Libro Polochic Matemática 1er. Sem

8/20/2019 Libro Polochic Matemática 1er. Sem

http://slidepdf.com/reader/full/libro-polochic-matematica-1er-sem 1/326

1.³ M`hjbnngr`tf - Irupf [fnfhjbh[rbegr sgegstrg - BIG]0;

E`tgeàtbh`





1.³ M`hjbnngr`tf ‘ Irupf [fnfhjbh[rbegr sgegstrg ‘ BIG]

E`tgeàtbh` 0;



E`tgeàtbh` 0;

[rbegr sgegstrg

]gsgrvàfs tfafs nfs agrghjfs. Puga` rbiurfsègltg prfjbmba` n` rgprfauhhbøl tft`n f p`rhb`nag gstg e`tgrb`n gauh`tbvf, pfr hu`nqubgr egabf f prfhgabebgltf, sbl n` ùtfrbz`hbøl agn BlstbtutfIu`tge`ntghf ag Gauh`hbøl ]àbfdølbh`, BIG]. Xgiól `rtåhunf 1= ag n` Hflstbtuhbøl [fnåtbh` agIu`tge`n` qug sg rgdgrg n` utfrå`.

© Blstbtutf Iu`tge`ntghf ag Gauh`hbøl ]àbfdølbh`, BIG].Gs ul` fmr` prfauhba` pfr gn Agp`rtègltf ag ]ga`hhbøl y Absgðf, p`r` gn

Blstbtutf Iu`tge`ntghf ag Gauh`hbøl ]àbfdølbh`, BIG].

00 vglba` 0:-17, Hbuaà Lugv`, zfl` =Hbuaà ag Iu`tge`n`.

[MZ3 =10= 4444D`x3 =10= 49;1

Hfrrgf gnghtrølbhf3 bigrKbigr.gau.it[àibl` wgm3 www.bigr.gau.it

Gabhbøl =;04Beprgsf gl BIG] t`nngrgs iràdhfs

Høabif3 0000;;1=;0

BXML 29:22=240170=



Ålabhg

Ålabhg ............................................................................................................................................................... B

¡Mbglvglba` y mbglvglbaf! .................................................................................................................... 0

Xge`l` 0Nøibh` B .................... ...................... ..................... ..................... ..................... ..................... ..................... ................. 08

¡[`r` hfeglz`r! Gn huepng`ðfs ag @ngc`lar` ..................... ..................... ..................... ..................... ............... 01

Gn eulaf ag n` e`tgeàtbh`

0. Nøibh` prfpfsbhbfl`n .................... ..................... ..................... ..................... ..................... ..................... .......... 07

=. [rfpfsbhbflgs sbepngs ..................... ..................... ..................... ..................... ..................... ..................... ...... 07

=.0 Lgi`hbøl nøibh` (~) ................... ..................... ..................... ..................... ..................... ..................... .. 09

8. Nøibh` ag prgabhàfs.................... ..................... ..................... ..................... ..................... ..................... ......... 0: 8.0 Hu`ltbhàfr ulbvgrs`n (∋) .................... ..................... ..................... ..................... ..................... ........ 0:

8.= Hu`ltbhàfr gxbstglhb`n (∈) ..................... ..................... ..................... ..................... ..................... ..... 0:

]gsuegl ................... ..................... ..................... ..................... ..................... ..................... ..................... ...................... .... 02

@utfhfltrfn .................... ..................... ..................... ..................... ..................... ..................... ..................... .................. =;

@ibnbaà ag hànhunf eglt`n ................... ..................... ..................... ..................... ..................... ..................... ......... ==

]`zflèbgltf nøibhf ..................... ..................... ..................... ..................... ..................... ..................... ................. =8

Ags`rrfnng lugv`s j`mbnbaàgs ................... ..................... ..................... ..................... ..................... ..................... .. =1

Xge`l` =Nøibh` BB ..................... ....................... ..................... ..................... ..................... ..................... ..................... ............. =7

¡[`r` hfeglz`r! Nfs hflghtbvfs ................... ..................... ..................... ..................... ..................... ..................... .. =4


0. [rfpfsbhbflgs hfepugst`s .................... ..................... ..................... ..................... ..................... ................... =9

=. Hflghtbvfs nøibhfs ..................... ..................... ..................... ..................... ..................... ..................... ............. =9

=.0 Hflculhbøl (∩) ..................... ..................... ..................... ..................... ..................... ..................... .......... =9

=.= Absyulhbøl (∪) ................... ..................... ..................... ..................... ..................... ..................... ............. =:

=.8 Hflabhbfl`n (↝) .................. ..................... ..................... ..................... ..................... ..................... .......... =:

=.1 Mbhflabhbfl`n (↑) ................... ..................... ..................... ..................... ..................... ..................... ...... =:8. ]gin`s ag nfs hflghtbvfs nøibhfs .................... ..................... ..................... ..................... ..................... ........ =2

8.0 ]gin` ag n` hflculhbøl .................... ..................... ..................... ..................... ..................... ................ =2

8.= ]gin` ag n` absyulhbøl ..................... ..................... ..................... ..................... ..................... ................ 8;

8.8 ]gin` agn hflabhbfl`n .................... ..................... ..................... ..................... ..................... ................... 80

8.1 ]gin` agn mbhflabhbfl`n ................... ..................... ..................... ..................... ..................... ................ 8=

]gsuegl ................... ..................... ..................... ...................... ..................... ..................... ..................... ..................... .... 87

Blvgstbiug gl n` rga ................... ..................... ..................... ..................... ..................... ..................... ..................... ... 87

BE`tgeàtbh` ∝ Ålabhg



@utfhfltrfn .................. ..................... ..................... ..................... ..................... ..................... ..................... .................... 84

@ibnbaà ag hànhunf eglt`n .................... ..................... ..................... ..................... ..................... ..................... ........ 8:

]`zflèbgltf nøibhf ................... ..................... ..................... ..................... ..................... ..................... ................... 82

Ags`rrfnng lugv`s j`mbnbaàgs .................... ..................... ..................... ..................... ..................... ..................... . 1;

Xge`l` 8Lóegrfs rg`ngs .................. ..................... ..................... ..................... ..................... ..................... ........ 10

¡[`r` hfeglz`r! Nfs pbtìørbhfs y nfs lóegrfs brr`hbfl`ngs .................. ..................... ..................... ............ 1=


0. Lóegrfs rg`ngs (]) ..................... ..................... ..................... ..................... ..................... ..................... ............ 18

0.0 H`r`htgråstbh s ag nfs lóegrfs rg`ngs ..................... ..................... ..................... ..................... ...... 18

0.= T`nfr `msfnutf ag ul lóegrf rg`n ................... ..................... ..................... ..................... ................ 11

=. Fpgr`hbflgs hfemblà`s .................... ..................... ..................... ..................... ..................... ..................... 17

8. [rfpbgaàgs ag nfs lóegrfs rg`ngs .................. ..................... ..................... ..................... ..................... ... 14

1. [ftglhb hbøl ag lóegrfs rg`ngs ..................... ..................... ..................... ..................... ..................... ....... 1:

1.0 [rfpbgaàgs ag nfs gxpflgltgs p`r` lóegrfs rg`ngs ..................... ..................... .................. 1:

7. ]àbh`hbøl ag lóegrfs rg`ngs .................. ..................... ..................... ..................... ..................... .............. 7;

7.0 [rfpbgaàgs ag nfs ràbh`ngs p`r` lóegrfs rg`ngs .................... ..................... ..................... .... 7;

]gsuegl ..................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... ... 70

@utfhfltrfn .................. ..................... ..................... ..................... ..................... ..................... ..................... .................... 7=

@ibnbaà ag hànhunf eglt`n .................... ..................... ..................... ..................... ..................... ..................... ........ 71


Ags`rrfnng lugv`s j`mbnbaàgs .................... ...................... ..................... ..................... ..................... ..................... 74

Xge`l` 1 Lóegrfs hfepngcfs ..................... ..................... ..................... ..................... ..................... .......... 79

¡[`r` hfeglz`r! ]`åhgs p`rgs g bep`rgs .................. ..................... ..................... ..................... ..................... ....... 7:


0. Lóegrfs hfepngcfs (H) .................... ..................... ..................... ..................... ..................... ..................... .... 72

0.0 [ftglhb`hbøl ag b .................. ..................... ..................... ..................... ..................... ..................... ........ 40

0.= Dfre` mblfeb`n ag ul lóegrf hfepngcf ................... ..................... ..................... ..................... .. 40

=. Xue` y rgst` ag lóegrfs hfepngcfs ................... ..................... ..................... ..................... ..................... 4=

8. Euntbpnbh`hbøl ag lóegrfs hfepngcfs ..................... ..................... ..................... ..................... ................. 48

1. Abvbsbøl ag lóegrfs hfepngcfs .................. ..................... ..................... ..................... ..................... ........... 47

1.0 ]`hbfl`nbz`hbøl agn aglfeblàfr egab`ltg gn hflcuiàf .................. ..................... ........... 47

]gsuegl ..................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... ... 49

Blvgstbiug gl n` rga .................... ..................... ..................... ..................... ..................... ..................... ..................... .. 49

@utfhfltrfn ..................... ..................... ..................... ..................... ..................... ..................... ..................... ................. 4:

@ibnbaà ag hànhunf eglt`n .................... ..................... ..................... ..................... ..................... ..................... ........ 9;


Ags`rrfnng lugv`s j`mbnbaàgs .................... ..................... ..................... ..................... ..................... ..................... . 9=

BBBIG] ∝ [fnfhjbh



Xge`l` 7Xue` y rgst` ag e`trbhgs .................. ..................... ..................... ..................... ........... 98

¡[`r` hfeglz`r! Xuafou3 ul cugif ag bliglbf .................... ..................... ..................... ..................... ............... 91


0. E`trbhgs .................. ..................... ..................... ..................... ..................... ..................... ...................... .............. 97

=. Qbpfs ag e`trbhgs ................... ..................... ..................... ..................... ..................... ..................... ................. 94

=.0 E trbz tr`lspugst` ..................... ..................... ..................... ..................... ..................... ..................... .. 94

=.= E`trbz sbeâtrbh` ................... ..................... ..................... ..................... ..................... ..................... .......... 94

=.8 E`trbz bagltbaà ................... ..................... ..................... ..................... ..................... ..................... ......... 94

=.1 E`trbz blvgrtbmng f lf sbliun`r .................... ..................... ..................... ..................... ..................... . 94

=.7 E trbz sbliun`r .................. ..................... ..................... ..................... ..................... ..................... ............. 94

8. Fpgr`hbflgs gltrg e`trbhgs .................. ..................... ..................... ..................... ..................... ................... 99

8.0 Xue` y rgst` ................... ..................... ..................... ..................... ..................... ..................... ................ 99

8.= [rfpbgaàgs ag n` sue` ag e`trbhgs ................... ..................... ..................... ..................... ........... 92

]gsuegl .................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... .... :0

Blvgstbiug gl n` rga ................... ..................... ..................... ..................... ..................... ..................... ..................... ... :0

@utfhfltrfn .................... ..................... ..................... ..................... ..................... ..................... ..................... .................. :=

@ibnbaà ag hànhunf eglt`n ................... ..................... ..................... ..................... ..................... ..................... ......... :1

]`zflèbgltf nøibhf ..................... ..................... ..................... ..................... ..................... ..................... ................. :7

Ags`rrfnng lugv`s j`mbnbaàgs ................... ..................... ..................... ..................... ..................... ..................... .. :4

Xge`l` 4Euntbpnbh`hbøl ag e`trbhgs .................... ..................... ..................... ..................... ....... :9

¡[`r` hfeglz`r! Gn `cgargz y n` ngygla` ag Xbs` ..................... ..................... ..................... ..................... ........... ::


0. Euntbpnbh`hbøl ag e`trbhgs .................... ..................... ..................... ..................... ..................... ................... :2

0.0 [rfauhtf ag ul lóegrf rg`n pfr ul` e`trbz ................... ..................... ..................... ................. :2

0.= [rfauhtf ag e`trbhgs .................... ..................... ..................... ..................... ..................... ................... 20

]gsuegl ................... ...................... ..................... ..................... ..................... ..................... ..................... ..................... .... 27

Blvgstbiug gl n` rga ................... ..................... ..................... ..................... ..................... ..................... ..................... ... 27

@utfhfltrfn .................... ..................... ..................... ..................... ..................... ..................... ..................... .................. 24

@ibnbaà ag hànhunf eglt`n ................... ..................... ..................... ..................... ..................... ..................... ......... 2:

]`zflèbgltf nøibhf ..................... ..................... ..................... ..................... ..................... ..................... ................. 22Ags`rrfnng lugv`s j`mbnbaàgs ................... ..................... ..................... ..................... ..................... ..................... .. 0;;

BBBE`tgeàtbh` ∝ Ålabhg



Xge`l` 9Xuhgsbflgs e`tgeàtbh`s ................... ..................... ..................... ..................... ............. 0;0

¡[`r` hfeglz`r! N` suhgsbøl ag Dbmfl`hhb ................... ...................... ..................... ..................... ..................... .. 0;=


0. Xuhgsbflgs e`tgeàtbh s ..................... ..................... ..................... ..................... ..................... ..................... . 0;8 0.0 Xuhgsbflgs `rbteâtbh`s .................. ..................... ..................... ..................... ..................... .................. 0;8

0.= Xuhgsbflgs igfeâtrbh`s.................................. ..................... ..................... ..................... ..................... 0;9

]gsuegl ..................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... ... 000

Blvgstbiug gl n` rga .................... ..................... ..................... ..................... ..................... ..................... ..................... .. 000

@utfhfltrfn .................. ..................... ..................... ..................... ..................... ..................... ..................... .................... 00=



Ags`rrfnng lugv`s j`mbnbaàgs .................... ..................... ..................... ..................... ..................... ..................... . 004

Xge`l` :]gp`sf3 sge`l`s 0 ` 9 .................... ..................... ..................... ...................... ................ 009


Nøibh` B .................. ..................... ..................... ..................... ..................... ..................... ..................... ..................... ... 002

Nøibh` BB ..................... ..................... ..................... ..................... ..................... ..................... ..................... .................... 0=0

Lóegrfs rg`ngs ..................... ..................... ..................... ..................... ..................... ..................... ..................... ..... 0=8

Lóegrfs hfepngcfs ................... ..................... ..................... ..................... ..................... ..................... ................... 0=7

Xue` y rgst` ag e`trbhgs .................... ..................... ..................... ..................... ..................... ..................... ........ 0=9

Euntbpnbh`hbøl ag e`trbhgs .................. ..................... ..................... ..................... ..................... ..................... ........ 0=2

Xuhgsbflgs e`tgeàtbh s .................. ..................... ..................... ..................... ..................... ..................... ............ 080


Frbglt`hbflgs sfmrg n` prugm` p`rhb`n .............................................................................................................. 081

Xge`l` 2[ftglhb`hbøl ag gxprgsbflgs `nigmr`bh`s .................................. 087

¡[`r` hfeglz`r! ]gp`sf ag n` pftglhb`hbøl ag lóegrfs rg`ngs ................... ..................... ..................... .... 084


0. ]gin`s ag n` pftglhb`hbøl ................... ..................... ..................... ..................... ..................... ..................... . 089 0.0 ]gin` agn gxpflgltg hgrf........................... ..................... ..................... ..................... ..................... .... 089

0.= ]gin` agn prfauhtf ag pftglhb`s hfl biu`n m`sg .................... ..................... ..................... ........ 089

0.8 ]gin` agn hfhbgltg ag pftglhb`s ag biu`n m`sg .................... ..................... ..................... ............ 08:

0.1 ]gin` ag ul` pftglhb` gngvà` ftr` pftglhb` ..................... ..................... ..................... ........ 08:

0.7 ]gin` ag n` pftglhb` ag ul` dr`hhbøl `nigmr`bh` ................... ..................... ..................... ........ 08:

0.4 ]gin` agn gxpflgltg lgi`tbvf ................... ..................... ..................... ..................... ..................... ... 01;

=. Xbepnbh`hbøl ag gxprgsbflgs nigmr`bh`s ..................... ..................... ..................... ..................... ......... 010

BTBIG] ∝ [fnfhjbh



]gsuegl ................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... ..... 018

Blvgstbiug gl n` rga... ................... ..................... ..................... ..................... ..................... ..................... ..................... 018

@utfhfltrfn .................... ..................... ..................... ..................... ..................... ..................... ..................... .................. 011

@ibnbaà ag hànhunf eglt`n ................... ..................... ..................... ..................... ..................... ..................... ......... 01:

]`zflèbgltf nøibhf ................... ..................... ..................... ..................... ..................... ..................... ..................... 012

Ags`rrfnng lugv`s j`mbnbaàgs ................... ..................... ..................... ..................... ..................... ..................... .. 07;

Xge`l` 0;]àbh`hbøl ag gxprgsbflgs `nigmr`bh`s ......................................... 070

¡[`r` hfeglz`r! Biu`n lóegrf ag dbn`s y hfnuel`s .................. ..................... ..................... ..................... ....... 07=


0. ]gin`s ag nfs ràbh`ngs .................... ..................... ..................... ..................... ..................... ..................... ...... 078

=. Hflvgrsbøl ag ul` r`åz ` ul` pftglhb` dr`hhbfl`rb` ..................... ..................... ..................... .......... 071

8. ]àbh`ngs sgegc`ltgs ................... ..................... ..................... ..................... ..................... ..................... .......... 077

8.0 Xue` y rgst` ag ràbh`ngs sgegc`ltgs ..................... ..................... ..................... ..................... ....... 077 8.= Euntbpnbh`hbøl ag ràbh`ngs .................... ..................... ..................... ..................... ..................... ........ 074

8.8 Abvbsbøl ag ràbh`ngs .................. ..................... ..................... ..................... ..................... ..................... .. 07:

8.1 Gnbebl`hbøl ag d`htfrgs gl nfs ràbh`ngs .................... ..................... ..................... ..................... .... 072

1. ]`hbfl`nbz`hbøl ag aglfeblàfrgs .................... ..................... ..................... ..................... ..................... . 04;

]gsuegl ................... ..................... ..................... ..................... ..................... ..................... ..................... ...................... .... 04=

Blvgstbiug gl n` rga... ................... ..................... ..................... ..................... ..................... ..................... ..................... 04=

@utfhfltrfn .................... ..................... ..................... ..................... ..................... ..................... ..................... .................. 048

@ibnbaà ag hànhunf eglt`n ................... ..................... ..................... ..................... ..................... ..................... ......... 044

]`zflèbgltf nøibhf ................... ..................... ..................... ..................... ..................... ..................... ..................... 049Ags`rrfnng lugv`s j`mbnbaàgs ................... ..................... ..................... ..................... ..................... ..................... .. 04:

Xge`l` 00Fpgr`hbflgs pfnblfeb`ngs .................. ..................... ..................... ..................... ........... 042

¡[`r` hfeglz`r! N` bepfrt`lhb` agn ànigmr` .................... ..................... ..................... ..................... .................. 09;


0. [fnblfebfs ..................... ..................... ..................... ..................... ..................... ..................... ..................... ....... 090

0.0 Hn`sbh`hbøl ag nfs pfnblfebfs ................... ..................... ..................... ..................... ..................... . 090

=. Xue` y rgst` ag pfnblfebfs .................... ..................... ..................... ..................... ..................... ................ 09=

8. Euntbpnbh`hbøl ag pfnblfebfs .................. ..................... ..................... ..................... ..................... ................ 091

8.0 Euntbpnbh`hbøl ag ul lóegrf pfr ul pfnblfebf ................... ..................... ..................... .......... 091

8.= Euntbpnbh`hbøl ag ul eflfebf pfr ul pfnblfebf ................... ..................... ..................... ...... 091

8.8 Euntbpnbh`hbøl gltrg pfnblfebfs .................... ...................... ..................... ..................... .................. 097

1. Abvbsbøl ag ul pfnblfebf gltrg ul eflfebf ................... ..................... ..................... ..................... .... 094

TE`tgeàtbh` ∝ Ålabhg



]gsuegl ..................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... ... 099

Blvgstbiug gl n` rga... ..................... ..................... ..................... ..................... ..................... ..................... ................... 099

@utfhfltrfn .................. ..................... ..................... ..................... ..................... ..................... ..................... .................... 09:

@ibnbaà ag hànhunf eglt`n .................... ..................... ..................... ..................... ..................... ..................... ........ 0:;

]`zflèbgltf nøibhf ..................... ..................... ..................... ..................... ..................... ..................... ................... 0:0

Ags`rrfnng lugv`s j`mbnbaàgs .................... ..................... ..................... ..................... ..................... ..................... . 0:=

Xge`l` 0=[rfauhtfs lft`mngs .................. ..................... ..................... ..................... ..................... .............. 0:8

¡[`r` hfeglz`r! @n-Cw`rbzeb, gn hrgàfr agn ànigmr` .................... ..................... ..................... ..................... .. 0:1


0. [rfauhtfs lft`mngs ..................... ..................... ...................... ..................... ..................... ..................... ........... 0:7

0.0 Huàràf ag n` sue` ag ul mblfebf (` + m)= .................... ..................... ..................... ............... 0:7

0.= Huàràf ag n` abdgrglhb` ag ul mblfebf (` ‘ m)= .................. ..................... ..................... ........ 0:4

0.8 [rfauhtf ag n` sue` pfr n` abdgrglhb` (` + m)(` ‘ m) ..................... ..................... ..................... . 0:4 0.1 [rfauhtf ag afs mblfebfs hfl ul târeblf hfeól y sbilfs biu`ngs .................. ............. 0:9

0.7 Humf ag n` sue` ag ul mblfebf (` + m)8 ..................... ..................... ..................... ..................... .. 0:2

0.4 Humf ag n` abdgrglhb` ag ul mblfebf (` ‘ m)8 .................... ..................... ..................... ............... 02;

]gsuegl ..................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... ... 020

Blvgstbiug gl n` rga... ..................... ..................... ..................... ..................... ..................... ..................... ................... 020

@utfhfltrfn .................. ..................... ..................... ..................... ..................... ..................... ..................... .................... 02=


]`zflèbgltf nøibhf ..................... ..................... ..................... ..................... ..................... ..................... ................... 027

Ags`rrfnng lugv`s j`mbnbaàgs .................... ..................... ..................... ..................... ..................... ...................... 024

Xge`l` 08D`htfrbz`hbøl B .................. ..................... ......................... ..................... ..................... ..................... ........ 029

¡[`r` hfeglz`r! D`htfrgs prbefs ..................... ..................... ..................... ..................... ..................... .................. 02:


0. D`htfrbz hbøl ag pfnblfebfs .................. ..................... ..................... ..................... ..................... .................. 022

0.0 D`htfr hfeól eflfebf ................... ..................... ..................... ..................... ..................... .............. 022

0.= D`htfr hfeól pfr ìrup`hbøl ag târeblfs ................... ..................... ..................... ................... =;0

0.8 Qrblfebf huàràf pgrdghtf.............................. ..................... ..................... ..................... ................. =;8

0.1 Abdgrglhb` ag huàràfs ................... ..................... ..................... ..................... ..................... .............. =;7

]gsuegl ..................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... ... =;9

Blvgstbiug gl n` rga... ..................... ..................... ..................... ..................... ..................... ..................... ................... =;9

@utfhfltrfn .................. ..................... ..................... ..................... ..................... ..................... ..................... .................... =;:

@ibnbaà ag hànhunf eglt`n .................... ..................... ..................... ..................... ..................... ..................... ........ =0;

]`zflèbgltf nøibhf ..................... ..................... ..................... ..................... ..................... ..................... ................... =00

Ags`rrfnng lugv`s j`mbnbaàgs .................... ..................... ..................... ..................... ..................... ...................... =0=

TBBIG] ∝ [fnfhjbh





Blvgstbiug gl n` rga... ..................... ..................... ..................... ..................... ..................... ..................... ................... =7;

@utfhfltrfn .................. ..................... ..................... ..................... ..................... ..................... ..................... .................... =70

@ibnbaà ag hànhunf eglt`n .................... ..................... ..................... ..................... ..................... ..................... ........ =71

]`zflèbgltf nøibhf ..................... ..................... ..................... ..................... ..................... ..................... ................... =77

Ags`rrfnng lugv`s j`mbnbaàgs .................... ..................... ..................... ..................... ..................... ..................... . =74

Xge`l` 09]gp`sf3 sge`l`s 2 ` 04 ................... ..................... ..................... ..................... ............. =79


[ftglhb hbøl ag gxprgsbflgs nigmr`bh`s ................... ..................... ..................... ..................... ..................... =72

]àbh`hbøl ag gxprgsbflgs `nigmr`bh`s ................... ..................... ..................... ..................... ..................... ... =40

Fpgr`hbflgs pfnblfeb`ngs .................. ..................... ..................... ..................... ..................... ..................... ........ =48

[rfauhtfs lft`mngs ..................... ..................... ..................... ..................... ..................... ..................... ................... =47

D`htfrbz hbøl B .................... ..................... ..................... ..................... ..................... ..................... ..................... ......... =49

D`htfrbz hbøl BB ................... ..................... ..................... ..................... ..................... ..................... ..................... ......... =42Dr`hhbflgs `nigmr`bh`s B ................... ..................... ..................... ..................... ..................... ..................... ............ =90

Dr`hhbflgs `nigmr`bh`s BB ................... ..................... ..................... ..................... ..................... ..................... ........... =98

@ibnbaà ag hànhunf eglt`n .................... ..................... ..................... ..................... ..................... ..................... ........ =97

Frbglt`hbflgs sfmrg n` prugm` dbl`n ................................................................................................................... =94

Hn`vgs ............................................................................................................................................................... =99

Q`nngr e`tgeàtbhf[rugm` @ .................. ..................... ..................... ..................... ..................... ..................... ..................... .................... =22

[rugm` M ................... ..................... ..................... ..................... ..................... ..................... ..................... .................... 8;9

Mbmnbfir`då` ................................................................................................................................................... 808

TBBBBIG] ∝ [fnfhjbh



‘0 5 b ‘0 5 b

0 = 8 1

8

1

7

=

0

;

x

y

0E`tgeàtbh` ∝ ¡Mbglvglba` y mbglvglbaf!

¡Mbglvglba` y mbglvglbaf!

Blbhbèfs hfl gltusb`sef gn hursf ag e`tgeàtbh` ag 1.» m`hjbnngr`tf. Gstg nbmrf hfrrgspflag`n prbegr sgegstrg agn irupf [fnfhjbh y hflst` ag abghbsbgtg sge`l`s, gl n`s qug sg ags`-rrfnn`ràl hblhf hfepgtglhb`s e`rhà`s pfr gn Hurråhunf L`hbfl`n M`sg (HLM).

Gn hursf gstà gstruhturàf ag t`n dfre` qug nfs hfltglbafs sfl iràu`ngs y afsbhàfs p`r`dfrt`nghgr y glrbqughgr n`s rè`s sbiubgltgs3

• @rbteâtbh`3 pgrebtg rg`nbz`r fpgr`hbflgs `rbteâtbh`s hfl nfs lóegrfs rg`ngs.

• Ànigmr`3 `yua` ` nggr, fpgr`r y rgprgsglt`r bldfre`hbøl gl gn ngliu`cg sbemønbhf ag nè`tgeàtbh`.

• Igfegtrå`3 rgdbgrg n`s prfpbgaàgs ag n`s dbiur`s igfeâtrbh`s hfef àrg`s y vfnóeglgsag pfnåiflfs. Qèmbâl rgn`hbfl` n`s prfpbgaàgs gltrg rght`s.

• Qrbiflfegtrå`3 gst` rè` ag n` e`tgeàtbh` sg glh`ri` agn gstuabf ag nfs nàfs y àliu-nfs ag trbàliunfs rghtàliunfs, ` tr`vâs ag n`s r`zflgs trbiflfeâtrbh`s.

• Hànhunf abdgrglhb`n g bltgir`n3 pgrebtg rgn`hbfl`r n`s dulhbflgs y sus abdgrgltgs `pnbh`-hbflgs hfef r`zøl ag hèmbf.

@ltgs ag gltr`r `n agt`nng ag n`s hfepgtglhb`s p`r` gstg hursf, fmsgrvgefs n` pfrtà`.

Gn eulaf ag n` pfrtà` lfs rghugra` qug sfefs p`rtg agn pn`lgt` Qbgrr` y qug gl ân, nè`tgeàtbh` gs ul` jgrrèbglt` qug lfs `yua` ` agshumrbr, `prglagr y absdrut`r n`s `rtgs yn` huntur`.

Nfs lóegrfs e`y`s lfs rghugra`l qug pgrtglghgefs ` ul pugmnf euntbâtlbhf g bltgr-huntur`n, hfl ul hflfhbebgltf e`tgeàtbhf qug

agmgefs v`nfr`r y hflsgrv`r.

N` e`lf a`laf ul hnbh lfs blvbt` ` gxpnfr`r,agshumrbr y hfep`rtbr hflfhbebgltfs hfl gnrgstf agn eulaf ` tr`vâs ag bltgrlgt.

N` beìgl ag ul irupf ag e`lfs `nrgagafrag ul hfr`zøl lfs j`mn` ag gst`mnghgr rgn`-hbflgs hfl n`s pgrsfl`s qug lfs rfag`l.


http://slidepdf.com/reader/full/libro-polochic-matematica-1er-sem 14/326=BIG] ∝ [fnfhjbh

¶Høef `nh`lz`rà n`s hfepgtglhb`s e`tgeàtbh`s6

Lfs gldfh`rgefs gl hblhf hfepgtglhb`s gspghåh`s. [`r` s`mgr sb n`s j` `nh`lzàf, gn HurråhunfL`hbfl`n M`sg prfpflg blabhàfrgs ag nfirf, gstfs hrbtgrbfs sfl hfef ul tgreøegtrf qug ebagsu agsgepgðf gl hà` hfepgtglhb`. Brgefs `v`lz`laf p`sf ` p`sf.

T`y` càlafsg quâ sghhbflgs agn nbmrf j`hgl pfsbmng qug ustga ags`rrfnng hà` hfepgtglhb` qugprgsgltèfs ` hfltblu`hbøl.

Hfepgtglhb 0 Xghhbøl agn nbmrfXge`l`

0 = 8 1 7 4 9 : 2 0; 00 0= 08 01 07 04 09

[rfauhg p`trflgs`rbteâtbhfs,`nigmr`bhfs yigfeâtrbhfs

`pnbh`laf prfpbg-aàgs, rgn`hbflgs,iur`s igfeâ-trbh`s, såemfnfsy sgð`ngs agdgløeglfs l`tu-r`ngs, sfhb`ngs yhuntur`ngs.

¡[`r`hfeglz`r! Z Z Z Z Z

Gn eulaf agn` e`tgeàtbh` Z Z Z Z Z Z Z Z Z

]gsuegl

Blvgstbiug gln` rga Z Z Z Z Z Z

@utfhfltrfn Z Z Z Z Z Z Z Z

@ibnbaà aghànhunf eglt`n

Z Z Z Z Z Z Z

]`zflèbgltfnøibhf Z Z Z Z Z Z Z Z Z Z Z Z

Ags`rrfnng lugv`sj`mbnbaàgs Z Z Z Z Z Z Z Z Z Z Z

Hfepgtglhb = Xghhbøl agn nbmrf Xge`l`0 = 8 1 7 4 9 : 2 0; 00 0= 08 01 07 04 09

]gsugnvgsbtu`hbflgs,prfmnge`s agh`ràhtgr dfre`nqug age`la`l gnafeblbf agn pgl-sèbgltf nøibhfe`tgeàtbhf yn`s fpgr`hbflgs

e`tgeàtbh`sag `rbteâtbh` yànigmr` gl nfshflcultfs lu-eârbhfs rg`ngs yhfepngcfs.

¡[`r`hfeglz`r! Z Z Z Z Z Z Z Z Z Z

Gn eulaf agn` e`tgeàtbh` Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z

]gsuegl

Blvgstbiug gln` rga Z Z Z Z Z Z Z Z Z Z Z Z Z

@utfhfltrfn Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z


Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z

]`zflèbgltfnøibhf Z Z Z Z Z Z Z Z Z Z Z Z Z Z

Ags`rrfnng lugv`sj`mbnbaàgs Z Z Z Z Z Z Z Z Z Z Z Z Z


http://slidepdf.com/reader/full/libro-polochic-matematica-1er-sem 15/3268E`tgeàtbh` ∝ ¡Mbglvglba` y mbglvglbaf!

Hfepgtglhb` 8 Xghhbøl agn nbmrfXge`l`

0 = 8 1 7 4 9 : 2 0; 00 0= 08 01 07 04 09@pnbh` hflf-hbebgltfs sfmrgdulhbflgs, e`-trbhgs, igfegtrå`y vghtfrgs, glsbtu`hbflgs qugprfeugvgl gnegcfrèbgltf ytr`lsdfre`hbølagn egabf l`tur`n,

sfhb`n y huntur`nag su hfltgxtf.

¡[`r`hfeglz`r!

Gn eulaf agn` e`tgeàtbh` Z Z Z

]gsuegl

Blvgstbiug gln` rga Z

@utfhfltrfn Z


]`zflèbgltfnøibhf

Ags`rrfnng lugv`sj`mbnbaàgs


0 = 8 1 7 4 9 : 2 0; 00 0= 08 01 07 04 09

Wtbnbz` tâhlbh`sag suhgsbflgs

y sgrbgs p`r`bltgrprgt`rjghjfs sfhb`ngs,ghfløebhfs yigfiràhfs.

¡[`r`hfeglz`r! Z

Gn eulaf ag

n` e`tgeàtbh` Z Z Z

]gsuegl

Blvgstbiug gln` rga Z

@utfhfltrfn Z


]`zflèbgltfnøibhf Z

Ags`rrfnng lugv`sj`mbnbaàgs Z


http://slidepdf.com/reader/full/libro-polochic-matematica-1er-sem 16/3261BIG] ∝ [fnfhjbh

Hfl su gsdugrzf, pfarà hflqubst`r gst`s hfepgtglhb`s ` tr`vâs ag trgs tbpfs ag hfltglbafs3

Aghn`r`tbvfs3 Nfs hfltglbafs aghn`r`tbvfs ng `pfrt`ràl gn hflfhbebgltf ag nfsabstbltfs `spghtfs tgørbhfs, hflhgptu`ngs y hbgltådbhfs agn àrg`.

[rfhgabeglt`ngs3 Hfef blabh` su lfemrg, nfs hfltglbafs prfhgabeglt`ngs sgags`rrfnn`ràl ` m`sg ag gcgrhbhbfs, prfhgabebgltfs f `lànbsbs ag prfmnge`s gl nfshu`ngs puga` agefstr`r gn afeblbf y n` pugst` gl pràhtbh` ag nfs hflfhbebgltfsaghn`r`tbvfs.

@htbtuabl`ngs3 Nfs hfltglbafs `htbtuabl`ngs sfl tfaf `qugnnf qug tbglg qug vgrhfl su `htbtua `ltg gn gstuabf y `ltg n` vba`, gl iglgr`n. Gl gn nbmrf glhfltr`rà `htb-vbaàgs y rgdngxbflgs sfmrg sbtu`hbflgs hftbab`l`s aflag bltgrvbglg n` e`tgeàtbh`.


0 = 8 1 7 4 9 : 2 0; 00 0= 08 01 07 04 09Gepng` n`s tgfrå`sag igfegtrå` ytrbiflfegtrå`p`r` bltgrprgt`rbldfre`hbflgs ygn`mfr`r bldfregssfmrg sbtu`hbflgsrg`ngs.

¡[`r`hfeglz`r! Z

Gn eulaf agn` e`tgeàtbh`

]gsuegl

Blvgstbiug gln` rga

@utfhfltrfn

@ibnbaà ag

hànhunf eglt`n ]`zflèbgltf

nøibhf Z Z Z Z Z

Ags`rrfnng lugv`sj`mbnbaàgs Z Z Z Z












http://slidepdf.com/reader/full/libro-polochic-matematica-1er-sem 22/3260;BIG] ∝ [fnfhjbh

Nbmrf, hn`sg ràb`n y hårhunf ag gstuabf

¡Xu gqubpf ag tr`m`cf!

Gn nbmrf, hfl sgr ul` mugl` jgrrèbglt`, lf nf gs tfaf. [`r` qug ustga `nh`lhg gn lbvgn aghfepgtglhb` agsgàf, lugstrf sbstge` pflg ` su abspfsbhbøl3 gn nbmrf, n` hn`sg ràb`n y n`

blvbt`hbøl ` p`rtbhbp`r gl ul hårhunf ag gstuabf.

• N` hn`sg ràb`n tbglg hfef dulhbøl prblhbp`ngxpnbh`r y d`hbnbt`r n` hfeprglsbøl ag nfs tge`s

tr`tàfs gl gn nbmrf.

[ugag gshuhj`r n` hn`sg ràb`n gl ul` gebsfràg su nfh`nbaà, agsh`ri`rn`s gl lugstr` pà-ibl` www.bigr.gau.it f àqubrbrn`s gl HA gl n`hffrabl`hbøl rgibfl`n.

• Gn hårhunf ag gstuabf gs gn nui`rp`r` hfep`rtbr y `prglagr cultfs.

@prfvghjg gstfs rghursfs y pøygsg gl pgrsfl`s ag su hfeulbaà p`r` rgsfnvgr sus aua`s.

• Gn nbmrf huepng hu`trf dulhbflgs3

`. Qgxtf, gl gn qug glhugltr` n` bldfre`hbøl y gnags`rrfnnf ag nfs hfltglbafs ` gstuab`r.

m. [bz`rrøl, p`r` qug aur`ltg n` hn`sg ràb`nsumr`yg bag`s bepfrt`ltgs f rg`nbhg abstblt`s`htbvbaàgs.

h. Huàgrlf ag tr`m`cf, hfl gcgrhbhbfs p`r` pr`h-tbh`r nf `prglabaf.

a. Jgrrèbglt` ag ùtfgv`nu`hbøl, hu`laf rg-sugnvg su ùtfhfltrfn hà` sge`l`.

D`nt` hèmb`rpfrtàbt`


http://slidepdf.com/reader/full/libro-polochic-matematica-1er-sem 23/32600E`tgeàtbh` ∝ ¡Mbglvglba` y mbglvglbaf!

Lugstr` egtfafnfiå` p`sf ` p`sf

[`r` d`hbnbt`r su `prglabz`cg y `prfvghj`r eàs y egcfr gn gstuabf hà` sge`l`, sbi` gstfsp`sfs. ¡Lf sg s`ntg lbliulf!

Ng` gn hfltglbaf ag n` sge`l`

Nggr gn hfltglbaf lfs pgrebtg tglgr ul` bag` iglgr`n agn tge`3 quâ s`-

mgefs, hfl quâ nf rgn`hbflèfs, gth. Gstg prbegr hflt`htf tèmbâl lfs

j`rà h`gr gl n` huglt` agn gsdugrzf rg`nbz`r p`r` prglagr nf lugvf y lfs

pflarà "gl fla`" p`r` n` hn`sg ràb`n.

0

Gshuhjg n` hn`sg ràb`nHfl nfs 7 sgltbafs

N` hn`sg ràb`n gs lugstr` e`gstr`. Ag `jå qug gn prfirè` sg nnèg "GnE`gstrf gl H`s`". N`s e`gstr`s y e`gstrfs nfhutfrgs gxpnbh`l gn hfltglb-

af, prfpflgl gcgrhbhbfs y ftrfs gcgepnfs p`r` èpnb`r gn tge`.

=

Agspuâs ag n` hn`sg ràb`n, su tr`m`cf pgrsfl`n Gstuabf y ùtfhfltrfn

Dbl`nbzà` n` hn`sg ràb`n gs gn efegltf ag su tr`m`cf pgrsfl`n. Abstrbmuy`

su tbgepf3 gs egcfr ul pfhf hà` aå`, qug tfaf n` våspgr`.

8

Hflsuntg sus aua`sWl gstuab`ltg bltgnbigltg s`mg huàlaf pgabr `yua`

Hflsuntg nfs tge`s qug lf ng j`l qugaàf hn`rfs gl ftrfs nbmrfs, gl bltgr-

lgt, hfl dèbnb`rgs f èbifs. Xgiurf qug glhfltr`rà pgrsfl`s abspugst`s

` `yua`rng.

1

[`rtbhbpg gl ul hårhunf ag gstuabf

@prglagr cultfs[øli`sg ag hugraf hfl ftrfs gstuab`ltgs y fri`lbhgl ul hårhunf ag gstu-

abf. Xfnbhbtgl n` `yua` ag `niul` pgrsfl` vfnult`rb` ag n` hfeulbaà. Gsf

ngs `yua`rà ` rgsfnvgr aua`s y rgdfrz`r nf `prglabaf. @ageàs, tglaràl n`

fpfrtulbaà ag bltgrhèmb`r `prglabz`cgs, bag`s y sgltbebgltfs.

]ghugrag qug sbgeprg pugag `huabr ` su tutfr `sbilàf.

7


http://slidepdf.com/reader/full/libro-polochic-matematica-1er-sem 24/3260=BIG] ∝ [fnfhjbh

¶Høef `prfvghj`r egcfr su gstuabf6

• Musqug ul nui`r høefaf y hfl mugl` bnuebl`hbøl. Gs bepfrt`ltg qug sg`ngcg agn rubaf y ag n`s abstr`hhbflgs.

• Gnbc` ul jfr`rbf p`r` tr`m`c`r y gstuab`r. N` hflst`lhb` y n` abshbpnbl` sflsus egcfrgs hfep`ðgr`s ag gstuabf.

• Ng` hfl `tglhbøl n`s blstruhhbflgs ag nfs gcgrhbhbfs `ltgs ag rgsfnvgrnfs.

• Hflsuntg sus aua`s hfl ftr`s pgrsfl`s ag su hfeulbaà qug puga`l`yua`rng.



00

= 8 1

8

1

7

=

0

;

x

y

‘ 0 5 b

x = +

y = 5

r =

Nøibh` B

¶Puâ glhfltr`rà gst` sge`l`6

Gn huepng`ðfs ag @ngc`lar`

Nøibh` prfpfsbhbfl`n

@ibnbaà ag hànhunf hfl euntbpnbh`hbflgs y abvbsbflgs

[rfmnge`s ag nøibh`

Gst` sge`l` nfir`rà3

Hflfhgr y hfeprglagr gn hèpf ag gstuabf ag n` nøibh`.

Bagltbdbh`r y hflstrubr prfpfsbhbflgs sbepngs.

Agtgrebl`r gn v`nfr ag vgraà ag prfpfsbhbflgs sbepngs. Gcgrhbt`r n` lgi`hbøl ag prfpfsbhbflgs sbepngs.

Hfeprglagr y pr`htbh`r gn usf ag hu`ltbdbhàfrgs.

]gsfnvgr fpgr`hbflgs `rbteâtbh`s màsbh`s hfl ìbnbaà.

08E`tgeàtbh` ∝ Xge`l` 0



¡[`r` hfeglz`r!Gn huepng`ðfs ag @ngc`lar`

Tgèfs y sfnuhbflgefs ul pgqugðf prfmnge` gepng`laf lugstr` nøibh`.

Gn sbiubgltg gcgrhbhbf dug pn`ltgàf gl n`s fnbepbà`s ag e`tgeàtbh`s ag @sb` yXbli`pur, gl `mrbn ag =;07. Nâ`nf, pfli` ` tr`m`c`r sus lgurfl`s y rgsuânv`nf3

@nmgrtf y Mgrl`raf sg `h`m`l ag j`hgr èbifs ag @ngc`lar` y qubgrgl s`mgr

huàlaf gs su huepng`ðfs.

@ngc`lar` ngs a` ul` nbst` hfl 0; pfsbmngs dghj`s3

¡@ tr`m`c`r!

Hfl n` bldfre`hbøl aà`, rgspfla` n`s prgiult`s sbiubgltgs.

0) ¶Huàlaf gs gn huepng`ðfs ag @ngc`lar`6

=) Gxpnbqug høef j`nnø n` rgspugst`, ¶huàn dug su r`zflèbgltf6

Nugif, @ngc`lar` ngs abhg n` dghj` ag su huepng`ðfs, pgrf pfr sgp`ràf3 `

@nmgrtf ng abhg gn egs y ` Mgrl`raf, gn aå`.

@emfs cøvglgs sg pn`ltg`l n` sbiubgltg bldfre`hbøl3

• @nmgrtf3 "Lf sâ huàlaf gs gn huepng`ðfs ag @ngc`lar`, pgrf sâ qug

Mgrl`raf tèpfhf nf s`mg".

• Mgrl`raf3 "@n prblhbpbf lf s`må` huàlaf gr` gn huepng`ðfs ag @ngc`lar`,

pgrf `jfr` y` nf sâ".

• @nmgrtf3 "Gltflhgs, yf tèmbâl sâ huàlaf gs su huepng`ðfs".

@a`ptàf ag iff.in/=D2Icx

e`yf 07 e`yf 04 e`yf 02

culbf 09 culbf 0:

cunbf 01 cunbf 04

ìfstf 01 ìfstf 07 ìfstf 09

01BIG] ∝ [fnfhjbh



0. Nøibh` prfpfsbhbfl`n@ltgs ag blbhb`r, rghfragefs qug hfl n` `yua` ag n` nøibh` gstuabèfs nfs r`zf-

lèbgltfs y agauhbefs hflhnusbflgs ` p`rtbr ag supugstfs pn`ltgàfs.Wl` rè` ag n` hbglhb` ltgrbfr gs n` nøibh` prfpfsbhbfl`n, qug gstuab` glulhb`-afs y gv`nó` su lbvgn ag vgraà, gs aghbr, agtgrebl` sb nf qug sg `re` gs d`nsff vgraàgrf.

=. [rfpfsbhbflgs sbepngsNfs glulhbàfs qug gstuab` n` nøibh` prfpfsbhbfl`n sg aglfebl`l prfpfsbhbf-lgs, gst`s pugagl tfe`r afs v`nfrgs nøibhfs f ag vgraà3 vgraàgrf f d`nsf. Xblf pfagefs hflhnubr sb ul glulhbàf gs d`nsf f vgraàgrf, gltflhgs lf gs ul`prfpfsbhbøl.

N`s prfpfsbhbflgs sbepngs sfl `qugnn`s qug lf sg pugagl abvbabr sbl `dght`rsu sgltbaf. Xg rgprgsglt`l utbnbz`laf ngtr`s eblóshun`s. Tgèfs ulfs gcgepnfs3

q3 Rf l`hå gl Jugjugtgl`lif.

r 3 Iu`tge`n` gs p`rtg ag Hgltrfèârbh`.

w 3 Nfs eèådgrfs sfl vbvåp`rfs.

Hfef vgefs, q, r y w sfl prfpfsbhbflgs sbepngs pfrqug lf sg pugagl sgp`r`r.

Gl tfa`s sg pugag gv`nu`r su v`nfr ag vgraà, gl n` prbegr`, pugag sgr d`nsf fvgraàgrf, agpglabglaf sb n` pgrsfl` l`hbø gl Jugjugtgl`lif f gl niól ftrfnui`r. Ebgltr`s t`ltf, n` sgiula` y n` tgrhgr` prfpfsbhbøl tfe`l gn v`nfr nøibhfag vgraàgrf.

Lf tfafs nfs glulhbàfs sfl prfpfsbhbflgs. Hfef eglhbflèfs `ltgrbfr-egltg, sb lf pfagefs agtgrebl`r gn v`nfr nøibhf ag ul` fr`hbøl, lf gs ul`prfpfsbhbøl.

Gstf suhgag hfeólegltg gl glulhbàfs bepgr`tbvfs, ag prgiult` f `qugnnfsqug prgsglt`l ul cubhbf ag v`nfr.

Tgèfs `niul`s dr`sgs qug lf sfl prfpfsbhbflgs3

¡Lf tbrg m`sur` gl n` h`nng!

¶Puâ gstfy j`hbglaf6

Gs bepfrt`ltg gst`r gl n` rgulbøl.

Hfef vgefs, ` nfs glulhbàfs `ltgrbfrgs lf sg ngs pugag agtgrebl`r ul v`nfrnøibhf.

Cubhbf ag v`nfr3glulhbàf qug

gxprgs` ul` fpblbøl sumcgtbv`.





Gcgrhbhbf 0@. Ng` n`s fr`hbflgs sbiubgltgs. Nugif, e`rqug hfl ul` "Z" gn huàrf, sgiól sg` ul` prfpfsbhbøl

f lf. Ng aèfs ul gcgepnf.

Xå Lf

;) Iu`tge`n` gs p`rtg ag @eârbh`.

0) ¶Puâ jfr` gs6

=) [fr d`vfr, `vås`ng ` ân.

8) = gs ul abvbsfr ag 01.

1) Ebiugn Àlign @sturb`s dug ul gshrbtfr.

7) ¶Puâ eósbh` gshuhj`6

4) Gn tâ gs egcfr qug gn h`dâ.

9) Gn `zun gs ul hfnfr sghula`rbf.

:) Gn è`rbnnf gs ul hfnfr prbe`rbf.

2) N` Qbgrr` gs pn`l`.

0;) Nggr gs euy bepfrt`ltg.

M. @jfr` gs su turlf. [bglsg y gshrbm` gl n`s nålg`s sbiubgltgs fhjf prfpfsbhbflgs. Nugif, agtgreblgsu v`nfr nøibhf hfnfh`laf ul` T (vgraàgrf) f ul` D (d`nsf) gl gn gsp`hbf ` n` agrghj`. Iuågsg pfr

gn gcgepnf.

;)

0)

=)

8)

1)

7)

4)

9)

:)

N`s e`rbpfs`s pugagl vfn`r ir`lags abst`lhb`s. T

x

04BIG] ∝ [fnfhjbh





8. Nøibh` ag prgabhàfsN` nøibh` ag prgabhàfs pgrebtg j`hgr `re`hbflgs hfl rgspghtf ` rgn`hbflgsgltrg fmcgtfs qug s`tbsd`hgl gngegltfs ag ul hflcultf ulbvgrsf gst`mnghbaf.

Gl gst` nøibh` sg gepng`l nfs hu`ltbhàfrgs, nfs hu`ngs sfl såemfnfs utbnbz`-

afs p`r` gst`mnghgr huàltfs gngegltfs ag ul hflcultf agtgreblàf huepnglhfl hbgrt` prfpbgaà.

Nfs hu`ltbhàfrgs pgrebtgl n` hflstruhhbøl ag prfpfsbhbflgs ` p`rtbr ag dul-hbflgs prfpfsbhbfl`ngs, gs aghbr, ag cubhbfs qug gst`mnghgl p`ràegtrfs hfl nfshu`ngs agmgl huepnbr nfs gngegltfs ag ul hflcultf ulbvgrsf.

Nfs hu`ltbhàfrgs eàs utbnbzàfs sfl gn ulbvgrs`n y gn gxbstglhb`n.

8.0 Hu`ltbdbhàfr ulbvgrs`n (∋)

Xg gepng` p`r` `re`r qug tfafs nfs gngegltfs ag ul hflcultf huepngl hflul` hflabhbøl f p`ràegtrf agtgreblàf. Xg rgprgsglt` hfl gn såemfnf ∋ y sgngg "p`r` tfaf…". Tgèfs ul gcgepnf p`r` gltglagr egcfr3

∋ x ∊ ] , = x ∊ ]

N` gxprgsbøl `ltgrbfr sg ngg3 "p`r` tfa` x qug pgrtglghg ` ] (hflcultf ag nfslóegrfs rg`ngs), sg huepng qug = x pgrtglghg ` ] "

8.= Hu`ltbdbhàfr gxbstglhb`n (∈)

Xg utbnbz` p`r` blabh`r qug j`y n eglfs ul gngegltf gl ul hflcultf qug huepnghfl ul` hflabhbøl agtgreblà`. Xg rgprgsglt` hfl gn såemfnf ∈ y sg ngg "gxbstg`n eglfs ul…". Tgèfs ul gcgepnf.

Qfe`laf hfef rgdgrglhb` nfs hflcultfs sbiubgltgs3 @ 5 {0, =, 8, 1, 7}, M 5 {1, 7,4, 9, :}

∈ x ∊ M ∩ x ∊ @

N` gxprgsbøl `ltgrbfr sg ngg3 "gxbstg n eglfs ul gngegltf x qug pgrtglghg ` My pgrtglghg tèmbâl ` @".

Gcgrhbhbf 8@. Ng` n`s gxprgsbflgs sbemønbh`s sbiubgltgs. Nugif, rggshråm`n`s gl ngliu`cg hfeól. Iuågsg pfr gn

gcgepnf.

;) ∈ x ∊ M ∩ x ∊ P

]ghugrag qug x gsul` v`rb`mng y pugagrgprgsglt`r ` hu`nqubgrlóegrf.

Gn såemfnf ∩ sg ngg "y".

Gxbstg `n eglfs ul` x qug pgrtglghg ` M y tèmbâl pgrtglghg ` P.

0:BIG] ∝ [fnfhjbh



0) ∋ x ∊ W, 7 x ∊ W

=) ∈ x ∊ D ∩ x ∊ E

M. Ng` n`s gxprgsbflgs sbiubgltgs. Nugif, tr`lsdøregn`s ngliu`cg sbemønbhf y rggshråm`n`s. Ng aèfsul gcgepnf.

;) [`r` tfa` x qug pgrtglghg ` nfs lóegrfs l`tur`ngs, sg huepng qug : x pgrtglghg tèmbâl ` nfslóegrfs l`tur`ngs.

0) Gxbstg `n eglfs ul gngegltf x qug pgrtglghg ` @ y pgrtglghg tèmbâl ` M.

=) [`r` tfa` x qug pgrtglghg ` nfs lóegrfs rg`ngs, sg huepng qug 4 x gs ul lóegrf rg`n.

8) Gxbstg `n eglfs ul` x qug pgrtglghg ` F y pgrtglghg tèmbâl ` [.

∋ x ∊ L, : x ∊ L

0. N` nøibh` prfpfsbhbfl`n gstuab` glulhbàfs y gv`nó` su v`nfr ag vgraà, gs aghbr, agtgrebl` sbsfl vgraàgrfs f d`nsfs.

=. N`s prfpfsbhbflgs sfl glulhbàfs qug pugagl sgr `l`nbzàfs y `så agtgrebl`r su v`nfr nøibhf.

Wl glulhbàf lf gs ul` prfpfsbhbøl sb lf sg pugag agtgrebl`r su v`nfr ag vgraà. Gstf suhgaghfeólegltg hfl glulhbàfs bepgr`tbvfs, ag prgiult` f `qugnnfs qug prgsglt`l ul cubhbf agv`nfr.

N`s prfpfsbhbflgs sbepngs sfl qugnn`s qug lf pugagl sgr abvbaba`s sbl dght`r su sgltbaf. Gst`ssfl rgprgsgltà`s hfl ul` ngtr` eblóshun`.

=.0 N` lgi`hbøl gs ul` fpgr`hbøl qug hflsbstg gl blvgrtbr gn v`nfr nøibhf ag ul` prfpfsbhbøl? sgblabh` `ltgpflbglaf gn sbilf "~" ` n` prfpfsbhbøl.

8. Nfs hu`ltbdbhàfrgs sfl såemfnfs qug sg utbnbz`l p`r` gst`mnghgr huàltfs gngegltfs ag ul hflcultf agtgreblàf huepngl hfl hbgrt` prfpbgaà.

8.0 Gn hu`ltbdbhàfr ulbvgrs`n (∋) sg gepng` p`r` dbre`r qug tfafs nfs gngegltfs ag ul hflcultfhuepngl hfl ul` hflabhbøl aà`.

8.= Gn hu`ltbdbhàfr gxbstglhb`n (∈) sg utbnbz` p`r` blabh`r qug `n eglfs ul gngegltf ag ul hflcultf huepng hfl ul` hflabhbøl aà`.

]gsuegl




@htbvbaà 0. Ageugstrg nf prglabaf

@. Ng` nfs glulhbàfs sbiubgltgs. Nugif, gshrbm` ` n` agrghj` sb gs ul` prfpfsbhbøl f lf. Ng aèfs ulgcgepnf.

;) Gn tbgepf hfrrg ràpbaf.

0) 8 gs ul d`htfr ag 2.

=) Gn abhhbfl`rbf sbrvg p`r` nggr hugltfs ag lbðfs.

8) ¶[fr quâ lf gstàs gl h`s`6

1) Gn fhâ`lf [`hådbhf hfnbla` hfl n`s hfst`s n sur ag Iu`tge`n`.

7) Gn dåsbhf eàs bepfrt`ltg agn sbinf ZZ dug @nmgrt Gblstgbl.

4) ¡[fr d`vfr, j`z tu t`rg`!

9) Gn `zóh`r gs aunhg.

:) Iu`tge`n` gs gn egcfr p`ås.

M. Ng` nfs glulhbàfs sbiubgltgs. Nugif, tr`lsdøregnfs ` ngliu`cg sbemønbhf. ]ghugrag nf qug `prglabøsfmrg nfs hu`ltbdbhàfrgs.

0) [`r` tfa` x qug pgrtglghg ` nfs lóegrfs rg`ngs, sg huepng qug 0; x gs ul lóegrf rg`n.

=) Gxbstg `n eglfs ul gngegltf z qug pgrtglghg ` H y pgrtglghg tèmbâl ` Y.

8) [`r` tfa` x qug pgrtglghg ` @, gxbstg `n eglfs ul` y qug pgrtglghg ` M.

1) [`r` tfa` a qug pgrtglghg ` Z, gxbstg `n eglfs ul` m qug pgrtglghg ` S.

7) Gxbstg `n eglfs ul gngegltf x qug pgrtglghg ` ] y pgrtglghg tèmbâl ` L.

@utfhfltrfn

Lf gs prfpfsbhbøl.

=;BIG] ∝ [fnfhjbh



@htbvbaà =. [r`htbqug nf prglabaf

@. [bglsg y gshrbm` fhjf prfpfsbhbflgs sbepngs. Nugif, gl n` nålg` hfrrgspflabgltg, hfnfqug ul` T, sbgs vgraàgr` f ul` D, sb gs d`ns`. Iuågsg pfr gn gcgepnf.

;)

0)

=)

8)

1)

7)

4)

9)

:)

M. Ng` n`s prfpfsbhbflgs ag n` t`mn` sbiubgltg, gv`nógn`s y agtgreblg su v`nfr nøibhf. Nugif, gshrbm`n` lgi`hbøl ag hà` ul` gl gn gsp`hbf hfrrgspflabgltg. ]ghugrag gshrbmbr n` lgi`hbøl gl dfre`sbemønbh` y gn v`nfr ag vgraà qug tfe` n` prfpfsbhbøl. Ng aèfs ul gcgepnf.

;) t3 Nfs pgrrfs tbglgl pgnf.

0) a3 Nfs h`m`nnfs hfrrgl euy ràpbaf.

=) i3 Nfs iu`tge`ntghfs sfefs lfrtgègrbh`lfs.

8) y3 Wl` p`n`mr` tbglg ngtr`s.

1) p3 Nfs `lbe`ngs sfl sgrgs vbvfs.

7) q3 Gn agp`rtègltf ag Cutb`p` pgrtglghg ` Jflaur`s.

4) z3 9 gs ul d`htfr ag 74.

Gn rgnfc blabh` n` jfr`. T

~t3 Nfs pgrrfs lf tbglgl pgnf. D

T

=0E`tgeàtbh` ∝ Xge`l` 0



[`r` i`l`r ìbnbaà eglt`n y r`pbagz, tbglg qug pr`htbh`r hflst`ltgegltg. Ng prfpflgefs rgsfnvgrnfs gcgrhbhbfs ag gstg `p`rtàf. Xu gsdugrzf y hflst`lhb` ng i`r`ltbz`ràl muglfs rgsuntàfs.

@. Xue`s y rgst`s.0) 0= + 77 5

=) 84 ‘ =7 5

8) =0 + 7= 5

1) 4: ‘ =8 5

7) 97 + =7 5

4) 7= + 10 5

9) 84 + 10 5

:) 97 ‘ 07 5

2) 7; ‘ =7 5

0;) 01 + 88 5

00) 02 ‘ 09 5

0=) =0 + 89 5

08) == + 88 5

01) 1= ‘ 8= 5

07) 19 ‘ 88 5

M. Euntbpnbh`hbflgs.0) = Ù : 5

=) 8 Ù = 5

8) 9 Ù 7 5

1) 4 Ù 1 5

7) : Ù 9 5

4) 7 Ù = 5

9) 8 Ù 7 5

:) 0; Ù = 5

2) : Ù 4 5

0;) 2 Ù 1 5

00) 9 Ù 9 5

0=) 7 Ù 4 5

08) ; Ù 9 5

01) 1 Ù = 5

07) 8 Ù 4 5

04) : Ù 7 5

09) 2 Ù 0; 5

0:) 7 Ù 2 5

02) : Ù = 5

=;) 0 Ù 9 5

=0) 2 Ù 8 5

==) 7 Ù 1 5

=8) ; Ù = 5

=1) 4 Ù 7 5

H. Abvbsbflgs.

0) 0 ù 0 5

=) 88 ù 00 5

8) ; ù 2 5

1) : ù 1 5

7) 2 ù 8 5

4) 22 ù 8 5

9) 22 ù 00 5

:) 71 ù 4 5

2) 84 ù 2 5

0;) 87 ù 7 5

00) 48 ù 2 5

0=) 0; ù 0; 5

08) 0;; ù 0 5

01) 9= ù 2 5

07) 1= ù 4 5

04) 17 ù 2 5

09) 84 ù 1 5

0:) ; ù 4 5

02) 2 ù 8 5

=;) =9 ù 2 5

=0) 74 ù : 5

==) 84 ù 8 5

=8) 84 ù 4 5

=1) 1: ù : 5

==BIG] ∝ [fnfhjbh

@ibnbaà ag hànhunf eglt`n



[fli` ` tr`m`c`r sus lgurfl`s. ]gsugnv` nfs sbiubgltgs prfmnge`s ag nøibh`.

0) N`s afs pugrt`s. Wstga gstà glhgrràf gl ul` j`mbt`hbøl gl n` qug j`y afs pugrt`s vbibnà`s pfr

afs iu`rab`s. Wl` nngv` ` n` nbmgrtà y n` ftr` ` sgiubr glhgrràf. [ugag gngibr ul` pugrt` y `ltgspugag j`hgr ul` prgiult` ` ulf ag nfs iu`rab`s.

J`y ul prfmnge`3 ulf ag gnnfs sbgeprg abhg n` vgraà, pgrf gn ftrf sbgeprg ebgltg.

¶Puâ prgiult` j`rå` p`r` s`nbr n` nbmgrtà6 [bglsg su rgspugst` y gshråm`n` gl n`s nålg`s sbiubgltgs.

=) ¶Ag quâ hfnfr gs gn sfemrgrf6 Beìblg qug j`y hu`trf rgfs gl ul` prbsbøl y qug tfafs gnnfsgstàl gltgrràfs gl gn sugnf j`st` n` ntur` agn hugnnf, hfl nf qug sfnf ngs sfmrgs`ng n` h`mgz` p`r`pfagr rgspbr`r. J`y ul eurf qug sgp`r` ` trgs ag nfs prbsbflgrfs hfl rgspghtf `n hu`rtf, hfefeugstr` n` bnustr`hbøl.

@ M H A

Gn iu`rab` qug nfs gstà vbibn`laf ngs abhg3

‑ "Hà` ulf ag ustgags nngv` pugstf ul sfemrgrf. Ag nfs hu`trf sfemrgrfs afs sfl mn`lhfs yafs sfl lgirfs. Xbl j`mn`r gltrg så, tbglgl abgz eblutfs p`r` qug ulf ag ustgags eg abi` ag quâhfnfr gs su sfemrgrf. Xb nf `hbgrt`, nfs nbmgr`râ? y sb lf, sg quga`l gltgrràfs m`cf gn `rabgltg sfn".

Dbl`negltg, n h`mf ag ul eblutf, ulf ag nfs prbsbflgrfs hflsbiug vgrbiu`r n` rgspugst` hfrrght`.

¶Puâ prbsbflgrf gs gn qug abhg gn hfnfr ag su sfemrgrf6 ¶Høef nf j` `vgrbiuàf6


]`zflèbgltf nøibhf



E`rqug hfl ul hjgqug n` h`sbnn` qug egcfr blabqug su rglabebgltf.nfiràf gl

prfhgsflf

nfiràf

A g s p u â s a g g s t u a b

` r . . .

Hflfzhf y hfeprglaf gn hèpf ag gstuabf ag n` nøibh`.

Bagltbhf y hflstruyf prfpfsbhbflgs sbepngs.Agtgreblf gn v`nfr ag vgraà ag prfpfsbhbflgs sbepngs.

Gcgrhbtf n` lgi`hbøl ag prfpfsbhbflgs sbepngs.

Hfeprglaf y pr`htbhf gn usf ag hu`ltbhàfrgs.

]gsugnvf fpgr`hbflgs `rbteâtbh`s màsbh`s hfl ìbnbaà.

Fmsgrvg `tgltègltg hà` luegr`n y rfagg n` fphbøl qug hfepngt` n` sgrbghfrrghtègltg.

0) . . .

`. m. h. a.

=) ...

`. m. h. a.

8) ...

`. m. h. a.

1) ...

`. m. h. a.

7) . . .

`. m. h. a.

=1BIG] ∝ [fnfhjbh

]gvbsg su `prglabz`cg

Ags`rrfnng lugv`s j`mbnbaàgs





¡[`r` hfeglz`r!Nfs hflghtbvfs

Gl gn babfe` gsp`ðfn, nfs hflghtbvfs sfl p`n`mr`s utbnbzà`s hfef lgxfs qug

sbrvgl p`r` ulbr ftr`s p`n`mr`s, dr`sgs f bag`s. @ulqug lf lfs agefs huglt`,

nfs gepngèfs hà` vgz qug j`mnèfs f gshrbmbefs.

J`y ir`l lóegrf ag hflghtbvfs, niulfs ag gnnfs qug pfagefs eglhbfl`r sfl3

y, gltflhgs, àgeàs, f, u, f sg`, tèmbâl.

Tgèfs `niul`s fr`hbflgs hfef gcgepnf3

J`hgr `htbvbaà dåsbh` y hfegr s`nua`mng lfs `yua` ` pgre`lghgr s`lfs.

Xb gstuabèfs abshbpnblàègltg, gltflhgs pfargefs `prfm`r gn iràf.

Dubefs ` n` pn`y`, àgeàs vbsbtèfs ` ulfs èbifs.

Hfef pfagefs bagltbdbh`r, nfs hflghtbvfs `p`rghgl rgs`ntàfs y ulgl n`s bag`squg sg gxpflgl gl hà` fr`hbøl.

Gl gn huàrf sbiubgltg efstrèfs `niulfs ag nfs hflghtbvfs eàs hfeulgs3

¡@ tr`m`c`r!

Ng` n`s fr`hbflgs sbiubgltgs y sumr`yg gn hflghtbvf qug `p`rghg gl hà` ul`. Ng aèfs ul gcgepnf.

;) Xb sfefs tfngr`ltgs, gltflhgs hflvbvbrgefs gl `reflå`.

0) Nfs cìu`rgs s`nt`l ntf y sfl euy àibngs.

=) N`s pn`lt`s lghgsbt`l iu` p`r` vbvbr, gltflhgs lghgsbt`l sgr rgià`s.

8) Xb hfef euhj`s hjuhjgrå`s, gltflhgs tglarâ prfmnge`s ag s`nua.

1) Grlgstf pugag cui`r y hfrrgr euy ràpbaf.

]gn`hbøl Hflghtbvf

@abhbøl y, ageàs, tèmbâl, så ebsef, pfr ðàbaur`, eàs ul

Hfltr`stgpgrf, sbl gem`rif, lf fmst`ltg, gl hbgrtf efaf, gl

hbgrt` egaba`, ùlqug, gl hèmbf, `n hfltr`rbf

Hùs` ‘ gdghtf

pfr t`ltf, ag `jå qug, gl hflsghuglhb`, `så pugs, pfr

hflsbiubgltg, pfr nf t`ltf, pfr gsf, pfr nf qug sbiug, pfr

gst` r`zøl, gltflhgs, gltflhgs rgsunt` qug, ag e`lgr`

qug, nugif, `så qug

Hfep`r`tbvfsagn ebsef efaf, biu`negltg, ag efaf sbebn`r, hfef,

`så hfef

=4BIG] ∝ [fnfhjbh



0. [rfpfsbhbflgs hfepugst`sN`s prfpfsbhbflgs hfepugst`s sfl qugnn`s qug sg dfre`l pfr afs f eàs prf-

pfsbhbflgs sbepngs ulba`s ` tr`vâs ag hflghtbvfs nøibhfs, pgrf ¶quâ sfl gstfsóntbefs6 [flièfs `tglhbøl.

Hfef ngåefs gl n` pàibl` `ltgrbfr, gl gn babfe` gsp`ðfn j`y p`n`mr`s, nnèà`shflghtbvfs, qug sbrvgl ag gln`hg gltrg bag`s u ftr`s p`n`mr`s. Gl n` nøibh`, tè-mbâl gxbstgl gstfs lgxfs y sg nnè`l hflghtbvfs nøibhfs. Nfs eàs hfeulgs sflhu`trf3 y , f , sb… gltflhgs y sb y sfnf sb.

Tgèfs ulfs gcgepnfs3

Gn qugtz`n tbglg pnue`s y pugag vfn`r.

Gl gst` prfpfsbhbøl hfepugst` sg ulgl afs sbepngs3 Gn qugtz`n tbglg pnue`s? gn

qugtz`n pugag vfn`r . Gst`s sg ulgl ` tr`vâs agn lgxf "y".

@arb`l` pugag br `n p`rtbaf ag dutmfn f `n hblg.

N` prfpfsbhbøl `ltgrbfr sg hfepflg ag afs sbepngs3 @arb`l` pugag br `n p`rtbaf

ag dutmfn, @arb`l` pugag br `n hblg. Gst`s sg ulgl pfr egabf agn lgxf "f".

[`r` agtgrebl`r gn v`nfr ag vgraà ag ul` prfpfsbhbøl hfepugst` gs lghgs`rbfhflfhgr nfs hflghtbvfs nøibhfs y sus rgin`s.

=. Hflghtbvfs nøibhfsNfs hflghtbvfs nøibhfs sfl p`n`mr`s y såemfnfs qug sbrvgl hfef gln`hg p`rùlbr afs f eàs prfpfsbhbflgs sbepngs y `så dfre`r ul` prfpfsbhbøl hfepugst`.

Gstfs hflghtbvfs, cultf ` nfs hu`ltbhàfrgs qug vbefs n` sge`l` p`sà`, hfls-tbtuygl afs àrg`s dulaèglt`ngs gl n` nøibh`.

Lfsftrfs gstuab`rgefs nfs eàs utbnbzàfs3 hflculhbøl, absyulhbøl, hflabhbfl`ny mbhflabhbfl`n.

=.0 Hflculhbøl (∩)

Gl nøibh`, n` hflculhbøl hfrrgspflag n` hflculhbøl hfpun`tbv` " y". Qbglg hfefdulhbøl ulbr afs prfpfsbhbflgs y su såemfnf gs ∩.

Gcgepnf Xg gshrbmg Xg ngg

@ndrgaf gs `rqubtghtf y Jâhtfr gs eâabhf. p ∩ q p y q

=9E`tgeàtbh` ∝ Xge`l` =




=.= Absyulhbøl (∪)

N` absyulhbøl sg hfepfrt` hfef n` hflculhbøl "f" qug sg utbnbz` gl gn babfe`gsp`ðfn. Xu dulhbøl gs ulbr prfpfsbhbflgs p`r` a`rngs ul sgltbaf ag gnghhbølgltrg afs f eàs gngegltfs y su såemfnf gs ∪.


I`mrbgn` pugag sgr bliglbgr` f ghflfebst`. p ∪ q p f q

=.8 Hflabhbfl`n ( )

Qèmbâl hflfhbaf hfef bepnbh`hbøl, gstà hfldfreà` pfr n`s p`n`mr`s "Xb...gltflhgs". Xg hfepflg ag afs prfpfsbhbflgs. N` prbegr` gs rgqubsbtf blabspgl-s`mng p`r` qug sg huepn` n` sgiula`. Xu såemfnf gs ul` ghj` `pult`laf ` nàgrghj`3

Gcgepnf Xg gshrbmg Xg nggXb ân gstuab` euhjf, gltflhgs pfarà rgsfn-vgr gn gxègl sbl abdbhuntà.

p q Xb p gltflhgs q

=.1 Mbhflabhbfl`n ( )

Gstg hflghtbvf gxprgs` n` rgn`hbøl ag gqubv`nglhb` gltrg afs prfpfsbhbflgs. N`prbegr` sg huepng sfnf sb sg huepng n` sgiula` y vbhgvgrs`. N` dfre`l n`s p`n`-mr`s "sb y sfnf sb" y su såemfnf gs ul` afmng ghj`3


Nfs `lbe`ngs vbvbràl s`lfs sb y sfnf sb nfshuba`l mbgl.

p q p sb y sfnf sb q

Gcgrhbhbf 0]gprgsgltg sbemønbhègltg n`s prfpfsbhbflgs sbiubgltgs, jài`nf hfl n`s ngtr`s p y q. Wtbnbhg gn såemfnfagn hflghtbvf qug hfrrgspfla`. Ng aèfs ul gcgepnf.

;) A`lbgn m`rrg y Ibna` s`huag.

0) Xb hfrrg ràpbaf, gltflhgs i`l`rà n` hfepgtglhb`.=) [ugag nnè`rnf f glvb`rng ul egls`cg.

8) Gn dugif glhglagrà sb y sfnf sb n` ngð` gstà sgh`.

1) [fargefs huntbv`r e`åz y drbcfn.

7) N` h`s` sgrà dbreg sb y sfnf sb tbglg muglfs hbebgltfs.

4) Ngflgn pugag gstuab`r pgrbtf f m`hjbnngr`tf.

p ∩ q

=:BIG] ∝ [fnfhjbh



8. ]gin`s ag nfs hflghtbvfs nøibhfs[`r` gv`nu`r y agtgrebl`r gn v`nfr ag vgraà ag ul` prfpfsbhbøl hfepugst`,agmgefs m`s`rlfs gl n`s rgin`s ag nfs hflghtbvfs nøibhfs. [`r` gstg `lànbsbs,utbnbzèfs n`s t`mn`s ag vgraà. Tgèfs.

8.0 ]gin` ag n` hflculhbøl

Gn v`nfr ag vgraà ag ul` hflculhbøl gs vgraàgrf sfnf sb èm`s prfpfsbhbflgssfl vgraàgr`s, y gs d`nsf ag hu`nqubgr ftr` dfre`.

p q p ∩ q

v v vv d d d v d d d d

Hfeprfmgefs n` rgin` hfl ul gcgepnf3

p ∩ q3 [`r` n` hfemustbøl sg lghgsbt` hfemustbmng y fxåiglf.

Gxtr`gefs afs prfpfsbhbflgs sbepngs3

p3 [`r` n` hfemustbøl sg lghgsbt` hfemustbmng.

q3 [`r` n` hfemustbøl sg lghgsbt` fxåiglf.

p3 Xg lghgsbt` hfemustbmng. (T)q3 Xg lghgsbt` fxåiglf. (T)

N` prfpfsbhbøl hfepugst` gs vgraàgr` pfrqug sg huepng hfl èmfs hrbtgrbfs.J`y hfemustbmng y j`y fxåiglf.

p3 Xg lghgsbt` hfemustbmng. (T)

~q3 Lf sg lghgsbt` fxåiglf. (D)N` prfpfsbhbøl gs d`ns`, pfrqug sbl fxå-iglf lf pugag j`mgr hfemustbøl.

~ p3 Lf sg lghgsbt` hfemustbmng. (D)

q3 Xg lghgsbt` fxåiglf. (T)N` prfpfsbhbøl gs d`ns`, pfrqug sblhfemustbmng lf pugag j`mgr dugif.

~ p3 Lf sg lghgsbt` hfemustbmng. (D)

~q3 Lf sg lghgsbt` fxåiglf. (D)

N` prfpfsbhbøl gs d`ns`. Lf sg huepnglbliulf ag nfs hrbtgrbfs, lb j`y hfemus-tbmng lb j`y fxåiglf.

Lftg høef ul` prfpfsbhbøl hfepugst` ulba` pfr n` hflculhbøl tfe` v`nfr agvgraàgrf ólbhègltg sb n`s afs prfpfsbhbflgs sbepngs sfl hbgrt`s.

Q`mn` ag vgraà3 gsul` rgprgsglt`hbøliràh` ag nfs v`nfrgsag vgraà f d`nsgaàag ul` prfpfsbhbøl.

=2E`tgeàtbh` ∝ Xge`l` =



8.= ]gin` ag n` absyulhbøl

Gl ul` absyulhbøl, gn v`nfr ag n` vgraà rgsunt` d`nsf sfnf sb em`s prfpfsbhbflgssfl d`ns`s, y vgraàgrf ag hu`nqubgr ftr` dfre`.

p q p ∪ qv v vv d vd v vd d d


p ∪ q3 [`r` hfeulbh`rsg pugag nnè`rng f glvb`rng ul egls`cg.


p3 [`r` hfeulbh`rsg pugag nnè`rng.

q3 [`r` hfeulbh`rsg pugag glvb`rng ul egls`cg.

p3 [ugag nnè`rng. (T)

q3 [ugag glvb`rng ul egls`cg. (T)

N` prfpfsbhbøl hfepugst` gs vgr-aàgr` pfrqug sg pugag hfeulbh`rpfr èm`s vå`s.

p3 [ugag nnè`rng. (T)~q3 Lf pugag glvb`rng ul egls`cg. (D)

N` prfpfsbhbøl gs vgraàgr , pfrqugùlqug lf sg pugag hfeulbh`r pfregls`cg, pugag j`hgrnf nnè`laf.

~p3 Lf pugag nnè`rng. (D)

q3 [ugag glvb`rng ul egls`cg. (T)

N` prfpfsbhbøl gs vgraàgr , pfrqugùlqug lf sg pugag hfeulbh`r pfrnnèà`, pugag j`hgrnf glvb`laf ulegls`cg.

~p3 Lf pugag nnè`rng. (D)~q3 Lf pugag glvb`rng ul egls`cg. (D)

N` prfpfsbhbøl gs d`ns`. Lf pugag

j`hgr ul` nnèà` lb glvb`r ul egl-s`cg, gltflhgs lf j`y hfeulbh`hbøl.

Lftg hfef ul` prfpfsbhbøl hfepugst` ulba` pfr n` absyulhbøl tfe` v`nfr agd`nsf ólbhègltg sb n`s afs prfpfsbhbflgs sbepngs sfl d`ns`s.

8;BIG] ∝ [fnfhjbh



8.8 ]gin` agn hflabhbfl`n

Hfl gstg hflghtbvf, ul` prfpfsbhbøl gs d`ns` sb p gs vgraàgr` y q gs d`ns`. [`r`hu`nqubgr ftrf h`sf rgsunt` vgraàgr`.

p q p q

v v vv d d d v vd d v


p q3 Xb jfy gs vbgrlgs, gltflhgs e`ð`l` sgrà sàmàf.


p3 Jfy gs vbgrlgs.

q3 E`ð`l` sgrà sàmàf.

p3 Jfy gs vbgrlgs. (T)

q3 E`ð`l` sgrà sàmàf. (T)

N` prfpfsbhbøl hfepugst` gs vgra`-agr` pfrqug nugif ag vbgrlgs sbiuggn sàmàf.

p3 Jfy gs vbgrlgs. (T)

~q3 E`ð`l` lf sgrà sàmàf. (D)

N` prfpfsbhbøl gs d`ns`. Xb gn aå``htu`n gs vbgrlgs, blaua`mngegltge`ð`l` sgrà sàmàf.

~ p3 Jfy lf gs vbgrlgs. (D)

q3 E`ð`l` sgrà sàmàf. (T)

Gn glulhbàf gs vgraàgrf.N` prbegr` prfpfsbhbøl gs d`ns`, nfqug blabh`rå` qug så gs vbgrlgs, y pfrt`ltf, e`ð`l` så sgrà sàmàf.

~ p3 Jfy lf gs vbgrlgs. (D)

~q3 E`ð`l` lf sgrà sàmàf. (D)N` prfpfsbhbøl gs vgraàgr`, pugagsgr hu`nqubgr ftrf aå` ag n` sge`l`.

Hfl gn hflabhbfl`n gxbstg ul prfmnge` aglfeblàf hfef p`ràfc`s ag n`

bepnbh`hbøl. Gst`s p`ràfc`s sg a`l hu`laf ul` prfpfsbhbøl gs vgraàgr` ag`hugraf n` nøibh`, pgrf gxprgs` ul pn`ltgèbgltf qug lf tbglg sgltbaf hfeól.[fr gcgepnf3

Xb n` Nul` gstà jghj` ag qugsf, gltflhgs = + = 5 1.

N` prfpfsbhbøl `ltgrbfr gs vgraàgr`, ` pgs`r ag qug lf tbglg sgltbaf gl gnngliu`cg hfeól. Hfef lfsftrfs gstèfs `l`nbz`laf n`s prfpfsbhbflgs agsaggn pultf ag vbst` nøibhf, lf sg prgfhupg sb `niulfs glulhbàfs ng rgsunt`l hfl-tr`rbfs ` n` r`zøl.

80E`tgeàtbh` ∝ Xge`l` =



8.1 ]gin` agn mbhflabhbfl`n

Gn v`nfr ag vgraà ag ul mbhflabhbfl`n gs vgraàgrf hu`laf èm`s prfpfsbhbf-lgs tbglgl gn ebsef v`nfr ag vgraà, gs aghbr, èm`s sfl vgraàgr`s f d`ns`ssbeuntàlgègltg? gl h`sf hfltr`rbf, gs d`nsf.

p q p qv v vv d d d v d d d v

Fmsgrvgefs egcfr n` rgin` hfl ul gcgepnf3

p q3 @eånh`r pugag sgr prgsbagltg sb y sfnf sb gs hbuaà`lf iu`tge`ntghf.


p3 @eånh`r pugag sgr prgsbagltg.

q3 @eånh`r gs hbuaà`lf iu`tge`ntghf.

p3 @eånh`r pugag sgr prgsbagltg. (T)

q3 @eånh`r gs hbuaà`lf iu`tge`ntghf. (T)

N` prfpfsbhbøl hfepugst` gsvgraàgr , pfrqug @eånh`rgs hbuaà`lf iu`tge`ntghfy pfr t`ltf pugag fpt`r sgrprgsbagltg.

p3 @eånh`r pugag sgr prgsbagltg. (T)

~q3 @eånh`r lf gs hbuaà`lf iu`tge`ntghf. (D)

N` prfpfsbhbøl gs d`ns`. @n lfsgr hbuaà`lf iu`tge`ntghf,@eånh`r lf pugag fpt`r `npugstf ag prgsbagltg.

~ p3 @eånh`r lf pugag sgr prgsbagltg. (D)

q3 @eånh`r gs hbuaà`lf iu`tge`ntghf. (T)

N` prfpfsbhbøl gs d`ns`.@eånh`r så pugag fpt`r `nh`rif ag prgsbagltg pfrquggs hbuaà`lf iu`tge`ntghf.

~ p3 @eånh`r lf pugag sgr prgsbagltg. (D)

~q3 @eånh`r lf gs hbuaà`lf iu`tge`ntghf. (D)

N` prfpfsbhbøl gs vgraàgr .@eånh`r lf gs iu`tge`ntghfy pfr t`ltf lf puag pfstu-n`rsg p`r` prgsbagltg.

Agmgefs tglgr hubaàf ag lf hfldulabr n`s rgn`hbflgs mbhflabhbfl`ngs hfl n`sólbhègltg hflabhbfl`ngs.

8=BIG] ∝ [fnfhjbh



Gcgrhbhbf =@. Ng` nfs glulhbàfs sbiubgltgs, pfli` `tglhbøl `n hflghtbvf nøibhf qug nfs ulg. Gshrbm` gn v`nfr

nøibhf ag p y q gl gn huàrf hfrrgspflabgltg. Nugif, tr`lsdfreg n` prfpfsbhbøl hfepugst` `ngliu`cg sbemønbhf y agtgreblg su v`nfr ag vgraà. Fmsgrvg y iuågsg pfr gn gcgepnf.

;) Xb Ànv`rf pugag vft`r, gltflhgs gs e`yfr ag gaà. p3 T q3 T

0) Xb H`rnfs l`hbø gl Bz`m`n, gltflhgs gs jflaurgðf. p3 q3

=) Gn : gs ul lóegrf bep`r y gs eóntbpnf ag =.

p3 q3

8) Gn Xfn s`ng pfr gn Gstg y n` Nul` gs eàs ir`lag qug n` Qbgrr`. p3 q3

1) N` Qbgrr` gs gsdârbh` sb y sfnf sb gn Xfn gs ul` gstrgnn`. p3 q3

7) Xb gn ìu` e`rbl` gs s`nà`, gltflhgs gs rghfegla`mng mgmgr ìu` s`nà`. p3 q3

4) Iu`tge`n` tbglg =8 agp`rtègltfs f su h`pbt`n gs E`z`tgl`lif. p3 q3

9) Gn = lf gs abvbsfr ag 0= f gn 0= gs eóntbpnf ag 8. p3 q3

:) Nfs h`l`rbfs lf j`mn`l sb y sfnf nfs nfrfs vugn`l. p3 q3

p q3 T




M. [bglsg y gshrbm` fhjf prfpfsbhbflgs hfepugst`s. Nugif, agtgreblg su v`nfr ag vgraà. Iuågsgpfr gn gcgepnf.

;)

0)

=)

8)

1)

7)

4)

9)

:)

H. ]g`nbhg nfs prfhgabebgltfs qug sg pbagl gl hà` hfnuel` ag n`s t`mn`s ag vgraà. ]ghugrag nfqug `prglabø n` sge`l` `ltgrbfr sfmrg n`s lgi`hbflgs. Iuågsg pfr gn gcgepnf.

;) p ~ p q p ∪ q ~ p q

v d v v v

v d d v v

d v v v v

d v d d d

8) p q ~q p ~q p ∩ q

v v

v d

d v

d d

0) p ~ p q ~q ~ p ∪ ~q ~ p ∩ q

v v

v d

d v

d d

1) p q ~q p ∪ ~q p ∩ ~q

v v

v d

d v

d d

=) p ~ p q ~ p q q ∪ ~ p

v v

v d

d v

d d

7) p ~ p q ~q ~q ∩ ~ p p ~q

v v

v d

d v

d d

Gn lóegrf 8 gs ul lóegrf bep`r y gs eóntbpnf ag 8. T

81BIG] ∝ [fnfhjbh



0. N`s prfpfsbhbflgs hfepugst`s sfl `qugnn`s dfreà`s pfr afs f eàs prfpfsbhbflgs sbepngs qugsg ulgl ` tr`vâs ag hflghtbvfs nøibhfs.

=. Nfs hflghtbvfs nøibhfs sfl p`n`mr`s y såemfnfs qug sbrvgl hfef lgxf p`r` ulbr afs f eàsprfpfsbhbflgs sbepngs.

Nfs eàs hfeulgs sfl n` hflculhbøl, n` absyulhbøl, gn hflabhbfl`n y gn mbhflabhbfl`n.

=.0 N` hflculhbøl (∩) hfrrgspflag ` n` "y".

=.= N` absyulhbøl (∪) hfrrgspflag ` n` "f" y sg utbnbz` p`r` a`r sgltbaf ag gnghhbøl gltrg afs feàs gngegltfs.

=.8 Gn hflabhbfl`n ( ) f bepnbh`hbøl tfe` dfre` gl n`s p`n`mr`s "Xb… gltflhgs". Gstg hflghtbvfblabh` qug n` prbegr` prfpfsbhbøl gs rgqubsbtf p`r` qug sg huepn` n` sgiula`.

N`s p`ràfc`s ag n` bepnbh`hbøl sfl sbtu`hbflgs gl n`s qug ul` prfpfsbhbøl tbglg v`nfrnøibhf vgraàgrf, pgrf lf gxprgs` ul` bag` hfjgrgltg sgiól gn sgltbaf hfeól.

=.1 Gn mbhflabhbfl`n ( ) sg dfre` hfl n`s p`n`mr`s "sb y sfnf sb" g blabh` ul` rgn`hbøl ag agpgl-aglhb` eutu`. Gs aghbr, n` prbegr` sg huepng sfnf sb sg huepng n` sgiula` y vbhgvgrs`.

8. [`r` agtgrebl`r gn v`nfr ag vgraà ag ul` prfpfsbhbøl hfepugst` sg gepng`l n`s rgin`s ag nfshflghtbvfs nøibhfs. Xg prgsglt`l gl n`s t`mn`s ag vgraà sbiubgltgs3

]gsuegl

p q p ∩ q

v v vv d d d v d d d d

Hflculhbøl

p q p ∪ qv v vv d vd v vd d d

Absyulhbøl

p q p qv v vv d d d v vd d v

Hflabhbfl`n

p q p qv v vv d d d v d d d v

Mbhflabhbfl`n

Ng blvbtèfs ` vgr nfs vbagfs qug sg eugstr`l gl n`s abrghhbflgs sbiubgltgs, gstfs ng `yua`ràl ` prfdul-

abz`r gl `niulfs `spghtfs sfmrg nfs hflghtbvfs nøibhfs.• Hflghtfrgs nøibhfs y t`mn`s ag vgraà3 iff.in/njucRy

• ¶Høef hflvgrtbr ul` prfpfsbhbøl gl ngliu`cg sbemønbhf6 iff.in/07OjGC

• Q mn`s ag vgraà p`r` eàs ag afs hflghtfrgs nøibhfs3 iff.in/lszj`m

Q embâl ng prfpflgefs pr`htbh`r ul pfhf eàs y rgsfnvgr `niulfs gcgrhbhbfs gxtr`, qug pfarà glhfltr`rgl gn gln`hg sbiubgltg.

• Gcgrhbhbfs prfpugstfs ag hflghtbvfs nøibhfs y t`mn`s ag vgraà3 iff.in/YXAMeN


Blvgstbiug gl n` rga...




@. Ng` hà` agdblbhbøl ` n` bzqubgra`. Nugif, rgn`hbølgn` hfl gn hflghtbvf nøibhf qug ng hfrrgspflaghfnfh`laf gn lóegrf gl gn p`râltgsbs hfrrghtf.

0) Blabh` rgn`hbøl ag agpglaglhb` eutu`. Gs aghbr, n` prbegr` prfpfsb-hbøl sg huepng sfnf sb sg huepng n` sgiula` y vbhgvgrs`.

=) A` sgltbaf ag gnghhbøl gltrg afs f eàs gngegltfs. Xu v nfr ag vgraàrgsunt` d`nsf sfnf sb èm`s prfpfsbhbflgs sfl d`ns`s.

8) Gqubv`ng ` n` p`n`mr` "y" gl gn babfe` gsp`ðfn. Xu v`nfr nøibhf gs vgr-aàgrf sfnf sb èm`s prfpfsbhbflgs sfl hbgrt`s.

1) Q embâl nnèà` bepnbh`hbøl? blabh` hflabhbøl agn tbpf "Xb… gltflhgs".]gsunt` gl d`nsf hu`laf p gs vgraàgr` y q gs d`ns`, y gs vgraàgrf gl

hu`nqubgr ftrf h`sf.

M. Ng` nfs glulhbàfs sbiubgltgs y sumr`yg gn hflghtbvf qug `p`rghg gl hà` ulf. Nugif, tr`lsdfregn`s prfpfsbhbflgs ` ngliu`cg sbemønbhf y gshråm`n`s gl n` nålg` ag n` agrghj`. Iuågsg pfr gn gcgepnf.

;) Xb gn ìu` agn fhâ`lf gs s`nà` gltflhgs lf sg pugag mgmgr.

0) Gn h`htus tbglg gspbl`s y lghgsbt` pfh` ìu`.

=) N` nnuvb` h`g agn hbgnf y E`rtg gs ul pn`lgt`.

8) Xb gn cìu`r tbglg e`lhj`s, gltflhgs sg pugag hèudn`r.

1) Xb gn iu` gs blhfnfr`, gltflhgs sg pugag mgmgr.

7) Xb jfy gs sàmàf, gltflhgs e`ð`l` sgrà afeblif.

4) Gn r`tøl hfrrg ràpbaf f gn i`tf lf pugag s`nt`r.

9) Hfnfemb` gstà gl Xurèârbh` y [frtui`n gl Gurfp`.

:) Gl Zgn` j`y dråf sb y sfnf sb gl S`h`p` j`y h`nfr.

H. @jfr` gs su turlf. Gshrbm` hu`trf prfpfsbhbflgs hfepugst`s y gst`mngzh` su v`nfr ag vgraà.]ghugrag nf `prglabaf sfmrg n`s rgin`s ag nfs hflghtbvfs nøibhfs.

0)

=)

8)

1)

@utfhfltrfn

( ) hflculhbøl

( ) mbhflabhbfl`n

( ) hflabhbfl`n

( ) absyulhbøl

r s

84BIG] ∝ [fnfhjbh




@. ]g`nbhg nfs prfhgabebgltfs qug sg pbagl gl hà` hfnuel` ag n`s t`mn`s ag vgraà. ]ghugrag nf qug`prglabø sfmrg n`s lgi`hbflgs.

0) r ~r s ~s ~r ~s r ∪ s ~r s

v vv d

d v

d d

8) ` ~` m ~` ∪ m ` ∩ m

v vv d

d v

d d

=) p ~ p q ~q ~ p ~q ~q ∩ ~ p

v v

v d

d v

d d

1) r s ~s r ~s r s

v

v

d

d

M. Qr`lsdfreg hà` prfpfsbhbøl sbiubgltg ` ngliu`cg sbemønbhf y agtgreblg su v`nfr ag vgraà.

0) Gn ngøl gs jgrmåvfrf y n` fvgc` gs h`rlåvfr`. p3 q3

=) Glgrf gs gn prbegr egs agn `ðf y abhbgemrg gs gn óntbef.

p3 q3

8) Nfs h`m`nnfs sfl `lbe`ngs sb y sfnf sb Tglus lf gs ul pn`lgt`. p3 q3

1) Xb gn e`rtgs v` nugif agn nulgs, gltflhgs gn vbgrlgs v` `ltgs agn cugvgs. p3 q3

7) Gn ìu`h`tg vbglg ag n` e`zfrh` f n` Qbgrr` gs ul pn`lgt`. p3 q3

4) 02 gs eóntbpnf ag 7 sb y sfnf sb = gs abvbsfr ag :. p3 q3




@. ]gsugnv` eglt`negltg n`s euntbpnbh`hbflgs sbiubgltgs. Gshrbm` n` rgspugst` hfrrght` sfmrg n` nålg`hfrrgspflabgltg.

0) = Ù 8 Ù 0 5

=) 7 Ù 1 Ù : 5

8) 4 Ù 8 Ù = 5

1) 7 Ù 8 Ù 9 5

7) 0 Ù 7 Ù 2 5

4) = Ù 8 Ù ; 5

9) 7 Ù 4 Ù : 5

:) = Ù 8 Ù : 5

2) 0 Ù = Ù 7 5

0;) 7 Ù 1 Ù 9 5

00) : Ù 7 Ù ; 5

0=) 7 Ù 4 Ù 7 5

08) 7 Ù 1 Ù 4 5

01) 2 Ù : Ù 9 5

07) 8 Ù 7 Ù 1 5

04) 7 Ù 8 Ù 1 5

09) : Ù 9 Ù 1 5

0:) 4 Ù 2 Ù 7 5

02) 2 Ù 7 Ù 7 5

=;) : Ù 2 Ù 1 5

=0) : Ù 2 Ù 7 5

==) 8 Ù = Ù ; 5

=8) 9 Ù : Ù 0 5

=1) : Ù 7 Ù = 5

=7) 1 Ù 0 Ù 8 5

=4) : Ù 1 Ù : 5

=9) : Ù : Ù 9 5

=:) 2 Ù 4 Ù 2 5

=2) 9 Ù 1 Ù 0 5

8;) ; Ù 0 Ù ; 5

M. ]gsugnv` eglt`negltg n`s euntbpnbh`hbflgs y abvbsbflgs sbiubgltgs. Gshrbm` n` rgspugst` hfrrght`sfmrg n` nålg` hfrrgspflabgltg.

0) 7 Ù = ù = 5

=) = Ù : ù 1 5

8) : Ù 9 ù 1 5

1) 2 Ù 2 ù 8 5

7) 9 Ù 9 ù 0 5

4) 9 Ù 1 ù = 5

9) 0 Ù 0 ù 0 5

:) 4 Ù 2 ù 8 5

2) 7 Ù 4 ù 0; 5

0;) : Ù 8 ù = 5

00) 1 Ù 0 ù 1 5

0=) 9 Ù : ù : 5

08) 2 Ù 4 ù = 5

01) 1 Ù 4 ù 8 5

07) 8 Ù : ù = 5

04) 9 Ù 0 ù 0 5

09) : Ù 2 ù 4 5

0:) ; Ù 0 ù 8 5

02) : Ù : ù 1 5

=;) 9 Ù 2 ù 9 5

=0) : Ù 0 ù 1 5

==) 1 Ù 7 ù 0; 5

=8) 7 Ù : ù =; 5

=1) 0 Ù = ù 0 5

=7) 8 Ù = ù 0 5

=4) 1 Ù 8 ù 4 5

=9) 4 Ù 0 ù 8 5

=:) 8 Ù 2 ù 0 5

=2) ; Ù 0 ù 0 5

8;) 8 Ù 8 ù 2 5

8:BIG] ∝ [fnfhjbh




@. H`nhung gn v`nfr ag hà` ul` ag n`s ngtr`s sbiubgltgs, m`sàlafsg gl n`s fpgr`hbflgs blabhà`s.Qgli` gl huglt` qug n`s fpgr`hbflgs gstàl rgn`hbflà`s, pgrf pugagl lf tglgr ul fragl nblg`n.

P 5 N 5 Q 5 C 5 @ 5

J 5 A 5 D 5 E 5 ] 5

0) 7 + A 5 7

=) D ‘ A 5 0

8) 8C + =D 5 :

1) =A + D + 0;C 5 ]

7) ] ù 8 5 J

4) ‘=J + ] + 9A + 4 5 N

9) P 5 D + C + D

:) 0;J ‘ (D + P) + 8N ‘ 8P 5 E

2) Q 5 8;C + N + (@ + D) ‘ J

0;) E ‘ 7P 5 Q

M. Hflsbagr`laf nfs lóegrfs qug fmtuvf `ltgrbfregltg, h`nhung gn v nfr qug tbglgl n`s ngtr`ssbiubgltgs.

I 5 O 5 T 5 B 5 G 5

L 5 M 5 [ 5 H 5 X 5

0) (P ù =) Ù 1 5 H

=) N ‘ 8D + A 5 T

8) J + M 5 0=

1) (8T) ù = + GM 5 4G

7) Q ù = ‘ L + H + T 5 1=

4) G + 7M 5 B

9) 4B + 00] ‘ (G + =D) 5 [

:) B ‘ =G + (P Ù H) 5 I

2) 4@ + (P Ù E) ‘ [ + (C Ù X) + B 5 ;

0;) 4J + (G Ù T) + (=M + N) ‘ (8P + D) 5 O






prfhgsflf

nfiràf

A g s p u â s a g g s t u a b `

r . . .

Hfeprglaf g bagltbhf n`s prfpfsbhbflgs hfepugst`s.

H`tgifrbzf n`s rgin`s ag nfs hflghtbvfs nøibhfs eàs utbnbzàfs.

Agtgreblf gn v`nfr ag vgraà ag prfpfsbhbflgs hfepugst`s.

Hflstruyf t`mn`s ag vgraà.

]gsugnvf fpgr`hbflgs hfemblà`s hfl ìbnbaà.

@jfr` pflargefs ` prugm` lugstr`s lgurfl`s. Hfl nfs afebløs agmgefs ag-tght`r gn fragl nøibhf qug sbiugl n`s hj`s prgsgltà`s. Gstg prfhgabebgltf lfs`yua` ` ags`rrfnn`r lugstr` j`mbnbaà ag agauhhbøl.

[`r` nfs gcgrhbhbfs sbiubgltgs, e`rqug hfl ul` "x" n` fphbøl qug hfepngt` n`sgrbg3

=)

`. h.m. a.

`. m. h. a.

0)

1)

h.m. a.`.

8)

`. h.m. a.

1;BIG] ∝ [fnfhjbh





80

= 8 1

8

1

7

=

0

;

x

y

‘ 0 5 b

x = +

y = 5

r =

Lóegrfs rg`ngs¶Puâ glhfltr`rà gst` sge`l`6

Nfs pbtìørbhfs y nfs lóegrfs brr`hbfl`ngs

Fpgr`hbflgs hfemblà`s

Nfs lóegrfs rg`ngs, sus prfpbgaàgs, rgin`s ag n` pftglhb`hbøl y ag n` ràbh`hbøl [rfmnge`s ag `pnbh`hbøl hfl lóegrfs rg`ngs

Euntbpnbh`hbøl hfl gn eâtfaf Qzgnt`n


@pnbh`r n`s prfpbgaàgs ag nfs lóegrfs rg`ngs.

]gsfnvgr fpgr`hbflgs hfemblà`s.

]gp`s`r n`s ngygs ag n` pftglhb`hbøl y ag n` ràbh`hbøl ag nfs lóegrfs rg`ngs. @pnbh`r nfs lóegrfs rg`ngs gl n` rgsfnuhbøl ag prfmnge`s.

[r`htbh`r n` euntbpnbh`hbøl pfr gn eâtfaf Qzgnt`n.

Ags`rrfnn`r ìbnbaà ag hànhunf hfl fpgr`hbflgs hfemblà`s.






0. Lóegrfs rg`ngs (] )

]

B

P

S

L

...‘1, ‘8, ‘=, ‘0 ;, 0, =, 8, 1 ...0.=0=0=...

;.7

78

81

= ϋ

Gst` sge`l` hfl n` jbstfrb` ag nfs pbtìørbhfs blbhbèfs gn gstuabf agn hflcultfag nfs lóegrfs brr`hbfl`ngs.

]ghfragefs qug ul lóegrf brr`hbfl`n gs `qugn qug lf pugag sgr gxprgsàfhfef dr`hhbøl y su p`rtg aghbe`n lf gs pgrbøabh`, gs aghbr, sus aghbe`ngs lf sb-iugl ul p`trøl p`r` rgpgtbrsg. Nfs lóegrfs brr`hbfl`ngs (B), cultf nfs l`tur`ngs(L), nfs gltgrfs (S) y nfs r`hbfl`ngs (P) dfre`l gn hflcultf ag nfs lóegrfs rg`ngs(] ), hfef sg bnustr` gl n` beìgl ag `rrbm`.

Nfs lóegrfs rg`ngs sfl nfs qug pugagl gxprgs`rsg pfr ul lóegrf gltgrf f pfrul aghbe`n, sg` pgrbøabhf f lf. Gstg hflcultf sg rgprgsglt` hfl n` ngtr` ] y sgaglg sbemønbhègltg hfef3 ] 5 P ∣ B

0.0 H`r`htgråstbh`s ag nfs lóegrfs rg`ngs

• Nfs lóegrfs rg`ngs hflstbtuygl ul hflcultf bldblbtf pfrqug gstà dfreàfpfr n` ulbøl ag nfs hflcultfs ag lóegrfs r`hbfl`ngs (P) g brr`hbfl`ngs (B).

• ] gs ul hflcultf aglsf, gstf qubgrg aghbr qug gltrg afs lóegrfs r`hbfl`ngsj`y bldblbtfs brr`hbfl`ngs.

• ] gs ul hflcultf hfepngtf y hfltbluf, agmbaf qug tfaf pultf gl n` rght`lueârbh` sg pugag rgprgsglt`r hfl ul lóegrf rg`n. Nfs pultfs nbmrgs qugj`y gltrg afs lóegrfs r`hbfl`ngs sfl nnglàfs pfr nfs brr`hbfl`ngs, hfef

vgefs gl n` rght` qug gstà `m`cf.

N` hfltblubaà ag nfs lóegrfs rg`ngs pgrebtg qug j`y` lóegrfs hà` vgz eàshgrh`lfs ulfs ag ftrfs.

[fr gcgepnf3 =.2, =.22, =.222, =.2222… f 8.;;;0, 8.;;0, 8.;0, 8.0...

= 18

=.22

8.0

8.;0

=.2





0.= T`nfr `msfnutf ag ul lóegrf rg`n

Gl e`tgeàtbh` sg ng nnè` v`nfr msfnutf ag ul lóegrf rg`n, ` n` abst`lhb` qugj`y gltrg gn hgrf u frbigl y gstg lóegrf sbl bepfrt`r su sbilf, sg` pfsbtbvf (+)f lgi`tbvf (‘). Nf bagltbhèfs gshrbmbglaf gn lóegrf gltrg m`rr`s |`| y sg ngg"v`nfr `msfnutf ag `".

[fr gcgepnf3

`. Xb gn lóegrf gs pfsbtbvf f hgrf, su v`nfr `msfnutf gs gn ebsef lóegrf.

|81| 5 81

| 0

1| 5 0

1

m. Xb gn lóegrf gs lgi`tbvf, su v`nfr `msfnutf gs gn fpugstf agn lóegrf.

|‘8| 5 ‘(‘8) 5 8

|‘;.=7| 5 ‘(‘;.=7) 5 ;.=7

Ag gstfs rgsuntàfs pfagefs hflhnubr qug gn v`nfr `msfnutf ag ul lóegrfsbgeprg gs pfsbtbvf, pugs nf qug sg tfe` agn lóegrf gs n` abst`lhb` gltrg ân ygn hgrf.

|‘7| 5 7 |7| 5 7

‘4 ‘7 ‘1 ‘8 ‘= ‘0 ; 0 = 8 1 7 4

@quå lftèfs qug ‘7 y 7 tbglgl gn ebsef v`nfr `msfnutf pfrqug èmfs sgglhugltr`l ` n` ebse` abst`lhb` agn frbigl f hgrf.

Gcgrhbhbf 0Gshrbm` gn v`nfr `msfnutf ag nfs lóegrfs sbiubgltgs. Qbglg ul gcgepnf.

;) |‘ 1 | 5 9) |1.84=1| 5

0) |‘:| 5 :) |;| 5

=) |‘=7| 5 2) |‘8.=:41| 5

8) | 7

2| 5 0;) |

8 :| 5

1) |‘04| 5 00) |‘1.897| 5

7) |‘ =2| 5 0=) |‘ =7| 5

4) | ‘ :

09| 5 08) | ‘

8

7| 5

1

11BIG] ∝ [fnfhjbh



=. Fpgr`hbflgs hfemblà`sHfl nfs lóegrfs rg`ngs sg pugagl rg`nbz`r abdgrgltgs fpgr`hbflgs y n`s qugrgólgl v`rb`s fpgr`hbflgs gl ul` sfn` sg ngs nnè` fpgr`hbflgs hfemblà`s.

Gl n`s fpgr`hbflgs hfemblà`s pugagl p`rghgr sue`s, rgst`s, euntbpnbh`hbflgs,

abvbsbflgs, pftglhb`s, r`åhgs y sbilfs ag irup`hbøl. [`r` rg`nbz`rn`s, gs lghgs`rbfrgspgt`r n` cgr`rquå` ag n`s fpgr`hbflgs. [rgstg `tglhbøl ` nfs h`sfs sbiubgltgs3

`. Xb n`s fpgr`hbflgs lf blhnuygl sbilfs ag ìrup`hbøl, prbegrf sg rgsugnvgln`s euntbpnbh`hbflgs y abvbsbflgs gl gn fragl qug p`rghgl, nugif sg gdghtó`ln`s àbhbflgs y sustr`hhbflgs hfrrgspflabgltgs.

Gcgepnf3

• [rbegrf, rg`nbzèfs n`s euntbpnbh`hbflgsy abvbsbflgs.

• Nugif, n`s àbhbflgs y sustr`hhbflgs.

Ftrf gcgepnf3

• [rbegrf, rg`nbzèfs n` euntbpnbh`hbøl yabvbsbøl.

• Nugif, n` àbhbøl y sustr`hhbøl.

m. Xb n`s fpgr`hbflgs blhnuygl sbilfs ag ìrup`hbøl, gstfs sg gnbebl`l ag

àgltrf j`hb` dugr` y sg gdghtó`l n`s fpgr`hbflgs blabhà`s agltrf ag gnnfs.Gcgepnf3

• [rbegrf, rg`nbzèfs n` fpgr`hbøl quggstà agltrf ag nfs p`râltgsbs.

• Nugif, n` sustr`hhbøl qug gstà gl nfshfrhjgtgs.

• R pfr óntbef n` àbhbøl.

Ftrf gcgepnf3

• [rbegrf, rg`nbzèfs n` fpgr`hbøl quggstà agltrf ag nfs p`râltgsbs.

• Nugif, n` qug gstà agltrf ag nfs hfrhjg-tgs.

• Agspuâs n`s fpgr`hbflgs qug gstàlagltrf ag n`s nn`vgs.

• [fr óntbef, n` sustr`hhbøl.

7 • = + 8 ‘ = • 8 + 9 ‘ 1 ù = 5

0; + 8 ‘ 4 + 9 ‘ = 5 0=

0

1 •

7

8 +

7

2 ù

0

= ‘

7

4 5

7

0= +

0;

2 ‘

7

4 5

07 + 1; ‘ 8;

84 5

=7

84

07 + _: ‘ (= • 8)U 5

07 + _: ‘ 4U 5

07 + = 5 09

7 ‘ {0; ‘ _4 ‘ (07 ù 8)U + :} 5

7 ‘ {0; ‘ _4 ‘ 7U + :} 5

7 ‘ {0; ‘ 0 + :} 5

7 ‘ 09 5 ‘0=

]ghugrag gn prfhgab-ebgltf p`r` sue`r frgst`r dr`hhbflgs hflabdgrgltg aglfeblàfr.




8. [rfpbgaàgs ag nfs lóegrfs rg`ngsNfs lóegrfs rg`ngs huepngl hfl hbgrt`s h`r`htgråstbh`s f rgin`s gl sus abdgrgl-tgs fpgr`hbflgs nnèà`s prfpbgaàgs. [`r` hfeprglagr gst`s prfpbgaàgs, gslghgs`rbf hflsbagr`r qug n`s ngtr`s `, m y h pugagl rgprgsglt`r hu`nqubgr lóegrfrg`n, gs aghbr, tfe`r hu`nqubgr v`nfr. Tgèfs n`s prblhbp`ngs prfpbgaàgs ag nfs

lóegrfs rg`ngs.

Hnùsur`tbv`

Gn rgsuntàf ag n` fpgr`hbøl gltrg lóegrfs rg`ngs a`hfef rgsuntàf ftrf lóegrf rg`n.

` + m 5 h

=7 + 7 5 8;

Gngegltf lgutrf abtbvf

@n sue`r ul lóegrf rg`n hfl gn ; a` hfef rgsuntàf

gn ebsef lóegrf.

` + m 5 `

=.44 + ; 5 =.44

Blvgrsf àbtbvf u fpugstf

[`r` tfaf lóegrf ` gxbstg ftrf lóegrf ‘ nnèàfblvgrsf àbtbvf u fpugstf t`n qug3

` + (‘ ) 5 ;

8

1 + ( ‘

8

1 ) 5 ;

N` àbhbøl gs hfleut`tbv`

Gn fragl gl qug sg fpgrgl nfs sue`lafs lf `ntgr` gnrgsuntàf.

` + m 5 m + `

0= + 8 5 8 + 0=

07 5 07

N` euntbpnbh`hbøl gs hfleut`tbv`

Gn fragl gl qug sg fpgrgl nfs d`htfrgs lf `dght` gn

prfauhtf.

` • m 5 m • `

2 • 4 5 4 • 271 5 71

N` euntbpnbh`hbøl gs `sfhb`tbv`

N` e`lgr` gl qug sg ìrup`l nfs d`htfrgs lf `ntgr`gn prfauhtf.

(m • h) 5 ( • m)h

1(= • :) 5 (1 • =):

1(04) 5 (:):

41 5 41

N` àbhbøl gs `sfhb`tbv`

Gn fragl gl qug sg ìrup`l nfs sue`lafslf `ntgr` gn rgsuntàf.

( + m) + h 5 ` + (m + h)

(=.7 + 4) + 0; 5 =.7 + (4 + 0;)

:.7 + 0; 5 =.7 + 04

0:.7 5 0:.7

14BIG] ∝ [fnfhjbh



Blvgrsf euntbpnbh`tbvf

[`r` tfaf lóegrf ` gxbstg ftrf lóegrf0

` nnèàf

blvgrsf euntbpnbh`tbvf f sbeâtrbhf euntbpnbh`tbvf t`nqug3

` •0

` 5 0

7 •0

7 5 0

7

7 5 0

0 5 0

Abstrbmutbv` agn prfauhtf rgspghtf ` n` sue`

Gn prfauhtf ag ul lóegrf pfr ul` sue`

gs biu`n ` euntbpnbh`r gn lóegrf pfr hàùlf ag nfs sue`lafs.

`(m + h) 5 `m + `h

7(1 + 4) 5 (7 • 1) + (7 • 4)7(0;) 5 =; + 8;

7; 5 7;

Abstrbmutbv` agn prfauhtf rgspghtf ` n` rgst`

Gn prfauhtf ag ul lóegrf pfr ul` rgst`gs biu`n ` euntbpnbh`r gn lóegrf pfr gnebluglaf y pfr gn sustr`glaf.

`(m ‘ h) 5 `m ‘ `h

8(: ‘ =) 5 (8 • :) ‘ (8 • =)

8(4) 5 =1 ‘ 4

0: 5 0:

Gngegltf lgutrf ag n` euntbpnbh`hbøl

@n euntbpnbh`r ul lóegrf rg`n pfr 0 lf `ntgr` gnrgsuntàf.

` • 0 5 `

‘;.=7 • 0 5 ‘;.=7

Gcgrhbhbf =Fmsgrvg n` prfpbgaà qug gcgepnbh` hà` gcgrhbhbf. Nugif, gshrbm` gn lfemrg sfmrg n` nålg` ag nàgrghj`. Qbglg ul gcgepnf.

;) : • 7 5 7 • :

0) 0; + 17 5 77

=)8

1 + ( ‘

8

1 ) 5 ;

8) 0= • 4 5 4 • 0=

1) 04 •0

04 5 0

7) ‘8.0104 • 0 5 ‘8.0104

4) (0

1 + =) + 7 5

0

1 + (= + 7)

9) 1 (8 + 0) 5 ( 1 • 8) + ( 1 • 0)

hfleut`tbv`




1. [ftglhb`hbøl ag lóegrfs rg`ngsN` pftglhb`hbøl ag nfs lóegrfs rg`ngs sbrvg p`r` gshrbmbr ag dfre` ràpba` ullóegrf qug sg euntbpnbh` v`rb`s vghgs pfr så ebsef. Wl` pftglhb`, hfef sg nghflfhg, gstà dfreà` pfr nfs gngegltfs sbiubgltgs3

`lm`sg gxpflgltg

• N` m`sg gs gn lóegrf qug sg euntbpnbh` pfr så ebsef.

• Gn gxpflgltg blabh` gn lóegrf ag vghgs qug n` m`sg sg euntbpnbh` pfr såebse`. Ag dfre` sbemønbh` sg rgprgsglt` `så3

`l 5 ` • ` • ` . . . • `

1.0 ]gin`s ag nfs gxpflgltgs p`r` lóegrfsrg`ngs

[`r` `pnbh`r n` pftglhb`hbøl ` nfs lóegrfs rg`ngs, gs lghgs`rbf hflfhgr n`s prf-pbgaàgs qug nfs rbigl. Gst`s sfl3

Gn gxpflgltg hgrf

Qfaf lóegrf gngvàf n gxpflgltg hgrf a` hfefrgsuntàf 0.

`; 5 0

7; 5 0

(‘1); 5 0

Gn gxpflgltg lgi`tbvf

[`r` gngv`r ul lóegrf gltgrf ` ul` pftglhb`lgi`tbv`, `pnbhèfs su blvgrsf, pgrf hflgxpflgltg pfsbtbvf.

`‘l 50

`l

2 ‘8 50

28

(‘=) ‘7 50

(‘=)7

[`r` gngv`r ul` dr`hhbøl ` ul` pftglhb` lgi`tbv`,blvgrtbefs n` dr`hhbøl y gn gxpflgltg sg hflvbgrtggl pfsbtbvf.

( `

m )‘l

5 ( m

` )l

( =

1 ) ‘8

5 ( 1

= )8

( :

8 ) ‘=

5 ( 8

: )=

1:BIG] ∝ [fnfhjbh



N` euntbpnbh`hbøl ag pftglhb`s hfl biu`n m`sg

[`r` euntbpnbh`r pftglhb`s ag biu`n m`sg, sghfpb` n` m`sg y sg sue`l nfs gxpflgltgs.

`l • è 5 `l + e

1= • 18 5 1= + 8 5 17

(‘=)7 • (‘=)8 5 (‘=)7 + 8 5 (‘=):

[ftglhb` ag ul` pftglhb`

[`r` gngv`r ul` pftglhb` ` ftr` pftglhb`,hfpbèfs n` m`sg y euntbpnbhèfs nfs gxpf-lgltgs.

( l)e 5 ( )l • e

(7=)1 5 (7)= • 1 5 (7):

(0;8)7 5 (0;)8 • 7 5 (0;)07

[ftglhb` ag ul prfauhtf

[`r` gngv`r ul prfauhtf ` ul` pftglhb`,gngvèfs hà` d`htfr `n gxpflgltg aàf.

( • m)l 5 `l • ml

(= • :)4 5 =4 • :4

(4 • 1)8 5 48 • 18

[ftglhb` ag ul hfhbgltg hfl m`sgs biu`ngs

[`r` rgauhbr ul` dr`hhbøl hfl biu`n m`sg, sghfpb` n` m`sg y sg rgst`l nfs gxpflgltgs.

`l

è 5 `l ‘ e

47

48 5 47 ‘ 8 5 4=

29

24 5 29 ‘ 4 5 2

[ftglhb` ag ul` dr`hhbøl

[`r` gngv`r ul hfhbgltg ` ul` pftglhb`, sggngv` gn luegràfr y gn aglfeblàfr n gxpf-lgltg blabhàf.

( `

m )l

5`l

ml

( 7

= ):

57:

=:

( 2

9 )1

521

91

Gcgrhbhbf 8@pnbqug n`s rgin`s ag n` pftglhb`hbøl p`r` rgsfnvgr nfs gcgrhbhbfs sbiubgltgs. Qbglg ul gcgepnf.

;) :8 • :4 5 7) (‘8); 5

0) 4; 5 4) 29

24 5

=) ( 0

= )=

( 0

= )=

5 9) ( =

7 )8

5

8) ( 8

9 )1

5 :) : ‘= 5

1)7:

78 5 2) (‘=) ‘1 5

:8 + 4 5 :2




7. ]àbh`hbøl ag lóegrfs rg`ngsN` ràbh`hbøl gs n` fpgr`hbøl blvgrs` ` n` pftglhb`hbøl, hflsbstg gl j`nn`r n`r`åz ag ul lóegrf aàf, t`n qug, `n gngv`rnf ` n` pftglhb` blabhà` sg fmtgli`lugvègltg n` ebse` h`ltbaà. Nfs gngegltfs ag n` ràbh`hbøl sfl3

l

` 5 m r`åz

ràbh`laf

sbilf ràbh`n

ålabhg

• Gn ålabhg blabh` gn iràf ag n` r`åz qug sg agmg gxtr`gr.

• Gn sbilf ràbh`n blabh` n` h`ltbaà ` n` qug sg ng gxtr`g n` r`åz.

• Gn ràbh`laf gs gn lóegrf `n qug sg ng gxtr`g n` r`åz.

• N` r`åz gs gn rgsuntàf ag n` ràbh`hbøl.

7.0 [rfpbgaàgs ag nfs ràbh`ngs p`r` lóegrfsrg`ngs

N` ràbh`hbøl tèmbâl huepng hfl hbgrt`s prfpbgaàgs qug sfl m`st`ltg p`rg-hba`s ` n`s prfpbgaàgs ag n` pftglhb`hbøl, y` qug ul` r`åz gs ul` pftglhb` hflgxpflgltg r`hbfl`n.

]`åz ag ul prfauhtf

N` r`åz ag ul prfauhtf gs biu`n `n prfauhtf

ag n`s r`åhgs ag nfs ràbh`lafs hfl gn ebsefålabhg.

` • m 5 ` • m

1 • =7 5 1 • =78 : • : 5

8 : •

8 :

]`åz ag ul hfhbgltg

N` r`åz ag ul` dr`hhbøl gs biu`n `n hfhbgltg agn` r`åz agn luegràfr gltrg n` r`åz agn aglf-eblàfr.

l`

m 5

l `

l m

41

1 5 41

1

8

:

=9

5

8 :

8 =9

]`åz ag ftr` r`åz

Hu`laf ul` r`åz gstà `dghtà` pfr ftrf ràb-h`n, sg euntbpnbh`l nfs ålabhgs ag n`s r`åhgs, sghflsgrv` gn ràbh`laf y sg agc` hfef ul sfnfràbh`n.

l e 5

l • e `

8 9 5

= • 89 5

4 9

1 7 0; 5

1 • 70; 5

=; 0;

7;BIG] ∝ [fnfhjbh



]gsuegl

]`åz ag ul` pftglhb`Hu`laf gn ràbh`laf hfltbglg ul` pftglhb` hfl lóegrfs rg`ngs sg pugaglprgsglt`r afs h`sfs.

`. Xb gn ålabhg agn ràbh`n gs biu`n qug gn gxpflgltg agnràbh`laf, èmfs sg gnbebl`l.

8

78

58

78

5 7

4= 5 4= 5 4

l`l 5

l`l 5 `

m. Xb gn ålabhg gs abdgrgltg `n gxpflgltg agn ràbh`laf,sg pugag tr`lsdfre`r gl ul` dr`hhbøl y sb gs pfsbmngsg pugag sbepnbdbh`r.

e l 5 `l/e

81 5 81/= 5 8=

1 20= 5 20=/1 5 28

Gcgrhbhbf 1@pnbqug n`s prfpbgaàgs ag n` ràbh`hbøl p`r` rgsfnvgr nfs gcgrhbhbfs sbiubgltgs. Aflag sg` pfsbmng

gxprgsg n` rgspugst` hfef ul lóegrf gltgrf. Qbglg ul gcgepnf.

;)8 44 5 1)

88 5

0)41

1 5 7)

04

1 5

=) 0;= 5 4)7 17 5

8)1 04 • 04 5 9) :4 5

4 4/8 5 4 = 5 84

0. Gn hflcultf ag nfs lóegrfs rg`ngs sg rgprgsglt` hfl n` ngtr` ] y sg agdblg sbemønbhègltg hfef] 5 P ∣ B. Gstfs lóegrfs sg pugagl gxprgs`r pfr ul gltgrf f ul aghbe`n, sg` pgrbøabhf f lf.

0.0 Nfs lóegrfs rg`ngs sg h`r`htgrbz`l pfr sgr ul hflcultf dblbtf, aglsf, hfepngtf y hfltbluf.

0.= Gn v`nfr `msfnutf ag ul lóegrf rg`n gs n` abst`lhb` qug j`y gltrg gn hgrf u frbigl y gstglóegrf. Xu rgsuntàf sbgeprg gs pfsbtbvf.

=. N`s fpgr`hbflgs hfemblà`s sfl v`rb`s fpgr`hbflgs qug sg rgólgl gl ul` sfn`. [`r` rgsfnvgrn`s,gs lghgs`rbf rgspgt`r n` cgr`rquå` ag n`s fpgr`hbflgs.

8. Gn hflcultf ag nfs lóegrfs rg`ngs huepng hfl abdgrgltgs prfpbgaàgs.

1. N` pftglhb`hbøl ag lóegrfs rg`ngs gs n` dfre` ràpba` ag gshrbmbr ul lóegrf qug sg euntbpnbh`v`rb`s vghgs pfr så ebsef.

7. N` ràbh`hbøl gs n` fpgr`hbøl blvgrs` ` n` pftglhb`hbøl, hflsbstg gl j`nn`r n` r`åz ag ul lóegrfaàf.







@. @pnbqug n`s prfpbgaàgs ag n` pftglhb`hbøl gl nfs gcgrhbhbfs sbiubgltgs y gxprgsg su rgspugst` hfeful lóegrf rg`n, gl `niulfs h`sfs gs lghgs`rbf `pnbh`r eàs ag ul` rgin`. Qbglg ul gcgepnf.

;) ( 7

= ) ‘8

5 00) ‘94

‘9: 5

0) 8; 5 0=):0=

:0; 5

=) 2 ‘= 5 08)0;4

0;1 5

8)1:

14 5 01) ( =

1 ) ‘=

5

1) 0;; 5 07) ( ‘ 0

7 )8

5

7) =; ‘= 5 04) ‘0=8

‘0== 5

4) (7=)= 5 09) 8= • 8= 5

9) ( 8

= )1

5 0:) (8 • 7)= 5

:) (‘8) ‘8 5 02) (;.=7)8 5

2) (‘07); 5 =;) (‘7 • 1)8 5

0;) (‘88)= 5 =0) ( 8

= )=

( 8

= )8

5

M. @pnbqug n`s prfpbgaàgs ag n` r abh`hbøl gl nfs gcgrhbhbfs sbiubgltgs. Xbepnbdbqug n` rgspugst` aflagsg` pfsbmng. Qbglg ul gcgepnf.

;)8 : • 41 5 2)

7 07 5

0)4 88 5 0;)

=7

04 5

=): 8: 5 00)

4 1 0; 5

8)1

8 5 0=) 8=04

: 5

1) 0=1 5 08) ( 81 )

;

5

7):0

2 5 01)

12

0;; 5

4)2

04 5 07)

8 1 0; 5

9) (0=.7)= 5 04)8 :4 5

:) 0;; • 1 5 09)8 =9 • 0=7 5

8 : •

8 41 5 = • 1 5 :

( = 7 )8

5=8

78 5

:

0=7




]gsugnv` n`s fpgr`hbflgs rgspgt`laf n` cgr`rquå` ag n`s fpgr`hbflgs. Qbglg ul gcgepnf p`r` hà`h`sf.

@.;) 7 Ù : ù 1 5

0) 2 Ù = ù 4 5

=) 8 Ù 1 ù 8 5

8) 4 Ù 8 ù 2 5

1) 2 Ù 4 ù 8 5

7) 1 Ù 4 ù : 5

4) 8 Ù 1 ù 4 5

9) = Ù : ù = 5

:) 9 Ù 1 ù 1 5

2) 0= ù 4 Ù 8 5

0;) =1 ù : Ù 7 5

00) 4; ù 4 Ù 1 5

0=) 07 ù 8 Ù : 5

08) =1 ù 4 Ù ; 5

01) 1: ù = Ù = 5

07) 0: ù 4 Ù 2 5

04) 07 ù 7 Ù 9 5

09) =7 ù 7 Ù ; 5

M.

;) 1 + = Ù 8 5

0) : + 7 Ù = 5

=) 4 + 1 Ù 1 5

8) 7 + 9 Ù 8 5

1) 0 + : Ù 9 5

7) 4 + 9 Ù = 5

4) 8 + = Ù 8 5

9) 9 + : Ù 7 5

:) 2 + 8 Ù ; 5

2) : + 7 Ù 1 5

0;) 4 Ù 7 + 2 5

00) 1 Ù 9 + 0 5

0=) 9 Ù 8 + 9 5

08) 0; Ù = + : 5

01) 07 Ù 0 + 4 5

07) =7 Ù 1 + = 5

04) 7; Ù 0 + 2 5

09) =; Ù 8 + 7 5

0:) :; Ù = + : 5

02) 8; Ù 8 + 1 5

=;) 4; Ù 7 + 2 5

H.

;) 1 + 1 ù = 5

0) : + 2 ù 8 5

=) 2 + 7 ù 7 5

8) 1 + : ù 1 5

1) 0 + 4 ù 8 5

7) 7 + 8 ù 8 5

4) 1 + 0= ù 4 5

9) 9 + =; ù 1 5

:) 2 + 0: ù 2 5

2) 0 + 8; ù 8 5

0;) 0; ‘ 1 ù = 5

00) 0= ‘ 4 ù 8 5

0=) 07 ‘ 2 ù 8 5

08) 04 ‘ : ù : 5

01) =; ‘ 4 ù = 5

07) 0: ‘ 0= ù 1 5

04) 8; ‘ 0: ù 2 5

09) 0; ‘ =7 ù 7 5

0:) :; ‘ 07 ù 8 5

02) 7; ‘ =7 ù 7 5

=;) 14 ‘ 0: ù 4 5

0;

0;

4

71BIG] ∝ [fnfhjbh




]gsugnv` gl su huàgrlf nfs prfmnge`s sbiubgltgs. ]g`nbhg n`s fpgr`hbflgs `rbteâtbh`s qug lghgsbtg.

0) Wl ùtfmós vb`c` ` ul` vgnfhbaà ag 92 oe pfr jfr`. @n h`mf ag 1 jfr`s, ¶huàltfs obnøegtrfs ng

d`nt`ràl p`r` dbl`nbz`r ul vb`cg ag 104 oe6 =) @vgrbiþg huàltfs eblutfs j`l tr`lshurrbaf agsag n`s 0; jfr`s y 07 eblutfs j`st` n`s 04 jfr`s y

=; eblutfs agn ebsef aå`.

8) Nubs vglabø gn nulgs =9 nbmr`s ag e`lz`l`s, gn e`rtgs gn afmng qug gn nulgs, y gn ebârhfngs n`tgrhgr` p`rtg qug gn nulgs y gn e`rtgs cultfs. ¶Huàlt`s nbmr`s ag e`lz`l`s vglabø6

1) Wl` pgrsfl` qug tr`m c` gl ul supgregrhàf gltr` gl gn hu`rtf ag n`s vgraur`s qug gstà ` 1 »H,nugif p`s` `n hu`rtf ag nfs jgnàfs qug sg glhugltr` ` ‘=; »H. ¶Huàn gs n` v`rb`hbøl gl n`tgepgr`tur`6

7) Gl gn t`lqug ag ìu` ag n` hfeulbaà j`y 0:;; nbtrfs ag ìu`. Xb pfr ul tumf blirgs`l 7; nbtrfspfr eblutf y `n ebsef tbgepf pfr ftrf tumf s`ngl 8; nbtrfs pfr eblutf, ¶huàltfs nbtrfs ag ìu`j`mrà gl gn agpøsbtf agspuâs ag 8; eblutfs6

4) Wl prgebf ag P=,4;;.;; dug rgp`rtbaf gltrg trgs pgrsfl`s ag n` e`lgr` sbiubgltg3 ` n` prbegr` nghfrrgspflabø 1/2 ag gs` h`ltbaà, ` n` sbiubgltg 0/8 y ` n` óntbe` gn rgstf. ¶Puâ h`ltbaà rghbmbøhà` ul` y quâ dr`hhbøl agn ablgrf rghbmbø n` tgrhgr`6

9) [`r` n` blst`n`hbøl gnâhtrbh` ag ul` h`s` sg hfeprø ul rfnnf ag `nèmrg ag 7; egtrfs ag nflibtua.Xb sg hfrt`rfl 0;, 7.4, =.7 y :.97 egtrfs, ¶quâ h`ltbaà ag `nèmrg qugaø6

:) Wl lóegrf ag nftgrå` gstà dfreàf pfr 0; h`hjbtfs. Xb gn prgebf e`yfr ag n` nftgrå` gs P7;,;;;.;;y dug i`làf pfr ]utj qug tbglg 1, C`hbltf 7 y Qfe`s` 0 h`hjbtf, ¶huàltf ablgrf ng hfrrgspflag` hà` ulf, sgiól gn lóegrf ag h`hjbtfs qug tbglgl6

2) Wl` mfem` gxtr`g ìu` ag ul pfzf ` 7; e ag prfdulabaà y n` gngv j`st` ul agpøsbtf qug sgglhugltr` ` =; e ag `ntf. ¶Huàltfs egtrfs gs gngvà` gn ìu`6

0;) N` dèbnb` [ârgz rg`nbz` ul `lànbsbs ag su ghflfeå` hfl nfs blirgsfs sbiubgltgs3 afð` H`regnbl`tr`m`c` trgs jfr`s ab`rb`s nulgs, ebârhfngs y vbgrlgs? nfs e`rtgs y cugvgs sfnf tr`m`c` afs jfr`s.[fr hà` jfr` tr`m`cà` ng pì`l P=;.;;.

Gn gspfsf ag afð` H`regnbl` tbglg ul sugnaf ag P8,4;;.;; n egs y su jbcf Cfsâ, qug sfnf tr`m`c`trgs aå`s ` n` sge`l`, i`l` n` ebtà qug su pàrg. Nfs trgs `pfrt`l n` tft`nbaà ag sus s`n`rbfs.Hfl n` bldfre`hbøl prfpfrhbflà` rgspfla` n`s prgiult`s sbiubgltgs3

`. ¶Huàltf ablgrf blirgs` ` n` dèbnb` gl afs sge`l`s6

m. N` dèbnb` agstbl` 0/1 ag nfs blirgsfs ag 07 aå`s p`r` pì`r gn `nqubngr ag n` h`s`? y p`r`gn i`stf ag n` hfeba` agstbl`l P=7;.;; sge`l`ngs. ¶Huàltf j`mràl `jfrràf hà` afssge`l`s6







1 0

= 8 1

8

1

7

=

0

;

x

y

‘ 0 5 b

x = +

y = 5

r =

Lóegrfs hfepngcfs


]`åhgs p`rgs g bep`rgs

Lóegrfs hfepngcfs y fpgr`hbflgs `rbteâtbh`s

Hànhunf eglt`n ag fpgr`hbflgs hfemblà`s

[rfmnge`s ag sghuglhb`s nøibh`s

Gst` sge`l` nfir`rà3 Bagltbh`r gn hflcultf ag nfs lóegrfs hfepngcfs.

Fmtglgr r`åhgs p`rgs g bep`rgs ag lóegrfs pfsbtbvfs y lgi`tbvfs.

Gshrbmbr lóegrfs hfepngcfs purfs y gl dfre` mblfeb`n.

Agtgrebl`r gn hflcuiàf ag lóegrfs hfepngcfs.

]g`nbz`r fpgr`hbflgs `rbteâtbh`s hfl lóegrfs hfepngcfs.

]gsfnvgr fpgr`hbflgs hfemblà`s hfl ìbnbaà.




¡[`r` hfeglz`r!]`åhgs p`rgs g bep`rgs

l

`

N` sge`l` `ltgrbfr `prglabefs sfmrg nfs lóegrfs rg`ngs y n` ràbh`hbøl. Gs-

tuabèfs n`s r`åhgs huàrà`s y hómbh`s, pgrf nf hbgrtf gs qug n`s gxprgsbflgs

ràbh`ngs pugagl tglgr hfef ålabhg hu`nqubgr lóegrf gltgrf pfsbtbvf.

N`s gxprgsbflgs ràbh`ngs qug tbglgl hfef ålabhg =, 1, 4,… f hu`nqubgr lóegrf

gltgrf p`r pfsbtbvf rghbmgl gn lfemrg ag r`åhgs p`rgs.

Gcgepnfs ag r`åhgs p`rgs

=7 5 7 pfrqug 7= 5 7 • 7 5 =7

1 =74 5 1 pfrqug 11 5 1 • 1 • 1 • 1 5 =74

40

41 5 0

= pfrqug (0

= )4

5 (0

= )(0= )(0= )(0= )(0= )(0= ) 5 0

41

Gstf lfs ageugstr` qug n`s r`åhgs p`rgs ag hu`nqubgr lóegrf rg`n pfsbtbvf a`

hfef rgsuntàf ul lóegrf rg`n lf lgi`tbvf.

[fr ftr` p`rtg, n`s gxprgsbflgs ràbh`ngs qug tbglgl hfef ålabhg 8, 7, 9,… f

hu`nqubgr lóegrf gltgrf bep`r pfsbtbvf, sg ng nnè`l r`åhgs bep`rgs.

Gcgepnfs ag r`åhgs bep`rgs

8 : 5 = pfrqug =8 5 = • = • = 5 :

8 ‘: 5 ‘= pfrqug (‘=)8 5 (‘=)(‘=)(‘=) 5 ‘:

7 =18 5 8 pfrqug (8)7 5 8 • 8 • 8 • 8 • 8 5 =18

7 ‘=18 5 ‘8 pfrqug (‘8)7 5 (‘8)(‘8)(‘8)(‘8)(‘8) 5 ‘=18

[fagefs lft`r qug n` r`åz bep`r ag ul lóegrf pfsbtbvf gs ul lóegrf pfsbtbvf

y n` r`åz bep`r ag ul lóegrf lgi`tbvf a` hfef rgsuntàf ul lóegrf lgi`tbvf.

¡@ tr`m`c`r!]gspfla`3 ¶Puâ p`s`rå` sb ` ul lóegrf lgi`tbvf sg bltglt`r` gxtr`grng ul` r`åz p`r6 [rugmg hflhu`nqubgr h`ltbaà y gshrbm` su rgsuntàf.

7:BIG] ∝ [fnfhjbh





Gcgrhbhbf 0@. ]gsugnv` n`s r`åhgs p`rgs ag lóegrfs pfsbtbvfs y jì` n` hfep`r`hbøl hfl n` pftglhb`hbøl.

Qbglg ul gcgepnf.

;) =74 5 ? 5 4) ==7 5 ? 5

0) 2 5 ? 5 9) 1;; 5 ? 5

=) 04 5 ? 5 :) 2;; 5 ? 5

8) :0 5 ? 5 2) 0=0 5 ? 5

1) 011 5 ? 5 0;) 84;; 5 ? 5

7) 8=1 5 ? 5 00) =7;; 5 ? 5

M. ]gsugnv` n`s r`åhgs bep`rgs ag lóegrfs pfsbtbvfs y lgi`tbvfs. Jì` n` hfep`r`hbøl hfl n`

pftglhb`hbøl. Qbglg ul gcgepnf.

;) 8=9 5 ? 5 4) 8

‘=9 5 ? 5

0) 841 5 ? 5 9) 8

‘41 5 ? 5

=) 78= 5 ? 5 :) 7

‘8= 5 ? 5

8) 80=7 5 ? 5 2) 8

‘0=7 5 ? 5

1) 80;;; 5 ? 5 0;) 8

‘0;;; 5 ? 5

7) 9=0:9 5 ? 5 00) 9

‘=0:9 5 ? 5

H. @pnbqug sus hflfhbebgltfs sfmrg r`åhgs p`rgs lgi`tbv`s gl nfs h`sfs sbiubgltgs.Iuågsg pfr gn gcgepnf.

;) ‘1;; 5

0) ‘04 5

=) ‘=7 5

8) ‘84 5

1) ‘41 5

7) ‘042 5

4) ‘0;; 5

9) ‘024 5

04 04 = =74

8 88 =9

1;;(‘0) 5 1;;b= 5 1;; • b= 5 =;b

4;BIG] ∝ [fnfhjbh





=. Xue` y rgst` ag lóegrfshfepngcfs

Hfl nfs lóegrfs hfepngcfs, `n biu`n qug hfl nfs lóegrfs rg`ngs, tèmbâl sgpugagl rg`nbz`r n`s fpgr`hbflgs `rbteâtbh`s màsbh`s. @prglaèfs n` sue` y n`

rgst`. [rgstg `tglhbøl ` nfs p`sfs y ` nfs gcgepnfs.

0. Gshrbm` tfafs nfs lóegrfs beìbl`rbfs gl n` dfre` mb.

=. Xueg f rgstg n`s p`rtgs rg`ngs ag nfs lóegrfs hfepngcfs.

8. Xueg f rgstg n`s p`rtgs beìbl`rb`s ag nfs lóegrfs hfepngcfs.

1. Gshrbm` gn rgsuntàf ag n` dfre` ` + mb.

Xuegefs (8 + =b) + (: + 1b) 5

• Gnbeblèfs nfs p`râltgsbs. 8 + =b + : + 1b 5

• Fraglèfs nfs târeblfs. 8 + : + =b + 1b 5

• ]gauhbefs târeblfs sgegc`ltgs. 00 + 4b

Xuegefs (: + 7b) + (‘= ‘ 0;b) 5

• Gnbeblèfs nfs p`râltgsbs. : + 7b ‘ = ‘ 0;b 5

• Fraglèfs nfs târeblfs. : ‘ = + 7b ‘ 0;b 5

• ]gauhbefs târeblfs sgegc`ltgs. 4 ‘ 7b

]gstgefs (‘2 + 8b) ‘ (4 ‘ :b) 5

• Gnbeblèfs nfs p`râltgsbs. ‘2 + 8b ‘ 4 + :b 5

• Fraglèfs nfs târeblfs. ‘2 ‘ 4 + 8b + :b 5

• ]gauhbefs târeblfs sgegc`ltgs. ‘07 + 00b

]gstgefs (0

1 ‘0

8b ) ‘ (0

4 ‘0

=b ) 5• Gnbeblèfs nfs p`râltgsbs. 0

1 ‘

0

8b ‘

0

4 +

0

=b 5

• Fraglèfs nfs târeblfs. 0

1 ‘

0

4 ‘

0

8b +

0

=b 5

• ]gauhbefs târeblfs sgegc`ltgs. 0

0= +

0

4b

4=BIG] ∝ [fnfhjbh



8. Euntbpnbh`hbøl ag lóegrfshfepngcfs

@jfr` `prglaèfs ` euntbpnbh`r lóegrfs hfepngcfs. @n biu`n qug gl n` sue` ygl n` rgst`, agmg sgiubr `niulfs p`sfs3

0. Gshrbm` tfafs nfs lóegrfs beìbl`rbfs gl n` dfre` mb.

=. @pnbqug n` prfpbgaà abstrbmutbv` ag n` euntbpnbh`hbøl.

8. Xustbtuy` b= pfr ‘0 aflag `p`rgzh`.

1. ]gauzh` nfs târeblfs sgegc`ltgs y gn rgsuntàf gshråm`nf ag n`dfre` ` + mb.

Tgèfs nfs gcgepnfs3Euntbpnbqugefs 9b(1 + =b) 5

• @pnbhèfs n` prfpbgaà abstrbmutbv` ag 9b(1) + 9b(=b) 5

n` euntbpnbh`hbøl. =:b + 01b= 5

• Xustbtubefs b= pfr ‘0. =:b + 01(‘0) 5

• ]g`nbzèfs n` fpgr`hbøl. =:b ‘ 01 5

• Gshrbmbefs gn rgsuntàf ag n` dfre` ` + mb. ‘01 + =:b

Euntbpnbqugefs ‘2(7 ‘ 8b) 5

• [rbegrf, fmtglgefs n` r`åz ag ‘2 . 8b(7 ‘ 8b) 5

• @pnbhèfs n` prfpbgaà abstrbmutbv` ag07b ‘ 2b= 5 n` euntbpnbh`hbøl.

• Xustbtubefs b= pfr ‘0. 07b ‘ 2(‘0) 5

• ]g`nbzèfs n` fpgr`hbøl. 07b + 2 5

• Gshrbmbefs gn rgsuntàf ag n` dfre` ` + mb. 2 + 07b

Euntbpnbqugefs(= + 7b) (1 ‘ :b) 5

• @pnbhèfs n` prfpbgaà abstrbmutbv` ag : ‘ 04b + =;b ‘ 1;b= 5

n` euntbpnbh`hbøl.

• ]gauhbefs târeblfs sgegc`ltgs. : + 1b ‘ 1;b= 5

• Xustbtubefs b= pfr ‘0. : + 1b ‘ 1;(‘0) 5

: + 1b + 1; 5

• Gshrbmbefs gn rgsuntàf ag n` dfre` ` + mb. 1: + 1b

@tglhbøl3 gl n`s r`åhgsp`rgs ag lóegrfslgi`tbvfs lf sghuepng n` prfpbgaàagn prfauhtf.

‘2 • ‘1 ≫ ‘2 • ‘1




Euntbpnbqugefs

• @pnbhèfs n` prfpbgaà abstrbmutbv` ag nèuntbpnbh`hbøl.

• ]gauhbefs târeblfs sgegc`ltgs.

• Xustbtubefs b= pfr ‘0.

• Gshrbmbefs gn rgsuntàf ag n` dfre` ` + mb.

( 7

1 + =b)( 0

= ‘ 8b) 5

7

: ‘

07b

1 +

=b

= ‘ 4b= 5

7

: ‘

00b

1 ‘ 4b= 5

7

: ‘

00b

1 ‘ 4(‘0) 5

7

: ‘

00b

1 + 4 5

78

: ‘

00b

1

Gcgrhbhbf 8]g`nbhg n`s fpgr`hbflgs blabhà`s hfl lóegrfs hfepngcfs y gxprgsg su rgspugst` gl dfre` mblfeb`n.

0) (: + 8b) + (= + 9b) 5 =) (8 + 7b) ‘ (2 ‘ =b) 5

8) (‘1 + 9b) ‘ (0 ‘ 8b) 5 1) 7b(8 ‘ :b) 5

7) (1 ‘ 4b)(: + 0;b) 5 4) ( 0=

+ =b)( 18

‘ 7b) 5

41BIG] ∝ [fnfhjbh



1. Abvbsbøl ag lóegrfs hfepngcfsNfs lóegrfs hfepngcfs tèmbâl sg pugagl abvbabr, pgrf `ltgs ag gepgz`r àgs`rrfnn`r gst` fpgr`hbøl gs lghgs`rbf qug hflfzhèfs ul p`sf bepfrt`ltg3`prglagr ` r`hbfl`nbz`r gn aglfeblàfr egab`ltg gn hflcuiàf. Dbcâeflfs glquâ hflsbstg gstg prfhgabebgltf.

1.0 ]`hbfl`nbz`hbøl agn aglfeblàfr egab`ltggn hflcuiàf

]`hbfl`nbz`r ul` gxprgsbøl dr`hhbfl`rb` gs ul` tâhlbh` e`tgeàtbh` qug tfe` gnaglfeblàfr ag n` dr`hhbøl, sg glhugltr` su hflcuiàf y sg euntbpnbh` pfr gnluegràfr y aglfeblàfr ag n` gxprgsbøl blbhb`n.

Gl gn h`sf ag nfs lóegrfs hfepngcfs, gn hflcuiàf sg fmtbglg hèmb`laf gnsbilf ag n` p`rtg beìbl`rb`. Tgèfs `niulfs gcgepnfs3

Lóegrf hfepngcf Hflcuiàf

9b ‘9b

8 + =b 8 ‘ =b

‘1 ‘ 7b ‘1 + 7b

R s`mbafs ag høef sg fmtbglg gn hflcuiàf ag lóegrfs hfepngcfs, vgèfs nfsp`sfs p`r` rg`nbz`r n` abvbsbøl.

0. Hèmbg nfs lóegrfs beìbl`rbfs ` n` dfre` mb.

=. ]`hbfl`nbhg gn aglfeblàfr, euntbpnbh`laf gn luegràfr yaglfeblàfr pfr gn hflcuiàf.

8. Xustbtuy` b= pfr ‘0 aflag `p`rgzh`.

1. ]g`nbhg n`s fpgr`hbflgs lghgs`rb`s y gshrbm` n` rgspugst` ag n`dfre` ` + mb.

Abvbaèfs

• Euntbpnbhèfs gn luegràfr y aglfeblàfrpfr gn hflcuiàf.

• Xustbtubefs b= pfr ‘0.

• Gshrbmbefs gn rgsuntàf ag n` dfre` ` + mb.

0

8 + =b

08 + =b

( 8 ‘ =b8 ‘ =b ) 5 8 ‘ =b

2 ‘ 1b=

8 ‘ =b

2 ‘ 1( ‘0) 5

8 ‘ =b

2 + 1 5

8 ‘ =b

08 5

8

08 ‘

=b

08

]ghugrag qug3

(8 + =b)(8 ‘ =b) 5 2 ‘ 1b=









@. Fmsgrvg nfs lóegrfs hfepngcfs. Nugif, hfepngtg n` t`mn` gshrbmbglaf n` p`rtg rg`n y n` p`rtgbeìbl`rb` gl hà` h`sbnn`.

Lóegrf hfepngcf [`rtg rg`n [`rtg beìbl`rb`

‘01 + :b

0.7 ‘ 0=b

‘7 + 0:b

‘;.=7 ‘ 9b

M. Gshrbm` gn hflcuiàf p`r` nfs lóegrfs hfepngcfs sbiubgltgs. Qbglg ul gcgepnf.;) 7 ‘ :b 5

0) 1b 5

=) 4b 5

8) ‘1b 5

1) ‘0;b 5

7) ‘=8b 5

4) 8 + 4b 5

9) ‘7 ‘ 1b 5

:) 04 ‘ 2b 5

2) ‘07 ‘ 4b 5

0;) ‘02 + 01b 5

00) ‘=1 ‘ 0=b 5

@htbvbaà =. [r`htbqug nf `prglabaf

@. Gshrbm` hà` gxprgsbøl hfef ul lóegrf hfepngcf ag n` dfre` ` + mb.

0) = + ‘1 5 :) 12 + ‘12 5

=) ‘: + ‘2 5 2) 7

9 + ‘2 5

8) 0

1 + ‘:0 5 0;) ‘ 12 + ‘12 5

1) ;.=7 ‘ ‘1 5 00) ‘07 ‘ ‘042 5

7) ‘07 + ‘0;; 5 0=) 8: + ‘0=0 5

4) 7

= ‘ ‘=74 5 08) 8

‘=9 + ‘=7 5

9) 2 + ‘41 5 01) 78= ‘ ‘04 5

@utfhfltrfn

7 + :b

4:BIG] ∝ [fnfhjbh





]gsugnv` n`s fpgr`hbflgs nf eàs ràpbaf qug puga`. ]ghugrag qug p`r` nngi`r ` n` rgspugst` hfrrght`gs lghgs`rbf qug rgspgtg n` cgr`rquå` ag n`s fpgr`hbflgs. J`y ul gcgepnf p`r` hà` h`sf.

@. ;) =(1 + 8) 5

0) 4(7 + 7) 5

=) :(= + 9) 5

8) 8(9 + :) 5

1) 9(1 + 0) 5

7) 1(: + 1) 5

4) 4(2 + ;) 5 9) 9(= + 8) 5

:) 2(1 + =) 5

2) 7(9 + 7) 5

0;) 0;(4 + 1) 5

00) ‘1(: ‘ =) 5

0=) ‘=(07 ‘ 1) 5

08) ‘7(0; ‘ 8) 5

01) ‘2(=; ‘ ;) 5

07) ‘4(0: ‘ 8) 5

04) ‘:(01 ‘ 9) 5

09) ‘=(=7 ‘ 7) 5 0:) ‘1(04 ‘ 1) 5

02) ‘=;(: ‘ 8) 5

=;) ‘=7(0: ‘ 04) 5

=0) ‘1;(== ‘ 0:) 5

M.

;) =1 ù (8 + 7) 5

0) 7; ù (= + 8) 5

=) 12 ù (1 + 8) 5

8) 8; ù (= + 0) 5

1) 41 ù (1 + 1) 5

7) 44 ù (0 + =) 5

4) :0 ù (7 + 1) 5

9) 48 ù (1 + 8) 5 :) 1; ù (8 + 7) 5

2) 0: ù (4 + 8) 5

0;) 74 ù (1 + 8) 5

00) 84 ù (8 + 0) 5

0=) :: ù (4 + =) 5

08) 0= ‘ (84 ù 4) 5

01) 0; ‘ (=1 ù :) 5

07) 02 ‘ (71 ù 4) 5

04) 0: ‘ (8; ù 7) 5

09) =; ‘ (1: ù 4) 5

0:) 7; ‘ (7; ù 7) 5

02) =7 ‘ (12 ù 9) 5

=;) 07 ‘ (:0 ù 2) 5 =0) =2 ‘ (48 ù 9) 5

==) :; ‘ (2; ù 2) 5

=8) 47 ‘ (17 ù 7) 5

=1) 27 ‘ (1; ù :) 5

=7) 07 ‘ (=0 ù 8) 5

01

8

9;BIG] ∝ [fnfhjbh







prfhgsflf

nfiràf

A g s p u â s a g g s t u a b ` r . . .

Bagltbhf gn hflcultf ag nfs lóegrfs hfepngcfs.

Fmtglif r`åhgs p`rgs g bep`rgs ag lóegrfs pfsbtbvfs y lgi`tbvfs.Gshrbmf lóegrfs hfepngcfs purfs y ag n` dfre` mblfeb`n.

Agtgreblf gn hflcuiàf ag lóegrfs hfepngcfs.

]g`nbzf fpgr`hbflgs ag sue`, rgst`, euntbpnbh`hbøl y abvbsbøl aglóegrfs hfepngcfs.

]gsugnvf fpgr`hbflgs hfemblà`s hfl ìbnbaà.

Huàrf eàibhf

[`r` sgiubr gcgrhbt`laf su j`mbnbaà ag hànhunf, ng prfpflgefs hfepngt`r gnsbiubgltg huàrf eàibhf.

Agmg rgnngl`r n`s h`sbnn`s hfl nfs lóegrfs agn 0 n 2, sbl rgpgtbrsg y ag t`n e`lgr`qug n`s fpgr`hbflgs hfemblà`s huepn`l hfl n` biu`naà. Ng `yuaèfs hfl n`prbegr` n`.

: Ù = + 1 5 =;

+ Ù Ù

Ù ‘ 5 =;+ + Ù

+ + 5 =;

5 =; 5 =; 5 =;

9=BIG] ∝ [fnfhjbh





70

= 8 1

8

1

7

=

0

;

x

y

‘ 0 5 b

x = +

y = 5

r =

Xue` y rgst` ag e`trbhgs


Xuafou3 ul cugif ag bliglbf

E`trbhgs y su àbhbøl

Euntbpnbh`hbflgs y sbepnbh`hbøl ag dr`hhbflgs

Cugifs ag suafou


Hflhgptu`nbz`r quâ gs ul` e`trbz.

Hflfhgr g bagltbh`r n`s abeglsbflgs ag ul` e`trbz.

Fpgr`r sue`s y rgst`s ag e`trbhgs.

Hflfhgr n`s prfpbgaàgs ag n` sue` ag e`trbhgs.

]gsfnvgr euntbpnbh`hbflgs y sbepnbh`r dr`hhbflgs hfl ìbnbaà.




¡[`r` hfeglz`r!Xuafou3 ul cugif ag bliglbf

Pubzàs nf j`y` vbstf `n gst`r jfcg`laf ul pgrbøabhf, qubzàs y` nf j`y` rgsugntf

`niul` vgz, lfs rgdgrbefs `n suafou, gn pfpun`r cugif qug j`hg dulabr lugstr`s

lgurfl`s.

Gn suafou gs ul glbie` e`tgeàtbhf pumnbhàf pfr prbegr` vgz ` dbl`ngs ag

nfs `ðfs sgtglt`, sg vfnvbø pfpun`r gl C`pøl j`hb` 02:4 y sg abf ` hflfhgr

bltgrl`hbfl`negltg hgrh` ag =;;7.

Gn fmcgtbvf agn suafou gs rgnngl`r ul` huàråhun` ag 2 Ù 2 hgna`s, abvbaba` gl

sumhuàråhun`s ag 8 Ù 8, nnèà`s "h`c`s" f "rgibflgs", hfl nfs lóegrfs agn 0 n

2, p`rtbglaf ag lóegrfs qug y` gstàl hfnfhàfs gl `niul`s hgna`s.

Gn truhf gstà gl qug ul ebsef lóegrf lf sg agmg rgpgtbr gl ul` ebse` dbn`,

hfnuel` f h`c`. Tgèfs ul gcgepnf y su sfnuhbøl3

¡@ tr`m`c`r!

@jfr` ng tfh` ` ustga. ]gsugnv` gn suafou sbiubgltg. Ng rghfeglaèfs gshrbmbr hfl nàpbz pfr sb lghg-sbt` hfrrgibr `niól a`tf.

8 1

4 9 7

7

: 7 4 =

9 1 7 :

8 = 1

4 2 9 8 1

1 = 81 : 4 0 9

8 = 1 7 :

4 9 : 2 = 8

1 4 8 9 74 = 2 8 : 9

8 9 1 : 2 4

1 : 2 9 0 8

7 4 : 2

1 7

2 9 8 4 =

Blbhbf

0 9 : 8 2 7 = 1 4

8 = 1 0 : 4 9 2 7

2 7 4 = 9 1 8 : 0

: 2 7 4 8 9 0 = 1

9 1 0 7 = : 2 4 8

4 8 = 1 0 2 : 7 9

= 4 2 9 7 8 1 0 :

7 0 9 : 1 = 4 8 21 : 8 2 4 0 7 9 =

Xfnuhbøl

91BIG] ∝ [fnfhjbh



0. E`trbhgsXg lfemr` e`trbz ` tfaf hflcultf ag lóegrfs f gxprgsbflgs abspugstfs gl

dfre` rght`liun`r, dfre`laf n`s y hfnuel`s? gstf qubgrg aghbr qug gn suafouhfl qug lfs gltrgtglåèfs gl n` sghhbøl `ltgrbfr gs ul` e`trbz.

Gn fragl f abeglsbøl ag ul` e`trbz sg agtgrebl` pfr su lóegrf ag n`s yhfnuel`s. Wl` e`trbz ag abeglsbøl e Ù l tbglg e n`s y l hfnuel`s.

Ag gst` dfre`, ul` e`trbz pugag sgr ag fragl3 = Ù 8 (= n`s y 8 hfnuel`s), 1 Ù 4 (1 n`s y 4 hfnuel`s), gth. Xb n` e`trbz tbglg gn ebsef lóegrf ag n`s qug aghfnuel`s, aghbefs qug gs ul` e`trbz huàrà` y sg aglfebl` ag fragl3=, 8, 1...

Hà` ulf ag nfs lóegrfs qug hfldfre` n` e`trbz sg aglfebl` gngegltf. Wl

gngegltf sg abdgrglhb` ag ftrf pfr n` pfsbhbøl qug fhup`, gs aghbr, n` n` yn` hfnuel` ` n` qug pgrtglghg. Afs e`trbhgs sfl biu`ngs sb tbglgl n` ebseàbeglsbøl y nfs gngegltfs qug fhup`l gn ebsef nui`r gl èm`s, sfl biu`ngs.

Wl gngegltf hu`nqubgr` ag n` e`trbz, qug sg glhugltr` gl n` n` b y gl n` hfnuel` c, sg aglft` pfr `bc. Tgàefsnf iràhègltg p`r` gltglagr egcfr3

`00 `0= ... `0c ... `0l

`=0 `== ... `=c ... `=l

... ... ... ... ... ...`b0 `b= ... `bc ... `bl

... ... ... ... ... ...è0 è= ... èc ... èl

@ 5

Gl n` e`trbz @ lftèfs su abeglsbøl ag e Ù l y sus gngegltfs gl n`s b yhfnuel`s c.

Wl` e`trbz sglfemr` hfl ul`ngtr` e`yóshun`.

Gcgrhbhbf 0Gst`mngzh` n` abeglsbøl ag n`s e`trbhgs aà`s y gshråm`n` gl n` nålg` hfrrgspflabgltg. Iuågsg pfr gngcgepnf.

;) ‘0 ; =

9 ‘0 2 ‘4 1 =

=) 0 ‘0

: ;

1) 9 =

0 ‘7= ‘8

0) 7 =

9 :

1 9

8) 2 9 7 1 7) ‘9 0 1 7

8 4 = ‘7

: ‘8 = ‘=

1 8 ‘0 ;

8 Ù 8







8. Fpgr`hbflgs gltrg e`trbhgs

N`s fpgr`hbflgs qug sg pugagl rg`nbz`r gltrg e`trbhgs sfl n` sue`, n` rgst` y nèuntbpnbh`hbøl. Gst` sge`l` prglagrgefs gn prfhgabebgltf p`r` sue`r y rgst`rafs f eàs e`trbhgs.

8.0 Xue` y rgst`

Xfnègltg n`s e`trbhgs qug tbglgl n` ebse` abeglsbøl sg pugagl sue`r frgst`r. N` e`trbz sue` sg fmtbglg sue`laf nfs gngegltfs ag n`s e`trbhgs qugfhup`l n` ebse` pfsbhbøl? p`r` rgst`r sg sbiug gn ebsef prfhgsf.

Tgèfs hfl agtglbebgltf3

Xb @ 5 (`bc)eÙl y M 5 (mbc)eÙl gltflhgs su sue` gs @ + M 5 (` bc + mbc)eÙl

@ 5= ; 08 ; ;

7 0 0

M 50 ; 00 = 0

0 0 ;

@ + M 5= + 0 ; + ; 0 + 0

8 + 0 ; + = ; + 0

7 + 0 0 + 0 0 + ;

58 ; =

1 = 0

4 = 0

Gn prfhgabebgltf p`r` rgst`r gs euy sbebn`r, fmsgrvgefs3

@ ‘ M 5

= ‘ 0 ; ‘ ; 0 ‘ 0

8 ‘ 0 ; ‘ = ; ‘ 07 ‘ 0 0 ‘ 0 0 ‘ ;

5

0 ; ;

= ‘= ‘01 ; 0

Lf gs lghgs`rbf qug n`s e`trbhgs sg`l huàrà`s p`r` pfagr sue`rn`s frgst`rn`s, n` ólbh` hflabhbøl gs qug tgli`l n`s ebse`s abeglsbflgs.

Q embâl pfagefs fpgr`r eàs ag afs e`trbhgs, vgèfs ftrf gcgepnf3

Z 58 = 7

7 ‘0 ‘= R 5

; ‘9 0

8 ; 0 S 5

‘0 8 0

: 4 ;

Z + R + S 58 + ; + (‘0) = + (‘9) + 8 7 + 0 + 0

7 + 8 + : (‘0) + ; + 4 (‘=) + 0 + ; 5

= ‘= 9

04 7 ‘0

Z ‘ R ‘ S 58 ‘ ; ‘ (‘0) = ‘ (‘9) ‘ 8 7 ‘ 0 ‘ 0

7 ‘ 8 ‘ : (‘0) ‘ ; ‘ 4 (‘=) ‘ 0 ‘ ; 5

1 4 8

‘4 ‘9 ‘8




Gcgrhbhbf =Xbglaf n`s e`trbhgs3

@ 5= 7

8 ‘1 M 5

‘8 ‘2

4 = H 5

7 ‘9

‘0 ; A 5

4 1

; ‘2

G 5 ‘7 4 =

0 ‘0 ;

8 ; 1

D 54 ; ;

‘2 ‘= 1

0 1 =

I 57 0 09 ‘0 1

8 ‘= =

J 59 ‘9 24 8 ;

= 0 ;

]gsugnv` n`s sbiubgltgs sue`s y rgst`s ag e`trbhgs. [ugag us`r jfc`s `p`rtg sb nf lghgsbt`. ]ghugragtr`m`c`r gl fragl.

0) @ + M 4) D + I

=) @ + A 9) G + J + I

8) @ + H ‘ A :) I + J

1) H + A + @ 2) G + D + I ‘ J

7) G + D 0;) G + I ‘ J

9:BIG] ∝ [fnfhjbh



8.= [rfpbgaàgs ag n` sue` ag e`trbhgs

N` sue` gltrg e`trbhgs tbglg prfpbgaàgs qug sfl euy sbebn`rgs ` n`s qugpfsggl nfs lóegrfs rg`ngs. [flièfs `tglhbøl.

Bltgrl`

N` sue` f rgst` ag afs f eàs e`trbhgs ag fragl e Ù l gs ftr` e`trbz ag fragle Ù l. Gs aghbr, n` e`trbz rgsuntàf sgrà ag n`s ebse`s abeglsbflgs qug n`se`trbhgs qug sg fpgr`rfl.

(e Ù l) ° (e Ù l)5 e Ù l

9 ‘=

; 0

= ‘1

+; ‘=

8 ‘0

= =

5

9 ‘1

8 ;

1 ‘=

8 n`s 8 n`s

= hfnuel`s = hfnuel`s

8 Ù = 5 8 Ù =

@sfhb`tbv`

N` `sfhb`tbvbaà lfs abhg qug gn rgsuntàf ag n` fpgr`hbøl sgrà gn ebsef sblbepfrt`r gn ìrupèbgltf ag nfs târeblfs.

@ + (M + H) 5 (@ + M) + H

: 40 0

0 ‘9

+ ‘8 9 ‘1 ‘=

0 ;

+

= 1

; 0

0 0

5

9 09

‘8 ;

8 ‘4

: 4

0 0

0 ‘9

+

‘8 9

‘1 ‘=

0 ;

+

= 1

; 0

0 0

5

9 09

‘8 ;

8 ‘4

Gngegltf lgutrf

N` e`trbz hgrf, aglftà` hfl n` ngtr` F, gs ul` e`trbz gl n` qug hà` ulf ag sus

gngegltfs gs hgrf. Gst` e`trbz gs gn gngegltf lgutrf.

@ ° F 5 @

0 =

‘8 0

1 ‘=

+; ;

; ;

; ;

5

0 =

‘8 0

1 ‘=




Gngegltf fpugstf

N` e`trbz fpugst` gs `qugnn` gl n` qug tfafs nfs gngegltfs tbglgl gn sbilfhfltr`rbf rgspghtf ` ul` e`trbz frbibl`n. Xg rgprgsglt` `ltghgabglaf ul sbilfeglfs `n lfemrg ag n` e`trbz (‘M, ‘@, ‘S, gth).

@ + (‘@) 5 F

9 ‘1

‘1 0

4 ‘=

+ ‘9 1

1 ‘0

‘4 =

5

; ;

; ;

; ;

Hfleut`tbv`

Gst` prfpbgaà gst`mnghg qug `n hèmb`r gn fragl ag nfs sue`lafs, gn rgsuntàfpgre`lghg blv`rb`mng, gs aghbr, lf sg efabh`.

@ + M 5 M + @

= 4

0 ‘8

0 0

+7 ‘:

2 9

= ‘2

5

9 ‘=

0; 1

8 ‘:

57 ‘:

2 9

= ‘2

+= 4

0 ‘8

0 0

Gcgrhbhbf 8]gsugnv` n`s fpgr`hbflgs sbiubgltgs. Nugif, gshrbm` gl gn gsp`hbf hfrrgspflabgltg quâ prfpbgaà sggvbaglhb`. Iuågsg pfr gn gcgepnf.

[rfpbgaà

;) ; 0 ;

8 7 ‘4

7 : ‘9

+

= ‘1 :

; 1 ‘8

‘8 0 ‘4

+

2 ; ‘2

9 ‘= :

0 0 0

5

00 ‘8 ‘0

0; 9 ‘0

8 0; ‘0=

`sfhb`tbv`

0) = ‘1 :

; 1 ‘8

‘8 0 ‘4

+

‘= 1 ‘:

; ‘1 8

8 ‘0 4

5

=) 9 : ‘1

0 ; 7

8 4 9

+

; ; ;

; ; ;

; ; ;

5

:;BIG] ∝ [fnfhjbh



]gsuegl

0. Wl` e`trbz gs ul hflcultf ag lóegrfs f gxprgsbflgs abspugstfs gl dfre` rght`liun`r, dfre`lafdbn`s y hfnuel`s. Gn fragl f abeglsbøl ag ul` e`trbz sg agtgrebl` pfr gn lóegrf ag dbn`s yhfnuel`s qug n` hfepflgl.

Wl` e`trbz e Ù l tbglg e dbn`s y l hfnuel`s. Xb tbglg gn ebsef lóegrf ag dbn`s qug ag hfnuel`s,aghbefs qug gs ul` e`trbz huàrà`. @ hà` lóegrf f gxprgsbøl qug dfre` n` e`trbz ng aglfeb-lèfs gngegltf.

=. J`y `niulfs tbpfs ag e`trbhgs hfl h`r`htgråstbh`s p`rtbhun`rgs3

=.0 Wl` e`trbz tr`lspugst` (@t ) gs n` qug sg dfre` `n hèmb`r fraglàègltg n`s dbn`s pfr n`shfnuel`s ag n` e`trbz aà`.

=.= Wl` e`trbz gs sbeâtrbh` sb gs huàrà` y gs biu`n ` su tr`lspugst`.

=.8 Wl` e`trbz bagltbaà gs ul` e`trbz gl huy` abìfl`n prblhbp`n tfafs sus gngegltfs sflbiu`ngs ` 0 y nfs ageàs gngegltfs sfl ; (hgrf). Xg lfemr` hfl n` ngtr` B.

=.1 Wl` e`trbz lf sbliun`r f blvgrtbmng gs `qugnn` qug tbglg e`trbz blvgrs`.=.7 Wl` e`trbz sbliun`r lf tbglg blvgrs`.

8. N`s fpgr`hbflgs qug sg pugagl j`hgr hfl e`trbhgs sfl n` àbhbøl, n` rgst` y n` euntbpnbh`hbøl3

8.0 [`r` sue`r f rgst`r e`trbhgs gs lghgs`rbf qug tgli`l n` ebse` abeglsbøl. [`r` fpgr`r, sgagmgl sue`r nfs gngegltfs ag n`s e`trbhgs qug fhup`l n` ebse` pfsbhbøl . Xg pugaglfpgr`r afs f eàs e`trbhgs.

8.= N` sue` ag e`trbhgs tbglg n`s prfpbgaàgs sbiubgltgs3

• Gs bltgrl` pfrqug n` e`trbz rgsuntàf sgrà ag n`s ebse`s abeglsbflgs qug n`s e`trbhgsqug sg fpgr`rfl.

• Gs `sfhb`tbv pfrqug gn rgsuntàf gs gn ebsef sbl bepfrt`r høef irupgefs nfs târeblfs.

• Qbglg gngegltf lgutrf, hflfhba` hfef e`trbz hgrf (F).

• Qbglg gngegltf fpugstf, gn hu`n gs n` e`trbz hfl nfs ebsefs gngegltfs qug n` e`trbzfrbibl`n, pgrf hfl nfs sbilfs hfltr`rbfs.

• Gs hfleut`tbv pfrqug gn rgsuntàf pgre`lghg blv`rb`mng sbl bepfrt`r gn fragl gl qugfpgrèfs nfs târeblfs.

Ng blvbtèfs ` qug vbsbtg nfs sbiubgltgs sbtbfs ag bltgrlgt. Gl gn prbegrf, pfarà ags`rrfnn`r su pgls`-ebgltf `l`nåtbhf `n ebsef tbgepf qug sg gltrgtbglg cui`laf suafou. Gl n` sgiula` y tgrhgr` abrghhbøl,glhfltr`rà gcgrhbhbfs bltgr`htbvfs ag n` sue` ag e`trbhgs.

• Xuafou gl nålg`3 iff.in/oS=RDx

• Gcgrhbhbfs. @abhbøl y sustr`hhbøl ag e`trbhgs3 iff.in/w;7jZJ

• Xue` y rgst` ag e`trbhgs3 iff.in/H8rxjZ


:0E`tgeàtbh` ∝ Xge`l` 7




@. Bagltbdbqug n` abeglsbøl ag n`s e`trbhgs sbiubgltgs. Gshrbm` su rgspugst` gl n` nålg` hfrrgspflabgltg.

0) 8 =; ‘2

8 ‘:

8) 8 1 4 ‘9 ‘7 9 = ;

0 ; ; ‘8

=) 0 ‘9

0 7

1) 8 1 :

‘9 ; 0

M. Dfreung y gshrbm` ul` e`trbz hfl gn fragl blabhàf.0) 8 Ù 8 1) 0 Ù 8

=) = Ù 1 7) 8 Ù 0

8) 8 Ù = 4) = Ù 0

H. Dfreung, gshrbm` g bagltbdbqug n`s e`trbhgs qug sg pbagl gl hà` blhbsf. Iuågsg pfr gn gcgepnf.

@utfhfltrfn

E`trbz tr`lspugst` E`trbz huàrà` E`trbz bagltbaà

Mt 5 ‘0 8 ‘:

; ‘0 :

2 8 ;

:=BIG] ∝ [fnfhjbh




]gsugnv` n`s sbiubgltgs sue`s y rgst`s ag e`trbhgs. [ugag us`r jfc`s `p`rtg sb n`s lghgsbt`. ]ghugragtr`m`c`r fraglàègltg. Xu frbgltàfr pugag sfnbhbt`rng hflst`lhb` ag su prfhgabebgltf.

Aà`s n`s e`trbhgs sbiubgltgs3

Y 5 ‘1 = 0 :; 0 0 ‘1

R 5 7 ‘: 9 =0 ‘0 ; :

S 5 ‘1 : ‘: 02 9 ‘= ‘0

Fpgrg3

0) Y + R =) (Y + S) ‘ R

8) R + S 1) Y ‘ R + S

Aà`s n`s e`trbhgs3

A 5 ‘7 : 4

; 0 1

‘0 = 0

X 5 ‘= 1 ‘0

9 ‘0 :

= 8 ;

G 59 ‘7 ;7 = ;

‘9 : 2

7) A + G 4) A + X + G

9) X ‘ G :) G ‘ A




@. Gl gn gsp`hbf hfrrgspflabgltg, gshrbm` gn târeblf d`nt`ltg p`r` qug n`s euntbpnbh`hbflgs sg`lhfrrght`s.

0) = Ù 5 :

=) 8 Ù 5 2

8) : Ù 5 1;

1) 8 Ù 5 =9

7) 7 Ù 5 07

4) 2 Ù 5 0:

9) 1 Ù 5 0=

:) 4 Ù 5 71

2) 2 Ù 5 84

0;) 8 Ù 5 0=

00) Ù 9 5 1=

0=) Ù 2 5 17

08) Ù = 5 4

01) Ù 9 5 9

07) Ù 4 5 1=

04) Ù = 5 1

09) Ù : 5 9=

0:) Ù 1 5 ;

02) Ù 9 5 48

=;) Ù 0 5 0

=0) 1 Ù 5 04

==) 8 Ù 5 2

=8) : Ù 5 41

=1) 0 Ù 5 ;

=7) 9 Ù 5 01

=4) 8 Ù 5 =0

=9) ; Ù 5 ;

=:) 7 Ù 5 =7

=2) = Ù 5 0;

8;) 4 Ù 5 0:

M. Xbepnbdbqug n`s dr`hhbflgs sbiubgltgs j`st` su eålbe` gxprgsbøl. Wsg dr`hhbflgs beprfpb`s ygltgrfs ag sgr lghgs`rbf.

0) 4

:

5

=) 0;

=; 5

8) ==

88 5

1) =;

:; 5

7) 01

9

5

4) 01

=; 5

9) 7

=7 5

:) 0=

84 5

2) 0;

0;;

5

0;) =

4 5

00) 4

2 5

0=) 2

07 5

08) 07

0=

5

01) 9

12 5

07) 1

0; 5

04) :

9= 5

09) 84

4

5

0:) 4

:= 5

02) 07

9= 5

=;) =

: 5

=0) 1

0:

5

==) 2;

0=; 5

=8) 00

22 5

=1) 0;

:7 5

:1BIG] ∝ [fnfhjbh




@n blbhbf ag gst` sge`l` `prglabefs n`s rgin`s p`r` cui`r suafou, cugif qug sg j` hflvgrtbaf gl ulgltrgtglbebgltf euy pfpun`r gl nfs óntbefs `ðfs.

Gstg cugif lf sfnf gstbeun` n` nøibh`, n` egefrb` y n` h`p`hbaà ag hflhgltr`hbøl? rghbgltgegltg, ulgstuabf ag n` Wlbvgrsbaà ag Gabemurif, gl Blin`tgrr`, j` agefstràf qug gn suafou tèmbâl gcgrhbt`lugstr`s lgurfl`s, prfnfli` su tbgepf ag vba` hgnun`r y n`s j`hg eàs rgsbstgltgs ` gldgregaàgshfef gn @nzjgbegr.

Hfl tfa`s gst`s vglt`c`s, ¶quâ ng p`rghg sb rgsfnvgefs ulfs cugifs eàs6 ¡Àlbef!, qug gstàl dàhbngs.

7 9 = 1

4 1 7

1 8 : 7

: = 4 9 8 1 2 7

8 1 = 7 2 4

: 8 =

9 4 = 7 1

4 8 1 :

0 9

9 : 7 = 4 2

4 2 9 1 : 8

2 = 0 : 4

0 4 1 :

9 8 2 : 7 17 : 4 0 9

4 0 2 =

1 0 = 4 :

= 0 9 7






prfhgsflf

nfiràf


Hflhgptu`nbzf quâ gs ul` e`trbz.Hflfzhf g bagltbhf n`s abeglsbflgs ag ul` e`trbz.

Fpgrf sue`s y rgst`s ag e`trbhgs.

Hflfzhf n`s prfpbgaàgs ag n` sue` ag e`trbhgs.

]gsugnvf euntbpnbh`hbflgs y abvbsbflgs hfl ìbnbaà.

Gn ags`rrfnnf ag n` bltgnbiglhb` gsp`hb`n gs gsglhb`n p`r` pfagr e`lgc`r y e`-lbpun`r hfl agstrgz` nf rgn`hbflàf `n hfnfr, dfre`, nålg`, iur`, gsp`hbf, y n`rgn`hbøl qug gxbstg gltrg tfafs nfs `ltgrbfrgs.

@jfr` pflargefs ` tr`m`c`r lugstr`s j`mbnbaàgs gsp`hb`ngs. Fmsgrvg n`bnustr`hbøl sbiubgltg, nugif, hfepnâtgn` abmuc`laf n` ebtà qug d`nt` ag ul`dfre` sbeâtrbh`.

:4BIG] ∝ [fnfhjbh





40

= 8 1

8

1

7

=

0

;

x

y

‘ 0 5 b

x = +

y = 5

r =

Euntbpnbh`hbøl ag e`trbhgs


Gn `cgargz y n` ngygla` ag Xbs`


Ghu`hbflgs ag prbegr iràf

Xgrbgs ag iur`s rgn`hbflà`s


Hflfhgr y `pnbh`r n`s rgin`s p`r` euntbpnbh`r ul` e`trbz pfr ul lóegrf rg`n.

[r`htbh`r n` euntbpnbh`hbøl ag ul` e`trbz pfr ul lóegrf rg`n.

Hflfhgr y `pnbh`r n`s rgin`s p`r` euntbpnbh`r e`trbhgs gltrg så.

[r`htbh`r n` euntbpnbh`hbøl gltrg e`trbhgs.




¡[`r` hfeglz`r!Gn `cgargz y n` ngygla` ag Xbs`

Gn `cgargz gs ul cugif gltrg afs pgrsfl`s, hà` ul` huglt` hfl 04

pbgz`s qug sg eugvgl sfmrg ul t`mngrf gl dfre` e`trbhb`n f ag e`trbz

ag :Ù:, hfl 41 huàrfs. Xg tr`t` ag ul cugif ag gstr`tgib` gl gn qug

gn fmcgtbvf gs vglhgr f "agrrfh`r" `n "rgy" agn fpflgltg.

Xfmrg gn frbigl agn `cgargz j`l suribaf euhj`s ngygla`s, ul` ag gnn`s

gs n` agn s`mbf Xbs`, ngèfs3

Gl tbgepfs rgeftfs, `n lfrfgstg ag n` `ltbiu` Blab`, gl nf qug jfy sgrå`

[òbstàl f @di`lbstàl, vbvå` ul pfagrfsf y rbhf rgy nnèàf ]`b Mj`nbt.

@ pgs`r ag su hflabhbøl, gn rgy j`må` pgrabaf tfa` dgnbhbaà agmbaf ` n`

eugrtg ag su jbcf gl n` iugrr`.

¡@ tr`m`c`r!

]gspfla`3 ¶Puâ ng j`mrå` pgabaf ustga `n rgy ]`b Mj`nbt hfef rghfepgls`6 ¶[fr quâ6

Agv`stàf pfr gst` trìgab`, gn rgy sg `m`laflø ` så ebsef y agshubaø su rgblf. Xus eàs hgrh`-

lfs hflsgcgrfs y sbrvbgltgs sg gsdfrz`m`l pfr `lbe`rnf, blvbt`m`l mudflgs, m`bn`rblgs y h`lt`ltgs,

pgrf là` dulhbfl`m`.

Wl vbgcf s`mbf nnèàf Xbs` aghbabø hrg`r ul cugif p`r` agvfnvgr n` dgnbhbaà ` su rgy y pfagr gl-

trgtglgrng. Nugif ag euhjf tr`m`cf y rgdngxbøl, Xbs` sg prgsgltø drgltg `n sfmgr`lf p`r` efstr`rng

gn lugvf cugif. @mrbø ul` h`c` y `p`rghbø ul t`mngrf ag eàgr` hfl 41 h`sbnn`s y 8= dbiur`s agn

ebsef e`tgrb`n. Qr`s gxpnbh`r n`s rgin`s `n rgy, sg pusbgrfl ` cui`r.

@n ifmgrl`ltg ng glh`ltø gn p`s`tbgepf y cuiø aur`ltg jfr`s y aå`s hfltr` tfaf `qugn qug nf

rgt`r`.

@iràghbaf hfl Xbs`, gn rgy ng fdrghbø n` rghfepgls` qug gn s`mbf qubsbgsg. Gl ul prblhbpbf, gn s`mbf

sg lgi`m` ` `hgpt`r rgi`nfs, pgrf nugif ag euhj` blsbstglhb`, abcf3 ‑"Xgðfr, eg hfldfre`rå`

hfl qug eg pìuâbs ul ir`lf ag trbif pfr gn prbegr huàràf, afs pfr gn sgiulaf, hu`trf gl gn

tgrhgrf, gth."

Gn rgy `hhgabø ag blegab`tf, y e`laf ` sus `yua`ltgs ` h`nhun`r gn tft`n ag trbif y ` gltrgiàrsgnf `

Xbs`. Xbl gem`rif, nugif ag v`rbfs hànhunfs, n` hbdr` tft`n gr` ag ¡0:.1 trbnnflgs ag ir`lfs ag trbif!

Gn rgy qugaø sfrprglabaf pfrqug lb hfl tfa`s n`s hfsghj`s agn rgblf pfarå` rgulbr t`n h`ltbaà.

Gn s`mbf, s`tbsdghjf pfrqug gn rgy gr` dgnbz ag lugvf, hfl ul` sflrbs` rglulhbø `n rgi`nf.

::BIG] ∝ [fnfhjbh



0. Euntbpnbh`hbøl ag e`trbhgsGl "N` ngygla` ag Xbs`" sg j`hg gvbagltg qug n` dfre` e`trbhb`n tbglg v`rbgaà

ag `pnbh`hbflgs y usfs, ulf ag gstfs gs gn t`mngrf ag `cgargz.N` sge`l` p`sà` `prglabefs sfmrg n`s e`trbhgs, høef sue`rn`s y rgst`rn`s?`jfr` vgrgefs gn prfhgabebgltf p`r` euntbpnbh`r ul` e`trbz pfr ul lóegrf rg`ny pfr ftr` e`trbz. Tèfs, qug gs dàhbn, pflièfs `tglhbøl.

0.0 [rfauhtf ag ul lóegrf rg`n pfr ul` e`trbz

Xb tglgefs ul` e`trbz @ 5 (`bc) y n` euntbpnbhèfs pfr ul lóegrf rg`n o , nè`trbz rgsunt`ltg sgrà ul` e`trbz ag n` ebse` abeglsbøl qug @ y gl n` qug hà`gngegltf gstà euntbpnbhàf pfr o .

Gs aghbr, sbepngegltg agmgefs euntbpnbh`r gn lóegrf o pfr hà` ulf ag nfsgngegltfs ag n` e`trbz.

[`r` gst` fpgr`hbøl, sg pugagl euntbpnbh`r e`trbhgs ag hu`nqubgr fragl.

o @ 5 o ` m h

a g d

i j b

5ò mo ho

ao go do

io jo bo

¶R lftø nf sglhbnnf qug gs6 Tgèfs ulfs gcgepnfs3

=J 5 = = ; 08 ; ;

7 0 0

5= • = = • ; = • 0= • 8 = • ; = • ;

= • 7 = • 0 = • 0

51 ; =4 ; ;

0; = =

‘:Y 5 ‘: ‘= 8 ;

: 2 ‘0

0 ; 9

5(‘:) • (‘=) (‘:) • 8 (‘:) • ;

(‘:) • : (‘:) • 2 (‘:) • (‘0)

(‘:) • 0 (‘:) • ; (‘:) • 9

504 ‘=1 ;

‘41 ‘9= :

‘: ; ‘74

7M 5 7 4 ‘7 0

; 1 ‘=

0 0 0

57 • 4 7 • (‘7) 7 • 0

7 • ; 7 • 1 7 • (‘=)

7 • 0 7 • 0 7 • 0

58; ‘=7 7

; =; ‘0;

7 7 7

Wl óntbef gcgepnf `ltgs ag p`s`r ` n` pràhtbh`3

‘8Q 5 ‘8 1 7

8 1

‘8 ‘0

5(‘8) • 1 (‘8) • 7

(‘8) • 8 (‘8) • 1

(‘8) • (‘8) (‘8) • (‘0)

5 ‘0= ‘07

‘2 ‘0=

2 8





Gcgrhbhbf 0Gs su turlf, rgsugnv` n`s euntbpnbh`hbflgs sbiubgltgs.

0) 7 ‘= 2 ‘0 ; =)1 ‘= 2

0 1

8) ‘1

0 7

‘9 ;

1 8

1) ‘0

02 0;

; ‘0

‘01 =

7)= 1 0 ; =

‘= 8 0 0

4)4 4 : ‘0; 8

0 9 ; 0

9) ‘4

‘0 7 ‘=

0 ; 4

9 1 2

:)2

2 ‘1 4

; 0 ‘:

0 ‘0; 9

2;BIG] ∝ [fnfhjbh



0.= [rfauhtf ag e`trbhgs

[rbegrf, agmgefs tglgr hn`rf qug p`r` qug afs e`trbhgs (@ y M) puga`l euntb-pnbh`rsg, gn lóegrf ag hfnuel`s ag @ agmg hfblhbabr hfl gn lóegrf ag n`s

ag M. Xb sg huepng gst` rgstrbhhbøl, gstèfs nbstfs p`r` hfeglz`r.

Gn gngegltf hbc ag n` e`trbz prfauhtf sg fmtbglg euntbpnbh`laf hà` gngegltfag n` n` b ag n` e`trbz @ pfr hà` gngegltf ag n` hfnuel` c ag n` e`trbz M ysueàlafnfs. Lf sg prgfhupg, lf gs hfepnbhàf, vgèfs3

H 5 @ • M 5` m h

a g d

i j b

•r s t

u v w

x y z

5`r + mu + hx `s + mv + hy `t + mw + hz

ar + gu + dx as + gv + dy at + gw + dz

ir + ju + bx is + jv +by it + jw + bz

Hfef vgefs gl n`s e`trbhgs `ltgrbfrgs, sg euntbpnbh` hà` gngegltf ag n` prb-egr` n` ag @ pfr hà` gngegltf ag n` prbegr` hfnuel` ag M. @n tgrebl`rgstf, sg sue`l nfs prfauhtfs y `så sg fmtbglg gn gngegltf h00 ag n` e`trbz H.

Nugif, vfnvgefs ` tfe`r n` prbegr` n` ag @, pgrf `jfr` n` euntbpnbhèfs pfrn` sgiula` hfnuel` ag M ag n` ebse` dfre` qug nf jbhbefs ltgrbfregltg. Hflgstf fmtglgefs gn gngegltf h0=. Hfltbluèfs `så j`st` n` óntbe` hfnuel` ag M.

Hu`laf j`yèfs `h`màf hfl n` prbegr` n` ag @, p`sèfs n` sbiubgltg n` y rgpgtbefs gn ebsef prfhgsf. @så j`st` tgrebl`r hfl tfa`s n`s n`s, fmtgl-argefs gltflhgs lugstr` e`trbz rgsuntàf. Fmsgrvgefs ul gcgepnf3

I • A 5= 8

0 7 •

0 7 8

1 = =

Hfeprfmèfs qug gn lóegrf ag hfnuel`s ag n` e`trbz I (= hfnuel`s) hfblhbaghfl gn lóegrf ag n`s ag n` e`trbz A (= n`s), pfr t`ltf, sfl euntbpnbh`mngs.

0. Euntbpnbhèfs n` prbegr` dbn` ag I pfr n` prbegr` hfnuel` ag A (gngegltfpfr gngegltf) y sueèfs nfs rgsuntàfs, hfl gstf fmtglargefs gn gngegltfh00 ag n` lugv` e`trbz.

I • A 5= 8

0 7 • 0 7 8

1 = = 5

(= • 0) + (8 • 1)

=. Xgiubefs gn prfhgabebgltf euntbpnbh`laf n` prbegr` dbn` ag I hfl n` sg-iula` hfnuel` ag A. Hfl gstf fmtglgefs gn gngegltf h0= ag n` e`trbzrgsuntàf.

= 8

0 7 •

0 7 8

1 = = 5

(= • 0) + (8 • 1) (= • 7) + (8 • =)

8. Euntbpnbhèfs, ag lugvf, n` prbegr` dbn` ag I jfr` hfl n` tgrhgr` hfnuelàg A, n` óntbe`. @n fmtglgr h08 hfepngtèfs n` prbegr` dbn` ag n` lugv` e`trbz.

= 8

0 7 •

0 7 8

1 = = 5

(= • 0) + (8 • 1) (= • 7) + (8 • =) (= • 8) + (8 • =)

@ • M 5 H@e Ù l • Ml Ù p

5




1. @jfr` p`sèfs ` n` sgiula` dbn` ag I, qug n` euntbpnbhèfs pfr n` prbegr`hfnuel` ag A, fmtglbglaf gn gngegltf h=0.

= 8

0 7 •

0 7 8

1 = = 5

(= • 0) + (8 • 1) (= • 7) + (8 • =) (= • 8) + (8 • =)

(0 • 0) + (7 • 1)

7. Euntbpnbhèfs n` sgiula` dbn` ag I pfr n` sgiula` hfnuel` ag A. Fmtglg-efs gn gngegltf h==.

= 8

0 7 •

0 7 8

1 = = 5

(= • 0) + (8 • 1) (= • 7) + (8 • =) (= • 8) + (8 • =)

(0 • 0) + (7 • 1) (0 • 7) + (7 • =)

4. ¡R h`sb tgreblèfs! Euntbpnbhèfs n` sgiula` dbn` ag I pfr n` tgrhgr`hfnuel` ag A, n` óntbe`. Fmtglgefs gn gngegltf h=8.

= 8

0 7 • 0 7 8

1 = = 5

(= • 0) + (8 • 1) (= • 7) + (8 • =) (= • 8) + (8 • =)

(0 • 0) + (7 • 1) (0 • 7) + (7 • =) (0 • 8) + (7 • =)

9. [`r` dbl`nbz`r, fpgrèfs n`s euntbpnbh`hbflgs y n`s sue`s. Hfl gstf nngièfs` lugstr` e`trbz rgsuntàf.

(= • 0) + (8 • 1) (= • 7) + (8 • =) (= • 8) + (8 • =)

(0 • 0) + (7 • 1) (0 • 7) + (7 • =) (0 • 8) + (7 • =) 5

01 04 0=

=0 07 08

I • A 5= 8

0 7 •

0 7 8

1 = = 5

01 04 0=

=0 07 08

Tgèfs ftrf gcgepnf, `jfr` hfl afs e`trbhgs ag fragl 8. Gs gx`htègltg gnebsef prfhgabebgltf3

D • ] 5; ‘= 0

8 1 ‘0

= ; =

•1 8 7

7 ‘7 ;

8 1 0

0. Euntbpnbhèfs n` prbegr` dbn` ag D pfr hà` ul` ag n`s hfnuel`s ag ] ysueèfs sus prfauhtfs. Hfl gstf fmtglargefs n` prbegr` dbn` ag n` lugvè`trbz.

D • ] 5; ‘= 0

8 1 ‘0

= ; =

•1 8 7

7 ‘7 ;

8 1 0

5

(; • 1) + (‘= • 7) + (0 • 8) (; • 8) + (‘= • ‘7) + (0 • 1) (; • 7) + (‘= • ;) + (0 • 0)

=. Euntbpnbhèfs n` sgiula` dbn` ag D pfr hà` ul` ag n`s hfnuel`s ag ] ysueèfs sus prfauhtfs. ]g`nbz`laf nf `ltgrbfr, fmtglargefs n` sgiula`dbn` ag n` lugv` e`trbz.

D • ] 5; ‘= 0

8 1 ‘0

= ; =

•1 8 7

7 ‘7 ;

8 1 0

5

(; • 1) + (‘= • 7) + (0 • 8) (; • 8) + (‘= • ‘7) + (0 • 1) (; • 7) + (‘= • ;) + (0 • 0)

(8 • 1) + (1 • 7) + (‘0 • 8) (8 • 8) + (1 • ‘7) + (‘0 • 1) (8 • 7) + (1 • ;) + (‘0 • 0)

2=BIG] ∝ [fnfhjbh



8. Fpgrèfs n` euntbpnbh`hbøl ag n` tgrhgr` y óntbe` dbn` ag D pfr hà` ulàg n`s hfnuel`s ag ] y sueèfs sus prfauhtfs. Hfl gstf fmtglaråèfs n`óntbe` dbn` ag n` e`trbz rgsuntàf.

D • ] 5; ‘= 0

8 1 ‘0

= ; =

•1 8 7

7 ‘7 ;

8 1 0

5

(; • 1) + (‘= • 7) + (0 • 8) (; • 8) + (‘= • ‘7) + (0 • 1) (; • 7) + (‘= • ;) + (0 • 0)

(8 • 1) + (1 • 7) + (‘0 • 8) (8 • 8) + (1 • ‘7) + (‘0 • 1) (8 • 7) + (1 • ;) + (‘0 • 0)

(= • 1) + (; • 7) + (= • 8) (= • 8) + (; • ‘7) + (= • 1) (= • 7) + (; • ;) + (= • 0)

1. [`r` dbl`nbz`r, rgsfnvgefs n`s fpgr`hbflgs qug quga`l y nngièfs ` lugstrfrgsuntàf dbl`n.

(; • 1) + (‘= • 7) + (0 • 8) (; • 8) + (‘= • ‘7) + (0 • 1) (; • 7) + (‘= • ;) + (0 • 0)

(8 • 1) + (1 • 7) + (‘0 • 8) (8 • 8) + (1 • ‘7) + (‘0 • 1) (8 • 7) + (1 • ;) + (‘0 • 0)

(= • 1) + (; • 7) + (= • 8) (= • 8) + (; • ‘7) + (= • 1) (= • 7) + (; • ;) + (= • 0)

5 ‘9 01 0

=2 ‘07 01

01 01 0=

D • ] 5

; ‘= 0

8 1 ‘0

= ; = •

1 8 7

7 ‘7 ;

8 1 0 5

‘9 01 0

=2 ‘07 01

01 01 0=

¡Wl hflsgcf!

Xb `n blbhbf sg ng abhunt` umbh`rsg gl n`s n`s y hfnuel`s qug agmg fpgr`r p`r`h`nhun`r `niól gngegltf ag n` e`trbz rgsuntàf, pugag utbnbz`r n` tâhlbh` qug ngefstr`rgefs ` hfltblu`hbøl. Fmsgrvg gn gcgepnf.

@ • M 57 4

= 8

4 =

•7 4 2

0 = ‘=

Nf qug j`hgefs gs dfre`r 8 t`mn`s, ul` p`r`n` e`trbz @, ftr` p`r` n` e`trbz M y ul` p`r` nè`trbz rgsuntàf (H). N`s dfreèfs hfl n`s n`sy hfnuel`s nblgà`s, t`n hfef sg eugstr` gl nìur` ag n` agrghj`.

[fr gcgepnf, sb qugrgefs j`nn`r gn gngegltf H00,agmgefs fpgr`r n` n` y hfnuel` qug sg nblg`lhfl gst` h`sbnn`. Gl gstg h`sf sgrå` n` prbegr` nàg @ y n` prbegr` hfnuel` ag M.

Gst` iur` lfs `yua` ` iub`rlfs sfmrg quâ hfnuel`s y quâ n`s agmgefs fpg-r`r p`r` fmtglgr ul agtgreblàf gngegltf ag H. Hfl n` pràhtbh`, sg vfnvgrà eàsàibn y y` lf lghgsbt`rà iub`rsg ag gst` dfre`.

7 4 2

0 = ‘=

M

H

@7 4

= 8

4 =

7 4 2

0 = ‘=

M

H

@80 1= 88

08 0: 0=

8= 1; 7;

7 4

= 8

4 =




Gcgrhbhbf =@jfr` ng tfh` pr`htbh`r ` ustga. ]gsugnv` n`s euntbpnbh`hbflgs sbiubgltgs. Wtbnbhg jfc`s `p`rtg y gshrbm`su rgspugst` gl gn gsp`hbf hfrrgspflabgltg. Qr`m`cg gl fragl y nbepbgz`, su frbgltàfr pfarå` pgabrnghflst`lhb` ag sus prfhgabebgltfs.

0) 8 = ‘0 ;

• 8 70 7

=)= ‘8 1 •

‘1 ;0 =

‘= 0

8) ; 0

‘= 0

•

7 ‘0

0 0

1) ; 0

= 1

8 ‘=

•

1 ‘0

‘8 8

7) 8 1

4 ‘=

‘8 ;

• ‘1 ‘=

; 8

4)= 8 ‘=

‘7 ; ‘0

•

= ‘0

‘0 0

‘1 8

9) ; 0 7

= 4 0

; 8 ‘0

•

‘0 = 9

0 1 0

= ‘= 2

:) 8 ‘7 0

7 ; ‘0

8 ‘= 0

•

8 ‘7 0

1 ; 4

0 ‘1 =

2) 0 0 0

= ; 0

8 1 ‘0

•

‘= 8 ‘1

; 1 ‘7

9 ; ‘1

0;) = ‘= :

0 ; 9

; = ;

•

0 4 ‘8

‘= ‘0 ;

9 = 0

21BIG] ∝ [fnfhjbh



]gsuegl

0. N` euntbpnbh`hbøl f prfauhtf ag e`trbhgs gs ul` fpgr`hbøl qug pugag gdghtu`rsg gltrg afse`trbhgs f mbgl gltrg ul` e`trbz y ul lóegrf rg`n f gsh`n`r. [`r` rgsfnvgr gstg prfhgabebgltfagmgefs iub`rlfs pfr agtgreblà`s rgin`s.

0.0 [`r` euntbpnbh`r ul` e`trbz pfr ul lóegrf rg`n sg agmg euntbpnbh`r gsg lóegrf pfr hàùlf ag nfs gngegltfs ag n` e`trbz. Xg pugag euntbpnbh`r ul` e`trbz ag hu`nqubgr fragl.

o @ 5 o ` m h

a g d

i j b

5ò mo ho

ao go do

io jo bo

0.= [`r` euntbpnbh`r afs e`trbhgs gltrg så , lfs `pfyèfs gl gn prfhgabebgltf sbiubgltg3

• Xg hfeprugm` qug gn lóegrf ag hfnuel`s ag n` prbegr` e`trbz (@) hfblhba` hfl gn ló-egrf ag dbn`s ag n` sgiula` (M). Ag nf hfltr`rbf, lf sg pugagl fpgr`r.

• Xg euntbpnbh` n` prbegr` dbn` ag @ pfr n` prbegr` hfnuel` ag M. [rbegr gngegltf pfrprbegrf, sgiulaf pfr sgiulaf, gth. y sueèfs nfs prfauhtfs. Hfl gstf hflsgiubefs gnprbegr gngegltf ag n` e`trbz rgsuntàf.

• Xgiubefs euntbpnbh`laf n` prbegr` dbn` ag @ pfr n`s sbiubgltgs hfnuel`s ag M.

• ]gpgtbefs gn prfhgabebgltf hfl n`s sbiubgltgs dbn`s ag @ y hà` hfnuel` ag M.

• Nugif ag tgrebl`r hfl tfa`s n`s dbn`s ag @, nngièfs `n rgsuntàf dbl`n.

H 5 @ • M 5` m h

a g d

i j b

•r s t

u v w

x y z

5`r + mu + hx `s + mv + hy `t + mw + hz



Ng `lbeèfs ` qug fmsgrvg nfs vbagfs sbiubgltgs sfmrg n` euntbpnbh`hbøl ag e`trbhgs, gstfs ng pugagl

sgrvbr p`r` `hn`r`r `niul` aua`. ]ghugrag qug sb ng quga` `niul` blqubgtua, sbgeprg pugag hflsunt`rn`hfl su frbgltàfr gl gn hårhunf ag gstuabf.

• Euntbpnbh`hbøl ag ul` e`trbz pfr ul lóegrf rg`n3 iff.in/vZxT2]

• Euntbpnbh`hbøl ag e`trbhgs3 iff.in/7rp;vh iff.in/H=Ca7S

Q embâl ng prgsgltèfs `niulfs gcgrhbhbfs gxtr` p`r` qug pr`htbqug nf qug j` `prglabaf gst` sge`l`3

• [rfauhtf ag e`trbhgs ag fragl 83 iff.in/jxoy;X






Gshrbm` "d`nsf" f "vgraàgrf" sgiól hfrrgspfla` gl hà` glulhbàf.

0) Wl lóegrf rg`n pugag sgr euntbpnbhàf pfr ul` e`trbz ag hu`nqubgr fragl.

=) Xb ul` e`trbz @ tbglg = hfnuel`s y ul` e`trbz M tbglg = dbn`s, sg pugagl euntbpnbh`r.

8) Xb ul` e`trbz ] tbglg 8 hfnuel`s y ul` e`trbz E tbglg 1 dbn`s, sg pugagl euntbpnbh`r.

1) Xb ul` e`trbz Q tbglg = hfnuel`s y ul` e`trbz J tbglg = dbn`s, lf sg pugagl euntbpnbh`r.


@. ]gsugnv` n`s euntbpnbh`hbflgs sbiubgltgs. Qr`m`cg gl fragl.

0) ‘=

8 =

‘1 0

=) 7 0 = ‘7 1

8) ‘8

; 0

= 1

8 ‘=

1)9 ‘9 8 ; =

0 1 ‘= 8

7)1

‘0 ; =

1 ‘7 4

0 ; 8

4) ‘:

7 ; 0

‘: = ;

0 0 2

@utfhfltrfn

24BIG] ∝ [fnfhjbh



M. ]gsugnv` n`s euntbpnbh`hbflgs sbiubgltgs. Qr`m`cg gl fragl, pugag utbnbz`r jfc`s `p`rtg sb nf lghgsbt`]ghugrag qug su frbgltàfr pugag sfnbhbt`rng hflst`lhb` ag sus prfhgabebgltfs.

0) 8 =

‘0 ;

•

8 7

0 7

=) ‘= :

8 ;

•

0 =

; ‘=

8) ‘1 ; = •

; =

0 ‘8

8 ‘0

1) ; ‘0

1 :

•

0 ‘4

7 7

7) 8 ‘=

; ‘0

1 8

•

8 0 ‘=

; 0 =

4)8 7 ‘0

= 0 0

•

8 9

‘0 0

; ‘8

9) = 8 ‘=

; 0 ;

9 ‘1 ;

•

= 0

; ‘1

; =

:) ‘: ; 2

0 7 ‘9

8 8 =

•

1 = ‘=

0 0 0

0 8 ;

2) = ‘0 0

8 9 1

0 ‘= 0

•

; 9 =

‘0 0 =

= 8 ‘7

0;) 2 ‘0 0

‘1 : =

; ; 8

•

; ; 0

2 ‘9 :

‘= 8 ‘0




@. ]gsugnv` n`s ghu`hbflgs nblg`ngs sbiubgltgs. Gshrbm` gn v`nfr ag n` blhøilbt` gl n` nålg`hfrrgspflabgltg.

0) x + 8 5 2 x 5

=) x ‘ : 5 7 x 5

8) x + 9 5 07 x 5

1) x + 0; 5 ‘= x 5

7) x ‘ : 5 00 x 5

4) x ‘ 9 5 ; x 5

9) x ‘ ; 5 0 x 5

:) x + 2 5 2 x 5

2) x + 1 5 00 x 5

0;) x + : 5 04 x 5

00) : ‘ x 5 = x 5

0=) 07 ‘ x 5 2 x 5

08) 0; + x 5 0= x 5

01) ‘= + x 5 = x 5

07) 0 + x 5 7 x 5

04) ‘4 ‘ x 5 ‘8 x 5

09) ; ‘ x 5 0 x 5

0:) 0; + x 5 0; x 5

02) = ‘ x 5 ; x 5

=;) ‘: ‘ x 5 1 x 5

=0) 1 + x 5 0 x 5

==) 2 ‘ x 5 ‘9 x 5

M. ]gsugnv` n`s ghu`hbflgs nblg`ngs sbiubgltgs. Gshrbm` gn v`nfr ag n` blhøilbt` gl n` nålg`hfrrgspflabgltg.

0) : y 5 04 y 5

=) 1 y 5 =; y 5

8) ‘2 y 5 71 y 5

1) 8 y 5 0= y 5

7) 7 y 5 ‘=7 y 5

4) ‘= y 5 = y 5

9) 9 y 5 87 y 5

:) ‘1 5 =; y 5

2) 00 y 5 ‘22 y 5

0;) ‘: y 5 ‘1; y 5

00) 4 y 5 0: y 5

0=) ‘9 y 5 12 y 5

08) : y 5 41 y 5

01) ‘= y 5 ‘: y 5

07) ‘ y 5 ‘0 y 5

04) 0; y 5 ; y 5

09) = y 5 ‘0= y 5

0:) ‘8 y 5 ‘2 y 5

02) 8 y 5 ‘2 y 5

=;) 9 y 5 ‘74 y 5

=0) ‘4 y 5 1: y 5

==) 7 y 5 87 y 5

2:BIG] ∝ [fnfhjbh




@ hfltblu`hbøl glhfltr`rà flhg åtges. Gl hà` ulf p`rghgl hu`trf iur`s, trgs ag gnn`s iu`ra`l ul`rgn`hbøl ag niól tbpf. Wstga agmg bagltbh`r n` iur` qug lf gstà rgn`hbflà` y hbrhun`rn`. Iuågsg pfrgn gcgepnf. Gst` prugm` ng `yua`rà ` ags`rrfnn`r su h`p`hbaà ag `lànbsbs y r`zflèbgltf.

;) `. m. h. a.

0) `. m. h. a.

=) `. m. h. a.

8) `. m. h. a.

1) `. m. h. a.

7) `. m. h. a.

4) `. m. h. a.

9) `. m. h. a.

:) `. m. h. a.

2) `. m. h. a.

0;) `. m. h. a.






prfhgsflf

nfiràf

A g s p u â s a g g s t u a b ` r . . . Hflfzhf y pnbhf n`s rgin`s p`r` euntbpnbh`r ul` e`trbz pfr ul lóegrf

rg`n.

[r`htbhf n` euntbpnbh`hbøl ag ul` e`trbz pfr ul lóegrf rg`n.

Hflfzhf y `pnbhf n`s rgin`s p`r` euntbpnbh`r e`trbhgs gltrg så.

[r`htbhf n` euntbpnbh`hbøl gltrg e`trbhgs.

[`r` gn ags`rrfnnf agn r`zflèbgltf `mstr`htf y nøibhf sg gepng`l n`s prugm`shfl e`trbhgs, pugs rgqubgrgl gn hfrrghtf e`lgcf g bagltbh`hbøl ag nfs gngegl-tfs qug dfre`l n` e`trbz y gn `lànbsbs p`r` agshumrbr gn gngegltf qug sbiug gnp`trøl.

@ hfltblu`hbøl, ustga glhfltr`rà hu`trf blhbsfs hfl e`trbhgs ag 8 Ù 8? agmgràgngibr ag gltrg n`s fphbflgs gn gngegltf qug j`hg d`nt` gl n` h`sbnn` v`hå` yhbrhun`rnf.

0)0 1 0

8 0 8

1 0

=)

8)

1)9 04

: 7 08

1 1 :

`. 0 h. 7

m. 1 a. =

`. h.

m. a.

`. h.

m. a.

`. 4 h. 8

m. =0 a. 2

0;;BIG] ∝ [fnfhjbh





90

= 8 1

8

1

7

=

0

;

x

y

‘ 0 5 b

x = +

y = 5

r =

Xuhgsbflgs e`tgeàtbh`s¶Puâ glhfltr`rà gst` sge`l`6

N` suhgsbøl ag Dbmfl`hhb

Xuhgsbflgs `rbteâtbh`s y igfeâtrbh`s

Ghu`hbflgs ag prbegr iràf

Xgrbgs lueârbh`s


Hfeprglagr n`s h`r`htgråstbh`s ag ul` suhgsbøl `rbteâtbh`.

H`nhun`r gn târeblf d`nt`ltg gl ul` suhgsbøl `rbteâtbh`.

Hfeprglagr n`s h`r`htgråstbh`s ag ul` suhgsbøl igfeâtrbh`. H`nhun`r gn târeblf d`nt`ltg gl ul` suhgsbøl igfeâtrbh`.

]gsfnvgr prfmnge`s gepng`laf n`s suhgsbflgs e`tgeàtbh`s.

]gsfnvgr ghu`hbflgs ag prbegr iràf hfl ìbnbaà.

0;0E`tgeàtbh` ∝ Xge`l` 9



¡[`r` hfeglz`r!N` suhgsbøl ag Dbmfl`hhb

Ngfl`raf ag [bs`, tèmbâl hflfhbaf hfef Dbmfl`hhb, dug ul agst`hàf e`tg-

eàtbhf bt`nb`lf qug vbvbø gl gn sbinf ZBBB. Gltrg sus hfltrbmuhbflgs `n hèpf ag

nfs lóegrfs sg glhugltr` n` suhgsbøl ag Dbmfl`hhb, ul` sghuglhb` ag lóegrfs

qug blbhb` gl 0 y sg gxtbglag j`st` gn bldblbtf. Tgèfs `niulfs ag sus prbegrfs

gngegltfs3

0, 0, =, 8, 7, :, 08, =0, 81, 77, :2, 011, =88...

Hfef eglhbflèfs, n` suhgsbøl hfebglz` hfl nfs lóegrfs 0 y 0, y ` p`rtbr ag

gstfs, hà` târeblf gs n` sue` ag nfs afs `ltgrbfrgs.

0 + 0 5 =, 0 + = 5 8, = + 8 5 7, 8 + 7 5 :, 7 + : 5 08,

: + 08 5 =0, 08 + =0 5 81 ...

Nf sfrprglagltg ag gst` hflstruhhbøl e`tgeàtbh` gs qug sg prgsglt` rghurrgl-

tgegltg gl n` l`tur`ngz`. @p`rghg gl n` abstrbmuhbøl ag n`s rè`s ag nfs àrmfngs,

gl n` dfre`hbøl ag n`s jfc`s ag ul t`nnf, gl n` rgprfauhhbøl ag nfs hflgcfs, gl

nfs drutfs ag n` pbð` trfpbh`n, gl n` abspfsbhbøl ag n`s sgebnn`s gl luegrfs`s

dnfrgs, gltrg euhjfs ftrfs h`sfs.

[fr gcgepnf, n`s jfc`s ag ul` pn`lt` lulh` hrghgl abrghtègltg ul` `m`cf ag

ftr`, mrft`l gl dfre` ag gspbr`n hfl ul` hfldbiur`hbøl qug sbiug n` sghuglhb`

Dbmfl`hhb.

Ftrf gcgepnf gs gn p`trøl gl qug sg rgprfauhgl nfs hflgcfs, ul` p`rgc` aggstfs `lbe`ngs t`ra` ul egs gl nngi`r ` su gaà dârtbn, ` p`rtbr ag gsg efegltf

hà` egs gliglar`rà ` ftr` p`rgc` ag hflgcfs, qug ` su vgz, nugif ag nngi`r `

n` gaà ag dgrtbnbaà, tglarà ftr` p`rgc` eàs, y `så suhgsbvègltg.

¡@ tr`m`c`r!

Gepng`laf gn bltgrlgt f `niól ftrf rghursf, blvgstbiug ftrf nfirf agn e`tgeàtbhf Ngfl`raf ag [bs`.Gshrbm` sus j`nn`zifs gl n`s nålg`s sbiubgltgs3

0;=BIG] ∝ [fnfhjbh



0. Xuhgsbflgs e`tgeàtbh`s@så hfef puabefs lft`r gl n` sghhbøl ltgrbfr, ul` suhgsbøl e`tgeàtbh` gs ul

hflcultf fraglàf ag lóegrfs. Hà` ulf ag gstfs lóegrfs sg aglfebl` "târ-eblf", "gngegltf" f "ebgemrf", y `n lóegrf tft`n ag gngegltfs qug hfepflgn` suhgsbøl sg ng nnè` "nflibtua", gst` pugag sgr lbt` f bllbt`.

0;

0.³ târeblf

=;

=.³ târeblf

8;

8.³ târeblf

1;

1.³ târeblf

7;

7.³ târeblf

J`y v`rbfs tbpfs ag suhgsbflgs e`tgeàtbh`s, qug sbiugl abdgrgltgs p`trflgs,lfsftrfs gstuab`rgefs afs3 n`s suhgsbflgs `rbteâtbh`s y n`s suhgsbflgsigfeâtrbh`s.

0.0 Xuhgsbflgs `rbteâtbh`sWl` suhgsbøl f prfirgsbøl rbteâtbh` gs ul` sgrbg ag lóegrfs t`ngs qug hàùlf ag gnnfs, gxhgptu`laf gn prbegr târeblf, sg fmtbglg sue`laf f rgst`laful` h`ltbaà c` `n târeblf `ltgrbfr, ` gst` h`ltbaà hflst`ltg sg ng hflfhghfef abdgrglhb` y sg rgprgsglt` hfl n` n` ngtr` a .

`0 `= `8 `1 `7 `l...

°a °a °a °a ...

Gl aflag n` ngtr` "`" rgprgsglt` hà` târeblf y su sumålabhg blabh` n` pfsbhbøl

qug fhup` gl n` prfirgsbøl. Fmsgrvgefs gn gcgepnf sbiubgltg3

1 : 0= 04 =; =1

+1 +1 +1 +1 +1

[`r` h`nhun`r n` abdgrglhb` agmgefs rgst`r afs târeblfs hflsghutbvfs. Agpgl-abglaf sb n` abdgrglhb` (a ) ag ul` prfirgsbøl `rbteâtbh` sg` pfsbtbv , lun` flgi`tbv`, aghbefs3

• a < ;3 prfirgsbøl hrghbgltg. Hà` gngegltf gs e`yfr qug gn `ltgrbfr.

Gcgepnf3 0;, =;, 8;, 1;... (a 5 0;)

• a 5 ;3 prfirgsbøl hflst`ltg. Qfafs nfs târeblfs sfl biu`ngs.

Gcgepnf3 2, 2, 2, 2, 2... (a 5 ;)

• a > ;3 prfirgsbøl aghrghbgltg. Hà` gngegltf gs eglfr qug gn `ltgrbfr.

Gcgepnf3 0;, :, 4, 1, =, ;, ‘=, ‘1, ‘4... (a 5 ‘=)





Qâreblf iglgr`n ag ul` suhgsbøl rbteâtbh`

[`r` h`nhun`r gn v`nfr ag ul gngegltf agshflfhbaf gl ul` suhgsbøl `rbteâtbh`,utbnbzèfs n` døreun` agn târeblf iglgr`n, n` hu`n lfs pgrebtg `vgrbiu`rhu`nqubgr târeblf ag ul` suhgsbøl sb hflfhgefs gn prbegrf ag gnnfs y suabdgrglhb`.

N` døreun` agn târeblf iglgr`n ag ul` prfirgsbøl `rbteâtbh` gs3`l 5 `0 + (l ‘ 0)a

Gl n` hu`n `l rgprgsglt` gn târeblf qug qugrgefs `vgrbiu`r ag `hugraf ` supfsbhbøl? l gs n` pfsbhbøl qug fhup` `l y `0 gs gn prbegr târeblf ag n` suhgsbøl.

Tgèfs ul gcgepnf.

Glhfltrgefs quâ lóegrf fhup` n` pfsbhbøl 12 gl n` suhgsbøl sbiubgltg3

02, 8;, 10, 7=, 48... `12

• [`r` j`nn`r gn târeblf iglgr`n, sustbtubefs n` abdgrgl-hb` (a 5 00) y gn prbegr târeblf ( 0 5 02) gl n` døreun`qug vbefs `rrbm`. Nugif, fpgrèfs.

• Glhfltrèfs `så gn târeblf iglgr`n p`r` gst` sgrbg?hfl gst` gxprgsbøl pfagefs `vgrbiu`r hu`nqubgrtâreblf hfl t`l sfnf rgepn`z`r l pfr gn lóegrf agpfsbhbøl qug fhup`.

• Hfef qugrgefs glhfltr`r gn gngegltf `12 , j`hgefsn`s sustbtuhbflgs lghgs`rb`s y fpgrèfs.

Gn gngegltf qug fhup` n` pfsbhbøl 12 gs 719.

Hfl n` døreun` agn târeblf iglgr`n tèmbâl pfagefs h`nhun`r n` abdgrglhb` y gnprbegr târeblf ag ul` prfirgsbøl, sfnègltg agmgefs agspgc`r n` v`rb`mng quglfs bltgrgs`.

Tgèfs ul óntbef gcgepnf.

Glhfltrgefs gn prbegr târeblf ag ul` suhgsbøl huyf târeblf `:2 gs 0;9: y suabdgrglhb` gs 0=.

• Agspgcèfs n` døreun` agn târeblf iglgr`n gl

dulhbøl ag `0.

• Xustbtubefs nfs a`tfs y fpgrèfs.

Glhfltrèfs qug gn prbegr târeblf gs ==.

`l 5 `0 + (l ‘ 0)a `l 5 02 + (l ‘ 0)00

`l 5 02 + 00l ‘ 00

`l 5 : + 00l

`12 5 : + 00(12)

`12 5 719

`0 5 `l ‘ (l ‘ 0)a

`0 5 `:2 ‘ (:2 ‘ 0)0=

`0 5 0;9: ‘ 0;74

`0 5 ==

0;1BIG] ∝ [fnfhjbh



Xue` ag nfs târeblfs gl ul` prfirgsbøl `rbteâtbh`

[`r` d`hbnbt`r gn prfhgsf ag sue`r nfs gngegltfs ag ul` suhgsbøl `rbteâtbh`,gepngèfs n` døreun` sbiubgltg3

s 5 (`0 + `l )l

=

Gl aflag `0 gs gn prbegr târeblf ag n` sgrbg, l gs n` pfsbhbøl j`st` n` hu`nqugrgefs sue`r y `l gs gn v`nfr agn târeblf qug fhup` gs` pfsbhbøl.

[flièfs `tglhbøl `n gcgepnf.

Xuegefs j`st` gn târeblf `7; ; ag ul` suhgsbøl huyf prbegr gngegltf gs 07; ysu abdgrglhb` gs =7.

• [rbegrf, glhfltrèfs gn v`nfr agn târeblf`7; ;

• Nugif, `pnbhèfs y fpgrèfs n` døreun`p`r` sue`r tfafs nfs târeblfs j`st` `7; ;

N` sue` tft`n j`st` gn gngegltf 7;; ag n` suhgsbøl gs 8 028 97;.

`7; ; 5 `0 + (l ‘ 0)a

`7; ; 5 07; + (7;; ‘ 0)(=7)

`7;; 5 0= 4=7

s 5(07; + 0= 4=7)(7;;)

=

s 5 8 028 97;

Gcgrhbhbf 0@. H`nhung n` abdgrglhb` ag n`s suhgsbflgs sbiubgltgs. Nugif, gshrbm` sb sfl hrghbgltgs, hflst`ltgs f

aghrghbgltgs. Iuågsg pfr gn gcgepnf.

;) 09, 00, 7, ‘0, ‘9, ‘08, ‘02...

0) 88, 14, 72, 9=, :7, 2:...

=) =.7, =, 0.7, 0, ;.7, ;, ‘;.7, ‘0...

8) :, :, :, :, :, :, :, :...

1) 087, 002, 0;8, :9, 90...

7) 0;;, 0;9, 001, 0=0, 0=:...

4) 18, 18, 18, 18, 18...

9) 84, 8;.7, =7, 02.7, 01...

:) 0:, =1, 8;, 84, 1=, 1:...

a 5 ‘4 , gs aghrghbgltg




M. H`nhung gn târeblf qug sg ng pbag gl hà` ul` ag n`s suhgsbflgs sbiubgltgs.

0) `18 ag n` suhgsbøl3 =1, =:, 8=, 84, 1;... `18 5

=) `22 ag n` suhgsbøl3 79, 19, 89, =9, 09... `22 5

8) `0;0 ag n` suhgsbøl3 07, 0;.7, 4, 0.7, ‘8... `0;0 5

1) `0 ag ul` suhgsbøl hfl a 5 =8 y `87 5 0;1 `0 5

7) a ag ul` suhgsbøl hfl `0 5 8 y `74 5 42;.7 a 5

4) `49 ag ul` suhgsbøl hfl `0 5 81 y a 5 9 `49 5

9) a ag ul` suhgsbøl hfl `0 5 74 y `024 5 ‘07;1 a 5

:) `0 ag ul` suhgsbøl hfl a 5 ‘2.7 y `07; 5 =27 `0 5

H. H`nhung n` sue` ag nfs gngegltfs gl n`s prfirgsbflgs sbiubgltgs. [ugag tr`m`c`r gl jfc`s `p`rtg

sb gs lghgs`rbf. ]ghugrag tr`m`c`r gl fragl y agc`r hflst`lhb` ag su prfhgabebgltf.

0) Xue`tfrb` j`st` gn gngegltf `=;;, `0 5 =7 y a 5 0=

`=;; 5 s 5

=) Xue`tfrb` j`st` gn gngegltf `07;, `0 5 7 y a 5 0;

`07; 5 s 5

8) Xue`tfrb` j`st` gn gngegltf `97, `0 5 08 y a 5 07

`97 5 s 5

1) Xue`tfrb` j`st` gn gngegltf `8;;, `0 5 =1 y a 5 :

`8;; 5 s 5

7) Xue`tfrb` j`st` gn gngegltf `0=7, `0 5 8= y a 5 =

`0=7 5 s 5

4) Xue`tfrb` j`st` gn gngegltf `7;, `0 5 7 y a 5 02

`7; 5 s 5

9) Xue`tfrb` j`st` gn gngegltf `==7, `0 5 8 y a 5 8

`==7 5 s 5

:) Xue`tfrb` j`st` gn gngegltf `0;;, `0 5 0; y a 5 9

`0;; 5 s 5

0;4BIG] ∝ [fnfhjbh



0.= Xuhgsbflgs igfeâtrbh`s

Wl` suhgsbøl igfeâtrbh` gs ul` suhgsbøl gl n` qug hà` târeblf sg fmtbglgeuntbpnbh`laf `n `ltgrbfr pfr ul` h`ltbaà c` aglfeblà` r`zøl (r ).

= 4 0: 71 04= 1:4 ...

Ù8 Ù8 Ù8 Ù8 Ù8

N` r`zøl sg fmtbglg abvbabglaf afs gngegltfs hflsghutbvfs ag n` sgrbg. Gl gstgtbpf ag suhgsbøl, y` lf gs pfsbmng h`nbh`r ul` suhgsbøl igfeâtrbh` hfef hrg-hbgltg, aghrghbgltg f hflst`ltg ag `hugraf `n sbilf ag su r`zøl.

@jfr`, sb n` r`zøl gs ul` dr`hhbøl prfpb`, n` prfirgsbøl gs aghrghbgltg? sb sur`zøl tfe` gn v`nfr ag n` ulbaà, gs hflst`ltg.

Xb r gs ul lóegrf gltgrf pfsbtbvf, n` suhgsbøl sgrà hrghbgltg y sb gs ul lóegrfgltgrf lgi`tbvf, n` prfirgsbøl brà v`rb`laf ag sbilf hfl hà` gngegltf.

Qâreblf iglgr`n ag ul` suhgsbøl igfeâtrbhÀg dfre` sbebn`r qug hfl n`s prfirgsbflgs rbteâtbh`s, sg pugag fmtglgr gn v`nfrag ul gngegltf agtgreblàf ag n` sghuglhb` egab`ltg n` døreun` agn târeblfiglgr`n3

`l 5 `0 • r (l ‘ 0)

Gl aflag `l gs gn gngegltf qug sg agsg` hflfhgr, l gs n` pfsbhbøl qug fhup`gstg târeblf? `0 gs gn prbegr gngegltf y r , n` r`zøl.

Nfs p`sfs p`r` h`nhun`r ul târeblf agshflfhbaf sfl euy p`rghbafs ` nfs qugsgiubefs hfl n`s prfirgsbflgs `rbteâtbh`s.

[flièfs `tglhbøl ` ul gcgepnf.

Glhfltrgefs quâ lóegrf fhup` n` pfsbhbøl 08 gl n` suhgsbøl sbiubgltg3

=, 4, 0:, 71, 04=, 1:4...

• H`nhunèfs n` r`zøl (r ) p`r` gst` sgrbg abvb-abglaf afs târeblfs hflsghutbvfs.

• @jfr`, rgepn`zèfs gn prbegr târeblf (`0)

y n` r`zøl gl n` døreun`.Hfl gstf fmtglgefs gn târeblf iglgr`n.

• [fr óntbef, sustbtubefs l, gs aghbr, gn lóeg-rf ag pfsbhbøl qug fhup` gn târeblf qugqugrgefs j`nn`r y fpgrèfs.

Gn târeblf qug fhup` n` pfsbhbøl 08 gs 0 ;4= ::=.

4

= 5 8

r 5 8

`l 5 `0 • r (l ‘ 0)

`l 5 = • 8(l ‘ 0)

`l 5 = • 8(l ‘ 0)

`08 5 = • 8(08 ‘ 0)

`08 5 0 ;4= ::=

]ghugrag qug hfl n`døreun` agn târeblfiglgr`n tèmbâl pfag-efs h`nhun`r n` r`zøly gn prbegr târeblf agul` prfirgsbøl, sfnfm`st` hfl agspgc`rn` v`rb`mng qug lfsbltgrgs`. [fr gcgepnf3

r 5 l ‘ 0`l

`0




Gcgrhbhbf =@. H`nhung n`s r`zflgs p`r` n`s suhgsbflgs sbiubgltgs. Nugif, gshrbm` sb sfl hrghbgltgs, aghrghbgltgs f

hflst`ltgs. Iuågsg pfr gn gcgepnf.

;) 7;;, 0;;, =;, 1, ;.:...

0) =, :, 8=, 0=:, 70=, =;1:...

=) 8, 2, =9, :0, =18, 9=2...

8) 0;;;, 7;;, =7;, 0=7...

1) 0, 7, =7, 0=7, 4=7, 80=7...

7) 08, 08, 08, 08, 08, 08...

4) 0;;;, =;;, 1;, :...

9) :, 1:, =::, 09=:...

:) 7;, 7, ;.7, ;.;7, ;.;;7...

Xue` ag nfs târeblfs gl ul` prfirgsbøl igfeâtrbh`

@n biu`n qug hfef `prglabefs hfl n`s suhgsbflgs `rbteâtbh`s, sb agsgèfssue`r nfs târeblfs ag ul` prfirgsbøl igfeâtrbh`, pfagefs d`hbnbt`r lugstrftr`m`cf gepng`laf n` døreun` sbiubgltg3

s 5 (`l • r ) ‘ `0

r ‘ 0

Gl aflag `l gs gn v`nfr agn târeblf qug fhup` n` pfsbhbøl j`st` aflag qugrg-efs sue`r? r gs n` r`zøl igfeâtrbh` y `0

gs gn prbegr târeblf.

Hfeprglagrgefs egcfr hfl ul gcgepnf.

Xuegefs j`st` gn târeblf `07 ag ul` suhgsbøl huyf prbegr gngegltf gs 8 y sur`zøl gs =.

• [rbegrf agmgefs glhfltr`r gn v`nfr agngngegltf `07

• @jfr`, `pnbhèfs y fpgrèfs n` døreun`p`r` sue`r tfafs nfs gngegltfs j`st` `07

N` sue`tfrb` ag nfs gngegltfs j`st` `07 gs 2: 8;0. ¶R` vg qug gs euy dàhbn6

`07 5 `0 • r (l ‘ 0)

`07 5 8 • =(07 ‘ 0)

`07 5 12 07=

s 5 (`07 • r ) ‘ `0

r ‘ 0

s 5(12 07= • =) ‘ 8

= ‘ 0

s 5 2: 8;0

r 5 0

7 , aghrghbgltg

0;:BIG] ∝ [fnfhjbh



M. H`nhung gn târeblf qug sg ng pbag gl hà` ul` ag n`s suhgsbflgs sbiubgltgs. [ugag `prfxbe`r susrgspugst`s sgiól hrg` hflvglbgltg.

0) `0; ag n` suhgsbøl3 =, 1, :, 04, 8=... `0; 5

=) `07 ag n` suhgsbøl3 0, 8, 2, =9, :0... `07 5

8) `9 ag n` suhgsbøl3 7;;, 0;;;, =;;;... `9 5

1) `7 ag n` suhgsbøl3 2;, 8;, 0;... `7 5

7) `08 ag n` suhgsbøl3 8, ‘2, =9, ‘:0, =18... `08 5

4) `0 ag ul` suhgsbøl hfl r 5 7 y `9 5 0;2 897 `0 5

9) N` r`zøl ag ul` suhgsbøl hfl `0 5 1 y `7 5 8=1 r 5

:) N` r`zøl ag ul` suhgsbøl hfl `0 5 : y `7 5 0; 84: r 5

2) `9 ag n` suhgsbøl3 9, ‘74, 11:, ‘87:1... `9 5

0;) `0 ag ul` suhgsbøl hfl r 5 4 y `: 5 0 822 4:; `0 5

H. H`nhung n` sue` ag nfs gngegltfs gl n`s prfirgsbflgs sbiubgltgs. [ugag tr`m`c`r gl jfc`s `p`rtgsb gs lghgs`rbf. ]ghugrag tr`m`c`r gl fragl y agc`r hflst`lhb` ag su prfhgabebgltf.

0) Xue`tfrb` j`st` gn gngegltf `2, `0 5 9 y r 5 8

`2 5 s 5

=) Xue`tfrb` j`st` gn gngegltf `:, `0 5 0;; ;;; y r 5 0

=

`: 5 s 5

8) Xue`tfrb` j`st` gn gngegltf `07, `0 5 2 y r 5 =

`07 5 s 5

1) Xue`tfrb` j`st` gn gngegltf `7, `0 5 =7 y r 5 9

`7 5 s 5

7) Xue`tfrb` j`st` gn gngegltf `4, `0 5 9 y r 5 :

`4 5 s 5

4) Xue`tfrb` j`st` gn gngegltf `0=, `0 5 1 y r 5 8

`0= 5 s 5




@pnbh`hbflgs ag n`s suhgsbflgs `rbteâtbh`s y igfeâtrbh`s

N`s àrg`s gl n`s qug pfagefs `pnbh`r lugstrfs hflfhbebgltfs sfmrg suhgsbflgse`tgeàtbh` sfl euy v`rbà`s. [fr gcgepnf3

Gl n` m`lh`, nfs blvgrsbflbst`s y n`s pgrsfl`s qug tbglgl huglt`s m`lh`rb`s,pugagl h`nhun`r n`s i`l`lhb`s pftglhb`ngs sfmrg ul lgifhbf gepng`laf n`s

prfirgsbflgs `rbteâtbh`s.Gl nfs agpfrtgs, gl abvgrs`s abshbpnbl`s, hfef gn tglbs, sg gepng`l n`s suhgsbflgsigfeâtrbh`s y `rbteâtbh`s p`r` `l`nbz`r n`s pfsbmngs gstr`tgib`s qug ul cuiàfrpugag tfe`r gl ul p`rtbaf.

Gl n` bliglbgrå`, nfs lóhngfs ag ul e`tgrb`n ràb`htbvf sg agsbltgir`l gl dfreùlbdfreg gl ul agtgreblàf pgråfaf ag tbgepf, gstg hànhunf pugag sgr agshrbtfpfr ul` prfirgsbøl igfeâtrbh`.

[fr t`ltf, pfagefs rgsfnvgr euhjfs prfmnge`s hfl t`l sfnf pfagr bagltbh`rhfrrghtègltg nfs a`tfs ag n` prfirgsbøl y pnbh`r n`s døreun`s qug pr`htbhèfs.

Gcgrhbhbf 8]gsugnv` nfs prfmnge`s sbiubgltgs gepng`laf nf prglabaf sfmrg prfirgsbflgs rbteâtbh`s y igfeâ-trbh`s. Ng `hflsgcèfs prbegrf bagltbh`r y hn`sbh`r nfs a`tfs agn prfmnge`.

0) N` afsbs ag ul egabhègltf gs ag 7;; ei gn prbegr aå` y 07 ei eglfs hà` ulf ag nfs aå`ssbiubgltgs.

`. Xb gn tr`tèbgltf aur` =7 aå`s, ¶huàltfs ebnbirèfs tfe`rà gn gldgref gl gn óntbef aå`6

m. ¶Huàltfs ebnbirèfs j`mrà tfeàf gn gldgref ` nf n`rif ag tfaf gn tr`tèbgltf6

=) Wl tbpf ag m`htgrb` sg rgprfauhg aupnbhàlafsg hà` egab` jfr`. Xb gl n` prbegr` egab` jfr` sgfmsgrv`l 0;;; m`htgrb`s, ¶huàlt`s j`mràl nugif ag 7 jfr`s6

8) Nfs vghblfs ag ul` hfeulbaà aghbagl `jfrr`r p`r` hflstrubr ul s`nøl ag usfs eóntbpngs. Gnprbegr aå` agpfsbt`l P07;.;; gl ul` `nh`lhå` y hà` aå` aupnbh`l n` h`ltbaà agpfsbtà` gn aå``ltgrbfr. ¶Huàltf ablgrf tglaràl qug agpfsbt`r gn sâptbef aå`6

1) Xb hfnfhàrèfs P1,;;;.;; gl ul` huglt` ag `jfrrf gl n` qug i`l`råèfs P1.7; pfr hà` aå` qugp`sg gn ablgrf agpfsbtàf, ¶huàltf ablgrf j`mrå` nugif ag 0:; aå`s6

7) N` pfmn`hbøl ag pfnnfs ag ul` ir`lc` sg trbpnbh` hà` `ðf. Xb agspuâs ag 7 ðfs j`y 1784 lbe`ngs,¶huàltfs j`må` gl gn prbegr `ðf6

4) Xb gn nulgs ustga i`lø P77.;; y gl nfs aå`s sbiubgltgs i`lø gn afmng agn `ltgrbfr.

`. ¶Huàltf i`lø gn sàmàf6

m. ¶Huàltf i`lø gl tft`n ag nulgs ` sàmàf6

00;BIG] ∝ [fnfhjbh



]gsuegl

0. Wl` suhgsbøl e`tgeàtbh` gs ul hflcultf fraglàf ag lóegrfs. Hà` ulf ag gstfs lóegrfssg aglfebl` "târeblf", "gngegltf" f "ebgemrf", y `n lóegrf tft`n ag gngegltfs qug hfepflgn` suhgsbøl sg ng nnè` „nflibtua―, gst` pugag sgr dblbt` f bldblbt`.

0.0 Wl` suhgsbøl f prfirgsbøl `rbteâtbh` gs ul` sgrbg ag lóegrfs t`ngs qug hà` ulf ag gnnfs,gxhgptu`laf gn prbegr târeblf, sg fmtbglg sue`laf f rgst`laf ul` h`ltbaà dbc` `n târeblf`ltgrbfr, ` gst` h`ltbaà hflst`ltg sg ng hflfhg hfef abdgrglhb` y sg rgprgsglt` hfl n` n` ngtr` a .

08 =1 87 14 79 4: ...

+00 +00 +00 +00 +00

N` døreun` agn târeblf iglgr`n p`r` ul` suhgsbøl `rbteâtbh` lfs pgrebtg `vgrbiu`r hu`n-qubgr târeblf ag ul` suhgsbøl sb hflfhgefs gn prbegrf ag gnnfs y su abdgrglhb`.

`l 5 `0 + (l ‘ 0)a

[`r` d`hbnbt`r gn prfhgsf ag sue`r nfs gngegltfs ag ul` suhgsbøl `rbteâtbh`, gepngèfs n`døreun` sbiubgltg3

s 5 (`0 + `l )l

=

0.= Wl` suhgsbøl igfeâtrbh` gs ul` suhgsbøl gl n` qug hà` târeblf sg fmtbglg euntbpnbh`laf `n `ltgrbfr pfr ul` h`ltbaà dbc` aglfeblà` r`zøl (r ).

: =1 9= =04 41: 0211 ...

Ù8 Ù8 Ù8 Ù8 Ù8

N` døreun` agn târeblf iglgr`n p`r` ul` suhgsbøl igfeâtrbh` lfs pgrebtg `vgrbiu`r hu`n-

qubgr târeblf ag ul` suhgsbøl sb hflfhgefs gn prbegrf ag gnnfs y n` r`zøl.`l 5 `0 • r (l ‘ 0)

[`r` d`hbnbt`r gn prfhgsf ag sue`r nfs gngegltfs ag ul` suhgsbøl igfeâtrbh`, gepngèfs n`døreun` sbiubgltg3

s 5 (`l • r ) ‘ `0

r ‘ 0

Xb agsg` s`mgr ul pfhf eàs sfmrg n` suhgsbøl Dbmfl`hhb, ng blvbtèfs ` qug vbsbtg n`s pàibl`s wgmsbiubgltgs3

• ¶Puâ gs n` suhgsbøl ag Dbmfl`hhb6 iff.in/xnuDA[• N` sghuglhb` ag Dbmfl`hhb3 iff.in/cTfNxt

[`r` hflfhgr eàs sfmrg n`s prfpbgaàgs ag n`s suhgsbflgs `rbteâtbh`s y igfeâtrbh`s, pugag vbsbt`r3• [rfirgsbflgs `rbteâtbh`s3 iff.in/mXz1L;• [rfirgsbflgs igfeâtrbh`s3 iff.in/;tIZHf






@. H`nhung n` abdgrglhb` gl n`s suhgsbflgs sbiubgltgs. Nugif, gshrbm` sb sfl hrghbgltgs, hflst`ltgs faghrghbgltgs.

0) 877, 100, 149, 7=8, 792...

=) =7, 0;8, 0:0, =72, 889...

8) 22, 98, 19, =0, ‘7, ‘80...

1) 49, 49, 49, 49, 49...

7) ‘0=, ‘18, ‘91, ‘0;7, ‘084...

4) 22, 0;;, 0;0, 0;=, 0;8...

M. H`nhung n` r`zøl ag n`s suhgsbflgs sbiubgltgs.

0) =, 0=, 9=, 18=, =72=...

=) 00, ==, 11, ::, 094...

8) =777, 700, 0;=.=, =;.11...

1) :, ‘::, 24:, ‘0;41:...

7) 0;9:, 782, =42.7, 081.97...

4) 8, ‘=0, 019, ‘0;=2, 9=;8...


@. H`nhung nf qug sg ng sfnbhbt` ` hfltblu`hbøl.

0) `49 ag n` suhgsbøl3 99, 008, 012, 0:7... `49 5

=) `81 ag n` suhgsbøl3 02, 4, ‘9, ‘=;, ‘88... `81 5

8) `=;= ag n` suhgsbøl3 7, 00, 09, =8, =2... `=;= 5

1) `024 ag ul` suhgsbøl hfl a 5 01 y `0 5 =8 `024 5

7) Xue`tfrb` j`st` gn gngegltf `=;7, `0 5 87 y a 5 9 s 5

4) Xue`tfrb` j`st` gn gngegltf `08=, `0 5 01 y a 5 0= s 5

@utfhfltrfn

00=BIG] ∝ [fnfhjbh



M. H`nhung nf qug sg ng sfnbhbt` hfltblu`hbøl.

0) `0= ag n` suhgsbøl3 7, 07, 17, 087... `0= 5

=) `2 ag n` suhgsbøl3 8, ‘0=, 1:, ‘02=... `2 5

8) `01 ag n` suhgsbøl3 4, 0=, =1, 1:... `01 5

1) N` r`zøl ag ul` suhgsbøl hfl `0 5 = y `7 5 080== r 5

7) N` r`zøl ag ul` suhgsbøl hfl `0 5 222 y `1 5 89 r 5

4) Xue`tfrb` j`st` gn gngegltf `07, `0 5 0 y r 5 8 s 5

H. ]gsugnv` nfs prfmnge`s sbiubgltgs, rghugrag nf qug pr`htbhø sfmrg n`s prfirgsbflgs `rbteâtbh`s yigfeâtrbh`s.

0) Wl gqubpf ag ruimy 0 pn`lg` aupnbh`r su pultgf hfl hà` s`qug. Xb gl su prbegr s`qug tbglgl 8pultfs, ¶huàltfs pultfs tglarå`l nugif ag 7 s`qugs6

=) Wl àtfef ag pnutflbf pbgrag =: qu`ros= hà` egs. Xb `n h`mf ag 84 egsgs ng quga`l =719qu`ros, ¶huàltfs qu`ros tglå` gl gn prbegr egs6

8) Gl ul gabdbhbf, gn prbegr pbsf sg glhugltr` ` 2.1; egtrfs ag `ntur`, y n` abst`lhb` gltrg hà` pbsfgs ag =.2; egtrfs3

`. Fmtgli` n` døreun` iglgr`n qug lfs blabqug n` `ntur` ag ul agtgreblàf pbsf.

m. ¶@ quâ `ntur` gst`rà gn 0=.³ pbsf6

1) Gl gn prbegr egs, gn `nqubngr ag ul tgrrglf hugst` P=,8;;.;;? sb pfr hà` egs qug p`s` ng a`l ulagshugltf ag P087.;; gl gn `nqubngr3

`. ¶Huàltf agmgrà pì`r gl gn 2.³ egs6

m. ¶Huàltf sgrà gn ablgrf qug j`mrà i`stàf gl tft`n nugif ag 9 egsgs6

7) N` pfmn`hbøl ag ul p`ås ùeglt` r`zøl ag 0.;= `lu`negltg, sb htu`negltg tbglg 4 ebnnflgs agj`mbt`ltgs3

`. ¶Huàltfs j`mbt`ltgs tglarà nugif ag 2 `ðfs6

m. ¶Hu`ltfs j`mbt`ltgs tglarà nugif ag =7 `ðfs6

4) Xb agpfsbtèfs P=,8;;.;; gl ul` huglt` gl n` qug sg euntbpnbh` ` r`zøl ag 0.;7 eglsu`n, ¶huàl-

tf ablgrf tglargefs nugif ag 84 egsgs6

9) Wl` pgrsfl` j` i`làf gl hà` egs 08

ag nf qug i`lø gn egs `ltgrbfr.

`. Xb gn prbegr egs i`lø P4,;;;.;;, ¶huàltf i`l`rà gl gn tgrhgr egs6

m. ¶Huàltf j`mrà i`làf gl tft`n gl gsfs 8 egsgs6

0 ruimy3 agpfrtg ag hflt`htf frbibl`rbf ag Blin`tgrr`. Xg cugi` hfl ul m`nøl fv`nàf.= qu`ros3 p`rtåhun`s dåsbh`s dfreàfr`s ag nfs àtfefs. Xfl nfs hflstbtuygltgs dulaèglt`ngs ag n` e`tgrb`.




@. ]gsugnv` n`s ghu`hbflgs sbiubgltgs. Gshrbm` su rgspugst` gl n` nålg` hfrrgspflabgltg.

0) ‘8 ‘ t 5 4 t 5

=) 2 + t 5 08 t 5

8) 1 ‘ t 5 =8 t 5

1) 9 ‘ t 5 ‘== t 5

7) ‘2 + t 5 =0 t 5

4) 00 ‘ t 5 00 t 5

9) =8 + t 5 ‘0 t 5

:) 0; + t 5 02 t 5

2) 09 + t 5 ‘= t 5

0;) =0 ‘ t 5 = t 5

00) 7 + t 5 ‘7 t 5

0=) ‘08 ‘ t 5 ‘0 t 5

08) t + 0= 5 =1 t 5

01) ‘ t ‘ 0 5 ; t 5

07) t + =9 5 8 t 5

04) t + 02 5 ‘4 t 5

09) t ‘ 0; 5 ‘0= t 5

0:) ‘ t + 00 5 04 t 5

02) ‘ t ‘ 08 5 0: t 5

=;) ‘ t + 9 5 ‘=2 t 5

=0) t + =7 5 ‘=7 t 5

==) t + 08 5 08 t 5

=8) t + 02 5 ‘8 t 5

=1) ‘ t ‘ 09 5 ‘=2 t 5

M. ]gsugnv` n`s ghu`hbflgs sbiubgltgs. Gshrbm` su rgspugst` gl n` nålg` hfrrgspflabgltg.

0) ‘08o 5 47 o 5

=) ‘:o 5 ‘9= o 5

8) 2o 5 :0 o 5

1) ‘ o 5 4; o 5

7) ‘=o 5 ‘4; o 5

4) 0=o 5 ‘4; o 5

9) 00o 5 ‘22 o 5

:) 7o 5 17 o 5

2) 8o 5 ‘0= o 5

0;) 1o 5 =; o 5

00) 0;o 5 ‘7; o 5

0=) ‘ o 5 ‘0 o 5

08) ‘:o + = 5 44 o 5

01) =o ‘ 9 5 89 o 5

07) 00o + 02 5 24 o 5

04) 0=o + 7 5 :2 o 5

09) 8o + : 5 =4 o 5

0:) ‘0;o ‘ = 5 1: o 5

02) 0;o ‘ 0 5 22 o 5

=;) ==o + 08 5 92 o 5

=0) ‘09o + 01 5 22 o 5

==) 9o ‘ 04 5 88 o 5

=8) ‘08o ‘ 09 5 87 o 5

=1) ‘=o ‘ 00 5 10 o 5

001BIG] ∝ [fnfhjbh




[`r` tr`m`c`r hfl sghuglhb`s lueârbh`s f p`r` euhj`s ftr`s àrg`s ag n`s e`tgeàtbh`s, lghgsbtèfsul` ir`l h`p`hbaà ag `lànbsbs? hfl gn gcgrhbhbf sbiubgltg, pflargefs ` prugm` gst` j`mbnbaà.

Gl n`s sgrbgs sbiubgltgs, gnbc` y rfagg, ag gltrg n`s fphbflgs, gn lóegrf qug sbiug n` sghuglhb`.

0) ; = 1 4 : 0; 0= ...

`. 08 m. = h. 01 a. 0;

=) 4 2 0= 07 0: =0 =1 ...

`. =7 m. =: h. =4 a. =9

8) 2 : 9 4 7 1 8 ...

`. 0 m. = h. 1 a. ;

1) 02 09 07 08 00 2 9 ...

`. : m. 7 h. 4 a. 0;

7) 89 81 80 =: =7 == 02 ...

`. 07 m. 09 h. =0 a. 04

4) 0: =; =0 =8 =1 =4 =9 ...

`. 8; m. =2 h. =: a. =7

9) 02 =0 =0 =8 =7 =7... =9

`. =1 m. == h. =8 a. =4

:) 42 :2 22 002 0=2... 0820;2

`. 42 m. 92 h. 22 a. ::

2) 82 12 7111 42... 9172

`. 47 m. 92 h. 71 a. 41

0;) 00 11 ::== 9;1... 01;:87=

`. 22 m. 0== h. 071 a. 094







:0

= 8 1

8

1

7

=

0

;

x

y

‘ 0 5 b

x = +

y = 5

r =

]gp`sf3 sge`l`s 0 ` 9

Gst` sge`l` nfir`rà3 ]gp`s`r nfs hfltglbafs ag n` sge`l` 0 n` 9.

]gsfnvgr nfs gcgrhbhbfs ag rgp`sf p`r` gv nu`rsg gl n` prugm`

p`rhb`n.

[rgp`r`rsg mbgl p`r` n` prugm` p`rhb`n.

009E`tgeàtbh` ∝ Xge`l` :



Pugrba` y qugrbaf gstuab`ltg3

Xg `prfxbe` n` prbegr` prugm` p`rhb`n y agmg prgp`r`rsg àghuàègltg, rgp`s`laf nfs hfltg-

lbafs ag n`s sge`l`s 0 ` n` 9.

[`r` `prfvghj`r gstg rgp`sf, ng rghfeglaèfs3

• Jì` ul pn`l ag nf qug gstuab`rà hà` aå` y tr`tg ag huepnbrnf. Agabqug eàs tbgepf `

nfs tge`s qug ng rgsuntgl abdåhbngs.

• Musqug ul nui`r tr`lqubnf, bnueblàf y sbnglhbfsf p`r` gstuab`r.

• Ng` nfs rgsóeglgs ag hà` sge`l` y gshrbm` n`s bag`s eàs bepfrt`ltgs gl su huàgrlf.

• Gshuhjg n` hn`sg ràb`n. Xus e`gstrfs nfhutfrgs ng `hfep`ð`ràl gl gstg rgp`sf y ng yu-

a`ràl ` rgsfnvgr `niulfs gcgrhbhbfs.

• Hfeprugmg qug j`y` rg`nbzàf mbgl nfs ùtfhfltrfngs. Xb tbglg aua`s, vugnv` ` nggr n`s

sge`l`s, `jå glhfltr`rà gxpnbh`hbflgs y gcgepnfs.

¶Høef sgrà n` prugm` ag gv`nu`hbøl6

N` prugm` p`rhb`n gv`nó` nfs ebsefs hfltglbafs y ag n` ebse` dfre` gl qug nfs j` tr`m`càf

sge`l` ` sge`l`.

Gl n` prugm` glhfltr`rà3

• Wl` sgrbg ag ìbnbaà eglt`n p`r` egabr su agstrgz` y r`pbagz ag hànhunf gl ul tbgepf

nåebtg ag trgs eblutfs.

• Abdgrgltgs gcgrhbhbfs qug gv`nó`l nf prglabaf gl n`s sbgtg sge`l`s. Gstfs gcgrhbhbfs sgràl

sgegc`ltgs ` nfs qug ustga rgsfnvbø gl n`s `htbvbaàgs agn ùtfhfltrfn. Xg ng pgabrà3

rgspflagr prgiult`s,

e`rh`r hfl ul` gqubs "Z" gn huàrf ag n` fphbøl hfrrght`,

rgsfnvgr fpgr`hbflgs y

rgsfnvgr prfmnge`s.

• Hu`laf rgsugnv` gcgrhbhbfs y prfmnge`s, agmg agc`r hflst`lhb` ag su prfhgabebgltf.

• Euy bepfrt`ltg3 hà` sgrbg hfltbglg blstruhhbflgs gx`ht`s ag nf qug agmg rg`nbz`r gl

hà` `p`rtàf, `så hfef n` v`nfr`hbøl `sbilà`.

Xb ustga sg prgp`r` hfl tbgepf y agabh`hbøl, gn rgsuntàf sgrà s`tbsd`htfrbf.

Ng rghfraèfs qug agmg prgp`r`rsg p`r` n` prugm` p`rhb`n y p`r` n`

prugm` agn EBLGAWH pr`htbh`laf hfl nfs gcgrhbhbfs qug sg glhugltr`l

`n l`n ag su nbmrf gl gn Q`nngr e`tgeàtbhf.

00:BIG] ∝ [fnfhjbh



Nøibh` B

0. N` nøibh` prfpfsbhbfl`n gstuab` glulhbàfs y gv`nó` su v`nfr ag vgraà, gs aghbr, agtgrebl` sb sfl

vgraàgrfs f d`nsfs.

=. N`s prfpfsbhbflgs sfl glulhbàfs qug pugagl sgr `l`nbzàfs y `så agtgrebl`r su v`nfr nøibhf. N`s

prfpfsbhbflgs sbepngs sfl `qugnn`s qug lf pugagl sgr abvbaba`s sbl `dght`r su sgltbaf. Gst`s sfl

rgprgsgltà`s hfl ul` ngtr` eblóshun`.

r 3 N` h`dgål` tbglg ul gdghtf gstbeun`ltg.

t 3 Gn ìu` gs lghgs`rb` p`r` n` vba` agn sgr jue`lf.

Wl glulhbàf lf gs ul` prfpfsbhbøl sb lf sg pugag agtgrebl`r su v`nfr ag vgraà. Gstf suhgag

hfeólegltg hfl glulhbàfs bepgr`tbvfs, ag prgiult` f qugnnfs qug prgsglt`l ul cubhbf ag v`nfr

Gn dutmfn gs egcfr qug gn m`nflhgstf.

¶Puâ j`rgefs jfy6

=.0 N` lgi`hbøl gs ul` fpgr`hbøl qug hflsbstg gl blvgrtbr gn v`nfr nøibhf ag ul` prfpfsbhbøl? sg

blabh` `ltgpflbglaf gn sbilf "~" ` n` prfpfsbhbøl.

a 3 ]bh`raf @rcfl` gs iu`tge`ntghf. T Gs vgraàgrf. Gn h`lt`ltg l`hbø gl Cfhftgl`lif,

X`h`tgpâqugz.

~a 3 ]bh`raf @rcfl` lf gs iu`tge`ntghf. DN` lgi`hbøl ag n` prfpfsbhbøl a hèmb` ag v`nfr

ag vgraà, `jfr` gs d`ns`, pfrqug @rcfl` så gs

iu`tge`ntghf.

8. Nfs hu`ltbdbhàfrgs sfl såemfnfs qug sg utbnbz`l p`r` gst`mnghgr huàltfs gngegltfs ag ul hflcultf

agtgreblàf huepngl hfl hbgrt` prfpbgaà.

8.0 Gn hu`ltbdbhàfr ulbvgrs`n (∋) sg gepng` p`r` `dbre`r qug tfafs nfs gngegltfs ag ul hflcultf

huepngl hfl ul` hflabhbøl aà`. Xg ngg "p`r` tfaf…".

∋ x ∊ @, = x ∊ M3 [`r` tfa` x qug pgrtglghg `n hflcultf @, sg huepng qug = x pgrtglghg `n

hflcultf M.

8.= Gn hu`ltbdbhàfr gxbstglhb`n (∈) sg utbnbz` p`r` blabh`r qug n eglfs ul gngegltf ag ul hflcultf

huepng hfl ul` hflabhbøl aà`. Xg ngg "gxbstg `n eglfs ul…".

∈ x ∊ H ∩ x ∊ M3 Gxbstg `n eglfs ul` x qug pgrtglghg ` H y pgrtglghg tèmbâl ` M.





Gcgrhbhbf 0@. Ng` n`s prfpfsbhbflgs sbiubgltgs y agtgreblg su v`nfr nøibhf. Nugif, pbglsg y rggshrbm` su lgi`-

hbøl, rghugrag gn sbilf "~". Iuågsg pfr gn gcgepnf.

;) r3 Pubhjâ gs ul agp`rtègltf ag Iu`tge`n`.

~r3 Pubhjâ lf gs ul agp`rtègltf ag Iu`tge`n`.

0) w3 Hum` gs ul p`ås qug pgrtglghg ` Gurfp`.

=) j3 Gn ìu` agn e`r gs aunhg.

8) i3 Gn babfe` fdbhb`n ag Iu`tge`n` gs gn `ngeàl.

1) s3 8 gs abvbsfr ag 2.

7) p3 Gn vgrag gs ul hfnfr sghula`rbf.

4) q3 Eâxbhf sg glhugltr` `n sur ag Iu`tge`n`.

M. Ng` n`s gxprgsbflgs sbiubgltgs. Nugif, tr`lsdøregn`s ` ngliu`cg hfeól. Ng aèfs ul gcgepnf.

;) ∈ x ∊ Q ∩ x ∊ I

Gxbstg `n eglfs ul` x qug pgrtglghg ` Q y pgrtglghg tèmbâl ` I.

0) ∋ x ∊ G, : x ∊ J

=) ∈ x ∊ H ∩ x ∊ M

8) ∋ x ∊ @, x + 8 ∊ M

1) ∈ x ∊ ] ∩ x ∊ Y

7) ∈ x ∊ O ∩ x ∊ H

T

D

0=;BIG] ∝ [fnfhjbh



Nøibh` BB

0. N`s prfpfsbhbflgs hfepugst`s sfl qugnn`s dfreà`s pfr afs f eàs prfpfsbhbflgs sbepngs qug sg

ulgl ` tr`vâs ag hflghtbvfs nøibhfs.

Gn cìu`r gs ul eèådgrf ègrbh`lf y gstà gl pgnbirf ag gxtblhbøl.

Gl gn gcgepnf `ltgrbfr sg ulgl afs prfpfsbhbflgs sbepngs ` tr`vâs agn lgxf "y"3 Gn cìu`r gs ul

eèådgrf ègrbh`lf? gn cìu`r gstà gl pgnbirf ag gxtblhbøl.

=. Nfs hflghtbvfs nøibhfs sfl p`n`mr`s y såemfnfs qug sbrvgl hfef lgxf p`r` ulbr afs f eàs prfpfsb-

hbflgs sbepngs. Nfs eàs hfeulgs sfl n` hflculhbøl, n` absyulhbøl, gn hflabhbfl`n y gn mbhflabhbfl`n.

=.0 N` hflculhbøl (∩) hfrrgspflag ` n` "y".


I`mrbgn` gstà gstuab`laf y Nfrglzf gstà tr`m`c`laf. p ∩ q p y q

=.= N` absyulhbøl (∪) hfrrgspflag n` "f" y sg utbnbz` p`r` a`r sgltbaf ag gnghhbøl gltrg afs f eàs

gngegltfs.


@l` pugag gstuab`r e`tgeàtbh` f gstuab`r dåsbh`. p ∪ q p f q

=.8 Gn hflabhbfl`n ( ) f bepnbh`hbøl tfe` dfre` gl n`s p`n`mr`s "Xb… gltflhgs". Gstg hflghtbvf

blabh` qug n` prbegr` prfpfsbhbøl gs rgqubsbtf p`r` qug sg huepn` n` sgiula`.


Xb rgspgtèfs n`s ngygs, gltflhgs sgrgefs muglfs

hbuaà`lfs. p q Xb p gltflhgs q

=.1 Gn mbhflabhbfl`n ( ) sg dfre` hfl n`s p`n`mr`s "sb y sfnf sb" g blabh` ul` rgn`hbøl ag agpgl -

aglhb` eutu`. Gs aghbr, n` prbegr` sg huepng sfnf sb sg huepng n` sgiula` y vbhgvgrs`.


E`ð`l` sgrà nulgs sb y sfnf sb jfy gs afeblif. p q p sb y sfnf sb q

8. [`r` agtgrebl`r gn v`nfr ag vgraà ag ul` prfpfsbhbøl hfepugst` sg gepng`l n`s rgin`s ag nfshflghtbvfs nøibhfs. Xg prgsglt`l gl n`s t`mn`s ag vgraà sbiubgltgs3

p q p ∩ q

v v v

v d d

d v d

d d d

Hflculhbøl p q p ∪ qv v v

v d v

d v v

d d d

Absyulhbøl p q p q

v v v

v d d

d v v

d d v

Hflabhbfl`n p q p q

v v v

v d d

d v d

d d v

Mbhflabhbfl`n

0=0E`tgeàtbh` ∝ Xge`l` :





Lóegrfs rg`ngs

0. Gn hflcultf ag nfs lóegrfs rg`ngs sg rgprgsglt` hfl n` ngtr` ] y sg agdblg sbemønbhègltg hfef

] 5 P ∣ B. Gstfs lóegrfs sg pugagl gxprgs`r pfr ul gltgrf f ul aghbe`n, sg` pgrbøabhf f lf.

]

B

P

S

L

...‘1, ‘8, ‘=, ‘0 ;, 0, =, 8, 1 ...0.=0=0=...

;.7

78

81

= ϋ

Nfs lóegrfs rg`ngs (] ) glinfm`l ` nfs brr`hbfl`ngs (B), l`tur`ngs (L), gltgrfs (S) y r`hbfl`ngs (P).

0.0 Nfs lóegrfs rg`ngs sg h`r`htgrbz`l pfr sgr ul hflcultf dblbtf, aglsf, hfepngtf y hfltbluf.

0.= Gn v`nfr `msfnutf ag ul lóegrf rg`n gs n` abst`lhb` qug j`y gltrg gn hgrf u frbigl y gstg

lóegrf. Xu rgsuntàf sbgeprg gs pfsbtbvf. Gstg v`nfr sg bagltbh` gshrbmbâlafnf gltrg m`rr`s |`|.

=. N`s fpgr`hbflgs hfemblà`s sfl v`rb`s fpgr`hbflgs qug sg rgólgl gl ul` sfn`.

• Xb n`s fpgr`hbflgs lf blhnuygl sbilfs ag irup`hbøl, sg rgsugnvgl rgspgt`laf n` cgr`rquå` ag n`s

fpgr`hbflgs, prbegrf n`s euntbpnbh`hbflgs y abvbsbflgs y nugif n`s sue`s y rgst`s.

• Xb n`s fpgr`hbflgs tbglgl sbilfs ag ìrup`hbøl, gstfs sg gnbebl`l ag àgltrf j`hb` `dugr`.[rbegrf p`râltgsbs, nugif hfrhjgtgs y pfr óntbef nn`vgs.

8. Gn hflcultf ag nfs lóegrfs rg`ngs huepng hfl abdgrgltgs prfpbgaàgs.

1. N` pftglhb`hbøl ag lóegrfs rg`ngs gs n` dfre` ràpba` ag gshrbmbr ul lóegrf qug sg euntbpnbh`

v`rb`s vghgs pfr så ebsef.

7. N` ràbh`hbøl gs n` fpgr`hbøl blvgrs` n` pftglhb`hbøl, hflsbstg gl j`nn`r n` r`åz ag ul lóegrf aàf.

Gcgrhbhbf 8@. ]gsugnv` n`s fpgr`hbflgs hfemblà`s sbiubgltgs.

0) 07 + 10 Ù = ‘ 24 + 0;; 5 7) :7 + _(8 Ù 01) ù (8 Ù 9)U 5

=) 7 Ù =8 + (‘=:) ‘ 09 5 4) {_(8 + :) ‘ 84 Ù =U ‘ :} ‘ = Ù : 5

8) 0;0 ‘ =7 + 84 ù 4 + = 5 9) (: + 4) ù 9 + 2 Ù = 5

1) 09 + =7 ‘ _8 Ù (= + 70)U 5 :) _0=7 ù (7 Ù 7)U ‘ =7 + 0 5




M. Bagltbdbqug y gshrbm` n` prfpbgaà agn hflcultf ag lóegrfs rg`ngs qug sg gcgepnbdbh` gl hà`

blhbsf.

0) ;.97 • 0 5 ;.97

=) ( 8

4 ) • ( 4

8 ) 5 0

8) : + ; 5 :

1) 0: • 7 5 7 • 0:

7) 7(8 • 8) 5 (7 • 8)8

4) 1(7 + 1) 5 (1 • 7) + (1 • 1)

9) 87 + 9 5 1=

:) ( 9

4 ) + ( ‘ 9

4 ) 5 ;

H. @pnbqug n`s rgin`s ag n` pftglhb`hbøl p`r` rgsfnvgr nfs gcgrhbhbfs sbiubgltgs.0) 022; 5

=) 7= • 78 5

8) ( 8

= ) ‘8

5

1) (‘8)8 • (‘8)= 5

7) 7 ‘8 5

4) (8=)8 5

9) (= • 8)8 5

:) ( ‘ 0

= )7

5

2) 47

44 5

0;) (‘1) ‘= 5

00) (‘9); 5

0=) ( 4

8 )8

5

08) ( 7

= ) ‘=

5

01) (=8)8 5

A. @pnbqug n`s rgin`s ag n` ràbh`hbøl p`r` rgsfnvgr nfs gcgrhbhbfs sbiubgltgs.

0) 0: • = 5

=)41

1 5

8)

8 =98 : 5

1) 8 9=2 5

7)1 2= 5

4)2 0=78 5

9)8

1;24 5

:) 8= • = 5

2)1 04

1 =74

5

0;) 4=7 5

00)8 0=8 5

0=)0=7

7 5

08)1 4: 5

01) = • 12 5

0=1BIG] ∝ [fnfhjbh



Lóegrfs hfepngcfs

0. Gn hflcultf ag nfs lóegrfs hfepngcfs gs ul` gxtglsbøl ag nfs lóegrfs rg`ngs y sbrvg p`r` a`rng

sfnuhbøl ` n`s r`åhgs p`rgs ag lóegrfs lgi`tbvfs.

[`r` gnnf, nfs e`tgeàtbhfs bltrfaucgrfl ul lugvf såemfnf aglftàf pfr b , nnèàf ulbaà beìbl`-

rb` y qug huepng hfl n` hflabhbøl sbiubgltg3

‘0 5 b y qug b = 5 ‘0.

Tgèfs ul gcgepnf ràpbaf, rgsfnvèfs3

• Euntbpnbhèfs gn ràbh`laf pfr ‘0.

• [fr agdblbhbøl, sustbtubefs ( ‘0) pfr b = .

• @pnbhèfs n` prfpbgaà agn prfauhtf ag ul` r`åz y rgsfnvgefs.

0.0 N` pftglhb`hbøl ag b sbiug ul p`trøl qug sg rgpbtg ` p`rtbr ag b7. Xus pftglhb`s màsbh`s sfl3

b

0

5 b , b=

5 ‘0, b8

5 ‘ b , b1

5 0, b7

5 b .

0.= Nfs rgsuntàfs ag n`s r`åhgs p`rgs ag lóegrfs lgi`tbvfs rghbmgl gn lfemrg ag lóegrfs hfe-pngcfs purfs (mb) y ` nfs qug sg hfembl`l hfl nfs lóegrfs rg`ngs sg ngs nnè` dfre` mblfeb`nag ul lóegrf hfepngcf (` + mb).

=. [`r` sue`r y rgst`r lóegrfs hfepngcfs gs lghgs`rbf fpgr`r n`s p`rtgs rg`ngs, nugif sue`r f rgst`r

n`s p`rtgs beìbl`rb`s y gn rgsuntàf agmg gshrbmbrsg gl dfre` mblfeb`n.

8. [`r` euntbpnbh`r lóegrfs hfepngcfs sg agmg `pnbh`r n` prfpbgaà abstrbmutbv` ag n` euntbpnbh`hbøl,

sustbtubr b= pfr ‘0, rgauhbr târeblfs sgegc`ltgs y gshrbmbr gn rgsuntàf gl dfre` mblfeb`n.

1. [`r` abvbabr lóegrfs hfepngcfs gs lghgs`rbf r`hbfl`nbz`r gn aglfeblàfr egab`ltg gn hflcuiàf,

rg`nbz`r n`s fpgr`hbflgs hfrrgspflabgltgs, sustbtubr b= pfr ‘0 y gshrbmbr gn rgsuntàf ag n` dfre`

mblfeb`n.

Gcgrhbhbf 1@. @pnbqug sus hflfhbebgltfs sfmrg r`åhgs p`rgs ag lóegrfs lgi`tbvfs y rgsugnv` nfs h`sfs sbiubgltgs.

0) ‘2 5

=) ‘41 5

8) ‘0;; 5

1) ‘1;; 5

7) ‘:0 5

4) ‘12 5

9) ‘0 5

:) ‘= 5

2) ‘84 5

0;) ‘=7 5

00) ‘011 5

0=) ‘0=0 5

‘04

04 (‘0)

04b=

04 • b= 5 1b




M. ]g`nbhg n`s fpgr`hbflgs blabhà`s hfl lóegrfs hfepngcfs y gxprgsg su rgspugst` gl dfre`

mblfeb`n.

0) (9 ‘ 0=b) + (8 + b) =) (0; + 00b) + (09 ‘ 7b)

8) (0;b) ‘ (: + 9b) 1) 4b(00 + 0;b)

7) ‘0=b(9 + 1b) 4) (== + 7b)(0: ‘ 1b)

9) ( ‘ 0

8 ‘ 2b)( 0

00 + =b) :) ( 8

1 + 8b)( ‘

=

7 ‘ b)

2) =(7 + 8b) 0;) ‘8(= ‘ b)

0=4BIG] ∝ [fnfhjbh



Xue` y rgst` ag e`trbhgs

0. Wl` e`trbz gs ul hflcultf ag lóegrfs f gxprgsbflgs abspugstfs gl dfre` rght`liun`r, dfre`laf

dbn`s y hfnuel`s. Gn fragl f abeglsbøl ag ul` e`trbz sg agtgrebl` pfr gn lóegrf ag dbn`s y hfnue-

l`s qug n` hfepflgl.

`00 `0= ... `0c ... `0l

`=0 `== ... `=c ... `=l

... ... ... ... ... ...`b0 `b= ... `bc ... `bl

... ... ... ... ... ...è0 è= ... èc ... èl

@ 5

Wl` e`trbz e Ù l tbglg e dbn`s y l hfnuel`s. Xb tbglg gn ebsef lóegrf ag dbn`s qug ag hfnuel`s,

aghbefs qug gs ul` e`trbz huàrà`. @ hà` lóegrf f gxprgsbøl qug dfre` n` e`trbz ng aglfebl`-

efs gngegltf.

M 5

8 ‘=

; 0

7 :

8 n`s

= hfnuel`s

E`trbz M3 8 Ù =

=. J`y `niulfs tbpfs ag e`trbhgs hfl h`r`htgråstbh`s p`rtbhun`rgs3

=.0 Wl` e`trbz tr`lspugst` (@t) gs n` qug sg dfre` `n hèmb`r fraglàègltg n`s dbn`s pfr n`s

hfnuel`s ag n` e`trbz aà`.

=.= Wl` e`trbz gs sbeâtrbh` sb gs huàrà` y gs biu`n ` su tr`lspugst`.

=.8 Wl` e`trbz bagltbaà gs ul` e`trbz gl huy` abìfl`n prblhbp`n tfafs sus gngegltfs sfl biu`-

ngs ` 0 y nfs ageàs gngegltfs sfl ; (hgrf). Xg lfemr` hfl n` ngtr` B.=.1 Wl` e`trbz lf sbliun`r f blvgrtbmng gs `qugnn` qug tbglg e`trbz blvgrs`.

=.7 Wl` e`trbz sbliun`r lf tbglg blvgrs`.

8. N`s fpgr`hbflgs qug sg pugagl j`hgr hfl e`trbhgs sfl n` àbhbøl, n` rgst` y n` euntbpnbh`hbøl3

8.0 [`r` sue`r f rgst`r e`trbhgs gs lghgs`rbf qug tgli`l n` ebse` abeglsbøl. [`r` fpgr`r, sg

agmgl sue`r nfs gngegltfs ag n`s e`trbhgs qug fhup`l n` ebse` pfsbhbøl. Xg pugagl

fpgr`r afs f eàs e`trbhgs.

8.= N` sue` ag e`trbhgs tbglg n`s prfpbgaàgs sbiubgltgs3

• Gs bltgrl` pfrqug n` e`trbz rgsuntàf sgrà ag n`s ebse`s abeglsbflgs qug n`s e`trbhgs qug

sg fpgr`rfl.

• Gs `sfhb`tbv pfrqug gn rgsuntàf gs gn ebsef sbl bepfrt`r høef ìrupgefs nfs târeblfs.

• Qbglg gngegltf lgutrf, hflfhba` hfef e`trbz hgrf (F).

• Qbglg gngegltf fpugstf, gn hu`n gs n` e`trbz hfl nfs ebsefs gngegltfs qug n` e`trbz frbib-

l`n, pgrf hfl nfs sbilfs hfltr`rbfs rgspghtf ` n` e`trbz frbibl`n.

• Gs hfleut`tbv pfrqug gn rgsuntàf pgre`lghg blv`rb`mng sbl bepfrt`r gn fragl gl qug

fpgrèfs nfs sue`lafs.




Gcgrhbhbf 7@. [bglsg y dfreg nfs tbpfs ag e`trbhgs qug sg ng pbagl.

0) Wl` e`trbz ag 1 Ù 8 =) N` e`trbz bagltbaà

8) Wl` e`trbz sbeâtrbh` 1) Wl` e`trbz hfl su tr`lspugst`

M. Xbglaf n`s e`trbhgs3

@ 5 2 ‘7 4 2

1 9 ; ‘9 M 5 ‘8 4 7 ‘1

1 : ‘2 ‘7 H 5 4 7 1 ;

; 7 ‘2 9

Z 5; ‘0 :

‘7 1 4

0 2 ‘9

R 5 ‘: 4 ‘1

0 9 1

; ; :

S 51 7 2

0 ; ‘0

: 2 0

]gsugnv` n`s sbiubgltgs sue`s y rgst`s ag e`trbhgs. [ugag tr`m`c`r gl jfc`s `p`rtg sb gs lghgs`rbf.

0) @ + M =) Z ‘ S

8) M ‘ H + @ 1) R ‘ Z

7) H ‘ M 4) Z + R + S

0=:BIG] ∝ [fnfhjbh




0. N` euntbpnbh`hbøl f prfauhtf ag e`trbhgs gs ul` fpgr`hbøl qug pugag gdghtu`rsg gltrg afs

e`trbhgs f mbgl gltrg ul` e`trbz y ul lóegrf rg`n f gsh`n`r. [`r` rgsfnvgr gstg prfhgabebgltf

agmgefs iub`rlfs pfr agtgreblà`s rgin`s.

0.0 [`r` euntbpnbh`r ul` e`trbz pfr ul lóegrf rg`n sg agmg euntbpnbh`r gsg lóegrf pfr hà` ulfag nfs gngegltfs ag n` e`trbz. Xg pugag euntbpnbh`r ul` e`trbz ag hu`nqubgr fragl.

o @ 5 o

` m h

a g d

i j b

5

ò mo ho

ao go do

io jo bo

0.= [`r` euntbpnbh`r afs e`trbhgs gltrg så , lfs `pfyèfs gl gn prfhgabebgltf sbiubgltg3

• Xg hfeprugm` qug gn lóegrf ag hfnuel`s ag n` prbegr` e`trbz (@) hfblhba` hfl gn lóegrf

ag dbn`s ag n` sgiula` (M). Ag nf hfltr`rbf, lf sg pugagl fpgr`r.

• Xg euntbpnbh` n` prbegr` dbn` ag @ pfr n` prbegr` hfnuel` ag M. [rbegr gngegltf pfr prbeg-

rf, sgiulaf pfr sgiulaf, gth. y sueèfs nfs prfauhtfs. Hfl gstf hflsgiubefs gn prbegr

gngegltf ag n` e`trbz rgsuntàf.

• Xgiubefs euntbpnbh`laf n` prbegr` dbn` ag @ pfr n`s sbiubgltgs hfnuel`s ag M.

• ]gpgtbefs gn prfhgabebgltf hfl n`s sbiubgltgs dbn`s ag @ y hà` hfnuel` ag M.

• Nugif ag tgrebl`r hfl tfa`s n`s dbn`s ag @, nngièfs `n rgsuntàf dbl`n.

H 5 @ • M 5

` m h

a g d

i j b

•

r s t

u v w

x y z

5

`r + mu + hx `s + mv + hy `t + mw + hz



Gcgrhbhbf 4@. ]gsugnv` n`s euntbpnbh`hbflgs sbiubgltgs. [ugag utbnbz`r su huàgrlf p`r` rg`nbz`r gn prfhgabebgltf.

0) : : 0; ‘00

‘2 ; 0

9 1 ‘8

=) ‘8 ; ‘2 ‘1

‘2 7 0=

= ; ‘0




M. ]gsugnv` n`s euntbpnbh`hbflgs sbiubgltgs. [ugag utbnbz`r su huàgrlf p`r` rg`nbz`r gn prfhgabebgltf.

0)= ‘2

•

‘7 9

7 4

=) ‘0 7 1 •

: 4

7 ‘7

0; ‘0

8) ; ‘:

; 0

•

2 4

0 ‘0

1)9 ; :

‘8 7 =

•2 ‘7

4 8

‘9 ;

7)= 1 ‘=

; ‘0 0

•: ‘0

7 ;

; ‘2

4)2 7 ‘8

0 ‘0 :

• ‘9 = 8

0 ; ‘=

= 9 ‘2

9) 0 8 ‘:

; 2 ‘0

2 0; 00

•

: 8 =

7 1 ‘4

0 0 =

:) ‘: 4 7

1 0 ‘8

1 ‘4 ;

•

4 : 9

‘2 ; =

00 0; 0

2) ‘0 ‘8 48 : 0

; : 2

• ‘2 8 =

: 1 4

8 ‘00 =

0;) 0= ‘: 29 9 :

0 8 08

• 00 : 80 0 0

8 ; ‘:




Xuhgsbflgs e`tgeàtbh`s

0. Wl` suhgsbøl e`tgeàtbh` gs ul hflcultf fraglàf ag lóegrfs. Hà` ulf ag gstfs lóegrfs

sg aglfebl` "târeblf", "gngegltf" f "ebgemrf", y `n lóegrf tft`n ag gngegltfs qug hfepflg n`

suhgsbøl sg ng nnè` "nflibtua", gst` pugag sgr dblbt` f bldblbt`.

=. Wl` suhgsbøl f prfirgsbøl rbteâtbh` gs ul` sgrbg ag lóegrfs t`ngs qug hà` ulf ag gnnfs, gxhgp-tu`laf gn prbegr târeblf, sg fmtbglg sue`laf f rgst`laf ul` h`ltbaà dbc` `n târeblf `ltgrbfr, `

gst` h`ltbaà hflst`ltg sg ng hflfhg hfef abdgrglhb` y sg rgprgsglt` hfl n` n` ngtr` a .

08 =1 87 14 79 4: ...

+00 +00 +00 +00 +00

`. N` døreun` agn târeblf iglgr`n p`r` ul` suhgsbøl `rbteâtbh` lfs pgrebtg `vgrbiu`r hu`n-qubgr târeblf ag ul` suhgsbøl sb hflfhgefs gn prbegrf ag gnnfs y su abdgrglhb`.

`l 5 `0 + (l ‘ 0)a

[`r` d`hbnbt`r gn prfhgsf ag sue`r nfs gngegltfs ag ul` suhgsbøl `rbteâtbh` gepngèfs n`døreun` sbiubgltg3

s 5 (`0 + `l )l

=

8. Wl` suhgsbøl igfeâtrbh` gs ul` suhgsbøl gl n` qug hà` târeblf sg fmtbglg euntbpnbh`laf `n

`ltgrbfr pfr ul` h`ltbaà dbc` aglfeblà` r`zøl (r ).

: =1 9= =04 41: 0211 ...

Ù8 Ù8 Ù8 Ù8 Ù8

`. N` døreun` agn târeblf iglgr`n p`r` ul` suhgsbøl igfeâtrbh` lfs pgrebtg `vgrbiu`r hu`n-qubgr târeblf ag ul` suhgsbøl sb hflfhgefs gn prbegrf ag gnnfs y n` r`zøl.

`l 5 `0 • r (l ‘ 0)

[`r` d`hbnbt`r gn prfhgsf ag sue`r nfs gngegltfs ag ul` suhgsbøl igfeâtrbh` gepngèfs n`

døreun` sbiubgltg3

s 5 (`l • r ) ‘ `0

r ‘ 0

N`s àrg`s gl n`s qug pfagefs `pnbh`r lugstrfs hflfhbebgltfs sfmrg suhgsbflgs e`tgeàtbh` sfl euy

v`rbà`s. [fr gcgepnf3

Gl n` m`lh`, nfs blvgrsbflbst`s y n`s pgrsfl`s qug tbglgl huglt`s m`lh`rb`s, pugagl h`nhun`r n`s i`l`l-

hb`s pftglhb`ngs sfmrg ul lgifhbf gepng`laf n`s prfirgsbflgs `rbteâtbh`s.

Gl n` bliglbgrå`, nfs lóhngfs ag ul e`tgrb`n ràb`htbvf sg agsbltgir`l gl dfre` ulbdfreg gl ul agtgreb-

làf pgråfaf ag tbgepf, gstg hànhunf pugag sgr agshrbtf pfr ul` prfirgsbøl igfeâtrbh`.

[fr t`ltf, pfagefs rgsfnvgr euhjfs prfmnge`s hfl t`l sfnf pfagr bagltbh`r hfrrghtègltg nfs a`tfs

ag n` prfirgsbøl y `pnbh`r n`s døreun`s qug ustga y` afebl`.




Gcgrhbhbf 9@. H`nhung nf qug sg sfnbhbt` gl hà` ulf ag nfs blhbsfs sbiubgltgs. [ugag utbnbz`r jfc`s `p`rtg.

]ghugrag tr`m`c`r gl fragl.

0) `18 ag n` suhgsbøl3 :, 0, ‘4, ‘08, ‘=;… `18 5

=) `74 ag n` suhgsbøl3 0, =.7, 1, 7.7, 9… `74 5

8) `81 ag n` suhgsbøl3 :0, :4, 20, 24, 0;0… `81 5

1) a ag ul` suhgsbøl hfl `0 5 08 y `1= 5 8;; a 5

7) a ag ul` suhgsbøl hfl `0 5 179 y `40 5 ‘=48 a 5

4) `0 ag ul` suhgsbøl hfl a 5 ‘: y `:: 5 ‘48: `0 5

9) `0 ag ul` suhgsbøl hfl a 5 =.7 y `40 5 =81.7 `0 5

:) Xue`tfrb` j`st` gn gngegltf `44, `0 5 19 y a 5 2

`44 5 s 5

M. H`nhung nf qug sg sfnbhbt` gl hà` ulf ag nfs blhbsfs sbiubgltgs. [ugag utbnbz`r jfc`s `p`rtg.

]ghugrag tr`m`c`r gl fragl.

0) `0= ag n` suhgsbøl3 4, 0=, =1, 1:… `0= 5

=) `00 ag n` suhgsbøl3 1, ‘04, 41, ‘=74… `00 5

8) `07 ag n` suhgsbøl3 :, 0=, 0:, =9, 1;.7… `07 5

1) r ag ul` suhgsbøl hfl `0 5 ‘8 y `9 5 ‘14:97 r 5

7) r ag ul` suhgsbøl hfl `0 5 9 y `00 5 108818 r 5

4) `0 ag ul` suhgsbøl hfl r 5 7.7 y `1 5 0224.7 `0 5

9) `0 ag ul` suhgsbøl hfl r 5 =.7 y `7 5 82;.4=7 `0 5

:) Xue`tfrb` j`st` gn gngegltf `0;, `0 5 = y r 5 1

`0; 5 s 5

H. ]gsugnv` nfs prfmnge`s sbiubgltgs. [ugag j`hgrnf gl jfc`s `p`rtg.

0) Gn `nqubngr ag ul` eftfhbhngt` hugst` P8;.;; n` prbegr` jfr` y P:.;; eàs hà` lugv jfr`.

`. ¶Huàn gs gn prghbf tft`n ag `nqubngr ag 4 jfr`s6

m. Glhugltrg ul` døreun` qug lfs blabqug gn prghbf ag nqubngr p`r` niul` jfr` gl gspghådbhf.

=) Wl` eàqubl` hfstø blbhb`negltg P1:,;;;.;;. Nugif ag ulfs `ðfs sg vglabø ` n` ebtà ag su

prghbf. [`sàfs ulfs `ðfs, vfnvbø ` vglagrsg pfr n` ebtà, y `så suhgsbvègltg.

`. ¶Huàltf ng hfstø n` eàqubl` `n sgxtf prfpbgt`rbf6

m. Xb gn tft`n ag prfpbgt`rbfs j` sbaf fhjf, ¶huàn gs n` sue` tft`n pìà` pfr gs` eàqubl`6




@. ]gsugnv` n`s fpgr`hbflgs hfemblà`s sbiubgltgs. ]ghugrag sbgeprg n` cgr`rquå` ag n`s fpgr`hbflgs.

0) ‘2 + 7 Ù 0 + 8 5

=) 7 Ù 4 ù (‘=) + 4 5

8) 4 Ù 7 ‘ =; ù 7 5

1) 1 Ù 4 ù = + 0 5

7) 8 ù 8 + 1 ‘ 4 5

4) ‘8 ‘ 4 ù = ‘ 9 5

9) 2 Ù : ‘ 9 Ù 4 5

:) 8 + ; ‘ 7 ‘ 0 5

2) 9 Ù : ‘ 1 ‘ 0 5

0;) ‘= + 0; ù 7 5

00) = ‘ 8 + : ù (‘:) 5

0=) 1 ‘ 0 Ù ; ‘ 8 5

08) ‘: ‘ 1 + 4 ù = 5

01) ‘0 Ù 4 + = ‘ 8 5

07) 2 Ù 9 + 2 ù 8 5

04) ; ù 0 + : Ù 2 5

M. ]gsugnv` n`s fpgr`hbflgs hfemblà`s sbiubgltgs. ]ghugrag n` cgr`rquå` ag nfs sbilfs ag irup`hbøl.

0) _(2 + :) ‘ 1U Ù 8 5

=) (=; ù 1) ‘ 4 Ù = 5

8) 0 + 8 Ù (0 + 1) 5

1) 0; ù 7 ‘ (7 ‘ 1) 5

7) _(4 ‘ 9) Ù 7U + 1 5

4) (8 Ù 0) + = ‘ 0 5

9) ; ‘ _(1 + 1) ‘ 4U 5

:) = Ù 0 + (8 ‘ =) 5

2) (= Ù 8) ‘ (= + 8) 5

0;) _(2 ‘ 4) + 9U Ù 1 5

00) (7 ‘ 4) + 1 ‘ = 5

0=) _7 + (: ù 1)U ‘ 1 5

08) 9 ‘ _2 ù (4 ‘ 8)U 5

01) = Ù = ‘ (0 + :) 5

07) 7 Ù 1 ‘ (; + 0) 5

04) (9 + 8) Ù (0; ù 7) 5

H. ]gsugnv` n`s ghu`hbflgs sbiubgltgs. Gshrbm` su rgspugst` gl n` nålg` hfrrgspflabgltg.

0) ‘8 + e 5 0; e 5

=) : ‘ e 5 ‘0: e 5

8) 2 + e 5 ; e 5

1) 0 ‘ e 5 04 e 5

7) : + e 5 : e 5

4) ‘2 ‘ e 5 =0 e 5

9) 2 + e 5 02 e 5

:) ‘9 ‘ e 5 ‘09 e 5

2) ‘ e + 0: 5 01 e 5

0;) e ‘ 9 5 ‘=8 e 5

00) e + 00 5 == e 5

0=) ‘ e ‘ 4 5 =9 e 5

08) e + : 5 0= e 5

01) e + ; 5 ‘2 e 5

07) ‘ e + : 5 ‘= e 5

04) e ‘ : 5 08 e 5






prfhgsflf

nfiràf

A g s p u â s a g

g s t u

a b ` r . . . ]gp`sf nfs hfltglbafs ag n` sge`l` 0 ` n` 9.

]gsugnvf nfs gcgrhbhbfs ag rgp`sf p`r` gv`nu`reg gl n` prugm` p`rhb`n.

Eg sbgltf mbgl prgp`ràf p`r` n` prugm` p`rhb`n.

Frbglt`hbflgs sfmrg n` prugm` p`rhb`n

¡Nngiø gn efegltf ag n` prugm`!

R` gstà nbstf p`r` su prbegr` prugm` ag E`tgeàtbh`. Ng prgsgltèfs n`s óntbe`s rghfegla`hbflgs

qug pugagl `yua`rng ` n` jfr` agn gxègl.

Lf sg "`t`squg" gl lbliól gcgrhbhbf. Gepbghg pfr n`s prgiult`s qug sgp` egcfr y ng quga`rà eàs

tbgepf p`r` pgls`r gl n`s qug tgli` aua`s.

@n dbl`nbz`r su gxègl, rgng` tfa`s sus rgspugst`s y vg` sb `nif sg ng p`sø pfr `ntf.

[rgsgltg su prugm` nbepb` y fraglà`.

¡Àlbef! Gn rgsuntàf ag su gxègl sgrà gn prfauhtf ag su gsdugrzf.

Irupf3 [fnfhjbh [rugm`3 p`rhb`n @ ‘ =;04E`tgrb`3 E`tgeàtbh` [rbegr sgegstrg

Lfemrg3

H`rlâ3

Hårhunf ag gstuabf L³3 Dghj`3

[ultgf3

B XG]BG 0 pultf hà` rgspugst` hfrrght`. Qft`n 7 pultfs.

BLXQ]WHHBFLGX3 E`rqug hfl ul` "Z" gn huàrf ag n` fphbøl hfrrght`.

0) ¶Huàn ag n`s gxprgsbflgs sbiubgltgs gs ul`prfpfsbhbøl6

Jgrefs` lfhjg ¶Puâ t`n tg j` baf6 9 gs ul lóegrf prbef Gn cìu`r gs ul `lbe`n mgnnf.

Ng` `tgltègltg n`s

blstruhhbflgs ltgs

ag hfltgst`r. Xb tbglg

aua`, hflsuntg ` sufrbgltàfr(`).

@n rghbmbr n` prugm`,y `ltgs ag gepgz`r

` rgsfnvgrn`, gshrbm`

su lfemrg, lóegrf

ag h`rlâ, lóegrf ag

hårhunf ag gstuabf y

dghj`.





20

= 8 1

8

1

7

=

0

;

x

y

‘ 0 5 b

x = +

y = 5

r =

[ftglhb`hbøl ag gxprgsbflgs `nigmr`bh`s


]gp`sf ag n` pftglhb`hbøl ag lóegrfs rg`ngs

]gin`s ag n` pftglhb`hbøl y sbepnbh`hbøl ag gxprgsbflgs `nigmr`bh`s

[ftglhb`hbøl ag lóegrfs rg`ngs y ag gxprgsbflgs `nigmr`bh`s

Àrg`s ag iur`s igfeâtrbh`s


@pnbh`r n`s rgin`s ag pftglhb`hbøl ` gxprgsbflgs `nigmr`bh`s.

Xbepnbh`r gxprgsbflgs `nigmr`bh`s. [r`htbh`r n` ibnbaà ag hànhunf hfl pftglhb`s ag lóegrfs rg`ngs

y gxprgsbflgs `nigmr`bh`s.

Wtbnbz`r n`s gxprgsbflgs `nigmr`bh`s p`r` h`nhun`r àrg`s ag iur`s igfeâtrbh`s.




¡[`r` hfeglz`r!]gp`sf ag n` pftglhb`hbøl ag lóegrfs rg`ngs

@prglabefs gl n` sge`l` 8 sfmrg gn hflcultf ag nfs lóegrfs rg`ngs y n`s rgin`s

ag n` pftglhb`hbøl, rghfragefs `niul`s ag gnn`s.

Wl` pftglhb` sg rgprgsglt` ag dfre` sbemønbh` pfr3

`l 5 ` • ` • ` . . . • `

Aflag3

• ` rgprgsglt` n` m`sg y gs gn lóegrf qug sg

euntbpnbh` pfr så ebsef.

• l gs gn gxpflgltg g blabh` gn lóegrf ag

vghgs qug n` m`sg sg euntbpnbh` pfr så ebse`.

[fr gcgepnf3 81 5 8 • 8 • 8 • 8 5 :0

(‘7)8 5 (‘7)(‘7)(‘7) 5 ‘0=7

[rfauhtf ag pftglhb`s ag biu`n m`sg

Hu`laf sg euntbpnbh`l pftglhb`s ag biu`n m`sg, sg hfpb` n` m`sg y sg sue`l nfs

gxpflgltgs.

[fr gcgepnf3 4= • 41 5 4= + 1 5 44

(‘;.7)1(‘;.7)8 5 (‘;.7)1 + 8 5 (‘;.7)9

Hfhbgltg ag pftglhb`s ag biu`n m`sg

[`r` abvbabr pftglhb`s qug tbglgl biu`n m`sg, sg hfpb` n` m`sg y sg rgst`l nfs

gxpflgltgs.

[fr gcgepnf30;4

0;1 5 0;4 ‘ 1 5 0;=

(‘;.7)7

(‘;.7)= 5 (‘;.7)7 ‘ = 5 (‘;.7)8

¡@ tr`m`c`r!

@pnbqug n`s rgin`s ag n` pftglhb`hbøl qug jgefs rgp`sàf y gxprgsg hà` fpgr`hbøl hfef ul` sfn`pftglhb`. Qbglg ul gcgepnf.

;) 78 • 71 5 =) (‘:)4(‘:)1 5 1)(2)4

(2)8 5

0) 87 • 87 5 8) (‘=)8(‘=)7 5

`lm`sggxpflgltg

79




0. ]gin`s ag n` pftglhb`hbølN`s rgin`s ag n` pftglhb`hbøl p`r` gxprgsbflgs `nigmr`bh`s sfl n`s ebse`s

qug vbefs gl gn hflcultf ag nfs lóegrfs rg`ngs, n` abdgrglhb` gs qug `jfr` glnui`r ag lóegrfs sg utbnbz`l ngtr`s y gl `niulfs h`sfs n` hfembl`hbøl ag ngtr`sy lóegrfs. @ltgs ag `v`lz`r hfl n`s rgin`s ag n` pftglhb`hbøl, jìèfs ul`hfep`r`hbøl gltrg n`s pftglhb`s hfl lóegrfs rg`ngs y gxprgsbflgs `nigmr`bh`s.

[ftglhb`hbøl ag lóegrfsrg`ngs

[ftglhb`hbøl aggxprgsbflgs nigmr`bh`s

18 5 1 • 1 • 1 x8 5 x • x • x

(‘4)1 5 (‘4)(‘4)(‘4)(‘4) (‘ e)1 5 (‘ e)(‘ e)(‘ e)(‘ e)

(07 )8

5 (07 )(07 )(07 ) ( y

x )8

5 ( y

x )( y

x )( y

x )

0.0 ]gin` agn gxpflgltg hgrf

N` rgin` agn gxpflgltg hgrf blabh` qug hu`nqubgr m`sg gngvà` n gxpflgltg hgrfgs biu`n ` n` ulbaà (0), gxhgptf gl gn h`sf qug n` m`sg sg` hgrf.

x ; 5 0

Gcgepnf3

o ; 5 0

( xy); 5 0

(‘1wz ); 5 0

0.= ]gin` agn prfauhtf ag pftglhb`s hfl biu`nm`sg

[`r` euntbpnbh`r gxprgsbflgs `nigmr`bh`s qug tbglgl n` ebse` m`sg, sg hfpb` n`m`sg y sg sue`l nfs gxpflgltgs.

x e • x l 5 x e + l

Gcgepnf3`7 • `8 5 `7 + 8 5 `:

x9 • x: 5 x9 + : 5 x 07

(8 y)=(8 y)= 5 (8 y)= + = 5 (8 y)1





0.8 ]gin` agn hfhbgltg ag pftglhb`s ag biu`nm`sg

[`r` abvbabr gxprgsbflgs `nigmr`bh`s qug tbglgl n` ebse` m`sg y abdgrgltg gxpf-lgltg, sg e`ltbglg n` m`sg y sg rgst`l nfs gxpflgltgs.

x e

x l 5 x e ‘ l

Gcgepnf3m:

m7 5 m: ‘7 5 m8

(‘7 z )4

(‘7 z )1 5 (‘7 z )4 ‘ 1 5 (‘7 z )= 5 =7 z =

(= )7

(= )9 5

0

(= )9 ‘ 7 5

0

(= )= 5

0

1`=

0.1 ]gin` ag ul` pftglhb` gngvà` ` ftr`pftglhb`

Xb n` gxprgsbøl nigmr`bh` sg glhugltr` agltrf ag sbilfs ag irup`hbøl, sg hfpb`n` m`sg y sg euntbpnbh`l nfs gxpflgltgs.

( x e)l 5 x e • l

Gcgepnf3( 1)= 5 `1 • = 5 `:

(w8

y1

z =

)1

5 (w8

)1

( y1

)1

( z =

)1

5 w0=

y04

z :

(‘7`8m:h=)8 5 (‘7)8( 8)8(m:)8(h=)8 5 ‘0=7`2m=1h4

0.7 ]gin` ag n` pftglhb` ag ul` dr`hhbøl`nigmr`bh`

Xb n` gxprgsbøl `nigmr`bh` gs ul` dr`hhbøl, sg hfpb` n` m`sg y sg euntbpnbh` gngxpflgltg agn luegràfr y agn aglfeblàfr pfr gn gxpflgltg gxtgrlf.

( x y )e 5

x e

ye

Gcgepnf3

( `x

my )4

5`4 x 4

m4 y4

( ‘8e=l1

=w8 z = )=

5(‘8)=(e=)=(l1)=

(=)=(w8)=( z =)= 5

2e1l:

1w4 z 1

( 7w: x= y7

8 p4t 8u9 )8 5

(7)8(w:)8( x=)8( y7)8

(8)8( p4)8(t 8)8(u9)8 5

0=7w=1 x 4 y07

=9 p0:t 2u=0




Gcgrhbhbf 0Ng` n` rgin` blabhà` gl hà` p`rtàf y pnåqugn` n`s gxprgsbflgs nigmr`bh`s ag dfre` abrght`. J`yul gcgepnf p`r` hà` h`sf.

@. Gxpflgltg hgrf

;) 7(= y); 5 8) ‘7(= x); 5

0) (= z ); 5 1) ( ‘8 x7

y1 );

5

=) (‘7 p=q=r ); 5 7) :j(7 x0; y8); 5

M. [rfauhtf ag pftglhb`s ag biu`n m`sg

;) `9 • `= • `8 5 8) (el)8(el)1 5

0) x7 • x8 • x 5 1) (=wy)8(=wy) 5

=) e7 • e7 • e7 5 7) (rs)=(rs)7(rs) 5

H. Hfhbgltg ag pftglhb`s ag biu`n m`sg

;)e0;

e4 5 1)

(‘1 z )1

(‘1 z )1 5

0)m2

m8 5 7)

( pqr )4

( pqr )1 5

=)w07

w: 5 4)

(7el)4

(7el)1 5

8) y=1

y0= 5 9) (‘8 xyz )

7

(‘8 xyz )07 5

A. [ftglhb` ag ftr` pftglhb`

;) ( =m7h4)= 5 1) ( p7r 9)= 5

0) ( x:)= 5 7) (‘7`m:h=); 5

=) (el1)1 5 4) ‘1(e=j8o 1)= 5

8) (w= y4)8 5 9) ‘7`8(m7ha =)8 5

G. [ftglhb` ag ul` dr`hhbøl

;) ( 8`

=m )1

5 8) (7e8l=

=w1 z 8 )=

5

0) ( x y)4

5 1) ( 8j1o 1

7 p1q4 )=

5

=) (`m

ha )4

5 7) ( `1m9h8

w= x8 y= )1

5

7(0) 5 7

`0=

e1

`1m0;h0=

:0`1

04m 1




0.4 ]gin` agn gxpflgltg lgi`tbvf

`. Wl` nbtgr`n hfl gxpflgltg lgi`tbvf sg hflvbgrtg gl ul` pftglhb` pfsbtbv`gshrbmbglaf gn blvgrsf ag n` m`sg hfl gn ebsef gxpflgltg.

x ‘ e 5 0

x

e

Gcgepnf3i ‘7 5

0

i 7

(1`m) ‘= 50

(1`m)= 5

0

04`=m=

: p ‘4 5 : •0

p4 5

:

p4

m. Wl` dr`hhbøl `nigmr`bh` hfl gxpflgltg lgi`tbvf sg hflvbgrtg gl ul` pf-tglhb` pfsbtbv` gshrbmbglaf gn blvgrsf ag n` m`sg hfl gn ebsef gxpflgltg.

( `

m ) ‘ e

5 ( m

` )e

Gcgepnf3

( x y ) ‘7

5 ( y x )7

5 y7

x7

(=m

8` ) ‘=

5 (8`

=m )=

52`=

1m=

(7w8 x=

1 y= z = ) ‘8

5 ( 1 y= z =

7w8 x= )8

541 y4 z 4

0=7w2 x4

Gcgrhbhbf =@pnbqug n` rgin` agn gxpflgltg lgi`tbvf p`r` rgsfnvgr nfs gcgrhbhbfs sbiubgltgs.

0) ( `m ) ‘7

5 =) ( 8 x

1 y ) ‘8

5

8)

(7m=

8h8

)

‘=

5 1)

(`=m8

h1a 7 )

‘8

5

7) (1e8l=

= p1q8 ) ‘8

5 4) ( ‘7w 1 x7

y8 z 9 ) ‘8

5




=. Xbepnbdbh`hbøl ag gxprgsbflgs`nigmr`bh`s

Xbepnbh`r ul` gxprgsbøl `nigmr`bh` dfreà` hfl pftglhb`s ag lóegrfs rg`ngsgs tr`lsdfre`rn` ` ul` gxprgsbøl, gl n` qug hà` lóegrf rg`n `p`rgzh` sfnf ul`

vgz y tfafs nfs gxpflgltgs sg`l pfsbtbvfs.Xbepnbqugefs

• @pnbhèfs n` rgin` ag ul` pftglhb` gngvà`` ftr` pftglhb` gl gn d`htfr agrghjf.

• @pnbhèfs n` rgin` agn prfauhtf.

• Fmtglgefs gn rgsuntàf.

Xbepnbqugefs

• @pnbhèfs n` rgin` agn hfhbgltg ag pftglhb`sag biu`n m`sg.

• Hfef x: tbglg gn gxpflgltg e`yfr, sg quga`gl gn aglfeblàfr y sg ng rgst` gn gxpflgl-tg ag x4 y fmtglgefs gn rgsuntàf.

Xbepnbqugefs

• @pnbhèfs n` rgin` agn hfhbgltg ag pftglhb`sag biu`n m`sg gl n` dr`hhbøl.

• @pnbhèfs n` rgin` ag n` pftglhb` ag ul`pftglhb`.

• Fmtglgefs gn rgsuntàf.

Xbepnbqugefs

• @pnbhèfs n` rgin` agn prfauhtf ag pftgl-

hb`s ag biu`n m`sg.• Fmtglgefs gn rgsuntàf.

Gl `niulfs h`sfs, p`r` sbepnbh`r ul` gxprgsbøl `nigmr`bh` gs pfsbmng `pnbh`reàs ag ul` rgin` sbl cgr`rquå` niul`, gs aghbr, lf gxbstg ul fragl aglbaf p`r`fpgr`r.

(8`=h1a 7)(=`=ha 8)1 5

(8`=h1a 7)(04`:h1a 0=) 5

8 • 04`= + :h 1 + 1a 7 + 0= 5

1:`0;h:a 09

=7 x4 y2 z :

7 x:

y4

z =

5

7 y2 ‘ 4 z : ‘ =

x: ‘ 4 5

7 y8 z 4

x =

(1w: x1 y7

=w7 x= y8 )1

5

(=w:

‘ 7

x1

‘ =

y7

‘ 8

)1

5

(=w8 x= y=)1 5

04w0= x : y:

(0

= x1 y4 z :)(8

1 x= y8 z 7)

8

: x1 + = y4 + 8 z : + 7 5

8

: x 4 y2 z 08




Gcgrhbhbf 8Xbepnbqug n`s gxprgsbflgs `nigmr`bh`s ag dfre` abrght`. Qbglg ul gcgepnf.

;) ( 04w7 x4 y2

1w= x1 y8 )8

5 0) (8`=m1h8)(=`1m7h=) 5

(1w8 x= y4)8 5

41w2 x 4 y0:

=) (=`1m8h=)(8`8mh7)8 5 8) (‘= x8 y1)(4 x= y4) 5

1) (7w8 x= y1)(8 x= y8)8 5 7)0: p:q4r 7

4 p7q=r 5

4)41`0;m:h0=

:`1m8h9 5 9) (07 x4 y2 z 8

7 x1 y7 z 8 )1

5

:) ( w4 x9 y1

w1 x8 y= )7 5 2) ( 7

1 x4 y1 z 4 )( =8 x: y= z ) 5




]gsuegl

0. N`s rgin`s ag n` pftglhb`hbøl ag gxprgsbflgs `nigmr`bh`s sfl n`s ebse`s qug sg gepng`l gl gnhflcultf ag nfs lóegrfs rg`ngs. [`r` `pnbh`rn`s, agmgefs prgst`r `tglhbøl ` nfs gxpflgltgs qugn`s `hfep`ð`l.

0.0 ]gin` agn gxpflgltg hgrf

Blabh` qug hu`nqubgr m`sg gngvà` n gxpflgltg hgrf gs biu`n n` ulbaà(0), gxhgptf gl gn h`sf ag qug n` m`sg sg` hgrf.

0.= ]gin` agn prfauhtf ag pftglhb`s hfl biu`n m`sg

[`r` euntbpnbh`r gxprgsbflgs `nigmr`bh`s qug tbglgl n` ebse` m`sg,hfpbèfs n` m`sg y sueèfs nfs gxpflgltgs.

0.8 ]gin` agn hfhbgltg ag pftglhb`s ag biu`n m`sg

[`r` abvbabr gxprgsbflgs nigmr`bh`s qug tbglgl n` ebse` m`sg y abdgrgltg

gxpflgltg, sg e`ltbglg n` m`sg y sg rgst`l nfs gxpflgltgs.

0.1 ]gin` ag ul` pftglhb` gngvà` ` ftr` pftglhb`

Xb n` gxprgsbøl nigmr`bh` sg glhugltr` agltrf ag sbilfs ag irup`hbøl,sg hfpb` n` m`sg y sg euntbpnbh`l nfs gxpflgltgs.

0.7 ]gin` ag n` pftglhb` ag ul` dr`hhbøl `nigmr`bh`

Xb n` gxprgsbøl nigmr`bh` gs ul` dr`hhbøl, sg hfpb` n` m`sg y sg euntbpnbh`gn gxpflgltg agn luegràfr y agn aglfeblàfr pfr gn gxpflgltggxtgrlf.

0.4 ]gin` agn gxpflgltg lgi`tbvf`. Wl` nbtgr`n hfl gxpflgltg lgi`tbvf, sg hflvbgrtg gl ul` pftglhb`

pfsbtbv` gshrbmbglaf gn blvgrsf ag n` m`sg hfl gn ebsef gxpflgltg.

m. Wl` dr`hhbøl `nigmr`bh` hfl gxpflgltg lgi`tbvf, sg hflvbgrtg glul` pftglhb` pfsbtbv` gshrbmbglaf gn blvgrsf ag n` m`sg hfl gn ebsefgxpflgltg.

=. Xbepnbdbh`hbøl ag gxprgsbflgs nigmr`bh`s

Xbepnbdbh`r ul` gxprgsbøl `nigmr`bh` dfreà` hfl pftglhb`s ag lóegrfs rg`ngs gs tr`lsdfre`rn`` ul` gxprgsbøl, gl n` qug hà` lóegrf rg`n `p`rgzh` sfnf ul` vgz y tfafs nfs gxpflgltgs sg`lpfsbtbvfs.

@låegsg ` vbsbt`r n` pàibl` wgm qug ng suigrbefs p`r` rgdfrz`r y èpnb`r sus hflfhbebgltfs sfmrgpftglhb`hbøl3 jttp3//iff.in/Zo1Bzrn

x ; 5 0

x e • x l 5 x e + l

x e

x l 5 x e ‘ l

( x e)l 5 x e • l

( x y )e 5

x e

ye

x ‘ e 5 0

x e

( `

m ) ‘ e

5 ( m

` )e






E`rqug hfl ul` "Z" gn huàrf ag n` fphbøl qug prgsglt` n` rgspugst` hfrrght` ` hà` prgiult`.

0) ¶Huàn gs gn gxpflgltg gl n` gxprgsbøl ‘1`=6

=) ¶Puâ rgsuntàf sg fmtbglg ag n` pftglhb` 0;;;6

8) ¶Huàn gs gn rgsuntàf ag (‘= y)

8

6

1) ¶Puâ rgsuntàf gxprgs` ul` hfrrght` `pnbh`hbøl ag n`s rgin`sag n` pftglhb`hbøl ag gxprgsbflgs `nigmr`bh`s6

7) ¶Huàn gs gn rgsuntàf sbepnbdbhàf ag

( x4

x7

)

=

6

4) ¶Puâ gxprgsbøl gs gqubv`ngltg ` n` sbiubgltg3 ` • ` • ` • ` • `6

9) ¶Puâ gxprgsbøl gs hfrrght`6

:) ¶Puâ gxprgsbøl huepng hfl n` rgin` agn gxpflgltg hgrf6

@utfhfltrfn

= `

1

‘1

;

0

0;

0;;

‘: y ‘= y8

‘4 y4

‘: y8

x ‘0 5 ‘ x

x8 ù x= 5 x7

x8 • x8 5 x4

(= x8)8 5 4 x4

x=

x0=

x0;

x ‘=

`

`7

7`

7`7

7 x; 5 x

7 x; 5 ;

7 x; 5 0

7 x; 5 7

(8 xy); 5 0

(8 xy); 5 ;

(8 xy); 5 8

(8 xy); 5 xy





@. @pnbqug n` rgin` agn gxpflgltg hgrf p`r` sbepnbdbh`r n`s gxprgsbflgs `nigmr`bh`s. J`y ul gcgepnf.

;) ( m); 5

0) (7e); 5

=) ‘7(= x); 5

8) (04 x1 y1); 5

1) ( 8j1

7o );

5

7) (:e7

2l= );

5

4)

(

07m:

0;m4

)

;

5

9) ( ‘ m7

0=`m );

5

:) 8 x(9 x7 y1); 5

2) 0;(1`:m8); 5

0;) xy(7 x0; y8); 5

00) ‘=7( p=q8); 5

0=) 7el(0;;); 5

08) 0:j=(ha 0=); 5

M. @pnbqug n` rgin` agn prfauhtf ag pftglhb`s ag biu`n m`sg p`r` sbepnbdbh`r n`s gxprgsbflgs nigmr`bh`sJ`y ul gcgepnf.

;) `1 • `7 • `= 5

0) x4 • x1 5

=) z : • z = 5

8) w: • w9 5

1) m7

• m8

• m4

5 7) j4 • j1 • j= 5

4) m7 • m8 • m4 5

9) a : • a 7 • a 4 5

:) h0; • h0= • h ‘1 5

2) ( xy)7( xy): 5

0;) ( m)1( m) 5

00) ( pq):

( pq)2

5 0=) (1w)1(1w)= 5

08) ( st )8( st )9( st )1 5

H. @pnbqug n` rgin` agn hfhbgltg ag pftglhb`s ag biu`n m`sg p`r` sbepnbdbh`r n s gxprgsbflgs `nigmr`bh`sJ`y ul gcgepnf.

;)(0; p)4

(0; p)= 5

0)

`0=

`0; 5

=)h4

h8 5

8)m:

m: 5

1)t 2

t 0; 5

7)o 2

o : 5

4)

z 04

z 4 5

9) y07

y: 5

:) x0;

x0= 5

2)(el)7

(el)8 5

0;)(wxy)2

(wxy)2 5

00)

(1wx)07

(1wx)7 5

0=)(=7rs)4

(=7rs)01 5

08)(‘7 m)0=

(‘7 m)0; 5

01)(‘07w)01

(‘07w)0= 5

0

`00

(0; p)1




A. @pnbqug n` rgin` ag ul` pftglhb` gngvà` ` ftr` pftglhb` p`r` sbepnbdbh`r n`s gxprgsbflgs `nigmr`bh`s.J`y ul gcgepnf.

;) (‘7 =m1h8)8 5

0) (e8)8 5

=) (w:)= 5

8) ( 1m8)1 5

1) (=w8 x=)8 5

7) (‘7e1l:)= 5

4) (8`1m8h=)= 5

9) (=7 x4 y: z 4); 5

:) ( `1

m7

h=a 8 )8

5

2) (w1 x:

y= z 1 )9

5

0;) ( ‘8e1

7l8 )=

5

00) (8wx1

= y= )8

5

G. @pnbqug n` rgin` ag n` pftglhb` ag ul` dr`hhbøl y n` pftglhb` ag ftr` pftglhb` p`r` sbepnbdbh`r n`sgxprgsbflgs `nigmr`bh`s. J`y ul gcgepnf.

;) (8e

=l )8

5

0) ( `

m )7

5

=) ( `8

m8 )1

5

8) ( p7

q8 )=

5

1) ( j=

o 8 )4 5

7) ( `=m1

h7a 8 )8

5

4) ( w7 x4

y1 z : )=

5

9) (=`=

8m8 )8

5

:) ( 1 x8

7 y1 )8

5

2) ( ‘=w

‘8 z 7 )=

5

0;) ( 7 x= y=

8w8 z 8 )= 5

00) (0;e1l7

0; p1q1 )8

5

D. @pnbqug n` rgin` agn gxpflgltg lgi`tbvf p`r` sbepnbdbh`r n`s gxprgsbflgs. J`y ul gcgepnf.

;) (8 x

= y ) ‘1

5

0) x ‘: 5

=) ` ‘8 5

8) m ‘= 5

1) (7 x) ‘= 5

7) (‘8 y) ‘8 5

4) 1 x(8 y=) ‘= 5

9) 7`m(=h1) ‘8 5

:) 1 x(= y= z 8) ‘= 5

2) (8m

1`) ‘=

5

0;) ( 7 y=

8 x8 ) ‘=

5

00) (=m8

8h1 ) ‘8

5

‘0=7`4m0=h2

=9e8

:l8

( = y

8 x )1

504 y1

:0 x1






@. Gngvg hà` lóegrf ` n` pftglhb` blabhà`. Qbglg ul gcgepnf.

;) ‘71 5

0) 18 5

=) 2= 5

8) 88 5

1) =; 5

7) := 5

4) 02 5

9) ‘1= 5

:) ‘2; 5

2) ‘9= 5

0;) ‘88 5

00) ‘4= 5

0=) ‘0: 5

08) ‘1; 5

01) (‘8)8 5

07) (‘9)= 5

04) (‘2)= 5

09) (‘=)7 5

0:) (‘7)8 5

02) (‘0); 5

=;) (‘1)8 5

M. Abvba` n`s gxprgsbflgs `nigmr`bh`s. Qbglg ul gcgepnf.

;)`:

`4 5

0)m7

m= 5

=)h:

h8 5

8) x2

x= 5

1) y:

y1 5

7) z 0=

z 0= 5

4)w4

w8 5

9) y7

y; 5

:) x1

x= 5

2)t 2

t 7 5

0;)h01

h2 5

00) z 0;

z 7 5

0=) p0;

p8 5

08)m0=

m1 5

01)`7

`8 5

07)h:

h4 5

04)o 2

o 1 5

09)q00

q: 5

0:) x:

x ‘= 5

02)j4

j ‘4 5

=;) y0;

y ‘4 5

H. Gngvg hà` gxprgsbøl `nigmr`bh` n` pftglhb` blabhà`. Qbglg ul gcgepnf.

;) (‘8 x8)1 5

0) (:m8)= 5

=) (= z 7)1 5

8) (7 y=)8 5

1) (‘4`1)= 5

7) (0;h=)8 5

4) (9a 7)= 5

9) (1 z 4)8 5

:) (8`4)1 5

2) (: y0;); 5

0;) (‘2 x1)= 5

00) (‘7h4)= 5

0=) (: y8)= 5

08) (4m1)= 5

01) (‘2a :); 5

07) (‘= x8)1 5

04) ( ‘2 z 8)= 5

09) (‘7w9)8 5

`=

‘4=7

:0 x0=





Glhugltrg ul` gxprgsbøl `nigmr`bh` p`r` h`nhun`r gn àrg` tft`n ag hà` iur`.

Gn gcgepnf qug gstà ` hfltblu`hbøl prgsglt` nfs p`sfs ` sgiubr. Wstga pugag rgsfnvgr nfs ageàs

gcgrhbhbfs ag dfre` abrght`.;)

x

x

Gn àrg` ag ulf ag nfs huàràfs gs3 @ 5 x • x 5 x=

[`r` glhfltr`r gn àrg` tft`n, agmgefs euntbpnbh`r gn àrg` agn huàràf pfr 4.

Gl dfre` `nigmr`bh` gs3 @tft`n 5 4 x =

0) y

y

=)

x

x

x

x x

8) x

x

x

y y

y







0;0

= 8 1

8

1

7

=

0

;

x

y

‘ 0 5 b

x = +

y = 5

r =

]àbh`hbøl ag gxprgsbflgs `nigmr`bh`s


Biu`n lóegrf ag n`s y hfnuel`s

]gin`s ag nfs ràbh`ngs p`r` gxprgsbflgs `nigmr`bh`s Xue`, rgst`, euntbpnbh`hbøl y abvbsbøl ag ràbh`ngs hfl gxprgsbflgs `nigmr`bh`s

Gnbebl`hbøl ag d`htfrgs gl nfs ràbh`ngs ]`hbfl`nbz`hbøl ag gxprgsbflgs `nigmr`bh`s

[rfmnge`s ag àrg`, pgråegtrf y vfnuegl, `pnbh`laf n` ràbh`hbøl


@pnbh`r n`s rgin`s ag nfs ràbh`ngs ` gxprgsbflgs `nigmr`bh`s.

]g`nbz`r fpgr`hbflgs `rbteâtbh`s hfl gxprgsbflgs qug hfltbglgl ràbh`ngs. Xbepnbh`r ràbh`ngs qug hfltbglgl gxprgsbflgs `nigmr`bh`s.

]`hbfl`nbz`r gxprgsbflgs `nigmr`bh`s.

@pnbh`r n` ràbh`hbøl ` n` sfnuhbøl ag prfmnge`s.

070E`tgeàtbh` ∝ Xge`l` 0;



¡[`r` hfeglz`r!Biu`n lóegrf ag dbn`s y hfnuel`s

Beìblg qug ustga tbglg ul lgifhbf ag `nqubngr ag sbnn`s y egs`s p`r` dbgst`s y

ul hnbgltg ng pbag qug, gl ul s`nøl ag dfre` huàrà`, hfnfqug sbnn`s ` n` ebse`

abst`lhb`, ag t`n e`lgr` qug, j`y` n` ebse` h`ltbaà ag dbn`s y hfnuel`s.

Gstfs sfl nfs a`tfs qug s`mg3

Gn `nqubngr ag hà` sbnn` hugst` P8.;;.

N` h`ltbaà eàxbe` qug gn hnbgltg qubgrg pì`r pfr gn `nqubngr ag n`s sbnn`s gs

P18=.;;.

Hfl n` bldfre`hbøl prgsgltà` hfeglhgefs ` rgsfnvgr gn prfmnge`.

[rbegrf, agmgefs hflfhgr gn lóegrf ag sbnn`s qug sg pugag utbnbz`r, p`r` gnnf

abvbabefs n` h`ltbaà ag ablgrf abspflbmng P18=.;; agltrf ag P8.;;, qug gs gn

prghbf agn `nqubngr ag hà` sbnn`. P18=.;; ù P8.;; 5 011

@jfr` qug y` hflfhgefs gn lóegrf ag sbnn`s ` utbnbz`r, pfagefs `vgrbiu`r gn

lóegrf ag dbn`s y hfnuel`s, hfef gn gsp`hbf gs huàràf h`mg n` ebse` h`l-

tbaà ag dbn`s y hfnuel`s. Wtbnbz`rgefs n` ngtr` x p`r` rgprgsglt`r n`s dbn`s y n`s

hfnuel`s qug gs gn a`tf agshflfhbaf.

Hfef s`mgefs qug gn àrg` qug fhup`ràl n`s 011 sbnn`s gs huàrà`, gst` sg

fmtbglg `n euntbpnbh`r x • x 5 x=

[`r` fmtglgr gn v`nfr ag x, gnbeblèfs n` pftglhb` qug`hfep`ð` ` n` v`rb`mng, `pnbh`laf r`åz huàrà` gl èmfs

ebgemrfs ag n` ghu`hbøl.

Gn v`nfr ag x gs 0=, pfr nf t`ltf sg agmgl hfnfh`r 0= dbn`s y 0= hfnuel`s hfl

0= sbnn`s gl hà` ul`.

[`r` rgsfnvgr gstg pn`ltgèbgltf dug lghgs`rbf j`hgr usf ag nfs ràbh`ngs, tge`

qug tr`t`rgefs gst` sge`l`.

¡@ tr`m`c`r!

Xupfli` qug ustga tbglg ul tgrrglf ag dfre` huàrà` hfl ul àrg` ag 4=7 e=, agstblàf p`r`sgemr`r h`dâ. ¶Huàlt`s pn`lt`s pugag sgemr`r, abstrbmuba`s gl n` ebse` h`ltbaà ag n`s y hfnuel`s,sb tfa`s gst`ràl ` 0 egtrf ag abst`lhb`6

x= 5 011

x= 5 011

x 5 0=






=. Hflvgrsbøl ag ul` r`åz ` ul`pftglhb` dr`hhbfl`rb`

Gst` hflvgrsbøl sg rg`nbz` gl `niulfs h`sfs p`r` d`hbnbt`r n`s fpgr`hbflgs hflràbh`ngs, y` qug rgsunt` eàs dàhbn tr`m`c`r hfl gxpflgltgs.

Wl ràbh`n gs gqubv`ngltg ` ul` pftglhb` dr`hhbfl`rb` hu`laf gn aglfeblàfrag n` dr`hhbøl gs gn ålabhg agn ràbh`n y gn luegràfr gs gn gxpflgltg agn ràb-h`laf. [rgstg `tglhbøl `n rghuàrf y ` nfs gcgepnfs.

e l 5 `l/e

`8 5 `8/=8 x2 5 x2/8 5 x8

1 m7 5 m7/1

7 z 0; 5 z 0;/7 5 z =

1

w

9

5 w9/1

( x

y )1

5 ( x

y )1/=

5 ( x

y )=

[`r` hflvgrtbr ul gxpflgltg dr`hhbfl`rbf ` ràbh`n, sfnf j`hgefs n` fpgr`hbølblvgrs` ` nfs h`sfs `ltgrbfrgs. Tgèfs `niulfs gcgepnfs.

x=/7 57 x= e9/: 5

: e9

y8/1 51 y8 ( pq )

8/1

5 ( pq )8

1

x4/2 52 x4 ( w

z )=/7

5 ( w

z )=

7

Gcgrhbhbf =@. Hflvbgrt` nfs ràbh`ngs gl pftglhb`s dr`hhbfl`rb`s y sbepnbdbqug gn rgsuntàf aflag sg` pfsbmng.

Qbglg ul gcgepnf.

;)8 4 5

0)7 : 5

=) a :

5

8)8 (=; x)1 5

1)1 (:l)= 5

7)

(

p

q )4 5

4) ( jo )=

5

9): ( yz )1 5

:)

7

( x=

y8

) 5 M. Hflvbgrt` nfs gxpflgltgs dr`hhbfl`rbfs gl ràbh`ngs. Qbglg ul gcgepnf.

;) ( m)=/7 5

0) x8/1 5

=) w7/9 5

8) z =/8 5

1) (el)=/2 5

7) ( pq)8/1 5

4) ( jo )7/2

5

9) ( r

s )8/7

5

:) ( xyz )0/8 5

`4/8 5 `=

7 ( m)=

[`r` gvbt`r nfs v`nfrgs`msfnutfs, tfa`sn`s nbtgr`ngs ag nfsràbh`lafs rgprgsglt`llóegrfs rg`ngspfsbtbvfs.






8.= Euntbpnbh`hbøl ag ràbh`ngs

[`r` euntbpnbh`r ràbh`ngs, hflsbagrgefs nfs h`sfs sbiubgltgs3

`. ]àbh`ngs qug tbglgl gn ebsef ålabhg

Gl gstg h`sf agmgefs `pnbh`r n` rgin` agn prfauhtf ag r`åhgs. [rgstg `tglhbøl `n

rghuàrf y ` nfs p`sfs ag nfs gcgepnfs.

l • l

m 5 l` • m

Euntbpnbqugefs

• Gshrbmbefs nfs ràbh`lafs gl ul sfnf ràbh`n.

• Euntbpnbhèfs nfs ràbh`lafs y hflvgrtbefs gnràbh`n gl ul gxpflgltg dr`hhbfl`rbf.

• Xbepnbdbhèfs gn rgsuntàf.

Euntbpnbqugefs

• Gshrbmbefs nfs ràbh`lafs gl ul sfnf ràbh`n.

• Euntbpnbhèfs nfs ràbh`lafs. Gl gstg h`sf, gnrgsuntàf lf sg pugag sbepnbdbh`r.

m. ]àbh`ngs qug tbglgl abdgrgltg ålabhg g biu`n ràbh`laf

Hu`laf nfs ålabhgs sfl abdgrgltgs, hà` ràbh`n sg hflvbgrtg gl gxpflgltg dr`h-

hbfl`rbf y sg `pnbh` n` euntbpnbh`hbøl ag gxpflgltgs hfl m`sgs biu`ngs. Xb n`lugv` pftglhb` gs ul` dr`hhbøl, sg hflvbgrtg lugvègltg gl ràbh`n, hfef sgeugstr` gl nfs gcgepnfs.

e •

l 5 `0/e • `0/l

Euntbpnbqugefs

• Hflvgrtbefs hà` ràbh`n gl gxpflgltg dr`hhbfl`rbf.

• Hfpbèfs n` m`sg y sueèfs nfs gxpflgltgs.

• Hflvgrtbefs gn lugvf gxpflgltg dr`hhbfl`rbf gl ràbh`n.

Euntbpnbqugefs

• Hflvgrtbefs hà` ràbh`n gl gxpflgltg dr`hhbfl`rbf.

• Hfpbèfs n` m`sg y sueèfs nfs gxpflgltgs.

• Hflvgrtbefs gn lugvf gxpflgltg dr`hhbfl`rbf gl ràbh`n.

18`7 •

1=9 8 5

18`7 • =9 8 5

1:0 : 5 8`:/1 5

8`=

:07 x= y8 •

:7 xy1 5

:07 x= y8 • 7 xy1 5

:97 x 8 y9

8`m • `m 5

( m)0/8 • ( m)0/= 5

( m)0/8 + 0/= 5 ( m)7/4 5

4

(`m)7

7( xy)8 • ( xy)7 5

( xy)8/7 • ( xy)7/= 5

( xy)8/7 + 7/= 5 ( xy)80/0; 5

0;( xy)80






8.8 Abvbsbøl ag ràbh`ngs

[`r` abvbabr ràbh`ngs qug tbglgl gn ebsef ålabhg y abdgrgltg ràbh`laf, j`hgefsusf ag n` rgin` agn hfhbgltg ag nfs ràbh`ngs? sg abvbagl nfs ràbhàfs y ag sgrpfsbmng sg sbepnbh` gn rgsuntàf. N` abvbsbøl ag ràbh`ngs sbiug gst` rgin`3

l

l m

5 l `m

Abvbaèfs

• Gxprgsèfs gn hfhbgltg hfef ul sfnf ràbh`n.

• Xbepnbdbhèfs gn ràbh`laf.

• Gshrbmbefs gn rgsuntàf.

Abvbaèfs

• Gxprgsèfs gn hfhbgltg hfef ul sfnf ràbh`n.

• Xbepnbdbhèfs gn ràbh`laf.


17`8

=7`=

5

1

=7

`8

`= 5

=7`8 ‘ = 5

=7`

4=; x7 y1

41 x= y8

5

4=; x7 y1

1 x= y8 5

47 x7 ‘ = y1 ‘ 8 5

47 x 8 y

Gcgrhbhbf 7[r`htbqug n` abvbsbøl ag ràbh`ngs hfl gn ebsef ålabhg y abdgrgltg ràbh`laf.

0)0:

2`4

42`=

5 =)4 x7 y1

4 x8 y=

5

8)9 =4e4l7

908e8l1

5 1): 04 p7q4r 1:1 p=q8r 1

5




8.1 Gnbebl`hbøl ag d`htfrgs gl nfs ràbh`ngs

Gl `niulfs h`sfs pfagefs sbepnbh`r ul ràbh`n gnbeblàf nfs d`htfrgs agnràbh`laf j`st` qug n`s pftglhb`s ag n` p`rtg lueârbh` y ag n`s nbtgr`ngs sg`leglfrgs qug gn ålabhg agn ràbh`n. Nfs gcgepnfs sbiubgltgs ageugstr`l gst`fpgr`hbøl.

Xbepnbqugefs

• Hflvgrtbefs gn ràbh`laf gl d`htfrgs.

• @pnbhèfs n` rgin` agn prfauhtf y ag n` r`åz agul` pftglhb`.


Xbepnbqugefs




Xbepnbqugefs

• Gshrbmbefs nfs d`htfrgs hfef ul sfnf ràbh`n.




0:`8 5

2 • = • `= • ` 5

(8 )=(= ) 5

(8 )= • (= ) 5

8` =`

841 x8 y7 z : 5

8

(1 xyz =)8 • ( y= z =) 58(1 xyz =)8 •

8( y= z =) 5

1 xyz =8 y= z =

Gcgrhbhbf 4Xbepnbqug nfs ràbh`ngs hfl nfs gcgrhbhbfs sbiubgltgs.

0) 1:`7 5 =) 804 x7 y1 z 4 5

8) =9e8l7 p9 5 1) 9 x1 y= • 9 x= y8 5

= x7 y8 • : x= y 5

04 x9 y1 5

(1 x8 y=)=( x) 5

(1 x8 y=)= • ( x) 5

1 x 8 y= x




1. ]`hbfl`nbz`hbøl ag aglfeblàfrgsWl` gxprgsbøl `nigmr`bh` gxprgsà` gl su dfre` eàs sbepng f sbepnbhà` lf agmg tglgr ràbh`ngs gl gn aglfeblàfr. [`r` gnnf, tr`m`c`rgefs gl n` r`hbfl`nb-z`hbøl, tge` qug y` vbefs gl nfs lóegrfs hfepngcfs y qug jfy èpnbèfs ` n`ràbh`hbøl. [rgstg `tglhbøl ` hà` h`sf.

`. Hu`laf sg euntbpnbh` gn luegràfr y gn aglfeblàfr pfr gn ràbh`n agn ag-lfeblàfr.

]`hbfl`nbhgefs

• Euntbpnbhèfs gn luegràfr y gn aglf-eblàfr pfr = x

• Xbepnbdbhèfs n`s gxprgsbflgs y fmtglg-efs gn rgsuntàf.

]`hbfl`nbhgefs

• Euntbpnbhèfs gn luegràfr y gn aglf-eblàfr pfr 8w


m. Hu`laf gn aglfeblàfr tbglg ul hfgdbhbgltg, sg euntbpnbh` gn luegràfr y gnaglfeblàfr pfr gn ràbh`n, sbl tfe`r gl huglt` gn hfgdbhbgltg.

]`hbfl`nbhgefs

• Euntbpnbhèfs gn luegràfr y gn aglf-eblàfr pfr 7 x


]`hbfl`nbhgefs

• Euntbpnbhèfs gn luegràfr y gn aglf-eblàfr pfr =l


1

= x 5

1

= x ( = x

= x ) 5 1 = x

( = x )= 5

1 = x

( = x )= 5

1 = x

= x 5

= = x

x

7 y

8w 5

7 y

8w ( 8w

8w ) 5 7 y 8w

( 8w)= 5

7 y 8w

( 8w)= 5

7 y 8w

8w

:e

1 =l 5

:e

1 =l ( =l

=l ) 5 :e • =l

1( =l)= 5

04el

1( =l)= 5

1 el

1(=l) 5

el

=l

9

8 7 x 5

9

8 7 x ( 7 x

7 x ) 5 9 • 7 x

8( 7 x)= 5

87 x

8( 7 x)= 5

87 x

8(7 x) 5

87 x

07 x




h. Hu`laf gn aglfeblàfr gs ul mblfebf sg euntbpnbh` gn luegràfr y gn ag-lfeblàfr pfr gn hflcuiàf agn aglfeblàfr.

]`hbfl`nbhgefs

• Euntbpnbhèfs gn luegràfr y gn aglf-eblàfr pfr gn hflcuiàf.

• Xbepnbdbhèfs n`s gxprgsbflgs.


]`hbfl`nbhgefs

• Euntbpnbhèfs gn luegràfr y gn aglf-eblàfr pfr gn hflcuiàf.

• Xbepnbdbhèfs n`s gxprgsbflgs.


7

8 x + = x

7

8 x + = x ( 8 x ‘ = x

8 x ‘ = x ) 5

7( 8 x ‘ = x )

( 8 x )= ‘ ( = x )= 5

7( 8 x ‘ = x )

( 8 x )= ‘ ( = x )= 5

7( 8 x ‘ = x )

8 x ‘ = x 5

7( 8 x ‘ = x )

x

8 x

x ‘ 4

8 x

x ‘ 4 ( x + 4

x + 4 ) 5

8 x( x + 4 )

( x )= ‘ ( 4 )= 5

8 x( x + 4 )

( x )= ‘ ( 4 )= 5

8 x ( x + 4 )

x ‘ 4

Gcgrhbhbf 9]`hbfl`nbhg n`s gxprgsbflgs `nigmr`bh`s y sbepnbqug gn rgsuntàf aflag sg` pfsbmng.

0)7

= 8 x 5 =)

2 y

y + 8 5




]gsuegl

0. N`s rgin`s ag n` ràbh`hbøl sfl n`s ebse`s qug utbnbzèfs hfl nfs lóegrfs rg`ngs. Ag dfre`sbemønbh` sg rgprgsglt`l `så3

l

l

5

l

l

5 `

l

` • m 5

l

•l

m

l

`

m 5

l

l m

l e` 5 l • e `

=. [`r` hflvgrtbr ul ràbh`n gl gxpflgltg dr`hhbfl`rbf, hfpb`-efs gn ràbh`laf y abvbabefs su pftglhb` gltrg gn ålabhgagn ràbh`n.

8. Afs f eàs ràbh`ngs sfl sgegc`ltgs sb gn ràbh`laf y gnålabhg sfl biu`ngs.

8.0 N` sue` y rgst` ag ràbh`ngs sg pugag rg`nbz`r sb y sfnfsb sfl ràbh`ngs sgegc`ltgs. [`r` rg`nbz`r gst`s fpgr`-hbflgs, sg sue`l f rgst`l nfs hfgdbhbgltgs y sg hfpb` gnràbh`n.

8.= Gl n` euntbpnbh`hbøl ag ràbh`ngs sg pugagl prgsglt`r afs h`sfs3 nfs qug tbglgl gn ebsefålabhg y nfs qug tbglgl abdgrgltg ålabhg.

`. Hfl gn ebsef ålabhg

m. Hfl abdgrgltg ålabhg g biu`n ràbh`laf

8.8 [`r` abvbabr r abh`ngs qug tbglgl gn ebsef ålabhg, sgabvbagl nfs ràbhàfs y ag sgr pfsbmng sg sbepnbdbh` gnrgsuntàf.

1. [`r` gnbebl`r d`htfrgs gl nfs ràbh`ngs, sg gxtr`gl nfs d`htfrgs ag n` p`rtg lueârbh` y ag n`s nbtgr`ngsj`st` qug n`s pftglhb`s sg`l eglfrgs qug gn ålabhg agn ràbh`n.

7. N` r`hbfl`nbz`hbøl pgrebtg gxprgs`r nfs ràbh`ngs gl ul` dfre` sbepnbdbhà`, `n lf tglgr ràbh`ngsgl gn aglfeblàfr. Xg agmgl tfe`r gl huglt` nfs h`sfs sbiubgltgs3

`. Gn aglfeblàfr hflst` ag ul sfnf târeblf ràbh`n y lf tbglg hfgdbhbgltgs.

m. Gn aglfeblàfr gs ul ràbh`n y tbglg ul hfgdbhbgltg.h. Gn aglfeblàfr gstà dfreàf pfr afs ràbh`ngs qug dfre`l ul` sue` f abdgrglhb`, gs ulmblfebf.

[`r` pr`htbh`r n` ràbh`hbøl gl n` rga, blirgsg ` gst` abrghhbøl gnghtrølbh`3 jttp3//iff.in/CALY`z

e l 5 `l/e

== y y 0: x == y

`l x + m

l x ‘ h

l x 5 (` + m ‘ h)

l x

l

l m

5 l`

m

e •

l 5 `0/e • `0/l

l • l

m 5 l` • m






E`rqug ul` "Z" gl gn huàrf ag n` fphbøl qug hfepngt` hà` glulhbàf.

0) N` gxprgsbøl gqubv`ngltg ` x7 gs…

=) Gn rgsuntàf qug eugstr` n` dfre` hfrrght` ag fpgr`r l e x gs…

8) N` fphbøl qug rgprgsglt` gn rgsuntàf sbepnbdbhàf ag8 x4 gs…

1) Wl ràbh`n sgegc`ltg ` ‘7 x = x7

gs…

7) N` fphbøl qug rgprgsglt` gn hflcuiàf ag 7 x ‘ = x gs…

4) N` fphbøl qug rgprgsglt` gn rgsuntàf hfrrghtf ag n` fpgr`hbøl( x ‘ 4 )( x + 4 ) gs…

@utfhfltrfn

x7

x7/=

x=/7

7 x=

l + e

x

l ‘ e

x

l • e

x

l ù e

x

x

x=

x8/4

x=

= x7

7 x7

x = x8

‘7 x8 = x7

= x ‘ 7 x

7 x + = x

‘ 7 x ‘ = x

‘ 7 x + = x

x ‘ 4

x + 4

x ‘ 4

x + 4










Gl gstg `p`rtàf pr`htbh`rà n` ràbh`hbøl ag lóegrfs rg`ngs. [`r` rgsfnvgr nfs gcgrhbhbfs, gs lghgs`rbfqug rghugrag n`s døreun`s p`r` gn àrg` ag huàràfs, rghtàliunfs y gn hårhunf, `så tèmbâl n` agnvfnuegl p`r` gn humf.

0) Gn àrg` ag ul tgrrglf ag dfre` huàrà` gs 042 e=. ¶Huàltf ebag gn pgråegtrf agn tgrrglf6

=) Gn vfnuegl ag ul humf gs 0;;; he8. ¶Huàn gs gn àrg` ag ul` ag sus h`r`s6

8) Gn àrg` ag ul tgrrglf huàràf gs 4=7 e=. ¶Huàn gs gn àrg` ag ftrf tgrrglf ag n` ebse` dfre`huyf nàf ebag gn trbpng agn qug tbglg gn àrg` ag 4=7 e=6

1) Xg qubgrg abstrbmubr ` 7=2 gstuab`ltgs gl ul s`nøl ag dfre` huàr a`. ¶Huàltfs gstuab`ltgsj`mrà gl hà` dbn`6

7) N` ebtà agn huàràf ag n` abst`lhb` qug rghfrrg ul hbhnbst` gl 8; eblutfs gs 04= oe.

¶Huàltf rghfrrgrà gl = jfr`s6 4) Xg qubgrg hflstrubr ul t`mngrf huàràf qug tgli` ul` supgrdbhbg ag ==7 he= y qug ` su vgz

hfltgli` 011 h`sbnn`s biu`ngs, sbl gsp`hbf gltrg ul` y ftr`. ¶Huàltf egabrà gn nàf ag hà`h`sbnn`6

9) Gn àrg` ag ul huàràf gs 1;24 he=. ¶Huàltf egabrà gn pgråegtrf ag ftrf huàràf huyfnàf gs n` r`åz hómbh` ag nf qug ebag ul nàf agn prbegr huàràf6

:) Gl ul` j`mbt`hbøl sg qubgrgl hfnfh`r 8 egs`s huàrà`s ag = e= hà` ul` y = egs`s,tèmbâl huàrà`s, ag : e= hà` ul`. [ugst`s ul` ` hfltblu`hbøl ag n` ftr`, ¶quâ nflibtuafhup`ràl tfa`s n`s egs`s6

2) Wl tgrrglf huàràf tbglg ul` supgrdbhbg ag 2;; e=. ¶Huàltf hfst`rà hgrh`rnf sb gn egtrf age`nn` tbglg ul v`nfr ag P97.;;6

0;) Gn augðf ag ul tgrrglf, huy`s abeglsbflgs sfl 8= e ag n`rif pfr : e ag `lhjf, qubgrgpgreut`rnf pfr ul tgrrglf huàràf ag n` ebse` supgrdbhbg. ¶Huàltf agmg egabr hà` nàfagn tgrrglf huàràf6

00) Wl hfegrhb`ltg hfeprø hbgrtf lóegrf ag pn`ygr`s pfr P=74.;;. Xb gn lóegrf y gn prghbf agn`s pn`ygr`s sfl biu`ngs, ¶huàlt`s pn`ygr`s hfeprø6

0=) Wl` h`c` gl dfre` hómbh` tbglg ul vfnuegl ag 0=7 ;;; he8. Xb sg hfrt` pfr n` ebtà, ¶huàngs

sgràl n`s abeglsbflgs ag n` h`c` rgsunt`ltg6

08) J`nng gn àrg` ag n` hfrfl` hbrhun`r hfeprglaba` gltrg n`shbrhuldgrglhb`s blshrbt` y hbrhulshrbt` gl ul huàràf ag4.=7 he= ag àrg`.

8 . 7 1






prfhgsflf

nfiràf

A g s p u â s a g g s t u a b ` r . . . @pnbhf n`s rgin`s ag nfs ràbh`ngs ` gxprgsbflgs `nigmr`bh`s.

]g`nbzf fpgr`hbflgs `rbteâtbh`s hfl gxprgsbflgs qug hfltbglglràbh`ngs.

Xbepnbhf ràbh`ngs qug hfltbglgl gxprgsbflgs `nigmr`bh`s.

]`hbfl`nbzf gxprgsbflgs `nigmr`bh`s.

@pnbhf n` ràbh`hbøl ` n` sfnuhbøl ag prfmnge`s.

Gn ebstgrbfsf lóegrf 4

[`r` hfeglz`r hflfzhèfs gn såemfnf ! Gl e`tgeàtbh` sg ng nnè` d`htfrb`n ysg aglg hfef gn prfauhtf ag tfafs nfs lóegrfs gltgrfs pfsbtbvfs agsag gn

lóegrf blabhàf j`st` 0. Tgèfs gn d`htfrb`n ag 1 y 7.1! 5 1 Ù 8 Ù = Ù 0 5 =1

7! 5 7 Ù 1 Ù 8 Ù = Ù 0 5 0=;

Gstg såemfnf gs gn qug lfs sgrvbrà p`r` hfepngt`r n` prbegr` n` agn huàrf ag`m`cf, aflag, hfl hu`nqubgr fpgr`hbøl e`tgeàtbh` y utbnbz`laf nfs såemfnfs

, +, 8 , ‘ , ù, Ù agmg a`r hfef rgsuntàf 4.

[rgstg `tglhbøl høef sg fmtbglg 4 gl n` prbegr` n`.

(0 + 0 + 0 )! 5 (8)! 5 8 Ù = Ù 0 5 4

@jfr` ng tfh` ` ustga hfepngt`r n`s ageàs.

(0 + 0 + 0)! 5 4

= = = 5 4

8 8 8 5 4

1 1 1 5 4

7 7 7 5 4

4 4 4 5 4

9 9 9 5 4: : : 5 4

2 2 2 5 4






000

= 8 1

8

1

7

=

0

;

x

y

‘ 0 5 b

x = +

y = 5

r =

Fpgr`hbflgs pfnblfeb`ngs


N` bepfrt`lhb` agn ànigmr`

[fnblfebfs Fpgr`hbflgs hfl pfnblfebfs

Fpgr`hbflgs hfl eflfebfs

H`nhun`r gn àrg` sfemrgà` gl iur`s pn`l`s.


Hflhgptu`nbz`r g bagltbh`r ul pfnblfebf. Hn`sbh`r pfnblfebfs ag `hugraf ` su lóegrf ag târeblfs.

Bagltbh`r y rgsfnvgr fpgr`hbflgs gltrg pfnblfebfs.

]gsfnvgr fpgr`hbflgs ag eflfebfs hfl ìbnbaà.




¡[`r` hfeglz`r!N` bepfrt`lhb` agn ànigmr`

Gl `niól pultf ag lugstrfs gstuabfs, sg lfs pugag hruz`r pfr gn pglsèbgltf

n` prgiult`3 ¶pfr quâ gs bepfrt`ltg `prglagr ànigmr`6

[ugs mbgl, gst` rè` ag n` e`tgeàtbh` gs ul` pfagrfs` jgrrèbglt` ag lànbsbs

y tbglg ir`l v`rbgaà ag `pnbh`hbflgs? gs utbnbzà` pfr prfdgsbfl`ngs qug v`l

agsag nfs gnghtrbhbst`s, nfs `rqubtghtfs y nfs bldfreàtbhfs.

@ hfltblu`hbøl, vgrgefs `niulfs h`sfs gspghådbhfs gl nfs qug sg gepng` gn

ànigmr`3

• [`r` h`nhun`r n` nblg`hbøl ag `ltgl`s gnghtrfeìlâtbh`s. Gn absgðf ag ul`

`ltgl` p`r` rgags ag bltgrlgt bl`nàemrbh` sg j`hg gepng`laf n` døreun`

ag Hjgvysjgv, ul e`tgeàtbhf rusf. Hfl m`sg gl nfs a`tfs h`nhunàfs, sg

gst`mnghgl n`s abeglsbflgs baølg`s p`r` e`ltglgr ul` sgð`n ulbdfreg y

hfl n` eàxbe` hfmgrtur` pfsbmng.

• Gl n`s hbglhb`s mbfnøibh`s, sg utbnbz` p`r` prfyght`r gn rbtef ag hrghbebgl-

tf ag ul` pfmn`hbøl ag m`htgrb`s f n` pfsbmbnbaà ag prfpì`hbøl ag ul`

gldgregaà.

• Gl n` egtgfrfnfiå`, gn ànigmr` `yua` p`r` n`s prgabhhbflgs agn hnbe`? gn

BLXBTWEGJ utbnbz` efagnfs e`tgeàtbhfs p`r` h`nhun`r n` h`ltbaà ag nnuvb`,

vbgltfs, gth. qug sg gspgr`l gl `niul` rgibøl.

• Gl ghflfeå , n` e`tgeàtbh` tbglg ul rfn dulaèglt n? jgrrèbglt`s yhflhgptfs agn ànigmr` sg utbnbz`l p`r` prflfstbh`r nfs hfepfrtèbgltfs

qug tglarà n` ghflfeå` gl gn duturf.

[fr gcgepnf, gn M`lhf ag Iu`tge`n`, n` blstbtuhbøl glh`rià` ag n` ghflf-

eå` l`hbfl`n, utbnbz` n`s prfyghhbflgs p`r` tfe`r n`s egaba`s lghgs`rb`s

y e`ltglgr `så n` gst`mbnbaà gl nfs prghbfs ag nfs prfauhtfs gl gn p`ås.

Hfef puabefs lft`r, gn ànigmr` pugag sgr `mstr`ht`, pgrf tbglg ir`l h`ltbaà

ag `pnbh`hbflgs gl n` vba` rg`n, pfr nf qug rgsunt` euy bepfrt`ltg su afeblbf.

¡@ tr`m`c`r!

[bglsg y gshrbm` niul` ftr` pnbh`hbøl p`r` n` qug gs lghgs`rb` gn ànigmr`. [ugag pfy`rsg gl bltgrlgtu ftrf rghursf ag blvgstbi`hbøl.




0. [fnblfebfsWl pfnblfebf gs ul` gxprgsbøl e`tgeàtbh` dfreà` ` p`rtbr ag n` ulbøl ag

afs f eàs v`rb`mngs y hflst`ltgs, qug sg vblhun`l ` tr`vâs ag n`s fpgr`hbflgs`rbteâtbh`s ag sue`, rgst` y euntbpnbh`hbøl. Gl ftr`s p`n`mr`s, ul pfnblfebf gs nùlbøl ag afs f eàs eflfebfs.

0.³ târeblf =.³ târeblf 8.³ târeblf

9 x= + 1 xy ∝ 8

0.0 Hn`sbdbh`hbøl ag nfs pfnblfebfs

Xgiól su lóegrf ag târeblfs, ul pfnblfebf pugag rghbmbr abstbltfs lfemrgs3

• Mblfebf3 gs gn pfnblfebf eàs rgauhbaf, hfl afs târeblfs.• Qrblfebf3 pfnblfebf hfl trgs târeblfs.

• Qgtr`lfebf3 pfnblfebf hfl hu`trf târeblfs.

Lfs rgdgrbefs ` n`s gxprgsbflgs hfl hblhf târeblfs gl àgn`ltg sbepngegltghfef pfnblfebfs.

Xgiól nfs târeblfs qug hfeprglag3

[fnblfebf hfepngtf3 gs `qugn qug tbglg tfafs nfs târeblfs, agsag gn târeblfag e`yfr iràf j`st` gn târeblf blagpglabgltg.

1 y8 ∝ = y= + 7 y + 0;

Hfef lftèfs, gn pfnblfebf `ltgrbfr v` agsag gn târeblf hfl gxpflgltg 8j`st` gn târeblf blagpglabgltg, gs aghbr, gn qug y` lf tbglg v`rb`mng.

[fnblfebf blhfepngtf3 gs `qugn qug lf tbglg tfafs nfs târeblfs agsag gntâreblf ag e`yfr iràf j`st` gn târeblf blagpglabgltg.

4 y8 ∝ 7 y + 8 5

[fagefs vgr qug gl n` gxprgsbøl `ltgrbfr j`hg d`nt` gn târeblf ag sgiulafiràf ( y =) p`r` qug sg` hfepngtf.

Xgiól su iràf3

@n biu`n qug hfl nfs eflfebfs, nfs pfnblfebfs tbglgl "iràf". Gn iràf ag ulpfnblfebf nf agtgrebl` gn eflfebf f târeblf hfl gn e`yfr gxpflgltg. Tgèfs3

7w8 + 4w= ‘ 0;w + 4

E`yfr gxpflgltg

Gl gn gcgepnf `ltgrbfr, gn e`yfr gxpflgltg gs 8, pfr nf qug aghbefs qug gs ulpfnblfebf ag "tgrhgr iràf" f "hómbhf"? sb su gxpflgltg e`yfr dugr` =, sgrå` ag"sgiulaf iràf" y `så hfltbló` hfl tfafs nfs lóegrfs.

]ghfragefs qug ul

eflfebf gs ul`gxprgsbøl `nigmr`bh`,dfreà` pfr ulhfghbgltg y ul`p`rtg nbtgr`n (v`rb`mng)qug sg euntbpnbh`l.





Gcgrhbhbf 0@. Gshrbm` gl n` nålg` hfrrgspflabgltg sb sg tr`t` ag ul mblfebf, trblfebf f ag ul tgtr`lfebf.

0) ‘0;t + 7t 1 + t ‘ 0; 1) w8 + =v

=) 1 x8 + 7 xy= 7) ‘ d + d =

8) 7` ‘ 1`m1 + 1m8 4) 8a = + a 8 ‘ 7

=. Xue` y rgst` ag pfnblfebfs[`r` n` àbhbøl y n` sustr`hhbøl gltrg afs f eàs pfnblfebfs, sg sue`l f rgst`lnfs hfghbgltgs ag nfs târeblfs sgegc`ltgs, gs aghbr, ag nfs târeblfs qug tgl-i`l n` ebse` v`rb`mng hfl gn ebsef iràf. Tèfs, lf sg prgfhupg, qug gs euysglhbnnf.

]gsfnvèfs n` sbiubgltg sue` pfnblfeb`n3

• Fraglèfs nfs pfnblfebfs sb lf nf gstàl.

• Gnbeblèfs nfs p`râltgsbs tglbglaf hub-aàf hfl n` ngy ag sbilfs.

• @irupèfs nfs eflfebfs f târeblfssgegc`ltgs.

• Fpgrèfs nfs hfgdbhbgltgs ag nfs târeb-lfs sgegc`ltgs. @qugnnfs eflfebfs quglf tbglgl lbliól târeblf sgegc`ltg `nhu`n sue`rsg sfnègltg sg hfpb`l.

N` rgst` sbiug ul prfhgsf euy sbebn`r, pflièfs `tglhbøl.

]g`nbhgefs n` rgst` sbiubgltg3

• Fraglèfs nfs pfnblfebfs sb lfnf gstàl.

• Gnbeblèfs nfs p`râltgsbs tglbgl-af hubaàf hfl n` ngy ag sbilfs.

• @irupèfs nfs eflfebfs ftâreblfs qug tbglgl n` ebse`v`rb`mng y gn ebsef iràf.

• Fpgrèfs nfs hfgdbhbgltgs ag nfstâreblfs sgegc`ltgs. @qugnnfseflfebfs qug lf tbglgl lblióltâreblf sgegc`ltg `n hu`n fpg-r`rsg sfnègltg sg hfpb`l.

(: x1 + 7 x8 ‘ 4) + (1 x ∝ 9 x8 + = x1)

(: x1 + 7 x8 ‘ 4) + (= x1 ∝ 9 x8 + 1 x) 5

: x1 + 7 x8 ‘ 4 + = x1 ∝ 9 x8 + 1 x 5

: x1 + = x1 + 7 x8 ∝ 9 x8 + 1 x ‘ 4 5

0; x 1 ∝ = x 8 + 1 x ‘ 4

(‘02 x8 + =0 x7 + 2) ‘ (01 x7 ‘ 2 x8 + : x= + 0)

(=0 x7 ‘ 02 x8 + 2) ‘ (01 x7 ∝ 2 x8 + : x= + 0) 5

=0 x7 ‘ 02 x8 + 2 ‘ 01 x7 + 2 x8 ‘ : x= ‘ 0 5

=0 x7 ‘ 01 x7 ‘ 02 x8 + 2 x8 ‘ : x= + 2 ‘ 0 5

9 x 7 ‘ 0; x 8 ‘ : x = + :




M. Gshrbm` gl n` nålg` hfrrgspflabgltg gn iràf ag hà` ulf ag nfs pfnblfebfs sbiubgltgs.

0) 8 x8 + = x= + x ‘ 2 4) 0= + y + x

=) y1 ‘ 0; y8 + y= + y 9) ‘== + 2r 8 + 1r = + r 1

8) ‘0;a 8 + a 1 ‘ 8a = :) 88 x8 ‘ x= ‘ 0; + 8 x

1) j= + j7 + 1 2) ‘=0 ‘ `= ‘ `

7) 4t 8 ‘ t = + 02 0;) r = + r

H. ]gsugnv` n`s sbiubgltgs sue`s y rgst`s ag pfnblfebfs. Xb lghgsbt` eàs gsp`hbf, pugag tr`m`c`r gljfc`s `p`rtg.

0) ( x8 + x=) + (00 x= ‘ x) 4) ‘(r + 0:r = + 08) + (‘:r = + 7r + 7)

=) (0; ‘ 0= x=) ‘ (7 x ‘ = x= + 7) 9) (18 x= y + 08 xy) ‘ (‘4 xy + 08 x= y + 08)

8) (4 y + 8 y8 ‘ 1) ‘ ( y + 09 ‘ 8 y8) :) (8=w8 ‘ 8w= ‘ w + =) + (0=w ‘ 8 + 1w8)

1) ( t 8 + t ) + (02t ‘ 00`t 8 + 0) 2) ‘(=8 i + 04 i = + =) + (‘: i = + 0)

7) (14`1 + =0`8) + ( = ‘ 01`1 + `8) 0;) ‘(:wy8 ‘ 1 y= + wy) + (=wy + 1w + 8wy8)




8. Euntbpnbh`hbøl ag pfnblfebfs[`r` euntbpnbh`r pfnblfebfs sg utbnbz` n` prfpbgaà abstrbmutbv`? gn prfauhtf pug-ag sgr ag ul lóegrf pfr ul pfnblfebf, ag ul eflfebf pfr ul pfnblfebf ygltrg afs pfnblfebfs.

8.0 Euntbpnbh`hbøl ag ul lóegrf pfr ulpfnblfebf

Xg euntbpnbh` gn lóegrf pfr hà` ulf ag nfs târeblfs ag n` gxprgsbøl pfnblfeb`n. Gnrgsuntàf gs ftrf pfnblfebf qug pfsgg gn ebsef iràf agn pfnblfebf qug sgfpgrø y hfef hfghbgltgs sg tbglgl nfs prfauhtfs ag nfs hfghbgltgs agn pfnb-lfebf pfr gn lóegrf.

Euntbpnbqugefs3

• @pnbhèfs n` prfpbgaàabstrbmutbv` `n euntbpnbh`r(‘1) pfr hà` ulf ag nfstâreblfs agn pfnblfebf.

• Fpgrèfs y fmtglgefs gnrgsuntàf dbl`n. Qglièfssbgeprg gl egltg n` ngy agsbilfs.

8.= Euntbpnbh`hbøl ag ul eflfebf pfr ulpfnblfebf

Xg euntbpnbh` gn eflfebf pfr tfafs y hà` ulf ag nfs târeblfs qug dfre`l gnpfnblfebf, gs aghbr, hfghbgltg pfr hfghbgltg y v`rb`mng pfr v`rb`mng. Gl gstgprfhgabebgltf sg gepng` n` prfpbgaà abstrbmutbv`.

[`r` gst` fpgr`hbøl agmgefs rghfra`r n`s ngygs ag nfs gxpflgltgs, sb ng surig`niul` aua`, pugag rgirgs`r y hflsunt`r n` sge`l` 2 ag su nbmrf.

Fpgrgefs3

• @pnbhèfs n` prfpbgaàabstrbmutbv` `n euntbpnbh`r(7wx8)

pfr hà` ulf ag nfstâreblfs agn pfnblfebf.

Hfef aghåèfs `rrbm`, `ntr`m`c`r hfl n`s v`rb`mngsagmgefs `pnbh`r n`s ngygsag nfs gxpflgltgs.

• ]gsfnvgefs y fmtglgefsgn rgsuntàf dbl`n.

(‘1)(‘9 x1 + 04 x8 y ‘ x= + 08)

(‘1)(‘9 x1 + 04 x8 y ‘ x= + 08) 5

(‘1)(‘9 x1) + (‘1)(04 x8 y) + (‘1)(‘ x=) + ( ‘1)(08) 5

=: x 1 ‘ 41 x 8 y + 1 x = ‘ 7=

(7wx8)(1 x1 + : x8 ∝ = x ∝ 00)

(7wx8)(1 x1 + : x8 ∝ = x ∝ 00) 5

(7wx8)(1 x1) + (7wx8)(: x8) + (7wx8)(‘= x) + (7wx8)(∝00) 5

=;wx 9 + 1;wx 4 ‘ 0;wx 1 ‘ 77wx 8




8.8 Euntbpnbh`hbøl gltrg pfnblfebfs

[`r` euntbpnbh`r afs pfnblfebfs gltrg så, sg euntbpnbh` hà` târeblf agn prbegrpfnblfebf pfr tfafs y hà` ulf ag nfs târeblfs agn sgiulaf pfnblfebf.

]gsfnvèfs3

• Ag lugvf, hfef gl nfs h`sfs `ltgrbf-rgs, `pnbhèfs n` prfpbgaà abstrbmu-tbv`.

• Euntbpnbhèfs gn prbegr târeblf agnprbegr pfnblfebf (8 x8) pfr hà` ulfag nfs târeblfs agn sgiulaf pfnblf-ebf.

• Euntbpnbhèfs gn sgiulaf târeblfagn prbegr pfnblfebf (1 x) pfr hàùlf ag nfs târeblfs agn sgiulafpfnblfebf.

• Fraglèfs nfs eflfebfs y sue`-efs nfs qug tgli`l gn ebsef iràf.

• Xg fmtbglg gn rgsuntàf dbl`n, ftrfpfnblfebf huyf iràf gs n` sue` agnfs iràfs ag nfs pfnblfebfs qug sgeuntbpnbh`l.

Gcgrhbhbf =Gl su huàgrlf, rgsugnv` n`s euntbpnbh`hbflgs pfnblfeb`ngs sbiubgltgs. Nugif, gshrbm` su rgspugst` gl

n` nålg` hfrrgspflabgltg.

0) 0;(7 x8 + 08 x ‘ 8) 5

=) (8`m= + 7`=)(8`8m= ‘ 1m) 5

8) (7w8 x)(9w= + 8 x= ‘ 7) 5

1) (‘:)(2 xy= + y8 + x8) 5

7) (2 i =j8 + 9ij= ‘ :j ‘ 8)(1 i =j= + 0= ij ‘ 07) 5

4) (01m=h8)(9 m8 + 00`h= ‘ :) 5

9) 09(8l8e ‘ :l=e= + 4le8 + :) 5

:) (0=w= xy ‘ 1w= x= y= + 2wx8 y8)(7w= x + : xy8 ‘ 9wxy) 5

2) (2 x8 y)(8 xy= ‘ 7 x= y + 9 x) 5

0;) (‘1w= z + 4wz)(:wy= ‘ 7 x8 z =) 5

(8 x8 + 1 x)(8 x1 ∝ 1 x= ∝ 2)

(8 x8 + 1 x)(8 x1 ∝ 1 x= ∝ 2) 5

(8 x8 + 1 x)(8 x1 ∝ 1 x= ∝ 2) 5

(8 x8)(8 x1) + (8 x8)(∝1 x=) + (8 x8)(∝2) 5

2 x 9 ‘ 0= x 7 ‘ =9 x 8

(8 x8 + 1 x)(8 x1 ∝ 1 x= ∝ 2) 5

(1 x)(8 x1) + (1 x)(∝1 x=) + (1 x)(∝2) 5

0= x 7 ‘ 04 x 8 ‘ 84 x

2 x 9 ‘ 0= x 7 + 0= x 7 ‘ 04 x 8 ‘ =9 x 8 ‘ 84 x 5

2 x 9 ‘ 18 x 8 ‘ 84 x

2 x 9 ‘ 18 x 8 ‘ 84 x




Gcgrhbhbf 8Gl su huàgrlf, rgsugnv` n`s abvbsbflgs pfnblfeb`ngs sbiubgltgs. Nugif, gshrbm` su rgspugst` gl n` nålg`hfrrgspflabgltg.

0) (4 x1 y ‘ 2 x8 y= + 0= x= y8 ‘ 4 xy1) ù (8 xy) 5

=) (1 x8 + 4 x= ‘ : x ‘ =) ù (= x) 5

8) (0=`8m= ‘ =;`=m + 04`m ‘ 1) ù (∝ 1`=m) 5

1) (=;wx=

+ 07w=

x ‘ =7w) ù (7wx) 5

7) (:0`8m= ‘ =9 =m8 + 71`m1 + m) ù (2`=m8) 5

4) (2 x= y= + 8 xy8 ‘ 0=) ù (8 xy=) 5

9) (1;m1h8 ‘ ==m=h= + 0:mh) ù (∝ =m=h8) 5

:) (0= xy8 + 0: x= y= ‘ 1= x8 y + 4 x1 ‘ 0=) ù (4 x= y8) 5

2) (1:`1m1 ‘ 8=`8m= + :`=m) ù (:`8m8) 5

1. Abvbsbøl ag ul pfnblfebf gltrg uleflfebf

[`r` abvbabr ul pfnblfebf gltrg ul eflfebf, sg abvbag hà` ulf ag nfs târeblfsagn pfnblfebf pfr gn eflfebf, tglbglaf gl huglt` n` ngy ag nfs sbilfs.

]gsfnvèfs3

• Abvbabefs hà` ulf ag nfs târeblfsagn pfnblfebf gltrg gn eflfebf.

• Fpgrèfs, abvbabglaf hfgdbhbgltgsgltrg hfgdbhbgltgs y v`rb`mngs gltrgv`rb`mngs.

• Fmtglgefs gn rgsuntàf dbl`n.

Tgèfs ul óntbef gcgepnf, fpgrgefs3• Abvbabefs hà` ulf ag nfs târeblfs agn

pfnblfebf gltrg gn eflfebf. Qglgefshubaàf hfl n` ngy ag sbilfs.

• Fpgrèfs, abvbabglaf hfgdbhbgltgsgltrg hfgdbhbgltgs y v`rb`mngs gltrgv`rb`mngs.

• Fraglèfs nfs târeblfs y fmtglgefsgn rgsuntàf dbl`n.

(8 x8

y=

+ 7 x=

y ‘ 4 xy=

+ 1 xy) ù (∝1 x=

y) 5

8 x8 y=

∝1 x= y +

7 x= y

∝1 x= y ‘

4 xy=

∝1 x= y +

1 xy

∝1 x= y 5

∝8 xy

1 ∝

7

1 +

8 y

= x ∝

0 x

5

∝ 8 xy

1 +

8 y

= x ‘

0 x

‘ 7

1

(=; x8 ∝ 07 x= + 1; x + 7) ù (7 x=) 5

=; x8

7 x= ∝07 x=

7 x= +1; x7 x= +

77 x= 5

=; x8

7 x= ∝07 x=

7 x= +1; x7 x= +

77 x= 5

1 x ‘ 8 +:

x + 0

x =




]gsuegl

0. Wl pfnblfebf gs ul` gxprgsbøl e`tgeàtbh` dfreà` ` p`rtbr ag n` ulbøl ag afs f eàs v`rb`mngsy hflst`ltgs, qug sg vblhun`l ` tr`vâs ag n`s fpgr`hbflgs `rbteâtbh`s ag sue`, rgst` y euntbpnbh`-hbøl. Gl ftr`s p`n`mr`s, ul pfnblfebf gs n` ulbøl ag afs f eàs eflfebfs.

0.0 Nfs pfnblfebfs sg pugagl hn`sbdbh`r sgiól su lóegrf ag târeblfs, sgiól nfs târeblfs qughfeprglag y sgiól gn iràf ag sus v`rb`mngs.

=. [`r` n` àbhbøl y n` sustr`hhbøl gltrg afs f eàs pfnblfebfs, sg sue`l f rgst`l nfs hfgdbhbgltgsag nfs târeblfs sgegc`ltgs, gs aghbr, ag nfs târeblfs qug tgli`l n` ebse` v`rb`mng hfl gn ebsefiràf.

8. [`r` euntbpnbh`r pfnblfebfs sg utbnbz` n` prfpbgaà abstrbmutbv`? gn prfauhtf ag pfnblfebfs pugagprgsglt`r abdgrgltgs h`sfs3

8.0 Euntbpnbh`hbøl ag ul lóegrf pfr ul pfnblfebf3 sg euntbpnbh` gn lóegrf pfr hà` ulf agnfs târeblfs ag n` gxprgsbøl pfnblfeb`n.

8.= Euntbpnbh`hbøl ag ul eflfebf pfr ul pfnblfebf3 sg euntbpnbh` gn eflfebf pfr tfafsy hà` ulf ag nfs târeblfs qug dfre`l gn pfnblfebf, gs aghbr, hfgdbhbgltg pfr hfgdbhbgltg yv`rb`mng pfr v`rb`mng.

8.8 Euntbpnbh`hbøl gltrg pfnblfebfs3 sg euntbpnbh` hà` târeblf agn prbegr pfnblfebf pfr tf-afs y hà` ulf ag nfs târeblfs agn sgiulaf pfnblfebf.

1. [`r` abvbabr ul pfnblfebf pfr ul eflfebf, sg abvbag hà` ulf ag nfs târeblfs agn pfnblfebfgltrg gn eflfebf.

Ng `lbeèfs ` fmsgrv`r nfs vbagfs sbiubgltgs sfmrg n`s fpgr`hbflgs hfl pfnblfebfs3

• Euntbpnbh`hbøl ag pfnblfebfs3 iff.in/tB=RoS

• Abvbsbøl ag pfnblfebfs3 iff.in/bvdOEo

[ugag glhfltr`r eàs gcgrhbhbfs p`r` pr`htbh`r gl nfs sbtbfs wgm sbiubgltgs3

• Xue`s y rgst`s3 iff.in/S@wD07

• Euntbpnbh`hbøl3 iff.in/t19YOQ






@. Fmsgrvg g blabqug sb nfs pfnblfebfs sfl hfepngtfs f blhfepngtfs. Iuågsg pfr gn gcgepnf.

;) ‘ 1 + 0=`8 + 0:`= ‘ 9 + 09

0) = x8 ‘ x= + 1 x ‘ 02

=) m8 + 4m= ‘ 1

8) ‘00r 7 + 02r 1 + =9r = ‘ =:

1) 0; y7 ‘ 1 y1 ‘ : y8 + 02 y= + y ‘ =0

M. Fmsgrvg g blabqug gn iràf ag nfs pfnblfebfs sbiubgltgs. Iuågsg pfr gn gcgepnf.

;) 8 x8 + =0 x1 ‘ 2 x + 09

0) ‘09`7 + 9 8 ‘ 0:`= + =:

=) =7e= ‘ 07e8 + 0=e= + 17

8) ‘01 x= + 0: x + 01 x ‘ ==

1) 7 y + 8 x ‘ 4 y + =7


@. ]gsugnv` n`s sue`s y n`s rgst`s ag pfnblfebfs sbiubgltgs. [ugag j`hgrnf gl su huàgrlf f gl jfc`s

`p`rtg.

0) (‘4m x= + : x1 ‘ 0= ‘ = x8 +0

=mx) + (‘0; x1 ‘ 7 + 4mx + 2 x8 + m x=)

=) (:`1 ‘ 01`4 ‘ 04`8 ‘ =` + 07) ∝ (∝8`4 ‘ =0`= + 01` ∝ =8)

8) (∝== x8 + =8 x= ∝ 07 x7 + =0 x1 + =7 x ∝ 09) ∝ (9 x8 ∝ =0 x1 + 09 x= ∝ 0: x7)

1) (=: x= z ∝ 10 x1 + 01 xz ∝ 2) ∝ (∝09 x1 ∝ 8 x= z ‘ : ∝ =9 xz )

7) (17 x9 + 09 x7 ∝ =7 x + 40 x8 ∝ 0=) + (=2 x8 ∝ 8: x7 + 7 x9 ∝ 0: x8)

4) (∝=:el= + =7l ‘ e=l + =:) ∝ (84l ∝ 10e=l + = ∝ :el= + 9e)

9) (7:el8 + 9e=l ‘ 10el ‘ 8e) ∝ (8el8 + =7el + 1 ‘ 4e=l)

:) (8= y8 ∝ 0= y1 + 81 y ‘ == + 8 y=) + (=1 y1 ∝ 1 y= ‘ 81 ∝ 09 y8 + 1 y)

2) (∝:=w= z = + 89w8 + 4w8 z ‘ 7wz =) + (4 z + =8w8 z ∝ 94 ‘ 8w= z = ‘ =w8)

0;) (∝08r = s + =1 s= ‘ 7rs8 + 1) ∝ (∝48r = s + 1=r + :0 s= ‘ :rs8)

@utfhfltrfn

hfepngtf

hu`rtf iràf




M. ]gsugnv` n`s euntbpnbh`hbflgs pfnblfeb`ngs sbiubgltgs. [ugag j`hgrnf gl su huàgrlf f gl jfc`s `p`rtg

0) (∝0=)(7 xy= + : x= y ‘ 0= x + : y + 0=)

=) (4`m8)(7`=m + 4`m8 ‘ 2 + 1`=)

8) (=o =v + 8ov= ∝ o )(00 + =ov ‘ v8)

1) (∝: y= z =)(9 yz 8 + : y8 z = ‘ 0; y)

7) (:)(‘1`m8 ‘ 4` + 2`=m)

4) (8 x ∝ 8 y=)( ‘7 y8 + 4 x= y ‘ :)

9) (∝2 xy= + 4 x= y8 ‘ 7 x8 y)(2 + 1 x= y ‘ x8 y=)

:) (0;e8l=)(7el= + 0;e=l8 ‘ 2)

2) (∝2)(‘ u8 + 4u= ‘ u)

0;) (0;`=m1 ‘ 7`8m + 4m)( = + =`m=)

00) (∝:m=h1)(0= + 1h8 ‘ m1)

0=) (∝02e= + 2e ∝ 9)( 2e8 ‘ 00e= + :e ∝ 00)

H. ]gsugnv` n`s sbiubgltgs abvbsbflgs ag pfnblfebfs gltrg eflfebfs. [ugag j`hgrnf gl su huàgrlf fgl jfc`s `p`rtg.

0) ( x= ‘ xz =) ù ( x)

=) (4`8m8 ‘ 2`= x1) ù (∝8`=)

8) (∝7 x8 + 0; xy= + 1`=m8) ù (∝= x)

1) (04 x7 ‘ : x1 + 9 x8 ∝ 0=) ù (1 x1)

7) (∝12`= + 8`=m ‘ =0`m=) ù (9`m=)

4) (0;e8l1 + e1l8 ‘ 4e=l=) ù (∝=el)

9) ( m8 ‘ 00`1 + 1`8 ∝ 7) ù ( 8)

:) (∝41 + 04 x1 ‘ = + 1 x8) ù (∝: x)

2) (84wz 8 + =w= z = ∝ 2w) ù (8w= z 1)

0;) (0:o 8e1 + 8e8 ‘ 2o 1) ù (2o 1e8)

00) (∝: s + 4r = s=) ù (∝:rs=)

0=) (0;;m=h + 7;mh= ‘ 7h1) ù (0;m8h8)




@. ]gsugnv` n`s sbiubgltgs sue`s y rgst`s ag eflfebfs. Gshrbm` su rgspugst` gl n` nålg`hfrrgspflabgltg.

0) 1 x

=

+ 9 x

=

5

=) 0; y1 + 0; y1 5

8) = x8 ‘ 2 x8 5

1) ‘0; x= ‘ 00 x= 5

7) 02 x + 8 x 5

4) 0=t 1 ‘ 0=t 1 5

9) ‘00 y8 + 07 y8 5

:) 7 y= + 8 y= 5

2) 2w8 ‘ 8w8 5

0;) ‘1r + 1r 5

00) ‘==w

4

‘ 09w

4

5

0=) 00e8 + =e8 5

08) 8 d 1 + 7 d 1 5

01) ‘ 1j7 + 8j7 5

07) i + 7 i 5

04) ‘ c + = c 5

09) 8 y8 + 1 y8 5

0:) 0= x1 + 9 x1 5

02) ‘8t 8 ‘ 2t 8 5

=;) w ‘ =w 5

M. ]gsugnv` n`s sbiubgltgs euntbpnbh`hbflgs ag eflfebfs. Gshrbm` su rgspugst` gl n` nålg`hfrrgspflabgltg.

0) 8 x1 • 8 x7 5

=) 7 xy • (‘8 x8) 5

8) (‘= y1) • 8 y7 5

1) (‘8 y8) • (‘1 xy=) 5

7) 1 xy • ‘9 x1 5

4) 7w8

• 4 y8

5

9) 1`8 • (‘7`1m) 5

:) (‘0;m=) • 0;m= 5

2) (‘8u8) • 2u= 5

0;) (‘1t 1) • 9t 8 y 5

00) 7y • 4 y 5

0=) 0= xy • 1 x8 5

08) 00r = • (‘9rt ) 5

01) 2w= • 2w7 5

07) (‘00 d 8) • 7 d = c 5

04) 1u1

• 1u=

5

09) (‘0= xy8 z ) • 0; x8 y= 5

0:) t = • =t 8 5

02) 9r 1 • 9r = p8 5

=;) = xyz 8 • (‘8 x8 y=) 5

0:;BIG] ∝ [fnfhjbh






E`rqug hfl ul hjgqug n` h`sbnn` qug egcfr blabqug su rglabebgltf.

nfiràf glprfhgsf

lfnfiràf

A g s p u â s a g

g s t u a b ` r . . .

Hflhgptu`nbzf g bagltbhf ul pfnblfebf.

Hn`sbhf pfnblfebfs ag `hugraf ` su lóegrf ag târeblfs.

Bagltbhf y rgsugnvf fpgr`hbflgs gltrg pfnblfebfs.

]gsugnvf fpgr`hbflgs ag eflfebfs hfl ìbnbaà.

@jfr` lfs gltrgtglargefs `n ebsef tbgepf qug `prglagefs. Qr`m`c`rgefshfl `niulfs `hgrtbcfs ag `rbteâtbh` hfl p`nbnnfs. [`r` gst` `htbvbaà lghgsbtèfs08 p`nbnnfs ag abgltgs, f `niól e`tgrb`n sbebn`r qug tgli` n` ebse` nflibtua.

Gstg tbpf ag `hgrtbcfs tbglg ulfs =;; `ðfs ag jbstfrb` y sfl bepfrt`ltgs pfrqug

eftbv`l `n gstuab`ltg ` j`hgr usf ag sus `prglabz`cgs ag `rbteâtbh` màsbh` p`r`nngi`r ` ul` sfnuhbøl s`tbsd`htfrb` agn gcgrhbhbf pn`ltgàf, `n ebsef tbgepf qugags`rrfnn` n` nøibh` y pr`htbh` sus hflfhbebgltfs ag n` luegr`hbøl rfe`l`.

]gsugnv` nf qug sg ng pbag3

0) Dfreg hfl p`nbnnfs n` dbiur` agn rghuàrf. Nugif, hèmbg ag nui`r 0 ag nfs 0=p`nbnnfs y jì` qug qugag dfreà` ul` biu`naà vgraàgr`, pfrqug 4 ‘ 1 lfgs biu`n 2. Gl gn gsp`hbf gl mn`lhf abmucg høef quga`rå` n` dbiur` hfrrght`.

Tn ‘ nT 5 nZ

=) Dfreg hfl p`nbnnfs n` dbiur` agn rghuàrf. Nugif, hèmbg ag nui`r 0 ag nfs 08p`nbnnfs y jì` qug qugag ul` biu`naà vgraàgr`, pfrqug = + : lf gs biu`n` 7. Gl gn gsp`hbf gl mn`lhf abmucg høef quga`rå` n` dbiur` hfrrght`.

T 5 nn + Tnnn

0:=BIG] ∝ [fnfhjbh





0=0

= 8 1

8

1

7

=

0

;

x

y

‘ 0 5 b

x = +

y = 5

r =

[rfauhtfs lft`mngs


@n-Cw`rbzeb, gn hrgàfr agn ànigmr`

Gn huàràf y gn humf ag n` sue` y ag n` abdgrglhb` ag ul mblfebf

Qâreblfs agn huàràf y agn humf ag ul mblfebf

Àrg` `nigmr`bh` ag huàràfs


Hflfhgr n` jbstfrb` ag n-Cw`rbzeb y su `pfrtg ` n` e`tgeàtbh`.

]gsfnvgr gn huàràf y gn humf ag n` sue` f abdgrglhb` ag ul mblfebf.

[r`htbh`r gn hànhunf eglt`n n hfepngt`r prfauhtfs lft`mngs.

H`nhun`r gl dfre` `nigmr`bh` gn àrg` ag iur`s igfeâtrbh`s.

0:8E`tgeàtbh` ∝ Xge`l` 0=



¡[`r` hfeglz`r!@n-Cw`rbzeb, gn hrgàfr agn ànigmr`

¡@ tr`m`c`r!

¶X`mg gn lfemrg ag ftr`s pgrsfl`s qug jbhbgrfl `pfrtgs bepfrt`ltgs ` n` e`tgeàtbh`6 Gshråm`nfs gln`s nålg`s.

Gs euy prfm`mng qug ustga rghflfzh` f j`y` vbstf gn rgtr`tf ag gstg e`tgeà-

tbhf, gs n` rgprgsglt`hbøl beìbl`rb` ag `n-Cw`rbzeb, ` qubgl sg ng hflsbagr` gn

dulaàfr agn ànigmr`.

[fhf sg s`mg ag n` vba` ag gstg `strølfef, igøir`df y e`tgeàtbhf eusuneàl,

`dbhbflàf ` n`s hbglhb`s gx`ht`s? qug vbvbø `prfxbeàègltg gltrg nfs `ðfs

9:; ‘ :7; y sg hrgg qug rgsbabø gl Brò.

Gl su Nbmrf ag n` rgauhhbøl, `n-Cw`rbzeb gshrbmbø n`s prbegr`s rgin`s agn hànhunf

`nigmr`bhf, n` tr`lspfsbhbøl ag târeblfs gl ul` ghu`hbøl, prgvbf hèmbf ag

sbilf, n` `lun`hbøl ag târeblfs baâltbhfs gl èmfs ebgemrfs, n`s ghu`hbflgs

ag sgiulaf iràf y ftrfs tge`s rgn`hbflàfs hfl gn ànigmr`.

Hfl su Obt`m `n-y`mrw`-n-euq`m`n` f Nbmrf agn ànigmr`, `n-Cw`rbzeb blbhbø n` nbtg-r`tur` e`tgeàtbh` gl n` qug ageugstr` n` sfnuhbøl ag ghu`hbflgs ag prbegr y

sgiulaf iràf, qug lf abdbgrg ag nfs prfhgabebgltfs gepngàfs jfy aå`.

Gl n` ulbvgrsbaà Fxdfra, Blin`tgrr`, j`y ul` hfpb` qug rgprfauhg gn frbibl`n agn

ànigmr` ag `n-Cw`rbzeb qug prfpfrhbfl` n` bldfre`hbøl eàs `ltbiu` `hgrh` agn

usf qug j`hå`l nfs àr`mgs agn sbstge` ag luegr`hbøl ag m`sg abgz, jfy usàf

gl tfaf gn eulaf.

(9:; ‘ :7;)

0:1BIG] ∝ [fnfhjbh



0. [rfauhtfs lft`mngsNfs prfauhtfs lft`mngs sfl h`sfs gspghb`ngs ag n` pftglhb`hbøl y ag n` euntb-

pnbh`hbøl ag pfnblfebfs qug huepngl hfl rgin`s c`s, y sg pugagl rgsfnvgr pfrsbepng blspghhbøl, gs aghbr, rgsfnvgr eglt`negltg n`s fpgr`hbflgs, sbl lghgsbaàag gshrbmbr tfaf gn prfhgabebgltf. Gst` sge`l` prgsgltèfs gcgepnfs gl nfsqug rg`nbzèfs n`s fpgr`hbflgs qug blabh` hà` døreun`. Nugif, ustga, hfl n`pràhtbh`, agmgrà gshrbmbr sfnf gn rgsuntàf.

0.0 Huàràf ag n` sue` ag ul mblfebf (` + m)=

Hfl nfs lóegrfs rg`ngs `prglabefs qug gngv`r ul` h`ltbaà `n huàràf gs nfebsef qug euntbpnbh`r n` m`sg pfr så ebse` afs vghgs. N` ebse` rgin` `pnbh` sbags`rrfnnèfs gn huàràf ag ul mblfebf.

( + m)= 5 ( + m)( + m)

5 `( + m) + m( + m)

5 `= + `m + `m + m=

5 `= + =`m + m=

Gl nui`r ag rg`nbz`r tfa`s n`s fpgr`hbflgs qug efstrèfs, ul hèblf eàs hfrtfp`r` fmtglgr gn ebsef rgsuntàf gs ` tr`vâs ag n` døreun` iglgr`n agn huàràfag ul mblfebf qug blabh`3

( + m)= 5 `= + =`m + m=

Xg ngg3 Gn huàràf ag n` sue` ag ul mblfebf gs biu`n `n huàràf agn prbegr

târeblf, eàs gn afmng prfauhtf agn prbegrf pfr gn sgiulaf, eàs gn huàràf agn

sgiulaf.

Gcgepnf3

( x + 7)= 5 ( x)= + =( x • 7) + (7)= 5 x = + 0; x + =7

( y + :)= 5 ( y)= + =( y • :) + (:)= 5 y= + 04 y + 41

(=`8 + 1m=)= 5 (=`8)= + =(=`8 • 1m=) + (1m=)= 5 1`4 + 04`8m= + 04m1

Gcgrhbhbf 0Ags`rrfnng gn huàràf ag n` sue` ag ul mblfebf. Iuågsg pfr gn gcgepnf.

;) ( x= + m8)= 5

0) ( + 8)= 5

=) (=h= + 1a =)= 5

( x=)= + =( x= • m8) + (m8)= 5 x1 + = x=m8 + m4





0.= Huàràf ag n` abdgrglhb` ag ulmblfebf (` ‘ m)=

Gn h`sf ag n` rgst` ag ul mblfebf `n huàràf gs euy p`rghbaf `n `ltgrbfr, hfln` abdgrglhb` qug nfs sbilfs sg `ntgrl`l gltrg pfsbtbvf y lgi`tbvf. Tgèfs nèuntbpnbh`hbøl y nugif su døreun`.

( ‘ m)= 5 ( ‘ m)( ‘ m)

5 `( ‘ m) ‘ m( ‘ m)

5 `= ‘ `m ‘ `m + m=

5 `= ‘ =`m + m=

N` døreun` iglgr`n agn huàràf ag n` abdgrglhb` ag ul mblfebf gs3

( ‘ m)= 5 `= ‘ =`m + m=

Xg ngg3 Gn huàràf ag n` abdgrglhb` ag ul mblfebf gs biu`n n huàràf agn prbegr

târeblf, eglfs gn afmng prfauhtf agn prbegrf pfr gn sgiulaf, eàs gn huàràfagn sgiulaf.

Gcgepnf3 ( ‘ 8)= 5 ( )= ‘ =( • 8) + (8)= 5 `= ‘ 4` + 2

(1 x ‘ 7)= 5 (1 x)= ‘ =(1 x • 7) + (7)= 5 04 x = ‘ 1; x + =7

(7 y1 ‘ = z 8)= 5 (7 y1)= ‘ =(7 y1 • = z 8) + (= z 8)= 5 =7 y: ‘ =; y1 z 8 + 1 z 4

0.8 [rfauhtf ag n` sue` pfr nàbdgrglhb` (` + m)(` ‘ m)

Gl gstg prfauhtf agmg fmsgrv`r qug nfs târeblfs ag nfs mblfebfs tbglgl n`sebse`s nbtgr`ngs, pgrf gstàl ulba`s pfr sbilfs abdgrgltgs.

( + m)( ‘ m) 5 `( ‘ m) + m( ‘ m)

5 `= ‘ `m + `m ‘ m=

5 `= ‘ ; ‘ m=

N` døreun` iglgr`n agn prfauhtf ag n` sue` pfr n` abdgrglhb` ag afs mblfebfs gs3

( + m)( ‘ m) 5 `= ‘ m=

Xg ngg3 Gn prfauhtf ag n` sue` pfr n` abdgrglhb` ag afs mblfebfs gs biu`n ` n`

abdgrglhb` ag sus huàràfs.

Gcgepnf3

( + 7)( ‘ 7) 5 ( )= ‘ (7)= 5 `= ‘ =7

(4 x + :)(4 x ‘ :) 5 (4 x)= ‘ (:)= 5 84 x = ‘ 41

(8 x + 1 y)(8 x ‘ 1 y) 5 (8 x)= ‘ (1 y)= 5 2 x = ‘ 04 y=

@tglhbøl3 `n ags`rrfnn`rn` abdgrglhb` ag ul

mblfebf ` hu`nqubgrpftglhb`, nfs sbilfssbgeprg v`l `ntgrlfs+, ‘, +, ‘, gth.

0:4BIG] ∝ [fnfhjbh



0.1 [rfauhtf ag afs mblfebfs hfl ul târeblfhfeól y sbilfs biu`ngs

Gl gstg prfauhtf sg pugagl prgsglt`r afs h`sfs3 qug nfs sbilfs sg`l pfsbtbvfsf lgi`tbvfs. [rgstg `tglhbøl ` hà` ulf.

`. (` + m)(` + h)

[`r` ags`rrfnn`r gn prfauhtf ag afs mblfebfs hfl ul târeblf hfeól y sbilfpfsbtbvf, sg prfhgag `så3

( + m)( + h) 5 `( + h) + m( + h)

5 `= + `h + `m + mh

5 `= + `(m + h) + mh

N` døreun` iglgr`n agn prfauhtf ag afs mblfebfs hfl ul târeblf hfeól ysbilfs pfsbtbvfs gs3

( + m)( + h) 5 `= + `(m + h) + mh

Xg ngg3 Gn prfauhtf ag afs mblfebfs hfl ul târeblf hfeól y sbilfs biu`ngs gs

biu`n `n huàràf agn târeblf hfeól, eàs gn târeblf hfeól pfr n` sue` ag nfs

târeblfs lf hfeulgs, eàs gn prfauhtf ag nfs târeblfs lf hfeulgs.

Gcgepnf3 ( + 1)(` + =) 5 ( )= + `(1 + =) + (1 • =) 5 `= + 4` + :

(1 y= + 4)(1 y= + 8) 5 (1 y=)= + 1 y=(4 + 8) + (4 • 8) 5 04 y1 + 84 y= + 0:

m. (` ‘ m)(` ‘ h)

[`r` ags`rrfnn`r gn prfauhtf ag afs mblfebfs hfl ul târeblf hfeól y sbilflgi`tbvf, sg prfhgag `så3

( ‘ m)( ‘ h) 5 `( ‘ h) ‘ m( ‘ h)

5 `= ‘ `h ‘ `m + mh

5 `= + `(‘ m ‘ h) + mh

N` døreun` iglgr`n agn prfauhtf ag afs mblfebfs hfl ul târeblf hfeól ysbilfs lgi`tbvfs gs3

( ‘ m)( ‘ h) 5 `=

+ `(‘ m ‘ h) + mh

Xg ngg3 Gn prfauhtf ag afs mblfebfs hfl ul târeblf hfeól y sbilfs biu`ngs gs

biu`n `n huàràf agn târeblf hfeól, eàs gn târeblf hfeól pfr n` sue` ag nfs

târeblfs lf hfeulgs, eàs gn prfauhtf ag nfs târeblfs lf hfeulgs.

Gcgepnf3

( ‘ 2)( ‘ 8) 5 ( )= + `(‘2 ‘ 8) + (‘2)(‘8) 5 `= ‘ 0=` + =9

(8m ‘ 1)(8m ‘ 4) 5 (8m)= + 8m(‘1 ‘ 4) + (‘1)(‘4) 5 2m= ‘ 8;m + =1




Gcgrhbhbf =Ags`rrfnng nfs prfauhtfs lft`mngs blabhàfs gl hà` `p`rtàf. J`y ul gcgepnf p`r` hà` h`sf.

@. Huàràf ag n` abdgrglhb` ag ul mblfebf.

;) ( x8 ‘ 4)= 5

0) ( x ‘ y)= 5

=) (w ‘ z )= 5

8) (8 x ‘ 1)= 5

1) (4 y ‘ 7)= 5

7) ( 0

=w ‘ = z )

=

5

4) ( x ‘ 0

7 z )

=

5

9) (8 x1 ‘ 1 y8)= 5

:) (= x= ‘ 8 y=)= 5

2) (: x7 ‘ 1 y8)= 5

0;) (1 x1 ‘ 7 y2)= 5

00) (8e=l8 ‘ : pq=)= 5

0=) ( 0

8w ‘ 2 z )

=

5

08) ( 1

8 x= ‘

7

= y8)

=

5

M. [rfauhtf ag n` sue` pfr n` abdgrglhb`.

;) (= x + 8 y)(= x ‘ 8 y) 5

0) ( p + q)( p ‘ q) 5

=) ( y + 1)( y ‘ 1) 5

8) (w + 4)(w ‘ 4) 5

1) (8 x + y)(8 x ‘ y) 5

7) ( =8

` + m)( =8

` ‘ m) 5

4) ( :

7e= + 8)( :

7e= ‘ 8) 5

9) (7 z + :)(7 z ‘ :) 5

:) (1 y= + z )(1 y= ‘ z ) 5

2) (= x= + 8)(= x= ‘ 8) 5

0;) (8 x + 1 y)(8 x ‘ 1 y) 5

00) (7 x + 0; y)(7 x ‘ 0; y) 5

0=) ( 0= x + 7 y)( 0

= x ‘ 7 y) 5

08) ( 7

1e= + 8l)( 7

1e= ‘ 8l) 5

H. [rfauhtf ag afs mblfebfs hfl ul târeblf hfeól.

;) ( + 7)( + 8) 5

0) ( x + :)( x + 2) 5

=) ( y + 8)( y + 1) 5

8) ( z + 9)( z + =) 5

1) ( + 1)( + 1) 5

7) (m + 4)(m + 4) 5

4) (o = + :)(o = + =) 5

9) (8w8 + 4)(8w8 + 7) 5

:) ( ‘ 4)(` ‘ 1) 5

2) (=h ‘ 8)(=h ‘ =) 5

0;) (7 x ‘ :)(7 x ‘ 1) 5

00) (8 y ‘ :)(8 y ‘ 9) 5

0=) (= x ‘ 7)(= x ‘ 8) 5

08) (0; x ‘ 8)(0; x ‘ 8) 5

01) (2 z = ‘ 4)(2 z = ‘ 0;) 5

07) (8 x= y ‘ =)(8 x= y ‘ 1) 5

x4 ‘ 0= x8 + 84

1 x= ‘ 2 y=

`= + :` + 07

0::BIG] ∝ [fnfhjbh



0.7 Humf ag n` sue` ag ul mblfebf (` + m)8

[`r` gngv`r ul mblfebf `n humf, nf agshfepflgefs gl afs d`htfrgs y ags`rrf-nnèfs n`s fpgr`hbflgs hfrrgspflabgltgs.

( + m)8 5 ( + m)=( + m)

5 (=

+ =`m + m=

)( + m)5 `=( + m) + =`m( + m) + m=( + m)

5 `8 + `=m + =`=m + =`m= + `m= + m8

5 `8 + 8`=m + 8`m= + m8

Wl hèblf eàs hfrtf p`r` fmtglgr gn ebsef rgsuntàf gs ` tr`vâs ag n` døreunìglgr`n agn humf ag ul mblfebf.

( + m)8 5 `8 + 8`=m + 8`m= + m8

Xg ngg3 Gn humf ag n` sue` ag ul mblfebf gs biu`n n humf agn prbegr târeblf, eàs

gn trbpng agn prbegrf `n huàràf pfr gn sgiulaf, eàs gn trbpng agn prbegrf pfr gnsgiulaf `n huàràf, eàs gn humf agn sgiulaf.

Gcgepnf3

(e + l)8 5 (e)8 + 8(e)=(l) + 8(e)(l)= + (l)8

5 e8 + 8(e=)l + 8e(l=) + l8

5 e8 + 8e=l + 8el= + l8

(1 x + y)8 5 (1 x)8 + 8(1 x)=( y) + 8(1 x)( y)= + ( y)8

5 41 x8 + 8(04 x=)( y) + 8(1 x)( y=) + y8

5 41 x 8

+ 1: x =

y + 0= xy=

+ y8

(8 p= + =t 8)8 5 (8 p=)8 + 8(8 p=)=(=t 8) + 8(8 p=)(=t 8)= + (=t 8)8

5 =9 p4 + 8(2 p1)(=t 8) + 8(8 p=)(1t 4) + :t 2

5 =9 p4 + 71 p1t 8 + 84 p=t 4 + :t 2

Gcgrhbhbf 8Gshrbm` nfs târeblfs qug hfepngt`l gn ags`rrfnnf agn mblfebf `n humf.

0) (1w + z )8 5 ( )8 + 8(1w)=( ) + 8( )( z )= + ( )8

5 + 8( )( z ) + 8(1w)( ) +

5 41w8 + 1:w= z + 0=wz = + z 8

=) (8 x= + = y=)8 5 ( )8 + 8( )=(= y=) + 8(8 x=)( )= + ( )8

5 =9 x4 + 8(2 x1)( ) + 8( )(1 y1) + : y4

5 + 71 x 1 y= + + : y4




0.4 Humf ag n` abdgrglhb` ag ul mblfebf (` ‘ m)8

Gl gstg h`sf, n` abdgrglhb` hfl n` sue` gs qug nfs sbilfs agn rgsuntàf sg ntgrl`l+, ‘, +, ‘ gl hà` ulf ag nfs târeblfs.

( ‘ m)8 5 ( ‘ m)=( ‘ m)

5 (=

‘ =`m + m=

)( ‘ m)5 `=( ‘ m) ‘ =`m( ‘ m) + m=( ‘ m)

5 `8 ‘ `=m ‘ =`=m + =`m= + `m= ‘ m8

5 `8 ‘ 8`=m + 8`m= ‘ m8

N` døreun` iglgr`n ag n` abdgrglhb` ag ul mblfebf `n humf gs3

( ‘ m)8 5 `8 ‘ 8`=m + 8`m= ‘ m8

Xg ngg3 Gn humf ag n` abdgrglhb` ag ul mblfebf gs biu`n `n humf agn prbegr târeblf,

eglfs gn trbpng agn prbegrf `n huàràf pfr gn sgiulaf, eàs gn trbpng agn prbegrf

pfr gn sgiulaf `n huàràf, eglfs gn humf agn sgiulaf.

Gcgepnf3

(h ‘ a )8 5 (h)8 ‘ 8(h)=(a ) + 8(h)(a )= ‘ (a )8

5 h8 ‘ 8(h=)a + 8h(a =) ‘ a 8

5 h8 ‘ 8h=a + 8ha = ‘ a 8

( x ‘ = y)8 5 ( x)8 ‘ 8( x)=(= y) + 8( x)(= y)= ‘ (= y)8

5 x8 ‘ 8( x=)(= y) + 8( x)(1 y=) ‘ : y8

5 x 8 ‘ 4 x = y + 0= xy= ‘ : y8

(1w ‘ 7)8 5 (1w)8 ‘ 8(1w)=(7) + 8(1w)(7)= + (7)8

5 41w8 ‘ 8(04w=)(7) + 8(1w)(=7) ‘ 0=7

5 41w8 ‘ =1;w= + 8;;w ‘ 0=7

Gcgrhbhbf 1Gshrbm` nfs târeblfs qug hfepngt`l gn ags`rrfnnf agn mblfebf `n humf.

0) (=h ‘ a )8 5 (=h)8 ‘ 8( )=( ) + 8(=h)( )= ‘ (a )8

5 ‘ 8(1h=)( ) + 8( )(a =) ‘

5 :h8 ‘ 0=h=a + 4ha = ‘ a 8

=) (1 y8 ‘ = z )8 5 ( )8 ‘ 8( )=(= z ) + 8( )(= z )= ‘ ( )8

5 41 y2 ‘ 8( )( ) + 8( )(1 z =) ‘

5 41 y2 ‘ 24 + 1: ‘ : z 8




]gsuegl

Nfs prfauhtfs lft`mngs sfl h`sfs gspghb`ngs ag n` pftglhb`hbøl y euntbpnbh`hbøl ag pfnblfebfs qughuepngl hfl rgin`s c`s y qug sg pugagl rgsfnvgr pfr sbepng blspghhbøl, gs aghbr, rgsfnvgr eglt`n-egltg n`s fpgr`hbflgs, sbl lghgsbaà ag gshrbmbr tfaf gn prfhgabebgltf.

Nfs prfauhtfs lft`mngs qug gstuabèfs gst` sge`l` sg rgsuegl gl n` t`mn` sbiubgltg3

[`r` rgdfrz`r sus hflfhbebgltfs sfmrg prfauhtfs lft`mngs, vbsbtg gst` abrghhbøl3 jttp3//iff.in/IbCH`9

Lfemrg [rfauhtf lft`mng Døreun` iglgr`n

Huàràf ag n` sue` ag ul mblfebf ( + m)= 5 `= + =`m + m=

Huàràf ag n` abdgrglhb` ag ul mblfebf ( ‘ m)= 5 `= ‘ =`m + m=

[rfauhtf ag n` sue` pfr n` abdgrglhb` ( + m)( ‘ m) 5 `= ‘ m=

[rfauhtf ag afs mblfebfs hfl ul târeblf

hfeól y sbilfs biu`ngs

( + m)( + h) 5

( ‘ m)( ‘ h) 5

`= + `(m + h) + mh

`= + `(‘ m ‘ h) + mh

Humf ag n` sue` ag ul mblfebf ( + m)8 5 `8 + 8`=m + 8`m= + m8

Humf ag n` abdgrglhb` ag ul mblfebf ( ‘ m)8 5 `8 ‘ 8`=m + 8`m= ‘ m8

020E`tgeàtbh` ∝ Xge`l` 0=





E`rqug ul` "Z" gl gn huàrf ag n` fphbøl qug rgspflag hfrrghtègltg hà` prgiult`.

0) ¶Puâ gxprgsbøl gs gqubv`ngltg ( + m)=6

=) ¶Huàn gs gn prbegr târeblf agn rgsuntàf ag (8 x + =)(8 x ‘ =)6

8) ¶Huàn gs gn prbegr târeblf agn rgsuntàf ag (1 x + =)=6

1) ¶Huàn gs ul gcgepnf agn prfauhtf ag n` sue` pfr n` abdgrglhb`6

7) ¶Huàn gs gn sgiulaf târeblf `n fpgr`r (= y + 8)(= y + 7)6

4) ¶Huàn gs gn rgsuntàf ag ( x + y)( x ‘ y)6

9) ¶Huàltfs târeblfs rgsunt`l ag gngv`r ( x + y)

8

6

:) ¶Huàn gs gn óntbef târeblf `n gngv`r (= x= + 8 y8)86

@utfhfltrfn

( + m)( ‘ m) ( + m)( + m)

( ‘ m)( ‘ m)

( ‘ m)( + m)

4 x

2 x

8 x=

2 x=

1 : x

04 x

04 x=

( + 8)(` + 1)

( + 7)( ‘ 7)

( ‘ 4)( ‘ =)

( ‘ 8)( ‘ 8)

: y 07

1 y=

04 y

x + y

x ‘ y

x= + y=

x= ‘ y=

= 8

1

4

: x4

: x7

=9 y2

=9 y4





@. Fpgrg gn prfauhtf ag n` sue` pfr n` abdgrglhb` pfr sbepng blspghhbøl y gshrbm` gn rgsuntàf gl n`nålg`. Qbglg ul gcgepnf.

;) (7` + 1)(7` ‘ 1) 5

0) (1m + 4)(1m ‘ 4) 5

=) (=h + 2)(=h ‘ 2) 5

8) (4 x + 1)(4 x ‘ 1) 5

1) ( z + 0;)( z ‘ 0;) 5

7) (7 y ‘ :)(7 y + :) 5

4) (9w ‘ 2)(9w + 2) 5

9) (0; x + =)(0; x ‘ =) 5

:) ( 0

=

x ‘ 1)( 0

=

x + 1) 5

2) ( 7

8 y + 2)( 7

8 y ‘ 2) 5

0;) ( 8

7` + 7)( 8

7` ‘ 7) 5

00) ( :

8a +

4

9 )( :

8a ‘

4

9 ) 5

0=) (8m + = )(8m ‘ = ) 5

08) (=`= + 4m)(=`= ‘ 4m) 5

01) (7j8 ‘ 8o )(7j8 + 8o ) 5

07) (4e ‘ 8l)(4e + 8l) 5

M. Fpgrg gn prfauhtf ag afs mblfebfs hfl ul târeblf hfeól y sbilfs biu`ngs pfr sbepng blspghhbøl.

;) ( + 7)( + :) 5

0) ( x ‘ 4)( x ‘ 9) 5

=) (h ‘ :)(h ‘ =) 5

8) ( z ‘ 4)( z ‘ :) 5

1) (a ‘ 1)(a ‘ 7) 5

7) (m ‘ 4)(m ‘ :) 5

4) (w + :)(w + 2) 5

9) ( y + 0;)( y + 7) 5

:) ( z ‘ 1)( z ‘ 4) 5

2) ( y ‘ 8)( y ‘ 7) 5

0;) ( x + =)( x + 9) 5

00) (w + :)(w + 9) 5

0=) (8 y ‘ 7)(8 y ‘ 8) 5

08) (7 p + 7)(7 p + 7) 5

H. Ags`rrfnng pfr sbepng blspghhbøl gn humf ag nfs mblfebfs sbiubgltgs.

;) ( + 1)8 5

0) (m + =)8 5

=) (h + 1)8 5

8) (a + 0)8 5

1) (= x + 8)8 5

7) (8m + 8)8 5

4) (7e + =)8 5

9) (= z ‘ 8)8 5

:) (1w ‘ 0)8 5

2) (1 x ‘ =)8 5

0;) (7 y= ‘ 0)8 5

00) (= z = ‘ 1)8 5

0=) ( =

8m ‘ 7)

8

5

08) ( 7

4o = ‘ 8)

8

5

=7`= ‘ 04

`= + 08` + 1;

`8 + 0=`= + 1:` + 41






[r`htbqug nfs prfauhtfs lft`mngs hfl n`s àrg`s ag hà` iur`. Iuågsg pfr gn gcgepnf.

;)

x + 8

x ‘ 8

@ 5 ( x + 8)( x ‘ 8)@ 5 x= ‘ 2

0)

x + 1

x ‘ 1

=)

1 x + 7

1 x + 8

8)

x + 0;

x + 7

1)

x= + 4

x= ‘ 1

7)= x + :

= x ‘ :





E`rqug hfl ul hjgqug n` h`sbnn` qug egcfr blabqug su rglabebgltf.nfiràf glprfhgsf lfnfiràf

A g s p u â s a g

g s t u a b ` r . . .

Hflfzhf n` jbstfrb` ag `n-Cw`rbzeb y su `pfrtg ` n` e`tgeàtbh`.

]gsugnvf gn huàràf y gn humf ag n` sue` f abdgrglhb` ag ul mblfebf.

[r`htbhf gn hànhunf eglt`n `n hfepngt`r prfauhtfs lft`mngs.

H`nhunf gl dfre` `nigmr`bh` gn àrg` ag iur`s igfeâtrbh`s.

Nfs prfauhtfs lft`mngs y n`s iur`s igfeâtrbh`s

[fagefs rgn`hbfl`r nfs prfauhtfs lft`mngs hfl gn hànhunf ag àrg`s ag iur`sigfeâtrbh`s.

[fr gcgepnf, gn huàràf ag n` agrghj` sg dfreø hfl n`s iur`s sbiubgltgs3

• Wl huàràf ag nàf `

• Wl huàràf ag nàf m

• Afs rghtàliunfs ag n`rif ` y `lhjf m

Xb qugrgefs glhfltr`r gn àrg` tft`n, sueèfs gn àrg` ag hà` iur`.

`= + `m + `m + m= 5 `= + =`m + m=

[grf sb j`hgefs usf ag nfs prfauhtfs lft`mngs, pfagefs h`nhun`rn` ag dfreàbrght`, `så3 ( + m)= 5 `= + =`m + m= y nngièfs `n ebsef rgsuntàf.

@jfr` ng hfrrgspflag ` ustga glhfltr`r gn àrg` agnrghtàliunf. ]g`nbhg nf qug sg pbag gl hà` blhbsf.

`. Fmtgli` gn àrg` ag hà` sghhbøl y sóegn`s p`r` fmtglgr gn àrg` tft`n.

+ + + 5m. @jfr` utbnbhg ul prfauhtf lft`mng p`r` glhfltr`r gn àrg` tft`n.

h. ¶Puâ prfauhtf lft`mng utbnbzø6

x x

x

x

m m

`

`

`

`

`

`

m m

m

m

`

`

`

`m m

m

m






080

= 8 1

8

1

7

=

0

;

x

y

‘ 0 5 b

x = +

y = 5

r =

D`htfrbz`hbøl B¶Puâ glhfltr`rà gst` sge`l`6

D`htfrgs prbefs

D`htfrbz`hbøl ag pfnblfebfs

Euntbpnbh`hbøl ag eflfebfs

[rfmnge` ag nøibh`


]ghflfhgr n`s h`r`htgråstbh`s ag ul pfnblfebf d`htfrbz`mng.

[r`htbh`r n` d`htfrbz`hbøl ag ul pfnblfebf pfr d`htfr hfeól.

[r`htbh`r n` d`htfrbz`hbøl ag ul pfnblfebf pfr ìrup`hbøl ag târeblfs.

[r`htbh`r n` d`htfrbz`hbøl ag ul trblfebf huàràf pgrdghtf.

[r`htbh`r n` d`htfrbz`hbøl ag ul` abdgrglhb` ag huàràfs.

[r`htbh`r n` euntbpnbh`hbøl ag eflfebfs.




¡[`r` hfeglz`r!

D`htfrgs prbefs

@ nf n`rif ag n`s sge`l`s `ltgrbfrgs jgefs tr`m`càf agshfepflbglaf nfs

lóegrfs gl sus d`htfrgs prbefs. Gstf nf j`hgefs hu`laf sg agsg` sbepnbdbh`r

`niul` gxprgsbøl f p`r` e`lbpun`r `niul` fpgr`hbøl ` lugstr` hflvglbglhb`.

@jfr`, lfs tfe`rgefs ul tbgepf p`r` hflfhgr gl prfdulabaà gstg tge`, pfr-

qug sgrà gsglhb`n p`r` tr`m`c`r hfl nf qug `prglagrgefs gst` sge`l`. Tgèfs

quâ sfl nfs d`htfrgs prbefs.

[rbegrf, agmgefs tglgr hn`rf n` agdblbhbøl ag ul lóegrf prbef3 gstg gs ul

lóegrf l`tur`n qug pugag agshfepflgrsg sfnègltg gl afs d`htfrgs abstbltfs3

ân ebsef y gn 0. Gstfs lóegrfs sfl hfltr`rbfs ` nfs hfepugstfs, gs aghbr,

`qugnnfs qug tbglgl `n eglfs ul abvbsfr l`tur`n abstbltf agn ulf y ag så ebsefs.

Nfs lóegrfs prbefs eglfrgs ` 8; sfl3

0, =, 8, 7, 9, 00, 08, 09, 02, =8...

Wl lóegrf hfepugstf sg pugag "agsinfs`r" f "agshfepflgr" gl sus d`htfrgs

prbefs, gs aghbr, gl `qugnnfs lóegrfs prbefs abvbsfrgs gx`htfs ag gsg lóegrf

gltgrf? gn prfhgsf p`r` glhfltr`r gstfs abvbsfrgs sg aglfebl` d`htfrbz`hbøl ag

gltgrfs. Gstg prfhgabebgltf ng rgsunt`rà dèbnb`r, vgèfs ul gcgepnf3

07; =

97 8

=7 77 7

0

Ag e`lgr` sbebn`r qug gl n` rbteâtbh`, gl gn ànigmr` sg utbnbz` n` d`htfrbz`hbøl p`r`

e`lbpun`r y agshfepflgr `niól pfnblfebf gl eflfebfs f gl ftrf pfnblfebf

eàs sbepnbdbhàf, hfl gn fmcgtbvf ag j`hgr eàs dàhbn lugstrf tr`m`cf.

¡@ tr`m`c`r!

H`nhung nfs d`htfrgs prbefs ag nfs lóegrfs sbiubgltgs.

2; 89: =:4




0. D`htfrbz`hbøl ag pfnblfebfsGspghåhègltg gl ànigmr`, hu`laf j`mnèfs ag d`htfrbz`r ul` gxprgsbøl lfs

rgdgrbefs n prfhgsf ag sbepnbh`rn` j`nn`laf afs f eàs d`htfrgs huyf prfauhtfgs biu`n ` n` gxprgsbøl prfpugst`. Iglgr`negltg sg gepng` gstg prfhgabebgltf

p`r` tr`m`c`r hfl pfnblfebfs.

J`y abdgrgltgs eâtfafs f "h`sfs" p`r` d`htfrbz`r, agpglabglaf ag n` gxprgsbøl

qug agsggefs sbepnbh`r? gst` sge`l`, lfsftrfs gstuab`rgefs nfs prbegrfs

hu`trf h`sfs ag ul tft`n ag sbgtg qug vgrgefs. Muglf, ¡e`lfs ` n` fmr`!

0.0 D`htfr hfeól eflfebf

Gstg eâtfaf hflsbstg gl gxtr`gr ag ul pfnblfebf, ul eflfebf hfef d`htfr

hfeól ` hà` ulf ag nfs târeblfs agn pfnblfebf gl hugstbøl. Nugif, sg agc`blabhà` n` euntbpnbh`hbøl gltrg gn d`htfr hfeól gxtr`åaf y nfs târeblfs qug

quga`l ag n` gxprgsbøl frbibl`n. [flièfs `tglhbøl `n gcgepnf qug sbiug.

D`htfrbhgefs gn pfnblfebf3

• [rbegrf, h`nhunèfs gn Eàxbef Hfeól Abvb-

sfr (EHA) ag nfs hfgdbhbgltgs agn pfnblfebf.

Gl gstg h`sf, fmtglgefs qug gn EHA ag 01,

=: y 74 gs 01. Hfl gstf glhfltrèfs gn hfgdb-

hbgltg agn d`htfr hfeól. Xb nfs hfgdbhbgltgs lf

tuvbgr`l EHA, lf sg prgfhupg, sglhbnnègltg

hfltbló` hfl gn prfhgabebgltf.

• Abvbabefs hà` ulf ag nfs târeblfs agn pfnb-

lfebf gltrg su EHA, hfl gstf nf "gxtr`gefs",

pfr nf qug agcèfs n` euntbpnbh`hbøl blabhà`.

• @jfr`, bagltbdbhèfs quâ v`rb`mngs f nbtgr`ngs

pfsggl gl hfeól tfafs nfs eflfebfs. Gl gstg

h`sf, n` v`rb`mng x gs n` qug sg rgpbtg f gs "hf-

eól" gl tfafs nfs târeblfs. Gngibefs n` x qug

tgli` gn eglfr gxpflgltg, gl gstg h`sf sgrå`

x =, y n` "gxtr`gefs", abvbabglaf hà` târeblf

gltrg n` ebse`.

• @n gxtr`gr n` x, hfepngtèfs gn d`htfr hfeól,

qug sgrå` 01 x=. Fraglèfs gn pfnblfebf y hfl

gstf quga`rå` n` gxprgsbøl d`htfrbzà`3

[fagefs hfeprfm`r qug ul` d`htfrbz`hbøl gs hfrrght` sb n fpgr`r n` euntbpnbh`-

hbøl blabhà`, fmtglgefs gn pfnblfebf frbibl`n. Hfeprfmgefs gn gcgepnf qug

`h`mèfs ag vgr3 01 x =( y = + 1 x = ‘ = x ) 5 01 x = y= ‘ =: x 8 + 74 x 1

01 x= y= ‘ =: x8 + 74 x1

01 =: 74 =

9 01 =: 9

0 = 1

= • 9 5 01

01(0 x= y= ‘ = x8 + 1 x1)

01( x= y= ‘ = x8 + 1 x1) 5

x= y=

x= ‘

= x8

x= +

1 x1

x= 5

01 x=( y= ‘ = x + 1 x=) 5

01 x =( y = + 1 x = ‘ = x )







0.= D`htfr hfeól pfr ìrup`hbøl ag târeblfs

Gepngèfs gn h`sf ag d`htfr hfeól pfr irup`hbøl ag târeblfs, sb nfs eflfebfs

qug hfepflgl gn pfnblfebf pugagl rgulbrsg gl irupfs hfl ul d`htfr hfeól

abdgrgltg gl hà` irupf. Nugif, gepngèfs n` ebse` tâhlbh` qug gl gn d`htfr

hfeól. Gltglagrgefs hfl ul gcgepnf3

D`htfrbhgefs n` gxprgsbøl sbiubgltg3

• Lftèfs qug pfagefs ìrup`r

‘4 xy y 1ey, pfrqug tbglgl gl hfeól

y. Ag biu`n dfre`, pfagefs ìrup`r

=ep= y ‘8 xp= pfrqug tbglgl gl

hfeól p=.

• @jfr`, utbnbz`laf gn prfhgabebgltf

qug `prglabefs gl n` sghhbøl 0.0,

prfhgagefs ` gxtr`gr gn d`htfr

hfeól ag hà` ulf ag nfs "irupfs"ag târeblfs.

• @n fragl`r n`s gxprgsbflgs agltrf

ag nfs p`râltgsbs, lfs aèfs huglt`

qug sfl biu`ngs, nf qug qubgrg aghbr,

qug gstfs sfl ftrf d`htfr hfeól.

• [fagefs gxtr`gr gn lugvf d`htfr

hfeól qug lfs qugaø y rgulbr nfs

eflfebfs qug fmtuvbefs dugr` ag

nfs p`râltgsbs. Hfl gstf fmtglarå`-efs lugstr` gxprgsbøl d`htfrbzà`.

¶Tg qug gs dàhbn6 Xfnf tglgefs qug br pfhf ` pfhf, sbiubglaf gn prfhgabebgltf.

Tgèfs ftrf gcgepnf, d`htfrbhgefs3

• @irupèfs 87 8 z y =;`8 y, pfrqug tbglgl

gl hfeól `8. @irupèfs ‘0=wy y ‘=0wz

pfrqug tbglgl gl hfeól w. Agmgefs

tglgr hubaàf hfl nfs sbilfs.

• @jfr`, gxtr`gefs gn d`htfr hfeól ag

hà` ulf ag nfs irupfs ag târeblfs. Gl gstg h`sf, gxtr`gefs ( ‘8w) agn

sgiulaf irupf p`r` qug n` gxprgsbøl

sg` biu`n ` n` agn prbegr irupf

• ]gulbefs nfs eflfebfs qug quga`-

rfl dugr` agn p`râltgsbs. Fmtglgefs

`så lugstr` gxprgsbøl d`htfrbzà`.

87`8 z ‘ =0wz ‘ 0=wy + =;`8 y 5

(87`8 z + =;`8 y) + (‘=0w z ‘ 0=w y) 5

7`8(9 z + 1 y) + (‘8w)(9 z + 1 y) 5

7`8(9 z + 1 y) ‘ 8w(9 z + 1 y) 5

(7`8 ‘ 8w)(9 z + 1 y)

‘4 xy + =ep= + 1ey ‘ 8 xp= 5

(‘4 x y + 1e y) + (=e p= ‘ 8 x p=) 5

= y(‘8 x + =e) + p=(=e ‘ 8 x) 5

= y(=e ‘ 8 x) + p=(=e ‘ 8 x) 5

(= y + p=)(=e ‘ 8 x )

=;0E`tgeàtbh` ∝ Xge`l` 08



Gcgrhbhbf =

D`htfrbhg nfs pfnblfebfs sbiubgltgs. [ugag tr`m`c`r gl jfc`s `p`rtg sb gs lghgs`rbf.

0) 4`8 ‘ 0;`x ‘ 7mx= + 8`=mx =) 0=my ‘ 4m + 4 x= y ‘ 8 x=

8) 9`=my + :mx= y= + 1m= x= + 01`= y8 1) ‘0=`z + 04`xy= + 8m= z ‘ 1m= xy=

7) ‘2mx= + 2`= x= ‘ 8my= + 8`= y= 4) 7ml ‘ 0;me + 0:e8 x ‘ 2e=lx ‘ =l1 y + 1el8 y

9) =e8 yz ‘ 4e=l8 y ‘ 07ml8 + 7mez :) ‘0: y8 z + 00`8 x= ‘ 2 xy= z + ==`8 xy

2) 2e=l8 + :`=ez ‘ 8`l8 z ‘ =1è8 0;) ‘2mxy + 0: x + 0;m8r ‘ 7m1ry

00) 07rx8 + =1 sz + =; x= z + 0:rsx ‘ 8; s1 y ‘ =7 s8 x= y 0=) : xy8 ‘ 8wz = ‘ 0; yz + 7 z 8 + 4wy ‘ 1 xy= z =

=;=BIG] ∝ [fnfhjbh





Gcgrhbhbf 8

D`htfrbhg nfs pfnblfebfs sbiubgltgs. @niul`s gxprgsbflgs pugagl gst`r agsfraglà`s, `så qug tglarà

qug fragl`rn`s `ltgs ag fpgr`r. [ugag tr`m`c`r gl jfc`s `p`rtg sb gs lghgs`rbf.

0) `= ‘ =`m + m= =) x= + 01 x + 12

8) `= ‘0;` + =7 1) 2 ‘ 4 x + x=

7) 2m= ‘ 8;`=m + =7`1 4) 1 x= + 1 x`8 + `4

9) 04 + 1; x= + =7 x1 :) x= ∝ =; x + 0;;

2) x4 + 0; x8 + =7 0;) :0 z = ‘ 0:; z + 0;;

00) 0;; x0; ‘ 4;`1 x7 y4 + 2`: y0= 0=) `= ‘ =1è= x= + 011e1 x1

=;1BIG] ∝ [fnfhjbh



0.1 Abdgrglhb` ag huàràfs

Xg ng hflfhg hfef abdgrglhb` ag huàràfs ` ul mblfebf gl gn qug sus

târeblfs tbglgl ul` rgst` ag pfr egabf y sg ngs pugag h`nhun`r r`åz huàrà`.

]ghfraèfs qug n` sge`l` `ltgrbfr `prglabefs sfmrg prfauhtfs lft`mngs y

tglåèfs qug3

( + m)( ‘ m) 5 `= ‘ m=

[`r` n` abdgrglhb` ag huàràfs, gn prfhgabebgltf gs gx`htègltg gn hfltr`rbf3

abdgrglhb` ag huàràfs

`= ‘ m= 5 ( + m)( ‘ m)

Hfef lftèfs, p`r` n` d`htfrbz`hbøl ag gstg h`sf agmgefs gxtr`gr n` r`åz hu`-

arà` ag nfs târeblfs. Nugif, agc`r blabhà` n` euntbpnbh`hbøl gltrg n` sue` y n`

abdgrglhb` ag gst`s r`åhgs. Gltglagrgefs hn`règltg hfl ul gcgepnf3

D`htfrbhgefs gn mblfebf3

• Gxtr`gefs n`s r`åhgs huàrà`s ag èmfs

târeblfs.

• @jfr`, agcèfs blabhà` n` euntbpnbh`hbøl ag

n` sue` pfr n` abdgrglhb` ag n`s r`åhgs qug gl-

hfltrèfs. [rbegrf n` r`åz agn prbegr târeblf y

nugif n` r`åz agn sgiulaf târeblf.

¡Nbstf!, y` tglgefs n` gxprgsbøl d`htfrbzà`.

Gl fh`sbflgs sg lfs pugagl prgsglt`r gxprgsbflgs qug lf sfl ul` abdgrglhb` ag

huàràfs gl så, pgrf qug n`s pfagefs d`htfrbz`r hfef t`n. Lf sg `sustg, sfnfagmgefs sgiubr gn ebsef eâtfaf, vgèfs.

D`htfrbhgefs gn pfnblfebf sbiubgltg3

• Lftèfs qug n` gxprgsbøl n` pfagefs tr`m`c`r

hfef ul` abdgrglhb` ag huàràfs, pfr nf qug

sgiubefs gx`htègltg gn ebsef prfhgabebgltf.

Gxtr`gefs n` r`åz huàrà` ag èmfs târeb-

lfs.

• Blabhèfs n` euntbpnbh`hbøl ag n` sue` pfr nàbdgrglhb` ag èmfs târeblfs.

• Xbepnbdbhèfs n` gxprgsbøl y ¡nbstf!, fmtglgefs

n` gxprgsbøl d`htfrbzà`.

¶Nf lftø6, sgiubefs n` ebse` tâhlbh` qug gn

gcgepnf `ltgrbfr.

:0e1 ‘ =7l=

:0e1 5 2e=

=7l= 5 7l

(2e= + 7l)(2e= ‘ 7l)

5 (2e= + 7l)(2e= ‘ 7l)

y= ‘ 2( x ‘ 0)=

y= 5 y2( x ‘ 0)= 5 8( x ‘ 0)

(

y + 8( x ‘ 0))( y ‘ 8( x ‘ 0))

( y + 8 x ‘ 8)(

y ‘ 8 x + 8)

5 ( y + 8 x ‘ 8)( y ‘ 8 x + 8)




Gcgrhbhbf 1

D`htfrbhg nfs pfnblfebfs sbiubgltgs. Gl `niul`s gxprgsbflgs agmgrà `pnbh`r sus hflfhbebgltfs sfmrg

lóegrfs r`hbfl`ngs. [ugag tr`m`c`r gl jfc`s `p`rtg sb gs lghgs`rbf.

0) x4 ‘ 1 =) m= ‘ 0

8) 04 ‘ e1 1) 1 ‘ x=

7) =7`0; y1 ‘ 12m0= x4 4) 04 x1 y: ‘ 0;;

9) 84 x= ‘ `4m1 :) :0m1 x= ‘ 84`=h=

2) 12 x1 y= ‘ 1`1m= 0;) `=

84 ‘ x4

=7

00) 01

‘ 2`= 0=) 0 ‘`=

=7

=;4BIG] ∝ [fnfhjbh



]gsuegl

0. D`htfrbz`r ul` gxprgsbøl sg rgdbgrg n prfhgsf ag sbepnbdbh`rn j`nn`laf afs f eàs d`htfrgs, huyf

prfauhtf gs biu`n ` n` gxprgsbøl prfpugst`. Iglgr`negltg sg gepng` gstg prfhgabebgltf p`r`

tr`m`c`r hfl pfnblfebfs.

J`y abdgrgltgs eâtfafs f "h`sfs" p`r` d`htfrbz`r, agpglabglaf ag n` gxprgsbøl qug agsggefs

sbepnbdbh`r. Gst` sge`l` gstuabèfs hu`trf3

0.0 D`htfr hfeól eflfebf3 gstg eâtfaf hflsbstg gl gxtr`gr ag ul pfnblfebf, ul eflfebf

hfef d`htfr hfeól ` hà` ulf ag nfs târeblfs agn pfnblfebf gl hugstbøl. Nugif, sg agc`

blabhà` n` euntbpnbh`hbøl gltrg gn d`htfr hfeól gxtr`åaf y nfs târeblfs qug quga`l ag n`

gxprgsbøl frbibl`n.

0.= D`htfr hfeól pfr ìrup`hbøl ag târeblfs3 gepngèfs gstg h`sf sb nfs eflfebfs qug

hfepflgl gn pfnblfebf pugagl rgulbrsg gl irupfs hfl ul d`htfr hfeól abdgrgltg gl hà`

irupf. Nugif, gepngèfs n` ebse` tâhlbh` qug gl gn d`htfr hfeól.

0.8 Qrblfebf huàràf pgrdghtf3 sg bagltbdbh` pfr tglgr trgs târeblfs, ag nfs hu`ngs afs tbglgl

r`åz huàrà` gx`ht`, y gn rgst`ltg gqubv`ng `n afmng agn prfauhtf ag n`s r`åhgs ag nfs ftrfs

eflfebfs.

( x + y)= 5 x = + = xy + y=

[`r` d`htfrbz`r, agmgefs rgfragl`r nfs târeblfs agc`laf ag prbegrf y ag tgrhgrf nfs târeb-

lfs qug tgli`l r`åz huàrà`, nugif gxtr`gefs n` r`åz huàrà` agn prbegr y tgrhgr târeblf

y nfs gshrbmbefs gl ul p`râltgsbs, sgp`ràlafnfs pfr gn sbilf qug `hfep`ð` `n sgiulaf

târeblf? `n hgrr`r gn p`râltgsbs gngvèfs tfaf gn mblfebf `n huàràf.

0.1 Abdgrglhb` ag huàràfs3 mblfebf gl gn qug sus târeblfs tbglgl ul` rgst` ag pfr egabf ysg ngs pugag h`nhun`r r`åz huàrà`.

[`r` n` d`htfrbz`hbøl ag gstg h`sf agmgefs gxtr`gr n` r`åz huàrà` ag nfs târeblfs. Nugif,

agc`r blabhà` n` euntbpnbh`hbøl ag n` sue` pfr n` abdgrglhb` ag gst`s r`åhgs.

`= ‘ m= 5 ( + m)( ‘ m)

[ugag vbsbt`r nfs sbiubgltgs sbtbfs wgm p`r` glhfltr`r eàs gcgepnfs sfmrg nfs h`sfs ag d`htfrbz`hbøl3

• H`sfs ag d`htfrbz`hbøl3 iff.in/QIvAiJ

• D`htfrbz`hbøl3 iff.in/uNBJYR





@htbvbaà 0. Hfeprugmg nf `prglabaf

Hfl sus p`n`mr`s, rgspfla` n`s prgiult`s sbiubgltgs.

0. ¶Puâ gs ul lóegrf prbef6

=. ¶Puâ sbilbdbh` "d`htfrbz`r"6


@. D`htfrbhg nfs pfnblfebfs sbiubgltgs. Gepngg gn prfhgabebgltf ag ul d`htfr hfeól eflfebf.

[ugag utbnbz`r jfc`s `p`rtg sb gs lghgs`rbf.

0) 0=h8a = + 4;h=a 8 =) 87`=m1 z = ‘ 77`=m8 x

8) =1`= xy= ‘ 84 x= y1 1) 07 y8 + =; y= ‘ 7 y

7) 01 x8 y= ‘ =: x= + 74 x1 y8 4) 1=`4 xz 8 ‘ =;`1 z = + 0:`8 x= z ‘ 1`= z 8

M. D`htfrbhg nfs pfnblfebfs sbiubgltgs. Gepngg gn prfhgabebgltf ag ul d`htfr hfeól pfr irup`hbøl

ag târeblfs. [ugag utbnbz`r jfc`s `p`rtg sb gs lghgs`rbf.

0) 8 x8 ‘ 2`x= ‘ x + 8` =) =`= x ‘ 7`= y + 07my ‘ 4mx

8) = x= y + = xz = + y= z = + xy8 1) l= x ‘ 7`= y= ‘ l= y= + 7`= x

7) 0= x= y8 + 1 x8 ‘ 8 x ‘ 2 y8 4) 1`8 x ‘ 1`=m + 8me ‘ 8èx

@utfhfltrfn

=;:BIG] ∝ [fnfhjbh



H. D`htfrbhg nfs pfnblfebfs sbiubgltgs. Gepngg gn prfhgabebgltf ag ul trblfebf huàràf pgrdghtf.

[ugag utbnbz`r jfc`s p`rtg sb gs lghgs`rbf.

0) 84 + 0=e= + e1 =) 0 + 12`= ‘ 01`

8) `0= + 0:`4 + :0 1) 0=0 + 02: x4 + :0 x0=

7) 2`1 z 4 + 0:`= x= y= z 8 + 2 x1 y1 4) 0 + 01 x= y= + 12 x1 y1

9) `1m: + 1`=m1h + 1h= :) 1`4 z 4 ‘ =1`8m= x= z 8 + 84m1 x1

2) 04`4m1 ‘ :`8m=a 8 d 1 + a 4 d : 0;) e: ‘ =;e1 s= x8 + 0;; s1 x4

A. D`htfrbhg nfs pfnblfebfs sbiubgltgs. Gepngg gn prfhgabebgltf ag ul` abdgrglhb` ag huàràfs. [ugag

utbnbz`r jfc`s `p`rtg sb gs lghgs`rbf.

0) 0;;e1l1 ‘ =7r 4 s: =) =74`0= x4 ‘ =:2m1l0;

8) 2`=m1h4a : ‘ 0 1) 2`1m4 ‘ 1v:

7) 04e=l1 ‘ 41a 1v4 4) =7h1e=l4 ‘ 2 x= y1

9) `1 d 4 i 1j0= ‘ c0=o 0;e4r = :) x=

0;; ‘

y= z 1

:0

2) x2

12 ‘ 1`0;

0=0 0;) 0

04 ‘ 1 x=

12




@. ]gsugnv` n`s euntbpnbh`hbflgs ag eflfebfs sbiubgltgs. ]ghugrag n`s rgin`s ag nfs gxpflgltgs. Xb

su rgsuntàf blhnuyg gxpflgltgs lgi`tbvfs, aâcgnfs ag gs` dfre`, lf tr`lsdfreg n` gxprgsbøl `

lóegrf r`hbfl`n.

0) ‘8`(4` ‘=) 5

=) 4`= • 1`8 5

8) :`8(‘8`1) 5

1) ‘1`1(2`1) 5

7) `= • 4`8 5

4) 2` ‘7 • 7`= 5

9) ‘8`1(2`4) 5

:) =` ‘= • 1`= 5

2) 4`8(‘9`) 5

0;) 1` ‘4 • 8`= 5

00) ‘9`7(=`1) 5

0=) (‘1`2)(‘7 ) 5

08) `8(‘1`=) 5

01) 4` ‘ 1 • :`= 5

07) ‘9`8(1`9) 5

04) ‘7`9(4`4) 5

09) =`4 • 2`= 5

0:) :`8(‘9`8) 5

02) 9` ‘8(‘4` ‘8) 5

=;) 4`9 • 4`1 5

=0) 7`8 • 4`8 5

==) 9` ‘4 • 9`1 5

M. ]gsugnv` n`s euntbpnbh`hbflgs ag eflfebfs sbiubgltgs. ]ghugrag n`s rgin`s ag nfs gxpflgltgs.

0) 8 x= y8 • 1`= y8 5

=) ‘7`8r =(7r =t 8) 5

8) :w= x1(‘7 x1 z 1) 5

1) 9 x8 y8 • 4`8 x1 5

7) ‘=`=m1(8m7h=) 5

4) :m

4

a

9

• 9`

=

a

=

5

9) ‘=h1t 8(8h1 z 1) 5

:) 4q=w= • 2w7 z 8 5

2) : ij7(‘1ì =) 5

0;) :u8v= • 2u= x8 5

00) 9el(‘=ew8) 5

0=) 8`=m • 7`h= 5

08) :w1v(‘2wz 1) 5

01) 4 x7 y7 • 1 xz 7 5

07) (‘9`4 z 8)(‘8w8 z =) 5

04) 2`

=

z (‘:`v

8

) 5

09) 8 d =m • 1`=m1 5

0:) ‘:q=r (2r = s) 5

02) 4t 8u • 7ty8 5

=;) (‘1w= z )(‘7 s= z ) 5

=0;BIG] ∝ [fnfhjbh




¡Agspbgrtg sus lgurfl`s! Ng` y rgsugnv` gn sbiubgltg prfmnge` ag nøibh`.

N` hàgl` ag frf

Wl cfvgl gstà gstuab`laf gl ul` hbuaà, ngcàf ag su dèbnb`. Hà` egs, sus pàrgs ng glvå`l ablgrf p`r` qug puga` humrbr sus i`stfs.

Gl hbgrt` fh`sbøl, pfr ul` abdhuntà ghfløebh`, sus pàrgs ng `vbs`l qug gn ablgrf sg v` ` rgtr`s`rul`s sge`l`s. Gn gstuab`ltg lghgsbt` glhfltr`r n` e`lgr` ag pì`r gn `nqubngr ag su j`mbt`hbøl,gltflhgs rghugra` qug tbglg ul` hàgl` ag frf hfl =8 gsn`mflgs. Xg ng fhurrg ul` bag` y j`mn` hfl nàugð` ag n` pglsbøl? èmfs hflhnuygl qug ul gsn`møl humrg gn hfstf ag ul aå` ag `nqubngr. R ag gs`dfre` pugag sfnvglt`r su gstàå` aur`ltg vgbltbtrâs aå`s.

Hfef gn cfvgl s`mg qug gn ablgrf nngi`rà gl niól efegltf gl gsg n`psf ag tbgepf, tbglg n` bltglhbølag rrubl`r su hàgl` nf eglfs pfsbmng j`hbglaf n` eglfr h`ltbaà ag hfrtgs, ag e`lgr` t`n qug hààå` n` sgðfr` tgli` gl su pfagr t`ltfs gsn`mflgs hfef aå`s ân ng àgua`.

[grdghhbfl` ul pfhf n` bag` y sg n` gxpnbh` ` n` augð`, gn `hugraf `n qug nngi` hfl n` augð` gs gnsbiubgltg3 ân pugag a`rng ul gsn`møl pfr aå`, f pugag a`rng ul gsn`møl gn aå` ulf , gn aå` afs pugag

pgabrng gsg gsn`møl y gltrgi`rng ` hèmbf ul` pgqugð` hàgl` hfepugst` pfr afs gsn`mflgs. Gn aå`trgs pugag a`rng ul gsn`møl sfnf (qug cultf hfl nfs afs qug gnn` tbglg ng sgrvbrå`l p`r` pì`r gn tgrhgraå`) f pugag pgabrng qug ng agvugnv` nfs afs qug gnn` y` tbglg y gltrgi`rng ul pgqugðf sgiegltf hfltrgs gsn`mflgs, y `så sbiubglaf, aå` pfr aå`. Nf ólbhf qug agmgrå` bepfrt`rng ` n` augð` gs tglgr gl su

pfagr hà` aå` n` h`ltbaà ag gsn`mflgs gqubv`ngltg ` n` h`ltbaà ag aå`s qug gn gstuab`ltg ng agmg pfr gst`r gl su jftgn.

@a`ptàf ag iff.in/ElTieM

@jfr`, rgspfla` n`s prgiult`s sbiubgltgs3

0) ¶Huàn gs gn eålbef lóegrf ag sgiegltfs gl qug tbglg qug p`rtbr n` hàgl` gn gstuab`ltg p`r`

`rrubl`rn` nf eglfs pfsbmng y huepnbr hfl su `hugraf nfs vgbltbtrâs aå`s6

=) ¶Høef pfarå` sgr gn prfhgabebgltf qug gn cfvgl agmg sgiubr p`r` huepnbr hfl su `hugraf y qug

n` augð` sbgeprg tgli` n` h`ltbaà ag gsn`mflgs gqubv`ngltg ` nfs aå`s qug ân ng àgua`6





E`rqug hfl ul hjgqug n` h`sbnn` qug egcfr blabqug su rglabebgltf.nfiràf

glprfhgsf

lfnfiràf


]ghflfzhf n`s h`r`htgråstbh`s ag ul pfnblfebf d`htfrbz`mng.

[r`htbhf n` d`htfrbz`hbøl ag ul pfnblfebf pfr d`htfr hfeól.

[r`htbhf n` d`htfrbz`hbøl ag ul pfnblfebf pfr irup`hbøl ag târeblfs.

[r`htbhf n` d`htfrbz`hbøl ag ul trblfebf huàràf pgrdghtf.

[r`htbhf n` d`htfrbz`hbøl ag ul` abdgrglhb` ag huàràfs.

[r`htbhf n` euntbpnbh`hbøl ag eflfebfs.

Hu`laf tr`m`cèfs hfl ànigmr`, agmgefs s`mgr høef rgprgsglt`r n`s egaba`s

ag iur`s pn`l`s. Gst` j`mbnbaà rgsunt` dulaèglt`n hu`laf sg lfs prgsglt`l

prfmnge`s qug rgqubgrgl h`nhun`r àrg`s y pgråegtrfs ag `niul` supgrhbg.

Gltflhgs, pr`htbqugefs hfl ul gcgrhbhbf p`r` ags`rrfnn`r gst` agstrgz`3

Agmgefs h`nhun`r y agc`r blabhà` n` døreun` p`r`

glhfltr`r gn àrg` ag nfs rghtàliunfs sbiubgltgs.

Nfs ólbhfs a`tfs hflfhbafs sfmrg n`s egaba`s

ag sus nàfs sg eugstr`l gl gn abìrè`.

[`r` h`nhun`r nf qug lfs pbagl, lfs aèfs

huglt` qug gn àrg` sbl sfemrg`r gs ul hu`-

aràf, pfrqug sus nàfs ebagl nf ebsef ( x),

pfr nf qug su àrg` sgrå` ag x • x 5 x=. [`r` gn

rghtàliunf sfemrgàf sgrå` ag 1 • x 5 1 x.

@jfr` ng tfh` ` ustga, gl su huàgrlf, h`nhung y agcg blabhà` n` døreun` p`r`

glhfltr`r gn àrg` y pgråegtrf ag n`s iur`s sbiubgltg. Nfs a`tfs hflfhbafs sg

eugstr`l gl n` bnustr`hbøl. Hà` pfnåiflf sg rgprgsglt` hfl ul` ngtr` e`yóshu-

n`, p`r` j`hgr eàs dàhbn su bagltbh`hbøl.

x

x

1

x

x

1

1 x x =

@ M

H A

7 y

7

x

=0=BIG] ∝ [fnfhjbh





01 0

= 8 1

8

1

7

=

0

;

x

y

‘ 0 5 b

x = +

y = 5

r =

D`htfrbz`hbøl BB


O`rn Drbgarbhj Iùss, gn prålhbpg ag nfs e`tgeàtbhfs

D`htfrbz`hbøl ag trblfebfs y ag humfs

Gn ngliu`cg `nigmr`bhf

Àrg` ag iur`s igfeâtrbh`s


D`htfrbz`r trblfebfs ag n` dfre` x= + mx + h y `x= + mx + h.

D`htfrbz`r n` sue` y abdgrglhb` ag humfs.

[r`htbh`r gn hànhunf eglt`n hfl n` hfepngt`hbøl ag sbilfs y d`htfrgs.

Hflvgrtbr gxprgsbflgs agn ngliu`cg hfeól `n ngliu`cg `nigmr`bhf.

Agtgrebl`r nfs d`htfrgs qug hfepngt`l gn àrg` ag iur`s igfeâtrbh`s.




¡[`r` hfeglz`r!O`rn Drbgarbhj Iùss, gn prålhbpg ag nfs e`tgeàtbhfs

¡@ tr`m`c`r!Gl n` sge`l` 9 gstuabø n`s prfirgsbflgs `rbteâtbh`s y n` døreun` p`r` n` sue` ag tfafs nfs târeblfs.Wtbnåhgn` p`r` hflfhgr gn rgsuntàf qug fmtuvf Iùss `n rgsfnvgr gn prfmnge` ag nfs lóegrfs

hflsghutbvfs3 s 5 (`0 + `l )l

=

(0999 ‘ 0:77)

O`rn Drbgarbhj Iùss, e`tgeàtbhf, dåsbhf y `strølfef `ngeàl, l`hbø gl gn sglf

ag ul` dèbnb` juebnag. Agsag euy tgepr`l` gaà, abf eugstr`s ag ul` prfab-

ibfs` h`p`hbaà p`r` n`s e`tgeàtbh`s. Huglt` ul` ngygla` qug ` nfs trgs `ðfsbltgrruepbø su pàrg hu`laf gst`m` fhupàf gl n` hflt`mbnbaà ag su lgifhbf

p`r` blabh`rng ul grrfr ag hànhunf.

@ nfs 0; `ðfs, gn e`gstrf ag n` gshugn` prfpusf gn prfmnge` ag sue`r nfs

lóegrfs hflsghutbvfs 0 + = +...+ 0;;. Xg abhg qug gn e`gstrf `pgl`s j`må`

tgreblàf ag glulhb`r gn prfmnge` hu`laf Iùss y` tglå` n` sfnuhbøl. @n h`mf

ag ul` jfr`, sus hfep`ðgrfs tgrebl`rfl gn hànhunf ag ir`lags sue`s, ebgl-

tr`s qug Iùss sfnf j`må` gshrbtf ul lóegrf y gr` n` ólbh` rgspugst` hfrrght`.

Xus hfltrbmuhbflgs ` n` e`tgeàtbh`, dåsbh` e`tgeàtbh`, `strflfeå`, gstàåstbh`

y ftr`s rè`s `pnbhà`s ag n` hbglhb`, ng abgrfl gl vba` gn lfemrg ag prålhbpg ag

nfs e`tgeàtbhfs.

Ng iust`m` pumnbh`r ul tr`m`cf j`st` gst`r sgiurf ag qug gst`m` pgrdghtègltg

gn`mfràf? nngiø ` aghbr3 "hu`laf sg dbl`nbz` ul lfmng gabdbhbf lf agmgl quga`r

vbsbmngs nfs `laèbfs". [grf Iùss lf sfnègltg rgtbrø nfs `laèbfs, sblf qug

agstruyø nfs pn`lfs, pfr nf hu`n jfy aå` lf j`y dfre` ag s`mgr høef fmtglå`

sus rgsuntàfs.Qgxtf a`ptàf ag3 jttp3//iff.in/4OBTgj

=01BIG] ∝ [fnfhjbh



0. Qrblfebf ag n` dfre` x = + mx + h

Gst` sge`l` hfltbluèfs hfl n` d`htfrbz`hbøl, fpgr`hbøl hfltr`rb` ` nfs prf-

auhtfs lft`mngs, y` qug n` d`htfrbz`hbøl pgrebtg gxprgs`r nfs pfnblfebfs aàfsgl târeblfs ag sus d`htfrgs.

[`r` d`htfrbz`r ul trblfebf ag n` dfre` x = + mx + h, rghfragefs gn prfauhtfag afs mblfebfs hfl ul târeblf hfeól. [fr gcgepnf3 ( x + =)( x + 1) 5 x= + 4 x + :.Gstg tbpf ag trblfebfs agmgefs nngv`rnfs ` ul prfauhtf ag afs mblfebfs, hfefgst`m` blbhb`negltg.

[`r` d`htfrbz`r ul trblfebf ag n` dfre` x = + mx + h, agmgefs sgiubr gstfs p`sfs3

`. Agshfepflgr gn trblfebf gl afs d`htfrgs hfl afs târeblfs hàùlf. Gn prbegr târeblf ag hà` d`htfr gs x , qug hfrrgspflag ` n` r`åzhuàrà` ag x=.

m. Hfpb`r gn sbilf agn sgiulaf târeblf agn trblfebf y gshrbmbrnf gl gnprbegr d`htfr. Nugif, euntbpnbh`r gn sbilf agn sgiulaf târeblf pfr gnsbilf agn tgrhgr târeblf. Gn rgsuntàf sg gshrbmg gl gn sgiulaf d`htfr.

h. [`r` j`nn`r nfs sgiulafs târeblfs, sg mush`l afs lóegrfs qug su-eàfs f rgstàfs agl gn hfgdbhbgltg m y euntbpnbhàfs agl h.

Xbièfs nfs p`sfs y d`htfrbhgefs

• Agshfepflgefs gn trblfebf gl afs d`htfrgs(mblfebfs), gxtr`yglaf n` r`åz huàrà` agnprbegr târeblf y gn rgsuntàf nf gshrbmbefs glhà` ulf ag nfs d`htfrgs.

• Hfpbèfs gn sbilf agn sgiulaf târeblf y nfgshrbmbefs gl gn prbegr d`htfr. Nugif, euntb-pnbhèfs nfs sbilfs agn sgiulaf târeblf pfrgn tgrhgrf y nf gshrbmbefs gl gn sgiulaf d`htfr.

• Mushèfs afs lóegrfs qug sueàfs agl ‘9

y qug euntbpnbhàfs, 0=.• Gn trblfebf d`htfrbzàf gs gn prfauhtf ag nfs

mblfebfs.

x= ‘ 9 x + 0=

x=

5 x

( x )( x )

x= ‘ 9 x + 0=

Ù

( x ‘ )( x ‘ )

x= ‘ 9 x + 0= 5

( x ‘ 1)( x ‘ 8)

@tglhbøl3 sb lf j`y ul` p`rgc` ag lóegrfs gltgrfs qug huepn`hfl n`s hflabhbflgs ag n` sue` f rgst` y n` euntbpnbh`hbøl, gn trblfebflf gs d`htfrbz`mng.





D`htfrbhgefs

• Agshfepflgefs gn trblfebf gl afs d`htfrgs (mblf-ebfs), gxtr`yglaf n` r`åz huàrà` agn prbegr târeblf ygn rgsuntàf nf gshrbmbefs gl hà` ulf ag nfs d`htfrgs.

• Hfpbèfs gn sbilf agn sgiulaf târeblf y nf gshrbmbefs

gl gn prbegr d`htfr. Nugif, euntbpnbhèfs nfs sbilfs agnsgiulaf târeblf pfr gn tgrhgrf y nf gshrbmbefs gl gnsgiulaf d`htfr.

• Mushèfs afs lóegrfs qug sueàfs agl : y qugeuntbpnbhàfs, 07.

• Gn trblfebf d`htfrbzàf gs gn prfauhtf ag nfs mblfebfs.

D`htfrbhgefs

• Agshfepflgefs gn trblfebf gl afs d`htfrgs (mblf-

ebfs) gxtr`yglaf n` r`åz huàrà` agn prbegr târeblf ygn rgsuntàf nf gshrbmbefs gl hà` ulf ag nfs d`htfrgs.

• Hfpbèfs gn sbilf agn sgiulaf târeblf y nf gshrbmbefsgl gn prbegr d`htfr. Nugif, euntbpnbhèfs nfs sbilfs agnsgiulaf târeblf pfr gn tgrhgrf y nf gshrbmbefs gl gnsgiulaf d`htfr.

• Mushèfs afs lóegrfs qug rgstàfs agl ‘0 y qugeuntbpnbhàfs agl 4 y=.

• Gn trblfebf d`htfrbzàf gs gn prfauhtf ag nfs mblfebfs.

x= ‘ xy ‘ 4 y=

x= 5 x

( x )( x )

x= ‘ xy ‘ 4 y=

Ù

( x ‘ )( x + )

x= ‘ xy ‘ 4 y= 5

( x ‘ 8 y)( x + = y)

x= + : x + 07

x= 5 x

( x )( x )

x= + : x + 07

Ù

( x + )( x + )

x= + : x + 07 5

( x + 7)( x + 8)

Gcgrhbhbf 0D`htfrbhg abrghtègltg nfs trblfebfs ag n` dfre` x= + mx + h. J`y ul gcgepnf.

;) x= ‘ 7 x + 1 5

8) x= ‘ = x ‘ 8 5

4) x= ‘ : xy + 07 y= 5

0) x= + 8 x + = 5

1) x= + 1 x + 1 5

9) x= + 1 xy ‘ 0= y= 5

=) x= ‘ 00 x + 8; 5

7) x= ‘ =7 x + :1 5

:) x= ‘ 1 xy ‘ 7 y= 5

( x ‘ 1)( x ‘ 0)




=. Qrblfebf ag n` dfre` `x = + mx + h

Gl gstg h`sf, gn hfghbgltg ` agn prbegr târeblf gs abdgrgltg ag 0.

[`r` d`htfrbz`r ul trblfebf ag gstg tbpf, agmgefs hflvgrtbrnf ` n` dfre` x= + mx + h, qug vbefs gl gn `p`rtàf `ltgrbfr. Tgèfs nfs p`sfs p`r` d`htfrbz`r

ul trblfebf ag n` dfre` `x=

+ mx + h.

`. Euntbpnbh`r gn trblfebf pfr gn hfgdbhbgltg (`) agn prbegr târeblf. `( x= + mx + h) 5 `= x= + `mx + `h

m. Gxprgs`r gn prbegr târeblf rgsunt`ltg, hfef ( x)= y gl gn sgiulaftâreblf hfleut`r gn fragl ag `mx pfr m`x. ( x)= + m( x) + `h

h. D`htfrbz`r gn pfnblfebf qug sg fmtuvf gl nfs p`sfs ltgrbfrgs hfefsg jbzf hfl n` dfre` x= + mx + h

a. Abvbabr nfs prfauhtfs rgsunt`ltgs gltrg nfs d`htfrgs ag `, qug gs gn

lóegrf pfr gn hu`n sg euntbpnbhø gn trblfebf.

D`htfrbhgefs

• Euntbpnbhèfs gn trblfebf pfr 8.

• Gxprgsèfs 2 x= hfef ul` sfn` pftglhb` yhfleutèfs 09 y 8 gl gn sgiulaf târeblf.

• D`htfrbzèfs gn trblfebf rgsunt`ltg, hfefnf jbhbefs hfl n` dfre` x= + mx + h.

• Gn rgsuntàf p`rhb`n nf abvbabefs gltrg nfsd`htfrgs ag 8 5 8 Ù 0.

• Gn trblfebf d`htfrbzàf gs gn prfauhtf ag nfsmblfebfs.

D`htfrbhgefs



• D`htfrbzèfs gn trblfebf rgsunt`ltg, hfefnf jbhbefs hfl n` dfre` x= + mx + h.

• Gn rgsuntàf p`rhb`n nf abvbabefs gltrg nfsd`htfrgs ag 4 5 8 Ù =.


8 x= ‘ 09 x + 0; 5

8(8 x= ‘ 09 x + 0;) 5

2 x= ‘ 8(09 x) + 8; 5

(8 x)= ‘ 09(8 x) + 8; 5

(8 x ‘ 07)(8 x ‘ =) 5

(8 x ‘ 07)

8

(8 x ‘ =)

0 5

( x ‘ 7)(8 x ‘ =)

4 x= ‘ 9 x ‘ 8 5

4(4 x= ‘ 9 x ‘ 8) 5

84 x= ‘ 4(9 x) ‘ 0: 5

(4 x)= ‘ 9(4 x) ‘ 0: 5

(4 x ‘ 2)(4 x + =) 5

(4 x ‘ 2)

8 (4 x + =)

= 5

(= x ‘ 8)(8 x + 0)




Gcgrhbhbf =D`htfrbhg abrghtègltg nfs trblfebfs ag n` dfre` `x= + mx + h. J`y ul gcgepnf.

;) = x= + 9 x + 8 5 0) 4 x= + 00 x + 8 5

1 x= + =(9 x) + 4 5

(= x)= + 9(= x) + 4 5

(= x + 4)(= x + 0) 5

(= x + 4)

= (= x + 0)

0 5

( x + 8)(= x + 0)

=) 4 x= + 09 x + 7 5 8) 0= x= ‘ 08 x + 8 5

D`htfrbhgefs



• D`htfrbzèfs gn trblfebf rgsunt`ltg hfef nfjbhbefs hfl n` dfre` x= + mx + h.

• Gn rgsuntàf p`rhb`n nf abvbabefs gltrg nfsd`htfrgs ag 1 5 1 Ù 0.


1 x= ‘ 00 x + 4 5

1(1 x= ‘ 00 x + 4) 5

04 x= ‘ 1(00 x) + =1 5

(1 x)= ‘ 00(1 x) + =1 5

(1 x ‘ :)(1 x ‘ 8) 5

(1 x ‘ :)

1 (1 x ‘ 8)

0 5

( x ‘ =)(1 x ‘ 8)

=0:BIG] ∝ [fnfhjbh



Gcgrhbhbf 8D`htfrbhg n` sue` ag humfs sbiubgltgs. Iuågsg pfr gn gcgepnf.

;) w8 + x8 5

0) y8 + z 8 5

=) x8 + : 5

8) =9 + j8 5

1) : + :w8 5

7) x8 + 41 5

8. Xue` ag humfs `8 + m8 Blbhbèfs gn gstuabf ag n` sue` ag humfs, j`hbglaf rgdgrglhb` ` n` euntbpnbh`-hbøl ag nfs pfnblfebfs. Tgèfs.

`= ‘ `m + m=

Ù ` + m

`=m ‘ `m= + m8

`8 ‘ `=m + `m=

`8 + m8

@så qug ( + m)( = ‘ `m + m=) 5 `8 + m8 y hfef blabhèfs `ltgrbfregltg, d`htfrbz`rgs n` fpgr`hbøl fpugst` nfs prfauhtfs lft`mngs? gltflhgs n` sue` ag humfs gspfsbmng gxprgs`rn` hfl n` døreun`3

`8 + m8 5 ( + m)( = ‘ `m + m=)

R sg d`htfrbz` hfl nfs p`sfs sbiubgltgs3

`. Gl gn prbegr d`htfr sg gshrbmg n` sue` agn rgsuntàf ag n` r`åz hómbhàgn prbegr y sgiulaf târeblf ( + m).

m. Gl gn sgiulaf d`htfr agmg br gn prbegr târeblf `n huàràf eglfs gnprbegrf pfr gn sgiulaf, eàs gn sgiulaf `n huàràf ( = ‘ `m + m=).

Gcgepnf3 x8 + y8 5 ( x + y)( x = ‘ xy + y=)

: z 8 + 41 5 (= z + 1)_(= z )= ‘ (= z • 1) + (1)=U

5 (= z + 1)(1 z = ‘ : z + 04)

:e8 + =9l8 5 (=e + 8l)_(=e)= ‘ (=e • 8l) + (8l)=U

5 (=e + 8l)(1e= ‘ 4el + 2l=)

(w + x)_(w)= ‘ (w • x) + ( x)=U 5 (w + x )(w= ‘ wx + x =)




Gcgrhbhbf 1D`htfrbhg n` abdgrglhb` ag humfs sbiubgltgs. Iuågsg pfr gn gcgepnf.

;) x8 ‘ 0 5

0) w8 ‘ x8 5

=) y8 ‘ : z 8 5

8) : x8 ‘ 41 5

1) =9 ‘ y8 5

7) 0=7 ‘ :w8 5

1. Abdgrglhb` ag humfs `8 ‘ m8 Gn prfhgabebgltf p`r` d`htfrbz`r ul` abdgrglhb` ag humfs gs p`rghbaf `n quggepngèfs gl n` sue` ag humfs, nf qug hèmb`l sfl nfs sbilfs. ]g`nbhgefs nèuntbpnbh`hbøl ag nfs pfnblfebfs.

`= + `m + m=

Ù ` ‘ m

‘ =m ‘ `m= ‘ m8

`8 + `=m + `m=

`8 ‘ m8

@så qug ( ‘ m)( = + `m + m=) 5 `8 ‘ m8 y su døreun` gs3

`8 ‘ m8 5 ( ‘ m)( = + `m + m=)

R sg d`htfrbz` hfl nfs p`sfs sbiubgltgs3

`. Gl gn prbegr d`htfr sg gshrbmg n` rgst` agn rgsuntàf ag n` r`åz hómbhàgn prbegr y sgiulaf târeblf ( ‘ m).

m. Gl gn sgiulaf d`htfr agmg br gn prbegr târeblf `n huàràf eàs gnprbegrf pfr gn sgiulaf, eàs gn sgiulaf `n huàràf ( = + `m + m=).

Gcgepnf3

x8 ‘ y8 5 ( x ‘ y)( x = + xy + y=)

m8 ‘ : 5 (m ‘ =)_(m)= + (m • =) + (=)=U

5 (m ‘ =)(m= + =m + 1)

0=7w8 ‘ z 8 5 (7w ‘ z )_(7w)= + (7w • z ) + ( z )=U

5 (7w ‘ z )(=7w= + 7wz + z =)

( x ‘ 0)_( x)= + ( x • 0) + (0)=U 5 ( x ‘ 0)( x = + x + 0)

==;BIG] ∝ [fnfhjbh



]gsuegl

0. D`htfrbz`r ul trblfebf ag n` dfre` x = + mx + h sbilbdbh` gxprgs`rnf hfef gn prfauhtf ag afsd`htfrgs ag n` dfre` ( x + )( x + m). [`r` d`htfrbz`r trblfebfs ag gst` dfre` agmgefs3

• Agshfepflgr gn trblfebf gl afs d`htfrgs (mblfebfs). [`r` gnnf fmtglgefs n` r`åz huàrà` agnprbegr târeblf.

• Hfpb`r gn sbilf agn sgiulaf târeblf gl gn prbegr d`htfr. Nugif, euntbpnbh`r gn sbilf agn sgiul-af târeblf pfr gn sbilf agn tgrhgr târeblf y gshrbmbrnf gl gn sgiulaf d`htfr.

• Gngibr afs lóegrfs qug sueàfs f rgstàfs agl hfef rgsuntàf gn târeblf m y n euntbpnbh`r-nfs agl hfef rgsuntàf gn târeblf h.

=. Nfs trblfebfs ag n` dfre` `x = + mx + h sg d`htfrbz`l hfl nfs p`sfs sbiubgltgs3

• Euntbpnbh`r gn trblfebf pfr gn hfgdbhbgltg ( ).

• Gxprgs`r gn prbegr târeblf rgsunt`ltg, hfef ( x)=

y gl gn sgiulaf târeblf hfleut`r gn fragl`mx pfr m`x.

• D`htfrbz`r gn pfnblfebf qug sg fmtuvf gl nfs p`sfs `ltgrbfrgs, hfef sg jbzf hfl gn trblfebfag n` dfre` x= + mx + h.

• Abvbabr nfs prfauhtfs gltrg nfs d`htfrgs ag `, qug gs gn lóegrf pfr gn hu`n sg euntbpnbhø gntrblfebf.

8. N` sue` ag humfs `8 + m8 sg d`htfrbz` ag `hugraf hfl n` døreun` sbiubgltg3

`8 + m8 5 ( + m)( = ‘ `m + m=)

1. N` abdgrglhb` ag humfs `8 ‘ m8 sg d`htfrbz` ag `hugraf hfl n` døreun` sbiubgltg3

`8 ‘ m8 5 ( ‘ m)( = + `m + m=)

[`r` rgdfrz`r sus hflfhbebgltfs sfmrg n` d`htfrbz`hbøl, vbsbtg3 jttp3//iff.in/LyY9mW

==0E`tgeàtbh` ∝ Xge`l` 01





E`rqug ul` "Z" gl gn huàrf ag n` fphbøl qug rgspflag hfrrghtègltg hà` prgiult`.

0) ¶Puâ gxprgsbøl rgprgsglt` n` d`htfrbz`hbøl hfrrght` ag x= + 7 x + 46

=) ¶Puâ sbilfs quga`l gl nfs d`htfrgs `n gxprgs` x= + = x ‘ :, hfeful` euntbpnbh`hbøl ag afs mblfebfs6

8) ¶Puâ gxprgsbøl rgprgsglt` n` d`htfrbz`hbøl hfrrght` ag x=

‘ x ‘ 0=6

1) ¶Huàn gs n` rgspugst` hfrrght` `n d`htfrbz`r 7 x= ‘ 4 x ‘ :6

7) @n d`htfrbz`r (: + `

8

), ¶huàn gs gn sgiulaf târeblf agn prbegr d`htfr6

4) ¶Puâ gxprgsbøl rgprgsglt` ul` abdgrglhb` ag humfs6

9) ¶Huàn gs gn óntbef târeblf qug rgsunt` `n d`htfrbz`r ( x8 ‘ 41)6

:) ¶Huàn gs gn fragl ag nfs sbilfs `n d`htfrbz`r ( x8 + y8)6

@utfhfltrfn

( x ‘ 4)( x ‘ 0) ( x ‘ 8)( x ‘ =)

( x + 4)( x + 0)

( x + 8)( x + =)

( ‘ )( ‘ )

( + )( ‘ )

( ‘ )( + )

( + )( + )

( x ‘ 4)( x ‘ =) ( x ‘ 8)( x ‘ 1)

( x ‘ 1)( x + 8)

( x ‘ 0=)( x ‘ 0)

( x ‘ =)(7 x ‘ 1)

( x + =)(7 x ‘ 1)

( x ‘ =)(7 x + 1)

( x + =)(7 x + 1)

`8 ‘ m8

`8 + m8

( ‘ m)8

( + m)8

( + )(+, +, +)

( + )(‘ , +, ‘)

( ‘ )(+, ‘ , +)

( + )(+, ‘ , +)

` `=

=

=`

1

04

x=

1 x

===BIG] ∝ [fnfhjbh




Fmsgrvg nfs h`sfs ag d`htfrbz`hbøl qug sg prgsglt`l gl hà` `p`rtàf. ]g`nbhg nfs gcgrhbhbfs ag dfreàbrght` y gshrbm` gn rgsuntàf sfmrg n` nålg`. Hu`laf sg` lghgs`rbf, rg`nåhgnfs gl su huàgrlf.

@. Qrblfebf ag n` dfre` x = + mx + h

;) x= + 9 x + 0= 5

0) x= ‘ x ‘ =; 5

=) x= ‘ 4 x ‘ 9 5

8) x= + 4 x + : 5

1) x= + 7 x + 1 5

7) x= ‘ 7 x ‘ 01 5

4) x= ‘ 8 x + = 5

9) x= ‘ 9 x ‘ : 5

:) x= ‘ 1 x ‘ 0= 5

2) x= + 1 x ‘ 17 5

0;) x= ‘ 0= x + =9 5

00) x= + 09 x ‘ 8: 5

M. Qrblfebf ag n` dfre` `x = + mx + h

;) 4 x= + 00 x + 8 5

0) = x= + x ‘ 8 5

=) 4 x= + x ‘ 0= 5

8) 7 x= + 8 x ‘ = 5

1) 4 x= + 00 x + 1 5

7) 8 x= + 04 x ‘ 0= 5

4) 8 x= ‘ = x ‘ 0 5

9) 0= x= ‘ x ‘ 4 5

:) = x= ‘ 7 x + 8 5

2) 8 x= ‘ 00 x + 4 5

0;) 4 x= + 00 x + 8 5

00) 8 x= ‘ 09 x + 0; 5

H. Xue` ag humfs x 8 + m

;) x8 + =9 5

0) x8 + : 5

=) x8 + 0 5

8) x8 + 41 5

1) : x8 + =9 y8 5

7) x8 + y8 5

4) 0=7 + x8 5

9) =9 y8 + : 5

:) =9 x8 + 41 5

2) 0=7 x8 + =9 y8 5

A. Abdgrglhb` ag humfs `8 ‘ m8

;) x8 ‘ =9 5

0) 41 ‘ x8 5

=) 0 ‘ =9 y8 5

8) x8 y8 ‘ 0 5

1) =9 ‘ x8 5

7) x8 ‘ y8 5

4) x8 ‘ 0 5

9) 41 y8 ‘ =9 x8 5

:) =9 x8 ‘ 0=7 5

2) 41 x8 ‘ 0=7 y8 5

( x + 1)( x + 8)

(= x + 8)(8 x + 0)

( x + 8)( x= ‘ 8 x + 2)

( x ‘ 8)( x= + 8 x + 2)




Gshrbm` sfmrg n` nålg` nfs târeblfs qug sg ng pbagl gl hà` `p`rtàf p`r` qug n` d`htfrbz`hbøl sg`hfrrght`. Dåcgsg gl nfs gcgepnfs.

@. Hfepngtg nfs sbilfs agn prbegr y sgiulaf târeblf.;) x= + 7 x + 4 5 ( x + 8)( x + =)

0) x= + 7 x ‘ =1 5 ( x :)( x 8)

=) x= ‘ = x ‘ =1 5 ( x 4)( x 1)

8) x= + 9 x + 0; 5 ( x 7)( x =)

1) x= ‘ 08 x + 84 5 ( x 2)( x 1)

7) x= + 0; x + =0 5 ( x 9)( x 8)

4) x= + 8 x ‘ 1 5 ( x 1)( x 0)

9) x= ‘ 4 x ‘ =9 5 ( x 2)( x 8)

:) x= ‘ : x ‘ =; 5 ( x 0;)( x =)

2) x= + 8 x ‘ 71 5 ( x 2)( x 4)

0;) x= ‘ : x + 0= 5 ( x 4)( x =)

00) x= ‘ 0= x ‘ 17 5 ( x 07)( x 8)

0=) x= ‘ 04 x + 41 5 ( x :)( x :)

08) x= + 08 x + 1= 5 ( x 9)( x 4)

01) x= + 00 x + =1 5 ( x :)( x 8)

07) x= ‘ 04 x + 1: 5 ( x 0=)( x 1)

M. Hfepngtg nfs târeblfs qug d`nt`l gl hà` mblfebf.

;) x= ‘ 1 x ‘ 8= 5 ( x ‘ :)( x + 1)

0) x= ‘ 9 x + 4 5 ( )( x ‘ 0)

=) x= + x ‘ 9= 5 ( )( x ‘ :)

8) x= ‘ 8 x ‘ =: 5 ( )( x + 1)

1) x= + 00 x + 8; 5 ( )( x + 7)

7) x= + 08 x + 1; 5 ( )( x + 7)

4) x= ‘ 08 x + 84 5 ( )( x ‘ 1)

9) x= ‘ 4 x ‘ 1; 5 ( )( x + 1)

:) x= + 8 x ‘ =: 5 ( x + 9)( )

2) x= + : x + 07 5 ( x + 7)( )

0;) x= ‘ 0: x + :0 5 ( x ‘ 2)( )

00) x= ‘ 0= x ‘ =: 5 ( x ‘ 01)( )

0=) x= ‘ 08 x ‘ 1: 5 ( x ‘ 04)( )

08) x= + 0; x + =7 5 ( x + 7)( )

01) x= ‘ 07 x + 11 5 ( x ‘ 00)( )

07) x= ‘ 07 x ‘ 0;; 5 ( x ‘ =;)( )

H. Gshrbm` gn mblfebf f trblfebf qug hfepngt` hà` d`htfrbz`hbøl.

;) `8 + : 5 ( + =)( = ‘ =` + 1)

0) =9 + m8 5 ( )(2 ‘ 8m + m=)

=) x8 + 41 5 ( )( x= ‘ 1 x + 04)

8) y8 + =9 5 ( )( y= ‘ 8 y + 2)

1) 41 + : x8 5 ( )(04 ‘ : x + 1 x=)

7) h8 ‘ =9 5 (h ‘ 8)( )

4) : x8 ‘ 0 5 (= x ‘ 0)( )

9) 0=7 ‘ : y8 5 (7 ‘ = y)( )

:) 41w8 ‘ z 8 5 (1w ‘ z )( )

2) x8 y8 ‘ : 5 ( xy ‘ =)( )

==1BIG] ∝ [fnfhjbh




Gn ngliu`cg `nigmr`bhf sg utbnbz` p`r` rgprgsglt`r v`nfrgs agshflfhbafs y pn`ltg`r ghu`hbflgs quggxprgsgl n` bldfre`hbøl prfpfrhbflà`. Gn rgtf qug ng prfpflgefs gs qug hflvbgrt` gxprgsbflgs agnngliu`cg hfeól, qug gstàl gl n` t`mn` ag `m`cf, gl ngliu`cg `nigmr`bhf.

Ngliu`cg hfeól Ngliu`cg nigmr`bhf

N` sue` ag afs lóegrfs abdgrgltgs.

Gn humf ag ul lóegrf.

N` sue` ag nfs huàràfs ag afs lóegrfs.

N` sue` ag afs lóegrfs abvbaba` gltrg su abdgrglhb`.

N`s afs qublt`s p`rtgs ag ul lóegrf ùegltàf gl ul tgrhbf.Wl lóegrf abseblubaf gl 4.

Wl lóegrf p`r hu`nqubgr`.

Wl lóegrf hu`nqubgr` ùegltàf gl abgz.

N` abdgrglhb` ag afs lóegrfs hu`ngsqubgr`.

N` sgebsue` ag afs lóegrfs.

Qrgs lóegrfs l`tur`ngs hflsghutbvfs.

Gn afmng ag ul lóegrf gxhgabaf gl fhjf.

Gn huàràf ag ul lóegrf eàs n` ebtà ag gsg lóegrf.

Gn humf ag ul lóegrf eàs gn trbpng agn huàràf ag abhjf lóegrf.

Gn aglfeblàfr ag ul` dr`hhbøl gs trgs ulbaàgs e`yfr qug suluegràfr.

Wl tgrrglf ag dfre` rght`liun`r, su `lhjf ebag n` ebtà ag sun`rif.

Gn `lhjf ag ul rghtàliunf gs biu`n ` n`s trgs hu`rt`s p`rtgs ag sunflibtua.

Gn luegràfr ag ul` dr`hhbøl gxhgag `n aglfeblàfr gl trgsulbaàgs.

Lóegrf qug `n ìrgi`rnf `n 2 sue` =1.

N` abdgrglhb` gltrg ul lóegrf y su `ltgrbfr.






prfhgsflf

nfiràf


D`htfrbzf trblfebfs ag n` dfre` x= + mx + h y `x= + mx + h.

D`htfrbzf n` sue` y abdgrglhb` ag humfs.

[r`htbhf gn hànhunf eglt`n hfl n` hfepngt`hbøl ag sbilfs y d`htfrgs.

Hflvbgrtf gxprgsbflgs agn ngliu`cg hfeól `n ngliu`cg `nigmr`bhf.

Agtgreblf nfs d`htfrgs qug hfepngt`l gn àrg` ag iur`s igfeâtrbh`s.

[`r` d`htfrbz`r glhugltrg gn nàf qug d`nt` gl hà` iur` y ageugstrg qug àrg`sg fmtbglg `n euntbpnbh`r nfs mblfebfs. [rgstg `tglhbøl `n gcgrhbhbf ;.

;)

x ‘ =

6

@ 5 x= + 0; x ‘ =1

Àrg` 5 x= + 0; x ‘ =1 5 ( x + 0=)( x ‘ =)

]/ Gn ftrf nàf agn rghtàliunf ebag x + 0=.

0)

x ‘ 1

6@ 5 x= ‘ 04 x + 1:

=)

: x

6

@ 5 1: x=

@ 5 =1 x=

8)

@ 5 7 x= + 08 xy + 4 y= x + = y

6

==4BIG] ∝ [fnfhjbh





070

= 8 1

8

1

7

=

0

;

x

y

‘ 0 5 b

x = +

y = 5

r =

Dr`hhbflgs `nigmr`bh`s B


]ghfragefs n`s fpgr`hbflgs hfl dr`hhbflgs

Dr`hhbflgs `nigmr`bh`s

T`nu`r gxprgsbflgs `nigmr`bh`s

E`tgeàtbh`s `pnbhà`s ` n` blvgstbi`hbøl ag h`sfs


Agshrbmbr g bagltbh`r quâ gs ul` dr`hhbøl `nigmr`bh`.

]gsfnvgr sbepnbh`hbflgs, euntbpnbh`hbflgs y abvbsbflgs hfl dr`hhbflgs `nigmr`bh`s.

H`nhun`r hfl ìbnbaà gn v nfr ag ul` gxprgsbøl `nigmr`bh` aàf gn v nfr ag su blhøilbt`.




¡[`r` hfeglz`r!]ghfragefs n`s fpgr`hbflgs hfl dr`hhbflgs

@ltgs ag hfeglz`r, agmgefs rghfra`r y tglgr hn`rf gn prfhgabebgltf qug sgiub-

efs p`r` rg`nbz`r niul`s fpgr`hbflgs hfl dr`hhbflgs. Gstf gs lghgs`rbf pfrqug

n`s dr`hhbflgs `nigmr`bh`s sbiugl ul hfepfrtèbgltf euy sbebn`r.

Xbepnbdbh`hbøl ag dr`hhbflgs3 sbepnbdbh`r ul` dr`hhbøl gs tr`lsdfre`rn` gl ul`

dr`hhbøl gqubv`ngltg eàs rgauhba`.

[`r` sbepnbdbh`r gepgzèfs abvbabglaf gltrg nfs prbegrfs lóegrfs prbefs3 =,

8, 7, 9... Gs aghbr, prfmèfs ` abvbabr luegràfr y aglfeblàfr gltrg = ebgl-

tr`s sg puga`, agspuâs p`sèfs `n 8 y `så suhgsbvègltg. Wl gcgepnf3

84

4; 5

0:

8; 5

2

07 5

8

7

Euntbpnbh`hbøl ag dr`hhbflgs3 p`r` glhfltr`r gn prfauhtf ag afs dr`hhbflgs,euntbpnbhèfs "gl nålg`", gs aghbr, luegràfr pfr luegràfr y aglfeblàfr pfr

aglfeblàfr. Nugif, sbepnbdbhèfs.

07

=7 •

4

2 5

07 • 4

=7 • 2 5

2;

==7

2;

==7 5

=

7

Abvbsbøl ag dr`hhbflgs3 p`r` abvbabr afs dr`hhbflgs, j`hgefs ul` euntbpnbh`hbøl

hruzà`, gn luegràfr ag n` prbegr` dr`hhbøl pfr gn aglfeblàfr ag n` sgiula`,

agspuâs, gn aglfeblàfr ag n` prbegr` dr`hhbøl pfr gn luegràfr ag n` sgiula`.

=

1 ù

7

0; 5

= • 0;

1 • 7 5

=;

=;=;

=; 5 0

¡@ tr`m`c`r!

]gsugnv` n`s sbiubgltgs euntbpnbh`hbflgs y abvbsbflgs ag dr`hhbflgs lueârbh`s.

0)1

4 •

7

: 5 =)

:

2 ù

1

7 5

8):

0; ù

8

4 5 1)

4

0= •

7

2 5

==:BIG] ∝ [fnfhjbh



0. Dr`hhbflgs `nigmr`bh`sWl` dr`hhbøl nigmr`bh` gs ul` gxprgsbøl dr`hhbfl`rb` gl n` qug gn luegràfr y

gn aglfeblàfr sfl gxprgsbflgs `nigmr`bh`s, iglgr`negltg pfnblfebfs. x= + x

1 x= ‘ 8?

1 y8 ‘ : y= + y ‘ 8

7 y= + y ‘ 1

Dr`hhbflgs nigmr`bh`s gqubv`ngltgs

Aghbefs qug afs dr`hhbflgs `nigmr`bh`s sfl gqubv`ngltgs sb gn prfauhtf agn lu-egràfr ag n` prbegr` pfr gn aglfeblàfr ag n` sgiula` gs biu`n `n prfauhtfagn aglfeblàfr ag n` prbegr` pfr gn luegràfr ag n` sgiula`.

[( x)P( x)

y]( x)X( x)

? sfl gqubv`ngltgs sb [( x) • X( x) 5 P( x) • ]( x)

Tgèfs ul gcgepnf3

• N`s dr`hhbflgs ltgrbfrgs sfl gqubv`ngltgs pfrquggn prfauhtf hruzàf gltrg gnn`s gs biu`n.

@jfr` qug y` s`mgefs quâ sfl n`s dr`hhbflgs `nigmr`bh`s pfagefs gstuab`r n`sfpgr`hbflgs qug sg pugagl rg`nbz`r hfl gnn`s.

0.0 Xbepnbdbh`hbøl ag dr`hhbflgs `nigmr`bh`s

N`s dr`hhbflgs `nigmr`bh`s sg sbepnbh`l ag dfre` euy p`rghba` ` n`s dr`hhbflgs`rbteâtbh`s.

[`r` sbepnbh`r ul` gxprgsbøl r`hbfl`n, d`htfrbzèfs luegràfr y aglfebl`-afr y nugif h`lhgnèfs d`htfrgs hfeulgs. [`r` gstf `pnbh`rgefs nfs h`sfs agd`htfrbz`hbøl qug jgefs `prglabaf gl n`s sge`l`s `ltgrbfrgs. [fli` `tglhbøl`n gcgepnf.

Xbepnbqugefs n` dr`hhbøl3

• Lftèfs qug gl gn luegràfr j`y ul trblfebf huàr`-

af pgrdghtf y gl gn aglfeblàfr j`y ul` abdgrglhb` aghuàràfs. @pnbh`laf nfs prfhgabebgltfs qug `prglab-efs gl n` sge`l` 08, d`htfrbzèfs nfs afs pfnblfebfs.

• @jfr`, h`lhgnèfs nfs d`htfrgs hfeulgs, gs aghbr, nfsqug sfl "biu`ngs". Gl gstg h`sf, sgrå` ( x ‘ =) qug sg rgpb-tg gl luegràfr y aglfeblàfr.

• Nngièfs ` n` gxprgsbøl sbepnbdbhà`.

x + =

x= ‘ 1 y 0

x ‘ =

( x + =)( x ‘ =) 5 ( x = ‘ 1)(0)

x = ‘ 1 5 x = ‘ 1

x= ‘ 1 x + 1

x= ‘ 1 5

( x ‘ =)=

( x ‘ =)( x + =) 5

( x ‘ =)=

( x ‘ =)( x + =) 5

( x ‘ =)

( x + =)







0.= Euntbpnbh`hbøl ag gxprgsbflgs `nigmr`bh`s

Gn prfauhtf ag afs dr`hhbflgs `nigmr`bh`s gs ftr` dr`hhbøl `nigmr`bh` gl n`qug gn luegràfr gs gn prfauhtf ag nfs luegràfrgs y gn aglfeblàfr gs gnprfauhtf ag nfs aglfeblàfrgs. Gs aghbr, agmgefs euntbpnbh`r "gl nålg`", `nbiu`n qug nf j`hgefs hfl n`s dr`hhbflgs lueârbh`s.

[( x)P( x)

• ]( x)X( x)

5 [( x) • ]( x)P( x) • X( x)

Gl aflag [( x), P( x), ]( x) y X( x) rgprgsglt`l `niul` gxprgsbøl pfnblfeb`n.

[`r` euntbpnbh`r afs f eàs gxprgsbflgs `nigmr`bh`s r`hbfl`ngs3

`. D`htfrbzèfs nfs luegràfrgs y nfs aglfeblàfrgs j`st` aflag sg`pfsbmng.

m. Blabhèfs n` euntbpnbh`hbøl agn luegràfr pfr luegràfr y aglfebl`-afr pfr aglfeblàfr.

h. Xbepnbdbhèfs gn rgsuntàf j`st` rgauhbrnf ` su eålbe` gxprgsbøl.

Gn prfhgabebgltf gs euy sglhbnnf, pflièfs `tglhbøl ` ul gcgepnf.

Euntbpnbqugefs n`s dr`hhbflgs sbiubgltgs3

• D`htfrbzèfs nfs luegràfrgs y nfs aglfebl`-afrgs ag èm`s dr`hhbflgs. [`r` gstf `pnbhèfsnfs prfhgabebgltfs qug jgefs `prglabaf gl n`ssge`l`s `ltgrbfrgs.

• Blabhèfs n` euntbpnbh`hbøl agn luegràfr pfrluegràfr y aglfeblàfr pfr aglfeblàfr.

• Xbepnbdbhèfs gn rgsuntàf h`lhgn`laf nfs d`h-tfrgs hfeulgs qug j`y gl gn luegràfr y gnaglfeblàfr.

Gl gstg h`sf, gstfs d`htfrgs sfl ( x ‘ 0) y ( x + =).

• Qgreblèfs ag rgsfnvgr sb quga`r` niul` fpgr`-hbøl pglabgltg.

• Fmtglgefs gn rgsuntàf dbl`n.

Hfef pfagefs lft`r, gs gsglhb`n gn afeblbf ag nfs abdgrgltgs h`sfs ag d`h-tfrbz`hbøl qug jgefs gstuabàf ` nf n`rif ag n`s sge`l`s `ltgrbfrgs. ]gsunt`bepfrt`ltg tèmbâl, pfagr bagltbh`r n`s h`r`htgråstbh`s ag hà` gxprgsbøl`nigmr`bh` p`r` `pnbh`r gn prfhgabebgltf hfrrgspflabgltg.

8 x ‘ 8

= x + 1 •

x= + 1 x + 1

x= ‘ x 5

8( x ‘ 0)

=( x + =) •

( x + =)=

x( x ‘ 0) 5

8( x ‘ 0)( x + =)=

= x( x + =)( x ‘ 0) 5

8( x ‘ 0)( x + =)=

= x( x + =)( x ‘ 0) 5

8( x + =)

= x 5

8 x + 4

= x




Gcgrhbhbf =]gsugnv` n`s euntbpnbh`hbflgs sbiubgltgs. [ugag utbnbz`r jfc`s `p`rtg f su huàgrlf sb gs lghgs`rbf.

0) x + 8

x ‘ = •

x= + = x ‘ :

x= ‘ 2 =)

x= ‘ = x + 0

x= ‘ 2 •

x= + x ‘ 4

x= ‘ 0

8)e + l

el ‘ l= •

l=

e= ‘ l= 1)

x= ‘ = x

x= ‘ 7 x + 4 •

x= + 1 x + 1

x= ‘ 1

7) xy ‘ = y=

x= + xy •

x= + = xy + y=

x= ‘ = xy 4)

2 ‘ 4 x + x=

x= ‘ 2 •

x= ‘ 7 x + 4

8 x= ‘ 2 x

9) x= ‘ 1 xy + 1 y=

x= ‘ = xy • x=

x= ‘ 1 y= :) =` ‘ =

=`= ‘ 7; • `= ‘ 1` ‘ 7

8` + 8

2) x= ‘ 2

x= ‘ 1 •

x= ‘ 9 x + 0;

x= ‘ : x + 07 0;)

= x= ‘ 8 x ‘ =

4 x + 8•

8 x + 4

x= ‘ 1

Ftrf gcgepnf, euntbpnbqugefs3

• D`htfrbzèfs nfs luegràfrgsy nfs aglfeblàfrgs ag n`strgs dr`hhbflgs.

• Blabhèfs n` euntbpnbh`hbøl ag

nfs d`htfrgs ag n`s dr`hhbflgs.

• Xbepnbdbhèfs gn rgsuntàfh`lhgn`laf nfs d`htfrgs hfeu-lgs. Gl gstg h`sf, nfs d`htfrgssfl ( + 0), ( ‘ 0), ( ‘ 8), ( + =) y (8` + 1).

• R lf j`y lbliul` fpgr`hbølsbl rgsfnvgr, pfr nf qug nngi`-efs `n rgsuntàf dbl`n.

`= ‘ 0

`= + =` •

`= ‘ ` ‘ 4

8`= + 9 + 1 •

8` + 1

`= ‘ 1` + 8 5

( + 0)( ‘ 0)

`( + =) •

( ‘ 8)( + =)

( + 0)(8` + 1) •

8` + 1

( ‘ 0)( ‘ 8) 5

( + 0)( ‘ 0)(` ‘ 8)( + =)(8` + 1)`( + =)(` + 0)(8` + 1)(` ‘ 0)(` ‘ 8)

5

( + 0)( ‘ 0)( ‘ 8)( + =)(8` + 1)

`( + =)( + 0)(8` + 1)(` ‘ 0)( ‘ 8) 5

0

`




0.8 Abvbsbøl ag dr`hhbflgs `nigmr`bh`s

[`r` abvbabr afs dr`hhbflgs `nigmr`bh`s, sgiubefs ul prfhgsf m`st`ltg sbebn`r `nqug j`hgefs hfl n`s dr`hhbflgs lueârbh`s.

Agmgefs euntbpnbh`r n` prbegr` dr`hhbøl pfr gn blvgrsf ag n` sgiula` dr`hhbøl.Aghbefs gltflhgs qug n` abvbsbøl gs ul` euntbpnbh`hbøl hruzà` y pfr t`ltf,utbnbzèfs n` ebse` tâhlbh` qug `prglabefs gl gn `p`rtàf `ltgrbfr.

[( x)P( x)

ù]( x)X( x)

5[( x) • X( x)P( x) • ]( x)

Gl aflag [( x), P( x), ]( x) y X( x) rgprgsglt`l `niul` gxprgsbøl pfnblfeb`n.

[fr gcgepnf.

]gsfnvèfs n` abvbsbøl3

• Blvgrtbefs n` sgiula` dr`hhbøl y hèmbèfs

gn sbilf ag abvbsbøl pfr gn ag euntbpnbh`hbøl.

• D`htfrbzèfs nfs luegràfrgs y nfs aglfeb-làfrgs.

• Blabhèfs n` euntbpnbh`hbøl ag nfs d`htfrgs.

• Xbepnbdbhèfs gn rgsuntàf h`lhgn`laf nfsd`htfrgs hfeulgs. Gl gstg h`sf, gstfs d`htfrgssfl ( x + =) y ( x ‘ =).

• Nngièfs `n rgsuntàf dbl`n. ¶Tg nf sglhbnnfqug gs6

Ftrf gcgepnf, abvbaèfs n`s gxprgsbflgs sbiubgltgs3

• Blvgrtbefs n` sgiula` dr`hhbøl y hèmbèfsgn sbilf ag abvbsbøl pfr gn ag euntbpnbh`hbøl.

• D`htfrbzèfs nfs luegràfrgs y nfs aglfeb-làfrgs. Gl gstg h`sf 8`= y 7`8 lf sg pugaglrgauhbr eàs, pfr nf qug nfs agcèfs `så.

• Blabhèfs n` euntbpnbh`hbøl ag nfs d`htfrgs.


• ]gsfnvgefs n`s fpgr`hbflgs qug quga`l ynngièfs `n rgsuntàf dbl`n.

x= + = x

x= ‘ 7 x + 4 ù

x= + 1 x + 1

x= ‘ 1 5

x= + = x

x= ‘ 7 x + 4 •

x= ‘ 1

x= + 1 x + 1 5

x( x + =)

( x ‘ =)( x ‘ 8)•

( x + =)( x ‘ =)

( x + =)= 5

x( x + =)( x + =)( x ‘ =)

( x ‘ =)( x ‘ 8)( x + =)= 5

x( x + =)( x + =)( x ‘ =)

( x ‘ =)( x ‘ 8)( x + =)= 5

x x ‘ 8

8`=

`= + 4`m + 2m= ù

7`8

`=m + 8`m= 5

8`=

`= + 4`m + 2m= •

`=m + 8`m=

7`8 5

8`=

( + 8m)= •`m( + 8m)

7`8 5

8`8m( + 8m)

7`8( + 8m)= 5

8`8m( + 8m)

7`8( + 8m)= 5

8m

7( + 8m) 5

8m

7` + 07m

]ghugrag3 gn blvgrsf

ag ul` dr`hhbøl gssbepngegltg n` dr`hhbøl"vfntgà`".

`m

m`




Gcgrhbhbf 8]gsugnv` n`s abvbsbflgs sbiubgltgs. [ugag utbnbz`r jfc`s `p`rtg f su huàgrlf sb gs lghgs`rbf.

0) x8 ‘ x

= x= + 4 x ù

7 x= ‘ 7 x

= x + 4 =)

: x= + =4 x + 07

04 x= ‘ 2 ù

4 x= + 08 x ‘ 7

2 x= ‘ 0

8)=; x= ‘ 8; x

07 x8 + 07 x= ù

1 x ‘ 4

x + 0 1)

`x= + 7

1`= ‘ 0 ù

`8 x= + 7`=

=` ‘ 0

7)`= ‘ 4` + 7

`= ‘ 07` + 74 ù

`= + =` ‘ 87

`= ‘ 7` ‘ =1 4)

`= ‘ 4`

`8 + 8`= ù

`= + 8` ‘ 71

`= + 2`




]gsuegl

0. Wl` dr`hhbøl nigmr`bh` gs ul` gxprgsbøl dr`hhbfl`rb` gl n` qug gn luegràfr y gn aglfeblàfrsfl gxprgsbflgs `nigmr`bh`s, iglgr`negltg pfnblfebfs.

[fr gcgepnf3 x= + 0; x + =7

x= ‘ y=

Afs dr`hhbflgs sfl gqubv`ngltgs sb gn prfauhtf hruzàf gltrg èm`s gs biu`n.

0.0 Xbepnbdbh`hbøl ag gxprgsbflgs nigmr`bh`s3 p`r` sbepnbdbh`r ul` gxprgsbøl r`hbfl`n, d`htfrb-zèfs gn luegràfr y gn aglfeblàfr, y nugif h`lhgnèfs d`htfrgs hfeulgs.

0.= Euntbpnbh`hbøl ag dr`hhbflgs nigmr`bh`s3 gn prfauhtf ag afs dr`hhbflgs nigmr`bh`s gs ftr`dr`hhbøl `nigmr`bh` gl n` qug gn luegràfr gs gn prfauhtf ag nfs luegràfrgs y gn aglfeb-làfr gs gn prfauhtf ag nfs aglfeblàfrgs. Gs aghbr, agmgefs euntbpnbh`r "gl nålg`".

Gn prfhgabebgltf gs gn sbiubgltg3`. D`htfrbzèfs nfs luegràfrgs y nfs aglfeblàfrgs j`st` aflag sg` pfsbmng.

m. Blabhèfs n` euntbpnbh`hbøl agn luegràfr pfr luegràfr y aglfeblàfr pfr aglfeb-làfr.

h. Xbepnbdbhèfs gn rgsuntàf j`st` rgauhbrnf ` su eålbe` gxprgsbøl.

0.8 Abvbsbøl ag dr`hhbflgs `nigmr`bh`s3 sg euntbpnbh` n` prbegr` dr`hhbøl pfr gn blvgrsf ag n`sgiula` dr`hhbøl. Aghbefs gltflhgs qug n` abvbsbøl gs ul` euntbpnbh`hbøl hruzà` y pfrt`ltf, utbnbzèfs n` ebse` tâhlbh` qug `prglabefs gl n` euntbpnbh`hbøl.

Ng gxjfrtèfs ` qug vg` y pfli` `tglhbøl ` nfs sbiubgltgs vbagfs. Gl gnnfs glhfltr`rà gcgepnfs ygxpnbh`hbflgs sfmrg høef tr`m`c`r hfl dr`hhbflgs `nigmr`bh`s.

• Xbepnbdbh`hbøl ag dr`hhbflgs `nigmr`bh`s3 iff.in/;YHtjh

• Euntbpnbh`hbøl y abvbsbøl ag dr`hhbflgs `nigmr`bh`s3 iff.in/0]2l@7

• Abvbsbøl ag dr`hhbflgs `nigmr`bh`s3 iff.in/Y=ZERA






Hfl sus p`n`mr`s rgspfla` n`s prgiult`s sbiubgltgs.

0) ¶Puâ sfl n`s dr`hhbflgs `nigmr`bh`s6

=) ¶Høef s`mgefs sb afs dr`hhbflgs `nigmr`bh`s sfl gqubv ngltgs6

@htbvbaà =. [r`htbqug nf `prglabaf@. Xbepnbdbqug n`s dr`hhbflgs `nigmr`bh`s sbiubgltgs. Jài`nf agsp`hbf y hfl h`ne`, sbgeprg pugag `pf-

y`rsg gl nfs prfhgabebgltfs ag d`htfrbz`hbøl qug jgefs vbstf gl n`s sge`l`s `ltgrbfrgs.

0)0= x= + 2 x

04 x= ‘ 2 =)

x= ‘ 12

x= ‘ 7 x ‘ 01

8) ex ‘ 7e + 1 x ‘ =;e= + :e + 04 1) 4 x

=

+ 7 x ‘ 407 x= ‘ 9 x ‘ =

7)=0 + 8m ‘ 9 ‘ `m

2 ‘ `= 4)

= x= + 9 x + 8

= x= ‘ 9 x ‘ 1

9)2 y= ‘ =7

8 y= + : y + 7 :)

8 z 8 + 2 z =

z = + 4 z + 2

@utfhfltrfn




M. ]gsugnv` n`s sbiubgltgs euntbpnbh`hbflgs hfl dr`hhbflgs `nigmr`bh`s. [ugag tr`m`c`r gl jfc`s `p`rtgsb gs lghgs`rbf.

0)81`x= y + 70`= y ‘ 4:`y=

4=7 x= ‘ 12 •

4=7 x= ‘ 87; x + 12

= x= ‘ 1 y + 8` =)

1 x= ‘ 2

= x= + 9 x + 4 •

1 x1 + 4 x8 + 2 x=

: x9 ‘ =9 x1

8)2 x= ‘ 1

8 x= + 7 x + = •

2 x1 ‘ 4 x8 + 1 x=

=9 x1 + : x 1)

1è8 ‘ 0=èl ‘ e= + 8l

e= + 0=e + 84 •

e= ‘ e ‘ 1=

e1 ‘ 2l=

7)w= ‘ 9w ‘ 0:

‘9 =w ‘ 01`= + 8w= + 4w •

8w ‘ 9 =

7w= ‘ 1=w ‘ =9 4)= x= + 00 x + 07

x= + 4 x + 2 • x= ‘ 2

9 x= + = xy= ‘ =0 x ‘ 4 y=

H. ]gsugnv` n`s sbiubgltgs abvbsbflgs hfl dr hhbflgs `nigmr`bh`s. [ugag tr`m`c`r gl jfc`s `p`rtg sb gslghgs`rbf.

0) y= ‘ 4 y + 7

y= ‘ 07 y + 74 ù

y= + = y ‘ 87

y= ‘ 7 y ‘ =1 =)

:0`= ‘ 1

74`8 + 87` ù

:0`= ‘ 84` + 1

48` ‘ 01

8)=1e= + 04e

1e ù

8e= + =;e + 0=

=e + = 1)

=7w= ‘ x=

0;w= x ‘ = x= ù

0=7w8 + x8

=7w1 ‘ 0;w= x + x=

7)e=l= ‘ =elxy + x= y=

el ‘ xy ù

el + xy

e=l= ‘ x= y= 4)

8`mx= ‘ = y= ‘ = x= + 8`my=

`= ù

07`8m ‘ 0;`=

x1 ‘ y1




@. Xustbtuy` gn v`nfr ag n`s blhøilbt`s gl n`s gxprgsbflgs sbiubgltgs y fpgrg p`r` glhfltr`r gn rgsun-tàf dbl`n. Gshrbm` su rgspugst` gl n` nålg` hfrrgspflabgltg.

0) ‘: x + 7, aflag x 5 7

=) 1 x ‘ 2, aflag x 5 ‘9

8) 4 x + 7, aflag x 5 8

1) : x ‘ 9, aflag x 5 9

7) ‘7 x ‘ 1, aflag x 5 8

4) 2 x + 8, aflag x 5 ‘0=

9) = x + ;, aflag x 5 8

:) 2 x + 4, aflag x 5 ‘7

2) 7 x ‘ :, aflag x 5 4

0;) 9 x + 0, aflag x 5 2

00) = x ‘ 8, aflag x 5 ‘9

0=) 7 x + 9, aflag x 5 ;

08) ‘2 x ‘ =, aflag x 5 1

01) 8 x + 1, aflag x 5 8

07) ‘7 x ‘ :, aflag x 5 ‘8

04) 2 x + 9, aflag x 5 ;

09) ‘8 x ‘ 1, aflag x 5 0

0:) 8 x + =, aflag x 5 1

02) ‘1 x + 8, aflag x 5 ‘9

=;) 1 x + 4, aflag x 5 0;

=0) 9 x ‘ 4, aflag x 5 00

==) ‘: x + 9, aflag x 5 ;

=8) ‘9 x + 4, aflag x 5 ‘0

=1) 4 x ‘ 8, aflag x 5 :

M. Xustbtuy` gn v`nfr ag n`s blhøilbt`s gl n`s gxprgsbflgs sbiubgltgs y fpgrg p`r` glhfltr`r gn rgsun-tàf dbl`n. Gshrbm` su rgspugst` gl n` nålg` hfrrgspflabgltg.

0) ‘1 z = ‘ 8, aflag z 5 7

=) = z = + 8, aflag z 5 ‘8

8) 9 z = + 7, aflag z 5 4

1) : z = ‘ 1, aflag z 5 ‘8

7) z = + 9, aflag z 5 1

4) = z = ‘ 4, aflag z 5 8

9) ‘1z= + 9, aflag z 5 1

:) 4 z = ‘ 0, aflag z 5 ‘0

2) ‘= z = + :, aflag z 5 8

0;) 9 z = + 2, aflag z 5 ;

00) z = ‘ 1, aflag z 5 ‘1

0=) 7 z = ‘ 9, aflag z 5 9

08) : z = + 8, aflag z 5 9

01) ‘ z = ‘ :, aflag z 5 ‘4

07) 8 z = + 2, aflag z 5 ‘:

04) 2 z = + :, aflag z 5 0

09) 8 z = + 1, aflag z 5 =

0:) ‘ z = ‘ 8, aflag z 5 ;

02) z = + 7, aflag z 5 7

=;) 4 z = + 1, aflag z 5 4

=0) ‘= z = ‘ 2, aflag z 5 ‘9

==) ‘8 z = + =, aflag z 5 :

=8) 8 z = + 8, aflag z 5 0

=1) ‘= z = ‘ =, aflag z 5 =





¡[fli` ` tr`m`c`r sus lgurfl`s! [bglsg y sfnuhbflg nfs prfmnge`s ag nøibh` sbiubgltgs.

N` hfsghj` agn ir`lcgrf

Wl ir`lcgrf tglå` ul tgrrglf gl gn qug hfsghj`m` tfe`tg. Gn tgrrglf gr` hustfabàf pfr hu`trf pgrrfsqug j`må`l sbaf hfnfhàfs gstr`tâibhègltg. Hà` pgrrf vbvå` gl ul` h`sbt` qug gn ir`lcgrf j`må`hflstrubaf, sg abstrbmuå`l ag n` dfre` gl qug eugstr` n` beìgl.

Gn ir`lcgrf tuvf mugl`s vglt`s y aghbabø ùeglt`r n`s abeglsbflgs ag su tgrrglf. [grf sg ng prgsgl-tø ul prfmnge`3 qugrå` qug nfs hu`trf pgrrfs sbiubgr`l vbibn`laf n` hfsghj` agsag sus h`sbt`s, sblhèmb`r su umbh`hbøl. ¶Høef nfir`r gstf6 Abmucg gl n` beìgl n` dfre` gl qug sg pugag gxp`labr gntgrrglf huepnbglaf hfl n` hflabhbøl eglhbflà`.

Blvgrsbøl hfeulbt`rb`

Nfs vghblfs @, M, H y A aghbabgrfl j`hgr ul `hugraf hfeulbt`rbf y sgemr`rfl e`åz gl ul` p`rhgn`

qug sg glhugltr` `n egabf ag sus h`s`s.[fhf agspuâs, hà` vghblf hflstruyø ul i`nnblgrf (G, D, I y J), tfafs gnnfs eàs hgrh` ag n` pn`lt`hbøl.Xbl gem`rif, n`s i`nnbl`s gepgz`rfl ` hfegrsg y ` `rrubl`r n`s e`zfrh`s.

@ltg gst` sbtu`hbøl, nfs vghblfs aghbabgrfl ngv`lt`r ul` v`nn` qug ngs pgrebt` `hhgagr `n e`bz`n, pgrfqug bepba` qug n`s i`nnbl`s puga`l gltr`r.

Fmsgrvg n` bnustr`hbøl y abmucg pfr aølag agmg p`s`r n` v`nn` p`r` huepnbr hfl gst` hflabhbøl.

M

A

H@ G I

J

D









¡[`r` hfeglz`r!

]ghfragefs n` sue` y rgst` ag dr`hhbflgs

Gst` sge`l` hfltbluèfs hfl n`s dr`hhbflgs `nigmr`bh`s, `jfr` hfl n` sue` y

rgst` ag dr`hhbflgs `nigmr`bh`s. @ltgs ag gepgz`r hfl gn tge`, rgdrgsh`rgefs

lugstrfs hflfhbebgltfs sfmrg n` sue` y rgst` ag dr`hhbflgs `rbteâtbh`s.

]ghfragefs qug sg pugagl prgsglt`r afs h`sfs3 hfl biu`n y hfl abdgrgltg

aglfeblàfr. [r`htbqugefs ul gcgepnf hfl hà` ulf.

• Dr`hhbflgs hfl biu`n aglfeblàfr

Hu`laf sueèfs f rgstèfs dr`hhbflgs hfl biu`n aglfeblàfr, fpgrèfs

nfs luegràfrgs y ólbhègltg hfpbèfs gn aglfeblàfr.

Gcgepnf37

8 +

0

8 5

4

8 5 =

4:

‘ 8:

5 8:

• Dr`hhbflgs hfl abdgrgltg aglfeblàfr

Hu`laf sueèfs f rgstèfs dr`hhbflgs hfl abdgrgltg aglfeblàfr,

rg`nbzèfs nfs p`sfs sbiubgltgs3

`. Agshfepflgefs nfs aglfeblàfrgs gl sus d`htfrgs prbefs p`r` j`nn`r

gn eålbef hfeól eóntbpnf (e.h.e.) qug sgrà gn aglfeblàfr hfeól.

m. Abvbabefs gn e.h.e. gltrg gn aglfeblàfr ag hà` dr`hhbøl y gn rgsuntàf

nf euntbpnbhèfs pfr su rgspghtbvf luegràfr.

[r`htbqugefs gstfs p`sfs hfl n` sue` sbiubgltg.

¡@ tr`m`c`r!

]g`nbhg n`s sue`s y n`s rgst`s dr`hhbfl`rb`s. Nugif, gshrbm` gn rgsuntàf sfmrg n` nålg`.

0)0

= +

7

= 5

=)2

9 ‘

8

9 5

8)4

2 ‘

00

2 +

8

2 5

1) 0

= +

0

8 5

7) 7

8 ‘

0

= +

9

4 5

4) =

8 ‘

8

: ‘

7

0= 5

7

8 +

=

2 5

8(7) + 0(=)

2 5

07 + =

2 5

09

2

e.h.e. 5 8 Å 8 5 8= 5 2

2 8 8

8 0 8

0 0




0. Xue` y rgst` ag dr`hhbflgs`nigmr`bh`s hfl biu`n aglfeblàfr

[`r` sue`r f rgst`r dr`hhbflgs `nigmr`bh`s hfl biu`n aglfeblàfr, sg `pnbh`n` ebse` rgin` ag n` sue` f rgst` ag dr`hhbflgs `rbteâtbh`s hfl aglfeblàfr

hfeól3 sueèfs f rgstèfs nfs luegràfrgs y hfpbèfs gn aglfeblàfr.

Tgèfs n` rgprgsglt`hbøl sbemønbh` gl gn rghuàrf ag m`cf y niulfs gcgepnfs.

`

h +

m

h 5

` + m

h

Xuegefs

• Xueèfs nfs luegràfrgs y hfpbèfs

gn aglfeblàfr.


]gstgefs

• ]gstèfs nfs luegràfrgs y hfpbèfs gn

aglfeblàfr.

• ]gauhbefs târeblfs sgegc`ltgs y gshrb-

mbefs gn rgsuntàf.

Xuegefs

• Xueèfs nfs luegràfrgs y hfpbèfs

gn aglfeblàfr.


• D`htfrbzèfs gn luegràfr y rgauhbefs

târeblfs.


7`=

1` +

9`=

1` 5

7`= + 9`=

1`5 0=`=

1` 5

8`

: x x + 7

‘ x + 1 x + 7

5

: x ‘ ( x + 1) x + 7

5: x ‘ x ‘ 1

x + 7 5

9 x ‘ 1 x + 7

x= + : x + 04( x + 8)( x ‘ 7)

+8 x + :

( x + 8)( x ‘ 7) 5

x= + : x + 8 x + 04 + :( x + 8)( x ‘ 7) 5

x= + 00 x + =1( x + 8)( x ‘ 7) 5

( x + :)( x + 8)( x + 8)( x ‘ 7)

5

x + : x ‘ 7





Gcgrhbhbf 0

]g`nbhg n`s fpgr`hbflgs `nigmr`bh`s sbiubgltgs. Qgli` gl huglt` qug gn rgsuntàf sg rgauhg ` su eålbe`

gxprgsbøl, gs aghbr, sg sbepnbdbh`.

0) x

x + = +

4 x

x + = 5 =)

7 x= ‘ 1

: x8 +

x= + 1

: x8 5

8)4 x=

4 x ‘ 9 ‘

9 x

4 x ‘ 9 5 1)

x= + 8 x

x= ‘ x ‘ 4 ‘

x= ‘ 4

x= ‘ x ‘ 4 5

]gstgefs

• ]gstèfs nfs luegràfrgs y hfpbèfs

gn aglfeblàfr.


• D`htfrbzèfs gn luegràfr y rgauhbefs

târeblfs.


7 x= + 1 x( x + 8)( x ‘ =)

‘8 x= ‘ = x

( x + 8)( x ‘ =) 5

7 x= + 1 x ‘ (8 x= ‘ = x)

( x + 8)( x ‘ =) 5

7 x= + 1 x ‘ 8 x= + = x

( x + 8)( x ‘ =)

5

= x= + 4 x

( x + 8)( x ‘ =) 5

= x ( x + 8)

( x + 8)( x ‘ =) 5

= x

x ‘ =




=. Xue` y rgst` ag dr`hhbflgs`nigmr`bh`s hfl abdgrgltg aglfeblàfr

[`r` sue`r y rgst`r dr`hhbflgs `nigmr`bh`s hfl abdgrgltg aglfeblàfr, agmgefs

sgiubr gstfs p`sfs3

`. D`htfrbz`r nfs aglfeblàfrgs p`r` j`nn`r gn eålbef hfeól eóntbpnf

(e.h.e.) qug sgrà gn aglfeblàfr hfeól.

m. Abvbabr gn e.h.e. gltrg gn aglfeblàfr ag hà` târeblf. Nugif, euntbpnbh`r

gn rgsuntàf pfr hà` luegràfr.

h. Xbepnbdbh`r gn rgsuntàf.

[r`htbh`rgefs gn e.h.e. ag pfnblfebfs, hfl nfs gcgepnfs sbiubgltgs.

0. Gl nfs pfnblfebfs3 y( y + =), ( y + 8)( y + =) y ( y + =)=

N` e`yfr pftglhb` ag y gs 0, ag ( y + =) gs = y n` ag ( y + 8) gs 0.

Gltflhgs gn e.h.e. gs3 y( y + =)=( y + 8)

=. Gl nfs pfnblfebfs3 x= + 1 x + 1 y = x= + 8 x ‘ =

D`htfrbzèfs x= + 1 x + 1 5 ( x + =)( x + =) 5 ( x + =)=

= x= + 8 x ‘ = 5 (= x ‘ 0)( x + =)

Gltflhgs gn e.h.e. gs3 ( x + =)= (= x ‘ 0)

@jfr` qug y` s`mgefs høef fmtglgr gn e.h.e. ag pfnblfebfs, pfagefs sue`r

y rgst`r dr`hhbflgs `nigmr`bh`s.

Xuegefs

• Abvbabefs gn e.h.e. ( x= y) gltrg hà` aglfebl`-

afr. Nugif, euntbpnbhèfs gn rgsuntàf pfr hà`

luegràfr.

• D`htfrbzèfs gn luegràfr y rgauhbefs târeblfs.


Xuegefs

• Abvbabefs gn e.h.e. (4 x=) gltrg hà` aglfeblàfr.

Nugif, euntbpnbhèfs gn rgsuntàf pfr hà`

luegràfr.


• D`htfrbzèfs gn luegràfr y sbepnbdbhèfs gn

rgsuntàf.

= x x=

+ 8 xy

5

= x( y) + 8( x)

x= y 5

= xy + 8 x

x= y

x (= y + 8)

x= y5

= y + 8 xy

8

= x +

x ‘ =

4 x=

5

8(8 x) + ( x ‘ =)(0)

4 x= 5

2 x + x ‘ =

4 x= 50; x ‘ =

4 x= 5

=(7 x ‘ 0)

4 x= 57 x ‘ 0

8 x =






8. Dr`hhbflgs hfepngc`s

Wl` dr`hhbøl hfepugst` f hfepngc gs hu`nqubgr dfre` dr`hhbfl`rb` qug tgli`

dr`hhbflgs gl gn luegràfr y gl gn aglfeblàfr, f gl èmfs. Xbemønbhègltg

sg rgprgsglt` `så3

` + m`

` ‘ m`

Nålg` prblhbp`n agn` dr`hhbøl

Luegràfr ag n`dr`hhbøl hfepngc`

Aglfeblàfr ag n`dr`hhbøl hfepngc`

Nf qug sg mush` gl tfa` dr`hhbøl hfepngc` gs sbepnbdbh`rn`, gs aghbr, tr`lsdfr-

e`rn` ` ul` dr`hhbøl sbepng, rgauhba` ` sus târeblfs eàs sglhbnnfs y qug sg`

gqubv`ngltg ` n` gxprgsbøl frbibl`n.

[`r` sbepnbdbh`r dr`hhbflgs hfepngc`s, agmgefs sgiubr gstfs p`sfs3

`. Glhfltr`r gn eålbef hfeól eóntbpnf e.h.e. ag tfafs nfs

aglfeblàfrgs.

m. Euntbpnbh`r gn luegràfr y gn aglfeblàfr ag n` dr`hhbøl

hfepngc` pfr gn e.h.e.

h. Xbepnbdbh`r gn rgsuntàf hu`laf sg` pfsbmng.

Xbepnbdbqugefs

• Gxprgsèfs tfafs nfs târeblfs hfef dr`hhbøl.

• Abvbabefs gn e.h.e. (0)( ) 5 ` gltrg hà` aglf-

eblàfr. Nugif, euntbpnbhèfs gn rgsuntàf pfr

hà` luegràfr.

• ]g`nbzèfs n` abvbsbøl ag dr`hhbflgs y sbepnbdb-

hèfs gn rgsuntàf.

• D`htfrbzèfs gn rgsuntàf p`rhb`n y lugvègltg

sbepnbdbhèfs.

0

`

+ 0

5` ‘ 0

`

0`

+00

5`0

‘ 0`

0(0) + 0( )

`

`( ) ‘ 0(0)

`

5

0 + `

`

`= ‘ 0

`

5

`(0 + )`( = ‘ 0)

5(0 + )( = ‘ 0)

5

(0 + )

( + 0)( ‘ 0)5

0

` ‘ 0

]ghugrag3

`m

ùha

5`m

•a h








]gsuegl

Gl n` sue` y rgst` ag dr`hhbflgs `nigmr`bh`s `n biu`n qug gl n`s dr`hhbflgs `rbteâtbh`s pfagefs

fpgr`r dr`hhbflgs hfl biu`n y abdgrgltg aglfeblàfr.

0. Gl n` sue` f rgst` ag dr`hhbflgs `nigmr`bh`s hfl biu`n aglfeblàfr, sg sue`l f rgst`l nfs

luegràfrgs y sg hfpb` gn aglfeblàfr hfeól.

=. [`r` sue`r f rgst`r dr`hhbflgs `nigmr`bh`s hfl abdgrgltg aglfeblàfr, prbegrf sg agmg

j`nn`r gn eålbef hfeól eóntbpnf (e.h.e.) ag nfs aglfeblàfrgs, d`htfrbzèfs hà` pfnblfebf y

tfeèfs tfafs nfs d`htfrgs abstbltfs gngvàfs ` n` eàxbe` pftglhb`. Nugif, abvbabefs gn e.h.e.

gltrg hà` aglfeblàfr y gn rgsuntàf sg euntbpnbh` pfr hà` luegràfr. Agspuâs sg sbepnbdbh`

n` dr`hhbøl.

8. N`s dr`hhbflgs hfepugst`s f hfepngc`s sfl n`s qug tbglgl dr`hhbflgs gl gn luegràfr f gn

aglfeblàfr, f gl èm`s p`rtgs ag n` dr`hhbøl. [`r` sbepnbdbh`rn` f rgauhbrn` ` sus târeblfs eàs

sglhbnnfs y qug sg` gqubv`ngltg ` n` gxprgsbøl frbibl`n, agmgefs sgiubr gstfs p`sfs3

`) Glhfltr`r gn eålbef hfeól eóntbpnf ag tfafs nfs aglfeblàfrgs.

m) Euntbpnbh`r gn luegr afr y gn aglfeblàfr ag n` dr`hhbøl hfepngc` pfr gn e.h.e.

h) Xbepnbdbh`r nfs rgsuntàfs hu`laf sg` pfsbmng.

[`r` `db`lz`r sus hflfhbebgltfs sfmrg dr`hhbflgs hfepngc`s, vbsbtg n` abrghhbøl sbiubgltg3

jttp3//iff.in/[uPHRA





@utfhfltrfn

@htbvbaà 0. Ageugstrg nf `prglabaf

E`rqug ul` "Z" gl gn huàrf ag n` fphbøl qug hfepngt` hà` glulhbàf.

0) N`s dr`hhbflgs dfreà`s pfr ftr` dr`hhbøl gl gn luegràfr f gl gnaglfeblàfr rghbmgl gn lfemrg ag…

=) Gn aglfeblàfr hfeól ag n`s dr`hhbflgs7 x

, :

x=,

4 x8

gs…

8) Gn rgsuntàf ag n` fpgr`hbøl :8

+=8

‘78

gs...

1) N` d`htfrbz`hbøl hfepngt` ag x8 ‘ x gs…

7) @n euntbpnbh`r gn mblfebf (0; ‘ y) pfr ‘0 sg fmtbglg…

4) N` d`htfrbz`hbøl hfrrght` agn trblfebf x= ‘ : x + 04 gs…

9) [fr sbepng blspghhbøl, gn târeblf qug sg gnbebl` gl n` abvbsbøl

` + m

`

` ‘ m

`

5 gs…

:

x

x=

x8

78

74

72

70=

( x= ‘ x)=

x( x= ‘ 0)

x( x=

+ x) x( x ‘ 0)( x + 0)

y ‘ 0;

y + 0;

‘ y + 0;

‘0; ‘ y

( x ‘ 1)=

( x + 1)=

( x ‘ 1)( x + 1)

( x ‘ =)( x + :)

`

`=

` + m

` ‘ m

prfpb`

sbepng

beprfpb`

hfepngc`





@. Xueg n`s dr`hhbflgs `nigmr`bh`s hfl biu`n aglfeblàfr y rgaózh`n`s ` su eålbe` gxprgsbøl.

0) x

= x ‘ 0 +

=

= x ‘ 0 5

8)8 x + =

= x + 8 +

x ‘ =

= x + 8 5

=)= x

8 x ‘ 7 +

7

8 x ‘ 7 5

1)= x ‘ 0

x= ‘ 8 x ‘ 1 +

8

x= ‘ 8 x ‘ 1 5

M. ]gstg n`s dr`hhbflgs nigmr`bh`s hfl biu`n aglfeblàfr y rgaózh`n`s su eålbe` gxprgsbøl.

0) x

= x + 9 ‘

=

= x + 9 5

8)8 x= ‘ 0

8 x8 ‘

4 x= ‘ 0

8 x8 5

=) x

x ‘ 7 ‘

7

x ‘ 7 5

1)9 x=

2 x= ‘ 1 ‘

4 x ‘ = x=

2 x= ‘ 1 5




H. @ltgs ag rgsfnvgr n`s fpgr`hbflgs, prgstg `tglhbøl ` nfs aglfeblàfrgs. Nugif, gl jfc`s `p`rtg

sueg n`s dr`hhbflgs `nigmr`bh`s, rgaózh`n`s ` su eålbe` gxprgsbøl y gshrbm` gn rgsuntàf sfmrg n`

nålg`. Qbglg ul gcgepnf.

;)= x + 7

4 x +

x + 4

1 x 5

0) x ‘ = x + 1

+ 4 x + 1

5

=) x ‘ 8

= x + 0 +

= x

= x + 0 5

8) x= + 1 x

x= ‘ x ‘ 4 +

1 x ‘ 1

x= ‘ x ‘ 4 5

1) x ‘ =

1 x +

x + 7

0; x 5

7) 8 x + =4 x

+ 1 x ‘ 0: x

5

4)8

x + 8 +

7

x ‘ 1 5

9)8 x + 4

x= + x ‘ =; +

x

x= ‘4 x + : 5

9x + =:

0=x

A. ]gstg n`s dr`hhbflgs `nigmr`bh`s hfl abdgrgltg aglfeblàfr y rgaózh`n`s ` su eålbe` gxprgsbøl

Qr`m`cg gl jfc`s `p`rtg. Nugif, gshrbm` n` rgspugst` gl n` nålg` hfrrgspflabgltg.

0) x + 0

x ‘ = ‘

x

x ‘ = 5

=) x

= x + 9 ‘

=

= x + 9 5

8)2 x= + 9

4 x8 ‘

9 ‘ 8 x=

4 x8 5

1) x8 ‘ 8

= x1 ‘

9 x8 ‘ 8

= x1 5

7) 8 x + 0

1 x ‘ = ‘

x + 0

1 x ‘ = 5

4) x + 8

0= x ‘

x + 1

04 x 5

9) x + 0

x= + x ‘ 0= ‘

=

x= +7 x ‘ =1 5

:)7 x ‘ 0

x= += x ‘ 8 ‘

= x + =

x= +7 x + 4 5

G. Xbepnbdbqug n`s dr`hhbflgs hfepngc`s gl jfc`s `p`rtg y gshrbm` gn rgsuntàf sfmrg n` nålg`.

0)

2`

+8`=

8 +0 x

5

=)8 y

+ 0= y

y + y

=

5

8) x +

0 x

0 x=

5

1)

01 x=

‘0=7

‘

0= x

‘07

5

7)4

x= ‘ 9

x ‘ =1

‘ 8

x= ‘

7 x

‘ :

5

4)

x + 8 +4

x ‘ 1

x + 2 +1=

x ‘ 1

5




Fmsgrvg hà` ghu`hbøl y rgsuânv`n` eglt`negltg. Nugif, gshrbm` su rgsuntàf. J`y ul gcgepnf gl

hà` `p`rtàf.

@. ;) : + y 5 07 y 5

0) 7 + y 5 08 y 5

=) 2 + y 5 04 y 5

8) 8 + y 5 0: y 5

1) 2 + y 5 =1 y 5

7) 0 + y 5 12 y 5

4) 0= + y 5 =7 y 5

9) =1 + y 5 8; y 5

:) 82 + y 5 71 y 5

2) 0; + y 5 1; y 5

0;) x ‘ 2 5 4 x 5

00) x ‘ 8 5 0= x 5

0=) x ‘ 4 5 07 x 5

08) x ‘ 1 5 =0 x 5

01) x ‘ : 5 0; x 5

07) x ‘ ; 5 8= x 5

04) x ‘ 01 5 4 x 5

09) x ‘ 0; 5 2; x 5

0:) x ‘ =7 5 7; x 5

02) x ‘ 4; 5 =; x 5

9

M.

;) 1 x 5 84 x 5

0) 7 x 5 0; x 5

=) 2 x 5 :0 x 5

8) 4 x 5 0: x 5

1) 9 x 5 ; x 5

7) 2 x 5 71 x 5

4) 8 x 5 8; x 5

9) 1 x 5 =: x 5

:) 0; x 5 0; x 5

2) 07 x 5 17 x 5

0;)

w

1 5 7

w 5

00) w

4 5 0: w 5

0=) w

07 5 = w 5

08) w

1 5 0; w 5

01) w

7 5 0= w 5

07) w

0 5 2; w 5

04) w

7 5 7 w 5

09) 1:w

5 =1 w 5

0:) 4;w

5 0= w 5

02) :;

w 5 =; w 5

2





@. [r`htbqug n` rgsfnuhbøl ag ghu`hbflgs ag prbegr iràf hfl nfs gcgrhbhbfs sbiubgltgs.

0) = x + 8 5 7

=) 1 x + 8 5 07

8) = x ‘ 2 5 ‘00

1) 9 x + 00 5 1

7) 7 x ‘ = x ‘ x 5 =;

4) = x + 7 x ‘ 8 x 5 8=

9) 9 x 5 4 + 1 x

:) 8 x 5 : ‘ 8 x

2) = x + 0= 5 9 x + =

0;) 00 x ‘ 4 x ‘ 4 5 =; ‘ : x

00)7 x

1 5 4;

0=) 4 x9

5 ‘1=

08) 8 x1

5 07=:

01) ‘ = x2

5 ‘ 90=

07) 7 x ‘: 5 ‘ 04=7

04) 8 x:

+ x1

5 0;

09) x8

‘ x:

5 7

0:) 8 x1

‘ : x2

5 70:

02) =( x + 1) 5 04

=;) 1(8 x + 9) + 7 5 88

=0) : + =(8 x ‘ 0) 5 =0

==) 8 ‘ =(8 x ‘ 1) 5 01

=8) = ‘ 7(= ‘ x) 5 ‘7

=1) 7(= x ‘ 0) 5 =7 + 8( x ‘ 8)

=7) 4(= x ‘ 8) 5 = ‘ 9(8 ‘ x)

=4) 8(7 ‘ = x) 5 : + 9(0 ‘ = x)

M. [n`ltgg ul` ghu`hbøl qug agshrbm` nf glulhbàf gl hà` prfmnge`. Nugif, rgsuânv`n` y hfeprugmg

su rgsuntàf.

0) @n trbpng ag ul lóegrf ng rgstèfs 04 y sg fmtbglg =;. ¶Huàn gs gsg lóegrf6

=) @ ul lóegrf ng sueèfs 0: y sg fmtbglg 20. ¶Huàn gs gsg gn lóegrf6

8) @n trbpng ag ul lóegrf ng rgstèfs =1 y sg fmtbglg =2. ¶Huàn gs gsg lóegrf6

1) N` sue` ag afs lóegrfs hflsghutbvfs gs 20. ¶Huàngs sfl gsfs lóegrfs6

7) N` sue` ag n`s gaàgs ag Gstgn` y E`ri`rbt` gs 11 `ðfs. Xb Gstgn` gs = `ðfs e`yfr qug

E`ri`rbt`, ¶quâ gaà tbglg hà` ul`6

4) J`nn`r n` egaba` ag nfs nàfs ag ul rghtàliunf qug tbglg =9 he ag pgråegtrf, y n` m`sg gs=9 ag n` `ntur`.

9) Xgm`stbàl i`st` P0=;.;; gl ul p`lt`nøl y ul` hèbs`. Lf s`mg gn prghbf ag hà` prgla`,

pgrf n` hèbs` v`ng afs hu`rt`s p`rtgs qug gn p`lt`nøl. ¶Huàltf v`ng hà` prgla`6

:) Gl gn s`nøl ag n` hfeulbaà j`y 7;; pgrsfl`s. Xb s`mgefs qug j`y 4: eucgrgs eàs qug

jfemrgs, ¶huàltfs jfemrgs y huàlt`s eucgrgs j`y6

2) @e`nb` i`stø gl n` tbgla` ul tgrhbf agn ablgrf qug nngv`m`. Xb sg qugaø hfl P=1.;;, ¶hfl

huàltf ablgrf nngiø ` n` tbgla`6





E`rqug hfl ul hjgqug n` h`sbnn` qug egcfr blabqug su rglabebgltf.nfiràf

glprfhgsf

lfnfiràf


Xuef y rgstf dr`hhbflgs `nigmr`bh`s hfl prghbsbøl.

Xbepnbhf dr`hhbflgs hfepngc`s.

]gsugnvf eglt`negltg ghu`hbflgs ag prbegr iràf.

]gsugnvf prfmnge`s hftbab`lfs hfl ghu`hbflgs ag prbegr iràf.

Qfe` ag aghbsbflgs

[r`htbqug n` ibnbaà ag hànhunf y n` tfe` ag aghbsbflgs hfl n` htbvbaà sbiubgltg.

Wstga j` sbaf agsbilàf prgsbagltg agn hfebtâ ag su hfeulbaà y ng prgfhup`

n` s`nua ag nfs lbðfs, pfr nf qug aghbag rg`nbz`r ul` glhugst` p`r` agtgrebl`r

n`s gldgregaàgs eàs drghugltgs. [`r` rg`nbz`r n` glhugst`, sfnbhbt` n` `yua` ag

84 pgrsfl`s ` n`s hu`ngs, àgeàs ag ftrfs ótbngs, ngs prfpfrhbfl`rà n`pbhgrfs.

Nfs prghbfs ag nfs n`pbhgrfs sg eugstr`l gl n` t`mn` ag `m`cf. Jì` nfs hànhunfs

sfnbhbtàfs p`r` agtgrebl`r n` h`ltbaà ag n`pbhgrfs ` hfepr`r y `n eglfr hfstf

pfsbmng.

0) H`nhung gn hfstf ag hà` fphbøl. Ng `yuaèfs hfl n` prbegr` dbn`.

=) Wstga hflsbagr` qug n` ebtà ag glhugstàfrgs lghgsbt`rà ul lugvf n`pbhgrf.

H`nhung lugvègltg gn hfstf y gshråm`nfs ag e`yfr ` eglfr gl n` nålg`.

Nugif, rfagg hfl ul hårhunf n` egcfr fphbøl.

8) ¶Puâ p`qugtg f p`qugtgs ag n`pbhgrfs agmg hfepr`r y pfr quâ6

N`pbhgrfs Hfstf pfr p`qugtg Hfstf tft`n

[`qugtg ag 0; ulbaàgs P00.;; P11.;;

[`qugtg ag =1 ulbaàgs P=1.;;

[`qugtg ag 1; ulbaàgs P8:.;;






090

= 8 1

8

1

7

=

0

;

x

y

‘ 0 5 b

x = +

y = 5

r =

]gp`sf3 sge`l`s 2 ` 04

Gst` sge`l` nfir`rà3 ]gp`s`r nfs hfltglbafs ag n` sge`l` 2 ` n` 04.

]gsfnvgr nfs gcgrhbhbfs ag rgp`sf p`r` gv`nu`rsg gl n` prugm`

l`n.

[rgp`r`rsg mbgl p`r` n` prugm` l`n.




Pugrba` y qugrbaf gstuab`ltg3

Xg `prfxbe` n` prugm` l`n y agmg prgp`r`rsg àghuàègltg, rgp`s`laf nfs hfltglbafs ag n`

sge`l` 2 ` n` 04.

[`r` `prfvghj`r gstg rgp`sf sbi` gst`s rghfegla`hbflgs3

• Jì` ul pn`l ag nf qug gstuab`rà hà` aå` y tr`tg ag huepnbrnf. Agabqug eàs tbgepf `

nfs tge`s qug ng rgsuntgl abdåhbngs.

• Musqug ul nui`r tr`lqubnf, bnueblàf y sbnglhbfsf p`r` gstuab`r.

• Ng` nfs rgsóeglgs ag hà` sge`l` y gshrbm` n`s bag`s eàs bepfrt`ltgs gl su huàgrlf.

• Gshuhjg n` hn`sg ràb`n. Xus e`gstrfs nfhutfrgs ng `hfep`ð`ràl gl gstg rgp`sf y ng `yu-

a`ràl ` rgsfnvgr `niulfs gcgrhbhbfs.

• Hfeprugmg qug j`y` rg`nbzàf mbgl nfs ùtfhfltrfngs. Xb tbglg aua`s, vugnv` ` nggr n`s

sge`l`s, `jå glhfltr`rà gxpnbh`hbflgs y gcgepnfs.

¶Høef sgrà n` prugm` ag gv`nu`hbøl6

N` prugm` l`n gv`nó` nfs ebsefs hfltglbafs y ag n` ebse` dfre` gl qug nfs j` tr`m`càf

sge`l` ` sge`l`.

Gl n` prugm` glhfltr`rà3

• Wl` sgrbg ag ìbnbaà eglt`n p`r egabr su agstrgz` y r`pbagz ag hànhunf gl ul tbgepf

nåebtg ag trgs eblutfs.

• Abdgrgltgs gcgrhbhbfs qug gv`nó`l nf `prglabaf gl n`s fhjf sge`l`s. Gstfs gcgrhbhbfs

sgràl sgegc`ltgs nfs qug ustga rgsfnvbø gl n`s htbvbaàgs agn utfhfltrfn. Xg ng pgabrà3

rgspflagr prgiult`s,

e`rh`r hfl ul` gqubs "Z" gn huàrf ag n` fphbøl hfrrght`,

rgsfnvgr fpgr`hbflgs y

rgsfnvgr prfmnge`s.

• Hu`laf rgsugnv` gcgrhbhbfs y prfmnge`s, agmg agc`r hflst`lhb` ag su prfhgabebgltf.

• Euy bepfrt`ltg3 hà` sgrbg hfltbglg blstruhhbflgs gx`ht`s ag nf qug agmg rg`nbz`r gl

hà` `p`rtàf, `så hfef n` v`nfr`hbøl `sbilà`.

Xb ustga sg prgp`r` hfl tbgepf y agabh`hbøl, gn rgsuntàf sgrà s`tbsd`htfrbf.

Ng rghfraèfs qug agmg prgp`r`rsg p`r` n` prugm` l`n y p`r` n`

prugm` agn EBLGAWH pr`htbh`laf hfl nfs gcgrhbhbfs qug sg glhugltr`l

`n l`n ag su nbmrf gl gn Q`nngr e`tgeàtbhf.






Gcgrhbhbf 0@. @pnbqug n`s rgin`s ag pftglhb`hbøl, ag dfre` abrght`. ]g`nbhg nf qug sg ng pbag gl hà` p`rtàf.

Agcg tfafs nfs gxpflgltgs pfsbtbvfs. Qbglg ul gcgepnf.

;) (e8l) ‘8(e8l)= 5

0) (‘=m8)8

(‘=m8)= 5

=) y ‘2 5

8) 2w8 z 1(4 y= z 8); 5

1) ( 9 x8 y= z =

: x8 z = );

5

7) 0:(m1h=); 5

4) x8

x= 5

9) m ‘8 • m7 • m= 5

:)(9e=l=)7

(9e=l=)8 5

2) (1`8m=)1(1`8m=)8 5

0;) (w= z =)(w= z =)8 5

00) (‘0=r 8

s1

);

5

0=) (8 x7 y8); 5

08)t 7

t 9 5

01) ( w=h=

1`l8 ) ‘=

5

07) (m9h4a =)4 5

04) (e1lw7

8r = z ) ‘0 5

09) ‘7(7mh8w1)7 5

0:) (0;e1l)2 5

02)`9

`= 5

=;) :`8m( x= y=)1 5

=0) (7r = s8

4e8l=u )=

5

==) ( x8 y= z 4)= 5

=8) m ‘0 5

=1) ( ‘=h= d =

2`m8 )8

5

=7) ( uw=

8 x7 y8 )1

5

=4) (e1l= s8

i =j7r 7 )1

5

=9) z 8 • z 4 • z ‘= 5

=:) ( ‘= x8 yz 1

rs=t = ) ‘7

5

=2) w ‘= 5

M. Xbepnbdbqug n`s gxprgsbflgs `nigmr`bh`s ag dfre` abrght`.

0)

(0:à 1 s8

8`=a = s )

8 5 4)

(8;m7a 9 d 4

4m9a 8 d )

‘= 5

=) (2`1mh1)(0=`8m8h=) 5 9)(00 x= yz 7)

(4 xy: z 9) 5

8)7m8h=v=

=7m=h=v= 5 :)

=`8e1l=

0;`7el= 5

1) (0:e=lr 8)(8e7l1r 7) 5 2) (9 =e8)(:`7e4) 5

7) ( :0h8 i

8h= i 8 )1

5 0;) (:`1m8)(8`=m=) 5

0

(e8l)






Gcgrhbhbf =@. @pnbqug n`s rgin`s ag n` ràbh`hbøl p`r` rgsfnvgr nfs gcgrhbhbfs sbiubgltgs.

0) (7 x= y8)= 5

=) 8 41 x4 y4 5

8)1`=m1

x= y4 5

1)4=7w1

:0 x: y0=1 5

7)7 (e7l7 x7)1 5

4)8

(=18e07l7) 5

9) (=74`:m1

04h0= y1 ) 5

:)1 04 x: 5

2)8 =9w0= x2 y4 z 8 5

M. Hflvbgrt` nfs gxpflgltgs dr`hhbfl`rbfs gl ràbh`ngs y sbepnbdbqug n` gxprgsbøl ag sgr pfsbmng.

0) ( xy)0/= 5

=) ( 1m:h4)=/1 5

8) e0/8 5

1) (m8 z 4)=/8 5

7) ( 1)0/1 5

4) (=9e4l8

w= )0/8 5

9) ( 2j8o 8e4)1/8 5

:) (rs)0/7 5

2) ( = x=

y4 z 8 )4/8 5

H. ]gsugnv` ag dfre` abrght` n`s sue`s y rgst`s ag ràbh`ngs sgegc`ltgs.

0) :8 00m ‘ 7

8 00m 5

=) 07` 0= xy + 7` 0= xy 5

8) 0=m18 04e=l= + 0:m18

04e=l= 5

1) =0m8h1 4`8 ‘ 8=m8h

1 4`8 5

7) ‘01 d =8wx + =7 d =8wx 5

4) =9e= 7 12r 8 s= ‘ 0:e= 7

12r 8 s= 5

A. ]gsugnv` n`s euntbpnbh`hbflgs y abvbsbflgs sbiubgltgs. Xbepnbdbqug su rgspugst`.

0)8 4m2h4

•7 4m2h4 5

=)7 8=m7 x7 z 0;

7 r 0; s07

5

8) `=ex8 •1 =ex8 5

1) x= y8 •

1 x= y8 5

7)2 x1 y4

41 x0= y: 5

4)8 41`8e4

8 70=`4l8

5

G. ]`hbfl`nbhg n`s gxprgsbflgs sbiubgltgs. Xbepnbdbqug su rgspugst`.

0): x

4`m 5

=)2 y

0;mex ‘ 1w 5

8):`

9mz + : x 5

1)01el

09`x 5

7)00 x

0=`y ‘ =h 5

4)0= xy

4w 5




Fpgr`hbflgs pfnblfeb`ngs

0. Wl pfnblfebf gs ul` gxprgsbøl e`tgeàtbh` dfreà` ` p`rtbr ag n` ulbøl ag afs f eàs v`rb`mngs y

hflst`ltgs, qug sg vblhun`l ` tr`vâs ag n`s fpgr`hbflgs `rbteâtbh`s ag sue`, rgst` y euntbpnbh`hbøl.

Gl ftr`s p`n`mr`s, ul pfnblfebf gs n` ulbøl ag afs f eàs eflfebfs.

0.³ târeblf =.³ târeblf 8.³ târeblf

2 x= ‘ 8; xy + =7 y=

0.0 Nfs pfnblfebfs sg pugagl hn`sbdbh`r sgiól su lóegrf ag târeblfs, sgiól nfs târeblfs qug

hfeprglag y sgiól gn iràf ag sus v`rb`mngs.

=. [`r` n` àbhbøl y n` sustr`hhbøl gltrg afs f eàs pfnblfebfs, sg sue`l f rgst`l nfs hfgdbhbgltgs ag

nfs târeblfs sgegc`ltgs, gs aghbr, ag nfs târeblfs qug tgli`l n` ebse` v`rb`mng hfl gn ebsef iràf.

8. [`r` euntbpnbh`r pfnblfebfs sg utbnbz` n` prfpbgaà abstrbmutbv`? gn prfauhtf ag pfnblfebfs pugag

prgsglt`r abdgrgltgs h`sfs3

8.0 Euntbpnbh`hbøl ag ul lóegrf pfr ul pfnblfebf3 sg euntbpnbh` gn lóegrf pfr hà` ulf ag nfs

târeblfs ag n` gxprgsbøl pfnblfeb`n.

8.= Euntbpnbh`hbøl ag ul eflfebf pfr ul pfnblfebf3 sg euntbpnbh` gn eflfebf pfr tfafs y hà`

ulf ag nfs târeblfs qug dfre`l gn pfnblfebf, gs aghbr, hfgdbhbgltg pfr hfgdbhbgltg y v`rb`mng pfr

v`rb`mng.

8.8 Euntbpnbh`hbøl gltrg pfnblfebfs3 sg euntbpnbh` hà` târeblf agn prbegr pfnblfebf pfr tfafs

y hà` ulf ag nfs târeblfs agn sgiulaf pfnblfebf.

1. [`r` abvbabr ul pfnblfebf pfr ul eflfebf, sg abvbag hà` ulf ag nfs târeblfs agn pfnblfebf

gltrg gn eflfebf.

Gcgrhbhbf 8@. ]gsugnv` n`s sbiubgltgs sue`s y rgst`s ag pfnblfebfs. Qgli` hubaàf hfl nfs sbilfs.

0) ‘(0=`1 + 9`= ‘ : ) + (:`1 + 2`= ‘ 08) =) (:e8 ‘ e= + =e) + (0:e1 + 09e8 ‘ =7e)

8) (: y8 ‘ 00 y= + 01) ‘ (: y8 + 0; y= ‘ y) 1) (=`x= + 01`8e) + ( x= ‘ 00`8e + e8)

7) (9 x= y= + 2 x8 y= + 00) ‘ (0= x= y= + 8 x8 y= ‘ y=) 4) ‘(=7o 7 + =0o 8 + 8=) ‘ (09o 7 ‘ :o 8 + 9o = + o )




M. ]gsugnv` n`s euntbpnbh`hbflgs pfnblfeb`ngs sbiubgltgs. Xbepnbdbqug sus rgspugst`s. [ugag utbnbz`r

jfc`s `p`rtg p`r` gn prfhgabebgltf.

0) (‘9)(: x8 + 4 x= ‘ x) 5

=) (08`m1)(:t = x + ty= + 2t ) 5

8) (=rt 8 + t 8w=)(:o =e ‘ 2l=) 5

1) (‘1)(00m8 x ‘ 08`= y ‘ = z ) 5

7) (7`=)(4`8 ‘ :`= ‘ 00` + =) 5

4) (:o 8 + 4o = ‘ 1o )(2o = + o ‘ 8) 5

9) 0=(:m8 + 9h= ‘ 0=m=h ‘ 0=) 5

:) :(0= x8 y= z + 8 x= y= ‘ 4 x + z ) 5

2) (:wx

8

)(2w

8

+ 9 x

=

+ 00wx ‘ =) 5

0;) (7 x1 ‘ 8 x= + = x)(01 x8 + 0= x ‘ 7) 5

00) (00m=h= ‘ 2m + 7h)(=m= + 7m + = ) 5

0=) (9e8l= + 2el)(0;ex8 ‘ =e8l= + el) 5

H. ]gsugnv` n`s abvbsbflgs pfnblfeb`ngs sbiubgltgs. Xbepnbdbqug sus rgspugst`s. [ugag utbnbz`r jfc`s

`p`rtg p`r` gn prfhgabebgltf.

0) (12m1 ‘ =0m8 + 9m) ù (9m=) 5

=) (2`h= + =1`= ‘ 4h=) ù (8`=h) 5

8) (0;l9 ‘ 0=l8 ‘ 71l= x) ù (=l=) 5

1) (‘84 p4 + 4 p1 + 1: p8 ‘ p) ù (4 p8) 5

7) (‘41a 4 + :à 8 ‘ 04a + 1) ù (:a 8) 5

4) (09e=l= ‘ 9l8 + 7e= ‘ :) ù (el=) 5

9) (8r 1 s + 1: s8 ‘ 71r 8 ‘ =9r = s=) ù (8r =s) 5

:) (‘0=7 y= z 8 ‘ 0;; yz = + =; yz ) ù (7 y= z =) 5

2) (0=e: ‘ 8e9 ‘ 9e7 + :e= + :) ù (e8) 5

0;) (12r 1w1 ‘ =:r 8w= + 01rw= + 4r =w) ù (9rw) 5

00) (07o 4e4 ‘ 02o 7e8 + ==o =e= ‘ o =) ù (oe=) 5

0=) (7;a 1 i 7 ‘ 41a 8 i 8 + 0=a = i = ‘ 011a = i ) ù (=a = i ) 5




[rfauhtfs lft`mngs

Nfs prfauhtfs lft`mngs sfl h`sfs gspghb`ngs ag n` pftglhb`hbøl y euntbpnbh`hbøl ag pfnblfebfs qug

huepngl hfl rgin`s c`s y qug sg pugagl rgsfnvgr pfr sbepng blspghhbøl, gs aghbr, rgsfnvgr eglt`negltg

n`s fpgr`hbflgs, sbl lghgsbaà ag gshrbmbr tfaf gn prfhgabebgltf.

Nfs prfauhtfs lft`mngs qug gstuabèfs sg rgsuegl gl n` t`mn` sbiubgltg3

Gcgrhbhbf 1@. Ags`rrfnng gn huàràf ag n` sue` y ag n` abdgrglhb` ag nfs mblfebfs ag dfre` abrght`. Ng aèfs

ul gcgepnf.

;) (7 = + 8m8)= 5

0) ( x + 2)= 5

=) ( y + 4)= 5

8) (7 x + 8)= 5

1) (= y + 8 z )= 5

7) (: x + 7 y)= 5

4) (2 x + = y)= 5

9) (1 x= + 7 y)= 5

:) (: x8 + 8 y=)= 5

2) ( 8

7j8 +

=

2o 8)

=

5

0;) ( 1

8w= +

2

= x=)

=

5

00) (9j ‘ o )= 5

0=) (m ‘ h)= 5

08) (7 x ‘ y)= 5

01) (2 ‘ 7 z )= 5

07) (4w ‘ 4)= 5

04) (7 p ‘ =r )= 5

09) (2 ‘ 7 z )= 5

0:) (=j= ‘ 8o =)= 5

02) (7 x= ‘ y8)= 5

=;) ( 7

8 x8 ‘

8

1 y7)

=

5

=0) ( 8

7w= ‘

9

2 x=) 5

Lfemrg [rfauhtf lft`mng Døreun` iglgr`n

Huàràf ag n` sue` ag ul mblfebf ( + m)= 5 `= + =`m + m=

Huàràf ag n` abdgrglhb` ag ul mblfebf ( ‘ m)= 5 `= ‘ =`m + m=

[rfauhtf ag n` sue` pfr n` abdgrglhb` ( + m)( ‘ m) 5 `= ‘ m=

[rfauhtf ag afs mblfebfs hfl ul târeblf

hfeól y sbilfs biu`ngs

( + m)( + h) 5

( ‘ m)( ‘ h) 5

`= + `(m + h) + mh

`= + `(‘ m ‘ h) + (‘ m)(‘ h)

Humf ag n` sue` ag ul mblfebf ( + m)8 5 `8 + 8`=m + 8`m= + m8

Humf ag n` abdgrglhb` ag ul mblfebf ( ‘ m)8 5 `8 ‘ 8`=m + 8`m= ‘ m8

=7`1 + 8;`=m8 + 2m4




M. Ags`rrfnng gn prfauhtf ag n` sue` pfr n` abdgrglhb` ag dfre` abrght`. Qbglg ul gcgepnf.

;) (7e + 8l)(7e ‘ 8l) 5

0) (7 + t )(7 ‘ t ) 5

=) (= x + y)(= x ‘ y) 5

8) (: y + x)(: y ‘ x) 5

1) (= x + 1)(= x ‘ 1) 5

7) (1 z + 1)(1 z ‘ 1) 5

4) (8 p + 4)(8 p ‘ 4) 5

9) (:e + 7)(:e ‘ 7) 5

:) ( 7

8e + 1)( 7

8e ‘ 1) 5

2) ( 0=

w + 8 x)( 0=

w ‘ 8 x) 5

0;) (: z + :)(: z ‘ :) 5

00) (7w + 4)(7w ‘ 4) 5

0=) (1q + 7)(1q ‘ 7) 5

08) (8 x= + y)(8 x= ‘ y) 5

01) (1 x= + 0)(1 x= ‘ 0) 5

07) (0; + 1 )(0; ‘ 1 ) 5

04) (8 x1 + = y=)(8 x1 ‘ = y=) 5

09) (7e= + :l=)(7e= ‘ :l=) 5

0:) ( 8

1e + 8l)( 8

1e ‘ 8l) 5

02) ( 12 y + 0

7 z )( 1

2 y ‘ 0

7 z ) 5

H. Ags`rrfnng gn prfauhtf ag afs mblfebfs hfl ul târeblf hfeól. Qbglg ul gcgepnf.

;) ( + :)( + 7) 5

0) (h + 2)(h + :) 5

=) ( x + :)( x + 4) 5

8) (m + 4)(m + 1) 5

1) ( y= + 8)( y= + 8) 5

7) (a + 1)(a + 0;) 5

4) ( z 8 + 0)( z 8 + 0;) 5

9) (w= + 7)(w= + 1) 5

:) (e ‘ 7)(e ‘ =) 5

2) (= x ‘ 1)(= x ‘ 0) 5

0;) (: y ‘ =)(: y ‘ 8) 5

00) (= x ‘ 7)(= x ‘ 9) 5

0=) (1 y ‘ 4)(1 y ‘ :) 5

08) (= x= ‘ 2)(= x= ‘ 0) 5

01) (9w= ‘ 1)(9w= ‘ =) 5

07) (8 y= ‘ 0)(8 y= ‘ 7) 5

A. Ags`rrfnng n` sue` y abdgrglhb` ag humfs. Qbglg ul gcgepnf.

;) (=` + 7)8 5

0) ( x + 8)8 5

=) (e + l)8 5

8) (w + 4)8 5

1) (7 y + =)8 5

7) (8 z + 8)8 5

4) (1l + 7)8 5

9) (1e + 0)8 5

:) (=r ‘ 0)8 5

2) (7 x ‘ =)8 5

0;) (8 y ‘ 7)8 5

00) (1 z = ‘ 1)8 5

0=) (=m ‘ 4)8 5

08) (7o = ‘ 1)8 5

=7e= ‘ 2l=

`= + 08` + 1;

:`8 ‘ 4;`= + 87;` + 0=7




D`htfrbz`hbøl B

0. D`htfrbz`r ul` gxprgsbøl sg rgdbgrg `n prfhgsf ag sbepnbdbh`rn j`nn`laf afs f eàs d`htfrgs, huyf

prfauhtf gs biu`n ` n` gxprgsbøl prfpugst`. Iglgr`negltg sg gepng` gstg prfhgabebgltf p`r` tr`-

m`c`r hfl pfnblfebfs.

J`y abdgrgltgs eâtfafs f "h`sfs" p`r` d`htfrbz`r, agpglabglaf ag n` gxprgsbøl qug agsggefs sbe-pnbdbh`r. Gstuabèfs hu`trf3

0.0 D`htfr hfeól eflfebf3 gstg eâtfaf hflsbstg gl gxtr`gr ag ul pfnblfebf, ul eflfebf

hfef d`htfr hfeól ` hà` ulf ag nfs târeblfs agn pfnblfebf gl hugstbøl. Nugif, sg agc`

blabhà` n` euntbpnbh`hbøl gltrg gn d`htfr hfeól gxtr`åaf y nfs târeblfs qug quga`l ag n`

gxprgsbøl frbibl`n.

0.= D`htfr hfeól pfr ìrup`hbøl ag târeblfs3 gepngèfs gstg h`sf sb nfs eflfebfs qug

hfepflgl gn pfnblfebf pugagl rgulbrsg gl irupfs hfl ul d`htfr hfeól abdgrgltg gl hà`

irupf. Nugif, gepngèfs n` ebse` tâhlbh` qug gl gn d`htfr hfeól.

0.8 Qrblfebf huàràf pgrdghtf3 sg bagltbdbh` pfr tglgr trgs târeblfs, ag nfs hu`ngs afs tbglglr`åhgs huàrà`s gx`ht`s, y gn rgst`ltg gqubv`ng `n afmng agn prfauhtf ag n`s r`åhgs ag nfs ftrfs

eflfebfs.

( x + y)= 5 x = + = xy + y=

[`r` d`htfrbz`r, agmgefs rgfragl`r nfs târeblfs agc`laf ag prbegrf y ag tgrhgrf nfs târeblfs

qug tgli`l r`åz huàrà`, nugif gxtr`gefs n` r`åz huàrà` agn prbegr y tgrhgr târeblf y nfs

gshrbmbefs gl ul p`râltgsbs, sgp`ràlafnfs pfr gn sbilf qug `hfep`ð` `n sgiulaf târeblf? `n

hgrr`r gn p`râltgsbs gngvèfs tfaf gn mblfebf `n huàràf.

0.1 Abdgrglhb` ag huàràfs3 mblfebf gl gn qug sus târeblfs tbglgl ul` rgst` ag pfr egabf y sgngs pugag h`nhun`r r`åz huàrà`.

[`r` n` d`htfrbz`hbøl ag gstg h`sf agmgefs gxtr`gr n` r`åz huàrà` ag nfs târeblfs. Nugif,

agc`r blabhà` n` euntbpnbh`hbøl gltrg n` sue` y n` abdgrglhb` ag gst`s r`åhgs.

`= ‘ m= 5 ( + m)( ‘ m)

Gcgrhbhbf 7@. Glhugltrg gn d`htfr hfeól ag nfs pfnblfebfs sbiubgltgs. Gshrbm` gn rgsuntàf sfmrg n` nålg`.

0) xy= ‘ y=w 5

=) 7 xy= ‘ 07 y 5

8) =1`8m= ‘ 0=`8m8 5

1) 07 y8 + =; y= ‘ 7 y 5

7) 1 x= y ‘ : xy= ‘ 0= xy8 5

4) =`= x + =`x= ‘ 8`x 5

9) 2 x8 ‘ 4 x= + 0= x7 ‘ 0: x9 5

:) 2 x=`m ‘ 8 x`=m8 + 4 x=`m 5

2) 04`1m7 ‘ =;`8m= ‘ =1`=m4 5

0;) =7 x9 ‘ 0; x7 + 0= x8 ‘ 0: x= 5




M. D`htfrbhg nfs pfnblfebfs pfr ìrup`hbøl ag târeblfs.

H. D`htfrbhg nfs pfnblfebfs sbiubgltgs. Gepngg gn prfhgabebgltf ag ul trblfebf huàràf pgrdghtf.

0) y= + 0= y + 84 5

=) 1`= + 0=` + 2 5

8) =7 x= ‘ 1; x + 04 5

1) 41 + 8= x + 1 x= 5

7) 1 x= ‘ 0= xy + 2 y= 5

4) 2e= ‘ 4ey + y= 5

9) 84 x= + =1 xy + 1 y= 5

:) 2e= ‘ 0=el + 1l= 5

2) 12r = + 00=rs + 41 s= 5

0;)2

1 x= ‘

07

1 x +

=7

04 5

00)=7

84 y= ‘

7

4 y +

0

1 5

0=) 84 x4 + =1 x8 y= + 1 y1 5

08) =7w: ‘ 0;;w1 + 0;; 5

01) 1 x1 y4 ‘ : x= y8 z = + 1 z 1 5

A. D`htfrbhg nfs pfnblfebfs sbiubgltgs. Gl `niulfs blhbsfs agmgrà gepng`r eàs ag ul h`sf ag

d`htfrbz`hbøl.

0) w= ‘ x= 5

=) x= ‘ 04 5

8) y= ‘ z = 5

1) z = ‘ 04 5

7) 1 x= ‘ 0;; 5

4) =7 y= ‘ 1 5

9) 1`= ‘ 04m= 5

:) x= y= ‘ =7 z = 5

2)2

1 x= ‘

0

=7 5

0;) 04w1 x= ‘ 41 5

00) y=

1 ‘ 04 z = 5

0=) `:

=7 ‘

m=h=

2 5

08) 0 ‘e=l=

84 5

01) :0w

1 x

= ‘ 12 y4 z = 5

0) `x ‘ x ‘ ` + 0 5

=) xy + 7 x ‘ y ‘ 7 5

8) xy + y= ‘ x ‘ y 5

1) xy ‘ x + 7 y ‘ 7 5

7) 8 x + 0= ‘ xy ‘ 1 y 5

4) 7`x + `y ‘ 7mx ‘ my 5

9) =er + es + 4lr + 8ls 5

:) 7 y ‘ 0; ‘ xy + = x 5

2) xy + 8 y + zx + 8 z 5

0;) 4e= x + 1e= y + 8lx + =ly 5

00) px ‘ pq ‘ = yx + = yq 5

0=) 4` ‘ 4m + 8`x ‘ 8 xm 5

08) 9 xe ‘ =0 x ‘ 1e + 0= 5

01) =lx + 01 x + 8lx= + =0 x= 5




D`htfrbz`hbøl BB

0. D`htfrbz`r ul trblfebf ag n` dfre` x = + mx + h sbilbdbh` gxprgs`rnf hfef gn prfauhtf ag afs d`htfrgs

ag n` dfre` ( x + `)( x + m). [`r` d`htfrbz`r trblfebfs ag gst` dfre` agmgefs3

`. Agshfepflgr gn trblfebf gl afs d`htfrgs (mblfebfs). [`r` gnnf fmtglgefs n` r`åz huàrà` agn

prbegr târeblf.

m. Hfpb`r gn sbilf agn sgiulaf târeblf gl gn prbegr d`htfr. Nugif, euntbpnbh`r gn sbilf agn sgiulaf

târeblf pfr gn sbilf agn tgrhgr târeblf y gshrbmbrnf gl gn sgiulaf d`htfr.

h. Gngibr afs lóegrfs qug sueàfs f rgstàfs agl hfef rgsuntàf gn târeblf m y `n euntbpnbh`rnfs

agl hfef rgsuntàf gn târeblf h.

=. Nfs trblfebfs ag n` dfre` `x = + mx + h sg d`htfrbz`l hfl nfs p`sfs sbiubgltgs3

`. Euntbpnbh`r gn trblfebf pfr gn hfgdbhbgltg ( ).

m. Gxprgs`r gn prbegr târeblf rgsunt`ltg, hfef ( x)= y gl gn sgiulaf târeblf hfleut`r gn fragl `mx

pfr m`x.

h. D`htfrbz`r gn pfnblfebf qug sg fmtuvf gl nfs p`sfs `ltgrbfrgs, hfef sg jbzf hfl gn trblfebf ag

n` dfre` x= + mx + h.

a. Abvbabr nfs prfauhtfs gltrg nfs d`htfrgs ag `, qug gs gn lóegrf pfr gn hu`n sg euntbpnbhø gn trblf-

ebf.

8. N` sue` ag humfs `8 + m8 sg d`htfrbz` ag `hugraf hfl n` døreun` sbiubgltg3

`8 + m8 5 ( + m)( = ‘ `m + m=)

`. Gl gn prbegr d`htfr sg gshrbmg n` sue` agn rgsuntàf ag n` r`åz hómbh` agn prbegr y sgiulaftâreblf ( + m).

m. Gl gn sgiulaf d`htfr agmg br gn prbegr târeblf `n huàràf eglfs gn prbegrf pfr gn sgiulaf,

eàs gn sgiulaf `n huàràf ( = ‘ `m + m=).

1. N` abdgrglhb` ag humfs `8 ‘ m8 sg d`htfrbz` ag `hugraf hfl n` døreun` sbiubgltg3

`8 ‘ m8 5 ( ‘ m)( = + `m + m=)

`. Gl gn prbegr d`htfr sg gshrbmg n` rgst` agn rgsuntàf ag n` r`åz hómbh` agn prbegr y sgiulaf

târeblf ( ‘ m).

m. Gl gn sgiulaf d`htfr agmg br gn prbegr târeblf `n huàràf eàs gn prbegrf pfr gn sgiulaf, eàsgn sgiulaf `n huàràf ( = + `m + m=).




Gcgrhbhbf 4@. D`htfrbhg nfs trblfebfs ag n` dfre` x= + mx + h y gshrbm` gn rgsuntàf sfmrg n` nålg`.

0) `= + 00` + =1 5

=) m= + 0=m + 87 5

8) w= ‘ 0;w + =1 5

1) x= ‘ 00 x + 0: 5

7) y= + 1 y ‘ 8= 5

4) z = + 9 z ‘ 4; 5

9) e= + 8e ‘ 71 5

:) l= ‘ 8l ‘ 9; 5

2) p= ‘ 2 p ‘ 84 5

0;) r = ‘ 9r ‘ 0: 5

00) s= ‘ 7 s ‘ 07; 5

0=) x= + = x ‘ 1: 5

M. D`htfrbhg nfs trblfebfs ag n` dfre` `x= + mx + h y gshrbm` gn rgsuntàf sfmrg n` nålg`.

0) :`= + 0:` + 8 5

=) 4e= + =8e + 9 5

8) 7l= + 09l + 4 5

1) 8w= + 1w ‘ 1 5

7) 4 p= + 09 p ‘ =: 5

4) 4 x= ‘ 08 x + 4 5

9) 7q= ‘ =8q ‘ 0; 5

:) 4 y= + 00 y + 8 5

2) :r = ‘ 89r ‘ 07 5

0;) : z = + 7 z + 0 5

00) 4j= ‘ 07j + 4 5

0=) 8o = + =8o + 8; 5

08) 0= x= ‘ 9 x + 0 5

01) 1 y= + : y + 8 5

07) =h= ‘ 7h + 8 5

04) 4`= + 9` + = 5

09) 7w= ‘ 9w + 0= 5

0:) 4 x= + 09 x ‘ 8 5

H. D`htfrbhg n` sue` y abdgrglhb` ag humfs y gshrbm` gn rgsuntàf sfmrg n` nålg`.

0) `8 + m8 5

=) : + h8 5

8) e8 + 41 5

1) : x8 + =9 y8 5

7) z 8 + 0=7 5

4) 41o 8 + 0 5

9) :w8 + x8 5

:) x8 y8 + =9 z 8 5

2) :`8m8 + 41h8a 8 5

0;) j8 ‘ o 8 5

00) =9 ‘ z 8 5

0=) :`8 ‘ m8h8 5

08) 0=7 ‘ `8 5

01) :e8 ‘ l8 5

07) 41 x8 ‘ : y8 5

04) =04 ‘ m8 5




Dr`hhbflgs `nigmr`bh`s B

0. Wl` dr`hhbøl `nigmr`bh` gs ul` gxprgsbøl dr`hhbfl`rb` gl n` qug luegràfr y aglfeblàfr sfl

gxprgsbflgs `nigmr`bh`s, iglgr`negltg pfnblfebfs.

[fr gcgepnf3 x= + 0; x + =7

x= ‘ y=

Afs dr`hhbflgs sfl gqubv`ngltgs sb gn prfauhtf hruzàf gltrg èm`s gs biu`n.

0.0 Xbepnbdbh`hbøl ag gxprgsbflgs nigmr`bh`s3 p`r` sbepnbdbh`r ul` gxprgsbøl r`hbfl`n, d`htfrbz`-

efs luegràfr y aglfeblàfr, y nugif h`lhgnèfs d`htfrgs hfeulgs.

0.= Euntbpnbh`hbøl ag dr`hhbflgs `nigmr`bh`s3 gn prfauhtf ag afs dr`hhbflgs `nigmr`bh`s gs ftr`

dr`hhbøl `nigmr`bh` gl n` qug gn luegràfr gs gn prfauhtf ag nfs luegràfrgs y gn aglfebl`-

afr gs gn prfauhtf ag nfs aglfeblàfrgs. Gs aghbr, agmgefs euntbpnbh`r "gl nålg`".

Gn prfhgabebgltf gs gn sbiubgltg3

• D`htfrbzèfs nfs luegràfrgs y nfs aglfeblàfrgs j`st` aflag sg` pfsbmng.

• Blabhèfs n` euntbpnbh`hbøl agn luegràfr pfr luegràfr y aglfeblàfr pfr aglfebl -

afr.

• Xbepnbdbhèfs gn rgsuntàf j`st` rgauhbrnf ` su eålbe` gxprgsbøl.

[( x)

P( x)•

]( x)

X( x)5

[( x) • ]( x)

P( x) • X( x)

0.8 Abvbsbøl ag dr`hhbflgs `nigmr`bh`s3 sg euntbpnbh` n` prbegr` dr`hhbøl pfr gn blvgrsf ag n` sg-

iula` dr`hhbøl. Aghbefs gltflhgs qug n` abvbsbøl gs ul` euntbpnbh`hbøl hruzà` y pfr t`ltf,

utbnbzèfs n` ebse` tâhlbh` qug `prglabefs gl n` euntbpnbh`hbøl.

[( x)

P( x)ù

]( x)

X( x)5

[( x) • X( x)

P( x) • ]( x)

Gcgrhbhbf 9@. Xbepnbdbqug n`s gxprgsbflgs r`hbfl`ngs gl su huàgrlf. Nugif, gshrbm` n` rgspugst` sfmrg n` nålg`.

0)1` + 1

1

5

=)8 y

8 y= + 4 y 5

8) x= ‘ 2 y=

( x + 8 y)= 5

1) ( ‘ m)=

`= ‘ m= 5

7) y= ‘ 8 y + 1 y ‘ 0=

y ‘ 8

5

4) e= + 8e ‘ 0:

8e ‘ 2 5

9) 4 z = ‘ 9 z ‘ 7

= z + 0 5

:) w= + =wx ‘ 8 x=

w= + 7wx + 4 x= 5




M. Euntbpnbqug n`s gxprgsbflgs r`hbfl`ngs y sbepnbdbqug gn rgsuntàf gl su huàgrlf. Nugif gshrbm` n`

rgspugst` sfmrg n` nålg`.

0)8 x ‘ =

8 x + = •

1 x ‘ 0

0 ‘ 1 x 5

8) x

x= ‘ y= •

x + y

x= + xy 5

7)

y= ‘ 84

y= + y ‘ 8; •

y ‘ 7

8 y 5

=) x8 + :

x= ‘ x ‘ 4 •

x + 8

x= ‘ = x + 2 5

1) x= + 8 x + =

x1 y •

x8 y=

x= + 1 x + 8 5

4)

x= + x ‘ 4

x= ‘ 7 x + 4 •

x= ‘ = x ‘ 8

x= ‘ 1 x ‘ 7 5

H. Abvba` n`s gxprgsbflgs r`hbfl`ngs y sbepnbdbqug gn rgsuntàf gl su huàgrlf. Nugif, gshrbm` n`

rgspugst` sfmrg n` nålg`.

0)

x= + = x ‘ :

x= ‘ 8 x ‘ 1 ù

x= ‘ 1 x + 1

x= + 4 x + : 5

8) x= ‘9 x + 0;

x= ‘ 4 x + 7 ù

x= + 7 x ‘ 01

x= + : x + 9 5

7) y= + 8 y + =

y= ‘ 7 y + 1 ù

y= + 4 y ‘ 04

y= + y ‘ =; 5

=)

x= ‘ 1 x + 8

8 x= ‘ x ‘ = ù

x= + 0; x + =1

x= + 8 x ‘ 0: 5

1)0

x= + 9 x ‘ 0: ù

0

x= ‘ 09 x + 8; 5

4)9 x= ‘ 07 x + =

x= + x ‘ 4 ù

x= ‘ 8 x ‘ 0;

x= ‘ = x ‘ 07 5




Dr`hhbflgs `nigmr`bh`s BB

Gl n` sue` y rgst` ag dr`hhbflgs nigmr`bh`s n biu`n qug gl n`s dr`hhbflgs rbteâtbh`s pfagefs fpgr`r

dr`hhbflgs hfl biu`n y abdgrgltg aglfeblàfr.

0. Gl n` sue` f rgst` ag dr`hhbflgs `nigmr`bh`s hfl biu`n aglfeblàfr, sg sue`l f rgst`l nfs

luegràfrgs y sg hfpb` gn aglfeblàfr hfeól.

`

h +

m

h 5

` + m

h

=. [`r` sue`r f rgst`r dr`hhbflgs `nigmr`bh`s hfl abdgrgltg aglfeblàfr, sgiubefs nfs p`sfs

sbiubgltgs3

`. D`htfrbz`r nfs aglfeblàfrgs p`r` j`nn`r gn eålbef hfeól eóntbpnf (e.h.e.) gn hu`n sgrà gn

aglfeblàfr hfeól.

m. Abvbabr gn e.h.e. gltrg gn aglfeblàfr ag hà` târeblf. Nugif, euntbpnbh`r gn rgsuntàf pfr hà`

luegràfr.

h. Xbepnbdbh`r gn rgsuntàf.

8. N`s dr`hhbflgs hfepugst`s f hfepngc`s sfl n`s qug tbglgl dr`hhbflgs gl gn luegràfr f gn

aglfeblàfr, f gl èm`s p`rtgs ag n` dr`hhbøl.

` + m`

` ‘ m`

Nålg` prblhbp`n agn` dr`hhbøl

Luegràfr ag n`dr`hhbøl hfepngc`

Aglfeblàfr ag n`dr`hhbøl hfepngc`

[`r` sbepnbdbh`r f rgauhbr ul` dr`hhbøl hfepngc` ` sus târeblfs eàs sglhbnnfs y qug sg` gqubv`ngltg

` n` gxprgsbøl frbibl`n, agmgefs sgiubr gstfs p`sfs3

`. Glhfltr`r gn eålbef hfeól eóntbpnf ag tfafs nfs aglfeblàfrgs.

m. Euntbpnbh`r gn luegràfr y gn aglfeblàfr ag n` dr`hhbøl hfepngc` pfr gn e.h.e.

h. Xbepnbdbh`r nfs rgsuntàfs hu`laf sg` pfsbmng.




Gcgrhbhbf :@. Xueg n`s gxprgsbflgs r`hbfl`ngs hfl biu`n y abdgrgltg aglfeblàfr gl su huàgrlf. Nugif,

gshrbm` n` rgspugst` sfmrg n` nålg`.

0) x

x + =

+8

x + =

5

=) x

x + 8 +

0

x + 8 5

8)= x ‘ 8

= x + 7 +

:

= x + 7 5

1)1 x= + 8 x

7 x + = +

x= ‘ x

7 x + = 5

7) = x ‘ 0

x= ‘ 8 x ‘ 1 + 8

x= ‘ 8 x ‘ 1 5

4)7 x ‘ 9

4 x= ‘ 00 x + 8 +

0 ‘ x

4 x= ‘ 00 x + 8 5

9)8

1 x ‘ 8

+0

8 x + 0

5

:) x

x + = +

8

= x ‘ 8 5

2) x

= x + 8 +

0

x ‘ = 5

0;) x ‘ =

1 x= + 7 x + 0 +

1

1 x= + 2 x + = 5

00) ‘1 x + 9

= x= + 7 x + = + x=

8 x= + 1 x ‘ 1 5

0=) x ‘ =

x= ‘ 7 x ‘ =1 +

8

x= + 00 x + =1 5

M. ]gstg n`s gxprgsbflgs r`hbfl`ngs hfl biu`n y abdgrgltg aglfeblàfr gl su huàgrlf. Nugif, gshrbm`

n` rgspugst` sfmrg n` nålg`.

0)1 x

1 x ‘ 8 ‘

8

1 x ‘ 8 5

=)= x + 0

8 x ‘ 9 ‘

x + :

8 x ‘ 9 5

8)= x= + 0

1 x= ‘

= x= ‘ 0

1 x= 5

1)4 x=

4 x ‘ 9

‘9 x

4 x ‘ 9

5

7) x=

1 x= ‘ 0 ‘

x ‘ x=

1 x= ‘ 0 5

4) x= + 8 x

x= + x ‘ 0= ‘

x ‘ 0=

x= + x ‘ 0= 5

9) x + 0

4 x ‘

= x + 8

1 x 5

:) x + 8

8 x ‘

7 x ‘ 0

7 x= 5

2)1 x ‘ 7

= x= ‘ 7 x ‘ 8 ‘

x

= x= ‘ 7 x ‘ 8 5

0;) x ‘ =

1 x=

+ 7 x + 0

‘1

1 x=

+ 2 x + =

5

00)8 x ‘ =

x= ‘ 8 x ‘ 1 ‘

x + =

x= ‘ 7 x + 1 5

0=)9 x + 01

8 x= + 0; x ‘ : ‘

x + 4

8 x= + x ‘ = 5




@. ]gsugnv` n`s fpgr`hbflgs hfemblà`s sbiubgltgs. ]ghugrag sbgeprg n` cgr`rquå` ag n`s fpgr`hbflgs.

0) ‘2 + 7 Ù 0 + 8 5

=) 7 Ù 4 ù (‘=) + 4 5

8) 4 Ù 7 ‘ =; ù 7 5

1) 1 Ù 4 ù = + 0 5

7) 8 ù 8 + 1 ‘ 4 5

4) ‘8 ‘ 4 ù = ‘ 9 5

9) 2 Ù : ‘ 9 Ù 4 5

:) 8 + ; ‘ 7 ‘ 0 5

2) 9 Ù : ‘ 1 ‘ 0 5

0;) ‘= + 0; ù 7 5

00) = ‘ 8 + : ù (‘:) 5

0=) 1 ‘ 0 Ù ; ‘ 8 5

08) ‘: ‘ 1 + 4 ù = 5

01) ‘0 Ù 4 + = ‘ 8 5

07) 2 Ù 9 + 2 ù 8 5

04) ; ù 0 + : Ù 2 5

M. ]gsugnv` n`s fpgr`hbflgs hfemblà`s sbiubgltgs. ]ghugrag n` cgr`rquå` ag nfs sbilfs ag irup`hbøl.

0) _(2 + :) ‘ 1U Ù 8 5

=) (=; ù 1) ‘ 4 Ù = 5

8) 0 + 8 Ù (0 + 1) 5

1) 0; ù 7 ‘ (7 ‘ 1) 5

7) _(4 ‘ 9) Ù 7U + 1 5

4) (8 Ù 0) + = ‘ 0 5

9) ; ‘ _(1 + 1) ‘ 4U 5

:) = Ù 0 + (8 ‘ =) 5

2) (= Ù 8) ‘ (= + 8) 5

0;) _(2 ‘ 4) + 9U Ù 1 5

00) (7 ‘ 4) + 1 ‘ = 5

0=) _7 + (: ù 1)U ‘ 1 5

08) 9 ‘ _2 ù (4 ‘ 8)U 5

01) = Ù = ‘ (0 + :) 5

07) 7 Ù 1 ‘ (; + 0) 5

04) (9 + 8) Ù (0; ù 7) 5

H. ]gsugnv` n`s ghu`hbflgs sbiubgltgs. Gshrbm` su rgspugst` gl n` nålg` hfrrgspflabgltg.

0) ‘8 + e 5 0; e 5

=) : ‘ e 5 ‘0: e 5

8) 2 + e 5 ; e 5

1) 0 ‘ e 5 04 e 5

7) : + e 5 : e 5

4) ‘2 ‘ e 5 =0 e 5

9) 2 + e 5 02 e 5

:) ‘9 ‘ e 5 ‘09 e 5

2) ‘ e + 0: 5 01 e 5

0;) e ‘ 9 5 ‘=8 e 5

00) e + 00 5 == e 5

0=) ‘ e ‘ 4 5 =9 e 5

08) e + : 5 0= e 5

01) e + ; 5 ‘2 e 5

07) ‘ e + : 5 ‘= e 5

04) e ‘ : 5 08 e 5







Hn`vgs

=99E`tgeàtbh` ∝ Hn`vgs



Xge`l` 0¡@ tr`m`c`r!0) 04 ag cunbf

=) Ng prgsgltèfs n` dfre` eàs abrght` ag rgsfnvgr gn prfmnge`3

@nmgrtf s`mg gn egs y Mgrl`raf gn aå`. Gl n` prbegr``dbre`hbøl, @nmgrtf gstà sgiurf qug Mgrl`raf lf s`mghuàlaf gs gn huepng`ðfs. Gstf pgrebtg agsh`rt`re`yf y culbf, pfrqug sb Mgrl`raf s`mg gn aå`, nfsólbhfs lóegrfs qug lf sg rgpbtgl sfl gn 02 y gn 0:,sb dugr` ulf ag gsfs aå`s Mgrl`raf så nf s`mrå`.

Hfl e`yf y culbf agsh`rtàfs, `jfr` Mgrl`rafabhg qug ân så s`mg huàlaf gs gn huepng`ðfs. Gstfagsghj` ul` dghj` qug nngvg gn 01, pfrqug `p`rghgafs vghgs, gn 01 ag cunbf y gn 01 ag ìfstf.

Gltflhgs n`s dghj`s pfsbmngs sfl gn 04 ag cunbf, 07 agìfstf, y 09 ag ìfstf.

Nugif, @nmgrtf abhg qug sb Mgrl`raf s`mg, ân tèmbâlnf s`mg. Gltflhgs sb èmfs gstàl sgiurfs lf pugagsgr gl ìfstf pfrqug pfarå` sgr gn 07 f gn 09.

[fr nf t`ltf, gn huepng`ðfs ag @ngc`lar` gs gn 04 ag cunbf.

Gcgrhbhbf 0@.;) Xå 4) Lf0) Lf 9) Xå

=) Lf :) Xå8) Xå 2) Xå1) Xå 0;) Lf7) Lf

M. ]gspugst`s abstblt`s. Hfeprugmg qug nfs glul-hbàfs gshrbtfs huepn`l hfl nfs hrbtgrbfs p`r` sgr

prfpfsbhbflgs. ]gvbsg tèmbâl nfs v`nfrgs nøibhfs.

Gcgrhbhbf =Gn gstuab`ltg puaf j`mgr gepngàf abvgrs`s dr`sgs

p`r` gxprgs`r n` lgi`hbøl (gs d`nsf qug, lf gs hbgrtfqug, gth). Ng prgsgltèfs ulfs gcgepnfs3

;) ~t3 N` nnuvb` lf h`g agn hbgnf. D

0) ~ c 3 Gn ìu` agn e r lf gs aunhg. T=) ~ p3 Gl gn `brg lf j`y fxåiglf. D8) ~v 3 Gs d`nsf qug ul` mbhbhngt` tbglg afs ruga`s. D1) ~s3 Nfs r`yfs ag sfn lf a`l h`nfr. D7) ~r 3 N`s rfh`s lf sfl sgrgs vbvfs. T

Gcgrhbhbf 8@.;) Gxbstg `n eglfs ul` x qug pgrtglghg ` M y tèmbâl

pgrtglghg ` P.0) [`r` tfa` x qug pgrtglghg ` W, sg huepng qug 7 x

pgrtglghg ` W.=) Gxbstg `n eglfs ul x qug pgrtglghg D y pgrtglghg

tèmbâl ` E.

M.;) ∋ x ∊ L , :x ∊ L0) ∈ x ∊ @ ∩ x ∊ M=) ∋ x ∊ ] , 4 x ∊ ] 8) ∈ x ∊ F ∩ x ∊ [

Ags`rrfnng lugv`s j`mbnbaàgs0) m=) a8) h1) m7) h






Xge`l` 8¡@ tr`m`c`r!0) Bldblbt`s hbdr`s aghbe ngs lf pgrbøabh`s.

=) Lf gs pfsbmng, pfrqug lf sg fmtbglg ul` r`åz hu`-arà` gx`ht`.

Gcgrhbhbf 0;) 1 9) 1.84=1

0) : :) ;

=) =7 2) 8.=:41

8) 72

0;) 8 :

1) 04 00) 1.897

7) =2 0=) =74) :

09

08) 8

7

Gcgrhbhbf =;) hfleut`tbv`0) hnùsur`tbv`=) blvgrsf àbtbvf u fpugstf8) hfleut`tbv` ag n` euntbpnbh hbøl1) blvgrsf euntbpnbh`tbvf7) gngegltf lgutrf ag n` euntbpnbh`hbøl4) `sfhb`tbv`9) abstrbmutbv` agn prfauhtf rgspghtf ` n` sue`

Gcgrhbhbf 8;) :8 + 4 5 :2

0) 0

=) ( 0= )

= + =

5 ( 0= )

1

8) 81

91

1) 7: ‘ 8 5 77

7) 0

4) 29 ‘ 4 5 2

9) =8

78

:) 0:=

2) 0(‘=)1

Gcgrhbhbf 1;) 4 4/8 5 4 = 5 84

0) 411 5 := 5 1

=) 0;= 5 0;

8) 1 04 •1 04 5 = • = 5 1

1) 8•= 8 5 4 8

7)041

5 1= 5 =

4) 7 17 5 1

9) :4/= 5 :8 5 70=


7

9

1 01Å 10 5 791

=:;BIG] ∝ [fnfhjbh



Xge`l` 1¡@ tr`m`c`r!N`s rgspugst`s pugagl v`rb`r.

Gcgrhbhbf 0@.;) 04? 04 = 5 =74

0) 8? 8= 5 2 =) 1? 1= 5 04 8) 2? 2= 5 :0 1) 0=? 0== 5 011 7) 0:? 0:= 5 8=1 4) 07? 07= 5 ==7 9) =;? =;= 5 1;; :) 8;? 8;= 5 2;; 2) 00? 00= 5 0=0

0;) 4;? 4;

=

5 84;;00) 7;? 7;= 5 =7;;

M.;) 8? 88 5 =9

0) 1? 18 5 41 =) =? =7 5 8= 8) 7? 78 5 0=7 1) 0;? 0;8 5 0;;; 7) 8? 89 5 =0:9 4) ‘8? (‘8)8 5 ‘=9 9) ‘1? (‘1)8 5 ‘41 :) ‘=? (‘=)7 5 ‘8=

2) ‘7? (‘7)8

5 ‘0=70;) ‘0;? (‘0;)8 5 ‘0;;;00) ‘8? (‘8)9 5 ‘=0:9

H.;) 1;;(‘0) 5 1;;b = 5 1;; • b = 5 =;b

0) 04(‘0) 5 04b= 5 04 • b= 5 1b

=) =7(‘0) 5 =7b= 5 =7 • b= 5 7b

8) 84(‘0) 5 84b= 5 84 • b= 5 4b

1) 41(‘0) 5 41b= 5 41 • b= 5 :b

7) 042(‘0) 5 042b= 5 042 • b= 5 08b

4) 0;;(‘0) 5 0;;b= 5 0;; • b= 5 0;b

9) 024(‘0) 5 024b=

5 024 • b=

5 01b

Gcgrhbhbf =0) 8 + 8b 7) = + 0;b

=) : + 4b 4) ‘ 0= + b

8) ‘7 + =b 9) ‘: ‘ 1b

1) 0; ‘ 4b :) ‘07 + =.:8b

Gcgrhbhbf 80) : + 8b + = + 9b 5 : + = + 8b + 9b 5

0; + 0;b

8) ‘1 + 9b ‘ 0 + 8b 5 ‘1 ‘ 0 + 9b + 8b 5 ‘7 + 0;b

7) 8= + 1;b ‘ 1:b ‘ 4;b= 5 8= ‘ :b ‘ 4;b= 5 8= ‘ :b ‘ 4;(‘0) 5 8= ‘ :b + 4; 5 8= + 4; ‘ :b 5 2= ‘ :b

Gcgrhbhbf 1

0) ( 2 + b

b )( b

b ) 5 (2b + b=

b= ) 5 2b + (‘0)

‘0 5 2b

‘0 ‘ 0

‘0 5

‘2b + 0 5 0 ‘ 2b

=) (0; ‘ 7b

b )( b

b ) 5 (0;b ‘ 7b=

b= ) 5

0;b + 7(‘0)‘0 5 0;b

‘0 ‘ 7

‘0 5

‘0;b + 7 5 7 ‘ 0;b


=) 8 + 7b ‘ 2 + =b 5 8 ‘ 2 + 7b + =b 5

‘4 + 9b

1) 07b ‘ 1;b= 5 07b ‘ 1;(‘0) 5 07b + 1; 5 1; + 07b

4) 14

‘ 7b

= + :b

8 ‘ 0;b= 5

=8

+ b

4 ‘ 0;b= 5

=8

+ b

4 ‘ 0;(‘0) 5

=8 + b4 + 0; 5

=8

+ 0; + b

4 5

8=8

+ b

4

: Ù = + 1 5 =;

+ Ù Ù

8 Ù 9 ‘ 0 5 =;

+ + Ù

2 + 4 + 7 5 =;

5 =; 5 =; 5 =;

=:0E`tgeàtbh` ∝ Hn`vgs



Xge`l` 7¡@ tr`m`c`r!

9 8 = 1 7 0 4 : 2

0 7 4 9 : 2 = 1 8

: 2 1 4 = 8 9 0 7

4 = 2 0 8 : 7 9 1

8 0 9 = 1 7 : 2 4

1 : 7 2 9 4 0 8 =

7 4 : 8 0 1 2 = 9

= 1 0 7 2 9 8 4 :

2 9 8 : 4 = 1 7 0

Gcgrhbhbf 0;) 8 Ù 80) 8 Ù ==) = Ù =8) 0 Ù 11) 8 Ù =7) 1 Ù 1

Gcgrhbhbf =

0) ‘0 ‘12 ‘=

=) : 28 ‘08

8) 0 ‘4= 7

1) 08 == ‘08

7) 0 4 =‘: ‘8 11 1 4

4) 00 0 0‘= ‘8 :1 = 1

9) 9 ; 0=

01 0 1: ‘0 4

:) 0= ‘4 0;

08 = 17 ‘0 =

2) ‘0 01 ‘4‘9 ‘9 :7 0 :

0;) ‘9 01 ‘4= ‘7 11 ‘8 4

Gcgrhbhbf 8[rfpbgaà

;) 00 ‘8 ‘00; 9 ‘0

8 0; ‘0=

`sfhb`tbv`

0) ; ; ;

; ; ;

; ; ;

Gngegltffpugstf

=) 9 : ‘1

0 ; 7

8 4 9

Gngegltflgutrf

Ags`rrfnng lugv`s j`mbnbaàgs]gvbsg qug j`y` abmucàf tfafs nfs agt`nngs ag n`bnustr`hbøl. N` beìgl hfepngt` agmgrå` vgrsg `så3

=:=BIG] ∝ [fnfhjbh



Xge`l` 4 Xge`l` 9¡@ tr`m`c`r!]gspugst`s abstblt`s. Gn gstuab`ltg puaf j`mgr gshrbtf

hu`nqubgr rghfepgls` ag su prgdgrglhb`.

Gcgrhbhbf 00) ‘0; 17 ‘7 ; =) ‘: 84

1 04

8) ‘1 ‘=;=: ;

‘04 ‘0=

1) ‘02 ‘0;; 0

01 ‘=

7) : = ; 1

‘1 4 = =

4) 84 1: ‘4; 0:

4 1= ; 4

9) 4 ‘8; 0=‘4 ; ‘84

‘1= ‘=1 ‘71

:) :0 ‘84 71; 2 ‘9=2 ‘2; 48

Gcgrhbhbf =0) 00 =7

‘8 ‘7

=) 02 ‘=

8) 0 0‘2 8

1) ‘8 8‘1 0;0: ‘2

7) ‘0= 4‘=1 ‘0:0= 4

4) 2 ‘7‘4 =

9) 00 ‘4 144 =4 =20 01 ‘4

:) ‘0; ‘02 ‘=701 ‘=0 8= ‘02 ‘9

2) 7 9 ‘088 4 ‘0=‘08 =7 ‘=:

0;) 4= 8; =7; =; 1‘1 ‘= ;

Ags`rrfnng lugv`s j`mbnbaàgs0) `=) h8) `1) a

¡@ tr`m`c`r!]gspugst`s abstblt`s. Gn gstuab`ltg puaf j`mgr gshrbtf

sfmrg gn nfirf ag Ngfl`raf ag [bs` qug eàs ng j`y`bltgrgsàf.

Gcgrhbhbf 0@.;) a 5 ‘4 , gs aghrghbgltg 7) a 5 9, gs hrghbgltg0) a 5 08, gs hrghbgltg 4) a 5 ;, gs hflst`ltg=) a 5 ‘ ;.7, gs aghrghbgltg 9) a 5 ‘7.7, gs aghrghbgltg8) a 5 ;, gs hflst`ltg :) a 5 4, gs hrghbgltg1) a 5 ‘04, gs aghrghbgltg

M.0) `18 5 02= 7) a 5 0=.7

=) `22 5 ‘2=8 4) `49 5 1248) `0;0 5 ‘187 9) a 5 ‘:

1) `0 5 ‘49: :) `0 5 090;.7

H.0) `=;; 5 =108? s 5 =18 :;;

=) `07; 5 0127? s 5 00= 7;;

8) `97 5 00=8? s 5 1= 4;;

1) `8;; 5 =104? s 5 844 ;;;

7) `0=7 5 =:;? s 5 02 7;;

4) `7; 5 284? s 5 =8 7=7

9) `==7 5 497? s 5 94 =97

:) `0;; 5 9;8? s 5 87 47;

Gcgrhbhbf =@.

;) r 50

7 , aghrghbgltg 7) r 5 0, hflst`ltg

0) r 5 1, hrghbgltg 4) r 5 0

7, aghrghbgltg

=) r 5 8, hrghbgltg 9) r 5 4, hrghbgltg

8) r 5 0

=, aghrghbgltg :) r 5 0

0;, aghrghbgltg

1) r 5 7, hrghbgltg

M.0) `0; 5 0;=1 4) `0 5 9

=) `07 5 1 9:= 242 9) r 5 8

8) `9 5 8= ;;; :) r 5 4

1) `7 50;2

≊ 0.00 2) `9 5 0 :87 ;;:

7) `08 50 721 8=8 0;) `0 5 7




Xge`l` :Gcgrhbhbf 0@. N` rga`hhbøl ag n`s prfpfsbhbflgs lgià`s pugag

v`rb`r. N` bag` iglgr`n agmg e`ltglgrsg.

;) ~r3 Pubhjâ lf gs ul agp`rtègltf ag Iu`tge`n`. D 0) ~w 3 Hum` gs ul p`ås qug lf pgrtglghg Gurfp`. T=) ~j3 Gn ìu` agn e`r lf gs aunhg. T8) ~ i3 Lf gs hbgrtf qug gn babfe` fdbhb`n ag Iu`tge`n`

gs gn `ngeàl. T1) ~s3 8 lf gs abvbsfr ag 2. D7) ~ p3 Gn vgrag lf gs ul hfnfr sghula`rbf. D4) ~q3 Gs d`nsf qug Eâxbhf sg glhugltr` `n sur

ag Iu`tge`n`. T

M.;) Gxbstg `n eglfs ul` x qug pgrtglghg ` Q y pgrtglghg

tèmbâl ` I.

0) [`r` tfa` x qug pgrtglghg ` G, sg huepng qug : xtèmbâl pgrtglghg J.

=) Gxbstg `n eglfs ul` x qug pgrtglghg ` H y tèmbâlpgrtglghg ` M.

8) [`r` tfa` x qug pgrtglghg ` @, sg huepng qug x + 8pgrtglghg ` M.

1) Gxbstg `n eglfs ul` x qug pgrtglghg ` ] y tèmbâlpgrtglghg ` Y.

7) Gxbstg `n eglfs ul` x qug pgrtglghg ` O y tèmbâlpgrtglghg ` H.

Gcgrhbhbf =@. [`r` rgprgsglt`r n`s prfpfsbhbflgs sg gepng`rfl n`s

ngtr`s p y q, sbl gem`rif gn gstuab`ltg puaf j`mgrutbnbzàf hu`nqubgr ftr` hfembl`hbøl ag ngtr`s.

0) p ∪ q=) p ↝ q8) p ∩ q1) p ↑ q7) p ↝ q

M. p q

;) T T p ↝ q3 T

0) T D p ↝ q3 D=) T D p ∩ q3 D8) T D p ∪ q3 D1) T T p ∩ q3 T7) T D p ∩ q3 D4) T T p ↑ q3 T

H.0) `2 5 17 2=9? s 5 4: ::9

=) `: 5 9:0.=7? s 5 022 =0:.97

8) `07 5 019 174? s 5 =21 2;8

1) `7 5 4; ;=7? s 5 9; ;=7

7) `4 5 ==2 894? s 5 =4= 018

4) `0= 5 9;: 7::? s 5 0 ;4= ::;

Gcgrhbhbf 8N` rga`hhbøl ag n`s rgspugst`s pfarà v`rb`r. N` rgspugstàgmg hfblhbabr.

0) `0 5 7;; ei? a 5 ‘07 ei `. `

=7 5 01; ei

Gn p`hbgltg tfe`rà 01; ei ag egabhègltfgl gn óntbef aå` agn tr`tèbgltf.

m. s 5 :;;; ei Gn p`hbgltg j`mrà bligrbaf :;;; ei ag egab-

hègltf ` nf n`rif ag tfaf gn tr`tèbgltf.

=) r 5 =? `0 5 0;;;? `

0; 5 70= ;;;

Nugif ag 7 jfr`s, j`mrà 70= ;;; m`htgrb`s.

8) `0 5 P07;.;; , r 5 =, `9 5 P2,4;;.;; Gl gn sâptbef aå` agmgràl agpfsbt`r P2,4;;.;;.

1) `0 5 P1,;;;.;;, a 5 P1.7;? `0:; 5 P1,:;7.7; Nugif ag 0:; aå`s, j`mrå` agpfsbtàfs P1,:;7.7;.

7) `7 5 1784, r 5 8? `0 5 74 Gl gn prbegr `ðf j`må` 74 pfnnfs .

4) `0 5 P77.;;? r 5 = `. `4 5 P0,94;.;; Gn sàmàf i`lâ P0,94;.;;.

m. s 5 P8,147.;; Gl tft`n, ag nulgs ` sàmàf, i`lâ P8,147.;;.

Ags`rrfnng lugv`s j`mbnbaàgs:

4 1

9 7 2

41

0

09

0==

9

:

00

0108

028

7

04

0; 2

07

0:

0) Gn gcgrhbhbf tbglg eàs ag ul`sfnuhbøl. Nf bepfrt`ltg gshuepnbr hfl n` hflabhbølaà`. Wl` ag n`s pfsbmngsrgspugst`s p`r` gstg gcgrhbhbfsg ng prgsglt` hfef gcgepnf.

=) Gn gcgrhbhbf tbglg eàs ag ul`sfnuhbøl. Nf bepfrt`ltg gshuepnbr hfl n` hflabhbølaà`. Wl` ag n`s pfsbmngsrgspugst`s p`r` gstg gcgrhbhbfsg ng prgsglt` hfef gcgepnf.

=:1BIG] ∝ [fnfhjbh



Gcgrhbhbf [email protected]) 0;0 7) :9

=) 9; 4) ‘:78) :1 9) =;1) ‘009 :) ‘02

M.0) gngegltf lgutrf euntbpnbh`tbvf=) blvgrsf euntbpnbh`tbvf8) gngegltf lgutrf àbtbvf1) hfleut`tbv`7) `sfhb`tbv`4) abstrbmutbv` agn prfauhtf rgspghtf ` n` sue`9) hnùsur`tbv`:) blvgrsf àbtbvf

H. @niulfs blhbsfs sg eugstr`l hfl afs rgspugst`s qugsfl gqubv`ngltgs. @em`s sfl hfrrght`s.

0) 0 :) ‘ 08=

=) 77 f 80=7 2) 4‘0 f 04

8) ( =8 )

8 f :

=9 0;) 0

041) (‘8)7 f ‘=18 00) 0

7) 078 f 0

0=7 0=) :

4) 84 f 9=2 08) 1=79) 48 f =04 01) =4 f 70=

A.0) 4 :) :

=) 1 2) 0=

8) 8=

0;) 7

1) 8 00) 0=7) 8 0=) 74) 7 08) 849) 1 01) 01

Gcgrhbhbf [email protected]) 8b 9) b

=) :b :) b =8) 0;b 2) 4b

1) =;b 0;) 7b

7) 2b 00) 0=b

4) 9b 0=) 00b

M.0) 0; ‘ 00b 4) 104 + =b

=) =9 + 4b 9)

728

88 ‘

12b

888) ‘: + 8b :) =9

0; ‘ 82b

=;

1) ‘4; + 44b 2) 709

‘ 8b

09

7) 1: ‘ :1b 0;) ‘ 47

‘ 8b

7

Gcgrhbhbf 7@.

0) [uaf j`mgr dfreàf hu`nqubgr e`trbz qug huepn`hfl n` hflabhbøl ag tglgr 1 dbn`s y 8 hfnuel`s.

=) B 50 ; ;; 0 ;; ; 0

8) [uaf j`mgr dfreàf hu`nqubgr e`trbz qug huepn` hfln` hflabhbøl ag sbegtrå`. Ng prgsgltèfs ul gcgepnf3

@ 57 4 94 : ‘09 ‘0 2

@t57 4 94 : ‘09 0 2

1) [uaf j`mgr dfreàf hu`nqubgr e`trbz y su rgspghtbvè`trbz tr`lspugst`. Ng aèfs ul gcgepnf3

E 5‘= 4 :1 ‘0 ;; ; ‘=

Et5‘= 1 ;4 ‘0 ;: ; ‘=

M.

0) 4 0 00 7: 07 ‘2 ‘0=

=) ‘1 ‘4 ‘0‘4 1 9‘9 ; ‘:

8) ; ‘1 9 7: 0; ; ‘02

1) ‘: 9 ‘0=4 8 ‘=

‘0 ‘2 07

7) 2 ‘0 ‘0 1‘1 ‘8 ; 0=

4) ‘1 0; 08‘8 00 22 0: =




Gcgrhbhbf [email protected]) 41 :; ‘::

‘9= ; :74 8= ‘=1

=) ; =9 0==9 ‘07 ‘84‘4 ; 8

M.0) ‘77 ‘1; =) 79 ‘87

8) ‘: :0 ‘0

1) 9 ‘87‘00 8;

7) 84 04‘7 ‘2

4) ‘41 ‘8 11: 7: ‘49

9) 07 9 ‘8=11 87 ‘74088 9: ‘=;

:) ‘19 ‘01 ‘82‘0: = =99: 8= 04

2) 8 ‘:0 ‘:1; 8; 7420 ‘49 44

0;) 070 :: ‘110;: 48 ‘8478 00 ‘2:

Gcgrhbhbf [email protected]) `18 5 ‘=:4=) `74 5 :8.78) `81 5 =14

1) a 5 97) a 5 ‘0=

4) `0 5 7:

9) `0 5 :1.7:) `44 5 48=?

s 5 == 1;9

M.0) `0= 5 0= =::=) `00 5 1 021 8;18) `07 5 =887.18

1) r 5 77) r 5 84) `0 5 0=

9) `0 5 0;:) `0; 5 7=1 =::? s 5 422 ;7;

H.0) `0 5 P8;.;;? a 5 P:.;;`. `4 5 P9;.;;? s 5 P8;;.;; N` sgxt` jfr` tglarà ul hfstf ag P9;.;; y gl tft`n,

nugif ag gs`s 4 jfr`s ag `nqubngr, agmgrgefs pì`r

P8;;.;;m. Døreun` agn târeblf iglgr`n3 `

l 5 == + :l

=) `0 5 P1:,;;;.;;? r 5 0=

`. `4 5 P0,7;;.;; @n sgxtf prfpbgt`rbf, n` eàqubl` ng hfstø P0,7;;.;;m. `: 5 P897.;;? s 5 P27,4=7.;; Xb gn prfpbgt`rbf lóegrf fhjf sg quga` hfl n`

eàqubl`, n` sue` tft`n pìà` gl tfa`s n`s vglt`ssgrà ag P27,4=7.;;

Xge`l` 2¡@ tr`m`c`r!;) 79 8) (‘=):

0) 80; 1) (2)8

=) (‘:)0;

Gcgrhbhbf 0@.;) 7 8) ‘7(0) 5 ‘7

0) 0 1) :j(0) 5 :j

=) 0

M.;) `0= 8) (el)9

0) x2 1) (=wy)1

=) e07 7) (rs):

H. ;) e1 1) 0

0) m4 7) ( pqr )=

=) w9 4) (7el)=

8) y0= 9) 0

(‘8 xyz )0;

A. ;) `1m0;h0= 1) p0;r 01

0) x04 7) 0

=) e1l04 4) ‘ 1e1j4o :

8) w4 y0: 9) ‘7`8m07h8a 4

G. ;) :0 1

04m01 8) =7e4l1

1w: z 4

0) x4

y4 1) 2j

:o :

=7 p:q

0=

=) `4m4

h

4a 4 7) `04m

=:h0=

w: x

0= y:

Gcgrhbhbf =0) ( m

` )7

5m7

`7

=) (1 y

8 x)8

541 y8

=9 x8

8) (8h8

7m=)= 5 2h4

=7m1

1) ( h1a 7

`=m8 )8

5h0=a 07

`4m2

7) ( = p1q8

1e8l= )8

50 p0=q2

:e2l4

4) ( y8 z 9

‘7w1 x7 )8

5 y2 z =0

‘0=7w0= x07

=:4BIG] ∝ [fnfhjbh



Gcgrhbhbf 8;) (1w8 x= y4 )8 5 41w2 x4 y0:

0) 4`4m2h7

=) (=`1m8h=)(=9`2m8h07) 5 71`08m4h09

8) ‘0= x7 y0;

1) (7w8 x= y1)(=9 x4 y2) 5 087w8 x: y0;

7) 8 p8q1r 1

4) :`4m7h7

9) (8 x= y1)1 5 :0 x: y04

:) (w= x1 y=)7 5 w0; x=; y0;

2) 0;

0= x01 y4 z 9 5

7

4 x01 y4 z 9

Ags`rrfnng lugv`s j`mbnbaàgs0) 08 5 0 Gn humf gstà dfreàf pfr ul` pbgz`.

=) =8 5 : Gn humf gstà dfreàf pfr fhjf pbgz`s.

8) 88 5 =9 Gn humf gstà dfreàf pfr vgbltbsbgtg pbgz`s.

1) 18 5 41 Gn humf gstà dfreàf pfr sgsglt` y hu`trf pbgz`s.

Xge`l` 0;¡@ tr`m`c`r!

@ 5 4=7e=

x= 5 4=7e=

x= 5 4=7e=

x 5 =7e

Xg pugagl sgemr`r =7 pn`lt`s gl hà` dbn` y hfnuel` `0 egtrf ag abst`lhb` hà` ul`.

Gcgrhbhbf 0;) ` • m

0)8

h •8

a

=) el

8) =; x

1) 8 e= •

8 l=

7) p

q

4)8 r 8 s

9) :

: 5 `

:) 4 4 5 `

Gcgrhbhbf =@. ;) `4/8 5 `=

0) `:/7

=) a :/= 5 a 1

8) (=; x)1/8

1) (:l)=/1 5 (:l)0/=

7) ( p

q )0/04

4) j

o

9) ( yz )1/: 5 ( yz )0/=

:) ( x= y8)0/7

M. ;) 7

(`m)= 7) 1 ( pq)8

0) 1 x8 4) ( jo )

72

=) 9 w7 9) ( r

s )8

7

8) 8 z = :) 8

( xyz )

1) 2 (el)=






Xge`l` 00¡@ tr`m`c`r!]gspugst`s v`rb`mngs. [uaf j`mgr gshrbtf n` `pnbh`hbølqug eàs ng j`y` bltgrgsàf. @niul`s àrg`s aflag sg

utbnbz` gn ànigmr` sfl n` bliglbgrå`, n` hflstruhhbøl, gl n`mfns` ag v`nfrgs, gltrg euhj`s ftr`s.

Gcgrhbhbf [email protected]) tgtr`lfebf 1) mblfebf=) mblfebf 7) mblfebf8) trblfebf 4) trblfebf

M. 0) tgrhgr iràf f hómbhf =) hu`rtf iràf 8) hu`rtf iràf 1) qubltf iràf 7) tgrhgr iràf f hómbhf 4) prbegr iràf f nblg`n 9) hu`rtf iràf :) tgrhgr iràf f hómbhf 2) sgiulaf iràf f huàràtbhf0;) sgiulaf iràf f huàràtbhf

H. 0) x8 + 0= x= ‘ x

=) ‘0; x= ‘ 7 x + 7

8) 4 y8 + 7 y ‘ =0

1) ‘0;`t 8

+ =;t + 0 7) 8=`1 + ==`8 + `=

4) ‘=4r = + 1r ‘ :

9) 8; x= y + 02 xy ‘ 08

:) 84w8 ‘ 8w= + 00w ‘ 0

2) ‘=1 i = ‘ =8 i ‘ 0

0;) ‘7wy8 + wy + 1w + 1 y=

Gcgrhbhbf = 0) 7; x8 + 08; x ‘ 8;

=) 07`7m= + 2`1m1 ‘ =;`=m ‘ 0=`m8

8) 87w7 x + 07w8 x8 ‘ =7w8 x

1) ‘: x8 ‘ 9= xy= ‘ : y8

7) 84 i 1j7 + 084 i 8j1 ‘ :8 i =j8 ‘ 0= i =j= ‘ =;0 ij= ‘ 84 ij +

0=;j + 17

4) 2:`m7h8 + 071`m=h7 ‘ 00=m=h8

9) 70l8e ‘ 084l=e= + 0;=le8 + 084

:) ‘=;w1 x8 y= + 4;w1 x= y + 17w8 x1 y8 + =:w8 x8 y8 ‘ :1w8 x= y=

‘ 48w= x1 y1 ‘ 8=w= x8 y7 + 24w= x= y1 + 9=wx1 y4

2) ‘17 x7 y= + =9 x1 y8 + 48 x1 y

0;) ‘8=w8 y= z + =;w= x8 z 8 + 1:w= y= z ‘ 8;wx8 z 8

Gcgrhbhbf 80) = x8 ‘ 8 x= y + 1 xy= ‘ = y8

=) = x= + 8 x ‘0

x ‘ 1

8) ‘ 8`m ‘1

` + 7

1) 8w + 1 x ‘7

x

7) 2`

m +

4m

2 ‘ 8

4) ‘ 1

( xy=) + 8 x + y

9) ‘=;m= ‘2

mh= +

00

h

:) x=

y8 ‘

=

x= y8 ‘

9 x

y= +

=

x +

8

y

2) 0`m=

+ 4`m ‘ 1m

Ags`rrfnng lugv`s j`mbnbaàgs0) TB + BT 5 Z=) Z 5 BB + TBB




Xge`l` 0=¡@ tr`m`c`r!N`s rgspugst`s pugagl v`rb`r. @niulfs gcgepnfs sfl3Guhnbags, [btàifr`s, Lgwtfl, gth.

Gcgrhbhbf 0;) (x

= )= + =(x= • m8 ) + (m8 )= 5 x1 + =x=m8 + m4

0) (`)= + =(` • 8) + (8)= 5 `= + 4` + 2

=) (=h=)= + =(=h= • 1a =) + (1a =)= 5 1h1 + 04h=a = +04a 1

Gcgrhbhbf =@. ;) x

4 ‘ 0= x

8 + 84

0) x

= ‘ = xy + y=

=) w= ‘ =wz + z =

8) 2 x

= ‘ =1 x + 04

1) 84 y

= ‘ 4; y + =7

7) 0

1w= ‘ =wz + 1 z =

4) x

= ‘=

7 xz +

0

=7 z =

9) 2 x

: ‘ =1 x

1 y8 + 04 y4

:) 1 x

1 ‘ 0= x

= y= + 2 y1

2) 41 x0; ‘ 41 x

7 y

8 + 04 y4

0;) 04 x

: ‘ 1; x

1 y2 + =7 y

0:

00) 2e1l4 ‘ 1:e=l8 pq= + 41 p=q1

0=) 0

2w= ‘ 4wz + :0 z =

08) 04

2

x1 ‘=;

8

x

= y8 +=7

1

y

4

M.

;) 1 x

= ‘ 2 y

=

0) p= ‘ q=

=) y

= ‘ 04

8) w= ‘ 84

1) 2 x= ‘ y

=

7) 12

`= ‘ m=

4) 41=7

e1 ‘ 2

9) =7 z = ‘ 41

:) 04 y

1

‘ z

=

2) 1 x

1 ‘ 2

0;) 2 x

= ‘ 04 y

=

00) =7 x

= ‘ 0;; y

=

0=) 01 x

= ‘ =7 y

=

08) =704

e

1 ‘ 2l

=

H. ;) `= + :` + 07

0) x= + 09 x + 9=

=) y=

+ 9 y + 0= 8) z = + 2 z + 01

1) `= + :` + 04

7) m= + 0=m + 84

4) o 1 + 0;o = + 04

9) 2w4 + 88w8 + 8;

:) `= ‘ 0;` + =1

2) 1h= ‘ 0;h + 4

0;) =7 x= ‘ 4; x + 8=

00) 2 y= ‘ 17 y + 74

0=) 1 x= ‘ 04 x + 07

08) 0;; x= ‘ 4; x + 2

01) :0 z 1 ‘ 011 z = + 4;

07) 2 x1 y= ‘ 0: x= y + :

Gcgrhbhbf 80) (1w + z )8 5

(1w)8 + 8(1w)=( z ) + 8(1w)( z )= + ( z )8 5

41w8 + 8(04w=)( z ) + 8(1w)( z =) + z 8 5

41w8 + 1:w= z + 0=wz = + z 8

=) (8 x= + = y

=)8 5

(8 x=)8 + 8(8 x=)=(= y=) + 8(8 x=)(= y=)= + (= y=)8 5

=9 x4 + 8(2 x

1)(= y

=) + 8(8 x

=)(1 y

1) + : y

4 5

=9 x 4 + 71 x 1 y

= + 84 x = y1 + : y4

Gcgrhbhbf 10) (=h ‘ a )8 5

(=h)8 ‘ 8(=h)=(a ) + 8(=h)(a )= ‘ (a )8 5

:h8 ‘ 8(1h=)(a ) + 8(=h)(a =) ‘ a 8 5

:h8 ‘ 0=h=a + 4ha = ‘ a 8

=) (1 y8 ‘ =z)8 5

(1 y

8)8 ‘ 8(1 y8)=(= z ) + 8(1 y8)(= z )= + (= z )8 5

41 y

2 ‘ 8(04 y

4)(= z ) + 8(1 y8)(1 z =) ‘ : z 8 5

41 y

2 ‘ 24 y4 z + 1: y

8 z = ‘ : z 8


`. x=

+ mx + `x + `m 5 x=

+ x(` + m) + `mm. ( x + `)( x + m)

h. [rfauhtf ag afs mblfebfs hfl ul târeblf hfeól ysbilfs biu`ngs.




Xge`l` 08¡@ tr`m`c`r!2; =

17 807 8

7 7

0

89: =

0:2 848 8

=0 8

9 9

0

=:4 =

018 0008 08

0

Gcgrhbhbf 00) `m=(‘ 8 z = ‘ `=m=hx= + ` + 0)

=) 4 x=(‘8 x7+ = x8 + = x ‘ 0)

8) y=(‘7w= y + 0; x= + 7= xyz 8)

1) 0= x=

(8 x1

‘ 7 x8

+ 1 x ‘ 4)7) 8 y(‘8`m8 + 1my + = yz )

4) =e=l(‘:` + 2ml ‘ 0;ez = + 00e)

9) ‘4(`8m= z ‘ =`= z 8 ‘ =`m8h ‘ 1mh)

:) 7e=(0;`=el1 ‘ m=el + 8ez ‘ 7l8r )

Gcgrhbhbf = 0) (8`= ‘ 7 x)(=` + mx)

=) (8 x= + 4m)(= y ‘ 0)

8) (= y= + m)(9`= y+1mx=)

1) (1 xy= ‘ 8 z )(‘ m= + 1`)

7) (8`= ‘ 8m)(8 x= + y=)

4) (=e ‘ l)( =l

8

y + 2e

=

x ‘ 7m) 9) (‘8l8 + ez )(=e= y + 7m)

:) ( x + = y)(00`8 x ‘ 2 y= z )

2) (8e= ‘ `z )(8l8 ‘ :è)

0;) (my ‘ =)(‘7m8r + 2 x)

00) (7 x= + 4 s)(8rx ‘ 7s8 y + 1 z )

0=) (‘ z = + = y)(1 xy= ‘ 7 z + 8w)

Gcgrhbhbf 8 0) (` ‘ m)=

=) ( x + 9)=

8) (` ‘ 7)=

1) ( x ‘ 8)=

7) (7`= ‘ 8m)=

4) (`8 + = x)=

9) (7 x= + 1)=

:) ( x ‘ 0;)=

2) ( x8 + 7)=

0;) (2 z ‘ 0;)=

00) (8`

1 y4 ‘ 0; x

7)=

0=) (` ‘ 0=e= x=)=

Gcgrhbhbf 1 0) ( x8 + =)( x8 ‘ =)

=) (m + 0)(m ‘ 0)

8) (1 + e=)(1 ‘ e=) 1) (= + x)(= ‘ x)

7) (7`7 y= + 9m4 x8)(7`7 y= ‘ 9m4 x8)

4) (1 x= y1 + 0;)(1 x= y1 ‘ 0;)

9) (4 x + `8m=)(4 x ‘ `8m=)

:) (2m= x + 4`h)(2m= x ‘ 4`h)

2) (9 x= y + =`=m)(9 x= y ‘ =`=m)

0;) (`

4 +

x8

7 )(`

4 ‘

x8

7 )00) (0

= + 8`)(0

= ‘ 8`)

0=)

(0 +

`

7 )(0 ‘

`

7 )Ags`rrfnng lugv`s j`mbnbaàgs[gråegtrfs3@ 5 x + x + 7 + 7 5 = x + 0;

M 5 x + x + y + y 5 = x + = y H 5 7 + 7 + 7 + 7 5 =;

A 5 y + y + 7 + 7 5 = y + 0;

Àrg`s3@ 5 7 x

M 5 xy

H 5 7 • 7 5 =7A 5 7 y




Xge`l` 01¡@ tr`m`c`r!X 5

(0 + 0;;)0;;=

5 0; ;;;=

5 7;7;

Gcgrhbhbf 0;) (x ‘ 1) (x + 0)

0) ( x + =)( x + 0)

=) ( x ‘ 4)( x ‘ 7)

8) ( x ‘ 8)( x + 0)

1) ( x + =)( x + =)

7) ( x ‘ =0)( x ‘ 1)

4) ( x ‘ 7 y)( x ‘ 8 y)

9) ( x + 4 y)( x ‘ = y)

:) ( x ‘ 7 y)( x + y)

Gcgrhbhbf =;) = x

= + 9 x + 8 5

1 x

= + =(9 x) + 4 5

(= x)= + 9(= x) + 4 5

(= x + 4)(= x + 0) 5

( = x + 4

= )( = x + 0

0 ) 5 ( x + 8)(= x + 0)

0) 4 x= + 00 x + 8 5

84 x

= + 4(00 x) + 0: 5

(4 x)= + 00(4 x) + 0: 5

(4 x + 2)(4 x + =) 5

(

4 x + 2

8

)(

4 x + =

= ) 5 (= x + 8)(8 x + 0)

=) 4 x

= + 09 x + 7 5

84 x

= + 4(09 x) + 8; 5

(4 x)= + 09(4 x) + 8; 5

(4 x + 07)(4 x + =) 5

( 4 x + 07

8 )( 4 x + =

= ) 5 (= x + 7)(8 x + 0)

8) 0= x

= ‘ 08 x + 8 5

011 x

= ‘ 0=( x) + 84 5

(0= x)= ‘ (0= x) + 84 5

(0= x ‘ 2)(0= x ‘ 1) 5

(0= x ‘ 2

8 )( 0= x ‘ 1

1 ) 5 (1 x ‘ 8)(8 x ‘ 0)

Gcgrhbhbf 8;) w

8 + x

8 5

(w + x)_(w)

= ‘ (w • x) + ( x)=U 5 (w + x )(w

= ‘ wx + x =)

0) y 8 + z 8 5

( y + z )_( y) = ‘ ( y • z ) + ( z )=U 5 ( y + z)( y

= ‘ yz + z=)

=) x 8 + : 5

( x + =)_( x) = ‘ ( x • =) + (=)=U 5 ( x + =)( x = ‘ = x + 1)

8) =9 + j 8 5

(8 + j)_(8) = ‘ (8 • j) + (j)=U 5 (8 + j)(2 ‘ 8j + j=)

1) : + :w 8 5

(= + =w)_(=) =

‘ (= • =w) + (=w)=

U 5 (= + =w)(1 ‘ 1w + 1w=

)7) x 8 + 41 5

( x + 1)_( x) = ‘ ( x • 1) + (1)=U 5 ( x + 1)( x = ‘ 1 x + 04)

Gcgrhbhbf 1;) x8 ‘ 0 5

( x ‘ 0)_( x)= + ( x • 0) + (0)=U 5 ( x ‘ 0)( x = + x + 0)

0) w8 ‘ x8 5

(w ‘ x)_(w)= + (w • x) + ( x)=U 5 (w ‘ x )(w= + wx + x =)

=) y8 ‘ : z 8 5

( x ‘ = z )_( y)= + ( y • = z ) + (= z )=U 5 ( y ‘ = z )( y= + = yz + 1 z =)

8) : x8 ‘ 41 5 (= x ‘ 1)_(= x)= + (= x • 1) + (1)=U 5 (= x ‘ 1)(1 x = + : x + 04)

1) =9 ‘ y8 5

(8 ‘ y)_(8)= + (8 • y) + ( y)=U 5 (8 ‘ y)(2 + 8 y + y=)

7) 0=7 ‘ :w8 5

(7 ‘ =w)_(7)= + (7 • =w) + (=w)=U 5 (7 ‘ =w)(=7 + 0;w + 1w=)

Ags`rrfnng lugv`s j`mbnbaàgs0) Àrg` 5 x= ‘ 04 x + 1: 5 ( x ‘ 1)( x ‘ 0=)

Gn ftrf nàf agn rghtàliunf ebag x ‘ 0=

=) @ tft n 5=1 x

= + 1: x

= 5 9= x

=

@ tft n 5 : x(2 x)

Gn ftrf nàf agn rghtàliunf ebag 2 x

8) @ 5 7 x= + 07 xy + 4 y= 5 ( x + = y) (7 x + 8 y)

Gn ftrf nàf agn røemf ebag 7 x + 8 y




Xge`l` 07¡@ tr`m`c`r!0) 7

0= 8) :

7

=) 0;

2 1) 7

0:

Gcgrhbhbf 0 0) x= ‘ 7 x + 4

=`x ‘ 4` 5

( x ‘ 8)( x ‘ =)

=`( x ‘ 8) 5 ( x ‘ 8)( x ‘ =)

=`( x ‘ 8) 5 x ‘ =

=`

=) z = ‘ 7 z + 4

z = ‘ 9 z + 0= 5 ( z ‘ 8)( z ‘ =)

( z ‘ 1)( z ‘ 8) 5 ( z ‘ 8)( z ‘ =)

( z ‘ 1)( z ‘ 8) 5 z ‘ =

z ‘ 1

8) :`8 + =9

1`= + 0=` + 25 (=` + 8)(1`= ‘ 4` + 2)

(=` + 8)= 5

(=` + 8)(1`=

‘ 4` + 2)(=` + 8)= 5 1`

=

‘ 4` + 2=` + 8

1) `= ‘ 1`m + 1m=

`8 ‘ :m8 5 (` ‘ =m)=

(` ‘ =m)( = + =`m + 1m=)5

( ‘ =m)=

( ‘ =m)( = + =`m + 1m=) 5 ` ‘ =m

`= + =`m + 1m=

7) y: ‘ =7 y

= y8 + : y= ‘ 0; y 5 y( y9 ‘ =7)

= y( y= + 1 y ‘ 7) 5

y( y9 ‘ =7)

= y( y

= + 1 y ‘ 7) 5 y9 ‘ =7

=( y= + 1 y ‘ 7) 5 y9 ‘ =7

= y= + : y ‘ 0;

4) x8

‘ 1 x=

‘ 0= x x8 ‘ 8 x= ‘ 0; x

5 x ( x + =)( x ‘ 4) x ( x + =)( x ‘ 7)

5 x( x + =)( x ‘ 4) x( x + =)( x ‘ 7)

5 x ‘ 4 x ‘ 7

9) w= ‘ 8w

w= + 8w 5

w(w ‘ 8)

w(w + 8) 5 w(w ‘ 8)

w(w + 8) 5 w ‘ 8

w + 8

:) 4 x= + 7 x ‘ 4

07 x= ‘ 9 x ‘ = 5 (8 x ‘ =)(= x + 8)

(7 x + 0)(8 x ‘ =) 5

(8 x ‘ =)(= x + 8)

(7 x + 0)(8 x ‘ =) 5 = x + 8

7 x + 0

2) x= + x ‘ =

x8 ‘ x= ‘ x + 0 5 ( x + =) ( x ‘ 0)

( x8 ‘ x=) ‘ ( x ‘ 0) 5 ( x + =) ( x ‘ 0)

x=( x ‘ 0) ‘ ( x ‘ 0) 5

( x + =)( x ‘ 0)

( x= ‘ 0)( x ‘ 0) 5 ( x + =)( x ‘ 0)

( x ‘ 0)( x + 0)( x ‘ 0) 5 ( x + =) ( x ‘ 0)

( x + 0)( x ‘ 0)=

( x + =) ( x ‘ 0)

( x + 0)( x ‘ 0)= 5 x + =

( x + 0)( x ‘ 0)

0;) = x= + 9 x + 8

= x= ‘ 9 x ‘ 1 5 (= x + 0)( x + 8)

(= x + 0)( x ‘ 1)5

(= x + 0)( x + 8)

(= x + 0)( x ‘ 1)5 x + 8

x ‘ 1

Gcgrhbhbf = 0) x + 8

x ‘ =•

x= + = x ‘ :

x= ‘ 2 5 ( x + 8)( x + 1)( x ‘ =)

( x ‘ =)( x ‘ 8)( x + 8) 5

( x + 8)( x + 1)( x ‘ =)( x ‘ =)( x ‘ 8)( x + 8)

5 x + 1 x ‘ 8

=) x= ‘ = x + 0

x= ‘ 2 •

x= + x ‘ 4

x= ‘ 0 5 ( x ‘ 0)=( x + 8)( x ‘ =)

( x + 8)( x ‘ 8)( x + 0)( x ‘ 0)5

( x ‘ 0)=( x + 8)( x ‘ =)

( x + 8)( x ‘ 8)( x + 0)( x ‘ 0)5 ( x ‘ 0)( x ‘ =)

( x ‘ 8)( x + 0)

8) e + l

el ‘ l= •

l=

e= ‘ l= 5

l=(e + l)

l(e ‘ l)(e ‘ l)(e + l)

l=(e + l)

l(e ‘ l)(e ‘ l)(e + l) 5 l

(e ‘ l)=

1) x= ‘ = x

x= ‘ 7 x + 4 •

x= + 1 x + 1

x= ‘ 1 5 x( x ‘ =)( x + =)=

( x ‘ =)( x ‘ 8)( x ‘ =)( x + =)5

x( x ‘ =)( x + =)=

( x ‘ =)( x ‘ 8)( x ‘ =)( x + =) 5 x ( x + =)

( x ‘ 8)( x ‘ =)

7) xy ‘ = y=

x= + xy •

x= + = xy + y=

x= ‘ = xy 5 y( x ‘ = y)( x + y)=

= x( x + y)( x ‘ = y)5

y( x ‘ = y)( x + y)=

= x( x + y)( x ‘ = y)5 y( x + y)

= x

4) 2 ‘ 4 x + x=

x= ‘ 2 •

x= ‘ 7 x + 4

8 x= ‘ 2 x 5 ( x ‘ 8)=( x ‘ =)( x ‘ 8)

8 x( x ‘ 8)( x + 8)( x ‘ 8) 5

( x ‘ 8)=

( x ‘ =)( x ‘ 8)8 x( x ‘ 8)( x + 8)( x ‘ 8) 5 ( x ‘ =)( x ‘ 8)

8 x ( x + 8)

9) x= ‘ 1 xy + 1 y=

x= ‘ = xy •

x=

x= ‘ 1 y

= 5

x =( x ‘ = y)=

x ( x ‘ = y)( x ‘ = y)( x + = y)5

x=( x ‘ = y)=

x( x ‘ = y)=( x + = y)5 x

x + = y

:) =` ‘ =

=`= ‘ 7; •

`= ‘ 1` ‘ 7

8` + 8 5 =( ‘ 0)(` + 0)( ‘ 7)

4( ‘ 7)( + 7)( + 0)5

=( ‘ 0)( + 0)(` ‘ 7)

4( ‘ 7)( + 7)( + 0)5 ` ‘ 0

8(` + 7)

2) x= ‘ 2 x= ‘ 1

• x= ‘ 9 x + 0; x= ‘ : x + 07

5 ( x ‘ 8)( x + 8)( x ‘ =)( x ‘ 7)( x ‘ =)( x + =)( x ‘ 8)( x ‘ 7)

5

( x ‘ 8)( x + 8)( x ‘ =)( x ‘ 7)

( x ‘ =)( x + =)( x ‘ 8)( x ‘ 7)5 x + 8

x + =

0;) = x= ‘ 8 x ‘ =

4 x + 8 •

8 x + 4

x= ‘ 1 5 8(= x + 0)( x ‘ =)( x + =)

8(= x + 0)( x ‘ =)( x + =) 5

8(= x + 0)( x ‘ =)( x + =)

8(= x + 0)( x ‘ =)( x + =)5 0




Gcgrhbhbf 80) x8 ‘ x

= x= + 4 x ù 7 x= ‘ 7 x

= x + 4 5 x8 ‘ x

= x= + 4 x • = x + 4

7 x= ‘ 7 x 5

= x ( x ‘ 0)( x + 0)( x + 8)0; x =( x + 8)( x ‘ 0)

5 = x( x ‘ 0)( x + 0)( x + 8)0; x=( x + 8)( x ‘ 0)

5

x + 0

7 x

=) : x= + =4 x + 07

04 x= ‘ 2 ù 4 x= + 08 x ‘ 7

2 x= ‘ 05

: x= + =4 x + 07

04 x= ‘ 2 • 2 x= ‘ 0

4 x= + 08 x ‘ 75

(1 x + 8)(= x + 7)(8 x ‘ 0)(8 x + 0)

(1 x ‘ 8)(1 x + 8)(8 x ‘ 0)(= x + 7) 5

(1 x + 8)(= x + 7)(8 x ‘ 0)(8 x + 0)

(1 x ‘ 8)(1 x + 8)(8 x ‘ 0)(= x + 7) 5 8 x + 0

1 x ‘ 8

8)=; x= ‘ 8; x

07 x8 + 07 x= ù1 x ‘ 4

x + 0 5

=; x= ‘ 8; x

07 x8 + 07 x= • x + 0

1 x ‘ 4 5

0; x (= x ‘ 8)( x + 0)

8; x =( x + 0)(= x ‘ 8) 5 0; x(= x ‘ 8)( x + 0)

8; x=( x + 0)(= x ‘ 8) 5 0

8 x

1)`x= + 7

1`= ‘ 0 ù

`8 x= + 7`=

=` ‘ 0 5

`x= + 7

1`= ‘ 0 •

=` ‘ 0

`8 x= + 7`= 5

(`x = + 7)(=` ‘ 0)

`=(= ‘ 0)(=` + 0)(`x = + 7) 5 (`x= + 7)(=` ‘ 0)

`=(= ‘ 0)(= + 0)(`x= + 7) 5

0

`=(=` + 0)

7)`= ‘ 4 + 7

`= ‘ 07 + 74 ù

`= + = ‘ 87

`= ‘ 7 ‘ =1 5

`= ‘ 4 + 7

`= ‘ 07 + 74 •

`= ‘ 7 ‘ =1

`= + = ‘ 87 5

( ‘ 7)( ‘ 0)( ‘ :)( + 8)

( ‘ :)( ‘ 9)( + 9)(` ‘ 7) 5

( ‘ 7)( ‘ 0)( ‘ :)( + 8)

( ‘ :)( ‘ 9)( + 9)(` ‘ 7) 5

(` ‘ 0)(` + 8)

(` ‘ 9)( + 9)

4)`= ‘ 4`

`8 + 8`= ù`= + 8` ‘ 71

`= + 2` 5

`= ‘ 4`

`8 + 8`= • `= + 2`

`= + 8` ‘ 71 5

`=(` ‘ 4)(` + 2)

`=(` + 8)(` + 2)(` ‘ 4) 5 `=(` ‘ 4)( + 2)

`=(` + 8)(` + 2)(` ‘ 4) 5

0` + 8

Ags`rrfnng lugv`s j`mbnbaàgsN` rga`hhbøl pugag v`rb`r, n` bag` prblhbp n agmg e`ltglgrsg3

N` `sbstgltg gstà ebltbglaf pfrqug n`s pàibl`s 022 y =;;hfrrgspflagl ul` ebse` jfc`.

Xge`l` 04¡@ tr`m`c`r!0) 4

= 5 8 1) 8 + =

45 7

4

=) 4

9 7) 0; ‘ 8 + 9

45 01

4 5 9

8

8) ‘

=

2 4) 04 ‘ 2 ‘ 0;

=1 5 ‘

8

=1 5 ‘

0

:

Gcgrhbhbf 00)

x

x + = +

4 x

x + = 5

x + 4 x

x + =5

9 x

x + =

=)7 x= ‘ 1

: x8 +

x= + 1

: x8 5

7 x= ‘ 1 + x= + 1

: x8 5

4 x=

: x85

8

1 x

8)4 x=

4 x ‘ 9 ‘

9 x

4 x ‘ 9 5

4 x= ‘ 9 x

4 x ‘9 5

x(4 x ‘ 9)

4 x ‘ 9 5 x

1) x= + 8 x

x= ‘ x ‘ 4 ‘

x= ‘ 4

x= ‘ x ‘ 4 5

x= + 8 x ‘ x= + 4

x= ‘ x ‘ 4 5

8 x + 4

x= ‘ x ‘ 45

8( x + =)

( x ‘ 8)( x + =)5

8

x ‘ 8

Gcgrhbhbf =

0)8 x + 0

7 x +

x ‘ =

0; x 5

(8 x + 0)(=) + ( x ‘ =)(0)

0; x5

4 x + = + x ‘ =

0; x5

9 x

0; x 5

9

0;

=) x + 8

8 x

‘7 x ‘ 0

7 x=

5( x + 8)(7 x) ‘ (7 x ‘ 0)(8)

07 x=

5

7 x= + 07 x ‘ 07 x + 8

07 x= 5

7 x = + 8

07 x =

Gcgrhbhbf 8

0)

x + 0

0

0 + x

0

5

x + 0

0

0 + x

0

50( x + 0)

0(0 + x)5

0( x + 0)

0( x + 0)5 0

=) x ‘

1

x

x + 1 +1

x

5

x

0 ‘

1

x x

0

+1

0

+1

x

5

x= ‘ 1

x

x= + 1 x + 1

x

5

x( x= ‘ 1)

x( x= + 1 x + 1)5

( x + =)( x ‘ =)

( x + =)( x + =) 5

x ‘ =

x + =

8) x +

00

x +

4

x=

8 + 1

x ‘

1

x=

5

x

0 +

00

x +

4

x=

8

0 +

1

x ‘

1

x=

5

x8 + 00 x + 4

x=

8 x= + 1 x ‘ 1

x=

5

x=( x8 + 00 x + 4)

x=(8 x= + 1 x ‘ 1) 5

x 8 + 00 x + 4

8 x = + 1 x ‘ 1




Xge`l` 09

1)

`= ‘ m=

`

` + m

`8

5

`=( =) ‘ `=m=

`8

` + m

`8

5

`1 ‘ `=m=

`8

` + m

`8

5

`8( 1 ‘ `=m=)

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Q`nngr e`tgeàtbhf

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Q`nngr e`tgeàtbhf

]gp`sf p`r` n` prugm` ag iràu`lafs agn EBLGAWH

Gstbeà` (f) gstuab`ltg agn irupf [fnfhjbh3

Ng s`nuaèfs hfl h`rbðf y gspgrèfs qug gstâ gstuab`laf hfl gltusb`sef gstg prbegr ðfag m`hjbnngr`tf.

Gn Eblbstgrbf ag Gauh`hbøl rg`nbz` hà` `ðf prugm`s ag e`tgeàtbh` y ag hfeprglsbøl nghtfr`` tfafs nfs gstuab`ltgs qug gstàl hurs`laf gn óntbef ðf agn hbhnf màsbhf f gn óntbef ðf ag

abvgrsbhàf. Gn prfpøsbtf ag gst`s prugm`s gs fmtglgr bldfre`hbøl qug pgrebt` egcfr`r n`h`nbaà gauh`tbv` ag lugstrf p`ås.

Hflsghugltgs hfl gst` egcfr` gauh`tbv`, jgefs blhnubaf, gl gstg nbmrf, ul` sghhbøl nnèà`Q`nngr e`tgeàtbhf qug hfltbglg afs prugm`s, p`r` qug sg v`y` prgp`r`laf pfhf ` pfhf.

¶Puâ hfltglbafs sg rgp`s`l gl gstg t`nngr6

[rugm` @

[rgsglt` 88 åtges qug `m`rh`l nfs hfltglbafs ag n`s sge`l`s 0 ` n` 9 y tge`s ag prfpfrhbf-l`nbaà, sghuglhb`s lueârbh`s, fpgr`hbflgs hfl nfs lóegrfs r`hbfl`ngs, sbilfs ag hfep`r`-hbøl, e.h.e., E.H.A., pfrhglt`cg y egaba`s ag vfnuegl.

[rugm` M

[rgsglt` 8= åtges qug `m`rh`l nfs hfltglbafs ag n`s sge`l`s 2 ` n` 04 y tge`s ag ghu`hbf-lgs ag prbegr iràf, ngliu`cg nigmr`bhf, rgin` ag trgs sbepng y pgråegtrf ag huàrbnàtgrfs.

¶Høef sg gv`nu`ràl gstfs hfltglbafs6

• Gl n` prbegr` prugm` p`rhb`n sg gv`nu`ràl nfs hfltglbafs ag n` prugm` @.

• Gl n` prugm` l`n sg blhnuygl nfs hfltglbafs ag n` prugm` M.

Nfs tge`s qug lf `p`rghgl gl gstg nbmrf nfs glhfltr`rà gl gn e`tgrb`n ag e`tgeàtbh` ag nfs

irupfs Pubrbiuà, Wt`tnàl y S`hungu f tèmbâl pugag blvgstbi`r gl bltgrlgt.


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Blstruhhbflgs3

0. Wsg gst`s pàibl`s sfnf p`r` nggr n`s prgiult`s y glulhbàfs. Lf n`s sumr`yg,lb jì` `lft`hbflgs f e`rh`s gl gnn`s.

=. N` jfc` ag rgspugst`s sg glhugltr` n l`n ag h a` prugm . ]ghørtgn .

8. Ng` hà` prgiult` f glulhbàf y n s pfsbmngs rgspugst`s u fphbflgs qugn` rgspflagl.

1. Xgnghhbflg n` rgspugst` hfrrght` y rgnnglg gn hårhunf hfrrgspflabgltg gl n`jfc` ag rgspugst`s. Xfnègltg ul` fphbøl rgspflag hfrrghtègltg ` n`prgiult`.

7. Lf `rruiug lb jì` tr`zfs gl ftrfs nui`rgs ag n` jfc` ag rgspugst`s.

4. [`r` rgspflagr gst` prugm` agmgrà utbnbz`r n`pbhgrf lgirf.

9. Wtbnbhg jfc s `p rtg p`r` rg`nbz`r gn prfhgabebgltf ag n`s fpgr`hbflgs.

:. Eba` gn tbgepf qug t`ra` gl rgsfnvgr n` prugm`. [r`htbqug j`st` qug hflsbi`j`hgrnf gl gn eglfr tbgepf pfsbmng.

Iuågsg hfl gn gcgepnf agn rghuàrf.

;) ¶Huàn gs gn rgsuntàf ag fpgr`r ‘0;= + 1;;6

`. =2:m. 1=;h. 8:;a. 10;=

Jfc` ag rgspugst`s

;. `. m. h. a.

0. `. m. h. a.

=. `. m. h. a.

Ag gst` dfre`sg agmg rgnngl`rgn hårhunf.

Gl su jfc` ag rgspugst`sagmgrà rgnngl`r gn hårhunfag n` rgspugst` hfrrght`.

@jfr` rgvbsg su jfc` ag rgspugst`s, n` fphbøl ; y` gstà e`rhà`.

Óntbe`s rghfegla`hbflgs3

• Xb lf s`mg ul` rgspugst`, lf sg agtgli`, sbi` àgn`ltg.

• Gstà prfjbmbaf gn usf ag h`nhunàfr`.

Blstruhhbflgs3 E`rqug gl n` jfc` ag rgspugst`s gn rgsuntàf hfrrghtf ag n`fpgr`hbøl.


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0) ¶Puâ lfemrg rghbmg gn hflcultf ag nfs lóegrfs gltgrfs pfsbtbvfs ag hgrf j`st` bllbtf6

`. l`tur`ngs m. r`hbfl`ngs h. lgi`tbvfs a. brr`hbfl`ngs

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`. ; m. ‘0 h.0

` a. 0

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`. 9 m. 0; h. ;.=7 a.7

1

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`. 0;; m. 0; h. 0 a. ;

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`. |‘07| 5 ; m. |‘07| 5 0 h. |‘07| 5 07 a. |‘07| 5 ‘07

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0 h. 0

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7 5

7

1 m.

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1 5

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: h.

7

8 5

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`. 0; m. 04 h. 8= a. 41

00) ¶Huàn gs n` pftglhb` ag (=9)= gxprgsà` gl m`sg 86

`. 84 m. 87 h. 81 a. 88

0=) ¶Huàn gs gn lóegrf eàs pgqugðf ag n` sgrbg ag lóegrfs rg`ngs 8.9, ‘:.4, ‘1.9, 2.8, ‘0.:6

`. 8.9 m. ;.0 h. ‘1.9 a. ‘:.4



08) Gl n` sgrbg3 1, 9, 00, 04, ¶huàn gs gn lóegrf sbiubgltg6

`. 02 m. =; h. =0 a. ==

01) Gl n` sgrbg3 0;, 0:, 81, 44, ¶huàn gs gn lóegrf sbiubgltg6

`. 0=: m. 0=2 h. 08; a. 080

07) ¶Huàn gs n` rgspugst` hfrrght` ag n` fpgr`hbøl =; ù 1 + 8 • = ‘ 06

`. 0; m. 07 h. =; a.=;

08

04) ¶Puâ lóegrf gs gn e.h.e. ag 0;, 07, =;6

`. 7 m. 8; h. 4; a. 0=;

09) ¶Huàn gs gn E.H.A. ag 0=, 0:, =16

`. 0 m. = h. 8 a. 4

0:) ¶Puâ agshfepfsbhbøl gl d`htfrgs prbefs gs hfrrght`6

`. =1 5 1 • 4 m. =1 5 : • 8 h. = • 8 • 1 a. =1 5 = • = • = • 8

02) ¶Huàn ag n s sbiubgltgs `re`hbflgs gs vgraàgr`6

`. 8 • _ 07 ‘ = • (4 ‘ = • =) U 5 ‘8 h. 8 • _ 07 ‘ = • (4 ‘ = • =) U 5 ‘82

m. 8 • _ 07 ‘ = • (4 ‘ = • =) U 5 88 a. 8 • _ 07 ‘ = • (4 ‘ = • =) U 5 08

=;) ¶Huàn ag n s gxprgsbflgs sbiubgltgs gs hfrrght`6

`. 8 < ‘8 m. 8 > ‘8 h. ‘8 < 8 a. ‘8 < ‘8

=0) ¶Huàn gs n` rgspugst` hfrrght` ag gngv`r ( 0;;

1 ); 6

`. =7 m. 0; h. 0 a. ;

==) ¶Huàn ag n s sbiubgltgs fpgr`hbflgs gs hfrrght 6

`. 1= • 18 5 044 m. 1= • 18 5 047 h. 1= • 18 5 17 a. 1= • 18 5 14

=8) ¶Puâ gxprgsbøl ag n`s sbiubgltgs gs gqubv`ngltg ` 2= ù 8=6

`. (2 ù 8)= ‘ = m. (2 ù 8)= ù = h. (2 ù 8)= a (2 ù 8)= • =

=1) ¶Puâ rgsuntàf sg fmtbglg `n sue`r8

: +

0

16

`.1

: m.

7

: h.

1

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1

0=


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=7) ¶Huàn gs gn rgsuntàf agn prfauhtf7

= •

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1 ag i`nøl ag pbltur` `zun,

0

= ag i`nøl ag pbltur` mn`lh` y

=

8 ag i`nøl ag pbltur` rfc`. ¶Huàltfs i`nflgs ag pbltur` sg utbnbz`rfl gl tft`n6

`. 8 i`nflgs `. 7 i`nflgs h.0:

7 ag i`nflgs a.

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0= ag i`nflgs

=2) Mglcèål tbglg trgs rfnnfs ag nèmrg hfl ul n`rif ag 07, =; y =7 egtrfs hà` ulf. Xb agsg` abvbabrtfaf gn `nèmrg gl pbgz`s biu`ngs, ag n` e`yfr nflibtua pfsbmng p`r` lf agspgrabhb`r `nèmrg, ¶quân`rif agmg tglgr hà` pbgz`6

`. 0 e m. 8 e h. 7 e a. 07 e

8;) ]gibl` vbsbt` ` su jgre`lf [garf hà` 4 aå`s y ` su jgre`l` ]fs`rbf hà` 1, sb jfy vbsbtø ` èmfs,¶huàltfs aå`s agmgl p`s`r p`r` qug vbsbtg ` nfs afs gn ebsef aå`6

`. 1 a. 4 h. 0= a. 0:

80) Afs pbltfrgs t`ra`l 2 aå`s gl pblt`r ul` h`s`. Xb hfltr`t`l ul tgrhgr pbltfr, ¶huàltfs aå`s t`ra`ràlgl pblt`r n` h`s` nfs trgs cultfs6

`. 8 aå`s m. 1 aå`s h. 7 aå`s a. 4 aå`s

8=) Gl ul hårhunf ag gstuabf ag BIG] nfs gstuab`ltgs ag màsbhf y m`hjbnngr`tf sue`l 0=;, ag nfs hu`ngs gn 9; % sfl eucgrgs y gn rgstf, jfemrgs. Ag nfs jfemrgs, gn =7 % gstuab` m`hjbnngr`tf. ¶Huàltfs jfemrgsgstuab`l m`hjbnngr`tf6

`. : m. 2 h. 0; a. 07

88) Gl ul rghbpbgltg j`y 7;;; he8 ag ul` sfnuhbøl. Xb 0 nbtrf gs biu`n ` 0;;; he8, ¶huàltfs nbtrfs tglgefsag n` sfnuhbøl6

`. 7 nbtrfs m. 8 nbtrfs h. 1 nbtrfs a. 4 nbtrfs







Q`nngr e`tgeàtbhf

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0) ¶Puâ rgsuntàf sg fmtbglg `n fpgr`r 7(= + 8 7 )6

`. =7 7 m. 9 + : 7 h. 0; + 07 7 a. 0; + 8 7

=) ¶Huàn gs n` hflst`ltg lueârbh` gl n` gxprgsbøl 7 xy86

`. x m. y h. 7 a. 8

8) ¶Puâ prfpbgaà eugstr` n` gxprgsbøl `(h + a ) 5 `h + à 6

`. hgrràur` m. abstrbmutbv` h. hfleut`tbv` a. gngegltf `msfrmgltg

1) ¶Puâ pfnblfebf rgprgsglt` gn pgråegtrf ag ul rghtàliunf qug ebag x + 0 ag `lhjf y = x ag n`rif6

`. 1 x m. 8 x + 0 h. 4 x + = a. ( x + 0) + (= x)

7) ¶Puâ gxprgsbøl gcgepnbh` hfrrghtègltg n` euntbpnbh`hbøl ag pftglhb`s ag biu`n m`sg6

`. è • `l 5 è + l h. è • `l 5 è • l

m. è • `l 5 è ‘ l a. è • `l 5 è ù l

4) ¶Huàn ag n`s `re`hbflgs sbiubgltgs gs vgraàgr`6

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m. 1 i • 1 i • 1 i 5 41 i 8 a. 1 i • 1 i • 1 i 5 1 i 8

9) ¶Høef sg gxprgs` 7 x= + 8 gl gn ngliu`cg usu`n6

`. Gn afmng ag ul lóegrf x eàs trgsm. Gn quåltupnf ag ul lóegrf x `n huàràf eàs trgsh. Wl lóegrf x `n huàràf eàs trgsa. Hblhf pfr x eàs trgs

:) ¶Huàn gs gn rgsuntàf hfrrghtf ag (: x= y8)(‘8 x4 y= z 8)6

`. ‘=1 x1 yz 8 m. ‘=1 x: y7 z 8 h. ‘00 x: y7 z 8 a. =1 x0= y4 z 8

2) ¶Huàltfs târeblfs dfre`l gn pfnblfebf =

8 z + 8 z 7 + 1 z = ‘ : y + 4 y=6

`. 7 m. 0; h. 00 a. 01

0;) ¶Høef sg gxprgs` gl ngliu`cg e`tgeàtbhf "Gn pgråegtrf ([) ag ul rghtàliunf gs gn afmng prfauhtfag n` sue` agn `lhjf (Y) y gn n`rif (N)"6

`. [ 5 =Y + N m. [ 5 Y + =N h. [ 5 =(Y + N) a. [ 5 =(Y • N)



00) ¶Huàn gs gn d`htfr hfeól gl n` gxprgsbøl = x8 ‘ 1 x= + : x6

`. = m. x h. x= a. = x

0=) ¶Huàn ag n`s gxprgsbflgs sbiubgltgs gs ul trblfebf huàràf pgrdghtf6

`. x= ‘ 04 m. x= ‘ 7 z + 01 h. x= ‘ : x + 41 a. x= + 0; x + =7

08) ¶Huàn gs gn rgsuntàf ag sbepnbh`r ‘8 x ‘ = + 8 x= y ‘ 1 + 7 x ‘ 4 x=6

`. 8 x= y ‘ : x= ‘ 4 m. 8 x= y ‘ 4 x= + = x ‘ 4 h. 8 x= y ‘ 4 x= + : x + 4 a. ‘9 x= y

01) ¶Puâ rgsuntàf sg fmtbglg ag (1 x= + 8 x + 0) + (9 x= ‘ 1 x + 4)6

`. 09 x m. 09 x= h. 0; x + 9 a. 00 x= ‘ x + 9

07) ¶Huàn gs n` abdgrglhb` ag (:`m= ‘ 4mh + 1) ‘ (=`m= ‘ 2mh + 0;)6

`. 4`m= + 8mh + 4 m. 4`m= + 8mh ‘ 4 h. 4`m= ‘ 8mh + 4 a. 4`m= ‘ 8mh ‘ 4

04) ¶Huàn gs gn rgsuntàf hfrrghtf ag (8 x4 y1)(‘1 x8 y= z 1)6

`. 0= x8 y= z 1 m. ‘0= x2 y4 z 1 h. ‘0= x= y= z 1 a. ‘9 x2 y4 z 1

09) ¶Huàn gs gn rgsuntàf hfrrghtf ag euntbpnbh`r ( x + 2)( x ‘ =)6

`. x= ‘ 0: m. = x + 9 h. x= + 9 x ‘ 0: a. x= ‘ 9 x + 0:

0:) ¶Puâ mblfebf gstà gxp`labaf hfrrghtègltg6`. (j ‘ o )= 5 j= ‘ o h. (q ‘ =)= 5 q= ‘ 1q + 1

m. ( x + 7)= 5 x= + =7 a. (m ‘ 1)= 5 m= + :m ‘ 04

02) ¶Huàn gs gn rgsuntàf ag abvbabr (07 x1 + 0= x8 ‘ 2 x=) ù 8 x6

`. 7 x8 + 1 x= ‘ 8 x m. 7 x1 + 1 x8 ‘ 8 x= h. 07 x7 + 1 x1 ‘ 8 x8 a. 07 x8 + 07 x= ‘ 07 x

=;) ¶Huàngs sfl nfs d`htfrgs ag x= ‘ 0= x + =96

`. ( x + :)( x ‘ 1) m. ( x ‘ 4)( x + 4) h. ( x ‘ 2)( x ‘ 8) a. ( x + 1)( x + 8)

=0) ¶Puâ rgsuntàf sg fmtbglg `n v`nu`r gn pfnblfebf = x= + 07 sb x 5 86

`. 88 m. =4 h. =0 a. 09

==) ¶Puâ v`nfr sg fmtbglg ag =`h • =`m sb ` 5 =, m 5 2 y h 5 0

86

`. 1 m. : h. =1 a. 0;

8


http://slidepdf.com/reader/full/libro-polochic-matematica-1er-sem 322/32680; BIG] ∝ [fnfhjbh

=8) ¶Puâ rgsuntàf sg fmtbglg `n sbepnbh`r0;w8 z 4

7w= z 6

`. 7wz 7 m. =wz 7 h. =w8/= z 4/0 a. 7w8/= z 4/0

=1) ¶Huàn gs n` rgspugst` hfrrght` ag x= + 4 x + :

x + =6

`. x + = m. x + 1 h. x ‘ = a. x ‘ 1

=7) ¶Huàn ag n s sbiubgltgs fpgr`hbflgs gs hfrrght 6

`. ( y ‘ 1)= 5 ‘04 y= h. ( y ‘ 1)= 5 y= ‘ 04

m. ( y ‘ 1)= 5 y= ‘ : y + 04 a. ( y ‘ 1)= 5 y= + 04

=4) ¶Huàn gs gn v nfr ag x gl n` ghu`hbøl = x ‘ 8 5 786

`. = m. 4 h. 0; a. =:

=9) ¶Huàn gs gn v nfr ag x gl n` ghu`hbøl 7 x + 4 5 0; x + 76

`. 0 m. = h. 0

= a. 0

7

=:) Gl agtgreblàf aå`, n` tgepgr`tur` gl Pugtz`ntgl`lif ` n`s 83;; ag n` e`ð`l` gr` ag ‘1 »H. Nugifn` tgepgr`tur` ùegltø 0 »H hà` jfr`, ¶huàn gr` n` tgepgr`tur` ` n`s 0;3;; ag n` e`ð`l` gsg aå`6

`. ; »H m. 8 »H h. 9 »H a. ‘=

=2) Wl` ltgl` ag 0= egtrfs ag `ntf prfyght` ul` sfemr` ag : egtrfs. ¶Huàn sgrà n` `ntur` ag ul gabhbf

qug ` n` ebse` jfr` prfyght` ul` sfemr` ag 4 egtrfs6`. 2 egtrfs m. 0; egtrfs h. 07 egtrfs a. =; egtrfs

8;) Qfafs nfs aå`s, X`lar` hèbl` 0

= oe pfr n` e`ð`l` y 8

1 ag oe pfr n` t`rag. ¶Huàltfs obnøegtrfs

hèbl` hà` aå`6

`. 8

= oe m. 1

4 oe h. 7

1 oe a. 8

: oe

80) Gl ul e`p` hà` 1 he rgprgsglt`l =0; oe ag abst`lhb` rg`n. Xb afs hbuaàgs gstàl sgp`rà`s pfr 0; he gl gn e`p`, ¶quâ abst`lhb` rg`n j`y gltrg gnn`s6

`. 1=; oe m. 7=7 oe h. :4; oe a. 0;7; oe

8=) Wl hèblf hfeul n gs hflstrubaf pfr 4; tr`m`càfrgs gl 7; aå`s. ¶Huàltfs tr`m`càfrgs sg jumbgr`llghgsbtàf p`r` hflstrubr gn ebsef hèblf gl =; aå`s6

`. 0;; tr`m`càfrgs m. 00; tr`m`càfrgs h. 0=; tr`m`càfrgs a. 07; tr`m`càfrgs







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Libro Polochic Matemática 1er. Sem