `ltivin`nks.

0) Kslrid` liflg fokrgs quk sk`f octipcgs nk < y nk ; ` c` vkz.

4) Kslrid` cgs nikz priokrgs octipcgs nk 2 quk sk`f p`rks.

) Kslrid` tgngs cgs octipcgs nk 1 o`ygrks quk 8= y okfgrks quk 04=.

L Tkspgfn` c`s prkbuft`s. kfb` prkskftk c`s prgpikn`nks nk cgs octipcgs y c`s prgpikn`nks nkcgs nivisgrks.

0) Luc ks kc o`ygr nivisgr nk 28>7

4) Luc ks kc okfgr nivisgr nk 0;017

;17

>) Luftgs octipcgs tikfk uf fokrg7

0=9O`tkotil` \ko`f` 1


122/350

@. Tksukcv` c`s gpkr`ligfks sif lgfsuct`r c`s t`dc`s. Hb`cg cg os rping quk pukn`. ^ikfk ufkekopcg.

=) 0 x 2 ? >) ; x 2 ? 8) 9 x < ?

0) 4 x 8 ? ;) 1 x 1 ? 9) 0= x 4 ?

4) < x 9 ? 1) 2 x ; ? 0=) 9 x 1 ?

x 8 ? 2) 8 x > ? 00) 8 x 2 ?

D. Ouctipciquk l`n` fokrg nkc 0 c 0= pgr kc fokrg quk tikfk c lkftrg y kslrid` kc rksuct`ngkf cgs lrlucgs kxtkrfgs. ]`r` ouctipcil`r sib` kc skfting nk c`s `bue`s nkc rkcge. Kc puftg rgegifnil` c` ouctipcil`lif pgr 0. C` rukn` nkc 1 ck sirvk nk kekopcg.

1

>4

;> 08

1 041=

2

20>

8

8

L. Kflukftrk kc nivisgr p`r` quk l`n` nivisif sk` lgrrklt`. ^ikfk uf kekopcg.

=) ;= ? ; 1) 24 ? 8

0) 24 ? 9 2) >; ? 9

4) >; ? ; 8) ;> ? 1

>) 4; ? ; 0=) 08 ? 9

;) ? 8

0=

2

00=IBKT Puiribu



123/350


Tksukcv` kf su lu`nkrfg cgs prgdcko`s. Kfskk ` su grikft`ngr(`) vgcuft`rig(`) sus prglknioikf-tgs p`r` quk pukn` rkvis`rcgs. ^gok kf lukft` quk `cbufgs prgdcko`s fklksit`f os nk uf`gpkr`lif.

0) Eu`f tikfk uf rgccg nk 08; y`rn`s nk tkc` quk nkdk rkp`rtir kf p`rtks ibu`cks kftrk c`s; tikfn`s quk `noifistr`. Luft`s y`rn`s nkdk o`fn`r ` l`n` tikfn`7

4) @f` O`r` lgopr 2 cidr`s nk o`fz`f`s pgr ;1 quktz`cks. @c lgft`r c`s o`fz`f`s,kflgftr quk tkf` 48 o`fz`f`s. Luftg ck lgst l`n` uf`7

nglkf`s, luftgs hukvgs cksgdr`f7

>) Kf uf aif nk sko`f`, Tit` vkfni kf su tikfn` < quksgs nk 4 cidr s l`n` ufg. \i kcprklig pgr cidr` ks nk 4; quktz`cks, luftg rklùn pgr c` vkft` nk cgs quksgs7

;) [f piftgr kst pift`fng c`s vkft`f`s nk uf kniailig nk ngs pisgs quk tikfk >= vkft`f`skf l`n` pisg. \i kf uf n` pift ; vkft`f`s nk l`n` pisg, luft`s vkft`f`s ck a`ct`fp`r` tkroif`r7

1) Kc nukg nk uf `co`lf lgopr 41 lo`r`s agtgbrail`s pgr uf tgt`c nk 0 lo`r`s `c prklig quk ck lgst`rgf, ` luftg nkdk vkfnkrc`s lo`r`s rkst`ftks si nksk` gdtkfkr uf dkfkailig tgt`c nk 88= quktz`cks7

2) C` nuk` nk uf` tikfn` nkpgrtiv` lgopr = quktz`cks l`n` uf` y kc rkstg c`s vkfni `c prklig quk c`slgopr. Luftg b`f pgr c` vkft` nk tgn`s c`s pkcgt`s7

8) Ngf Dktg, kc nukg nk uf nkpsitg, vkfni 840 p`quktks nk b`cckt`s kc sd`ng. \i pgr. \isk pinikrgf 1 c`t`s ` l`n` kstuni`ftk p`r` uf` lòp`` nk rklilcèk, luft`s c`t`srkufikrgf kf tgt`c7


T`zgfòikftg cbilg


124/350004IBKT Puiribu



prglksgfg

cgbr`ng

Nkspusnkkstuni`r...

Nkaifg qu ks uf octipcg y qu ks uf nivisgr nk uf fokrg f`tur`c.

Inkftiailg octipcgs y nivisgrks nk uf fokrg f`tur`c.

Agrog lgfeuftgs nk octipcgs y nivisgrks nk uf fokrg n`ng.

Inkftiailg c`s prgpikn`nks nk cgs octipcgs y nk cgs nivisgrks nkfokrgs f`tur`cks.

@pcilg ` situ`ligfks lgtini`f`s kc lgflkptg nk octipcg nk uffokrg f`tur`c.

Tksukcvg lgrrklt`okftk cgs kekrliligs nk c`s t`dc`s nk ouctipcil`r.Tksukcvg lgf xitg prgdcko`s o`tkotilgs.



125/350

2

Nivisidicin`n nkfokrgs f`tur`cks

00

Lgog c` nivisif ks kx`lt`, s`dkogs quk 4>8 ks nivisidck kftrk 4, pkrg h`y ufl`oifg os lgrtg quk lgfsistk kf `pcil`r uf lritkrig nk nivisidicin`n.

Cgs lritkrigs nk nivisidicin`n sgf rkbc`s ae`s quk fgs pkroitkf rklgfglkr,

si uf fokrg ks nivisidck kftrk gtrg, sif rk`ciz`r c` nivisif.

Kst` sko`f` lgfglkrkogs cgs lritkrigs nk nivisidicin`n kftrk cgs fokrgs 4, , 1, 8.

Tktgo`fng kc kekopcg `ftkrigr (4>8), lgf sgcg gdskrv`r quk c` liar` nk c`sufin`nks, g sk` c` ctio` liar`, ks p`r, s`dkogs quk 4>8 ks nivisidck kftrk 4.

Qk`ogs gtrgs kekopcgs nk fokrgs nivisidcks kftrk 45

>=, >;4, 22>, 011 ; 2=1 4 >=< 080 821

091 0L N [

001IBKT Puiribu



129/350

d. Nivisidicin`n kftrk ? 9

9 ? < x 81 ks nivisidck kftrk < pgrquk c` suo` nk sus liar`s ks octipcg nk 81> + 8 + 1 ? 08

08 ? < x 1


130/350

l. Nivisidicin`n kftrk ;

[f fokrg ks nivisidck kftrk ; g tikfk quift` p`rtk, lu`fng c` liar` nksus ufin`nks ks = ;.

Lgog kekopcg nk fokrgs nivisidcks kftrk ; tkfkogs5

2=, 1284== , 2;9004== , 9821;1;>;>1004; 121;>


131/350

Fokrg Nivisidck kftrk 0= Nivisidck kftrk 0== Nivisidck kftrk 0===

=) 82== x

0) ;===4) 228=

) 8>, 1, 8.

d. Nivisidicin`n kftrk ;1== ks nivisidck

1) Kc fokrg ;;>2; ks nivisidck kftrk

2) Kc fokrg ; ks nivisgr nk

8) Cgs fokrgs 4 y < nivinkf `c fokrg

9) [f fokrg fgnivisidck kftrk < ks

;;) Gdskrvk cgs sibuikftks fokrgs y kflikrrk cgs quk sk`f nivisidcks kftrk 4;.

; ;2; ;= 2 ; < 4 8 ; ; 0 2 8 0 9 > ; 1 2

8 0 < > 4 0 1 0 4 < ; > < 4

< > ; 0 < < 1 2 > ; < 8 9 2 1

N. Tk`cilk c`s `ltivin`nks `pcil`fng cgs lritkrigs nk nivisidicin`n.

0) Kslrid` liflg fokrgs quk sk`f nivisidcks kftrk 4.

, , , ,

4) Kslrid` liflg fokrgs quk sk`f nivisidcks kftrk


135/350

Fokrg Nivisidck kftrk 4 Nivisidck kftrk < Nivisidck kftrk ;

08 x

>4

1=

88

018

42=2

4


136/350

@. Kslrid` kc rksuct`ng nk c`s ouctipcil`ligfks g kc fokrg quk a`ct` p`r` quk c` ouctcipcil`lifsk` lgrrklt`. r`tk nk h`lkrcg cg os rping pgsidck y oin` luftg tikopg t`rn`. ]rgpfb`sk

h`lkrcg l`n` vkz os rping.

0) ; x 9 ?

4) 8 x ; ?

) 1 x ; ?

;) ; x ; ?

1) 9 x 1 ?

2) 1 x 8 ?

8) 2 x 1 ?

9) 1 x 1 ?

0=) ; x 1 ?

00) 2 x ; ?

04) 1 x 2 ?

0) 8 x 2 ?

0;) 2 x 9 ?

01) ; x 8 ?

02) 8 x 1 ?

08) 2 x 8 ?

09) 8 x 9 ?

4=) > x 8 ?

40) 2 x ? 18

4;) ; x ? >;

41) x 2 ? >4

42) 9 x < ?

48) > x ? 48

49) x 9 ? ;>

? >

0) 4 ? 2

)

;) >9 ? 2

1) >8 ? 1

2) 40 ? ? 9

9) 48 ? 2

0=) 1> ? 8

00) 8= ? 8

1

04>IBKT Puiribu



137/350


0) \kr pgsidck rkp`rtir kx`ltòkftk 0>2 cidrgs kftrk < kslukc`s7 ]`r` rkspgfnkr pciqukkc lritkrig nk nivisidicin`n kftrk trks.

4) Ks pgsidck rkp`rtir kx`ltòkftk >8; dgtks nk eubg kf ; lè`s7 Kxpciquk su rkspukst``pgy`ng kf kc lritkrig nk nivisidicin`n kftrk ;.

quktz`cks, luftg v`cnrf 9 nglkf`s7

>) ]knrg y O`rt` aukrgf ` c` aikst` ifa`ftic nk c` oufilip`cin`n. ]knrg rklgbi >< nuclksy O`rt` quktz`cks, luftg b`st`r go`s`7

9) [f l`rpiftkrg nkdk lgcgl`r 2 pukrt`s ` l`n` uf` nk c`s


138/350



prglksgfg

cgbr`ng

Nkspus

nkkstuni`r... Nkaifg cgs lgflkptgs5 nivisidicin`n y lritkrigs nk nivisidicin`n.

Kxpcilg y `pcilg cgs lritkrigs nk nivisidicin`n.

Tksukcvg kekrliligs lgf `bicin`n nk c`s t`dc`s nk ouctipcil`r nkc 0 `c 9.

Tksukcvg lgf xitg cgs prgdcko`s o`tkotilgs.


041IBKT Puiribu


139/350

8

Tkp`sg\ko`f`s 0-2



140/350

Cgs cgbrgs quk lgfskbuir kst` sko`f`sgf5 @ai`fz`r cgs lgftkfings nk c` sko`f` 0 ` c` 25

Nkaifir uf lgfeuftg, inkftiail`r sus kckokftgs ylc`siail`rcg skbf su l`rnif`cin`n.

Tkprkskft`r lgfeuftgs kf agro` nkslriptiv`, kfu-okr`tiv` y brail`.

Kst`dcklkr k inkftiail`r rkc`ligfks nk pkrtkfkfli`,fg pkrtkfkfli`, lgftkflif y fg lgftkflif.

Lgop`r`r y grnkf`r fokrgs f`tur`cks. Lgoprkfnkr y `pcil`r c`s prgpikn`nks nk c`s gpk-

r`ligfks kf kc lgfeuftg nk fokrgs f`tur`cks. Inkftiail`r octipcgs y nivisgrks nk uf fokrg

f`tur`c. Agro`r lgfeuftgs nk cgs priokrgs octipcgs y nk-

tgngs cgs nivisigrks nk uf fokrg f`tur`c.

Niakrkfli`r y `pcil`r cgs lritkrigs nk nivisidicin`nkftrk 4,


141/350

Pukrin` y qukring kstuni`ftk5

\k `prgxio` c` priokr` kv`cu`lif y tkfkogs quk prkp`r`rfgs `nklu`nòkftk.]`r` kccg, rkp`s`rkogs cgs lgftkfings nk c`s sko`f`s 0 ` c` 2.

Kflgftr`r kc rksuokf nk l`n` tko` y kekrliligs p`r` pr`ltil`r. Kstunik lgf tkf-lif y rksukcv` tgngs cgs kekrliligs quk sk ck prkskft`f.

]`r` `prgvklh`r okegr kstk rkp`sg ck rklgokfnògs5

0. Dusquk uf cub`r tr`fquicg p`r` kstuni`r.

4. Ck` cgs rksokfks nk l`n` sko`f` y kslrid` c`s ink`s os iopgrt`ftks kfsu lu`nkrfg. Kslridir sus lgfglioikftgs ck `yun`r ` rktkfkrcgs okegr.

. @fgtk cgs lgftkfings quk fg ck h`y`f qukn`ng lc`rgs. Fg sk quknk lgfnun`s. \u grikft`ngr` u grikft`ngr vgcuft`rig ck kxpcil`r lgf bustg.^òdif puknk lgfsuct`r lgf fgsgtrgs pgr lgrrkg kckltrfilg. Fukstr`nirkllif ks5 ibkrMibkr.knu.bt

;. Lgoprukdk quk sus ùtglgftrgcks kstf dikf hklhgs. \i tikfk nun`s, vukcv`` ckkr c`s sko`f`s. Kf kcc` kflgftr`r os kxpcil`ligfks y kekopcgs.

1. r`dèk lgf cpiz. \i sk kquivgl`, puknk dgrr`r y h`lkr v`rigs iftkftgs.

2. Kstunik uf pgquitg l`n` n`. O`ftkfb` su kstunig ni`rig lgog kf c`ssko`f`s ftkrigrks.

8. Lu`fng kst skburg nk quk ngoif` kc tko`, p`sk `c sibuikftk ùfquk fgh`y` lgopckt`ng tgngs cgs kekrliligs.

9. Ck` c`s ifnil`ligfks p`r` c` prukd` p`rli`c nk c` pbif` sibuikftk y `lun``c kxòkf lgf skburin`n y lgfai`fz`.

fiog y xitgs kf su kv`cu`lif

Log s`l`r prgvklhg nk kstk rkp`sg7


]`r` lgokfz`r!


142/350

[f`s rklgokfn`ligfks os...

Log skr c` prukd` nk kv`cu`lif7

C` prukd` p`rli`c kv`c` cgs oisogs lgftkfings y nk c` oiso` agro` kf quk cgs tr`dèògssko`f` ` sko`f`.

Kf c` prukd` kflgftr`r5

[f` skrik nk lclucg okft`c. Kf kcc` sk oink su nkstrkz` p`r` c` rk`ciz`lif nk gpk-r`ligfks dsil`s kf uf tikopg coitk nk trks oifutgs. Ks ioprkslifnidck quk skp` c`st`dc`s nk ouctipcil`r nk okogri`.

Niakrkftks kekrliligs quk kv`c`f cg `prkfning kf c`s siktk sko`f`s. Kstgs kekrliligsskrf skoke`ftks ` cgs quk ustkn kc`dgr kf c`s `ltivin`nks nkc ùtglgftrgc. \k ckpknir5

rkspgfnkr prkbuft`s,

rkcckf`r kc lu`nrg nk c` gplif lgrrklt`,

lgopckt`r t`dc`s y o`p`s lgflkptu`cks,

rksgcvkr gpkr`ligfks y

rksgcvkr prgdcko`s.

Lu`fng rksukcv` kekrliligs y prgdcko`s, ks fklks`rig quk nkek lgfst`fli` nk su tr`-dèg. Kslrid` tgn`s c`s gpkr`ligfks quk rk`cilk kf c`s pbif`s nk su kv`cu`lif. \ktgo` kf lukft` kc prglksg y c` rkspukst`.

Ouy iopgrt`ftk5L`n` skrik nk c` prukd` lgftikfk ifstrulligfks kx`lt`s nk cg quknkdk rk`ciz`r kf l`n` `p`rt`ng, `s lgog c` v`cgr`lif `sibf`n`.

\i kflukftr` uf` skrik quk fg s`dk log rk`ciz`r g quk ck kst cckv fng oulhgtikopg, lgftifk su prukd` y rkbrksk ` kcc` nkspus nk h`dkr tkroif`ng c`s gtr`sskriks. Tklukrnk quk tik

Libro Quiriguá Matemática 1er.sem.15

Documents

Transcript of Libro Quiriguá Matemática 1er.sem.15