cmnynkt; 01 emeronti2 tyeck nig BhuKuNrYm 08 emeronti3 KIM SOKUN RbPaK 27 emeronti5 cmnyntspak 38 emeronti6 PaKry 43 emeronti7 rgval; ;rgval

Átírás

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2 ;; ;; ;; ; ; emeronti cmnynkt; 0 emeronti tyeck nig BhuKuNrYm 08 emeronti tyeck nig BhuKuNrYm 6 emeronti TI4 RbPaK 7 emeronti5 cmnyntspak 8 emeronti6 PaKry 4 emeronti7 rgvas; ;rgval rgval; 46 emeronti8 kensambicknit 49 emeronti9 smikardwerktimyymanmyygbaøat 56 emeronti0 vismpab 7

3 emeronti lmhat;!>k-rkcmnynkt; n Edl 4 < n < 67. x-cmnynkt;ess a Edl 55 a < KNnaRbmaNviFIénkenSamxageRkam ³ k> x> ( ) K> 56 8 (47 7) 5 cmnynkt; X> [ (47 ) 6] g> #> enagnuvitüal½ymyymansissrbus ^&( nak; nig sissrsi %&*nak; kñúgmyyqñamsissedlganesovepaenakñúgbnñal½y)an $ k,al b ercincag $k,almancmnyn *@$ nak;. etisissedl)anganesovepaenakñúgbnñal½yticcag $ k,alman cmnynb:unµannak;? $> b unña)antijepøemon % cegáam EdlkñúgmYycegáam man!0 Epø. Kat;)anjaMGs; * Epø ehiyepøemonedlenasl;)anebgeck[b Ún² Kat; ^ nak;. etib ÚnKat;mñak;²TTYl)anemonb:unµanEpø? - -

4 emeronti cmnynkt; %> enakñúgfñak;eronmyymansissrbus!* nak; nig sissrsixøh. ehiyesovepaedl )antijtamggs;mancmnyn &^ k,al. rkcmnynsissrsi? ^>bgát;cmnynxagerkamykrtwmxþg;db; xþg;ry nig xþg;ban; ³ k>!$((0% x> #*$000 X> ^&%$%%0 &>efvikar):an;sµantémøén ³ k> x> K> *> KNnatémøén k> 5 x> 5 K> 8 X> x ebi x 5 (>srmyl k> 9 6 x> 4 4 K> X> 8 - -

5 emeronti cmnynkt; dmenahrsay!>k-rkcmnynkt; nedl 4 < n < 67 cmnynkt; n EdlenAcenøaH 4 nig 67 KW ³ 44,45,46,47,48, 49 50,5,5,5,54,55,56,57,58,59,60,6,6,6,64,65,66. x-cmnynkt;ess a Edl 55 a < 65 cmnynkt;ess a EdlenAcenøaH 55 nig 65 KW,57,59,6, KNnaRbmaNviFIénkenSamxageRkam ³ k> ( ) 0 (0 0 4) x> ( ) (00 5) K> 56 8 (47 7)

6 emeronti cmnynkt; X> [ (47 ) 6] [ ] (69 4) 65 0 g> #>cmnynsissedl)anganesovepaenakñúgbnñal½yticcag $ k,alman cmnyn ³ nak;. $> b unña)antijepøemon % cegáam EdlkñúgmYycegáam man!0 Epø. Kat;)anjaMGs; * Epø ehiyepøemonedlenasl;)anebgeck[b Ún² Kat; ^ nak;. etib ÚnKat;mñak;²TTYl)anemonb:unµanEpø? -cmnynepøemonsrub ³ Epø -cmnynepøemonenasl; ³ Epø -cmnynepøemonedlb Ún²TTYl)an ³ 4 6 7Epø

7 emeronti cmnynkt; %> enakñúgfñak;eronmyymansissrbus!* nak; nig sissrsixøh. ehiyesovepaedl )antijtamggs;mancmnyn &^ k,al. rkcmnynsissrsi? -cmnynsissrsi ³ nak; ^>bgát;cmnynxagerkamykrtwmxþg;db; xþg;ry nig xþg;ban; ³ k>!$((0% -ykrtwmxþg;db;kw: ³ ykrtwmxþg;rykw ³ ykrtwmxþg;ban;kw ³ x> #*$000 -ykrtwmxþg;db;kw: ³ ykrtwmxþg;rykw ³ ykrtwmxþg;ban;kw ³ ykrtwmxþg;db;kw: ³ ykrtwmxþg;rykw ³

8 emeronti cmnynkt; -ykrtwmxþg;ban;kw ³ X> ^&%$%%0 -ykrtwmxþg;db;kw: ³ ykrtwmxþg;rykw ³ ykrtwmxþg;ban;kw ³ &>efvikar):an;sµantémøén ³ k> x> K> *> KNnatémøén k> x> K> X> x ebi x 5 ek)an x (>srmyl k>

9 emeronti cmnynkt; x> K> X>

10 emeronti lmhat;!>curbba ak;fa ³ k> plbuk 78 0 Eckdac;nwg x> pldk Eckdac;nwg K> plbuk Eckdac;nwg 5 tyeck nig BhuKuN X> pldk 50 4 Eckdac;nwg ekmanmyycmnynmanelxr)amxþg; 74 0 bmebjelxkñúgrbgb;edim,i[cmnynenaheckdac; ³ k> nwg pg nig 5 pg x> nwg pg nig 9 pg. #> ekmancmnyn 660, 540, 645 nig 60. eticmnynnaxøhcabhukunén 0? $> ekmancmnyn, 5, 7, 8, 9, 0,,, nig9 k>currktyeckéncmnynnimyy² x>eticmnynnaxøhcacmnynbfm %> bmebkcmnyn 6, 00,9, 5, 567, 980, 097, 506 nig 7008 CaplKuNktþabfm

11 emeronti ^>rktyeckrymfmbmput PGCD éncmnynxagerkam ³ k> 9 nig 5 x> 0 nig 08 K> 8 nig 4 X> 9, 60 nig 96 g>, 7 tyeck nig BhuKuN 48 nig c>, 68, 44 &>rkbhukunrymtucbmput ( PPCM b ) k> 8 nig 5 x> 0 nig 5 K>, 40 6 nig 5 LCM éncmnynxagerkam ³ 4 nig 45 X> 70, nig 8 g>, 8 5 nig 0 c>, 0 *> rkcmnyn x nig y edaydwgfa (x, y) y Edl 0 < x < y. (>rkcmnyn x EdltUcCageKbMputedaydwgfa PPCM (6, x) 4 88 nig 58 PGCD nig x 7-9 -

12 emeronti dmenahrsay!>curbba ak;fa ³ k> plbuk 78 0 Eckdac;nwg ekman cugerkaycacmnynku. tyeck nig BhuKuN Eckdac;nwg erbahcmnynenhmanelx x> pldk Eckdac;nwg ekman elxlmdab; Eckdac;nwg. Eckdac;nwg erbahplbuk K> plbuk Eckdac;nwg 5 ekman edayelx 5. Eckdac;nwg 5 erbahelxcugerkaybba b; X> pldk 50 4 Eckdac;nwg 9 ekman Eckdac;nwg 9 erbahplbukelx lmdab; Eckdac;nwg

13 emeronti tyeck nig ekmanmyycmnynmanelxr)amxþg; 74 0 bmebjelxkñúgrbgb;edim,i[cmnynenaheckdac; ³ k> nwg pg nig 5 pg edaycmnynenhmanelxcugerkaycaelx 0 enahvaeckdac;nwg ehiynwg 5 Canic. ducenhelxedlrtuvbmebjkñúgrbgb;enahkw ³ 0,,,,4,5,6,7,8,9. x> nwg pg nig 9 pg. edayplbukelxlmdab; edim,i[cmnyn 74 0 Eckdac;nwg pg nig 9 pgluhrtaet vaeckdac;nwg 9. ducenhelxedlrtuvbmebjkw 4. #> ekmancmnyn 660, 540, 645 nig 60. eticmnynnaxøhcabhukunén 0? cmnynedlcabhukunén 0 KW 660, 540. $> ekmancmnyn, 5, 7, 8, 9, 0,,, nig9 k>currktyeckéncmnynnimyy² -cmnyn mantyeck nig -cmnyn 5 mantyeck,, 5 nig 5 - -

14 emeronti tyeck nig BhuKuN -cmnyn 7 mantyeck nig 7 -cmnyn 8 mantyeck,, 6, 9 nig 8 -cmnyn 9 mantyeck nig 9 -cmnyn 0 mantyeck,, 4,5, 0 nig 0 -cmnyn mantyeck,, 7 nig -cmnyn mantyeck,, nig -cmnyn 9 mantyeck nig 9 x>eticmnynnaxøhcacmnynbfm %> bmebkcmnyn 6, 00,9, 5, 567, 980, 097, 506 nig 7008 CaplKuNktþabfm

15 emeronti tyeck nig BhuKuN ^>rktyeckrymfmbmput PGCD éncmnynxagerkam ³ k> 9 nig 5 PGCD (9,5 ) x> 0 nig 08 PGCD (0,08) K> 8 nig 4 PGCD (8, 4 ) X> 9, 60 nig 96 4 PGCD (9,60,96 ) g> 48, 7 nig PGCD (48,7, ) c> 6, 68, 44 nig 5 PGCD (6,68,44, 5 ) - -

16 emeronti tyeck nig BhuKuN &>rkbhukunrymtucbmput ( PPCM b LCM ) éncmnynxagerkam ³ k> 8 nig 5 PPCM (8, 5) x> 0 nig 5 05 PPCM (0, 5) K> 4, 40 nig PPCM (4,40, 45) X> 70, nig 8 PPCM (70,, 8 ) g> 5, 8 nig PPCM (5,8, 0 ) c> 88, 0 nig PPCM (88, 0,58) 640 *> rkcmnyn x nig y edaydwgfa (x, y) Edl 0 < x < y. eday PGCD (x, y) enah x p, y q Edl (p,q) PGCD. PGCD nig x y 7-4 -

17 emeronti tyeck nig BhuKuN ekman x y 7 ek)an p q 7 b p q 6 eday 0 < x < y enah 0 < p < q ektaj p, q 5 ducenh x, y 60. (>rkcmnyn x EdltUcCageKbMputedaydwgfa PPCM (6, x) 4 eday ducenh x 4 CatémøtUcCageKEdl PPCM (6, x)

18 emeronti cmnynkt;rwlatihv lmhat;!>sresrcmnynkt;rwulatib 7,5, 0,0,,,8,,9,, 4,. k>tamlmdab;elig edacmnuc A( 4),B( 6),C() nig D (5) elibnþat;cmnyn rycknna RbEvg AB, AC nig BC. #> bmebjsmpabxagerkam ³ k> 4... x>... K> a 7 nig b CacMnYnpÞúyén a enah b... $> KNnaplbUkxageRkamedayeRbIbnÞat;cMnYn ³ k> 4 6 x> 5 ( ) K> 7 ( ) X> 0 5 g> ( 4) %> bmebjsmpabxagerkam ³ k> x> 5... K> 8... X> g> c> 7 ( )

19 emeronti ^>KNnapldk ENnaM ³ ( 6) k> 7 4 x> 7 K> 0 g> ( 0) cmnynkt;rwlatihv b 5 ( 9) 5 (9) X> 0 ( 5) c> 5 q> ( ) &>KNnakenSamxageRkam ³ k> 0 ( 7) x> 4 ( 6) K> 7 ( 6) 4 X> 8 ( ) g> 4 6 ( 5) c> 5 ( 8) *> KNnarYceRbobeFob k> 0 nig 0 x> ( 9) ( 5) nig ( 5) ( 9) etivifidkmanlkçn³rtlb;b et? K> [ 4 ( 7)] nig 4 [( 7) ] X> ( 7) nig (7 ) etivifidkmanlkçn³pþmúb et? - 7 -

20 emeronti (>KNnaplKuNxageRkam ³ k> ( 6)( 4) cmnynkt;rwlatihv x> ( )( ) K> 9)(7) ( X> ( 0)( 9) g> ( 500)( 0) q> (59)!0> bmebjcenøahkñúgsmpabxagerkam ³ c> ( 5)(400) k> 5 ( )(...) x> ( )(...) K> 4 ( )(...) X> ( 0)(...) 4 g> (...)( ) 78 c> ( )(...) q> ( 7)(...) 0!!> KNnaplEckxageRkam ³ k> ( 6) 9 C> (...)( 5) 0 x> 50 ( 5) K> ( 45) 9 44 g> q> X> c> ( 7 0) C> [ ( 8)] [ 5 ( 5)] 0-8 -

21 emeronti KNna k> 6 9 ( ) x> 0 8 K> X> 4 g> 7 ( 5) c> 8 q> [5 ( )] C> 50 [5 ( )] Q> !#> KNnakenSamelxxageRkam ³ k> 8 [7 ( 5)] x> [ 40 ( )] 6 K> [( 5) 7] ( 8 7) X> [ (88 )] (8 ) g> 00 {[50 (40 8)] [ 7 (9)]}!$> bunamanxøi % RKab; bgkat;[xøiefm # RKab;eTot. buna)an elgxøicamyysmcaj;gs;xøi!0rkab;. etibunartuvcmbak;xøismcmnynb:unµanrkab;? - 9 -

22 emeronti cmnynkt;rwlatihv dmenahrsay!>sresrcmnynkt;rwulatib 7,5, 0,0,,,8,,9,, 4,. k>tamlmdab;elig,, 0, 7,,0,,,5,8,9,,4. x>tamlmdab;cuh 4,,9,8,5,,,0,, edacmnuc A( 4),B( 6),C() nig D (5) elibnþat;cmnyn rycknna RbEvg AB, AC nig BC. ek)an AB 4 ( 6) 4 6 AC ( 4) 4 7 BC ( 6)

23 emeronti cmnynkt;rwlatihv #> bmebjsmpabxagerkam ³ k> 4 4 x> K> a 7 nig b CacMnYnpÞúyén a enah b 7 7 %> bmebjsmpabxagerkam ³ k> 9 ( 9) 0 x> 5 ( ) K> 8 ( 4) X> 5 ( ) 8 g> 6 5 ( ) c> 7 ( ) ( ). ^>KNnapldk ENnaM ³ ( 6) k> 7 4 x> 7 5 K> 0 X> 0 ( 5) 5 b 5 ( 9) 5 (9) - -

24 emeronti cmnynkt;rwlatihv g> ( 0) c> 5 8 q> ( ) 6 &>KNnakenSamxageRkam ³ k> 0 ( 7) x> 4 ( 6) K> 7 ( 6) X> 8 ( ) g> 4 6 ( 5) c> 5 ( 8)

25 emeronti *> KNnarYceRbobeFob k> 0 7 nig 0 7 ducenh 0 ( 0) x> ( 9) ( 5) nig ( 5) ( 9) ducenh ( 9) ( 5) [( 5) ( 9)] vifidkkµanlkçn³rtlb;et. K> [ 4 ( 7)] nig 4 [( 7) ] 4 ( 7 ) 4 7 ducenh [ 4 ( 7)] 4 [( 7) ] X> ( 7) 7 0 nig (7 ) 4 8 ducenh ( 7) (7 ) vifidkkµanlkçn³pþmúet. (>KNnaplKuNxageRkam ³ cmnynkt;rwlatihv k> ( 6)( 4) 4 x> ( )( ) 6 - -

26 emeronti cmnynkt;rwlatihv K> ( 9)(7) 6 X> ( 0)( 9) 90 g> ( 500)( 0) 5000 c> ( 5)(400) 6000 q> (59) 89!0> bmebjcenøahkñúgsmpabxagerkam ³ k> 5 ( )(5) x> ( )() K> 4 ( )(4) X> ( 0)(...) 4 mingacman g> ( 6)() 78 c> ( )(4) q> ( 7)(0) 0 C> ( 6)( 5) 0!!> KNnaplEckxageRkam ³ k> ( 6) 9 x> 50 ( 5) - 4 -

27 emeronti cmnynkt;rwlatihv K> ( 45) X> g> 4 [ ( 8)] c> ( 7 0) q> 9 [ 5 ( 5)] C> 0!@> KNna k> 6 9 ( ) 5 x> K> X> g> 7 ( 5) 7 ( 0) 70 c> 8 6 q> [5 ( )] ( 0) 0 C> 50 [5 ( )] 50 ( 0) 5 Q>

28 emeronti!#> KNnakenSamelxxageRkam ³ k> 8 [7 ( 5)] cmnynkt;rwlatihv 8 4 x> [ 40 ( )] 6 ( 40 ) K> [( 5) 7] ( 8 7) ( 46 7) ( 7) 84 X> [ (88 )] (8 ) ( ) g> 00 {[50 (40 8)] [ 7 (9)]} 00 [55 ( 6)] 00 ( 480) 5 4!$> bunamanxøi % RKab; bgkat;[xøiefm # RKab;eTot. buna)an elgxøicamyysmcaj;gs;xøi!0rkab;. etibunartuvcmbak;xøismcmnynb:unµanrkab;? cmnynxøiedlbunacmbak;sm KW. erbah

29 emeronti4 lmhat;!>bmebjcmnynkñúgrbgb;xagerkam ³ (...) k> 5 0 (...) 4 K> x> 4 8 (...) srmylrbpakxagerkam ³ 6 k> 9 8 x> X> 84 K> 96 g> 4464 #> KNna x nig y xagerkam ³ 6 k> (...) x 5 0 x> 7 y 8 8 x. X> x 7 K> $> erbobefob nig erobryrbpakxagerkamtamlmdab;cuh k> 7 5 4,,, x> 7 7,,,, RbPaK - 7 -

30 emeronti4 %> bmebjsbaøa (, > ) < kñúgrbgb;xagerkam ³ 8 7 k> 9 9 x> 6 0 K> 7 7 X> ^>KNnarYcsRmYl 4 7 k> x> K> g> c> q> C> &>KNnarYcsRmYllTæpl 4 k> K> g> q> X> x> X> ( 8. ) (. ) c> RbPaK - 8 -

31 emeronti4 RbPaK *> KNnarYcsRmYl k> x> K> X> (.. ) ( ) g> c> q> (>BinitülMnaMKMrU , tamlmnamkmruenhcurknna ³ 4,

32 emeronti4 RbPaK - 0 -!0> BinitülMnaMKMrU ³ 5 4 4, 5,, sresrrbpak 7, 6, 5 nig 8 tamlmnamkmruxageli?

33 emeronti4 dmenahrsay!>bmebjcmnynkñúgrbgb;xagerkam ³ k> x> K> 4 8 srmylrbpakxagerkam ³ 6 k> x> 84 K> X> g> 4464 #> KNna x nig y xagerkam ³ x 6 k> ek)an x. RbPaK - -

34 emeronti4 RbPaK x> 5 0 y y. ek)an 7 0 K> x ek)an x 9 8 X> x 8 8 ek)an x. $> erbobefob nig erobryrbpakxagerkamtamlmdab;cuh k>,,, ek)an < < < nig erobryrbpakxagerkamtamlmdab;cuh,,, x>,,,, ek)an < < < < nig erobryrbpakxagerkamtamlmdab;cuh,,, ,

35 emeronti4 %> bmebjsbaøa (, > ) < kñúgrbgb;xagerkam ³ 8 7 k> < 9 9 x> > 6 0 K> > 7 7 X> < ^>KNnarYcsRmYl k> x> K> X> g> c> q> C> (48 7) (4 4) RbPaK

36 emeronti4 &>KNnarYcsRmYllTæpl 4 80 k> ( ) () x> ( 5) ( ) (4) X> ( 8 ) ( ) g> c> ( ) K> 0 q> 4 *> KNnarYcsRmYl k> 5 8 ( ) 0 x> RbPaK - 4 -

37 emeronti4 RbPaK K> X> ( ) ( ) g> c>

38 emeronti4 RbPaK q> (>BinitülMnaMKMrU 4 4,, tamlmnamkmruenhcurknna ³ ) (... ) 5 4 ( ) 4 ( ) ( ) (!0> BinitülMnaMKMrU ³ , 5,, sresrrbpak 7, 6, 5 nig 8 tamlmnamkmruxageli

39 emeronti4 RbPaK ek)an

40 emeronti5 lmhat;!>sresrrbpaknimyy²xagerkamcacmnyntspak ³ 7 k> 0 x> K> 000 X> cmnyntspak g> sresrcmnyntspaknimyy²xagerkamcarbpakcrmuh CaTRmg;Edl brgym)an ³ k>. 8 0 x> K> X> g> 8. 7 #> KNnatémøxageRkam ³ k> x> K> X> g> c> ( ) ( ) - 8 -

41 emeronti5 $> rktémøxagerkam ³ k> 4.7 x> cmnyntspak K> 4. ( 0.8) (.4) X> [ 5.9 ( ) ] 5 g> 9. [( 4.) 0.7]. 6 c> 0. [ 99 ( 00) ] ( 0.005) q> ( 4 0.7) (0.06.) 4 %> KNnatémøxageRkam ³ k> x>.4 6 K> (.7) ( 0.5) X> g> 0.8 [.54 (.8)] ^>BIraCFanIPñMeBjeTAextþeBaFisat;mancm ay.65km extþkmbugqñamgsßitenacenøahpñmebj nig extþebafisat;. ebikmbugqñamgmancm ay 9.45km BIPñMeBj. etibiextþkmbg;qñamgetaextþebafisat;mancm ayb:unµan km?

42 emeronti5 dmenahrsay!>sresrrbpaknimyy²xagerkamcacmnyntspak ³ 7 k> x> K> X> g> sresrcmnyntspaknimyy²xagerkamcarbpakcrmuh CaTRmg;Edl brgym)an ³ 8 4 k> x> K> X> g>

43 emeronti5 cmnyntspak #> KNnatémøxageRkam ³ k> x> K> X> g> c> 5.8 ( 0.7) ( 9.6) $> rktémøxagerkam ³ k> x> K> 4. ( 0.8) (.4) 5. 4 X> [ 5.9 ( ) ] g> 9. [( 4.) 0.7] c> 0. [ 99 ( 00) ] ( 0.005). q> ( 4 0.7) (0.06.)

44 emeronti5 cmnyntspak %> KNnatémøxageRkam ³ k> x> K> (.7) ( 0.5) X> g> 0.8 [.54 (.8)] ( 0.6)(0.00) [ ] 0.8 ( 0.000) 500 ^>BIraCFanIPñMeBjeTAextþeBaFisat;mancm ay.65km extþkmbugqñamgsßitenacenøahpñmebj nig extþebafisat;. ebikmbg;qñamgmancm ay 9.45km BIPñMeBj. etibiextþkmbg;qñamgetaextþebafisat;mancm ayb:unµan km? 89. ek)an ³ 89.65km 9.45km 98.0km ducenhbiextþkmbg;qñamgetaextþebafisat;mancm ay 98.0km

45 emeronti6 lmhat;!>sresrpakrynimyy²carbpak nig CacMnYnTsPaK ³ k> 48 % x> 8 % K> 7.5% X> 66 % g> 99 % c> 0 bþúrrbpak nig cmnyntspaknimyy²capakry ³ k> 7 6 x> 0 X> g>. 5 7 K> c> 6 5 PaKry #> Rkumh unmyy)ankat;bnßybukálikcmnyn 4 nak;ecjbibukálik 400 nak;. etibukálikedl)ankat;bnßymanb:unµanpakry? $> enamnðle)aheqñatmyykenøgmangñkcuheqµahe)aheqñat 8500nak; ehiyenaéf e)aheqñatman 5% éngñkcuheqµahmin)anmke)aheqñat. KNnacMnYnmnusSEdl)anmke)aHeqñat? %> BUsM)anepJIR)ak; erolenAFnaKarmYyedayTTYl)an GRtakarR)ak; 7 %. etiryheblb:unµanetibkat;ttyl)anr)ak; TaMgedImTaMgkar)an erol? - 4 -

46 emeronti6 dmenahrsay!>sresrpakrynimyy²carbpak nig CacMnYnTsPaK ³ 48 k> 48 % x> 8 % K> 7.5% X> 66 % g> 99 % c> 0 % bþúrrbpak nig cmnyntspaknimyy²capakry ³ 6 k> 85.7% 7 7 x> 85% 0 K> % X> % g>.5.5% 6 c> 4% 5 5 PaKry

47 emeronti6 PaKry #> Rkumh unmyy)ankat;bnßybukálikcmnyn 4 nak;ecjbibukálik 400 nak;. etibukálikedl)ankat;bnßymanb:unµanpakry? 4 bukálikedl)ankat;bnßyman ³ % 400 $> enamnðle)aheqñatmyykenøgmangñkcuheqµahe)aheqñat 8500nak; ehiyenaéf e)aheqñatman 5% éngñkcuheqµahmin)anmke)aheqñat. KNnacMnYnmnusSEdl)anmke)aHeqñat? KNnacMnYnmnusSEdl)anmke)aHeqñat ³ % % 8075nak;. %> BUsM)anepJIR)ak; erolenAFnaKarmYyedayTTYl)an GRtakarR)ak; 7 %. etiryheblb:unµanetibkat;ttyl)anr)ak; TaMgedImTaMgkar)an erol? ryhebledlkat;epji ³ %

48 emeronti7 rgvas;rgval; lmhat;!>buplefvim úledrmyyedimmanrbevg 5 cm. k> ebikat;manedkrbevg 460mm etikat;efivm úl)anb:unµanedim? x> ebi 0% énedkrtuvxateblefvi etikat;efivm lkçikavas;kmnat;)anmyypab;edayerbiem:rteqimanrbevg 98 cm rycnagvas;pab;dedledayerbiem:rtsmbt;edlmanrbevg 99 cm kñúgkarvas;elikerkayenhnagexijfapab;kmnat;xøicagmun cm. etikmnat;bitr)akdmanrbevgb:unµanem:rt? #> mnussebjv½ymñak;manqam 5. 5 lit. ebiekdwgfaqam mm manekalikarkhm % lan. rkcmnynekalikarrkhmtamggs;? $> rfynþmyysiusamggs; 9 lit kñúg 00 km. ebifugsamgrfynþmanragrbelbieb:tekgedlmanvimart ³ 85 cm, 45cm, cm edayvas;bixagerka. ebicmnuh 0 % ticcagmadedl)anknnatamrubmnþ. k> etifugsamgrfynþenahmancmnuhbitb:unµanlit? x>etirfynþenahgacefvidmeni)anb:unµankilúem:rt?

49 emeronti7 rgvas;rgval; dmenahrsay!>buplefvim úledrmyyedimmanrbevg 5 cm. k> ebikat;manedkrbevg 460mm etikat;efivm úl)anb:unµanedim? x> ebi 0% énedkrtuvxateblefvi etikat;efivm úl)anb:unµanedim? cemøiy k> cmnynm úledlkat;efiv)an eday 460 mm 46cm nigm úledrmyyedimmanrbevg 5 cm 46 ducenh edim 5 x> ebi 0% énedkrtuvxateblefvi enahkat;efivm úl)ankw ³ % 46 80% 0 edim 5 5 #> mnussebjv½ymñak;manqam 5. 5 lit. ebiekdwgfaqam mm manekalikarkhm % lan. rkcmnynekalikarrkhmtamggs;? edayqam 5. 5lIt 5.5dm mm ducenhcmnynekalikarqamtamggs;man ³ lan lan

50 emeronti7 rgvas;rgval; $> rfynþmyysiusamggs; 9 lit kñúg 00 km. ebifugsamgrfynþmanragrbelbieb:tekgedlmanvimart ³ 85 cm, 45cm, cm edayvas;bixagerka. ebicmnuh 0 % ticcagmadedl)anknnatamrubmnþ. k> etifugsamgrfynþenahmancmnuhbitb:unµanlit? x>etirfynþenahgacefvidmeni)anb:unµankilúem:rt? cemøiy k> cmnuhbitkitcaliténfugsamgrfynþ ³ -madrbs;fugsamg ³ cm lit ebicmnuh 0 % ticcagmadedl)anknnatamrubmnþducenhcmnuhbit KitCalIténFugsaMgKW % 4. lit. x>cm ayedlrfynþenahgacefvidmenir)an ³ 4. 00km 459km

51 emeronti8 kensambicknit lmhat;!>sresrcakensambicknittaml,hxagerkam ³ k> byndk x x> y buk 0 K> kaerénplbuk x nig y X> b buknwgplkunmnign g>plkunrvag x buk y nig x dk k> plbukénbircmnynesµinwg 80. ebicmnyntimyyesµinwg x. cursresrkensambicknitsrmab;cmnyntibir? x> pldkénbircmnynesµinwg 0 ebicmnynedltuccagektagedayy sresrcakensambicknitsrmab;cmnynmyyetot? #> KNnatémøkenSamBICKNit x (y ) cmebah ³ k> x 7 nig y x> x nig y K> x 5 nig y 4 $> rfynþmyymanel,ónmfüm 60 km / h. sresrrubmnþcm aycr d kñúgryhebl t rycknnacm aycr kalnaekebikbrkñúgryhebl nig h5mn t. t h, t h0mn,t 0mn

52 emeronti8 kensambicknit %> ctuekanekgmyymanbenþayesµinwg 5 m nig TTwg x m. k> sresrrubmnþbrimart P énctuekanekg? x> KNnabrimaRtctuekaNEkgcMeBaH x 5m, x 8m, x 0m ^> KNna k> x 7x x> x 5x K> x 6x X> 4x ( 8x) &> BnøatkenSam ³ k> ( x )(x ) x> ( x )(x ) K> ( x )(5 x) X> ( x )(x ) g> ( x )(y ) c> 4 (x ) (x 5)(x ) q> ( x) (x )(x ) *> dak;caplkunktþa ³ k> ax bx x> ax ab K> a ab X> 5x 0x g> a b c c> 8x y 4z q> (x 4) x(x 4) C> ( x 5)(x ) (x )(x 5) Q> xy x 4y j> ab 7b ac 7c

53 emeronti8 dmenahrsay!>sresrcakensambicknittaml,hxagerkam ³ k> byndk x sresr 4 x x> y buk 0 sresr y 0 K> kaerénplbuk x nig y sresr ( x y) kensambicknit X> b buknwgplkunm nign sresr b mn g>plkunrvag x buk y nig x dk y sresr ( x y)(x k> plbukénbircmnynesµinwg 80. ebicmnyntimyyesµinwg x. cursresrkensambicknitsrmab;cmnyntibir? x> pldkénbircmnynesµinwg 0 ebicmnynedltuccagektagedayy sresrcakensambicknitsrmab;cmnynmyyetot? cemøiy k> sresrkensambicknitsrmab;cmnyntibir tag y CacMnYnTIBIr ek)an x y 80 ducenh y 80 x. x> sresrcakensambicknitsrmab;cmnynmyyetotkw y

54 emeronti8 kensambicknit #> KNnatémøkenSamBICKNit x (y ) cmebah ³ k> x 7 nig y ek)an x (y ) 7(6 ) x> x nig y ek)an x (y ) ( )( ) K> x 5 nig y 4 ek)an x(y ) (5)( ) 50 $> rfynþmyymanel,ónmfüm 60 km / h. sresrrubmnþcm aycr d kñúgryhebl t rycknnacm aycr kalnaekebikbrkñúgryhebl t h, t h0mn,t 0mn nig t h5mn. cemøiy sresrrubmnþcm aycr d kñúgryhebl t KW d 60t -ebi t h enah d 60 0km ebi t h0mn h enah d 60 40km 60 -ebi t 0mn h enah d 60 0km

55 emeronti8 kensambicknit %> ctuekanekgmyymanbenþayesµinwg 5 m nig TTwg x m. k> sresrrubmnþbrimart P énctuekanekg? x> KNnabrimaRtctuekaNEkgcMeBaH x 5m, x 8m, x 0m cemøiy k> sresrrubmnþbrimart P énctuekanekg ek)an P (5 x) 0 x x> KNnabrimaRtctuekaNEkgcMeBaH x 5m, x 8m, x 0m -ebi x 5m enah P m -ebi x 8m enah P m -ebi x 0m enah P m ^> KNna k> x 7x 0x x> x 5x x K> x 6x 8x X> 4x ( 8x) x &> BnøatkenSam ³ k> (x )(x ) x x x x - 5 -

56 emeronti8 kensambicknit x> (x )(x ) x x x 6 x x 6 K> (x )(5 x) 0x 6x 5 9x 6x 9x 5 X> (x )(x ) x x x 9 x 6x 9 g> ( x )(y ) xy 4x y 6 c> 4 (x ) (x 5)(x ) 4x x x 7 x 5x 5 q> ( x) (x )(x ) 4 x x x 5x 6 x 6x *> dak;caplkunktþa ³ k> ax bx x(a b) x> ax ab a(x b) K> ab a a(b ) X> 5x 0x 5x(x ) g> a b c (a b c) c> 8x y 4z 4(x y z) q> (x 4) x(x 4) (x 4)( x)

57 emeronti8 kensambicknit C> ( x 5)(x ) (x )(x 5) (x 5)(x x ) (x 5)( x ) Q> xy x 4y x(y ) 4(y ) (y )(x 4) j> ab 7b ac 7c b(a 7) c(a 7) (a 7)(b c)

58 emeronti9 smikardwerktimyymanmyygbaøat lmhat;!>ekmansmikar x, 5 (x ) 8, x 4x 0 nig x (x 5). edahrsaysmikarxagerkamrycepþógpþat;cemøiy ³ k> 5 x x> 40 n 70 K> m.. 5 X> y 4 9 g> 8 y 56 c> 6 x 48 q> 0 7x x C> 8 t Q> 5 0. #>edahrsaysmikarxagerkam k> 5(n ) 5 4 (n 5) x> x 5 6x K> 5 y 9y 0 y X> (4x ) x 6 x 40 g> 0.05x c> (y ) 5 4y q> 5( y) 7 4( y)

59 emeronti9 smikardwerktimyymanmyygbaøat $> edahrsaysmikarxagerkamrycepþógpþat;cemøiy ³ k> 5 x 0(x ) 40 x> 5 4(t ) (t 7) x x K> X> x 5 x 7 x x 4 6x g> c> x x x x x x x x q> C> %> ctuekanekgmyymanbenþayesµinwg 4 m ehiyépþrklaesµinwg épþrklakaeredlmanrcugesµinwg m. KNnaTTwgctuekaNEkgenaH? ^> m:asiunftmyymantémø 7duløa bnþab;bibba úhtémø 0 %. etitémøedimrbs;m:asiunftenahesµinwgb:unµan? &> FugBIrmansaMg 50 lit. ebiekyksamg litbifugtimyy nig yk 5 litecjbifugtibirenahfugtamgbirenasl;samgesµikña. etifugnimyy²mansamgb:unµanlit? *> KNnabrimaRtRtIekaNmYy ebirtiekanenahmanrgvas;rcugtimyy esµinwg 6 m rgvas;rcugtibiresµinwg 7 énbrimart nigrcugtibiesµinwg énbrimart

60 emeronti9 smikardwerktimyymanmyygbaøat (>LanmYyecjdMeNIrBIPñMeBjeTAkMBg;camEdlmancm ayesµinwg 4 km edayel,ón 65 km / h ehiylanmyyetotecjdmenir Rcas;KñaBIkMBg;cammkPñMeBjedayel,Ón 45 km / h. ekdwgfalantamgbirecjdmenirenaebletmyy. k> rkcm aybipñmebetakenøgcybkña? x> etib:unµanem:agerkaymketiblantamgbircybkña?!0> plbukbircmnynesµinwg 4 ehiybirdgéncmnynti! buknwgcmnynti@ esµinwg 6. KNnacMnYnTaMgBIrenaH?!!> vir³ekµgcag»bukrbs;kat; 4 qñam. ekdwgfaryhebl@qñametotpl bukgayugñktamgbiresµinwg 40 qñam. rkgayu»buk nig Gayurbs;vIr³ >

61 emeronti9 smikardwerktimyymanmyygbaøat dmenahrsay!>ekmansmikar x, 5 (x ) 8, x 4x 0 nig x (x 5). etismikarnaxøhcasmikardwerktimyymanmyygbaøat? smikaredlcasmikardwerktimyymanmyygbaøatkw³ x nig 5 (x ) edahrsaysmikarxagerkamrycepþógpþat;cemøiy ³ k> 5 x ek)an x 5 x 6 epþógpþat; 5 6. ducenh 6 Bit x. x> 40 n 70 ek)an n n 0 epþógpþat;

62 emeronti9 smikardwerktimyymanmyygbaøat Bit n. ducenh 0 K> m.. 5 ek)an m.5. m 0. epþógpþat; Bit ducenh m 0.. X> y 4 9 ek)an y 9 4 y epþógpþat; Bit ducenh y. g> 8 y 56 ek)an y 56 8 y

63 emeronti9 smikardwerktimyymanmyygbaøat epþógpþat; Bit y. ducenh 5 c> 6 x 48 ek)an x x 48 6 epþógpþat; 6 () 48 ducenh Bit x. q> 0 7x 0 ek)an x 7 x 6 epþógpþat; 0 7(6) ducenh Bit x

64 emeronti9 smikardwerktimyymanmyygbaøat x C> 8 7 ek)an x 8 7 x 6 6 epþógpþat; Bit x. ducenh 6 t Q> 5 0. ek)an t 5 0. t Bit t 0.. epþógpþat; 5 ducenh 55 #>edahrsaysmikarxagerkam k> 5(n ) 5 4 (n 5) 5n 0 5 5n 5 5n n 4 n 5 n

65 emeronti9 smikardwerktimyymanmyygbaøat 7n 4 4 n 7 n x> x 5 6x x 6x 5 4x 8 x x 8 4 K> 5 y 9y 0 y 5y 9y y 0 5y 5 5 y 5 y X> (4x ) x 6 x 40 8x x 6 x 40 8x x x 6 40 x x x - 6 -

66 emeronti9 smikardwerktimyymanmyygbaøat g> 0.05x x x 0.5 x x c> (y ) 5 4y 6y 5 4y 6y 8 4y 6y 4y 0y y y q> 5( y) 7 4( y) 5 0y 7 8 4y 0y 4y 0y 4y 6y 4 y

67 emeronti9 smikardwerktimyymanmyygbaøat $> edahrsaysmikarxagerkamrycepþógpþat;cemøiy ³ k> 5 x 0(x ) 40 5x 0x x x x> 5 4(t ) (t 7) K> 5 4t 8 4t 4t t 5 t 8 t t 4 8 t x x 7 x x 7 x x 7 7x 7 x 4x 7 x

68 emeronti9 smikardwerktimyymanmyygbaøat X> g> x 5 x 9 x 5 x x 5 x 4x 4 5 x 4x x 5 4 x 9 x 9 x x (x ) 4(x 4) 8 8 x 6 4x x 8 8 x x 9 x

69 emeronti9 smikardwerktimyymanmyygbaøat c> x ek)an 6x 4 x 4(x ) 6x x x 8x 4 6x 8x 6x 4 lkçxnð x 0 x x minyk x x x 4 5 0x 0x 6x 60 4x x 4 0 x 7 x x x 5 4 0(x ) 4(x ) 5(x ) q> C> b x 0x 0 4x 6x 5x 0 5x 0 x

70 emeronti9 smikardwerktimyymanmyygbaøat %> ctuekanekgmyymanbenþayesµinwg 4 m ehiyépþrklaesµinwg épþrklakaeredlmanrcugesµinwg m. KNnaTTwgctuekaNEkgenaH? cemøiy KNnaTTwgctuekaNEkg tag x CaTTwgrbs;ctuekaNEkgenH tambrmab;ek)ansmikar ³ 4x 4x 44 x x ^> m:asiunftmyymantémø 7duløa bnþab;bibba úhtémø % etitémøedimrbs;m:asiunftenahesµinwgb:unµan? cemøiy tag x CatémøedImrbs;m:asIunft 0 ek)ansmikar x x ektaj)an x 90 duløa

71 emeronti9 smikardwerktimyymanmyygbaøat &> FugBIrmansaMg 50 lit. ebiekyksamg litbifugtimyy nig yk 5 litecjbifugtibirenahfugtamgbirenasl;samgesµikña. etifugnimyy²mansamgb:unµanlit? cemøiy tag x nig y CacMNuHsaMgkñúgFugTImYy nig FugTIBIrerogKña tambrmab;ek)ansmikar ³ x y 50 () ehiy x y 5 nam[ y x 5 x () yksmikar ( ) CMnYskñúg ( ) ek)an x x 50 x 50 x 8 x 8 64 ehiy y ducenhfugtimyymansamg 64lIt nigfugtibirmansamg 86lIt

72 emeronti9 smikardwerktimyymanmyygbaøat *> KNnabrimaRtRtIekaNmYy ebirtiekanenahmanrgvas;rcugtimyy esµinwg 6 m rgvas;rcugtibiresµinwg 7 énbrimart nigrcugtibiesµinwg énbrimart. cemøiy KNnabrimaRténRtIekaN ³ tag p CabrimaRténRtIekaN ek)an p 6 p p p p 8p 7 6 6p 7p 6 p p cm ducenhbrimartrtiekankw 4 cm

73 emeronti9 smikardwerktimyymanmyygbaøat (>LanmYyecjdMeNIrBIPñMeBjeTAkMBg;camEdlmancm ayesµinwg 4 km edayel,ón 65 km / h ehiylanmyyetotecjdmenir Rcas;KñaBIkMBg;cammkPñMeBjedayel,Ón 45 km / h. ekdwgfalantamgbirecjdmenirenaebletmyy. k> rkcm aybipñmebjetakenøgcybkña? x> etib:unµanem:agerkaymketiblantamgbircybkña? cemøiy k> rkcm aybipñmebjetakenøgcybkña tag x Cacm aybipñmebjetakenøgrfynþtamgbircybkña ek)an 4 x Cacm aybikmbg;cametakenøgcyb. x 4 x ek)ansmikar x>cmebah x (4 x) 9x 6 x x 6 x km x t x ek)an h - 7 -

74 emeronti9 smikardwerktimyymanmyygbaøat!0> plbukbircmnynesµinwg 4 ehiybirdgéncmnynti! buknwgcmnynti@ esµinwg 6. KNnacMnYnTaMgBIrenaH? cemøiy KNnacMnYnTaMgBIr tag x CacMnYnTImYy nig 4 x CacMnYnTIBIr ek)ansmikar x 4 x 6 nam[ x. ducenhcmnyntimyyesµi nigcmnyntibir 4.!!> vir³ekµgcag»bukrbs;kat; 4 qñam. ekdwgfaryhebl@qñametotpl bukgayugñktamgbiresµinwg 40 qñam. rkgayu»buk nig Gayurbs;vIr³? cemøiy tag x CaGayurbs;vIrH nig x 4 CaGayurbs;»Buk ek)ansmikar ( x ) (x 4 ) 40 x 8 40 b x 6 ducenhvirhmangayu 6qñaM nig»bukmangayu qñam

75 emeronti0 lmhat;!>erciserisvismpabkñúgsmenrxagerkam ³ vismpab < ; 8 > 5 ; a b ; c < d ; a c < a d,a c a bmebjsbaøa > b <, enakñúgrbgb;xagerkam ³ k> 0 < 0 enah x> a < b enah a c K> a > b enah a c b c b c X> < enah 4 4 g> 0 < enah 0 ( 4) c> a < b enah a c b c ( 4) q> a > b enah a c b c cmebah c > 0 C> a > b enah a c b c cmebah c < 0 #>bgðajfaebi a > b nig c > d nam[ a c > b d ryc[]tahrn_caelxbba ak;. $> bgðajfaebi a > b nig c < d nam[ a c > b d ryc[]tahrn_caelxbba ak;

76 emeronti0 %>bgðajfaebi a > b nig b > c nam[ a > c ryc[]tahrn_caelxbba ak;. ^> erboefobcmnyn ³ k> nig x> nig K> nig 5 X> nig &> a 0, b. epþógpþat;fa a b a b? vismpab

77 emeronti0 dmenahrsay!>erciserisvismpabkñúgsmenrxagerkam ³ vismpab < ; 8 > 5 ; a b ; c < d ; a c < a d,a c a d cemøiy smenredlcavismpabman ³ <, 8 > 5, a < d, nig a c < a bmebjsbaøa > b <, enakñúgrbgb;xagerkam ³ k> 0 < 0 enah 0 4 < 0 4 x> a < b enah a c < K> a > b enah a c > b c b c X> < enah 4 < 4 g> 0 < enah 0 ( 4) c> a < b enah a c < b c > ( 4) q> a > b enah a c > b c cmebah c > 0 C> a > b enah a c > b c cmebah c <

78 emeronti0 #>bgðajfaebi a > b nig c > d nam[ a c > b d ryc[]tahrn_caelxbba ak;. cemøiy ekman a > b nam[ a b > 0 ehiy c > d nam[ c d > 0 ek)an ( a b) (c d) > 0 ducenh a c > b d. ]TahrN_Caelx ³ yk a 0, b 7, c 9, d 5 ek)an 0 > 7 nig 9 > 5 ducenh 0 9 > > Bit. $> bgðajfaebi a > b nig c < d nam[ a c > b d ryc[]tahrn_caelxbba ak;. cemøiy ekman a > b nam[ a b > 0 ehiy c < d nam[ d c > 0 vismpab

79 emeronti0 vismpab ek)an ( a b) (d c) > 0 ducenh a b > b d. ]TahrN_Caelx ³ yk a, b 7, c 9, d 5 ek)an > 7 nig 9 < 5 ducenh 9 > > Bit. %>bgðajfaebi a > b nig c ryc[]tahrn_caelxbba ak;. cemøiy ekman a > b nam[ a b > 0 b > nam[ a > c ehiy b > c nam[ b c > 0 ek)an ( a b) (b c) > 0 a c > 0 ducenh a > c. ]TahrN_CaelxbBa ak; ³ yk a 7, b 0, c 5 ekman 7 > 0 nig 0 > 5 ducenh 7 > 5 Bit

80 emeronti0 ^> erboefobcmnyn ³ k> 5 nig eday 5 < 7 ducenh <. 7 x> nig 7 8 ekman eday 8 > 7 ducenh K> nig ekman nig 55 eday 70 < 77 ducenh 55 7 X> nig 605 ekman <. 5 7 > nig >. 7 4 vismpab ducenh &> a 0, b. epþógpþat;fa a b a b eday a b 0 9, a b 0 ducenh a b a b