Libro Polochic Matemática 2 º Semestre 2016

7/25/2019 Libro Polochic Matemtica 2 Semestre 2016

1/367

4. Hmfbokkgrmt` - Nrup` U`k`fbofVgnuid` sgjgstrg - ONGX=5

Jmtgjtofm


2/367

4. Hmfbokkgrmt` Nrup` U`k`fbofVgnuid` sgjgstrg ONGX

Jmtgjtofm

=5


3/367

Jmtgjtofm =5

Vgnuid` sgjgstrg

Xgsgrvmd`s t`d`s k`s dgrgfb`s. ]ugdm ronur`smjgitg pr`bohodm km rgpr`duffoi t`tmk ` pmrfomkdg gstg jmtgromk gdufmtov`, p`r fumkquogr jgdo` ` pr`fgdojogit`, soi km mut`rozmfoi dgk Oistotut`Numtgjmktgf` dg Gdufmfoi Xmdoàiofm, ONGX. Vgni mrtfuk` 48 dg km Fìstotufoi U`ktofm dgNumtgjmkm qug sg rgagrg m km mut`rm.

Oistotut` Numtgjmktgf` dg Gdufmfoi Xmdoàiofm, ONGX.Gs uim `hrm pr`dufodm p`r gk Dgpmrtmjgit` dg Xgdmffoi y Dosg`, pmrm gk

Oistotut` Numtgjmktgf dg Gdufmfoi Xmdoàiofm, ONGX.

== mvgiodm =2-47, Foudmd Iugvm, zìm 8Foudmd dg Numtgjmkm.

UHR3 8657 =5=5

F`rrg` gkgftriof`3 ongrEongr.gdu.ntUnoim wgh3 www.ongr.gdu.nt

Gdofoi 85=1Ojprgs` gi ONGX tmkkgrgs nraf`s

Fdon`3 ====554858

OVHI ?>2??8?1=41==

ONGX


4/367

idofg

idofg ............................................................................................................................................................... O

Hogivgiodm y hogivgiod`! .................................................................................................................... =

Vgjmim =2Gfumfoìgs fumdrtofms..................... ..................... ..................... ..................... ..................... ..... =6

Umrm f`jgizmr! Km bost`rom dg k`s soni`s jmtgjtof`s..................... ..................... ..................... ................. =4

Gk juid` dg km jmtgjtofm

=. Gfumfoìgs fumdrtofms ................... ..................... ..................... ..................... ..................... ..................... ...... =7

=.= Fkmsofmfoi dg kms gfumfoìgs fumdrtofms..................... ..................... ..................... ................. =1

8. Xgs`kufoi dg gfumfoìgs fumdrtofms f`jpkgtms................... ..................... ..................... ................. =>

6. Xgs`kufoi dg gfumfoìgs fumdrtofms oif`jpkgtms.................... ..................... ..................... ............. 8=

4. Xgs`kufoi dg pr hkgjms fì gfumfoìgs fumdrtofms..................... ..................... ..................... ...... 86

Xgsujgi................... ..................... ..................... ..................... ..................... ..................... ..................... ...................... .... 87

Oivgstonug gi km rgd................... ..................... ..................... ..................... ..................... ..................... ..................... ... 87

Mut`fìtr`k.................... ..................... ..................... ..................... ..................... ..................... ..................... .................. 81

Mnokodmd dg fkfuk` jgitmk................... ..................... ..................... ..................... ..................... ..................... ......... 82

Xmzìmjogit` knof` ..................... ..................... ..................... ..................... ..................... ..................... ................. 8?Dgsmrr`kkg iugvms bmhokodmdgs................... ..................... ..................... ..................... ..................... ..................... .. 65

Vgjmim =?Oigfumfoìgs.................... ..................... ........................ ..................... ..................... ..................... .................. 6=

Umrm f`jgizmr! Kms dgsonumkdmdgs jmtgjtofms gi km vodm domrom..................... ..................... ................ 68


=. Oigfumfoìgs ..................... ..................... ..................... ..................... ..................... ..................... ..................... ... 66

=.= Fkmsofmfoi dg kms oigfumfoìgs.................... ..................... ..................... ..................... .................. 66

=.8 Xgs`kufoi dg oigfumfoìgs dg projgr nrmd`.................... ...................... ..................... ............. 64 =.6 Xgs`kufoi dg oigfumfoìgs dg sgnuid` nrmd`.................... ..................... ..................... .......... 6>

Xgsujgi................... ..................... ..................... ..................... ...................... ..................... ..................... ..................... .... 6?

Oivgstonug gi km rgd................... ..................... ..................... ..................... ..................... ..................... ..................... ... 6?

Mut`fìtr`k.................... ..................... ..................... ..................... ..................... ..................... ..................... .................. 45


Xmzìmjogit` knof` ..................... ..................... ..................... ..................... ..................... ..................... ................. 46

Dgsmrr`kkg iugvms bmhokodmdgs................... ..................... ..................... ..................... ..................... ..................... .. 44

OJmtgjtofm idofg


5/367

Vgjmim 85Vostgjms dg gfumfoìgs dg 88.................... ..................... ..................... .... 47

Umrm f`jgizmr! Kms vmromhkgs.................... ..................... ..................... ..................... ..................... ..................... .... 41


=. Vostgjms dg gfumfoìgs koigmkgs dg 88 .................... ..................... ..................... ..................... ............. 4>

=.= Jt`d` dg onumkmfoi................... ..................... ..................... ..................... ..................... .................. 4>

=.8 Jt`d` dg sustotufoi.................. ..................... ..................... ..................... ..................... .................. 42

=.6 Jt`d` dg rgduffoi.................... ..................... ..................... ..................... ..................... .................. 4?

8. Vostgjms dg gfumfoìgs i` koigmkgs dg 88 ..................... ..................... ..................... ..................... ..... 7=

6. Mpkofmfoìgs.................... ..................... ..................... ..................... ..................... ..................... ..................... ..... 78

Xgsujgi..................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... ... 76

Oivgstonug gi km rgd..................... ..................... ..................... ..................... ..................... ..................... ..................... . 76

Mut`fìtr`k.................. ..................... ..................... ..................... ..................... ..................... ..................... .................... 74

Mnokodmd dg fkfuk` jgitmk.................... ..................... ..................... ..................... ..................... ..................... ........ 71

Xmzìmjogit` knof` ................... ..................... ..................... ..................... ..................... ..................... ................... 7>Dgsmrr`kkg iugvms bmhokodmdgs..................... ..................... ..................... ..................... ..................... ..................... 72

Vgjmim 8=Vostgjms dg gfumfoìgs dg 66.................... ..................... ..................... .... 7?

Umrm f`jgizmr! @hcgt`s trodojgisoìmkgs.................. ..................... ..................... ..................... ..................... ... 15


=. Vostgjms dg gfumfoìgs koigmkgs dg 66 .................... ..................... ..................... ..................... ............. 1=

=.= Jt`d` dg rgduffoi.................... ..................... ..................... ..................... ..................... .................. 1=

=.8 Jt`d` dg sustotufoi.................. ..................... ..................... ..................... ..................... .................. 16

8. Mpkofmfoìgs.................... ..................... ..................... ..................... ..................... ..................... ..................... ..... 17

Xgsujgi..................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... ... 1>

Oivgstonug gi km rgd.................... ..................... ..................... ..................... ..................... ..................... ..................... .. 1>

Mut`fìtr`k.................. ..................... ..................... ..................... ..................... ..................... ..................... .................... 12

Mnokodmd dg fkfuk` jgitmk.................... ..................... ..................... ..................... ..................... ..................... ........ >5

Xmzìmjogit` knof` ................... ..................... ..................... ..................... ..................... ..................... ................... >=

Dgsmrr`kkg iugvms bmhokodmdgs.................... ..................... ..................... ..................... ..................... ...................... >8

Vgjmim 88Vostgjm dg f``rdgimdms rgftminukmrgs ......................................... >6

Umrm f`jgizmr! Umrmkgk`s y jgrodomi`s................... ..................... ..................... ..................... ..................... ...... >4


=. Vostgjm dg f``rdgimdms rgftminukmrgs.................... ..................... ..................... ..................... ................ >7

8. Dostmifom gitrg puit`s (d)................... ..................... ..................... ..................... ..................... ..................... >1

OO ONGX U`k`fbof


6/367

6. Uuit` jgdo` (J).................... ..................... ..................... ..................... ..................... ..................... ................. >2

4. Ugidogitg (j).................. ..................... ..................... ...................... ..................... ..................... ..................... ... 25

Xgsujgi................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... ..... 26

Oivgstonug gi km rgd................... ..................... ..................... ..................... ..................... ..................... ..................... ... 26

Mut`fìtr`k.................... ..................... ..................... ..................... ..................... ..................... ..................... .................. 24


Xmzìmjogit` knof` ..................... ..................... ..................... ..................... ..................... ..................... ................. 2>


Vgjmim 86Auifoìgs jmtgjtofms O..................... ..................... ..................... ..................... ........... 2?

Umrm f`jgizmr! Xgkmfoìgs f`todomims3 kms auifoìgs..................... ..................... ..................... ..................... ?5


=. Auifoi jmtgjtofm..................... ..................... ..................... ..................... ..................... ..................... .......... ?=

=.= Auifoìgs dg projgr nrmd`................... ..................... ..................... ..................... ..................... ........ ?8 =.8 Nrfm dg auifoìgs dg projgr nrmd .................... ..................... ..................... ..................... ....... ?8

=.6 Dgtgrjoimfoi dgk d`joio` g ojmngi dg auifoìgs................... ..................... ..................... .. ?4

=.4 Mpkofmfoìgs dg kms auifoìgs koigmkgs................... ..................... ..................... ..................... ........ ?7

Xgsujgi................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... ..... ?>

Oivgstonug gi km rgd.................. ..................... ...................... ..................... ..................... ..................... ..................... ... ?>

Mut`fìtr`k.................... ..................... ..................... ..................... ..................... ..................... ..................... .................. ?2

Mnokodmd dg fkfuk` jgitmk.................. ..................... ..................... ..................... ..................... ..................... .......... =55

Xmzìmjogit` knof` .................... ..................... ...................... ..................... ..................... ..................... ................. =5=

Dgsmrr`kkg iugvms bmhokodmdgs................... ..................... ..................... ..................... ..................... ..................... .. =58

Vgjmim 84Auifoìgs jmtgjtofms OO.................. ..................... ..................... ..................... ........... =56

Umrm f`jgizmr! Gk j`vojogit` pmrmhkof`................... ..................... ..................... ..................... ..................... . =54


=. Auifoìgs fumdrtofms ..................... ..................... ..................... ..................... ..................... ..................... ...... =57

=.= Nrfm dg auifoìgs fumdrtofms.................... ..................... ..................... ..................... .................. =56

=.8 Mpkofmfoìgs................... ..................... ..................... ..................... ..................... ..................... ................. =5?

Xgsujgi.................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... .... ===Oivgstonug gi km rgd................... ..................... ..................... ..................... ..................... ..................... ..................... ... ===

Mut`fìtr`k.................... ..................... ..................... ..................... ..................... ..................... ..................... .................. ==8

Mnokodmd dg fkfuk` jgitmk................... ..................... ..................... ..................... ..................... ..................... ......... ==4

Xmzìmjogit` knof` ..................... ..................... ..................... ..................... ..................... ..................... ................. ==7

Dgsmrr`kkg iugvms bmhokodmdgs................... ..................... ..................... ..................... ..................... ..................... .. ==1

OOOJmtgjtofm idofg


7/367

Vgjmim 87Xgpms`3 sgjmims =2 m 84..................... ..................... ...................... ..................... ..... ==>


Gfumfoìgs fumdrtofms ..................... ..................... ..................... ..................... ..................... ..................... ............ ==?

Oigfumfoìgs ................... ..................... ..................... ..................... ..................... ..................... ..................... ............. =8=

Vostgjms dg gfumfoìgs dg 88...................... ..................... ..................... ..................... ..................... .............. =86

Vostgjms dg gfumfoìgs dg 66...................... ..................... ..................... ..................... ..................... .............. =87

Vostgjms dg f``rdgimdms rgftminukmrgs..................... ..................... ..................... ..................... .................... =8>

Auifoìgs jmtgjtofms O..................... ..................... ..................... ..................... ..................... ..................... ........ =8?

Auifoìgs jmtgjtofms OO.................... ..................... ..................... ..................... ..................... ..................... ........ =6=

Mnokodmd dg fkfuk` jgitmk.................... ..................... ..................... ..................... ..................... ..................... ........ =66

@rogitmfoìgs s`hrg km prughm pmrfomk .............................................................................................................. =64

Tmkkgr jmtgjtof`

Urughm M.................. ..................... ..................... ..................... ..................... ..................... ..................... .................... =67

Vgjmim 81Gfumfoìgs dg km rgftm.................... ..................... ..................... ..................... .................... =47

Umrm f`jgizmr! Xgpms` dg km pgidogitg.................. ..................... ..................... ..................... ...................... ...... =41


=. Gfumfoìgs dg km rgftm ................... ..................... ..................... ..................... ..................... ..................... ........ =4>

8. Mpkofmfoìgs.................... ..................... ..................... ..................... ..................... ..................... ..................... ..... =7=

6. Xgftms pmrmkgkms y pgrpgidofukmrgs.................... ..................... ..................... ..................... ..................... ... =78

Xgsujgi.......................................................................................................................................................... =74Oivgstonug gi km rgd... ..................... ..................... ..................... ..................... ..................... ..................... ................... =74

Mut`fìtr`k.................. ..................... ..................... ..................... ..................... ..................... ..................... .................... =77

Mnokodmd dg fkfuk` jgitmk.................... ..................... ..................... ..................... ..................... ..................... ........ =72

Xmzìmjogit` knof`..................... ..................... ..................... ..................... ..................... ..................... ................... =7?

Dgsmrr`kkg iugvms bmhokodmdgs.................... ..................... ..................... ..................... ..................... ..................... . =15

Vgjmim 8>Km forfuiagrgifom y km pmrh`km................... ..................... ..................... ............ =1=

Umrm f`jgizmr! Fb`kq'oc, gk fmkgidmro` smnrmd` dg k`s jmyms..................... ..................... ..................... .... =18


=. Km forfuiagrgifom................... ..................... ..................... ..................... ..................... ..................... ................... =16

8. Km pmrh`km ..................... ..................... ..................... ..................... ..................... ..................... ..................... ..... =11

Xgsujgi..................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... ... =>=

Oivgstonug gi km rgd... ..................... ..................... ..................... ..................... ..................... ..................... ................... =>=

Mut`fìtr`k.................. ..................... ..................... ..................... ..................... ..................... ..................... .................... =>8

OQ ONGX U`k`fbof


8/367

Mnokodmd dg fkfuk` jgitmk................... ..................... ..................... ..................... ..................... ..................... ......... =>4

Xmzìmjogit` knof`................... ..................... ..................... ..................... ..................... ..................... ..................... =>7

Dgsmrr`kkg iugvms bmhokodmdgs................... ..................... ..................... ..................... ..................... ..................... .. =>1

Vgjmim 82

Vgffoìgs fiofms.................... ..................... ..................... ..................... ..................... ................ =>>

Umrm f`jgizmr! Aonurms fiofms..................... ..................... ..................... ..................... ..................... ................... =>2


=. Vgffoìgs fiofms................... ..................... ..................... ..................... ..................... ..................... ................. =>?

=.= Km gkopsg.................... ..................... ..................... ..................... ..................... ..................... ..................... ... =>?

=.8 Km boprh`km................... ..................... ..................... ..................... ..................... ..................... ................. =26

Xgsujgi.................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... .... =22

Oivgstonug gi km rgd... ................... ..................... ..................... ...................... ..................... ..................... .................... =22

Mut`fìtr`k.................... ..................... ..................... ..................... ..................... ..................... ..................... .................. =2?

Mnokodmd dg fkfuk` jgitmk................... ..................... ..................... ..................... ..................... ..................... ......... =?8Xmzìmjogit` knof`................... ..................... ..................... ...................... ..................... ..................... .................... =?6

Dgsmrr`kkg iugvms bmhokodmdgs................... ..................... ..................... ..................... ..................... ..................... .. =?4

Vgjmim 8?Tg`rgjm dg Uotn`rms.................. ..................... ..................... ..................... ..................... ..... =?7

Umrm f`jgizmr! Uotn`rms dg Vmj`s................... ..................... ..................... ..................... ..................... ............ =?1


=. Tg`rgjm dg Uotn`rms................... ..................... ..................... ..................... ..................... ..................... ......... =?>

=.= Dgj`strmfoi dgk tg`rgjm.................... ..................... ..................... ..................... ..................... ........ =?>

=.8 Mpkofmfoìgs................... ..................... ..................... ..................... ..................... ..................... ................. 855

Xgsujgi.......................................................................................................................................................... 856Oivgstonug gi km rgd... ................... ..................... ..................... ..................... ..................... ..................... ..................... 856

Mut`fìtr`k.................... ..................... ..................... ..................... ..................... ..................... ..................... .................. 854


Xmzìmjogit` knof`................... ..................... ..................... ..................... ..................... ..................... ..................... 85>


Vgjmim 65Tg`rgjm dg Tmkgs, fìnrugifom ysgjgcmizm dg troinuk`s................... ..................... ..................... ..................... ............... 85?

Umrm f`jgizmr! Xmzìgs y pr`p`rfoìgs.................... ..................... ..................... ..................... ..................... .... 8=5


=. Tg`rgjm dg Tmkgs.................... ..................... ..................... ..................... ..................... ..................... ................. 8==

=.= Tg`rgjm dg Tmkgs gi ui troinuk`..................... ..................... ..................... ..................... ............... 8=8

QJmtgjtofm idofg


9/367

8. Troinuk`s fìnrugitgs.................... ..................... ..................... ..................... ..................... ..................... .... 8=4

8.= Frotgro`s dg fìnrugifom dg troinuk`s..................... ..................... ..................... ..................... ..... 8=7

6. Troinuk`s sgjgcmitgs................... ..................... ..................... ..................... ..................... ..................... ........ 8=1

6.= Frotgro`s dg sgjgcmizm dg troinuk`s.................... ..................... ..................... ..................... ......... 8=>

6.8 Mpkofmfoìgs..................... ..................... ..................... ..................... ..................... ..................... ............... 8=>

Xgsujgi..................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... ... 8=?

Oivgstonug gi km rgd... ..................... ..................... ..................... ..................... ..................... ..................... ................... 8=?

Mut`fìtr`k.................. ..................... ..................... ..................... ..................... ..................... ..................... .................... 885


Xmzìmjogit` knof`..................... ..................... ..................... ..................... ..................... ..................... ................... 886

Dgsmrr`kkg iugvms bmhokodmdgs..................... ..................... ..................... ..................... ..................... ..................... 884

Vgjmim 6=Fugrp`s ng`jtrof`s O..................... ..................... ..................... ..................... .................... 887

Umrm f`jgizmr! Uorjodg dg Km Dmitm..................... ..................... ..................... ..................... ..................... ...... 881


=. Fugrp`s ng`jtrof`s O.................. ..................... ..................... ..................... ..................... ..................... ........ 88>

=.= Fkmsofmfoi dg k`s fugrp`s ng`jtrof`s................... ..................... ..................... ..................... .. 88>

=.8 Urosjms.................... ..................... ..................... ..................... ..................... ..................... ..................... ..... 882

=.6 Uorjodgs.................. ..................... ..................... ..................... ..................... ..................... ..................... .. 86=

Xgsujgi..................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... ... 866

Oivgstonug gi km rgd... ..................... ..................... ..................... ..................... ..................... ..................... ................... 866

Mut`fìtr`k.................. ..................... ..................... ..................... ..................... ..................... ..................... .................... 864


Xmzìmjogit` knof`..................... ..................... ..................... ..................... ..................... ..................... ................... 86>Dgsmrr`kkg iugvms bmhokodmdgs.................... ..................... ...................... ..................... ..................... ..................... 862

Vgjmim 68Fugrp`s ng`jtrof`s OO................... ..................... ..................... ..................... .................... 86?

Umrm f`jgizmr! Gk ijgr` po ()..................... ..................... ..................... ..................... ..................... .................. 845


=. Fugrp`s ng`jtrof`s OO..................... ..................... ..................... ..................... ..................... ..................... .... 84=

=.= Gk fokoidr` .................. ..................... ..................... ..................... ..................... ..................... ..................... .. 84=

=.8 Gk fì` ................... ..................... ..................... ..................... ...................... ..................... ..................... ..... 847

=.6 Km gsagrm.................... ..................... ..................... ..................... ..................... ..................... ..................... .. 842

Xgsujgi..................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... ... 875

Oivgstonug gi km rgd... ..................... ..................... ..................... ..................... ..................... ..................... ................... 875

Mut`fìtr`k..................... ..................... ..................... ..................... ..................... ..................... ...................... ................ 87=


Xmzìmjogit` knof`..................... ..................... ..................... ..................... ..................... ..................... ................... 876

Dgsmrr`kkg iugvms bmhokodmdgs.................... ..................... ..................... ..................... ..................... ..................... . 874

QO ONGX U`k`fbof


10/367

Vgjmim 66Xmzìgs tronì`jtrofms..................... ..................... ..................... ..................... ........... 877

Umrm f`jgizmr! Bopmrf` dg Iofgm y km tronì`jgtrm.................... ..................... ..................... ..................... 871


=. Xmzìgs tronì`jtrofms auidmjgitmkgs..................... ..................... ..................... ..................... ........... 87>

=.= Vgi` ................... ..................... ..................... ..................... ..................... ..................... ..................... ........... 87>

=.8 F`sgi` ..................... ..................... ..................... ..................... ...................... ..................... ..................... ... 87?

=.6 Tmingitg................... ..................... ..................... ..................... ..................... ..................... ..................... ... 815

8. Xmzìgs tronì`jtrofms rgfpr`fms.................... ..................... ..................... ..................... ..................... . 81=

6. Xmzìgs tronì`jtrofms oivgrsms.................... ..................... ..................... ..................... ..................... ..... 818

4. Mpkofmfoìgs..................... ..................... ..................... ..................... ...................... ..................... ..................... ... 817

Xgsujgi................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... ..... 81>

Oivgstonug gi km rgd... ................... ..................... ..................... ..................... ..................... ..................... ..................... 81>

Mut`fìtr`k.................... ..................... ..................... ..................... ..................... ..................... ..................... .................. 812

Mnokodmd dg fkfuk` jgitmk................... ..................... ..................... ..................... ..................... ..................... ......... 8>5Xmzìmjogit` knof`................... ..................... ..................... ..................... ..................... ..................... ..................... 8>=

Dgsmrr`kkg iugvms bmhokodmdgs................... ..................... ..................... ..................... ..................... ..................... .. 8>8

Vgjmim 64Xgpms`3 sgjmims 81 m 66.................... ..................... ..................... ..................... ....... 8>7


Gfumfoìgs dg km rgftm ..................... ..................... ..................... ..................... ..................... ..................... ............. 8>>

Km forfuiagrgifom y km pmrh`km.................... ..................... ..................... ..................... ..................... ................... 8>?

Vgffoìgs fiofms................... ..................... ..................... ..................... ..................... ..................... ..................... ... 82=

Tg`rgjm dg Uotn`rms.................. ..................... ..................... ..................... ..................... ..................... ................. 826

Tg`rgjm dg Tmkgs, fìnrugifom y sgjgcmizm dg troinuk`s.................... ..................... ..................... ...... 827

Fugrp`s ng`jtrof`s O.................... ..................... ..................... ..................... ..................... ..................... ............. 82>

Fugrp`s ng`jtrof`s OO..................... ..................... ..................... ..................... ..................... ..................... .......... 82?

Xmzìgs tronì`jtrofms................... ..................... ..................... ..................... ..................... ..................... .......... 8?=

Mnokodmd dg fkfuk` jgitmk................... ..................... ..................... ..................... ..................... ..................... ......... 8?6

@rogitmfoìgs s`hrg km prughm aoimk ................................................................................................................... 8?4

Tmkkgr jmtgjtof`

Urughm H.................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... .... 8?7

Hohko`nrmam ................................................................................................................................................... 657

QOOJmtgjtofm idofg


11/367


12/367

=:o=:o

= 8 6 4

6

4

7

8

=

5

x

y

=Jmtgjtofm Hogivgiodm y hogivgiod`!

Hogivgiodm y hogivgiod`!

Fìtoiumj`s fì gitusomsj` gk furs` dg jmtgjtofm dg 4. hmfbokkgrmt`. Gstg kohr` f`rrgs-

pìdg mk sgnuid` sgjgstrgdgk nrup` U`k`fbofy fìstm dg dogfosogtg sgjmims, gi kms qugsg dgsmrr`kkmri foif` f`jpgtgifoms jmrfmdms p`r gk Furrfuk` Imfoìmk Hmsg (FIH).

Gk furs` gst gstrufturmd` dg tmk a`rjm qug k`s fìtgiod`s sì nrmdumkgs y d`sofmd`s pmrm

a`rtmkgfgr y giroqugfgr kms rmjms sonuogitgs3

Mrotjtofm3 pgrjotg rgmkozmr `pgrmfoìgs mrotjtofms fì k`s ijgr`s rgmkgs.

knghrm3 myudm m kggr, `pgrmr y rgprgsgitmr oia`rjmfoi gi gk kginumcg sojhkof` dg kmjmtgjtofm.

Ng`jgtrm3 rgaogrg kms pr`pogdmdgs dg k`s fugrp`s ng`jtrof`s f`j` rgms y v`kjgigsdg p`kogdr`s. Tmjhoi rgkmfoìm kms pr`pogdmdgs gitrg rgftms.

Tronì`jgtrm3 gstm rmjm dg km jmtgjtofm sg gifmrnm dgk gstudo` dg k`s kmd`s y inu-k`s dg troinuk`s rgftinuk`s, m trmvs dg kms rmzìgs tronì`jtrofms.

Mitgs dg gitrmr mk dgtmkkg dg kms f`jpgtgifoms pmrm gstg furs`, `hsgrvgj`s km p`rtmdm.

Gk juid` dg km p`rtmdm i`s rgfugrdm qug s`j`s pmrtg dgk pkmigtm Togrrm y qug gi k, km

jmtgjtofm gs uim bgrrmjogitm qug i`s myudm m dgsfuhror, mprgidgr y dosarutmr kms mrtgs y

km fukturm.

K`s ijgr`s jmyms i`s rgfugrdmi qug pgrtgigfgj`s m ui pughk` juktotiof` g oitgrfukturmk,

fì ui fì`fojogit` jmtgjtof` qug dghgj`s vmk`rmr y fìsgrvmr.

Km jmi` dmid` ui fkof i`s oivotm m gxpk`rmr,

dgsfuhror y f`jpmrtor fì`fojogit`s fì gkrgst` dgk juid` m trmvs dg oitgrigt.

Km ojmngi dg ui nrup` dg jmi`s mkrgdgd`r

dg ui f`rmzi i`s bmhkm dg gstmhkgfgr rgkm-

foìgs fì kms pgrsìms qug i`s r`dgmi.


13/367

8 ONGX U`k`fbof

Fj` mkfmizmr kms f`jpgtgifoms jmtgjtofms0

I`s gia`fmrgj`s gi foif` f`jpgtgifoms gspgffms. Umrm smhgr so kms bm mkfmizmd`, gk Furrfuk`

Imfoìmk Hmsg pr`pìg oidofmd`rgs dg k`nr`, gst`s frotgro`s sì f`j` ui tgrjjgtr` qug jodg

su dgsgjpg` gi fmdm f`jpgtgifom. Orgj`s mvmizmid` pms` m pms`.

Qmym cid`sg qu sgffoìgs dgk kohr` bmfgi p`sohkg qug ustgd dgsmrr`kkg fmdm f`jpgtgifom qug

prgsgitmj`s m fìtoiumfoi.

F`jpgtgifom = Vgffoi dgk kohr`Vgjmim

=2 =? 85 8= 88 86 84 87 81 8> 82 8? 65 6= 68 66 64

Ur`dufg pmtrìgs

mrotjtof`s,

mknghrmof`s y

ng`jtrof`s

mpkofmid` pr`pog-dmdgs, rgkmfoìgs,

nurms ng`j-

trofms, sjh`k`s

y sgmkgs dg

agijgi`s imtu-

rmkgs, s`fomkgs y

fukturmkgs.

Umrmf`jgizmr! R R R R R R R R R R

Gk juid` dgkm jmtgjtofm R R R R R R R R R R R R R R R R

Xgsujgi

Oivgstonug gikm rgd R R R R R R R R R R R R R

Mut`fìtr`k R R R R R R R R R R R R R R R

Mnokodmd dgfkfuk` jgitmk R R R R R R R R R R R R R

Xmzìmjogit`knof` R R R R R R R R R R R

Dgsmrr`kkg iugvmsbmhokodmdgs R R R R R R R R R

F`jpgtgifom 8 Vgffoi dgk kohr` Vgjmim=2 =? 85 8= 88 86 84 87 81 8> 82 8? 65 6= 68 66 64

Xgsugkvg

sotumfoìgs,

pr`hkgjms dg

fmrftgr a`rjmk

qug dgjmidmi gk

d`joio` dgk pgi-

smjogit` knof`

jmtgjtof` y

kms pgrmfoìgs

jmtgjtofmsdg mrotjtofm y

knghrm gi k`s

fìcuit`s iu-

jrof`s rgmkgs y

f`jpkgc`s.

Umrmf`jgizmr! R R R

Gk juid` dgkm jmtgjtofm R R R R R R R R R R R R R R R R R

Xgsujgi

Oivgstonug gikm rgd R R R R R R R R R R

Mut`fìtr`k R R R R R R R R R R R R R R

Mnokodmd dgfkfuk` jgitmk R R R R R R R R R R R R R R R R R

Xmzìmjogit`knof` R R R R R R R R R R R

Dgsmrr`kkg iugvmsbmhokodmdgs R R R R R R R R R R R R


14/367

6Jmtgjtofm Hogivgiodm y hogivgiod`!

F`jpgtgifom 6 Vgffoi dgk kohr`Vgjmim

=2 =? 85 8= 88 86 84 87 81 8> 82 8? 65 6= 68 66 64Mpkofm fì`-

fojogit`s s`hrg

auifoìgs, jm-

trofgs, ng`jgtrm

y vgft`rgs, gi

sotumfoìgs qug

pr`jugvgi gk

jgc`rmjogit` y

trmisa`rjmfoi

dgk jgdo` imturmk,

s`fomk y fukturmkdg su fìtgxt`.

Umrmf`jgizmr! R R R R

Gk juid` dgkm jmtgjtofm R R R R R R R R R R R R

Xgsujgi

Oivgstonug gikm rgd R R R R R R R R R R

Mut`fìtr`k R R R R R R R R R R

Mnokodmd dgfkfuk` jgitmk R

Xmzìmjogit`knof` R R R R R

Dgsmrr`kkg iugvmsbmhokodmdgs R R R R


=2=?858=88868487818>828?656=686664

Wtokozm tfiofms

dg sufgsoìgs

y sgrogs pmrmoitgrprgtmr

bgfb`s s`fomkgs,

gfìjof`s y

ng`nrf`s.

Umrmf`jgizmr! R R R R

Gk juid` dg

km jmtgjtofm R R

Xgsujgi

Oivgstonug gikm rgd

Mut`fìtr`k R R

Mnokodmd dgfkfuk` jgitmk

Xmzìmjogit`knof` R R R R R R R

Dgsmrr`kkg iugvmsbmhokodmdgs R R R R R R


15/3674ONGX U`k`fbof

Fì su gsaugrz`, p`dr fìquostmr gstms f`jpgtgifoms m trmvs dg trgs top`s dg fìtgiod`s3

Dgfkmrmtov`s3 k`s fìtgiod`s dgfkmrmtov`s kg mp`rtmri gk fì`fojogit` dg k`s dos -toit`s mspgft`s tgrof`s, fìfgptumkgs y fogitaof`s dgk rgm.

Ur`fgdojgitmkgs3 f`j` oidofm su i`jhrg, k`s fìtgiod`s pr`fgdojgitmkgs sgdgsmrr`kkmri m hmsg dg gcgrfofo`s, pr`fgdojogit`s ` mikosos dg pr`hkgjms gi k`s

fumkgs pugdm dgj`strmr gk d`joio` y km pugstm gi prftofm dg k`s fì`fojogit`s

dgfkmrmtov`s.

Mftotudoimkgs3 k`s fìtgiod`s mftotudoimkgs sì t`d` mqugkk` qug togig qug vgr fìsu mftotud mitg gk gstudo` y mitg km vodm, gi ngigrmk. Gi gk kohr` gifìtrmr mftovodm-

dgs y rgakgxoìgs s`hrg sotumfoìgs f`todomims dìdg oitgrvogig km jmtgjtofm.


=2 =? 85 8= 88 86 84 87 81 8> 82 8? 65 6= 68 66 64Gjpkgm kms tg`rms

dg ng`jgtrm y

tronì`jgtrm

pmrm oitgrprgtmr

oia`rjmfoìgs y

gkmh`rmr oia`rjgs

s`hrg sotumfoìgs

rgmkgs.

Umrmf`jgizmr! R R R R R R R R

Gk juid` dgkm jmtgjtofm R R R R R R R R R R R

Xgsujgi

Oivgstonug gikm rgd R R R R R R R R

Mut`fìtr`k R R R R R

Mnokodmd dgfkfuk` jgitmk

R

Xmzìmjogit`knof` R R R

Dgsmrr`kkg iugvmsbmhokodmdgs R R


16/367

7Jmtgjtofm Hogivgiodm y hogivgiod`!

U`rtmdm

Km p`rtmdm dg fmdm sgjmim, f`j` km dgk pgrodof`,

oidofm gk ijgr` dg km sgjmim y miuifom gk ttuk`

dgk tgjm qug gstudomr.

Dghmc` dgk ttuk` vgr gstg mpmrtmd`. Kg sgrvor dg

rutm pmrm smhgr qu gifìtrmr3 kgfturm, fìtgiod`s

y mftovodmdgs. Vogjprg sg prgsgitmi fumtr` sgffo`-

igs gi gk josj` `rdgi.

K`nr`s dg km sgjmimM fìtoiumfoi mpmrgfgi k`s k`nr`s ` rgsuk-

tmd`s qug mkfmizmr mk imkozmr gk gstudo` dg

fmdm sgjmim.

Km kostm tgrjoim fì uim kigm pmrm qug ustgd

gsfrohm su k`nr` pgrsìmk.

Fì`zfmj`s iugstr` kohr`!

Gstg kohr` f`jogizm fì ui idofg dg fìtgiod`s ngigrmkgs y tgrjoim fì kms fkmvgs ` s`kufoìgs dgk`s gcgrfofo`s.

Wsmr kms fkmvgs fì rgspìsmhokodmd kg pgrjotor dgsmrr`kkmr mutì`jm gi su mprgidozmcg. Fmdm sgjmim

fìtogig fumtr` sgffoìgs proifopmkgs. Bmnmj`s ui rgf`rrod`3


17/367


18/367


19/367


20/367


21/367

=5 ONGX U`k`fbof

Kohr`, fkmsg rmdomk y frfuk` dg gstudo`

Vu gquop` dg trmhmc`!

Gk kohr`, fì sgr uim hugim bgrrmjogitm, i` k` gs t`d`. Umrm qug ustgd mkfmifg gk iovgk dg

f`jpgtgifom dgsgmd`,iugstr` sostgjm pìg m su dosp`sofoi3 gk kohr`, km fkmsg rmdomk y km

oivotmfoi m pmrtofopmr gi ui frfuk` dg gstudo`.

Km fkmsg rmdomk togig f`j` auifoi proifopmkgxpkofmr y amfokotmr km f`jprgisoi dg k`s tgjms

trmtmd`s gi gk kohr`.

Uugdg gsfufbmr km fkmsg rmdomk gi uim gjos`rm

dg su k`fmkodmd, dgsfmrnmrkms gi iugstrm p-

noim www.ongr.gdu.nt ` mdquororkms gi FDgi km

f``rdoimfoi rgnoìmk.

Gk frfuk` dg gstudo` gs gk kunmrpmrm f`jpmrtor y mprgidgr cuit`s.

Mpr`vgfbg gst`s rgfurs`s y mpygsg gi pgrsìms dg su f`juiodmd pmrm rgs`kvgr sus dudms.

Gk kohr`fujpkg fumtr` auifoìgs3

m. Tgxt`, gi gk qug gifugitrm km oia`rjmfoi y gk

dgsmrr`kk` dg k`s fìtgiod`s m gstudomr.

h. Uozmrri, pmrm qug durmitg km fkmsg rmdomk

suhrmyg odgms ojp`rtmitgs ` rgmkofg dostoitms

mftovodmdgs.

f. Fumdgri` dg trmhmc`, fì gcgrfofo`s pmrm prmf-

tofmr k` mprgidod`.

d. Bgrrmjogitm dg mut`gvmkumfoi, fumid` rg-

sugkvg su mut`fìtr`k fmdm sgjmim.

Amktm fmjhomrp`rtmdotm


22/367

==Jmtgjtofm Hogivgiodm y hogivgiod`!

Iugstrm jgt`d`k`nm pms` m pms`

Umrm amfokotmr su mprgidozmcg y mpr`vgfbmr js y jgc`r gk gstudo` fmdm sgjmim, sonm gst`s

pms`s. I` sg smktg ioinui`!

Kgm gk fìtgiod` dg km sgjmim

Kggr gk fìtgiod` i`s pgrjotg tgigr uim odgm ngigrmk dgk tgjm3 qu sm-

hgj`s, fì qu k` rgkmfoìmj`s, gtf. Gstg projgr fìtmft` tmjhoi i`s

bmr fmgr gi km fugitm dgk gsaugrz` m rgmkozmr pmrm mprgidgr k` iugv` y i`s

pìdr "gi ìdm" pmrm km fkmsg rmdomk.

=

Gsfufbg km fkmsg rmdomkFì k`s 7 sgitod`s

Km fkmsg rmdomk gs iugstrm jmgstrm. Dg mb qug gk pr`nrmjm sg kkmjg "Gk

Jmgstr` gi Fmsm". Kms jmgstrms y jmgstr`s k`fut`rgs gxpkofmi gk fìtgio-

d`, pr`pìgi gcgrfofo`s y `tr`s gcgjpk`s pmrm mjpkomr gk tgjm.

8

Dgspus dg km fkmsg rmdomk, su trmhmc` pgrsìmkGstudo` y mut`fìtr`k

Aoimkozmdm km fkmsg rmdomk gs gk j`jgit` dg su trmhmc` pgrsìmk. Dostrohuym

su togjp`3 gs jgc`r ui p`f` fmdm dm, qug t`d` km vspgrm.

6

Fìsuktg sus dudmsWi gstudomitg oitgkongitg smhg fuid` pgdor myudm

Fìsuktg k`s tgjms qug i` kg bmi qugdmd` fkmr`s gi `tr`s kohr`s, gi oitgr-

igt, fì amjokomrgs ` mjon`s. Vgnur` qug gifìtrmr pgrsìms dospugstms

m myudmrkg.

4

Umrtofopg gi ui frfuk` dg gstudo`

Mprgidgr cuit`sUinmsg dg mfugrd` fì tr`s gstudomitgs y rnmiofgi ui frfuk` dg gstu-

do`. V`kofotgi km myudm dg mknuim pgrsìm v`kuitmrom dg km f`juiodmd. Gs`

kgs myudmr m rgs`kvgr dudms y rga`rzmr k` mprgidod`. Mdgjs, tgidri km

`p`rtuiodmd dg oitgrfmjhomr mprgidozmcgs, odgms y sgitojogit`s.

Xgfugrdg qug sogjprg pugdg mfudor m su tut`r msonimd`.

7


23/367

=8 ONGX U`k`fbof

Fj` mpr`vgfbmr jgc`r su gstudo`0

Husqug ui kunmr fj`d` y fì hugim okujoimfoi. Gs ojp`rtmitg qug sg

mkgcg dgk ruod` y dg kms dostrmffoìgs.

Gkocm ui b`rmro` pmrm trmhmcmr y gstudomr. Km fìstmifom y km dosfopkoim sì

sus jgc`rgs f`jpmgrms dg gstudo`.

Kgm fì mtgifoi kms oistruffoìgs dg k`s gcgrfofo`s mitgs dg rgs`kvgrk`s.

Fìsuktg sus dudms fì `trms pgrsìms dg su f`juiodmd qug pugdmi

myudmrkg.

ONGX


24/367

=2=

8 6 4

6

4

7

8

=

5

x

y

=:o

x8+

y8:

r8

Gfumfoìgs fumdrtofms

]u gifìtrmr gstm sgjmim0

Km bost`rom dg k`s soni`s jmtgjtof`s

Gfumfoìgs fumdrtofms

Gfumfoìgs dg projgr nrmd`

Gfumfoìgs fumdrtofms y pr`hkgjms dg mpkofmfoi

Gstm sgjmim k`nrmr3

Fì`fgr km bost`rom dg k`s soni`s jmtgjtof`s.

Odgitoaofmr gk nrmd` dg kms gfumfoìgs.

Xgfì`fgr gfumfoìgs fumdrtofms f`jpkgtms g oif`jpkgtms.

Xgs`kvgr gfumfoìgs dg sgnuid` nrmd` p`r amft`rozmfoi y p`r arjukm.

Xgs`kvgr fì mnokodmd gfumfoìgs dg projgr nrmd`.

Bmkkmr km vmromfoi dgk rgm dg fumdroktgr`s.

=6Jmtgjtofm Vgjmim =2


25/367

Umrm f`jgizmr!Km bost`rom dg k`s soni`s jmtgjtof`s

Vg p`drm pgismr qug k`s soni`s utokozmd`s gi kms doagrgitgs `pgrmfoìgs jmtg-

jtofms bmi sod` k`s josj`s dg sogjprg, pgr` k` fogrt` gs qug bmi vmromd` fìgk pms` dgk togjp`. M pmrtor dgk sonk` RQ, p`f` m p`f`, sg augrì sustotuygid` kms

pmkmhrms, qug oidofmhmi kms `pgrmfoìgs, p`r k`s soni`s.

Fì`zfmj`s km bost`rom dg mknui`s soni`s jmtgjtof`s.

+ Vujm y rgstmVg dofg qug k`s otmkomi`s utokozmhmi uim py uim jpmrm oidofmr

km sujm y km rgstm (pkus y joius, gi kmti). Gstms kgtrms sg gko-

joimrì fumid` gk jmtgjtof` mkgji C`bmiigs ôdjmii

oitr`duc` k`s soni`s + y , qug utokozmj`s gi km mftumkodmd.

Ur`duft` ` juktopkofmfoiôkkomj @unbtrgd pr`pus` uim fruz pmrm dgsonimr km jukto-

pkofmfoi. U`stgro`rjgitg, gk jmtgjtof` N`ttarogd ôkbgkjKgohioz dgfodo pr`pìgr gk puit` sojpkg, p`rqug km fruz sg

fìauidm fì kmxdg kms gfumfoìgs.

mh m/h

DovosoiKm dovosoi gs km `pgrmfoi jmtgjtofm qug sg pugdg rgprg-

sgitmr fì js soni`s. Gitrg gkk`s gst km hmrrm b`rozìtmk,

soni` dg `rongi rmhg. Kugn`, gk jmtgjtof` Munustus Dg

J`rnmi km trmisa`rj gi `hkofum. U`stgro`rjgitg C`bmiiBgoirofb Xmbi oivgit gk soni` "". Kgohioz oitr`duc` k`s d`spuit`s "3"< k mfìsgcmhm su us` pmrm rgmkozmr km `pgrmfoi

gi uim s`km kigm y pmrm tgigr uim rgkmfoi fì gk soni` dg

juktopkofmfoi dg ui s`k` puit .

m

h m

3

h

:Onumk qugGstg soni` sg dghg m X`hgrt Xgf`rdg, qug gjpgz m utoko-

zmrk` gi =77>. k dgfm3 I` bmy d`s f`sms qug pugdmi sgr

js onumkgs qug d`s rgftms pmrmkgkms.

Gk soni` dg km onumkdmd onumk qug sgr dg jufbm utokodmd gi kms gfumfoìgs qug

gstudomrgj`s gstm sgjmim.

M trmhmcmr!

Kg oivotmj`s m imvgnmr p`r oitgrigt g oivgstonmr s`hrg k`s dgjs soni`s jmtgjtof`s y `trms bost`roms

dg oitgrs. Kg sungroj`s gstm dorgffoi3 bttp3//n``.nk/=8gn_B

=4ONGX U`k`fbof


26/367

=. Gfumfoìgs fumdrtofmsOiofogj`s iugstr` furs` dg jmtgjtofm rgf`rdmid` kms gfumfoìgs koigmkgs ` dg

projgr nrmd`, qug ym bgj`s trmhmcmd` gi k`s furs`s mitgro`rgs.

Wim gfumfoi koigmkgs km gxprgsoi mknghrmofm qug gst a`rjmdm p`r d`s jogj-hr`s, gi k`s qug pugdgi bmhgr `pgrmfoìgs mknghrmofms, sgpmrmd`s p`r gk soni`

onumk qug (:) y km vmromhkg ` oifniotm gst gkgvmdm m km p`tgifom ui` (x), p`r gkk`

sg kg fì`fg tmjhoi f`j` gfumfoi dg projgr nrmd`.

Ugr` i` t`dms kms gfumfoìgs sì dg projgr nrmd`, fì`zfmj`s qu dgtgrjoim

gk nrmd` dg uim gfumfoi.

Gk nrmd` dg uim gfumfoi k` gstmhkgfg gk jì`jo` ` trjoi` dg jmy`r nrmd`

mhs`kut`< gs dgfor, km vmromhkg gkgvmdm mk gxpìgitg jmy`r. Gi kms gfumfoìgs

sonuogitgs vgrgj`s fj` dgtgrjoimr gk nrmd` y su i`jhrg.

7x 6 + 4x: 1Gfumfoi dg projgr nrmd` ` koigmk, (gkgxpìgitg jmy`r dg km oifniotmxgs =).

8x8 >x + 6 : 5 Gfumfoi dg sgnuid` nrmd` ` fumdrtofm.

6x6 8x8+ =4x: 5 Gfumfoi dg tgrfgr nrmd` ` fhofm.

x4 =6x+ 61 : 5 Gfumfoi dg fumrt` nrmd`` hofumdrmdm.

Gstm sgjmim gstudomrgj`s km rgs`kufoi dg kms gfumfoìgs fumdrtofms ` dgsgnuid` nrmd`.

Wim gfumfoi fumdrtofmdg uim vmromhkg gs ui p`koi`jo` a`rjmd` p`r km sujmmknghrmofm dg trjoi`s fuy` nrmd` jxoj` gs d`s, gs dgfor, km oifniotm gst

gkgvmdm mk fumdrmd` ` su gxpìgitg gs d`s (x8).

Km gxprgsoi ngigrmk qug rgprgsgitm uim gfumfoi fumdrtofm gs3

mx8+hx+ f: 5

Dìdg3x3 gs km vmromhkg.

mx83 gs gk trjoi` fumdrtof`, dìdg gk f`gfogitg mi` pugdg sgr fgr`.

hx3 gs gk trjoi` koigmk ` dg projgr nrmd`.

f3 gs gk trjoi` oidgpgidogitg.

Kms gfumfoìgs fumdrtofms pugdgi tgigr3 d`s s`kufoìgs, uim s`kufoi, ` i`

tgigr s`kufoi dgitr` dg k`s ijgr`s rgmkgs.

=7Jmtgjtofm Vgjmim =2



27/367

=.= Fkmsoaofmfoi dg kms gfumfoìgs fumdrtofms

Kms gfumfoìgs fumdrtofms pugdgi sgr f`jpkgtms ` oif`jpkgtms. Qgmj`s kms

fmrmftgrstofms dg fmdm uim.

m. Gfumfoìgs fumdrtofms f`jpkgtms

Wim gfumfoi dg sgnuid` nrmd` sg fìsodgrm f`jpkgtm fumid` k`s f`gfogitgsiujrof`s m, hy gk trjoi` oidgpgidogitg f sì dostoit`s dg fgr`, p`r k` tmit`

fìtogigi k`s trgs trjoi`s. Mknui`s gcgjpk`s dg gfumfoìgs fumdrtofms

f`jpkgtms sì3

x8 8x+ 2 : 5 8x8+ 4x 1 : 5

F`gfogitg m: =

F`gfogitg h: 8

Trjoi` oidgpgidogitg f : 2

F`gfogitg m: 8

F`gfogitg h: 4

Trjoi` oidgpgidogitg f: 1

h. Gfumfoìgs fumdrtofms oif`jpkgtmsGi uim gfumfoi fumdrtofm i`pugdg amktmr gk trjoi` dg sgnuid` nrmd` mx8p`rqug dgcmrm dg sgr fumdrtofm, pgr` s pugdg amktmr gk trjoi` dg projgr nrmd`

hx, gk trjoi` oidgpgidogitg f` mjh`s. Gi fumkquogrm dg k`s fms`s sg trmtm dg

uim gfumfoi fumdrtofm oif`jpkgtm.

Km gxprgsoi jmtgjtofm qug gcgjpkofm fmdm uim dg gstms gfumfoìgs gs3

Fumid` amktm gk trjoi` koigmk hx.

Gcgjpk`3 4x8 ? : 5

Fumid` amktm gk trjoi` oidgpgidogitg f.

Gcgjpk`3 8x8+ 1x: 5

Fumid` amktm gk trjoi` koigmk hxy gk trjoi`

oidgpgidogitg f.

Gcgjpk`3 ?x8: 5

mx8+f: 5

mx8

+hx: 5

mx8: 5

Gcgrfofo` =Odgitoaoqug k`s vmk`rgs dg k`s f`gaofogitgs ijgrof`s y dgk trjoi` oidgpgidogitg gi kms gfumfoìgs

fumdrtofms sonuogitgs y gsfrhmk`s gi km f`kujim f`rrgspìdogitg. Togig ui gcgjpk`.

Gfumfoi F`gafogitgm h f

x8 ? : 5 = 5 ?

x8 2x: 5

4x8+ 8x+ = : 5

84x8 17x + 8= : 5

8x8 4 > x+ = : 5

=1ONGX U`k`fbof


28/367

8. Xgs`kufoi dg gfumfoìgs fumdrtofms f`jpkgtmsGi gstm sgffoi gstudomrgj`s km rgs`kufoi dg gfumfoìgs fumdrtofms f`jpkgtms.

K`s jt`d`s qug utokozmrgj`s sgri3 p`r amft`rozmfoi y p`r arjukm ngigrmk.

Wi pms` ojp`rtmitg pmrm rgs`kvgr gfumfoìgs fumdrtofms f`jpkgtms gs `rdgimr

km gfumfoi, dg tmk a`rjm qug t`d`s k`s trjoi`s qugdgi gi gk jogjhr` ozquogr-

d`, dg prgagrgifom, y gk kmd` dgrgfb` sg onumkm m 5< gs dgfor, kkgvmrkm m km a`rjm

mx8+ hx + f :5

m. V`kufoi p`r amft`rozmfoi

Umrm rgs`kvgr gfumfoìgs fumdrtofms p`r amft`rozmfoi, dghgj`s ms`fomrkms fì

gk troi`jo` dg km a`rjmx8 + hx + f` mx8+ hx + fy gxprgsmrkm f`j` ui pr`duft`

dg sus amft`rgs.

K`s pms`s pmrm rgs`kvgr kms gfumfoìgs fumdrtofms f`jpkgtms sì3

m) @rdgimr km gfumfoi g onumkmrkm m 5.

h) Amft`rozmr gk troi`jo` g onumkmrk` m 5.

f) T jmr fmdm amft`r p`r sgpmrmd` g onumkmrk`s m 5.

d) Dgspgcmr km vmromhkg.

g) U`r ktoj`, f`jpr`hmr gk ` k`s rgsuktmd`s gi km gfumfoi `ronoimk.

Xgs`kvmj`s3 x8+ 7x :1

@rdgimj`s km gfumfoi y km onumkmj`s m 5.

Amft`rozmj`s gk troi`jo`.

T jmj`s fmdm amft`r p`r sgpmrmd` y k`

onumkmj`s m 5.

Dgspgcmj`s km vmromhkg y `htgigj`s kms

s`kufoìgs.

Umrm f`jpr`hmr qug kms rgspugstms sì f`rrgftms, gi gstg gcgrfofo` sustotuorgj`s

km vmromhkg dg km gfumfoi `ronoimk p`r k`s vmk`rgs dgx=yx

8.

Vox: 6

x8+ 7x: 1

(6)8+ 7(6) : 1

? =7 :

1

1 : 1

Vox: 8

x8+ 7x: 1

(8)8+ 7(8) : 1

4 =5 :

1

1 : 1

Dg gstm jmigrm qugdm f`jpr`hmd` qug mjhms rgspugstms sì f`rrgftms.

x8+ 7x +1: 5

(x + 6)(x + 8) : 5

x + 6 : 5 x + 8 : 5

x=: 6 x

8: 8

Umrm rgs`kvgr gfum-foìgs fumdrtofmsfì mnokodmd, tginmgi fugitm k`s fms`sdg amft`rozmfoi qugvoj`s gi kms sgjmims=6 y =4.

=>Jmtgjtofm Vgjmim =2


29/367

Xgs`kvmj`s3

@rdgimj`s km gfumfoi y km onumkmj`s m 5.


T jmj`s fmdm amft`r p`r sgpmrmd` y

k` onumkmj`s m 5.

Dgspgcmj`s km vmromhkg y `htgigj`s

kms s`kufoìgs.

Xgs`kvmj`s3


T jmj`s fmdm amft`r p`r sgpmrmd` y

k` onumkmj`s m 5.

Dgspgcmj`s km vmromhkg y `htgigj`skms s`kufoìgs.

x8: 2x =1

6x8+ >x+ 8 : 5

Gcgrfofo` 8Xgsugkvm, gi gstg gspmfo`, kms gfumfoìgs fumdrtofms p`r amft`rozmfoi. Kugn`, gi su fumdgri`

f`jprughg fmdm rgsuktmd`.

x8 2x+ =1 : 5

(x 4)(x 4) : 5

x 4 : 5 x 4 : 5

x=: 4 x

8: 4

(6x+ =)(x+ 8) : 5

6x+ = : 5 x+ 8 : 5

6x: = x: 8

x=:

=

6 x

8: 8

=) x8 8x: 2 8) x8+ 87 : =5x

6) x8: 2x =7 4) x8 =1 : 1x

7) 8x8 >x+ 6 : 5 1) 8x8 4x 1 : 5

=2ONGX U`k`fbof


30/367

h. V`kufoi p`r arjukm ngigrmk

Gi gk mpmrtmd` mitgro`r mprgidoj`s m rgs`kvgr gfumfoìgs dg sgnuid` nrmd`

p`r dgsf`jp`sofoi gi amft`rgs, pgr` bmy gfumfoìgs fumdrtofms gi kms qug gsg

pr`fgdojogit` i` gs dg afok mpkofmfoi. Umrm gkk`, gxostg km arjukm fumdrtofm `

ngigrmk qug myudm m rgs`kvgr fumkquogr top` dg gfumfoi fumdrtofm.

Xgf`rdgj`s km arjukm fumdrtofm arjukm dg Qogtm y k`s gkgjgit`s qug km a`rjmi.

x:h h

8 4mf

8m

Dìdg3

x3 rgprgsgitm kms d`s s`kufoìgs dg km gfumfoi (x=yx

8).

h3 gs gk f`gfogitg dgk trjoi` koigmk hx.

3 gstg soni` a`rjmd` p`r k`s soni`s js y jgi`s sorvg pmrm `htgigr kms

d`s s`kufoìgs.

43 gs uim fìstmitg qug juktopkofm mk f`gfogitg my mk trjoi` f.

m3 gs gk f`gfogitg dgk trjoi` fumdrtof` mx8.

f3 gs gk trjoi` oidgpgidogitg.

83 gs uim fìstmitg gi gk dgi`joimd`r qug juktopkofm mk f`gfogitg m.

Umrm utokozmr km arjukm, odgitofmj`s k`s vmk`rgs dg k`s f`gfogitgs iujrof`s y dgk

trjoi` oidgpgidogitg gi km gfumfoi ronoimk. Kugn`, k`s sustotuoj`s gi km arjukm

dg mfugrd` fì kms fìstmitgs y soni`s oidofmd`s. Bmnmj`s gk projgr gcgjpk`.

Xgs`kvmj`s3 x8

:

4x + =8

@rdgimj`s km gfumfoi y rgkmfoìmj`s

fmdm kotgrmk fì su vmk`r.

Vustotuoj`s k`s vmk`rgs dg kms kotgrmkgs gi

km arjukm.

Xgmkozmj`s kms `pgrmfoìgs oidofmdms

dgitr` y augrm dgk rmdofmk.

Gxprgsmj`s gk rgsuktmd` fì x= dìdg

`pgrmj`s fì gk soni` + y x8 dìdg

`pgrmj`s fì gk soni`.

@htgigj`s k`s vmk`rgs dgx=yx

8.

x8+ 4x =8 : 5

m : =, h : 4, f: =8

x:(4) (4)8 4(=)(=8)

8(=)

x:

4 =1 +42

8 :

4 14

8

x=:

4 + 28

x8:

4 28

x=:

4

8: 8 x

8:

=8

8: 1

Xgfugrdg km kgy dg

soni`s< gi km projgrmsustotufoi(4) :4ydgitr` dgk rmdofmk4(=)(=8) : +42

=?Jmtgjtofm Vgjmim =2


31/367

Xgs`kvmj`s3

Xgkmfoìmj`s fmdm kotgrmk fì su vmk`r.

Vustotuoj`s k`s vmk`rgs dg kms kotgrmkgs

gi km arjukm.



Gxprgsmj`s gk rgsuktmd` fìx=dìdg

`pgrmj`s fì gk soni` +yx8dìdg



8.

Xgs`kvmj`s3

Xgkmfoìmj`s fmdm kotgrmk fì su vmk`r.

Vustotuoj`s k`s vmk`rgs dg kms kotgrmkgs

gi km arjukm.



Gxprgsmj`s gk rgsuktmd` fìx=dìdg`pgrmj`s fì gk soni` +yx

8dìdg



8.

Gcgrfofo` 6

Xgsugkvm gi su fumdgri` kms gfumfoìgs fumdrtofms f`jpkgtms. Wtokofg km arjukm ngigrmk yf`jprughg fmdm rgsuktmd`.

8x8 7x+ = : 5

m : 8, h : 7, f: =

x:

(7) (7)8 4(8)(=)

8(8)

x: 7 87 24

: 7 =>4

x=:

7 + =>4

x8:

7 =>4

x=:

7 + =>

48.82

x8:

7 =>

45.88

1x8+ 86x =2 : 5

m : 1, h : 86, f: =2

x:(86) (86)8 4(1)(=2)

8(1)

x:86 78? + 468

=8:

86 ?1=

=8

x=:

86 + 6=

=8 x8:

86 6=

=8

x=:

2

=8:

8

6 x

8:

74

=8:

?

8

=) x8 8x: 84

8) 6x8+ 7x: 5

6) 7x8 6x 8 : 5

4) x8 4x 8= : 5

7) 6x8 4x =7 : 5

1) ?x8: 6x+ 85

>) 8x8+ 61 : =>x

2) 4x8+ ? + =8x: 5

?) 1x8+ =5 : =?x

=5) 7x8+ 84x+ =1 : 5

==) ?x8+ 8?x+ 1 : 5

=8) =5x8+ ?x ? : 5

85ONGX U`k`fbof


32/367

6. Xgs`kufoi dg gfumfoìgs fumdrtofms oif`jpkgtms

m. Gfumfoìgs dg km a`rjm mx8+ hx: 5

F`j` docoj`s mitgro`rjgitg, kms gfumfoìgs fumdrtofms oif`jpkgtms sì

mqugkkms qug kgs amktm gk trjoi` hx` f. Mb`rm gstudomrgj`s kms qug kgs amktm gk

trjoi` f. K`s pms`s pmrm rgs`kvgr gstg top` gfumfoìgs sì3

m) Amft`rozmr p`r amft`r f`ji.

h) Onumkmr fmdm amft`r m5.

f) Xgs`kvgr kms d`s gfumfoìgs dg projgr nrmd` ` koigmkgs.

Xgs`kvmj`s3x8+ 8x: 5

Amft`rozmj`s gk hoi`jo`.

T jmj`s fmdm amft`r p`r sgpmrmd` y k` onumkmj`s m 5.

Dgspgcmj`s km vmromhkg y `htgigj`s kms s`kufoìgs.

Xgs`kvmj`s3 6x8 7x: 5




Xgs`kvmj`s3 2x8+ 8x: 5




Gcgrfofo` 4Xgsugkvm gi su fumdgri` kms gfumfoìgs fumdrtofms oif`jpkgtms y f`jprughg fmdm rgsuktmd` gi km

gfumfoi `ronoimk.

=) x8 8x: 5

8) 6x8+ 7x: 5

6) 8x8 6x: 5

4) x8: 8x

7) 8x8: 7x

1) ?x8: 6x

>) 8x8: =>x

2) 1x8: =2x

?) 4x8+ =8x: 5

x(x+ 8) : 5

x: 5 x+ 8 : 5

x=: 5 x

8: 8

x(6x 7) : 5

x : 5 6x 7 : 5

6x : 7

x=: 5 x

8: 7

6

8x(4x + =) : 5

8x : 5 4x + = : 5

x : 58

4x :=

x=: 5 x

8:

=

4

8=Jmtgjtofm Vgjmim =2


33/367

Xgs`kvmj`s3x8 4 : 5

@rdgimj`s km gfumfoi.

Gxtrmgj`s rmz fumdrmdm gi mjh`s jogjhr`s.

Gsfrohoj`s kms s`kufoìgs.

Xgs`kvmj`s3 ?x8 87 : 5



Dgspgcmj`s km vmromhkg y gsfrohoj`s kms

s`kufoìgs.

Xgs`kvmj`s3x8 =5 : 5




Xgs`kvmj`s3x8: =2



x8: 4

x8 : 4

x: 8

x=: 8 x8: 8

?x8: 87

?x8 : 87

6x: 7

x=:

7

6 x

8:

7

6

x8: =5

x8

: =5x: =5

x=: =56.=1

x8: =56.=1

x8 : =2

x: =2

x=: 6 8 4.84

x8: 6 8 4.84

Gcgrfofo` 7Xgsugkvm gi su fumdgri` kms gfumfoìgs fumdrtofms oif`jpkgtms y f`jprughg fmdm rgsuktmd` gi km

gfumfoi `ronoimk.

=) x8 14 : 5

8) x8 =55 : 5

6) 4x8: =1

4) ?x8: 87

7) x8 1 : 5

1) 1x8 84 : 5

h. Gfumfoìgs dg km a`rjm mx8+ f: 5

F`j` pugdg vgr gi gstms gfumfoìgs amktm gk trjoi` koigmk hx. K`s pms`s pmrm

rgs`kvgr gstg top` dg gfumfoìgs sì3

m) Gsfrohor gk trjoi` fumdrtof` gi gk projgr jogjhr y gk trjoi`

oidgpgidogitg gi gk sgnuid` jogjhr`.h) Gxtrmgr rmz fumdrmdm gi mjh`s jogjhr`s, mitgpìogid` gk

soni` mk rmdofmk dgk trjoi` oidgpgidogitg.

f) Gsfrohor kms d`s s`kufoìgs.

88ONGX U`k`fbof


34/367

4. Xgs`kufoi dg pr`hkgjms fì gfumfoìgs fumdrtofmsJufbms sotumfoìgs dg km vodm domrom y dg kms doagrgitgs fogifoms bmfgi us` dg

kms gfumfoìgs dg sgnuid` nrmd` pmrm kkgnmr m uim s`kufoi. M fìtoiumfoi

gjpkgmrgj`s gstg top` dg gfumfoìgs pmrm rgs`kvgr mknuims mpkofmfoìgs.

Xgs`kvmj`s3

Wim dg kms fmrms dg ui prosjm rgftminukmr jodg 4fgitjgtr`s js gi km hmsg

qug gi su mkturm y gk rgm gs dg =?8 fgitjgtr`s fumdrmd`s. Fukgs sì kms

dojgisoìgs dg gsm fmrm0

Dg sgr p`sohkg rgmkozmj`s uim dohuc` qug dgsfrohm k`s

dmt`s dgk pr`hkgjm.

Gsfrohoj`s k`s dmt`s dgk pr`hkgjm gi uim tmhkm.

Gsfrohoj`s km arjukm dgk rgm pmrm ui rgftinuk`.

Vustotuoj`s k`s vmk`rgs gi km arjukm.

Xgmkozmj`s kms `pgrmfoìgs.

@rdgigj`s km gfumfoi dg sgnuid` nrmd`.


@htgigj`s k`s rgsuktmd`s.

F`j` kms jgdodms dg km aonurm i` pugdgi sgr ignmtovms, dgsfmrtmj`s gkrgsuktmd`x8: =1, s`k` t`jmj`s gk vmk`r p`sotov`x

=: =8y k` sustotuoj`s gi

kms vmromhkgs oiofomkgs.

b :x: =8 fj< h :x+ 4 : =8 + 4 : =1 fj

Kms dojgisoìgs dg uim dg kms fmrms dgk prosjm sì =8 fjdg mkt` y =1 fjdg mifb`.

Mkturm Hmsg rgm

b:x h:x+ 4 M : =?8 fj8

x

x + 4

=?8 fj8

M:hb

=?8 : (x+ 4)(x)

=?8: x8+ 4x

x8+ 4x =?8 : 5

x:

(4) (4)8 4(=)(=?8)

8(=)

x:

4 =1 + >12

8:

4 >24

8

x=:

4 + 828

x8:

4 828

x=:

84

8:=8 x

8:

68

8: =1

86Jmtgjtofm Vgjmim =2


35/367

Gxprgsmj`s k`s dmt`s dgk pr`hkgjm

gi uim tmhkm.

Gsfrohoj`s km gfumfoi.

Dgsmrr`kkmj`s gk fumdrmd`.

Onumkmj`s km gfumfoi m 5.

Dovodoj`s km gfumfoi gitrg 8.

Amft`rozmj`s.

Onumkmj`s fmdm amft`r m 5.


Frostoim togig =5m`s y Mkhgrt` =7.

Dg sgr p`sohkg rgmkozmj`s ui dohuc`qug dgsfrohm k`s dmt`s dgk pr`hkgjm.

Gsfrohoj`s km arjukm dgk v`kujgi

pmrm ui fokoidr` forfukmr rgft`.

Dgspgcmj`s km vmromhkg qug i`s

oitgrgsm.


Xgmkozmj`s kms `pgrmfo`igs.


Xgs`kvmj`s3

Mkhgrt` gs 7m`s jmy`r qug su bgrjmim Frostoim. Km sujm dg kms gdmdgs mk

fumdrmd` gs 687m`s. Fuk gs km gdmd dg fmdm ui`0

Xgs`kvmj`s3

Wi amhrofmitg dg vms`s forfukmrgs rgft`s dgsgm gkmh`rmr ui vms` dg =5 fjdg mkt`

y f`i uim fmpmfodmd >27 fj6. ]u rmdo` oitgro`r dghg tgigr fmdm vms`0

Gdmd dgFrostoim

Gdmd dgMkhgrt`

Vujm dg kms gdmdgs mkfumdrmd`

x x+ 7 x8 + (x + 7)8

x8+ (x+ 7)8: 687

x8+x8+ =5x+ 87: 687

8x8+ =5x 655 : 5

x8+ 7x =75 : 5

(x+ =7)(x =5) : 5

x+ =7 : 5 x =5 : 5

x=: =7 x

8: =5

Gk vms` dghg tgigr ui rmdo` oitgri` dg 7fgitjgtr`s.

b

r

Q : r8b

Q

b: r

8

>27

(6.=4)(=5): r

8

>27

6=.4: r8

87: r8

7 : r 7 : r

84ONGX U`k`fbof


36/367

=. Gi kms gfumfoìgs fumdrtofmskm oifniotm gst gkgvmdm mk fumdrmd`. Gstms gfumfoìgs pugdgitgigr3 d`s s`kufoìgs, uim s`kufoi ` i` tgigr s`kufoi dgitr` dg k`s ijgr`s rgmkgs.

Km gxprgsoi ngigrmk qug rgprgsgitm uim gfumfoi fumdrtofm gs3

mx8+hx+ f: 5

=.= Fkmsoaofmfoi dg kms gfumfoìgs fumdrtofms3

m. Gfumfoìgs fumdrtofms f`jpkgtms3uim gfumfoi dg sgnuid` nrmd` sg fìsodgrm f`jpkgtmfumid` k`s f`gaofogitgs iujrof`s m, hy gk trjoi` oidgpgidogitg fsì dostoit`s dg fgr`.

h. Gfumfoìgs fumdrtofms oif`jpkgtms3 gi uim gfumfoi fumdrtofm i` pugdg amktmr gktrjoi` mx8, pgr` s pugdg amktmr gk trjoi` dg projgr nrmd` hx, gk trjoi` oidgpgidogitg

f` mjh`s. Gi fumkquogrm dg k`s fms`s sg trmtm dg uim gfumfoi fumdrtofm oif`jpkgtm.

8. Kms gfumfoìgs fumdrtofms f`jpkgtmssg pugdgi rgs`kvgr p`r amft`rozmfoi y p`r arjukm ngigrmk.

m. V`kufoi p`r amft`rozmfoi3 pmrm rgs`kvgr gfumfoìgs fumdrtofms p`r amft`rozmfoidghgj`s ms`fomrkms fì gk troi`jo` dg km a`rjmx8 + hx + f` mx8+ hx+ fy gxprgsmrkm f`j`

ui pr`duft` dg sus amft`rgs.

h. V`kufoi p`r arjukm ngigrmk3 km arjukm ngigrmk ` dg Qogtm i`s myudm m rgs`kvgrfumkquogr top` dg gfumfoi fumdrtofm sustotuygid` k`s vmk`rgs dg k`s f`gaofogitgs y dgk

trjoi` oidgpgidogitg gi km arjukm.

x:h h

8 4mf

8m

6. Xgs`kufoi dg gfumfoìgs fumdrtofms oif`jpkgtms< dg mfugrd` mk trjoi` qug kgs amktg, pugdgi

sgr dg km a`rjm3

m. mx8+ hx: 5 sg rgsugkvgi p`r amft`r f`ji.

h. mx8+ f: 5 sg rgsugkvgi `rdgimid` km gfumfoi y gxtrmygid` rmz fumdrmdm gi mjh`s

jogjhr`s.

Xgsujgi

Kg oivotmj`s m qug vosotg k`s sonuogitgs soto`s dg oitgrigt. Gi gkk`s p`dr rga`rzmr sus fì`fojogit`s

s`hrg kms gfumfoìgs fumdrtofms y rgmkozmr js gcgrfofo`s.

bttp3//n``.nk/Km]zsB

bttp3//n``.nk/m2RHCh

Oivgstonug gi km rgd...

87Jmtgjtofm Vgjmim =2


37/367

Mftovodmd =. Dgjugstrg k` mprgidod`M. @hsgrvg fì mtgifoi kms gfumfoìgs, dgtgrjoig so sì f`jpkgtms ` oif`jpkgtms y gsfrohm gk vmk`r

dg k`s f`gaofogitgs y dgk trjoi` oidgpgidogitg. Togig ui gcgjpk`.

5) x8+ 8x 2 : 5

m:

h:

f :

=) =5x8+ =1x 2 : 5

m:

h:

f :

8) 6x8+ 8x: 5

m:

h:

f :

f`jpkgtm

=

8

2 m

6) x8 =5x+ 84 : 5

m :

h:

f :

4) 1x8 2 : 5

m :

h:

f :

7) x8+ ==x+ 84 : 5

m :

h:

f :

H. Jmrqug fì uim "R" gk fumdr` dg km `pfoi qug rgspìdg f`rrgftmjgitg m fmdm prgnuitm.

=) Dg kms pfoìgs sonuogitgs, fuk gs uim gfumfoi fumdrtofm0 8x+ ? : =1

7 6 7 : =5

x4 8x8+ 7 : 5

6x8+ 8x 2 : 5

8) Fuk gs gk sgnuid` trjoi` gi km gfumfoix8 =>x+ >8 : 50 5

x8

>8

=>x

6) Fuk gs gk trjoi` oidgpgidogitg gi km gfumfoi 8x8+ 1x ? : 50 8

1?

1x

4) Fuk gs km amft`rozmfoi f`rrgftm dgx8+ 6x 740 (x+ ?)(x 1)

(x+ 1)(x ?)

(x+ ?)(x+ 1)

(x ?)(x 1)

Mut`fìtr`k

81ONGX U`k`fbof


38/367

Mftovodmd 8. Urmftoqug k` mprgidod`M. Xgsugkvm p`r amft`rozmfoi kms gfumfo`igs fumdrtofms f`jpkgtms y gsfrohm k`s rgsuktmd`s dgx

=yx

8.

=) x8+ 7x+ 1 : 5

x=:

x

8:

8) x8+ 8x 2 : 5

x=:

x

8:

6) x8+ 8x+ = : 5

x=:

x

8:

4) x8 2x+ =1 : 5

x=:

x

8:

7) x8+ 1x 8> : 5

x=:

x

8:

1) x8+ 2x+ =7 : 5

x=:

x

8:

>) 6x8+ 7x+ 8 : 5

x=:

x

8:

2) 8x8+ >x 4 : 5

x=:

x

8:

H. Xgsugkvm p`r amft`rozmfoi kms gfumfo`igs fumdrtofms oif`jpkgtms y gsfrohm k`s rgsuktmd`s dgx=yx

8

=) x8 6x: 5

x=:

x

8:

8) 4x8+ 85x: 5

x=:

x

8:

6) 6x8 =8x: 5

x=:

x

8:

4) x8 ? : 5

x=:

x

8:

7) x8 4? : 5

x=:

x

8:

1) x8 =7 : 5

x=:

x

8:

8>Jmtgjtofm Vgjmim =2


39/367

@hsgrvg fmdm gfumfoi dg projgr nrmd`, rgsukvmkm jgitmkjgitg y gsfrohm gk rgsuktmd` gi km kigm

f`rrgsp`idogitg. Kugn`, f`jprughg su rgsuktmd` gi km gfumfoi `ronoimk.

M.=) x+ ? : => x:

8) x+ =7 : =5 x:

6) x+ 87 : 75 x:

4) x+ =8 : 65 x:

7) x+ 47 : 15 x:

1) x+ =5 : ?5 x:

>) x+ =1 : 45 x:

2) x+ =2 : 65 x:

?) x+ =4 : 74 x:

=5) x+ 85 : 68 x:

==) x+ =5 : 77 x:

=8) x+ 87 : 75 x:

=6) x 2 : 4 x:

=4) x 1 : =8 x:

=7) x ? : =5 x:

=1) x 2 : 88 x:

=>) x 6 : =8 x:

=2) x 5 : 47 x:

=?) x =5 : =1 x:

85) x =7 : 47 x:

8=) x 15 : 65 x:

88) x 87 : 87 x:

86) x 85 : 68 x:

84) x 61 : 14 x:

H.

=) ?y: 61 y:

8) 4y : 84 y:

6) 2y : 14 y:

4) 2y: 45 y:

7) 6y : 61 y:

1) 7y: 47 y:

>) 6y :65 y:

2) 8y: 44 y:

?) 7y : 15 y:

=5) 2y : ?1 y:

==) =8y : 61 y:

=8) =7y :15 y:

=6)y

1: 4 y:

=4) y2 : 1 y:

=7)y

>: > y:

=1)y

6: =8 y:

=>)y

7: 4 y:

=2)y

=: =7 y:

=?)y

=8: 6 y:

85) 75y : 87 y:

8=) 42y : 4 y:

88) >7y : =7 y:

86) ?5y : 65 y:

84) 15y : 4 y:

82ONGX U`k`fbof

Mnokodmd dg fkfuk` jgitmk


40/367

M. Xgsugkvm gi su fumdgri` ` gi b`cms mpmrtg kms gfumfoìgs fumdrtofms. Wtokofg gk jt`d` qug dgsgg.

=) 8x8: 1x

8) x8

: =5x

6) 6x8: ?x

4) x8 2x: 5

7) 7x8 ?x: 5

1) 2x8+ 4x: 5

>) 4x8: 87

2) 2=x8: =1

?) x8 4 : 5

=5) =55x8: 14

==) x8 14 : 5

=8) =1x8 ? : 5

=6) x8+ 68 : =8x

=4) 8x8

+ =6x: 84

=7) x8 4x 68 : 5

=1) x8 6x =5 : 5

=>) 1x8x =8 : 5

=2) =5x8+x 6 : 5

=?) 4x8+ =>x: =2

85) 1x8+ =>x+ 7 : 5

8=) x(6x =7) : 48

88) x(x 8) : 67

86) (x+ 6)(=5 x) : 2x

84) (7x + ==) : =8x

H. Ukmitgg uim gfumfoi qug dgsfrohm k` giuifomd` gi fmdm pr`hkgjm. Kugn`, rgsukvmkm y f`jprughgsu rgsuktmd`. Xgmkofg gk gcgrfofo` gi su fumdgri`.

=) Wi rgftinuk` gs 7vgfgs js kmrn` qug gk d`hkg dg su mifb`. Vo su rgm gs ?5 fj8, fuit`

jodg gk kmrn` y gk mifb`0

8) Dgtgrjoig gk rgm dg ui fumdrmd` so su domnìmk jodg 75 fj.

6) Gk rgm dg uim jgsm rgftminukmr gs dg 8.>5 j8. Dgtgrjoig gk kmrn` y gk mifb` dg km jgsm, so gk

mifb` gs 5.>7vgfgs su kmrn`.

4) Km sujm dg d`s ijgr`s gs 7y su pr`duft` gs 24. Bmkkg dofb`s ijgr`s.

7) Dgtgrjoig kms dojgisoìgs dg ui tgrrgi` rgftminukmr qug togig ui pgrjgtr` dg 825 jy su

rgm gs dg 4555 j8.

1) Wi cmrdi rgftminukmr dg 75 j dg kmrn` p`r 64 jdg mifb` gst r`dgmd` p`r ui fmjoi` dg

mrgim uioa`rjg. Bmkkg km mifburm dg dofb` fmjoi` so sg smhg qug su rgm gs745 j

.>) D`s ijgr`s imturmkgs sg doagrgifomi gi d`s uiodmdgs y km sujm dg sus fumdrmd`s gs 725.

Bmkkg k`s ijgr`s.

2) Bmkkg km kìnotud dg k`s kmd`s dg ui troinuk` rgftinuk` fì ui rgm dg 1 fj8, pgrjgtr` dg =8

fjg bop`tgiusm dg 7 fj.

?) Gi tsgnuid`s km mkturm bgi pogs, s`hrg gk iovgk dgk sugk`, dg ui pr`ygftok gst dmdm p`r km

gfumfoi b: 845t =1t8. Fuit` togjp` kg t`jmr mk pr`ygftok kkgnmr m uim mkturm dg ?55pogs

s`hrg gk iovgk dgk sugk`0

8?Jmtgjtofm Vgjmim =2

Xmzìmjogit` knof`


41/367

Jmrqug fì ui fbgqug km fmsokkm qug jgc`r oidoqug su rgidojogit`.k`nrmd` gipr`fgs`

i`k`nrmd`

Dgspusdggstudo

mr...

Fì`zf` km bost`rom dg k`s soni`s jmtgjtof`s.

Odgitof` gk nrmd` dg kms gfumfoìgs.

Xgfì`zf` gfumfoìgs fumdrtofms f`jpkgtms g oif`jpkgtms.

Xgsugkv` gfumfoìgs dg sgnuid` nrmd` p`r amft`rozmfoi y p`r arjukm.

Xgsugkv` fì mnokodmd gfumfoìgs dg projgr nrmd`.

Bmkk` km vmromfoi dgk rgm dg fumdroktgr`s.

F`jpmrgj`s rgms

Gi gstm sgffoi i`tmrgj`s fj` uim pgqugm vmromfoi gi km jgdodm dg k`s

kmd`s dg ui fumdrmd` bmfg uim nrmi doagrgifom gi gk rgm. Urgstg mtgifoi mk

fms` sonuogitg.

Fìsodgrgj`s trgs fumdrmd`s fì kmd`sx,x+ 8yx+ 7qug sg jugstrmi m fìto-

iumfoi. Kugn`, rgmkofg k` qug sg kg podg gi fmdm oifos`.

x

=) Gsfrohm uim gfumfoi fumdrtofm pmrm gk rgm dg fmdm fumdrmd`.

M=: M8: M6:

8) Vo gk rgm dgk projgr fumdrmd` gs dg 61 fj8, fuit` jodg fmdm fumdrmd` p`r

kmd` y fukgs sì kms rgms0

=:

8:

6:

M=: M

8: M

6:

6) Vo gk rgm dgk tgrfgr fumdrmd` gs dg 887 fj8, fuit` jodg fmdm fumdrmd`

p`r kmd` y fuk gs gk rgm dg k`s `tr`s d`s0

=: 8: 6:

M=: M

8: M

6:

x + 8x + 7

65ONGX U`k`fbof

Xgvosg su mprgidozmcg

Dgsmrr`kkg iugvms bmhokodmdgs


42/367

=?=

8 6 4

6

4

7

8

=

5

x

y

=:o

x8+

y8:

r8

Oigfumfoìgs

]u gifìtrmr gstm sgjmim0

Kms dgsonumkdmdgs jmtgjtofms gi km vodm domrom

Oigfumfoìgs

Fkmsofmfoi dg uim sgrog dg ijgr`s dg mfugrd` m su rgkmfoi.

Gi hsqugdm dg dgtmkkgs

Gstm sgjmim k`nrmr3

F`jprgidgr y dgsfrohor qu gs uim oigfumfoi.

Doagrgifomr uim oigfumfoi dg uim dgsonumkdmd jmtgjtofm.

Xgs`kvgr f`rrgftmjgitg oigfumfoìgs.

Xgprgsgitmr km s`kufoi dg oigfumfoìgs gi km rgftm iujrofm.

@rdgimr y fkmsoaofmr uim sgrog dg ijgr`s rgspgft` m su rgkmfoi.

Urmftofmr km mnudgzm vosumk gi km hsqugdm dg dgtmkkgs.

6=Jmtgjtofm Vgjmim =?


43/367

Umrm f`jgizmr!

Kms dgsonumkdmdgs jmtgjtofms gi km vodm domrom

Kms dgsonumkdmdgs jmtgjtofms sg gjpkgmi t`d` gk togjp` gi gk juid` qug

i`s r`dgm. Vmhgr oitgrprgtmrkms gs ojp`rtmitg pmrm jmigcmr f`rrgftmjgitg km

oia`rjmfoi qug pgrfohoj`s dg iugstr` git`ri`.

Wstgd sg gifugitrm fì dgsonumkdmdgs jmtgjtofms gi jufbms `fmsoìgs,

pgr` quozs i` k` i`tm p`rqug ym kg sì amjokomrgs. U`r gcgjpk`, pogisg gi k`s

gsfgimro`s sonuogitgs3

Votumfoi Dgsonumkdmd jmtgjtofm

Mkturm jxojm dg ui fmjoiUmrm qug ui fmjoi pugdm pmsmr dghmc`

dg mkni pugitg, dghg tgigr uim mkturm

jgi`rm 4.=7 jgtr`s.

Tmrcgtm dg frdot`

Gk pmn` jioj` jgisumk dghg sgr jmy`r` onumkmk =5 % dg km dgudm t`tmk.

Mkturm jiojm pmrm ui cugn`Gi mknui`s cugn`s dg kms agroms, pmrm qug

ui io` pugdm suhorsg dghg tgigr uim

mkturmjmy`r ` onumkmk kjotg pgrjotod`.

Kjotg dg vgk`fodmd

Km vgk`fodmd pgrjotodm gi kms fmrrgtgrms

dghg sgr jgi`r ` onumkm >5 lj/b.

Joiut`s dg togjp` dg morg Gi mkni dgtgrjoimd` pkmi, gk togjp`

dg morg gi gk fgkukmr qug fìsujoj`s

mk jgs dghg sgr jgi`r ` onumk m =75joiut`s.

K` i`t0 T`dms kms sotumfoìgs mitgro`rgs sì dgsonumkdmdgs, y muiqug ustgd

i` sg d fugitm, ym utokozm gk rmzìmjogit` jmtgjtof` pmrm mimkozmr gstms y

`trms sotumfoìgs sojokmrgs. Qgrgj`s gstg tgjm gi km sonuogitg sgffoi.

M trmhmcmr!

Uogisg y gsfrohm uim forfuistmifom js gi km qug utokofg kms dgsonumkdmdgs jmtgjtofms. Gxpkoqug dg

a`rjm hrgvg su rgspugstm.

?5 fj

Lj/b

joi

68ONGX U`k`fbof


44/367

=. OigfumfoìgsMitgs dg gstudomr kms oigfumfoìgs, dghgj`s tgigr fkmr` qu sì kms dgsonumkdm-

dgs jmtgjtofms, dg kms fumkgs kgmj`s gi km sgffoi mitgro`r.

Wim dgsonumkdmd jmtgjtofmgs uim rgkmfoi dg `rdgi qug sg dm gitrg d`s

vmk`rgs fumid` gst`s sì dostoit`s< pmrm oidofmr gsg rdgi sg gjpkgmi k`s soni`s

dg rgkmfoi jmy`r qug (9), jgi`r qug (;), jmy`r ` onumk qug () y jgi`r ` onumk

qug (). Qgmj`s ui`s gcgjpk`s3

2?987, sg kgg3 "2?gs jmy`r qug 87"

61;8, sg kgg3 "61gs jgi`r qug 8"

Tgiogid` fkmr` k` mitgro`r, p`dgj`s pmsmr mk tgjm dg gstm sgjmim, kms oigfumfoìgs.

Wim oigfumfoigs uim dgsonumkdmd jmtgjtofm qug prgsgitm mk jgi`s uimvmromhkg

gi mknui` dg sus jogjhr`s, p`r gs` tmjhoi sg kg fì`fg f`j` dgsonumkdmd

mknghrmofm. Wi`s gcgjpk`s3

y 4==

x+ 4;2

y 11

=.= Fkmsoaofmfoi dg kms oigfumfoìgs

m. U`r gk soni` qug prgsgitmiVo km oigfumfoi prgsgitm gk soni` dg rgkmfoi "jmy`r qug" ` "jgi`r qug", sg

dgi`joim oigfumfoi dg sgitod` gstroft`, p`rqug iòifkuyg m sus gxtrgj`s.

x 9=5Gs gstroftm p`rqug gk vmk`r dg x sgpugdg mfgrfmr m =5k` js p`sohkg (=5.=,=5.5=, =5.55=), pgr` i k` oifkuyg.

y;6Gk vmk`r dg ysg pugdg mfgrfmr m6 k`js p`sohkg (8.?, 8.??, 8.???), pgrì`k` oifkuyg.

Vo km oigfumfoi prgsgitm gk soni` dg rgkmfoi "jmy`r ` onumk qug" ` "jgi`r `

onumk qug", sg dgi`joim oigfumfoi gi sgitod` mjpko`` i` gstroft`, p`rqug s

oifkuyg m sus gxtrgj`s.

y 2Gk vmk`r dgypugdg sgr 2` jgi`r qug2, gs dgfor, sk` oifkuyg.

x7Gk vmk`r dgxpugdg sgr7` jmy`r qug

7, gs dgfor, sk` oifkuyg.

66Jmtgjtofm Vgjmim =?



45/367

h. U`r su nrmd`

Mk onumk qug kms gfumfoìgs, gk nrmd` dg uim oigfumfoi gst dmd` p`r gk

gxpìgitg jmy`r dg km vmromhkg.

x 2 ;1, gs dg projgr nrmd` ` koigmk

y

8+ ? 88, gs dg sgnuid` nrmd` ` fumdrtofm

j6+ 86 965, gs dg tgrfgr nrmd` ` fhofm, gtf.

=.8 Xgs`kufoi dg oigfumfoìgs dg projgr nrmd`

Km s`kufoidg uim oigfumfoi gs gk fìcuit` dg vmk`rgs qug pugdg t`jmr km

vmromhkg y qug vgrofmi km oigfumfoi.

U`dgj`s gxprgsmr km s`kufoi dg d`s a`rjms, p`r jgdo` dg uim rgprgsgitmfoinrfm gi km rgftm iujrofm ` f`j` ui oitgrvmk`.

Gk pr`fgdojogit` pmrm rgs`kvgr uim oigfumfoi dg projgr nrmd` gs juy sojokmr mk

qug gjpkgmj`s pmrm rgs`kvgr uim gfumfoi. Uìnm mtgifoi mk gcgjpk` sonuogitg3

Xgs`kvmj`s km sonuogitg oigfumfoi3

Xgmkozmj`s km trmisp`sofoi dg trjoi`s.

Xgdufoj`s trjoi`s sgjgcmitgs y

gifìtrmj`s km rgspugstm.

Mb`rm, rgprgsgitmj`s km rgspugstm f`j` oitgrvmk` y gi a`rjm nrfm gi km rgftm

iujrofm.

(,8Zsg kgg3 km s`kufoi f`jprgidg t`d`s k`s ijgr`s dgsdg oiaiot` ignmtov`

bmstm 8, oifkuygid` gk 8.

K` vg0 Vg trmhmcm dg a`rjm juy pmrgfodm m kms gfumfoìgs.

Tginm gi fugitm qug, mk gxprgsmr km rgspugstm f`j` oitgrvmk`, fumid` ui kjotg `

gxtrgj` sg gxtogidg bmstm oiaiot`, sogjprg utokozmj`s pmritgsos.

8x + = x + 6

8x x6 =

x 8

Mk rgprgsgitmr kms`kufoi, utokozmj`spmritgsos( )so gs uimoigfumfoi gstroftm yf`rfbgtgs\ Zso gs uimoigfumfoi mjpkom.

1 7 4 6 8 = 5 = 8 6 4 7 1

Z

64ONGX U`k`fbof


46/367

Juy ojp`rtmitg!

Fumid` ui ijgr` ignmtov`juktopkofm ` dovodg ui kmd` dg km oigfumfoi y pmsm mk

`tr` bmfogid` km pgrmfoi fìtrmrom, gk soni` dg km dgsonumkdmd fmjhom dg sgitod`.

Xgs`kvmj`s3

Mpkofmj`s km pr`pogdmd dostrohutovm.

Xgmkozmj`s km trmisp`sofoi dg trjoi`s y

`pgrmj`s.

Dgspgcmj`s km vmromhkg y `pgrmj`s. F`j`

gk ijgr` qug pmsm m dovodor gs ignmtov`, gk

soni` dg dgsonumkdmd fmjhom dg sgitod`.

Vojpkoaofmj`s y bmkkmj`s km rgspugstm.

(, =6 ) sg kgg3 km s`kufoi f`jprgidg t`d`s k`s ijgr`s dgsdg oiaiotìgnmtov` bmstm =

6, soi oifkuor m =

6.

1 7 4 6 8 = 5 = 8 6 4 7 1

Tmjhoi sg pugdgi prgsgitmr oigfumfoìgs fì d`s soni`s dg rgkmfoi, kkmjmdms

d`hkgs ` dgtrgs f`jpìgitgs. Qgmj`s gk pr`fgs` pmrm rgs`kvgrkms.

Xgs`kvmj`s3

Dovodoj`s km gxprgsoi gi d`s pmrtgs. Kmprojgrm, a`rjmdm p`r gk projgr y sgnuid`

f`jpìgitg. Km i`jhrmj`s =. Km sgnuidm,

a`rjmdm p`r gk sgnuid` y tgrfgr f`jpìgitg.

Km i`jhrmj`s 8.

Kms trmhmcmj`s p`r sgpmrmd`. Urojgr`, rgs`kvgj`s =.

Odgitoaofmj`s su s`kufoi f`j` s=.

Kugn`, rgs`kvgj`s 8. Odgitoaofmj`s su s`kufoi

f`j` s8.

Fìtoiugj`s km s`kufoi dg gstm oigfumfoi d`hkg gi km pnoim sonuogitg.

6(4x+ =) 9=

=8x 69=

=8x 94

x ; 4=8

x ; =6

(

4 + 6x;7x+ 8 >x+ =

4 + 6x; 7x+ 8 >x+ =

4 + 6x; 7x+ 8

4 8; 7x 6x

8 ; 8x

= ;x(s=)

7x+ 8 >x+ =

7x >x= 8

8x=

x=

8(s8)

=

8

67Jmtgjtofm Vgjmim =?


47/367

Gcgrfofo` =

Xgsugkvm kms oigfumfoìgs sonuogitgs gi su fumdgri`. Xgfugrdg trmzmr km rgprgsgitmfoi nrfm.

=) 7x =8 9 6x 4 8) x 7;8x 1

6) 8(6x + =) 6(8x 8)9 x + 1 4) (x =)8

==;(x 8)8

7)(x+ 6)(x + 8)

6 4 ;

x(x + 8)

61)

6x + =

>

8 4x

6

7x 4

=4+

>x

1

>) 4x+ = 6 7x9=5 >x 2) 4x8x+ = 6x+ 8

Mb`rm, dghgj`s rgmkozmr uim oitgrsgffoigitrg k`s fìcuit`s dg vmk`rgs

qug f`jprgidg fmdm s`kufoi. Km rgspugstm pmrm km oigfumfoi d`hkg sgri

mqugkk`s vmk`rgs qug sg oifkuymi gi mjhms s`kufoìgs(s=y s8).

F`j` p`dgj`s vgr, mjhms s`kufoìgs (s=y s8) sg oitgrsgftmi gi gk oitgrvmk`

(=, ), p`r k` qug gstm sgrm km rgspugstm pmrm km oigfumfoi `ronoimk (4 + 6x;

7x+ 8 >x+ =). Gxprgsmj`s gstg rgsuktmd` f`j`3

(=, )sg kgg3km s`kufoi f`jprgidg t`d`s k`s ijgr`s dgsdg =bmstm oiaiot`

p`sotov`, soi oifkuor gk =.

6 8 = 5 = 8 6

(

(

\

s=

s8

=

8

61ONGX U`k`fbof


48/367

=.6 Xgs`kufoi dg oigfumfoìgs dg sgnuid` nrmd`

Umrm rgs`kvgr oigfumfoìgs dg sgnuid` nrmd` i`s hmsmj`s gi gk pr`fgdojogit`

dg km arjukm fumdrtofm, gk fumk mprgidoj`s km sgjmim mitgro`r.

Xgs`kvmj`s3

Onumkmj`s gk p`koi`jo` m fgr`.

Gjpkgmj`s km arjukm fumdrtofm pmrm

bmkkmr kms rmfgs dg km gfumfoi.

Whofmj`s k`s vmk`rgs dg kms rmfgs gi km

rgftm iujrofm3

4 6 8 = 5 = 8 6 4

I`tmj`s qug sg a`rjmi trgs sgnjgit`s3

(, 78 ), (7

8, 6)y(6, )

Mb`rm, gsf`ngj`s ui vmk`r dg prughm

pmrm fmdm sgnjgit` y k` gvmkumj`s gi

km gfumfoi `ronoimk. Fì gst` bmkkmj`sgk soni`dg fmdm sgnjgit`.

Km s`kufoi gst f`jpugstm p`r k`s sgnjgit s (` gk sgnjgit`) qug tginmi

gk josj` soni` qug gk projgr trjoi` dg km oigfumfoi ronoimk. Gi gstg fms`,

sgrm p`sotov`(8x8).

4 6 8 = 5 = 8 6 4

Km s`kufoi gs gk projgr y tgrfgr sgnjgit`.

Gitìfgs, gxprgsmj`s km rgspugstm gi a`rjm dg oitgrvmk` y gi km rgftm iujrofm3

(, 78 )W (6,) sg kgg3km s`kufoi f`jprgidg dgsdg oiaoiot` ignmtov`bmstm

7

8y dg 6m oiaoiot` p`sotov`.

4 6 8 = 5 = 8 6 4

8(4)8 (4) =7 : +8=

8(5)8 (5) =7 :=7

8(4)8 (4) =7 : +=6

8x8

x =7 9 5

8x8x =7 9 5

x:(=) (=)8 4(8)(=7)

8(8)

x:= ==

4

x=: 6 x

Libro Polochic Matemática 2 º Semestre 2016

Documents

Transcript of Libro Polochic Matemática 2 º Semestre 2016