SH. A. ALIMOV, O. R. XOLMUHAMEDOV, M. A. MIRZAAHMEDOV. Umumiy o rta ta lim maktablarining 7- sinfi uchun darslik

Transcript

1 SH A ALIMOV, O R XOLMUHAMEDOV, M A MIRZAAHMEDOV Umumiy o rt t lim mktlrining 7- sinfi uchun drslik Qyt ishlngn v to ldirilgn 5- nshri O zekiston Respuliksi Xlq t limi vzirligi tsdiqlgn O QITUVCHI NASHRIYOT-MATBAA IJODIY UYI TOSHKENT 07

2 UO K: 5(075) KBK 4 y 7 A-50 Aziz o quvchilrim! On yurtimiz mustqil O zekiston jhon ilm-u fnig, mdniytig yuzl uyuk olimlrni, shoirlrni, dvlt rolrini, musvvirlrni yetishtiri ergn Bilingki, siz ulrning ezgu ishlri dvomchisisiz! Yoshlik ilim olish dvridir Allomlr ytdi: Yoshlikd olingn ilim toshg itilgn yozuv ki o chmsdir Algerni, umumn, mtemtikni o rgnish qunt v izchillikni, ko pl msl v misollrni tushuni, idrok qili yechishni tl etdi Meni yxshi o rgni olsngiz, sizg umrod do st o li qolmn! Xulq-u odoingiz rkmol, ilmingiz ziyod o lishini ist Alger drsligingiz Drslikdgi shrtli elgilr: ¹ sosiy qoidlr v xosslr mtemtik tsdiqni soslsh yoki formulni keltiri chiqrish oshlndi soslsh yoki formulni keltiri chiqrish tugdi mslni yechish oshlndi mslni yechish tugdi qiziqrli msllr 5, 4, murkkroq msl sinov mshqlri sosiy mteril o yich ilimni tekshirish uchun mustqil ish trixiy m lumotlr Drslikdgi pul muomlsig doir mshqlrd nrxlr shrtli qul qilingn Respulik mqsdli kito jmg rmsi mlg lri hisoidn chop etildi ISBN ShA Alimov, OR Xolmuhmedov, MA Mirzhmedov Brch huquqlr himoylngn O qituvchi NMIU, 0 O qituvchi NMIU, qyt ishlngn v to ldirilgn nshri, 07

3 5 6- SINFLARDA O RGANILGAN MAVZULARNI TAKRORLASH Aziz o quvchi! Siz 5 6-sinflrd nturl sonlr, oddiy v o nli ksrlr, rtsionl sonlr ustid to rt mlg doir misol v msllrni yechgnsiz 5 6-sinflrd mtemtikdn olgn ilimlringizni yodg solish mqsdid Sizg ir necht mshqlr tklif etmiz Xlq iln muloqot v inson mnftlri yilid qurilgn zmonviy uylr shhrimizg ynd ko rk g ishldi Yngi qurilgn ko p qvtli uylrdn irining xondonlri,,,, 99, 00 sonlri iln nomerlngn Rqmlri yig indisi o zro teng o lgn xondonlr nechtdn? Ntijlrni jdvld v digrmmd ks ettiring Bir fermdgi sigirlr soni -fermdgig qrgnd % km Ammo -fermning hr ir sigiri -fermning hr ir sigirig qrgnd 7,5% ko p sut erdi Qysi ferm v nech foiz ko p sut oldi? 00 kg g ll m lum muddt quritilgch, uning msssi 0 kg g kmydi, nmligi es 0 % ni tshkil qildi Dstl g llning nmligi nech foiz edi? 4 Tenglmni yeching: 6 7 )5x + 48:4= 0:0+ 0 ) 4 x + 5= ) 7x + : = (7+ 8): 4) 6 x + = Ahmd velosiðedd sotig 0,8 km tezlik iln sot-u 5 minut yo l yurdi So ngr sotig,8 km tezlik iln,5 sot yo l yurdi Ahmd jmi nech kilometr yo l yurgn?

4 4 6 To g ri to rturchkning o yi 8 sm g teng Eni o yidn,5 sm qisq To g ri to rturchkning yuzini toping 7 To g ri to rturchkning yuzi 0,5 dm g, eni,4 dm g teng Shu to g ri to rturchkning perimetrini toping 8 Avtomoil 00km msofg 5l enzin srflydi Bu vtomoil: 50 km 60 km 70 km 80 km 0 km 50 km 60km yo lg qnch enzin srflydi? 9 Syyoh yo lning qismini o tdi Hisol ko rs, yo lning 7 yrmig yetishi uchun yn 9 km yurishi kerk ekn Syyoh jmi nech kilometr yo l yurishni mo ljllgn? 0 Bir vtomoil 00km msofg 8l, ikkinchi vtomoil es shunch msofg 0 l enzin srflydi Agr hr ir vtomoil kid l dn enzin o ls, u yonilg i ulr uchun nech kilometr yo lg yetdi? ) Mtoning nrxi 0 % psytirildi M lum vqtdn so ng, yngi nrx hm 5% psytirildi Mtoning nrxi jmi nech foiz kmygn? ) Gzlmning nrxi 0% ortdi M lum vqtdn so ng, yngi nrx hm 5% ortdi Gzlmning nrxi jmi nech foiz ortdi? Bug doyning nmligi % edi U quritilgch, nmligi % g tushdi Bug doyning msssi nech foizg kmydi? Tdirkor -v -nv mollrni soti, jmi so m foyd qildi -nv molning nrxi 0000 so m edi, tdirkor uni 5% foydsig sotdi -nv moldn 0% foyd ko rdi -nv molning nrxi nech so m? Ikkl nv molni soti, tdirkor nech foiz foyd ko rgn? 4 To g ri to rturchk sosining uzunligi 0 %, lndligi 5% orttirils, uning yuzi nech foiz ortdi? 5 To g ri to rturchk sosining uzunligi 0 %, lndligi 0% kmytirils, uning yuzi nech foiz kmydi?

5 6 Amllrni jring: ) ( 0): (( 8) ( ) + :( ) ) ( 48):( 6) ) ( 75) 4 04:( ) + ( 0):( 7) ) ( 0,5):(,6) + 90,7:( 4,5) 7,5, ) 5 ( 0,95) + ( 0,4) 8 : Tenglmni yeching: ) x+ x= 7 + ( 7) ),x,5x= ( 0,5) ) 6x 7x=,5( ) + 4 4) 4x x= :( ) 8 5 t sonning o rt rifmetigi 8,4 g teng Bu sonlrg yn itt son qo shi, o rt rifmetik qiymt hisolngn edi, u 0 g teng chiqdi Qo shilgn sonni toping 9 Krim ot 90 yoshd Uning nirlrining o rtch yoshi 0 d Nirlr yoshlrig Krim ot yoshini hm qo - shi, o rt rifmetik qiymt hisolngn edi, u g teng chiqdi Krim otning necht nirsi or? 0 Avtomoil 7 km/sot tezlik iln,5 sot, 60 km/sot tezlik iln,5 sot yurdi Avtomoil jmi nech kilometr yo l yurgn? Bu msofni u qndy o rtch tezlikd osi o tdi? Proporsiyning nom lum hdini toping: ),5: x =,4:4,8 ) 7,:,4 = x :4 ) x : = 9,:, 4) 4 : =,: x 7 7 5

6 ALGEBRAIK IFODALAR - Sonli ifodlr Alger so zi uyuk o zek mtemtigi v stronomi, vtndoshimiz Au Adulloh Muhmmd in Muso l-xorzmiyning Kito l-muxtsr fi hiso l-jr vl-muqol ( Al-jr vl-muqol ) sridgi l-jr (lotinchsig lger) so zidn olingn Bu srd l-xorzmiy dunyod irinchi mrt lger fnini izchillik iln yon qilgn Algerning sosiy mslsi lgerik ifodlr ustid mtemtik mllrni o rgnishdir Algerik ifodlrning eng sodd ko rinishi o lgn sonli ifodlr 5 6-sinf mtemtik kurslrid qrlgn edi Sonli ifod sonlrdn tuzili, mllr elgilri iln irlshtirilgn yozuv eknligini eslti o tmiz Msln, ,5 + 0: yozuvlr sonli ifodlrdir 5 Sonli ifodning qiymti de, shu sonli ifodd ko rstilgn mllrni jrish ntijsid hosil o lgn song ytildi Msln, +7 sonli ifodning qiymti soni, sonli ifodning qiymti 6 sonidir Sonli ifod itt sondn iort o lishi hm mumkin Uning qiymti shu sonning o zi o ldi B zn sonli ifodd sonlr v mllr elgilridn tshqri mllrning m lum trtid jrilishini ko rstuvchi qvslrdn foydlnildi Msln, (,5+,5), sonli ifodning qiymtini hisolshd vvl qvs ichidgi qo shish mli, keyin ko pytirish mli jrildi

7 Au Adulloh Muhmmd in Muso l-xorzmiy (78 850) uyuk o zek mtemtigi v stronomi (,5+,5), ifodning qiymtini hisol,,6 sonini hosil qilmiz Shuning uchun (,5+,5),=,6 tenglikni yozish mumkin = elgisi iln irlshtirilgn ikkit sonli ifod sonli tenglikni tshkil qildi Agr tenglikning chp v o ng qismlrining qiymtlri ir xil son o ls, u hold tenglik to g ri tenglik deyildi Msln, 5 = 8 to g ri tenglik, chunki uning ikkl qismining hm qiymti 7 sonig teng Sonli ifodlr v sonli tengliklrdn, hisolshlr iln ir qtord, sonlrning xosslrini yozishd hm foydlnildi 6 Msln, = tenglik ksrlrning sosiy xosssini, = +5 tenglik es qo shishning o rin lmshtirish qonunini ifodlydi Endi 6+ sonli ifodni qrylik 6+ =6+6=4 dn iort o lgn to g ri ntij mllrni qul qilingn jrish trtiig rioy qilingn holdgin hosil o ldi Agr qul qilingn hisolsh trtii uzils v vvl 6g ni qo shi, so ngr hosil o lgn yig indi g ko pytirils, u hold 54 dn iort noto g ri ntij hosil qilindi Bu ntij dstlki ifod (6 +) ki yozils, to g ri o - lr edi 6 + (6+) 7

8 Demk, hisolshning to g riligi sonli ifoddgi mllrning jrilish trtiig og liq ekn Sonlr ustid mllrning jrilish trtii lgerik ifodlrning son qiymtlrini topishg oid msllrni jrishd hm sqlni qoldi Qo shish v yirish irinchi osqich mllr, ko pytirish v o lish es ikkinchi osqich mllr deyilishini eslti o tmiz Kvdrt v kug ko trish uchunchi osqich mllr deyildi Sonli ifodning son qiymtini topishd mllr jrilishining quyidgi trtii qul qilingn: 8 ) Agr ifodd qvslr o lms, u hold vvl uchinchi osqich mllr, keyin ikkinchi osqich mllr v, nihoyt, irinchi osqich mllr jrildi, shu iln irg, ir xil osqich mllr ulr qndy trtid yozilgn o ls, xuddi shu trtid jrildi Msln, = = = 80+ 7= 87 ) Agr ifodd qvslr o ls, u hold vvl qvslr ichidgi sonlr ustid rch mllr, so ngr es qolgn rch mllr jrildi, und qvs ichidgi v undn tshqridgi rch mllr -ndd ko rstilgn trtid jrildi Msln, ( 4 5) 6 + (+ 4) = (8 4 5) 6 + (+ 4) = = ( 5) 6 + (+ 8) = = 6+ 0 = 7 ) Agr ksr ko rinishidgi ifodning qiymti hisolndign o ls, u hold ksrning surti v mxrjidgi mllr jrildi, so ngr irinchi ntij ikkinchisig o lindi Msln, = = = = ) Agr ifodd qvslr ichid oshq qvslr o ls, u hold vvl eng ichkridgi qvslr ichidgi mllr jrildi Msln, (8 (5 4)) = (8 (5 4)) = (8 ) = ( ) = 6

9 Amllrni jring: ),7 + (, 0,7) ), ) 9,49 (,5+ 0,99) 4) 6,4 Sonli ifodning qiymtini toping: ) ) 0, :,5 0 4 ) 7 4 4),7 : + 4,5 5 Qiymti: ) 8 ) 0 ) 4) 4 g teng ir necht sonli ifod yozing 4 Tenglik to g rimi: ) ),5 4, =,7+ 0,4 ) 4 0,75 0,5 = 0,5+ 0,5 4) Sonli tenglik shklid yozing (5 6):,+ 4, 5 = + + 7,58 4,5 4 8,9 6,6 = 5,8,55 9? 5 ) v 5 sonlri yig indisi v sonlri yirmsig teng 5 )40 v 0,0 sonlri ko pytmsi 6 sonining 5 g o linmsig teng 6 ) 0 v sonlri yirmsining ikkilngni shu sonlr yig indisidn uch mrt ktt ) v 6 sonlri yig indisining uchlngni shu sonlr ko - pytmsidn ikki mrt ortiq 7 Amllr trtiini ko rsting v hisolng: ),7 + 5 ) 48 0,05 54+,7 Mshqlr ) 7, ,4:0,8 4) (,5) + 5 0,4:0,6 5 9

10 8 Sonli ifodning qiymtini toping: ) ) ) ) 9 Amllrni jring: ) ) , 5 5 ) (8, ( + 6,)),5+ 4,:6 0, 4) ((7,8:0, ) +,):0,7 7,5:0,5 - Algerik ifodlr Quyidgi mslni qrymiz -msl Biror son o ylng, uni g ko pytiring, hosil o lgn ntijg 6 ni qo shing, topilgn yig indini g o ling v o ylngn sonni yiring Qndy son hosil o ldi? Aytylik, o ylngn son 8 o lsin Brch mllrni msl shrtid ko rstilgn trtid jrmiz: )8 =4 )4+6=0 )0:=0 4)0 8= soni hosil o ldi Bu yechimni qiymti g teng o lgn (8 +6) : 8 sonli ifod shklid yozish mumkin Bordi-yu, gr 5 soni o ylngn o ls, u hold qiymti yn g teng o lgn (5 +6): 5 sonli ifod hosil qilingn o lr edi Biz qndy sonni o ylmylik, ntijd soni hosil o lverr ekn-d, degn frz tug ildi Buni tekshiri ko rmiz O ylngn sonni hrfi iln elgilymiz v mllrni yn msl shrtid ko rstilgn trtid yozmiz: ( +6): Arifmetik mllrning izg m lum o lgn xosslridn foydlni, u ifodni soddlshtirmiz: ( + 6): = + = 0

11 Mslni yechishd istlgn sonni ildiruvchi hrfi, v 6 sonlri, mllr elgilri v qvslrdn iort ( + 6): ifod hosil qilindi Bu lgerik ifodg misoldir v u msl shrtini mtemtik tilg o tkzish nmunsidir Yn lgerik ifodlrg misollr keltirmiz: x+ y ( m+ n), + 7, ( + )( ), Algerik ifod sonlr v hrflrdn tuzili, mllr elgilri iln irlshtirilgn ifoddir Agr lgerik ifodg kirgn hrflr o rnig iror son qo yils v ko rstilgn mllr jrils, ntijd, hosil qilingn son erilgn lgerik ifodning son qiymti deyildi Msln, =, = o lgnd + 7 lgerik ifodning qiymti 5 g teng, chunki + 7= 5 shu lgerik ifodning qiymti = =0 o lgnd 4 g teng, chunki = 4 ning istlgn qiymtid ( + 6): lgerik ifodning qiymti g teng -msl ifodning qiymtini =0, =5 o lgnd toping (+ 7) ( 0+ 7) = = Algerik ifodning qiymtini toping: ), und = =, ) 0,5 4 c, und = 4, c = ) +, und =, = 4), und =, = 9 Mshqlr

12 Algerik ifodning qiymtini toping: 4 7, ) x y und x = 8, y = 4 4 ) x + y 5, und x = 9, y = 0 ) +, und = 4, = + c 4), c und =, c = Neft quvuridn sotd 7 t neft oqdi, m sotd quvurdn nech tonn neft oqi o tdi? Bir sutkd-chi? ) m sotd ) p sekundd ) m sot l minut v p sekundd nech minut or? 4 x v y sonlr yirmsining uchlngnini yozing Shu ifodning: ) x = 0,7, y = 0,4 ) x =,98, y = 4, ) x =, y = 4) = = 6 4 x 5, y 0,7 o lgndgi son qiymtini toping 5 x v y sonlr yig indisi iln ulr yirmsining ko pytmsini yozing Hosil o lgn lgerik ifodning: 6 ) 5 ) x =, y = ) x =, y = ) x = 0,5, y = 0,75 4) x =,, y =,8 o lgndgi son qiymtini toping Algerik ifodlrning son qiymtini toping (6 7): ) 7 ) mn( n+ k), und m = k =, n = n k (p+ ) p +, und = = p l p, l ( x y), und 8,,9,0 p + q x = y = p = q = ) 5( c + m), und = c = 6 q =, m = 5 q 4 + 4

13 8 Toq son formulsi n =k+ dn foydlni, k =0, k =, k =7, k =0 o lgnd n ning qiymtini toping 9 Algerik ifod shklid yozing: ) kichigi n g teng o lgn ikkit ketm-ket nturl sonning yig indisi ) kttsi m g teng o lgn ikkit ketmket nturl sonning ko pytmsi ) kichigi k g teng o lgn ucht ketm-ket juft nturl sonning yig indisi 4) kichigi p+ g teng o lgn ucht ketm-ket toq nturl sonning ko pytmsi 0 Shkllrning perimetri v yuzini lgerik ifod shklid yozing (-rsm):,5 c,5 d -rsm Uyni isitish uchun p tonn ko mir g mlndi shu zxirdn q tonn srf qilindi Nech tonn ko mir qoldi? ) p =0, q =5 o lgnd hisolng ) q son p sondn ktt o lishi mumkinmi? p g teng o lishi-chi? Kursh musoqsid hr iri 400 so mdn n t chiðt v hr iri 500 so mdn m t chiðt sotildi Hmm chiðtlr uchun qnch pul olingn? n =00, m =50 n =00, m =0 o lgnd hisolng Bitt lomning hosi 00 so m, itt dftrning hosi 40 so m, itt ruchkning hosi 60 so m c t lom, t dftr v t ruchkning umumiy (so mlrdgi) hosini p hrfi iln elgil, uni formul shklid yozing Agr c =9, =, =4 o ls, u formul o yich p ni hisolng 4 Issiqlik uztish stnsiysi uchun mo ljllngn gz quvuri orqli hr minutd 6 m gz o tdi 5 sutkd m sutkd quvurdn nech ku metr gz o tdi?

14 5 Geologlr o z yo nlishi o yich hrkt qili, otd sotig c kilometr tezlik iln sot-u 0 minut yurishdi oqimining tezligi sotig kilometr o lgn dryod oqim o yich sot-u 40 minut sold suzishdi v sotig kilometr tezlik iln sot-u 0 minut piyod yurishdi Yo nlishning (kmlrdgi) uzunligini s hrfi iln elgil, geologlr osi o tgn yo l formulsini yozing Agr =, km/sot, =5,7 km/sot, c =0,5 km/sot o ls, yo nlishning uzunligini hisolng - Algerik tengliklr, formullr Ko pgin mliy msllrni yechishd sonlrni elgilsh uchun hrflrdn foydlnish qulydir Msln, gr v to g ri to rturchk tomonlrining uzunliklri o ls, u hold uning yuzi, (+) uning perimetri Bu yerd v hrflri iln must sonlr to g ri to rturchkning tomonlrining uzunliklri elgilngn Agr to g ri to rturchk yuzini S hrfi iln, perimetrini es P ilnelgilsk, u hold quyidgi formullrni hosil qilmiz: S =, P = (+) Agr tomonlr uzunliklri sntimetrlrd o lchngn o ls, u hold S yuz kvdrt sntimetrlrd, P perimetr es sntimetrlrd ifodlndi Yozuvni qisqrtirish uchun ko pytirish elgisi nuqt ko pinch tushiri qoldirildi Msln, S=, P =( + ) de yozildi Hrflr iln, shuningdek, tenglmlrdgi nom lum sonlr hm elgilndi Msln: x +, = 95, tenglmdgi nom lum son x hrfi iln elgilngn, y+=7 tenglmdgi nom lum son es y hrfi iln elgilngn Hrflr iln rifmetik mllr qonunlri v xosslrini yozish hm qulydir Msln: 4 ( + c) = ( ) c = c, () ( + ) c = c+ c, () ( + ): c = : c+ : c ()

15 XVI srning tniqli mtemtigi Frnsu Viyet (540 60) lgerg hrfiy elgini kiritishning soschisi hisolndi Algerd irgin hrfning o zi hr xil sonli qiymtlr qul qilishi mumkin Jumldn, () v () tengliklrd,, c ixtiyoriy sonlr () tenglikd es, istlgn sonlr, lekin c 0, chunki nolg o lish mumkin ems Hrflr yordmid juft v toq nturl sonlr formulsini yozish mumkin Agr juft son o ls, u hold u son g o lindi v uni undy yozish mumkin: = n, u yerd n nturl son Agr toq son o ls, u hold uni g o lgndgi qoldiq g teng, inorin, sonni undy yozish mumkin: = n +, u yerd n nturl son yoki nol B zn, toq nturl sonlr formulsini quyidgich hm yozishdi: = k, u yerd k nturl son Formullr oshq fnlrd hm or H O suvning, Og + Ch + U () lol gulining formulsi eknini kimyo, otnik drslrid o rgngnsiz Hrflrdn foydlnish ir xil toifdgi ko pgin msllrni yechish yo lini yozishg imkon erdi Shung doir msllr qrylik -msl Fermerning og mydoni to g ri to rturchk shklid o li, uning o yi kilometrg, eni es kilometrg 5

16 teng Yngi yer o zlshtirilgndn keyin mydonning yuzi 0,88 km g ortdi Bog mydonining yuzi qnch o ldi? Hisolshlrni: ) =, v =0,8 ) =,4 v =4, uchun jring Dstl og ning yuzi km g teng edi, yngi yer ochilgndn keyin u ( +0,88) km g teng o ldi ) =, v =0,8 o lgnd,, 0,8+0,88=,64 ) =,4 v =4, o lgnd,,4 4,+0,88=6,9 -msl Syyoh qishloqdn chiqi, shhr tomon jo - ndi U kilometr piyod yurgnidn keyin vtousg o tirdi v vtousd t sotd shhrg yeti keldi Agr vtous 60 km/sot tezlik iln hrkt qilgn o ls: ) =5 v t =0,5 o lgnd qishloq iln shhr orsidgi s msofni hisolng ) s =70, =0 o lgnd t ni toping Syyoh vtousd t sotd 60 t kilometr yo l osgn Shuning uchun qishloq iln shhr orsidgi msof s = +60t formul iln ifodlndi ) =5 v t =0,5 o lgnd, s =5+60 0,5=5 km o ldi s ) s = +60t formuldn t ni topmiz: t = Bu yerdn s =70, =0 o lgnd, t = 0 = sot 60 Mshqlr 6 Jumllrni mtemtik tild yozing: ) m v n sonlrning yig indisini ) v sonlrning yirmsini ) v sonlr yirmsining ikkilngnini 4) m v n sonlr ko pytmsining ikkilngnini 5) n v m sonlr yig indisining ulr yirmsig o linmsini 6) v sonlr yig indisining ulr yirmsig ko pytmsini 7 Quyidgi ifodlrd hrflr qndy sonlrni ifodlshi mumkin: ) tnffus n minut dvom etdi ) sinfimizd y nfr o quvchi or 6

17 ) 7-sinfd x t o quv fni o qitildi 4) ir oyd k kun or? 8 Yerning sun iy yo ldoshi 9 km/s tezlik iln hrkt qildi Ushu jdvlni to ldiring: Bosi o tilgn msof, km Hrkt vqti, s 9 Sprk vtomoili 00 km yo lg litr yonilg i srf qildi Ushu jdvlni to ldiring: Bosi o tilgn msof, km s Yonilg i srfi, l Birinchi qopd m kilogrmm, ikkinchi qopd es irinchi qopdgidn n kilogrmm km un or Ikkinchi qopd nech kilogrmm un or? Mslni ) m =50 v n = ) m =45 v n =5 hollr uchun yeching Piyod sotd 5 km yo l osdi U: ) sotd nech kilometr yo l osdi? ) k sotd-chi? Do kong hr irid 50 kg dn un o lgn t qop keltirildi Do kong nech kilogrmm un keltirilgn? Bog onlr kund 5 gektr og g ishlov erishdi Ulr kund nech gektr og g ishlov erishdi? 4 Hr iri x so mdn 6 t dftr v hr iri y so mdn o rm qog oz soti olindi Hmm xrid qnch turdi? 5 Yuk mshinsi do kong omordn hr iri kilogrmmdn 5 yshik olxo ri v hr iri kilogrmmdn 0 yshik olm keltirdi Do kong nech kilogrmm mev keltirilgn? 6 Mshing hr iri m kilogrmmdn k qop ug doy v hr iri n kilogrmmdn c qop rp yuklndi Mshing nech kilogrmm don yuklngn? 7 To g ri to rturchk shklidgi tjri mydonining o yi metrg teng, eni es o yidn metr qisq Shu mydonning yuzi S ning formulsini yozing Alger, 7- sinf 7

18 8 Kinotetrd hr iri n t o rindiqq eg o lgn m t qtor v yn k t qo shimch o rindiq or Kinotetrd hmmsi o li necht o rindiq or? Mslni yechish formulsini tuzing v m =0, n =5, k =60 o lgnd hisolshlrni jring 9 Drs jdvlid 5 t drs, ikkit 5 minutlik v ikkit 0 minutlik tnffus o lgn kuni o quvchi mktd nech sot o ldi? ( drs 45 minut) 40 O lchmlri -rsmd ko rstilgn shkllrning perimetrlrini v yuzlrini hisolsh uchun formullr yozing: n m n y ) x ) m n n m x y m y x 8 d) e) -rsm 4 To g ri to rturchkning o yi kvdrtning tomonidn 8 m uzun, eni es shu kvdrt tomonidn 4 m qisq Kvdrt tomonini iror hrf iln elgil, to g ri to rturchk uchun: ) tomonlrning uzunligini ) perimetrini )yuzini yozing 4 Avtous t sotd s kilometr yo l osdi Avtomoil xuddi shu yo lni vtousdn sot oldin osi o tishi uchun qndy tezlikk eg o lishi kerk?

19 4 x =+ (km) formul vtousning hrkti hqidgi msl yechilishini ildirdi Msl shrtini tuzing 44 Mkt tjri mydoni kvdrt metr yuzg eg Bog yuzi 500 m o lgn joyni egllgn, qolgn mydon 0 t ir xil mydonchg o lingn Shu mydonchlrning hr iri qndy yuzg eg? 45 Bnkk so m pul qo yildi Bir yildn so ng jmg rm p % ko pydi Bir yildn keyin jmg rmning miqdori nech so mg yetdi? 46 Asosi detsimetr, perimetri es 4 dm o lgn to g ri to rturchkning yuzini hisolsh uchun ifod tuzing ning ushu jdvld keltirilgn qiymtlri uchun to g ri to rturchk yuzi S ning qiymtini (dm lrd) hisolng: 5 6 7,5 0,5 5 S ¹ Fqt 4 t 9 v rifmetik ml elgilri yordmid qiymti 00 g teng o lgn sonli ifod tuzing 47 Velosiðedchi sotig v kilometr tezlik iln hrkt qilmoqd U jo nsh joyidn s kilometr uzoqlikd o lgn qishloqq orishi kerk Agr u km yo lni osi o tgn o ls, ung qishloqq yeti orishi uchun yn qnch vqt tl qilindi? Agr u km yurgn v s = 6 km, v = km/sot o ls, qishloqq,5 sotd yeti or oldimi? 48 Bir vtomoil 00 km yo lg o rtch 5 l, ikkinchi vtomoil es 00 km yo lg o rtch 0 l enzin srflydi Hr ir vtomoil kid litr enzin o ls, ulr qndy msofg or oldi? Agr =0 l v vtomoillr Toshkentdn ir vqtd Smrqndg qr yo lg chiqishgn o ls, qysi mshin Smrqndg yeti kel oldi? (Toshkent v Smrqnd shhrlri orsidgi msof 00 km) 9

20 4- Arifmetik mllrning xosslri Algerni puxt o rgnish uchun rifmetik mllrning xosslrini yxshi ilish lozim Eslti o tylik, rifmetik mllr de qo shish, yirish, ko pytirish v o lish mllrig ytildi Sonlr ustidgi u mllrning xosslrini qisqch formullr ko rinishid yozmiz Amllrning sosiy xosslri, odtd, qonunlr de tldi Qonunlrdn foydlni, mllrning oshq xosslrini hm soslsh mumkin Qo shish v ko pytirish Qo shish v ko pytirishning sosiy qonunlrini sn o tmiz O rin lmshtirish qonuni: Guruhlsh qonuni: Tqsimot qonuni: +=+, = ( + ) + c = + ( + c), ( ) c = c ( ) ( + c) = + c Bu tengliklrd,, c ixtiyoriy sonlr Msln: ,+,5=,5+, = ( 8) (5+ 7) = ( 8) 5 + ( 8) 7 Qo shish v ko pytirish qonunlri yordmid mllrning oshq xosslrini hm hosil qilish mumkin Msln: + + c+ d = + ( + c+ d),( c) d = ( )( cd), ( + + cd ) = d + d + cd 0

21 -msl Hisolng: Hisolshlrni ko rstilgn trtid oli orish mumkin: 75 g 7 ni qo shi, ntijg 5 ni qo shish v oxirgi ntijg ni qo shish Lekin qo shishning xosslridn foydlni, hisolshlrni soddlshtirish mumkin: = (75+ 5) + (7+ ) = = 50 Bu misol shuni ko rstdiki, mllrning xosslridn foydlni, hisolshlrni eng sodd (oqilon) usuld jrish mumkin Amllrning xosslri lgerik ifodlrni soddlshtirish mqsdid jrildign lmshtirishlrd hm qo llnildi -msl Ifodni soddlshtiring: (+ 4 ) + 5(7 + ) (+ 4) + 5(7 + ) = = = = (6+ 5 ) + (+ 5) = (6+ 5) + (+ 5) = 4+ 7 Bu mslni yechish jryonid quyidgi ifod hosil o ldi: Bu ifodd 6 v 5 qo shiluvchilr o xshshdir, chunki ulr ir-iridn fqt koeffitsiyentlri ilngin frq qildi v 5 qo shiluvchilr hm o xshsh Shu sli ifod o rnig 4 +7 ifodni yozish, y ni o xshsh hdlrni ixchmlsh mumkin o ldi Orliq hisolshlrni og zki jri, lmshtirishlr yozuvini qisqrtirish mumkin Msln, 6(x + 4) + ( x + ) = 8x + 4+ x + = 0x + 6 Ayirish -msl Toshkent v Smrqnd shhrlri orsid Jizzx shhri joylshgn Toshkentdn Smrqndgch o lgn msof 00 km, Toshkentdn Jizzxgch o lgn msof es 80 km Jizzxdn Smrqndgch o lgn msofni toping Jizzxdn Smrqndgch o lgn msof x kilometr o lsin U hold 80+ x = 00, u yerdn x = 00 80= 0 Jvo: 0 km

22 80 +x=00 tenglikdn x qo shish mlig teskri de tluvchi yirish mli yordmid topildi sondn sonni yirish uchun song song qrmqrshi o lgn sonni qo shish kifoy: = + ( ) Shu sli yirish mlining xosslrini qo shish mlining xosslri orqli soslsh mumkin Msln: 5 + (49 ) = = 87, + ( c) = + c, (+ 9) = 9= 6, ( + c) = c, (8 77) = 8+ 77= 7, ( c) = + c 4-msl Ifodning qiymtini hisolng: 4(x 5 y) + 6( x y), und x =, y = Avvl erilgn ifodni soddlshtirmiz: 4(x 5 y) + 6( x y) = x 0y + 6x 6y = 8x 6 y Hosil o lgn ifodning x =, y = dgi qiymtini hisolymiz: 8 6 = 9 = 7 Amllrning xosslridn foydlnish lgerik ifodni vvl soddlshtiri, so ngr uning qiymtini oson yo l iln hisolsh imkonini erdi Bo lish 5-msl To g ri to rturchkning yuzi 80 sm, tomonlridn iri 95 sm To g ri to rturchkning ikkinchi tomoni uzunligini toping S = formuldn =95 sm o lgni uchun Jvo: 4 sm = S ni topmiz S =80sm, = = 4sm 80sm 95sm

23 = S tenglikdn ko pytirish mlig teskri de tluvchi o lish mli yordmid topildi sonni song o lish uchun sonni sonig teskri o lgn song ko pytirish kerk: = : = Shu sli o lish mlining xosslrini ko pytirishning xosslridn keltiri chiqrish mumkin 6-msl Tenglikni isotlng: + = +, c c c u yerd c 0 Bo lishni ko pytirish iln lmshtiri, quyidgini hosil qilmiz: + = ( + ) c c Tqsimot qonunini qo ll, ( + ) = + c c c ni topmiz Ko pytirishni o lish iln lmshtiri, ni hosil qilmiz + = + c c c c Mshqlr 49 Arifmetik mllr qonunlri v xosslrini qo ll, sonli ifodning qiymtini toping: ) 9 0,45+ 0,45 ) (5,8+ 44,+ 48, 4,) ) 4,07 5,49+ 8,9,5 4),40,7+ 4,40 0,8

24 50 O xshsh hdlrni ixchmlng: ) 4+ + ) 0,c 0,+ d c, d ) x y x + 5 y 4) 8,7 m+ n m+ n 5 O xshsh hdlrni ixchmlng: ),m+ n,n+ m+, m n ), 0,7+,6 4) y y + ) 0, ,5,7 5 ) x + x + 6) 5,7p,7q+ 0,p+ 0,8q+,9 q p Ifodni soddlshtiring: ) (x + ) + 5(+ x) ) 0( n+ m) 4(m+ 7 n) ) 4( + x) ( + x) 4) (5 c+ d) + ( d + c) 5 Ifodni soddlshtiring v son qiymtini toping: ) 5(x 7) + ( x), und x = 6 ) 7(0 x) + (x ), und x = 0,048 ) (6x ) + (5x 5), und x =,0 5 4) 0,0(,x 0,) + 0,( x 00), und x = 0 54 Arifmetik mllrning xosslridn foydlni hisolng: ) 6 (0,4+,,5) ) (8 + ): ) (4,8 0,4,) 4) 5 5 (5 + 0 ) Qvslrni ochish qoidlri Algerik yig indi -msl Yigirm qvtli inod lift ishlmoqd U 8-qvtdn 6 qvt pstg tushdi, so ngr qvt yuqorig ko trildi, 4 qvt pstg tushdi, 7 qvt yuqorig ko trildi, qvt pstg tushdi Lift qysi qvtd turidi? 4

25 Liftning qysi qvtd turgnligini topish uchun ifodning qiymtini hisolsh kerk Bu qiymt 4g teng Demk, lift 4-qvtd turidi Siz 6-sinf mtemtik kursidn ifod lgerik yig indi de tlishini ilsiz, chunki uni yig indi shklid undy yozish mumkin: 8 +( 6) + +( 4) +7+( ) Algerik yig indilrg oid yn misollr keltirmiz: ( 7) + ( ), + c d, + ( ) ( c) ( c) sonni yirish ( c) song qrm-qrshi sonni, y ni c sonni qo shishni ildirishini eslti o tmiz Shuning uchun oxirgi lgerik yig indini undy yozish mumkin: + ( ) + c Algerik yig indi u + v ishorlri iln irlshtirilgn ir necht lgerik ifodlrdn tuzilgn yozuvdir Odtd, ( 7) + ( ), + ( ) ( c ) ko rinishidgi lgerik yig indilr qisqch undy yozildi: ( 7) + ( ) = ( ) ( c) = + c +7 lgerik yig indid qo shiluvchilr, 7 v sonlri o ldi, chunki +7 =+7+( ) + c lgerik yig indid qo shiluvchilr,, c sonlr o ldi, chunki + c = + ( ) + c Qvslrni ochish v qvs ichig olish +(+c) ifodni qrymiz: qo shishning guruhlsh qonunini qo ll, uni undy yozish mumkin: + ( + c) = + + c Bu tenglikd c ni d iln lmshtirmiz: + ( d) = + d 5

26 Qvs oldid + ishorsi turgn ifodlrd lmshtirishlr jrish shu tengliklrg soslngn Bu tengliklr qvslrni ochishning quyidgi irinchi qoidsig oli keldi: Agr lgerik ifodg qvs ichig olingn lgerik yig indi qo shildign o ls, u hold shu lgerik yig indidgi hr ir qo shiluvchining ishorsini sqlgn hold qvslrni tushiri qoldirish mumkin Msln: ) 4 + (7 + ) = ) + ( + c d) = + + c d ) ( ) + c = + c Qvs oldid ishorsi turgn ifodlrd lmshtirishlr jrish yirish mlining quyidgi xosslrig soslngn: ( ) =, ( + ) =, ( + c) = c, ( c) = + c Bu tengliklrdn qvslrni ochishning quyidgi ikkinchi qoidsi keli chiqdi: Agr lgerik ifoddn qvs ichig olingn lgerik yig indi yirils, u hold shu lgerik yig indidgi hr ir qo shiluvchining ishorsini qrm-qrshisig o zgrtiri, qvslrni tushiri qoldirish mumkin 6 Msln: ) 4 (7 + ) = 4 7+ ) ( + c d) = c+ d ) ( ) + c = + + c -msl Qvslrni ochi soddlshtiring: x + (5 (8x + )) x + (5 (8x + )) = x + 5 (8x + ) = x + 5 8x = 5 x

27 B zn ir nech qo shiluvchini qvs ichig olish foydli o ldi Msln: ) = 08 (7 7) = 08 00= 8 ) + c+ d = + ( c+ d) Bu yerd qvs oldig + elgisi qo yilgn, shuning uchun qvs ichidgi rch qo shiluvchilrning ishorlri sqlni qoldi ) c+ d = ( + c d) Bu yerd qvs oldig elgisi qo yilgn, shuning uchun qvs ichig olingn rch qo shiluvchilrning ishorlri qrm-qrshisig o zgrtirildi 55 Algerik yig indini qvslrsiz yozing: ) ( + 4) + ( ) ( + 7) ) ( ) + ( 7) + c ) ( 4) + ( 9) ( ) 4) + ( ) 4c 56 Algerik yig indining qo shiluvchilrini yting: ) 5 c ) m 7 ) +47 4) 57 Algerik yig indi shklid yozing: ) + c ) + c ) 4) + c Qvslrni oching (58 59): 58 ) + ( ) c ) (+ ) c ) ( ) c 4) ( + ) c 59 ) + ( ( c d )) ) (( c) d) ) ( ( c d )) 4) ( + ( c ( d k))) 60 Qvslrni oching v soddlshtiring: Mshqlr ) ( + ) ) m (5 m (m )) ) 5 x (y x ) 4) 4 + ( (+ )) 7

28 6 m yoki ( m) sonlridn oshl, rch qo shiluvchilrni qvs oldig + ishorsini qo ygn hold qvs ichig oling: 8 ) + + m c ) m+ c+ 4 d ) + m+ c 4) m+ 6 m yoki ( m) sonlridn oshl, rch qo shiluvchilrni qvs oldig ishorsini qo ygn hold qvs ichig oling: ) + + m c ) c m + ) + + m+ c 4) m+ 6 ) + ifodni iri g teng o lgn ikkit qo shiluvchining yig indisi shklid yozing ) + ifodni kmyuvchisi o lgn yirm shklid yozing ) +4 ifodni kmyuvchisi o lgn yirm shklid yozing 4) +8 ifodni iri 8 g teng o lgn ikkit qo shiluvchining yig indisi shklid yozing 64 Tengliklrning chp qismlri ir xil Neg o ng qismlri hr xil? Qndy shrtlrd tenglik o rinli o ldi? ) :5 40 =0 ) :5 40 =90 ) :5 40 =40 4) :5 40 =0 5) :5 40 =70 6) :5 40 =407 7) :5 40 =50 65 Ko p nuqtlr o rnig + v ishorlrini shundy qo yingki, ntijd to g ri tenglik hosil o lsin: ) ( + c) = + ( c ) ) m ( n ) = m+ ( n ) ) c ( ) = c+ ( ) 4) n ( d l) = n+ ( d l)

29 O zingizni tekshiri ko ring! Hisolng: ) (7, 4,0+ 4,0,8): ) 5 0,0 4 Ifodni soddlshtiring v x = = 9, y 0,5 o lgnd uning son qiymtini toping: ( y x) ( y x) Bollr oromgohi uchun 0 t shxmt v 5 t koptok soti olishdi Bitt shxmt so m, itt koptok so m turdi Jmi xrid uchun qnch pul to lngn? 66 Soddlshtiring: ) (5 ) ( 5 ) ) 7x + y ( x + y) ) (6 ) (+ ) 4) 8 x (x y) 5 y 67 Tenglmni yeching: ) (x + ) + x = 6 ) ( x 5) (5 x) = ) ( x 4) + ( x + 6) = 4 4) ( x + 5) = 68 Ifodni vvl soddlshtiri, keyin uning son qiymtini toping: ) (c+ 5 d) ( c+ 4 d ), und c = 0,4, d = 0,6 ) ( 4 ) ( ), und = 0,, =,8 ) (7x + 8 y) (5x y ), und x = = 4, y 0,05 4) (5c 6) (c 5), und c = 0,5, = 9

30 I og doir mshqlr Algerik ifodning son qiymtini hisolng (69 75): 69 ) + c, und =, =, c = 0 ) c, und =, =, c = ) ( + c ), und =, =, c = 4) ( c ), und =, =,, c = 5 5) ( ) + ( c d), und = 4, =, c =, d = 6) ( ) ( c d), und = 0, = 4, c =, d = 7) ( c), und = 0,5, =, c =, 8) ( c) d, und = 5,, =,, c =,8, d =,8 70 ) 5( x y ) ) ( x + y ) ) (5 x y ) 4) ( x + y), und x =,5, y = 4,5 7 ) (( ) + ) ) (( ) 8): ) 4( ( )) 4) (5 ( + )):, und = 5, = 7 ) ( + ) ) ( ) + ) + 4) +, und =,, =,8 7 ) c, und =, c = ) 0,75 +, und =, = ) ( 6), und = 5 4) ( + 6), und = 74 Ifodlrning geometrik m nosini oching ), und v to g ri to rturchkning tomonlri ), und kvdrtning tomoni uzunligi ) (+), und v to g ri to rturchkning tomonlri uzunligi 4) 4, und kvdrtning tomoni 75 ) 4x, und ktt kvdrtning tomoni, x hr ir kichik kvdrtchning tomoni uzunligi (- rsm) 0

31 ) ) x y -rsm x ), und v ktt to g ri to rturchkning, x v x + y y es kichik to g ri to rturchklrning tomonlri (- rsm) ¹ 4-rsmd necht uchurchk, kvdrt v to g ri to rturchk or? 4-rsm 76 Bir gektr ko klmzor ir yil dvomid hvoni 70t chngdn tozlshg qodir 0 g 00 g m gektr ko - klmzor ir yild hvoni nech tonn chngdn tozlydi? Umumiy mydoni g o lgn ko klmzor hvoni nech tonn chngdn tozlydi? 77 Avtomoilning hrkt tezligi ikki mrt ortishi iln uning tormozlnish yo lining to rt mrt ortishi m lum Hrkt tezligi 0 km/sot o lgnd, tormozlnish yo lining uzunligi jdvld erilgn Tezlik 60 km/sot o lgnd, tormozlnish yo lining uzunligi qnch o ldi (5-rsm) Yuk mshinsi uchun v (km/sot) s (m) 0 9,5 Yengil mshin uchun v (km/sot) s (m) 0 7,5? m? m Tormoz yo li 5-rsm

32 78 (Au Ryhon Beruniy mslsi) Agr 0 dirhm pul ikki oyd 5 dirhm foyd keltirgn o ls, 8 dirhm puldn uch oyd qnch foyd olish mumkin? I og doir sinov mshqlri testlr = 5,, = 4,7 o ls, P = ( + ) ifodning son qiymtini toping A) 96 B) 9,6 C),96 D) 8,6 To g ri to rturchkning yuzi S g, sosi g teng Uning perimetrini topish uchun ifod tuzing S S A) + B) + C) S S + D) + Teng yonli uchurchkning perimetri P g, sosining uzunligi g teng Uchurchkning yon tomoni uzunligini topish uchun ifod tuzing A) P B) P C) P D) ( P ) 4 =,5, =,4 v c =,5 o ls, V = c ifodning son qiymtini toping A) 8, B) C), D), 5 = 5, = 6,4, c = 4,5 o ls, S = ( + c+ c ) ifodning son qiymtini toping A) 50,45 B) 8, C) 66,6 D) 09 6 On frzndlri uchun so mdn 8 t rsm dftr, so mdn 5 t ruchk, c so mdn 0 t dftr soti oldi Jmi xridni hisolsh uchun ifod tuzing A) 8+5+0c B) 8+5 (+c) C) 800c D) Qvslrni oching v soddlshtiring: 5 + ( (4+ )) A) 8 + B) 4 C) 4 D) 4

33 8 Ifodni soddlshtiring v uning =,4 =,5 o lgndgi qiymtini toping: 0,5 ( ) (4+,5 ) A) 7,4 B) 7,4 C),4 D),85 9 To g ri to rturchkning perimetri p g, sosi g teng Uning lndligini hisolsh uchun ifod tuzing A) p B) p C) p D) p 0 Ifodni soddlshtiring v uning à =,7, =4, o lgndgi son qiymtini toping: ( ) ( ) A) 4,6 B) 7,6 C) 8,7 D) 5 Uchurchk ir tomonining uzunligi g teng Ikkinchi tomoni uzunligi u tomonning 80 % ini tshkil qildi Uchinchi tomoni es irinchi v ikkinchi tomonlr yig indisining yrmig teng o ls, shu uchurchkning perimetrini toping A),8 B),7 C) D) +0,8 Agr h =6, r =, R =4 o ls, V = πhr ( + Rr + r ) ifodning son qiymtini toping A) 56π B) 55π C) 84π D) 8π Agr R =4,5, v H =6,5 o ls, S =π R(R+H) ifodning son qiymtini toping A) 00π B) 98π C) 99π D) 98,5π 4 Uchurchk ir tomonining uzunligi g teng o li, u ikkinchi tomonidn sm qisq, uchinchi tomonidn es sm uzun Shu uchurchkning perimetrini hisolsh uchun ifod tuzing A) B) 5 C) +5 D) Alger, 7- sinf

34 Trixiy m lumotlr Yurtdoshimiz uyuk mtemtik v stronom olim Au Adulloh Muhmmd in Muso l-xorzmiy (78 850) ning rifmetik ( Algorizmi hind hisoi hqid ) v lgerik ( Aljr vl-muqol ) srlri mtemtikning rivojig nihoytd kuchli t sir ko rstdi Bu srlr ko p tillrg trjim qilindi, srlr dvomid mtemtikdn sosiy qo llnm o - li xizmt qildi Algorizmi hind hisoi hqid risolsining XII sr oshidgi lotinch trjimsi Angliyning Kemrij universitetid sqlndi Al-Xorzmiyning u sri tufyli Yevropg o nli snoq sistemsi kiri orgn Muhmmd Muso Xorzmiyning o nlik snoq sistemsini, lgoritm v lger tushunchlrini dunyod irinchi o li ilm-fn sohsig joriy etgni v shu sosd niq fnlr rivoji uchun o z vqtid musthkm sos yrtgni umuminsoniy trqqiyot rivojid qndy ktt hmiytg eg o lgnini rchmiz yxshi ilmiz, de yozgn edi O zekiston Respuliksining irinchi Prezidenti IAKrimov o zining Yuksk m nviyt yengilms kuch srid Xorzmiy lgersi Al-jr vl-muqol hisoi hqid qisqch kito srining rch nusxsi Oksford universitetining Bodleyn kutuxonsid sqlndi Risol uch qismdn iort: ) lgerik qism ) geometrik qism ) vsiytlr hqidgi qism (Xorzmiy uni Vsiytlr kitoi de tgn) Al- Xorzmiy risolsid rch msllrning yoni v yechimlri so zlr iln erildi, hech qndy elgilshlr, hrfiy ifodlr ishltilmydi Al-Xorzmiy yozdi: Men rifmetikning oddiy v murkk msllrini o z ichig oluvchi Al-jr vl-muqol hisoi hqid qisqch kito ni t lif qildim, chunki meros tqsim qilishd, vsiytnom tuzishd, mol tqsimlshd v dliy ishlrid, svdod v hr qndy itimlrd v shuningdek, yer o lchshd, riqlr o tkzishd, muhndislikd v oshq shung o xshsh turlich ishlrd kishilr uchun u zrurdir Binorin, olim o zining u srini kundlik hyot tli v ehtiyojlrini hisog olgn hold yozgn 4

35 6- BIR NOMA LUMLI BIRINCHI DARAJALI TENGLAMALAR Tenglm v uning yechimlri Ushu mslni yechylik Msl Qlm v chizg ich irglikd 70 so m turdi Qlm chizg ichdn 90 so m rzon Chizg ichning hosini toping Aytylik, chizg ich x so m tursin, u hold qlm (x 90) so m turdi Mslning shrtig ko r x+( x 90)=70, undn x 90= 70, x = 460, x = 0 Jvo: Chizg ich 0 so m turdi x+ ( x 90) = 70 tenglikd x hrfi nom lum sonni yoki qisqch nom lumni ildirdi Hrf iln elgilngn nom lum son qtnshgn tenglik tenglm deyildi Tenglik elgisidn chp v o ngd turgn ifodlr tenglmning chp v o ng qismlri deyildi Tenglmning chp yoki o ng qismidgi hr ir qo shiluvchi tenglmning hdi deyildi x 90= 70 tenglmd chp qism x 90, o ng qism es 70 So ngr x =0 o lgnd shu tenglmning chp qismi 70 g teng, chunki 0 90=70 o ng qismi hm 70 g teng Demk, x =0 o lgnd u tenglm to g ri tenglikk ylndi: 0 90=70 Shu 0 soni erilgn tenglmning ildizi deyildi Tenglmning ildizi de, nom lumning shu tenglmni to g ri tenglikk ylntirdign qiymtig ytildi 5

36 6 Msln, soni x + = 5 tenglmning ildizi, chunki +=5 to g ri tenglik Tenglm ikkit, ucht v hokzo ildizlrg eg o lishi mumkin Msln, ( x )( x )=0 tenglm ikkit ildizg eg: v, chunki x = v x = d tenglm to g ri tenglikk ylndi ( x )( x+ 4)( x 5) = 0 tenglm es ucht ildizg eg:, 4 v 5 Tenglm ildizlrining soni cheksiz ko p o lishi mumkin Msln, ( x ) = x tenglmning ildizlri soni cheksiz ko p: x ning istlgn qiymti tenglmning ildizi o ldi, chunki hr ir x d tenglmning chp qismi o ng qismig teng Tenglm ildizlrg eg o lmsligi hm mumkin Msln, x + 5= x + tenglmning ildizlri yo q, chunki x ning istlgn qiymtid u tenglmning chp qismi o ng qismidn ktt o ldi Tenglmni yechish uning rch ildizlrini topish yoki ulrning yo qligini ko rstish demkdir Sodd hollrd x ning tenglmning ildizi o ldign qiymtini tnlsh oson o ldi Msln, x + = tenglmning ildizi soni eknligini osongin ko rish mumkin Biroq murkk hold ildizni irdnig topish oson o lmydi Msln, x ( x ) x = + 7 x x 5 0 tenglm x =7 o lgnd to g ri tenglikk ylnishini ilish nch qiyin Shuning uchun tenglmlrni yechishni o rgnish muhim

37 Ko pgin mliy msllrni yechish x = () ko rinishdgi tenglmg keltirildi, und v erilgn sonlr, x nom lum son () tenglm chiziqli tenglm de tldi Msln, x=, x =, x = chiziqli tenglmlrdir 5 Mshqlr 79 Tenglik shklid yozing: ) 4 soni x sondn 8 t ortiq ) 56 soni 4 sonidn x mrt ortiq ) x v sonlri yirmsining ikkilngni 4 g teng 4) x v 5 sonlri yig indisining yrmi ulrning ko pytmsig teng 80 sonlridn qysi iri tenglmning ildizi o ldi: ) x = 6 ) 4x 4= x+ 5 ) x + = 6 4) 5x 8= x + 4? 8 (Og zki) x ning qndy qiymtlrid tenglm to g ri tenglikk ylndi: ) x + 5= 6 ) 4 x = ) x = 0 4) x + = 0? 8 sonlri orsid tenglmning ildizi ormi: ) 4( x ) = x ) ( x+ ) = 4+ x ) 7( x+ ) 6x = 0 4) 5( x+ ) 4x = 4? 8 Ildizi: ) 5 soni ) soni ) 6 soni 4) 4 soni o lgn tenglm tuzing 84 sonni shundy tnlngki, 4x = x + tenglm ) x = ) x = ) x = 4) x =0, ildizg eg o lsin 7

38 7- Bir nom lumli irinchi drjli tenglmlrni yechish Al-Xorzmiyning Kito l-muxtsr fi hiso l-jr vlmuqol sridgi l-jr must hdlrni tiklsh, y ni mnfiy hdlrni tenglmning ir qismidn ikkinchi qismig must qili o tkzishni, vl-muqol es tenglmning ikkl qismidn teng hdlrni tshl yuorishni ildirgn Bu ir nom lumli tenglmlrni yechish to g ri tengliklrning sizlrg m lum xosslrig soslngn eknini ko rstdi Shu xosslrni eslti o tmiz Birinchi xossdn qo shiluvchilrni, ulrning ishorlrini qrm-qrshisig lmshtiri, tenglikning ir qismidn ikkinchi qismig oli o tish mumkinligi keli chiqdi 8 Xossning so z iln ifodlnishi Agr to g ri tenglikning ikkl qismig ir xil son qo shils yoki ikkl qismidn ir xil son yrils, u hold yn to g ri tenglik hosil o ldi Agr to g ri tenglikning ikkl qismi nolg teng o lmgn yni ir song ko pytirils yoki o lins, u hold yn to g ri tenglik hosil o ldi Xossning umumiy ko rinishd yozilishi Agr = o li, l ixtiyoriy son o ls, u hold +l=+l, l= = l o ldi Agr = o li, m 0 o ls, u hold m= m v : m= =:m o ldi Aytylik, = + m o lsin, u hold + ( m) = + m+ ( m) m= Tengliklrning u xosslri tenglmlrni yechishd qndy qo llnishini ko rylik -msl 9x = 5x tenglmni yeching Misol 7=7 7+=7+ 7 =7 7=7 7 =7 7 : =7 : x son erilgn tenglmning ildizi, y ni x shundy sonki, uni tenglmg qo yilgnd tenglm to g ri tenglikk ylndi, de frz qilmiz

39 Nom lum qtnshgn 5x hdni ishor iln tenglikning chp qismig, hdni + ishor iln o ng qismig oli o tmiz Ntijd, yn to g ri tenglik hosil o ldi: 9x 5x = Tenglmning ikkl qismidgi o xshsh hdlrni ixchml, 4x = tenglmni hosil qilmiz Bu tenglmning ikkl qismini 4 g o li, x = eknini topmiz Shundy qili, tenglm ildizg eg de frz qili, u ildiz fqt sonig teng o lishi mumkinligini ko rdik x = hqiqtn hm erilgn tenglmning ildizi o lishini tekshirmiz: 9 =5 Bu to g ri tenglik, chunki uning chp v o ng qismlri yni ir song 4 sonig teng Demk, erilgn tenglm fqt itt ildizg eg: x = Tekshirishni jrmslik hm mumkinligini t kidlymiz, chunki tenglikning foydlnilgn xosslri ir to g ri tenglikni ikkinchi to g ri tenglik iln lmshtirishg imkon erdi Yechishning u usulid hr doim to g ri ntij hosil qilindi (gr hisolshlrd xtolikk yo l qo yilms, ltt) x 5x 7=x+ +7 5x x = + 7 AL-JABR: x, chpg x o li o tsn! 7, sen o ngg +7 o li o tsn! 4x 5 + x =4x+8 5 x=8 VAL-MUQOBALA: chp v o ng qismdgi 5 lr-u, 4x lr, sizlr iln xyrlshmiz! 9

40 Tenglm yechilishini yozishd - mslni yechishdgidek tfsil yozm tushuntirishlrni jrish shrt ems Msln, 5x + 7= x + 5 tenglmning yechilishini undy yozish mumkin: 40 Jvo: x = 4 5x x = 5 7, x =, x = 4 -msl ( x+ ) ( x+ ) = 5 4( x + ) tenglmni yeching Tenglmning chp v o ng qismlrini soddlshtirmiz: qvslrni ochmiz v o xshsh hdlrni ixchmlymiz Ntijd, x+6 x 6=5 4x 4, x = 4x + tenglmni hosil qilmiz Demk, x=, undn x = Jvo: x = -msl 5 x x x = tenglmni yeching Tenglmning ikkl qismini ksrlrning umumiy mxrjig, y ni 6 g ko pytirmiz, u hold 5x x x = x ( x ) = 6 + ( x 5) Qvslrni ochmiz v o xshsh hdlrni ixchmlymiz: 5x x+ 6= 6+ x 5 x+ 6= x+, undn x = 5, x = 5 Jvo: x 5 = Shundy qili, tenglmni yechishd tenglmning quyidgi sosiy xosslridn foydlnildi -xoss Tenglmning istlgn hdi ishorsini qrmqrshisig o zgrtiri, uning ir qismidn ikkinchi qismig o tkzish mumkin -xoss Tenglmning ikkl qismini nolg teng o lmgn ir xil song ko pytirish yoki o lish mumkin Bu xosslr ir nom lumli istlgn tenglmni yechish imkonini erdi Buning uchun: ) nom lum qtnshgn hdlrni tenglikning chp

41 qismig, nom lum qtnshmgn hdlrni es o ng qismig o tkzish lozim (-xoss) ) o xshsh hdlrni ixchmlsh kerk ) tenglmning ikkl qismini nom lum oldid turgn koeffitsiyentg (gr u nolg teng o lms) o lish (- xoss) kerk Ko ri chiqilgn misollrd hr ir tenglm itt ildizg eg o ldi Ammo zi hollrd ir nom lumli tenglm ildizlrg eg o lmsligi mumkin yoki cheksiz ko p ildizg eg o lishi mumkin Shundy tenglmlrg misollr keltirmiz 4-msl ( x+ ) = ( x ) tenglm ildizlrg eg emsligini ko rsting undn Tenglmning ikkl qismini soddlshtirmiz: x+ = + x, x + = + x, x x=, 0x= Bu tenglm ildizlrg eg ems, chunki uning 0 x dn iort chp qismi nolg teng, o ng qismi es g teng, mmo 0 Jvo: tenglm yechimg eg ems 5-msl ( x) + = 5 x tenglm cheksiz ko p yechimg eg eknligini ko rsting Tenglmni soddlshtirmiz: x+ = 5 x 5 x = 5 x x + x = 5 5, 0 x = 0 Demk, x ning istlgn qiymti u tenglmning ildizi o - ldi Jvo: tenglm cheksiz ko p yechimg eg Tenglmni yeching (85 96): Mshqlr 85 ) x = 50 ) 9x = 4 ) 4x = 0,4 4) 7x = 7,06 86 ) 9x = 5 ) x = 7 ) x = 4) = 4 x 4

42 87 ) 0,x = 6 ),x =,69 ) 0,7x = 49 4) 0x = 0,5 88 ) 8x = 8 ) x = 6 ) 4 89 ) 5 4 5x = ) 4x = ) ) 5x = 9 ) x 5= 0 x ) 7x + 8= 4) 4x + 4= x + 5 x = 4 4) 6x = 6 0,x = 0 4) 0,x = 0 9 ) 5x+ (x+ 7) = 5 ) 8y 9 4y + 5= y 4 5 y ) 8 x (7x+ 8) = 9 4) 4+ 8y + 8= y 0 7y ) x = 7 5 ) x 6+ x = 5 x x 5 ) + = 8 y y 4 4) + = 4 9 ) 4 + 5= 4 9 y x y ) 5+ = 4+ x 4 ) 8 z = 6z 44 4) = 5+ x 94 ) 0,7x+,98= 0,7x,76 ) 0,8y 7,4= 0,05y 5,7 ) 5(5x ),7x+ 0,x = 6,5 0,5 x 4) 0,6x 0,6= 0,(0,4x,) 95 ) = + x x ) 6 x + 7 = 5 x 7 8 ) + y = x 6x + y 0 4) 0 = x 5 7 x x 5 ) 4 96 ) + = ) x 9x 5 + 5x 8x = 4 x 7 9x + x = 4) 4x 5 x x 4 = 4 8

43 ¹ Buvijon, nirngiz nech yoshd? Mening yoshim nechd o ls, nirm shunch oylik Buvijon, sizning yoshingiz nechd? Nirm yoshi iln mening yoshimni qo shsng, 65 chiqdi Nirmning yoshini endi o zing top qol 97 Tenglmning ildizlrg eg emsligini ko rsting: ) 8 0x = x + 5 6x 6x ) 5x 7 = 4x 5 x x ) x 5x + 5+ x x + 7x + 5 x + = 4) = x ning istlgn qiymti tenglmning ildizi o l olishini ko rsting: ) 0 4x+ = 9x 6x+ 9 7x+ 6 ) 9x + 4 5x = 8+ 7x 9 x + 5 ) 6(,x 0,5),x = 5,9x 4) 8(,x+ 0,5) 6,6x =,8x+ 99 Tenglmni yeching: ) ( x ) ( x + ) = 4x + 8 ) 4( x +,5) + ( x) = 0 ) 4(x + ) 7( x + ) = ( x ) 4),5(x + ) ( x +,5) =,5+ x 00 Tenglmni yeching: ) ) 96 4x + 00 = 7, x + 4,7 7,5 = 0,4 0 ) 4,:(x 7) = 0:7 7 4) 4 :0 = 4,5:(x ) 4

44 8- Msllrni tenglmlr yordmid yechish Tenglmlrni qo llsh ko pgin msllrni yechishni osonlshtirdi Bund mslni yechish, odtd, ikki osqichdn iort o ldi: ) mslning shrti o yich tenglm tuzish ) hosil o lgn tenglmni yechish Ushu mslni yechylik Msl Syyohlr tushgn kem sohildgi ektdn dryo oqimi o yich jo n, 5 sotdn keyin qyti kelishi kerk Dryo oqimining tezligi km/sot kemning turg un suvdgi tezligi 8 km/sot Agr syyohlr qytishdn oldin qirg oqd sot dm olgn o lslr, ulr sohildgi ektdn qnch msofg suzi orgnlr? ) Izlnyotgn msof x kilometr o lsin Kem u msofni oqim o yich 8 + = (km/sot) tezlik iln o tdi x v ung sot srf qildi Kem 8 = 5 (km/sot) tezlik iln orqsig qytdi v ung 5 x sot srf qildi Syyohlr x x qirg oqd sot dm oldilr Demk, syoht + + sot 5 dvom etdi, u es msl shrtig ko r 5 sotg teng Shundy qili, iz nom lum x msofni niqlsh uchun quyidgi tenglmni hosil qildik: x x + + = 5 5 ) endi x x + = 5 tenglmni yechmiz Bu tenglmning ikkl qismini 05 g ( v 5 sonlrining eng kichik umumiy o linuvchisig) ko - pytiri, 5x+7x=0, x =0 tenglikni hosil qilmiz, undn x =7,5 Jvo: kem sohildgi ektdn 7,5 km msofg suzi ordi 44

45 Mslni yechishning irinchi osqichid (y ni tenglm tuzishd) kem iln dryo oqimi tezliklri oqim o yich hrktd qo shilishi, oqimg qrshi hrktd es yirilishi v yo lning tezlikk nisti hrkt vqti eknligini ilish zrur o ldi Ikkinchi osqichd (y ni hosil o lgn tenglmni yechishd) tenglmlrning undn oldingi prgrfd o rgnilgn xosslrini qo llsh tl etildi Mtnli msl mzmunig mos tenglm tuzish msl shrtini mtemtik tili g o tkzish mslning mtemtik modelini tuzish demkdir Bitt mslni hl qilish uchun turli tenglm, turli mtemtik model tuzish mumkin 0 A v B shhrlri orsidgi msof 56 km A dn B g qr 66 km/sot tezlik iln yuk poyezdi yo lg chiqdi Ordn 0 minut o tgch, B dn A g qr 90 km/sot tezlik iln tezyurr poyezd yo lg chiqdi Yuk poyezdi yo lg chiqqnidn qnch vqtdn so ng tezyurr poyezd iln uchrshdi: Bu mslni yechish uchun tenglmlrni quyidgich tuzish mumkin: ) 66x + 90 x = 56 ) = x d) x 56 x = ( ) Mshqlr e) 56 90x = 66 x + )Hr ir tenglmd x nimni ildirdi? ) Hr ir tenglmd qndy miqdorlr tenglshtirilgn? 0 ) Belgilngn ishni 5 kishi kund jrishi mumkin 4 kun ishlgndn so ng, eshinchi kuni ulrg yordm 45

46 46 erish uchun 5 kishi keli qo shildi Qolgn ish nech kund tugtilgn? ) Ishchilr elgilngn vzifni 5 kund jr olishdi, 5 kundn so ng ulrg yn 8 kishi qo shildi v irglikd qolgn ishni 6 kund tugllshdi Ishchilr dstl nech kishi edilr? ) Bir ishni 0 kishi 8 kund jr oldi kundn so ng (uchinchi kuni) ulrg yordm erish uchun ir necht kishi keli qo shildi v qolgn ish 4 kund jrildi Necht kishi keli qo shilgn? 0 ) Ucht firmd 64 nfr ishchi or Ikkinchi firmd irinchisidgig qrgnd ishchilr 5 mrt ko p, uchinchi firmd es irinchi v ikkinchi firmlrd irglikd necht ishchi o ls, shunch ishchi or Hr ir firmd nechtdn ishchi or? ) Ucht kichik korxond 79 t mhsulot tyyorlndi Ikkinchi kichik korxond irinchi kichik korxong qrgnd mrt ko p, uchinchi kichik korxond es ikkinchisidgidn mrt km mhsulot tyyorlndi Hr ir kichik korxond nechtdn mhsulot tyyorlngn? 04 ) Teng yonli uchurchkning perimetri 5 sm g teng Agr uning yon tomoni sosidn 5 sm ortiq o ls, uchurchk tomonlri uzunliklrini toping ) Teng yonli uchurchkd sos yon tomonning 4 qismini tshkil etdi Agr uchurchkning perimetri sm g teng o ls, uning tomonlri uzunliklrini toping 05 ) Eni 00m o lgn to g ri to rturchk shklidgi mydonning chegrsi o yl riq qzildi Ariqning uzunligi km Mydonning o yini toping ) Bo yi enidn mrt uzun o lgn to g ri to rturchk shklidgi mydonni uzunligi 0 m o lgn pnjr iln o rshdi Mydonning o yi v enini toping 06 Yig indisi 8 g teng o lgn ucht ketm-ket toq sonni toping

47 07 To rtt ketm-ket juft son erilgn Agr chetki sonlr yig indisining ikkilngnidn o rtdgi sonlr must yirmsining uchlngni yirils, hosil o ldi Shu sonlrni toping 08 ) Yngi moslmni ishg tushirgch, ustning mo ljllngn ishni jrishg ketdign vqti 0% g kmydi Uning mehnt unumdorligi nech foizg oshgn? ) Frikg vtomt o rntildi U ir sotd ishchig qrgnd 8 t ortiq mhsulot ishl chiqrdi sotdn keyin vtomt ishchining 6 sotlik rejsini jrdi Avtomt ir sotd necht mhsulot ishl chiqrdi? ) Ustning mehnt unumdorligi 0 % g orts, uning ish rejsini jrishg ketdign vqti nech foizg qisqrdi? 09 Uzunligi 7 m o lgn mis simni msssi v ko ndlng kesimi mis simnikidek o lgn lumin sim iln lmshtirishmoqchi Nim de o ylysiz, lumin simning uzunligi nech metr o lrkin? 0 Bir necht do kon olm solingn 75 t yshikni teng o li olishmoqchi edi Ammo t do kon olmlrni olmsligini ildirdi Ntijd, qolgn hr ir do kong mo ljldgidn 0 yshik olm ortiqch erildi Do konlr necht ekn? ) Idishd qnchdir litr suv or Agr idishg l suv quyils, idishning yrmi to ldi Agr l suv to ki tshlns, qolgn suv idishning 8 qismini egllydi Dstl idishd nech litr suv o lgn? ) Idishning ichidgi suv iln irglikdgi msssi kg g teng Idishdgi suvning qismi gullrg quyilgch, 5 idishning msssi ichidgi suv msssidn mrt kmligi niqlndi Idishning msssi nech kilogrmm ekn? ) Neft omorid 640 t enzin or edi Ikkinchi kuni omor irinchi kundgidn 4 t ko p, uchinchi kuni es 47

48 ikkinchi kundgidn 04 t km enzin trqtdi Shundn so ng omord 96 t enzin qoldi Omor irinchi kuni nech tonn enzin trqtgn? ) Do kond uch kund 0 kg yog sotildi Ikkinchi kuni irinchi kundgining 7,5% qismich, uchinchi kuni es dstlki ikki kund qnch yog sotilgn o ls, shunch sotildi Do kond irinchi kuni nech kilogrmm yog sotilgn? ) Ust v o g li uyurtmni 0 kund jrishlri kerk edi Ulr yngi moslmni ishlti, hr kuni rejdn tshqri 7 t mhsulot tyyorl, 7 kund topshiriqni jrigin qolmsdn, lki ortiqch yn 54 t mhsulot tyyorlshdi Ust v o g li ir kund necht mhsulot tyyorlshgn? ) Zvod mshin ishl chiqrish o yich uyurtmni 5 kund jrishi kerk edi Zvod yngi texnologiyni joriy eti, hr kuni rejdn tshqri t ortiq mshin ishl chiqri, muddtg kun qolgnd fqt rejni jrigin qolmsdn, rejdn ortiq yn 6 t mshin ishl chiqrdi Zvod 5 kund rej o yich necht mshin ishl chiqrishi kerk edi? 48 O zingizni tekshiri ko ring! 0 4 sonlri ichid ( x 7) + 4= 7x tenglmning ildizi ormi? Tenglmni yeching: ) x ( x ) = 4+ ( x ) ) x x + + = 4 Sotuvchi molining 0% ini 40% foyd iln sotdi Jmi sotuvdn % foyd ko rish uchun u qolgn molini nech foiz foyd iln sotishi kerk?

49 II og doir mshqlr 4 ) kg i 00 so mdn olingn uzumning kg idn kg shrt olini, 70 so mg sotildi Uzumning nrxi 50so mg rzonlshdi Tijortchi vvlgi foydni sql qolmoqchi Shrtning yngi nrxi dstlkisidn nech so mg rzon qilinishi kerk? ) 0% li shrt hosil qilmoqchisiz Ayting-chi, nech litr qyngn suvg 00 grmm shkr qo shsiz? 5 ) Idishd dstl m lum miqdor suv or edi Agr idishg litr suv quyils, idishning 8 qismi to ldi Agr idishdgi dstlki suvdn litr oli tshlns, idishning qismi to l o ldi Dstl idishning qnch qismi to l 0 o lgn? ) Idishning qismi o sh Ahmd idishni to ldirmoqchi 5 U idishdgi suvning qnch qismi qdr suv quys, idish to ldi? Ung yordm ering 6 Yerning irinchi ikkit sun iy yo ldoshi msssi 59,4 kg ni tshkil qildi Birinchi sun iy yo ldosh uchinchisidn 4,4 kg yengil, ikkinchisi es 88, kg yengil Yerning irinchi ucht sun iy yo ldoshining hr irining msssini toping 7 Qyiq dryo oqimi o yich,4 sot v oqimg qrshi,sot suzdi Qyiqning oqim o yich osi o tgn yo li oqimg qrshi osi o tgn yo lidn, km ortiq o ldi Agr dryo oqimining tezligi,5 km/sot o ls, qyiqning turg un suvdgi tezligini toping 8 Bo ston v Guliston qishloqlri orsidgi msof 7km Bu qishloqlrdn ikkit syyoh ir vqtd yo lg chiqdi Birining tezligi sotig v kilometr, ikkinchisiniki es sotig u kilometr sotdn so ng ulr orsidgi msof nech kilometr o ldi? Hmm hollrni qrng v thlil qiling 4 Alger, 7- sinf 49

50 ¹4 Xodni o lkk rrlsh uchun minut kerk Shu xodni 4 o lkk rrlsh uchun nech minut kerk o ldi? 9 Idishning qismi suv iln to l Bu suvning 4 qismi ishltilgndn keyin ung 45 l suv solins, idishning 8 qismi o sh o ldi Idishg jmi nech litr suv ketdi? 0 Sinovd 60 t svol erildi, hr ir to g ri jvo 5 llg holndi 4 t noto g ri jvo uchun jrim siftid itt to g ri jvo ekor qilindi Bu sinovd hmm svollrni elgilgn ir o quvchi 5 ll olgn o ls, u necht svolg to g ri jvo ergn? II og doir sinov mshqlri testlr 5 x x = + + x tenglmning ildizi x0 o ls, x 0 + ifodning son qiymtini toping 8 4 A) 50 B) 0 C) 5 D) 7 x = x x + ifodni hisolng: tenglmning ildizi x 0 o ls, 8:x 0 A) 6 B) 7 C) 7 D) 46 7 ( x+ ):( x ) = 5: tenglmning ildizi x 0 o ls, x 0 +6 ifodning son qiymtini toping A) 80 B) 70 C) 80 D) 8 4 4:(x+ 5) = :(x ) tenglmning ildizi x 0 o ls, 4x 0 + ifodning son qiymtini toping A) 8 B) 0 C) 9 D) 0 5 0,8 (,5x ) 0,4x =0, (6x 5),6 tenglmning ildizi x 0 o ls, x 0 0,5 x 0 ifodning son qiymtini toping A) 5 B),5 C) 6,5 D) 5 50

51 6 Ucht jvond hmmsi o li 85 t kito or Birinchi jvond ikkinchisig qrgnd 8 t ko p, mmo uchinchi jvondgidn 9 t km kito or Hr ir jvond nechtdn kito or? A) B) C) 7 6 D) Teng yonli uchurchkning perimetri 5 sm g teng Asos yon tomondn 6 sm uzun Shu uchurchk yon tomonining sosig nistini toping A) 7:5 B) 5:7 C) : D) 0 :7 8 Teng yonli uchurchkning perimetri 4 sm g teng Yon tomon sosning qismini tshkil qildi Shu uchurchkning sosi yon tomonidn nech sntimetr uzun? A) 7,5 sm B) 6,5 sm C) 6sm D) 7sm 9 Ust uyurtmni 8 kund jrishi kerk edi U hr kuni rejdn tshqri 6t mhsulot tyyorl, uyurtmni 5 kund jrigin qolmsdn, lki ortiqch yn t mhsulot tyyorldi Ust rej o yich ir kund necht mhsulot tyyorlshi kerk edi? A) 6 B) 4 C) 5 D) 7 Tenglmni yeching (0 ): 0 8( x+ ) 5x = ( x+ 4,5) A) 5 B) 5 C) 6 D) 4,5 6(, x ),5x + 0,7x = 0,5( x 4) A) B) 0,5 C) D) 7 Uchurchkning ir tomoni ikkinchi tomonidn sm uzun, uchinchi tomonidn es 5 sm qisq Agr uchurchkning perimetri 4 sm o ls, uning eng uzun tomoni eng qisq tomonidn nech mrt uzun? A) B),5 C), D),8 5

52 Birinchi to pd 75 m, ikkinchi to pd 0 m tls or edi Ikkinchi to pdn irinchidn sotilgnig qrgnd mrt ko p tls sotildi Ntijd irinchi to pd ikkinchisig qrgnd mrt ko p tls qoldi Hr ir to pdn nech metrdn tls sotilgn? A) 4 m 7 m B) 0 m 90 m C) 5 m 45 sm D) m 99 m 4 Tenglmni yeching: ( x + ) ( x + ) = 7 5( x + ) A) B) C) D) Trixiy m lumotlr Muhmmd in Muso l-xorzmiy Al-jr vl-muqol hisoi hqid qisqch kito srid kiritilgn l-jr, vl-muqol qoidlrini iz 7- d tenglmning sosiy xosslri siftid yon qildik Algerd uch xil sonlr iln ish ko rildi, deydi l-xorzmiy Ulr: ildiz yoki nrs (tenglmdgi nom lum son x) kvdrt (mol) (nom lumning kvdrti x ) oddiy son (und nturl son nzrd tutildi) Xorzmiy shu uch xil miqdorlr orsidgi og lnishlrni thlil qildi v quyidgi tenglmlrni yechish usullrini ko rstdi: ) cx = x kvdrtlr ildizlrg teng ) cx = kvdrtlr sonlrg teng ) x = ildizlr song teng 4) cx + x = kvdrtlr v ildizlr sonlrg teng 5) cx + = x kvdrtlr v son ildizlrg teng 6) x + = cx ildizlr v son kvdrtlrg teng Biz 7- sinfd fqt chiziqli tenglmlrni o rgnmiz [) nddgi x = tenglm] Qolgnlri 8- sinfd o rgnildi Hr qndy chiziqli yoki kvdrt tenglm l-jr, vlmuqol lmshtirishlri ntijsid yuqoridgi 6 t tenglmning irig keltirilishi mumkin 5

53 BIRHADLAR VA KO PHADLAR 9- Nturl ko rstkichli drj Teng sonlrni qo shishni ko pytirish iln lmshtirish mumkin: = = n 5mrt n mrt Bir xil sonlrning ko pytmsini hm ko p hollrd ixchmroq yozuv iln lmshtirish mqsdg muvofiq o - ldi Tomonining uzunligi 5 irlikk teng kvdrtni qrylik (6-rsm) U 5 5=5 t irlik kvdrtdn iort Tomonining uzunligi 5 irlikk teng ku (7-rsm) es 5 5 5=5 t irlik kuni o z ichig oldi Sizg m lumki, 5 5 ko pytm 5 (o qilishi: eshning kvdrti ) ko pytm es 5 (o qilishi: eshning kui ) ki elgilndi: 5 5=5, 5 5 5=5 Xuddi shu ki, ko pytuvchilri ir xil sonlrdn iort ko pytmni yngi ml drjg ko trish mli iln lmshtirish mumkin: =, = 5 5mrt 0,4 =(0,4) mrt Umumn, n t teng ko pytuvchining ko pytmsini elgilsh uchun n yozuvidn foydlnildi: n = n mrt 9, 5

54 54 6- rsm 7- rsm U undy o qildi: sonning n ko rstkichli drjsi Odtd, qisqch qili: ning n-drjsi de ytildi sonning n nturl ko rstkichli drjsi de, hr iri g teng o lgn n t ko pytuvchining ko pytmsig ytildi: n = n mrt son (tkrorlnuvchi ko pytuvchi) drjning sosi, n son (ko pytuvchi nech mrt tkrorlnishini ko rstuvchi son) drj ko rstkichi deyildi Msln, 4 = =8, u yerd drjning sosi, 4 drj ko rstkichi, 8 es 4 drjning qiymti Xususn, sonning irinchi drjsi de, shu sonning o zig ytildi: = Msln, 5 = 5, 5 = 5, = 7 7 Drjning sosi istlgn son o lishi mumkinligini yti o tmiz, msln, 5 = = 8 = = ( ) = ( ) ( ) ( ) ( ) ( ) = 4 6 = = 8 0, = 0, 0, 0, = 0,008 6 ( ) = ( ) ( ) ( ) ( ) ( ) ( ) = 4 0 = 0 0 0= 0 0 = = 0000

55 Drjg ko trish mli uchinchi osqich ml Agr ifodd qvslr o lms, u hold vvl uchinchi osqich mllr, keyin ikkinchi osqich mllr (ko pytirish v o - lish), v nihoyt, irinchi osqich mllr (qo shish v yirish) jrilishini eslti o tmiz Msl Hisolng: = = 45 =67 Sonlrni drj yordmid yozishdn jud ko p hollrd, msln, nturl sonlrni xon qo shiluvchilri yig indisi shklid yozish uchun foydlnildi: 45= = Ktt sonlrni yozish uchun ko pinch 0 sonining drjlri qo llnildi Msln, Yerdn Quyoshgch o lgn msof txminn 50 mln km g teng o li, uni,5 0 8 km shklid yozildi: Yer shrining rdiusi tqrin 6,7 mlnm g teng, u 6,7 0 6 m ki yozildi Yerdn eng yqin yulduz (Sentvrning α si)gch o lgn msofni 4 0 km shklid yozildi 0 dn ktt o lgn hr ir sonni 0 n shklid yozish mumkin, und <0 v n nturl son Bundy yozuv sonning stndrt shkli deyildi Msln, 4578=4,578 0, 45,78=4,578 0, 0000=,0 0 5 Fizik v kimyo fnlrini o rgnishd, mikroklkultord hisolshlrd v oshq ko p hollrd sonning stndrt shkldgi yozuvidn foydlnildi Mshqlr Yig indini ko pytm shklid yozing ( ): ) ) c+ c+ c ) )

56 ) m+ m+ m 5) mrt ) ) mrt ) ( c d) + ( c d) 7) m + m+ + m n mrt 4) ( ) + ( ) + ( ) 8) k mrt Ko pytmni drj shklid yozing ( 5): ) ) ) ) (,7) (,7) (,7) (,7) 4 ) x x x x x ) ( )( )( ) ) m m m m m 4) ( )( )( )( ) x x m m m m m 4) n n n n n 5 ) ( x y)( x y)( x y ) ) ) ( + )( + ) Ko pytmning drj shklidgi yozuvidn foydlni, ifodni soddlshtiring (6 8): 6 ) 5 ) ) ) ),, 5 5 ) 0,5 0,5 0,5 4 4 ) 0, 0, ),, x x x 8 )9 9 9 ) ( ) ( ) y y y x y x y ) x x x x 4) (8 ) (8 ) (8 ) Ifodni soddlshtiring (9 0): 9 ) p p p p+q q ) + + ) + 4) x x x+x x x 56

57 0 ) c c+ c c+ + c c ) + k mrt n mrt m mrt ) ) n mrt k mrt 7 mrt Ifodni o qing, drjning sosini, drj ko rstkichini yting: ) ) ) ) (4 m+ n) 5 9 4) (,) 6) Hisolng ( 9): ) ) ) 4 4 4) 5 ) 5 ) ( ) 7 ) 0 5 4) ) ) 5 ) 7 4) 5 ) (,5) ) (,7) ) ( 0,) 4) ( 0,) 4 6 )( 5) ) 5 ) 4) ) 4 ( 0,) ) 5 (0,) (0,) 4 ( 0,) ) (,) (,6) 4) (,6) (,) 8 ) ( ) ) 5 ( ) ) ( 4) 4) ( ) 9 ) ( 5) ) ( ) 5 ) ( ) 4) ( ) ( ) 40 x ( x)( x) ifodning qiymtini x = 5 d toping 57

58 4 x ifodning qiymtini x ning jdvld keltirilgn qiymtlri uchun hisolng: x x x ifodning qiymtini x ning jdvld ko rstilgn qiymtlri uchun hisolng: x x 4 Quyidgi d volrning qysi iri to g ri, qysi iri noto g ri? Sini tushuntiring D vo noto g ri de ytsngiz, uni rd etuvchi misol toping ) ikkit sonning kvdrtlri teng o ls, u sonlrning o zlri hm teng ) ikkit sonning kulri teng o ls, u sonlrning o zlri hm teng ) gr mnfiy song uning kvdrti qo shils, must son hosil o ldi 4) gr mnfiy sondn uning kvdrti yirils, mnfiy son hosil o ldi 5) gr must sondn uning kvdrti yirils, must son hosil o ldi 58 Quyidgi d volrning qysi iri to g ri, qysi iri noto g - ri? Sini tushuntiring Mos misollr tuzing (44 45): 44 ) nturl sonning kvdrti ixtiyoriy rqm iln tugshi mumkin ) nturl sonning kui ixtiyoriy rqm iln tugshi mumkin 45 ) nturl sonning to rtinchi drjsi fqt rqmlridn iri iln tugshi mumkin ) nturl sonning eshinchi drjsi shu son qysi rqm iln tuggn o ls, o sh rqm iln tugydi

59 0- Nturl ko rstkichli drjning xosslri Drjg ko trish ir necht muhim xosslrg eg -xoss m n m+ n = Bir xil sosli drjlrni ko pytirishd sos o zgrmsdn qoldi, drj ko rstkichlri es qo shildi Nturl ko rstkichli drjning t rifig ko r m n = ( ) ( ) = = ( ) ( ) = mrt mrt m mrt ko pytirishning guruhlsh qonunig ko r = = 5mrt = = ( m+ n)mrt nturl ko rstkichli drjning t rifig ko r Shundy qili, = 5 = m + n = + m n = m + n -xoss m n m n n mrt : =, m > n, 0 Bir xil sosli drjlrni o lishd sos o zgrmsdn qoldi, drj ko rstkichlri es yirildi Shrtg ko r 5 > m > n, 0 Drjning irinchi xosssig ko r Shuning uchun 5 = 5 m n n = m 5 = 5 : m n = m : n 59

60 60 Shundy qili, n n 5 : = 5 m : n = m n, m > n, 0 =, 0 eknligini t kidlymiz -xoss m n mn ( ) = Drjni drjg ko trishd sos o zgrmsdn qoldi, drj ko rstkichlr es o zro ko pytirildi Nturl ko rstkichli drjning t rifig ko r m n m m m m ( ) = = ( ) = = drjning irinchi xosssig ko r n mrt n mrt = + = m+ m+ + m = = ko pytirishning t rifig ko r = = mn Shundy qili, ( ) = ( m ) n = mn 4-xoss ( ) n = n n Ko pytmni drjg ko trishd hr ir ko pytuvchi shu drjg ko trildi ( ) = ( ) ( ) ( ) = ( ) n = ( )( )( ) = mrt n mrt ko pytirishning guruhlsh v o rin lmshtirish qonunig ko r = ( ) ( ) = = ( )( ) = mrt mrt n mrt n mrt nturl ko rstkichli drjning t rifig ko r = = n n Shundy qili, ( ) = () n = n n

61 5-xoss n n = n 0 Ksrni drjg ko trishd uning surt v mxrji xuddi shu drjg ko trildi Nturl ko rstkichli drjning t rifig ko r n = = = = = + : = mrt n mrt ( m) n = mn ksrlrni ko pytirish qoidsig ko r mrt n mrt ( ) = = = = n = n = n mrt nmrt nturl ko rstkichli drjning t rifig ko r = = n n Shundy qili, - msl Hisolng: = = n, = = = 6 4 n msl Yorug likning trqlish tezligi 0 8 m/s g yqin, Yerdn Quyoshgch o lgn o rtch msof,5 0 m Yorug lik nuri Quyoshdn Yergch o lgn msofni qnch vqtd osi o tdi? Tekis hrktd osi o tilgn yo lning s = vt formulsig sosn:,5 0 = 0 8 t,,50 u yerdn t = = 0,50 = (s) 0 Jvo: 500 s = 8 min 0 s n m n m n m n m n n n n 6

62 6 Ko pytmni drj shklid yozing (46 5): 46 ) 5 4 )7 7 4 )6 6 4) ) c c ) 4 ) 48 )( ) ( ) )( 0,5) 4 ( 0,5) )( ) ( ) 4)(,) (,) 4 7 4)()() 6 49 ) 4 )( 5) 6 ( 5) ( 5) 4 ) 5 4)( 6) ( 6) ( 6) 7 50 )(,) (,) (,) 5 ) y 4 y y 7 4 ) 4) )(,5) (,5) 8 )(x ) 7 (x ) x 5x ) 6 6 4) (n+m) 5 (n+m) 5 5 ) ) 8 n ) c 8 c n 4) n (n nturl son) 5 Drjni ir xil sosli ikkit drjning ko pytmsi shklid yozing: ) 4 ) ) y 4) c 0 5)( x) 7 6)( ) 4 Sonlrni sosi o lgn drj shklid yozing (54 57): 54 ) ) 4 ) 4)8 Mshqlr 55 ) 6 )64 )56 4)04 56 ) 6 ) 4 7 ) 8 7 4) ) 7 8 ) n 8 ) ) 6 n (n nturl son)

63 Sonlrni sosi o lgn drj shklid yozing (58 6): 58 )9 ) )7 4)8 59 ) 79 )4 ) 4 4) 6 ¹5 Sonning o nli yozuvidgi oxirgi rqm nechg teng: ) ) ) ) ? 60 ) 5 7 ) 5 ) 5 7 4)8 6 ) n ) n+ 8 ) n 4)7 n (n nturl son) Bo linmni drj shklid yozing (6 64): 6 )7 0 :7 8 )4 :4 )(0,) 4 :(0,) 4)0 : ) : ) : ) x 7 7 : x 7 4) d 4 : d 6 64 ) y y : )( ) : ( ) 5 )() 5 : () 4)(m + n) 0 : (m + n) 5 Sonlrni sosi o lgn drj shklid yozing (65 66): 65 ) : ) 4 :4 )64:4 4): 66 )8: )56: ) 7 5 4) 0 Sonlrni sosi o lgn drj shklid yozing (67 68): 67 ) 5 : ) 4 : ) 4 : 9 4)7 : 68 )4 : 7 )8 : 9 ) Hisolng (69 7): 69 ) ) 4 ) 5 4) )

64 70 ) 8 ) 4 4 ) 7 ) 9 8 ( 5) 6 7 ) 5 7 ( 6) ) Tenglmni yeching (7 74): ) 6 4 4) )(c) 4 )(xyz) 7 )( 5 ) 8 4)( 4 9) ) x : = ) x : 4 = ) x 6 = 8 4) x 5 = 8 7 )5 5 x=5 7 )4 6 x=4 8 ) 8 :x= 8 4) : x = 9 x 74 ) = x ) = 8 5 ) = x 9 4) = x Ifodni sosi o lgn drj shklid yozing (75 77): 75 )( 5 ) 6 )( 8 ) 7 )( ) 5 8 4) 5 ( ) 8 76 ) 7 5 ( ) 4 ) ( ) )( ) 4 ( 4 ) 4) 5 ( ) 4 ( ) 77 )( 7 ) 5 :( ) 4 )( 6 ) 4 :( ) 5 ) 5 4 ( ) 78 n ning qndy qiymtid tenglik to g ri o ldi: 4) ( ) 4 ( ) ) n =9 )8= n )( ) n =6 4)( n ) =8? Sonlrni ko rstkichi o lgn drj shklid yozing (79 8): 79 )0,0 ) ) 4) 0, )5 4 )7 6 )( 0,7) 4 4) 8 ) 4 ) 6 ) c 0 4) x 0 Ko pytmni drjg ko tring (8 87): 8 ) ( 5) 4 )(7 6) 5 )(, 8) 5 4) ) () )(x) 4 )( 4x) 5 4)( 8) 84 ) (x) 7 )(6y) 6 )(,5cd) 4)(nm) 4

65 86 )(xy ) )( ) )( 4 ) 5 4)(0,c ) 87 )(0n m ) )(8 4 7 ) )(, 4 ) 4)( nm ) 4 Ko pytmni = () nmung qr drj shklid yozing (88 90): 88 )4 5 x 5 ) ) ) ) 5 )(,4) 4 4 )(,) y 4) 90 ) 6 )8r )9 7 n 7 m 7 4)5 Ifodni ko rstkichi o lgn drj shklid yozing (9 9): 9 ) c d 0 ) 4 6 )5 4 4)8m 9 ) 4 6 c ) x y 4 z 8 )49x 8 y 6 4)00c 8 x 6 9 )0,5 0 6 )0,49n m 0 ) x y 4) Ifodni ko rstkichi o lgn drj shklid yozing (94 97): 94 ) 6 ) 9 )5 5 4) )( 0,) ) 5 ) 0,5 4) 0,00 96 ) x y 9 ) 6 ) 9 c d 4) x y 9 z 6 97 ) 7 ) ) 5n 6 m 6 4) 0,008x y 9 Hisolng (98 0): )(0,5) ) 5 4 )( 0,5) 8 4)( 0,) )( 0,5) 9 ( 4) 9 6 ) (8,5) ) (,5) 4) (4,5) 5 Alger, 7- sinf

66 00 ) ) ) ) ) ) ) ) ) ) 8 4 (7 ) 4 7 ) Ksrni drjg ko tring (0 06): 0 ) m 04 ) ) 5 7 ) n ) 4) d ) 4) 9 5 ( ) 5 ( ) 8 4) 4 c 05 ) 4 ) 5c 4 7 ) 4) ) + ) 7 + c ) m+ n m n 5 4) + 7 Ksrni drj shklid yozing (07 09): 07 ) 08 ) 09 ) ) x y 6 6 ( ) ( ) ) ) ) 4 (4 x) 4 ( y) Hisolng (0 ): m 4) ) 5 6 4) ) 8 4) 7 0 A (x) nuqt koordintlr o qining qyerid o lishini chml ko rsting: B(x ) 0 C(x ) 0 C(x ) B(x ) 0 B(x )C(x ) ) ) d) x x x C (n ) nuqt koordintlr o qining qyerid o lishini chml ko rsting: 66

67 0 A(n) B(n ) 0 B(n )A(n) ) ) d) A(n) 0 B(n ) x x x ) Yerning msssi kg g teng Quyoshning msssi 0 0 kg Yerning msssi Quyoshning msssidn nech mrt km? ) Yerdn Sirius de nomlnuvchi yulduzgch o lgn msof km Yorug lik nuri Yerdn Siriusgch nech yild yeti orishini tqrin hisolng Ifodning son qiymtini toping: ),und = ) 4 Ifodni drj shklid yozing: n+ 4 n n+ ) 5 5 :5 ) n n n 4 + ) : 4),und = 6n 4 4n+ 5n 5n n+ 4n (n nturl son) 5 n ning qndy qiymtid tenglik to g ri o ldi: )(4 4 ) n =4 )(5 n ) =5 4 ) n =4 5 4)( ) n =? 6 Ko pytmni drjg ko tring: ) (8 4 c ) ) (9x 4 y z 7 ) ) (,x 5 y 7 z 7 ) 4) (, c 4 ) 5 7 Ifodni sosi o lgn drj shklid yozing: ) ) Sonlrdn qysi iri ktt: ) 4 4 ( )( ) 6 9 4) 6 5 ( ) 4 9 ( ) ) 54 4 mi yoki mi ) 00 0 mi yoki mi ) 0 0 mi yoki 0 0 mi 4) 6 0 mi yoki 40 mi? 9 To g ri tenglik hosil qiling Msl necht yechimg eg: ) () () = c ) () () = 8 5 c 7? 67

68 0 Tenglmni yeching: ) x :,75= 7,5 ) 8,9: x = 0, ) + = 8 x 4) 754,5:(7, + x ) = 5 Sonni stndrt shkld yozing: )6000 ) ) )Yerdn Quyoshgch o lgn msof km - Birhd v uning stndrt shkli Turli msllrni yechishd ko pinch, c, ko rinishdgi lgerik ifodlrg duch kelindi Msln, o lchm- lri 8-rsmd ko rstilgn sovitgichli mshin sig imi c g teng c ifod irinchisi rqm iln, qolgn uchtsi,, c hrflri iln elgilngn to rtt ko pytuvchining ko pytmsidir 68 Rqmlr iln yozilgn ko pytuvchilr sonli ko - pytuvchilr, hrflr iln elgilngn ko pytuvchilr es hrfiy ko pytuvchilr deyildi Sonli v hrfiy ko pytuvchilr ko pytmsidn iort lgerik ifod irhd deyildi Msln, ushu ifodlr irhdlrdir: c 8- rsm c,( 4), ( 0,) 4 Teng ko pytuvchilr ko pytmsini nturl ko rstkichli drj shklid yozish mumkin o lgnligi uchun sonning drjsi v sonlr drjlrining ko pytmsi hm irhdlr deyildi Msln, ushu ifodlr irhdlr o ldi: 5, 7, c,4, ( ) 4

69 Hr ir sonni shu son iln irning ko pytmsi shklid yozish mumkin o lgni uchun,, ko rinishdgi ifodlr 8 hm irhdlr de hisolndi Msl Birhdning qiymtini hisolng: und =, = 4, c = 9 7 ( ) ( ) 6c 0,5 0,5, Hrflrning qiymtlrini irhdg qo yi, uning qiymtini topmiz, y ni yettit sonning ko pytmsini hisolymiz: 9 6 0,5 0,5 4 7 Sonlrning irinchisini ikkinchisig, ulr qndy yozilgn o ls, xuddi shu trtid ko pytirish mumkin: = = = = = 4 = ,5 Ko pytirishning o rin lmshtirish v guruhlsh qonunlrini qo ll, hisolshni qisqch jrish hm mumkin: ( ) ( ) ( ) ( ) 6c 0,5 0,5 = 6 0,5 0,5 c = c Endi =, = 4, c = 9 o lgnd c irhdning qiymtini 7 topmiz: = = Mslni ikkinchi usul iln yechishd erilgn irhd nch sodd ko rinishd yozilgn edi: c Bu irhdning stndrt shklig misol Umumn, irinchi o rind turgn fqt itt son ko - pytuvchidn v hr xil sosli hrfiy drjlrdn tuzilgn irhdni stndrt shkldgi irhd deyildi 69

70 Hr qndy irhdni stndrt shkld yozish mumkin Buning uchun rch son ko pytuvchilrni o zro ko pytirish v ulrning ko pytmsini irinchi o ring yozish kerk So ngr ir xil hrfiy ko pytuvchilr ko pytmsini drj shklid yozish kerk Hrfiy ko pytuvchilr ko pinch, shrt o lms hm, lifo trtiid joylshtirildi Birhdning stndrt shklid ir xil hrflr yo qligini eslti o tmiz Stndrt shkld yozilgn irhdning son ko pytuvchisini shu irhdning koeffitsiyenti deyildi Msln, irhdning koeffitsiyenti g teng 5 irhdning koeffitsiyenti 5 g teng, ( 7) c irhdning koef- 6 6 fitsiyenti ( 7)g teng Oxirgi hold irhdni qvssiz yozish mumkin: ( 7) c = 7c g teng o lgn koeffitsiyent, odtd, yozilmydi, chunki irg ko pytirgn iln son o zgrmydi Msln, c = c, y ni c irhdning koeffitsiyenti irg teng Agr koeffitsiyent ( ) g teng o ls, u hold hm irni v qvslrni yozmsdn, fqt ishorsini qoldirish mumkin Msln, ( )c = c, y ni c irhdning koeffitsiyenti g teng Mshqlr So z orqli ytilgn fikrni lgerik ifod yordmid yozing ( 4): ) v sonlr ko pytmsining ikkilngni ) v c sonlr ko pytmsining uchlngni ) x v y sonlr kvdrtlrining ko pytmsi 4) son iln son kvdrtining ko pytmsi ) m sonning kui iln p sonning ko pytmsi ) sonning kvdrti iln son ko pytmsining uchlngni 4 ) t sotdgi sekundlr soni ) n metrdgi sntimetrlr soni 70

71 5 ) Berilgn o lchmlr o yich c c shtrixlngn yuzni hisolsh formulsini chiqring (9-rsm) c ) c +c( c)=c +c( c) tenglikning to g riligini shkl yordmid ko rsting ) Shtrixlngn yuzni ikkit to g ri to rturchk yuzlrining yirmsi siftid tsvirlng Bundn foydlni, 9- rsm ( c)( c)=c +c ( c)tenglikni isotlng 6 Birhdning son qiymtini toping: ),und = 4 ) 0,5,und = 4 ) c,und =, =, c = 4) 4pqr,und p =, q =, r = 6 7 6) y( 0,) x,und y = 5, x = 6 9 5) m ( 0,) n,und m =, n = 5 7 Birhdni stndrt shkld yozing: ) m m ) 0,5 5) pq ( ) 5 5 4) ( m ) ( m ) 6) qp ( ) ) z z z 5 4 pq 8 Birhdni stndrt shkld yozing v son qiymtini toping: ) cc,und =,c= 4 ),und = = 6 8 4, 9 (Qdimiy msl) Hovuzg 4 t quvur o tkzilgn o li, irinchi quvur hovuzni ir kund, ikkinchi quvur ikki kund, uchinchi quvur uch kund, to rtinchi quvur to rt kund to ldirdi To rtl quvur irglikd hovuzni qnch vqtd to ldirdi? 7 pq

72 - Birhdlrni ko pytirish Quyidgi mslni yechylik Msl To g ri urchkli prllelepiðedning hjmi V = c formul o yich hisolndi, u yerd prllelepiðedning o yi, eni v c lndligi Agr shu prllelepiðedning o yi 5 mrt, eni n mrt, lndligi n mrt uzytirils, yngi prllelepiðedning hjmi qndy o ldi? Yngi prllelepiðedning o lchmlrini topmiz: o yi 5, eni n, lndligi nc Bu hold uning hjmi o ldi V =(5) (n) (nc) (5) (n) (nc) ifod quyidgi ucht irhdning ko pytmsidir: 5, n, nc Sonlrni ko pytirish qoidlrig ko r undy tenglikni yozish mumkin: (5) (n) (nc)=5 n nc =(5 )(nnc)=0n c Birhdlrni ko pytirish ntijsid yn irhd hosil o ldi v uni stndrt shkld yozi, soddlshtirish lozim, msln, ( c) (4 )= c 4 = 5 c ( c) (4 ) = 5 c Ikki yoki ir necht ir xil irhdlrning ko pytmsini, y ni irhdning drjsini qrymiz, msln, (5 c) Bu irhd 5, c ko pytuvchilrning ko pytmsi o lgni uchun ko pytmni drjg ko trish xosssig ko r: (5 c) = 5 ( ) ( ) c = c Xuddi shu ki: (pq ) = p (q ) = 8p q 6 Birhdni nturl ko rstkichli drjg ko trish ntijsid yn irhd hosil o ldi 7

73 Mshqlr Birhdlrni ko pytiring (0 7): 0 ) ( ) ( ) ) ( ) ( ) ) ( ) ) ( p) ( c ) ) ( 4 ) ( 6 ) ) 5m ( 7 n) 4) ( ) ) 4 4) ( ) 8 0, ) ( p) ( q ) 0,, 5 8 0,5 4) c 7 6 ) m ( n) ) ( ) 8 ) c ) ( 4x y) ( 7 xy ) 4) ( ) 5 4 ) ( ) ( ) 4 6 c 4 c 6 c ) x x 7 4) xy y 4 ) ( c 5 ) ( 4 c) 5 ) xyz 5 6 ( xyz ) 0,4, ) x y z xy z ) ( nmr 4 5 ) ( nm r 5 ),5 4) c 5 c ) m ( 4n)( 4 mn) ) ( 8n) m ( 5mn) ) y xy ( 0, x ) 4) ( c) ( 5 c) ( 0,4c ) 4 6, ) ( ) ( ) 4 5 ) ( ) c ( c)( ) ) ( ) 5 ( ) ( ) 4) ( ), 5c c 7

74 Birhdni drjg ko tring (8 4): 4 8 ) ( ) ) (5 ) ) ( ) 4) ( ) ) ( ) ) ( 4 ) ) ( c ) 4) ( xyz) 40 ) ( ) ( c) ( xy) ( õ y ) ) 5 ) 4) 4 4 ) nm ) 4 nm ) ( 0, ) 4) (0,4 ) Amllrni jring (4 4): 4 ) ( ) ) ( ) ( ) ( ) ) ( 0, c ) ( 0 cx ) 4) c ( y ) ( 0, ) ) xy cx ) x y xy 4 ( c ) ) ( ) ( ) 4) ( ) 44 Birhdni oshq irhdning kvdrti shklid yozing: )9 )6x 4 ) )8x 6 y 5)6x 0 y 4 6), Birhdlrni ko pytiring v hosil o lgn ifodning qiymtini toping: ) 5, und =, = 7 ) mn 0 n,und m = 0,8, n = 4 5 ) 4 c, und = 4, = c = 6 4 4) 0,7mn 00 np,und m = 0,, n = 0,, p = 4 46 (Qdimiy msl) Bliqning uchdn ir qismi loyd, to rtdn ir qismi suv tgid v uch qrichi suv ustid Bliqning uzunligi nech qrich?

75 Algerd ko pinch irhdlrning yig indisi yoki yirmsidn iort o lgn lgerik ifodlr qrldi Msln, 0- rsmd tsvirlngn shklning shtrixlngn qismining yuzi c+ g teng, 0- rsmd tsvirlngn shklning yuzi es c g teng c+ ifod ushu ikkit irhdning yig indisi: c v c ifod v c irhdlrning yirmsi yoki v ( c ) irhdlrning yig indisi Bu ifodlr irhdlrning lgerik yig indisi o ldi Bundy ifodlr ko phdlr deyildi - Ko phdlr c c c ) ) 0-rsm Bir necht irhdning lgerik yig indisi ko phd deyildi Ko phdni tshkil qiluvchi irhdlr shu ko phdning hdlri deyildi Msln, 5nm m k 7nk +4 nm ko phdning hdlri 5nm, m k, 7nk, 4nm o ldi Ikkit hddn tuzilgn ko phd ikkihd deyildi, ucht hddn tuzilgn ko phd uchhd deyildi v hokzo Ikkihdg misollr:, 5 + 4c Birhdni hm ko phd de hisolymiz Uchhdg misollr: + c, c+ 75

76 Agr ko phdning zi hdlri stndrt shkld yozilmgn o ls, u hold shu ko phdning rch hdlrini stndrt shkld yozi, uni soddlshtirish mumkin Msl 4 5c+ 9 c c ko phdni soddlshtiring Berilgn ko phdning rch hdlrini stndrt shkld yozmiz: = c = c c c = c Demk, 4 5c+ 9c c = 8 5 c + c 47 Ko phdni tshkil qiluvchi irhdlrni yting: ) x + x ) 7 c 5 ) 4x x + 6 4) + 0,5x x 48 Ko phdni irhdlrning yig indisi shklid yozing: 4 ) ), ) 6x + x x + 5 4),5x 4 8xy 6xy xy 49 Birhdlrdn ko phd tuzing: 5 4 ) 6 x,7x v 9 4), v ) x, x v 5) 8,4, v 4 ) x, x v x 6) 4,, 5 50 Ko phdni, uning hr ir hdini stndrt shklg keltiri, soddlshtiring: ) + ) 4 8 ) ( ),5xy 4 xyz 4mnk5 mnk Mshqlr ) 4cc c c+ 5 xy xy

77 5 Ifodni, uning hr ir qo shiluvchisini stndrt shklg keltiri, soddlshtiring: ) + 4xxx xy,5yyy 4xyz 4mnk 5 mnk ) ( ) 4 9 ) ( ) ( ) Ko phdning son qiymtini toping: 4) ( ) ( ) ) + +, und =0,5, = ) 4 +, und =, = 0,5 ) x xy + y, und x = y = 4, 4) x +xy + y, und x =,, y=, 5 Ko phdni soddlshtiring v uning son qiymtini toping: ) + +4, und =, = ) 5 55, und = 5, = ) x yxy xy xy + xy, und x =, y = 4) xy x y xyxy, und x =, y= 4- O xshsh hdlrni ixchmlsh Ushu mslni yechylik -msl Hr ir shifsid ir xil sondgi hrflr o lgn ikkit kito or hr ir shifdgi strlr soni n t v hr ir strdgi hrflr soni m t Birinchi kito 00 shiflik, ikkinchisi 500 shiflik Ikkl kitod hmmsi o li necht hrf or? - usul Hr ir shifdgi hrflr soni mn t Birinchi kitod 00 nm t hrf, ikkinchisid 500 nm t hrf, ikklsid es t hrf or 00 nm nm = 800 nm 77

78 -usul Hr ir shifdgi hrflr soni mn g teng Ikkl kitodgi shiflr soni = 800 g, ulrdgi hrflr soni 800 nm g teng Ikkl jvo hm to g riligi ko rini turidi, shuning uchun nm +500 nm =800 nm Ammo hisolshlrd ikkinchi usul nch quly o ldi Msln, gr n =40, m =50 o ls, u hold nm =000 v 00 nm nm ifodni hisolsh uchun yn ucht hisolshni jrish kerk: = = nm ifodni hisolsh uchun es or-yo g i itt mlni jrish kerk, xolos: = Mn shuning uchun hm lgerik ifodlrni soddlshtirishni ilish muhim hmiytg eg 00 nm+500 nm ikkihd ikkit irhdning yig indisidn iort: 00 nm v 500 nm Bu irhdlr ir-iridn fqt koeffitsiyentlri iln frq qildi Bundy irhdlrni o xshsh irhdlr deyildi Msln, c v c irhdlr o xshsh, pq v 5q p irhdlr hm o xshsh, lekin v irhdlr o xshsh ems Bir xil irhdlrni hm o xshsh de hisolymiz Msln, v irhdlr o xshsh -msl c +4c + c + 4 ko phdni soddlshtiring O xshsh irhdlrni jrtmiz:,, 4 irhdlr o xshsh, ulrning tgig ittdn chiziq chizmiz, c v c o xshsh irhdlrning tgig ikkitdn chiziq chizmiz 4c irhdg o xshsh hd yo q, uning tgig chizmymiz, y ni c + 4c + c + 4 Ko phd hdlrining o rinlrini o xshsh hdlr yonm-yon turdign qili lmshtirmiz v o xshsh hdlrni qvs ichig olmiz: ( + 4) + ( c + c) + 4c

79 Ammo +4 =( +4) =6, c +c =( +)c = c o lgni uchun c +4c +c +4 =6 + c +4c Ko phdlrni o xshsh irhdlr lgerik yig indisi itt irhd iln lmshtirildign undy soddlshtirish o xshsh hdlrni ixchmlsh deyildi 6 + c +4 c ko phdd hr ir hd stndrt shkld yozilgn v ulr orsid o xshsh hdlr yo q Ko phdning undy shkli stndrt shkl deyildi Hr qndy ko phdni stndrt shkld yozish mumkin Buning uchun vvl ko phdning hr ir hdini stndrt shkld yozish v so ngr o xshsh hdlrni ixchmlsh kerk -msl Ko phdni stndrt shklg keltiring: + 6 c c 8 5 c+ c 5 6 c c c+ c = 5 = c c 4+ 5c+ c = =(c c) + ( c+ 5 c) + ( 4 + ) = = c+ c O xshsh hdlrni ixchmlng (54 55): Mshqlr 54 ) x + x + x 6 ) y 4 y 4 + y 4 y ) y y y 6 6 4) ) m+ q + q 4 m ) x + y + 4 x y ) + 4)

80 Ko phdni stndrt shklg keltiring (56 6): 56 ) x + 4x x 4 x ) y y + y y 0,c 0,c 0,5 c, +,4 0,8 ) 4) 57 ) x y + x + y ) ) + 0,7 5+, + 8 4) 5xy,5y xy+,y xy 5 58 ) xy + x y + xy x y xy 4 6 ) ) 9,87,89+ 8,97, 0,8 4) 8,5x 4,7y 5,x +,7y + 0,59 x 59 ) c + 4 c ) 8xy + 4x 5xy x + 4xy 9 xy 6 ) ) ) ( ) 5 4 c 5 c 4c c ) 8 8 c + 5c 5 c c ) ) x y + y x ) c 4 ) c 4 c c 80 4 ) 4x y 9+ 4 y x + 5 4) + ( ) 5 5 0,5 4 5

81 5- Ko phdlrni qo shish v yirish O lchmlri -rsmd ko rstilgn uchurchkni qrymiz Uning P perimetri tomonlr uzunliklrining yig indisig teng: P =( +)+(4 + )+( +4) Bu ifod quyidgi ucht ko phdning yig indisidir: +, 4+, +4 Qvslrni ochish qoidsig ko r undy yozish mumkin: P = O xshsh hdlrni ixchmlsk, P = tenglik hosil o ldi Ko phdlrning istlgn lgerik yig indisi hm xuddi shung o xshsh soddlshtirildi, msln, ( ) ( ) ( 4c) ( c ) ( c c) n m n m + q = n m n + m q = n q + = = 4ñ+ c c + c = c -rsm Bir necht ko phdlrni qo shish v yirish ntijsid yn ko phd hosil o ldi Bir necht ko phdning lgerik yig indisini stndrt shkldgi ko phd ko rinishid yozish uchun qvslrni ochish v o xshsh hdlrni ixchmlsh kerk B zi ko phdlrning yig indisi yoki yirmsini sonlrni qo shish v yirishg o xshsh ustun usulid topish quly o ldi Bund o xshsh hdlr irining ostig ikkinchisi turdign qili yozildi, msln, 6 Alger, 7- sinf 8

82 5 4c+ c ) + c 7c 5 c 4c ) 5c + 4c c c c+ c c+ + 5c 4c Mshqlr 8 Ko phdlrning lgerik yig indisini toping (6 67): 6 ) 8+ ( + 5 ) ) ( ) ( + ) ) 5x ( x y) 4) ( 4x+ ) + ( x ) ) x ( 4x + y) ) ( ) 64 ) ) ( ) 0,6 0,5 0,4 4) ,c 0,4c 0,c 0,5 c ) ( ) ( ) ) ( x y+ z) ( x + y z) 4) ( 7+ 4c) ( 0 4 c) ) ( 7m 4mn n ) ( m mn+ n ) ) ( ) + ( ) ) ( c+ c+ 7 ) ( 0c+ 0c ) 4) ( z + z + z ) ( z + z z ) ) ( + ) 0,, +, 0, ) ( ) ( ) ( ) ) ( p p ) ( p p ) + ( p p ) 4) ( 5x + 6x ) + ( x x ) ( x + 4 x ) 5 67 ) ( x + xy ) + ( xy ) + ( x y xy + x ) ) ( x + xy + x y) ( xy + x ) ( x y x )

83 ) ( ) + ( + ) ( + ) 4) ( 4 ) ( + ) + ( + ) Ko phdlrning yig indisi v yirmsini toping: ) 0,x + 0,0y v 0,7x 0,08 y ) 0,x 0,0y v 0,7x + 0,08 y ) 0, v 0,9 4) + 0, v 0, Ko phdlrning yig indisini ustun usulid toping: ) + v ) x + xy 4y v 4y xy + xy x 70 Ko phdlrning yirmsini ustun usulid toping: ) v ) + 4 v )Agr P =5 +, Q = 4 o ls, P + Q ifod nimg teng? ) Agr P =p q, Q =p 4q o ls, P Q ifod nimg teng? ) Agr A = +, B = + 5, C = o ls, A + B + C ni toping 4) Agr A = - +4, B = +4, C = o ls, A B + C ni toping 7 Isotlng: )esht ketm-ket nturl sonning yig indisi 5 g o lindi )to rtt ketm-ket nturl sonning yig indisi 4 g o linmydi )to rtt ketm-ket toq nturl sonning yig indisi 8 g o lindi 4)to rtt ketm-ket juft nturl sonning yig indisi 4 g o lindi 7 Avtousd n nfr yo lovchi or edi Dstlki ikki ektning hr irid m nfrdn yo lovchi vtousdn 8

84 tushdi, uchinchi ektd es hech kim tushmdi, lekin ir nech kishi vtousg chiqdi, shundn so ng vtousdgi yo lovchilr soni k nfr o ldi Uchinchi ektd vtousg nech kishi chiqqn? Ko phdni irhdg ko pytirish ntijsid yn ko phd hosil o ldi Hosil o lgn ko phdni uning rch hdlrini stndrt shkld yozi, soddlshtirish kerk Orliqdgi nti Ko phdni irhdg ko pytirish -rsm c O lchmlri -rsmd ko rstilgn to g ri urchkli prllelepiðedni qrymiz Uning hjmi sosining yuzi iln lndligining ko pytmsig teng: Istlgn ko phdni irhdg ko pytirish hm xuddi shundy jrildi, msln: ( nm nm ) ( 4nm) ( nm) ( 4nm) ( nm ) ( 4nm) = + = = 8nm + nm + = + ( 4 5c ) ( 5c) ( 5c) 4( 5c) ( ) ( + + c)() Bu ifod ++c ko phd iln irhdning ko pytmsi o ldi Ko pytirishning tqsimot qonunini qo ll, undy yozish mumkin: ( ++c)()=()+()+ +c()= +6 +c + 5c 5c = 5c+ 0 c 5 c Ko phdni irhdg ko pytirish uchun ko phdning hr ir hdini shu irhdg ko pytirish v hosil o lgn ko pytmlrni qo shish kerk

85 jlrni yozmsdn, irhdlrni og zki ko pytiri, irdnig jvoni yozish hm mumkin, msln, ( ) + = + 4 ( ) + = + 4 Birhdni ko phdg ko pytirish hm shung o xshsh jrildi, chunki ko pytuvchilrning o rinlrini lmshtirish iln ko pytm o zgrmydi, msln, 4pq (p q +) = =p q 4pq +8pq Ko phd v irhd ko pytmsini toping (74 78): 74 ) ( 5 0+ ) ( m) ) ( y ) 7-5 ) ( + x) 4) ( m n) ( ) ) ( ) n ) x( y x) ) ( 5x+ y) z x x + x 76 ) ( + ) ) ( + ) ) ( ) ) pq ( q p q ) 4) xy ( xy x ) 5 5 Mshqlr 77 ) 7( 5+ 6 ) ) xy( x + y + z) ) 8( c+ c ) 4) xyz( x + y + z ) ) )

86 Ifodni soddlshtiring (79 8): 6 x y 5 y x 79 ) ( t n) ( t n) ) ( ) ( ) ) 5( ) 4 ( ) 4) 7( 4p+ ) 6( 5+ 7 p) 80 ) ( x ) x ( x ) x ) ( ) ( ) 4 4 ) ( + ) + ( ) ( ) 4) ( x ) ( x ) + ( x ) ) ( 0,8y ) ( 4y+ ) + ( y) 5 0, 0,7 8 0,7 0, ) x + x + ) x x 4) 0,( 5y + 6) ( 0,5y ) + ( y ) 4, 5 0,,6 8 Algerik ifodning qiymtini toping: ,und =, = ) ( ) ( ) ) ( ) ( ) +,und =0, = ,und = 0, = 5 ) ( ) ( ) ,und =, = 4) ( ) ( ) 7- Ko phdni ko phdg ko pytirish 86 Ushu mslni qrylik Msl O lchmlri -rsmd ko rstilgn shkflr iln to silgn devor sirtining yuzini toping Shkflr iln nd o lgn devorning sirti tomonlri + c + =4 + c v + + = + o lgn to g ri to rturchkdn iort Bu to g ri to rturchkning yuzi S =(4 + c)( + ) g teng (4 + c)( + ) ifod (4 + c) v ( + ) ko phdlrning ko pytmsidir

87 Sonlrni ko pytirishning tqsimot qonunini qo ll, S = (4 + c)( + ) = 4( + ) + c( + ) ki yozish mumkin So ngr, 4( + ) = 8 +4 v c( + ) = c + c o lgni uchun S = c + c c - rsm Shundy qili, mzkur ko phdlrning ko pytmsini topish uchun 4+ c ko phdning hr ir hdini + ko phdning hr ir hdig ko pytirish v ntijlrni qo shishg to g ri keldi Ixtiyoriy ikkit ko phdni ko pytirish hm xuddi shundy jrildi, msln, ( ) ( ) ( ) ( ) 7n m n 5 m = (7 n)( n) + (7 n) 5m + m ( n) + ( ) ( ) + m 5m = n 5nm 6mn+ 0m = n 4nm+ 0 m ( ) ( ) 7n m n 5m = n 5nm 6mn + 0 m Ko phdni ko phdg ko pytirish uchun irinchi ko phdning hr ir hdini ikkinchi ko phdning hr ir hdig ko pytirish v hosil o lgn ko pytmlrni qo shish kerk Ko phdni ko phdg ko pytirish ntijsid yn ko phd hosil o - ldi Hosil qilingn ko phdni stndrt shkld yozish kerk 87

88 88 Msln, ( )( ) 4+ c 5 c = 0 c 0 + 4c + + 5c c = 0 c 0 + 9c c Bir necht ko phdni ko pytirishni nvtm-nvt jrish kerk, msln, ( )( )( ) ( )( ) + + = + + = = = 7 6 Ko phdlrni ko pytiring (8 9): 8 ) ( + ) ( + ) ) ( m+ 6) ( n ) ) ( z ) ( z + 4 ) 4) ( + 4) ( c+ 5 ) 84 ) ( c 4) ( d ) ) ( x+y) ( x+) ) ( 0) ( ) 4) ( p+ q) ( q) 85 ) ( x+ ) ( x+ 4) ) ( m ) ( m ) ) ( + ) ( 5 4) 4) ( 5p q) ( p q) 4 86 ) + ) + ) ( 0, m) ( m + 0, ) 4) ( 0,+ 0,5x) ( 0, 0,5x) 87 ) ( + ) ( + ) ) ( + ) ( + ) ) ( 5x 6y ) ( 6x 5 y ) 4) ( x + x + ) ( x + ) 88 ) ( ) ( ) ) ( ) ( + + ) ) ( 5x+ y) ( 5x 5 xy+9y ) 4) ( + ) ( ) Mshqlr

89 89 Nuqtlr o rnig qndy irhdlrni yozsngiz tenglik to g ri o ldi: )( 5)( )= )( )(6x 5y )=x +4x y 5y )( +4c)(+)=0c +8c )(+)( +5)=+5 +8c +0? 90 ) ( 0,x + 0,y z) ( x y ) ) ( x y+ z) ( x + y) 0, 0, 9 ) ( ) ( +) ( ) ) ( x+ ) ( x ) ( x + ) ) ( + ) ( ) ( + ) 4) ( x ) ( x + ) ( x ) 4 9 ) Tenglikning to g riligini isotlng: c + ( c)+(+d c)c+d( c)=(+d) )To g ri to rturchkning yuzini hisolsh uchun ikkit ifod tuzing (4-rsm) d II III c I IV 4- rsm c To g ri to rturchkning yuzi I, II, III, IV to g ri to rturchklr yuzlri yig indisig tengligidn foydlning v -tenglikk geometrik tlqin ering 9 ) Quyidgi shklning yuzini v perimetrini hisolsh uchun formullr tuzing (5- rsm): c d k l ) Shkl yordmid: ) (c + d)=c + d 5- rsm n 89

90 ) (k+l+n)=k + l + n tengliklrni isotlng Bu formullrning geometrik m nosini oching 94 ) ABCD to g ri to rturchkning (6- rsm) yuzi ( + ) ( c+ d) = c+ c+ d + d eknligini ko rsting ) ABFE to g ri to rturchkning (7-rsm) yuzi eknligini ko rsting B E C ( + ) ( c d) = c+ c d d M N K M c d F B d F c A K D A P E 6-rsm 7- rsm Bir necht irhd v ko phdlrni qo shish, yirish, ko - pytirish v nturl ko rstkichli drjg ko trish ntijsid yn ko phd hosil o lishi oldingi prgrflrd ko rstildi Sn o tilgn u mllr ichid o lish mli uchrmdi Bo - lish mlini o z ichig olgn ifodlr V od tfsil qrldi B zn o lish ntijsid hm ko phd hosil o ldi Birhdni irhdg o lish Msl irhdni 4 irhdg o ling Sonni sonlr ko pytmsig o lish xosssidn foydlnmiz: sonni ko pytmg o lishd shu sonni ko pytmning irinchi ko pytuvchisig o lish kerk, so ngr hosil o lgn ntijni ikkinchi ko pytuvchig o lish kerk v hokzo Ntijd, 8- Birhd v ko phdni irhdg o lish 90

91 ( ) ( ) = ( ) : 4 :4 : Endi ushu qoidni qo llymiz: ko pytmni song o lishd ko pytmning ko pytuvchilridn irini shu song o lish kerk U hold Shundy qili, ( ) ( ) ( ) ( ) :4 = :4 = 8 8 : = 8 : = 8 ( ) ( ) = : 4 8 Birhdlr oshq hollrd hm xuddi shundy o lindi, msln, : ( ) = ( c ) ( ) = ( ) ( ) : c 9 knm : kn m = k Bo lish ntijsini ko pytirish iln tekshirish mumkin: o linuvchi o luvchining o linmg ko pytmsig teng o lishi kerk 56 : c c = o lish to g ri j- Msln, ( ) ( ) rilgn, chunki = ( 7 ) 56c 5 c 8 Ko phdni irhdg o lish Msl +4 +8c ko phdni irhdg o ling Ushu qoiddn foydlnmiz: yig indini song o lishd hr ir qo shiluvchini shu song o lish kerk, y ni ( 4 8 c) : ( ) ( ) : ( ) ( 4 ) : ( ) ( 8 ) : ( ) = c = + + c Ko phdni irhdg oshq hollrd hm xuddi shundy o lindi, msln, ( 9 + ) : ( ) = ( ) ( ) ( ) ( ) ( ) ( ) = 9 : + : + : = + 9

92 9 Ko phdni irhdg o lish uchun ko phdning hr ir hdini shu irhdg o lish v hosil o lgn ntijlrni qo shish kerk Ko phdni irhdg o lish ntijsini ko pytirish iln tekshirish mumkin Msln, ( ) ( ) 6nm 45 nm : 9nm = 4n 5m 4 4 o lish to g ri jrilgn, chunki ( ) ( ) 4 4 6nm 45nm = 4n 5m 9 nm Ko rilgn misollrd irhd (ko phd)ni irhdg o lish ntijsid irhd (ko phd) hosil o ldi Bundy hollrd ko phd irhdg qoldiqsiz o lindi, deyildi Ammo ko phdni irhdg qoldiqsiz (utun) o lish hmm vqt hm mumkin o lvermydi Msln, + c ko phd irhdg qoldiqsiz (utun) o linmydi Birhd (ko phd)ni irhdg o lishd hrflr o luvchi nolg teng o lmydign qiymtlrni qul qildi, de frz qilindi Bo lishni jring (95 05): ) : ) y : y ) : 4) : 96 ) : 4 97 ) ( ) x ) ( 5 ) : 5 ) ( 8y ) : 6 4) 0c ( ) 8 c : ) :5 ) 98 ) x : ( ) ) ( m) 7 : ) 8 : 4) : ) 5 : ) : x Mshqlr : : 4) c : 8x ) 5 : ( ) 4) ( 7y) ( y) :

93 00 ) ( 6x) : ( x) ) ( 6xy) ( xy ) ) 5z : ( 5z) 4) : ( 4 ) 0 ) ) ( 5c) : ) : 5 c : : 4) (,69n) ( n) :, 0 ) 8 c : ( 4) ) ( 6,4xy) : ( 4x) ) ( 0 pq) : ( 6 q) 4) ( 0,4c) ( ) : 0,6 5 0 ) 4 : ( 7 ) ) ( 0, 0 ) : ( 0 ) ) ( 4 m ) : ( 6 m) 4) ( ) 04 ) : 05 ) : mnp mnp ) ( pq y ) (,7 py) 4 ) : 8,9 : c c 4) ( c) ( c ) 4 6 : 4 xy 5 0 m n : 5mn ) x y :, xy : 6,9 xy 4) c 5 : c 4 ) ( ) ( ) 06 Ifodni soddlshtiring: ) 4 : ) c 5 : c 4 : ) 9x y : ( xy ) 4) xyz ( xyz ) Bo lishni jring (07 0): 07 ) ( + 6 ) : ) ( m ) ( ) ) ( 0 5) :5 4) ( 6+ x) ( ) 08 ) ( ) 4 8 : 5mn 6 np : n ) ( x xy) : x ) ( ) 4 : : 4) ( cd d) ( d) : 9

94 09 ) ( ) ( ) 4 : 5 ) 7k 4 l 5 + kl : 0 kl ) ( c c 4 ) ( c 4 ) 4) : : 4 0 ) ( ) : ) ( 0 + 8) ( ) ) ( 8 x + y 6) : ( 4) 4) ( + ) ( ) Ifodni soddlshtiring: ) ( ) + ( + ) ( ) ) x x x ( x x) x 6 : 9 : 8 4 : 4 : ) ( ) ( + ) x xy : x xy xy : xy 4) ( ) + ( ) : 6 5 : : 6 4 : Dl hovli to g ri to rturchk shklid o li, uning o yi enidn,5 mrt uzun Knl qzish zrurti o lgni uchun hovlining o yini 6 m g kmytirishdi, enini es 6m g uzytirishdi Ntijd, dl hovlining yuzi vvlgi yuzig qrgnd 84 m g ortdi Dl hovlining vvlgi perimetri v yuzini toping 94 O zingizni tekshiri ko ring! Ifodni drj ko rinishid tsvirlng: : 6 ( ) Ifodni soddlshtiring: ( + c d) ( c d) Amllrni jring: ( 0,5 c)(5 c) (7m 0mn 0 m):(0 m) 4 Ifodni soddlshtiring v uning m = 0,5 o lgndgi son qiymtini toping: mm ( ) + ( m )( m+ ) + m

95 III og doir mshqlr Jumllrni mtemtik tild yozing: ) m sonning kvdrtini ) sonning kuini ) c v sonlr yig indisining kvdrtini 4) c v sonlr kvdrtlrining yig indisini 4 Jumllrni mtemtik tild yozing: ) n v m sonlr yirmsining kvdrtini ) n v m sonlr kvdrtlrining yirmsini ) n v m sonlr yirmsining kuini 4) v sonlr kulrining yirmsini 5 Kvdrtning tomoni c metrg teng Uning perimetri v yuzini yozing 6 To g ri to rturchk shklidgi oynning o yi enidn 0sm uzun Uni derz romig solish uchun o yi v enidn 0 sm dn kesishdi Oynning kesi tshlngn o - lklrining yuzi 400 sm g teng Oynning dstlki o lchmlrini toping 7 Bir tomoni ikkinchi tomonidn mrt ktt o lgn to g ri to rturchkning ir tomonini x iln elgil, uning yuzi formulsini yozing 8 Qirrsi m o lgn ku qirrsi sm o lgn kulrg jrtils v ulr ustm-ust qo yils, qndy lndlikdgi ustun hosil o ldi? 9 Agr odmning yurgi minutd o rtch 75 mrt urs, uning yurgi ir sutk dvomid nech mrt urdi? 0 O quvchi m po kkni ko tr oldimi? (sm po kkning msssi 0, g) Quyidgi sonlrni stndrt shkld yozing: )0 C v 760 mm sim ust osimli sm gzdgi molekullr soni g teng 95

96 ) prsek (stronomiyd qul qilingn uzunlik irligi) km g teng ) elektron hisolsh mshinsi sekundd t ml jrishi mumkin Yer shri sirti 50 mln km dn ortiq Yer hjmi 000 mlrd km dn ortiq Bu sonlrni stndrt shkld yozing l dengiz suvid o rtch 0,0000 mg oltin or km dengiz suvid qnch oltin or? 4 Ko phdni stndrt shklg keltiring: m 4n 0,n 5m + 5 5nm+ 8 ) ( ) ( ) ( ) ( ) ( ) ( ) ) ( ) ( ) + ( ) ( ) + ( ) 0,xy 5 6x 0,y ) c5+ c 8 9 4) nmk4n nm nk + nm 4 k 5 Ko phdning qiymtini toping: ) 0,08x + 7xy + 7 xy,und x = 4, y = 0, ) + 4+,und =, = 4 ) 5p p + p 7p 6p 7 p + p,und p = 4) 8x 7x + 6x 5x + x + x 8 x,und x = 6 Ko phdlrning lgerik yig indisini toping: ) ( x + xy ) + ( xy ) + ( xy xy + x ) ) ( x + 5xy + 7x y) ( 5xy + x ) ( 7x y x ) ) ( ) + ( + ) ( + ) 4) ( 4 + ) ( + ) + ( + )

97 ¹6 Yngi Sprk vtomoilining egsi yuri turgn v zõirdgi g ildirklrni rsmd ko rstilgn trtid lmshtiri turdi km yo l yurilgch, hmm g ildirklr ir xil yedirilgni m lum o ldi Hr ir g ildirk nech kilometr yo l osgn (8-rsm)? Ko phdlrni ko pytiring (7 8): 8- rsm (0, 0, )( ) ) m n+ p ( m+ ) ,5x 0,5 y + z ( x + y) 4) ( 0, 0,4+ ) ( 5 0 ) 7 ) x + y z x z ) ( ) 8 ) ( )( + )( ) ) ( x + )( x + )( x ) ) ( + )( )( + ) 4) ( x )( x + )( x ) 9 Bo lishni jring: ) ( 0,0 4 0, + 0,04 + 0,00 ) :( 0,0 ) ) ( x 5 x 4 x + x ) ( x ) ) 0,05 0,08 0,09 0,0 : 0, ) x + x x : x III og doir sinov mshqlri testlr Hisolng: 5 9 :8 A) B) C) 9 D) 7 7 Alger, 7- sinf 97

98 8 4 4 ( ) Hisolng: 6 ( ) ( ) ( ) A) B) C) D) Birhdning son qiymtini toping: c, und =, =, c = 0 5 A) 4 5 B) 4 5 C) -8 D) Birhdni stndrt shkld yozing: A) B) 4 C) 4 D) Birhdlrni ko pytiring: c c A) 0, c B) 0,( c) C) c D) c Ko phdni uning hr ir hdini stndrt shklg keltiri, soddlshtiring: A) 4 B) 4 C) 5 D) 5 7 Ko phdlrning lgerik yig indisini toping: 7 0,5+ + ( + ) A) + B) + C) D) 8 Ko phdni irhdg ko pytiring: 4 x ( x) A) x x B) x x C) x + x D) x x 4 ( ) C) + 9 Soddlshtiring: 5 (0,4 ) 4 A) ( ) B) + 9 D) + 9

99 0 Ko phdlrni ko pytiring: ( )( + )( + ) A) 4 4 B) + Ñ) 4 D) 4 Bo lishni jring: (6 4 + ):(4 ) A) B) C) D) : Ifodni soddlshtiring: ( ) ( ) A) 4 + B) 6 + C) 4 + D) 6 + Ko phdlrni ko pytiring: ( + )( )( + 4 ) A) 4 6 B) 8 C) 8 D) 4 4 Hisolng: (4 6): ( ) ( ) 0, : 0, A), B), C) 0,000 D) 0,000 5 ( ) A) B) C),7 D) 9 6 ( 5, ) :(, ) A) 8 B) 8, C) 8, D) 5, 7 Ko phdni irhdg ko pytiring: 8 + 0,6 ( 5 ) 5 7 A) 8 +0 B) C) D)

100 8 Hisolng: ( ) ( ) ( ) ( ) 6 8, 5,,69, A) 4 B),6 C) D),69 Trixiy m lumotlr Nom lum kttliklrni hrflr iln elgilsh mshhur yunon mtemtigi Diofnt (III sr) srlridyoq uchrydi Koeffitsiyentlrni hm, m lum miqdorlrni hm hrflr iln elgilshni F Viyet (540 60) irinchilrdn o li qo llgn Algerik tenglmlrni umumiy hold tdqiq qilish hrfiy koeffitsiyentlr kiritilgndn keyingin mumkin o ldi FViyet undosh osh lotin hrflri B, G, D, iln koeffitsiyentlrni, unli hrflri A, E, I, iln es nom lumlrni elgilgn Mshhur frnsuz mtemtigi v fylsufi R Dekrt ( ) koeffitsiyentlrni elgilsh uchun lotin lifosining dstlki (kichik) hrflri,, c, d, dn, nom lumlrni elgilsh uchun es lifoning oxirgi hrflri x, y, z lrdn foydlngn Drjning hozirgi zmonviy elgilnishi,,, n (n nturl son)ni hm Dekrt kiritgn (67-yil) Al-jr vl muqol srining Ko pytirish hqid o id l-xorzmiy irhdlrni ko pytirishg, ikkihdni ikki hdg ko pytirishg hmd soddlshtirishg doir misollrni qrydi Al-Xorzmiy misollridn zilrini keltirmiz: ) (0 xx ) ) (0 + x)(0 + x) ) (0 x)(0 x) 4) (0 x)(0 + x) 00 5) x x

101 6) (0 + x)( x 0) 7) (00+ x 0 x) (50+ 0x x ) 8) (00+ x 0 x) + (50+ 0x x ) Al-Xorzmiy, Ahmd Frg oniy, Beruniy, l-koshiy srlrid lgerik simvolik o lmgn Mtemtik Au Hsn Ali in Muhmmd l-klsdi (XV sr) srid lgerik simvolik elementlrini uchrtish mumkin Al-Klsdi tenglmlrd nom lumning irinchi drjsini shy so - zining irinchi hrfi iln, kvdrtini mol so zining, kuini k so zining irinchi hrflri iln elgilgn Tenglik = elgisi o rnig dl (tenglik) so zidgi hrfini ishltgn Biz o rgnyotgn Alger kursining simvoliksi (elgilshlr tizimi) XIV XVII srlrd shkllngn Al-Xorzmiy tenglmlrini yeching: ) 0 x + (0 + x) x = 4 x 4 ) 00 x + (00 0 x) 0 = x x ) 500 x + 00 x = 00 + x + x x x x x 4) 00 x + 00 x = 4 x + 0

102 9- KO PHADNI KO PAYTUVCHILARGA AJRATISH Umumiy ko pytuvchini qvsdn tshqrig chiqrish -msl -og tomoni 47 m o lgn kvdrt shklid Ung tutshgn - og to g ri to rturchk shklid o li, uning eni 47 m, o yi es 57 m Bog lrning mydoni irglikd nech gektrni tshkil etdi (9-rsm)? Agr =47 m, =57 m elgilsh kiritsk, izlnyotgn mydon S= + (m ) o ldi 9- rsm Bu ifodg v ning qiymtlrini qo yi hisolsh vqtni oldi Ammo ikkl og ning irglikdgi mydoni S ni (+) ko pytm hm ifodlydi, y ni += (+) (rsmg qrng) + ifod ung teng o lgn (+) ifodg lmshtirils, hisolsh ishi nch soddlshdi Chindn hm, += (+)=47 (47+57)=47000(m )=4,7 (g) 0 Jvo: 4,7 g Hisolshlrni soddlshtirish uchun + ko phd (+) ko pytm iln lmshtirildi Ko phdni ikkit yoki ir necht ko phdlr ko pytmsi shklid ifodlsh ko phdni ko pytuvchilrg jrtish (yoyish) deyildi Ko phdni ko pytuvchilrg jrtish lgerik ifodlr ustid mllr jrishd hm keng qo llnildi

103 - msl + c d ifodning =4, =6, c =7, d = o lgnd, son qiymtini toping Hisolshlrni quyidgich oli ormiz: =4 (6 +7 ) =4 0 =860 Bu yerd ko pytirishning tqsimot qonuni qo llnilgn: + c d = ( + c d) sonli ifodd umumiy ko pytuvchi 4 soni o ldi + c d lgerik ifodd es umumiy ko pytuvchi o ldi Agr ko phdning rch (son yoki hrfiy) hdlri umumiy ko pytuvchig eg o ls, u hold shu ko pytuvchini qvsdn tshqrig chiqrish mumkin Qvs ichid erilgn ko phdni shu umumiy ko pytuvchig o lish ntijsid hosil qilindign ko phd qoldi - msl Ushu ko phdni ko pytuvchilrg jrting: 6 + c Berilgn ko phdning rch hdlri umumiy ko - pytuvchig eg, chunki 6 =, =, c = (4c) Demk, 6 + c =( + 4c) Ko phdning umumiy hdini msl mzmunig qr, qvsdn tshqrig + ishorsi iln hm, ishorsi iln hm chiqrish mumkin Misollr keltirmiz: ) = ( )= ( ) ) 4 6 = ( ) yoki 4 6 = ( +) = ( ) Ko phdni umumiy ko pytuvchini qvsdn tshqrig chiqrish yo li iln ko pytuvchilrg jrtish uchun: ) shu umumiy ko pytuvchini topish ) uni qvsdn tshqrig chiqrish kerk Agr ko phd hdlrining koeffitsiyentlri nturl sonlr o ls, u hold umumiy ko pytuvchini topish uchun ko phd 0

104 hdlri koeffitsiyentlrining eng ktt umumiy o luvchisini topish, ir xil sosli drjlr orsidn es eng kichik ko rstkichli drjni topish lozimligini t kidl o tmiz Msln, 8x x ko phdni ko pytuvchilrg jrti, quyidgini hosil qilmiz: 04 7x (4 x) Bu yerd 7 soni 8 v sonlrining eng ktt umumiy o luvchisi, x v es x v ning eng kichik ko rstkichli drjlridir Ko phdni ko pytuvchilrg jrlgnligining to g riligini hosil o lgn ko phdlrni ko pytirish yo li iln tekshirish mumkin Msln, ko pytirishni jri, hosil qilmiz: 7x (4 x) = 8x x Umumiy ko pytuvchi ko phd o lishi hm mumkin, msln: ) 5( + )+x(+)=(+)(5+x) ) x( )+5y( )+( )=( )(x +5y+) B zn umumiy ko pytuvchini qvsdn tshqrig chiqrishdn oldin = ( ) tenglikni qo llsh foydli o - ldi, msln: ) ( )x ( )y = ( )x +( )y = ( )(x + y) ) 5 (x y) 0 (x y)+5(y x )=5 (x y) 0 (x y) 5(x y)=5(x y)( 4 5) 0 Sonlrni tu ko pytuvchilrg jrting: 70,, 40, 68, Ksrlrni qisqrtiring: Ko pytirishning tqsimot qonunini qo llng v hisolng: ) ) Mshqlr ) 4,78+ 4,78 4)

105 Ko pytmni ko phd shklid yozing: ) c ( c 5 ) ) ( + ) ( + ) ) ( ) x x 4) ( + ) ( ) 4 A ektdn B ektg tomon motorli qyiq 0 km/sot tezlik iln jo ndi Ordn ikki sot o tgndn keyin A dn B g tomon ikkinchi motorli qyiq 4 km/sot tezlik iln yo lg chiqdi Ikkl qyiq hm B g ir vqtd yeti keldi A dn B gch o lgn msofni toping 5 ) ifodning 0 g 90 g ) ifodning 49 g 50 g ) 8 6 ifodning 4 g 60 g krrli eknini isotlng Umumiy ko pytuvchini qvsdn tshqrig chiqring (6 44): 6 ) m+ n ) x ) 8 4 x 4) ) 9+ + ) 0x + 5y 5 z ) 8 4 4) 9x y + z 8 ) x y ) cd + c ) xy + x 4) x xy 9 ) 9mn+ 9 n ) d ) z yz 4) 6pk p 40 ) c+ ) 6 + ) xy x + xz 4) ) + ) + 4 ) 4) xy xy ) 8y + y 5 ) 5x 5 x 4 ) 6x 4 x 5 4) ) 9 ) 7c+ 4 c ) 0xy + 4 xy 4) 9xyz xy z 05

106 ) 6y + y y ) 4 5 ) ) xy xy + 6 xy 45 Hisolng: ) ) 0,7 + 0,7 9,5 ) ) 0,9 0,8,9 Ko pytuvchilrg jrting (46 49): 46 ) ( m+ n) + ( m+ n) ) ( ) ( ) ) ( + 5) c ( + 5 ) 4) ( y ) + ( y ) 47 ) ( ) ( ) ) ( x + y) ( x + y) ) n( m ) + 5m( m ) 4) ( c d) ( c d) 48 ) ( ) ( ) x y x y ) ( x y ) ( x y ) + + 4) x( ) y ( ) ) ( ) ( ) x y x y 49 ) ( x ) ( x ) + c( x ) ) c( p q) ( p q) + d( p q) ) x( + ) + y( + ) z( + ) 4) m( x + ) n( x + ) l( x + ) Ko pytuvchilrg jrting (50 5): ) c( ) + ( ) ) ( x y) + ( y x) ) ( c) c( c ) 4) ( x y) ( y x) 5 ) 7( y ) ( y) ) ( ) c( ) ) 6( ) + ( ) 4) ( m ) + ( m) 5 ) ( c) + ( c) ( c ) ) x( x y) + y( y x) ( x y) ) x( ) + y( ) + ( ) 4) ( ) + ( ) ( ) 7

107 5 Tenglmni yeching: 8 x x + = 0 x ) x :5= :4,5 ) ( ) ( ) ( ) x, x x 4 7x 5 4) =, 6 ) ( + ) ( ) = ( ) 54 It tulkining orqsidn quvdi It sekundig 8 m, tulki es 6 m tezlik iln chopmoqd Ulrning orsidgi msof dstl 60 m o lgn, tulkining o z uysig yeti olishi uchun es km qolgn edi Tulki o z uysig yeti olishg ulgurdimi? Guruhlsh usuli Guruhlsh usuli hmm hdlri uchun umumiy ko pytuvchi mvjud o lmgn ko phdlrg qo llnildi B zn, erilgn ko phdning ir necht hdlrini qvs ichig oli, umumiy ko pytuvchini niqlsh mumkin Ko phdni guruhlsh usuli qo shish v ko pytirishning guruhlsh, o rin lmshtirish v tqsimot qonunlrig soslngn Misollr qrymiz: ) ( + c) + + c = ( + c) + ( + c) = ( + c) ( + ) ) ( c) + c = ( c) ( c) = ( c) ( ) Birinchi misold ko phdning oxirgi ikkit hdini + ishorsi iln, ikkinchi misold ko phdning oxirgi ikkit hdini ishorsi iln qvs ichig olish yetrli o ldi ) m( x y) nx ny m( x y) ( nx ny ) = m( x y) + n( x y) = ( x y) ( m+ n) + = + = 4) mx my n( x y ) ( mx my ) n( x y ) = m( x + y ) + n( x + y ) = ( x + y ) ( n m) + + = + + = 07

108 Uchinchi v to rtinchi misollrd ko phdning ikkit hdini qvs ichig olishdn tshqri hosil qilingn hr ir guruhd umumiy ko pytuvchi qvsdn tshqrig: irinchi hold + ishorsi iln, ikkinchisid es ishorsi iln chiqrildi B zn ko phd hdlrini turli usullr iln guruhlsh mumkin Msln, m +n m n ko phdni ko pytuvchilrg ikki usul iln jrtish mumkin: I usul II usul m+ n m n = = (m+ n) (m+ n) = = m ( + n) m ( + n) = = ( m+ n)( ) m+ n m n = = (m m) + (n n) = = m( ) + n( ) = = ( )( m+ n) Oltit hddn iort ko phdni ko pytuvchilrg jrtishg doir misol qrymiz: x + x y y + z + z = ( x + x) ( y + y) + ( z + z) = = x ( + ) y ( + ) + z ( + ) = ( + )( x y + z) Bu yerd ko phdlr ikkitdn guruhlrg jrtilgn ulrni uchtdn guruhlsh hm mumkin edi: x + x y y + z + z = ( x y + z) + ( x y + z) = = x ( y + z) + x ( y + z) = ( + )( x y + z) 08 Ko phdni guruhlsh usuli iln ko pytuvchilrg jrtish uchun: ) ko phdning hdlrini, ulr ko phd shklidgi umumiy ko pytuvchig eg o ldign qili, guruhlrg irlshtirildi ) u umumiy ko pytuvchini qvsdn tshqrig chiqrildi

109 Ko pytuvchilrg jrting (55 60): 55 ) + + c ( + ) ) x + x ( + y) + y ) m n+ pm ( n) 4) x + x ( y) y 56 ) ( x + y) + ( x + y) ) m( m n) + ( m n) ) ( ) + 4) 4q( p ) + ( p ) 57 ) m( m n) m n ) 4q( p ) + p 4) q( p ) + ) mm ( n) n+ m 4 + p 58 ) ( x c) + c x ) ( ) ) ( + c) + d+ dc 4) + c c 59 ) c+ c d d ) x y 6 y + x ) c d + d c 4) 5y x + x 5 y 60 ) xy y x + + y ) x y x + cy + y cx 6 Hisolng: ) ) ) 4,7 4,7 + 5, 5, 4 4) 4 + 4, + +, Ifodning son qiymtini toping: ) 5 5x 7+ 7 x,und x =, = 4 ) m mn m+,und n m = 0,5, n = 0,5 Mshqlr ) + 5 5,und = 6,6, = 0,4 4) +,und =, = 0,

110 6 Hisolng: ) ) 7,4 + 7,47, 90,6 6,4 64 Tenglmni yeching: x x 4 x 4 0 ¹7 ) ( ) + = ) t( t ) + 7 4t 8= 0 Ali iln Vlining msssi irglikd 5 t trvuz msssig teng Vlining msssi t qovun msssidn 4 mrt ko p Vli iln t qovunning irglikdgi msssi t trvuz msssig teng Alining msssi necht qovunning msssig teng? - Yig indining kvdrti Ayirmning kvdrti Ikkit son yig indisining kvdrti ( + ) ni qrymiz Ko phdni ko phdg ko pytirish qoidsidn foydlni, hosil qilmiz: y ni 0 ( ) ( ) ( ) + = + + = = + +, ( ) + = + + () Ikki son yig indisining kvdrti irinchi son kvdrti, qo shuv irinchi son iln ikkinchi son ko pytmsining ikkilngni, qo shuv ikkinchi son kvdrtig teng () formulni 0-rsmd tsvirlngn kvdrtning yuzini ko zdn kechiri, osongin hosil qilish mumkinligini yti o tmiz Endi ikki son yirmsining kvdrtini qrymiz: y ni ( ) ( )( ) = = + = +, ( ) = + ()

111 Ikki son yirmsining kvdrti irinchi son kvdrti, yiruv irinchi son iln ikkinchi son ko pytmsining ikkilngni, qo shuv ikkinchi son kvdrtig teng () v () tengliklrd v istlgn sonlr yoki lgerik ifodlrdir () v () formullrni qo llshg doir misollr: 0-rsm ) ( ) ( ) ( ) m+ k = m + m k + k = 4m + mk + 9 k ) ( ) ( ) ) ( ) (( )( ) ) ( ) ( ) 4 5 = = = + = + = ( ) ( ) = + = + + = Zruriy hisolshlrni og zki jri, orliq ntijlrni yozmslik mumkin Msln, irdnig undy yozish mumkin: ( ) 5 7 = Yig indi yoki yirmning kvdrti formullri qisq ko pytirish formullri deyildi v zi hollrd hisolshlrni soddlshtirish uchun qo llnildi Msln: 99 = 00 = = 980 ) ( ) ) ( ) 5 = 50+ = = 704 () formul ( +) ifodning qiymtlrini tqriiy hisolshlrd hm qo llnildi son must yoki mnfiy son o li, uning moduli g nistn kichik o ls (msln, =0,00 yoki = 0,00), u hold son ynd kichik o ldi v shu sli (+) =+à+à tenglikni (+) + tqriiy tenglik iln lmshtirish mumkin Msln:

112 ) (,00) = ( +0,00) + 0,00 =,004 ) (0,997) = ( 0,00) 0,00 = 0,994 Yig indining kvdrti v yirmning kvdrti formullri ko phdni ko pytuvchilrg jrtishd hm qo llnildi, msln: ) x x x x ( x ) = = ) + = ( ) + ( ) = ( ) Msl Formulni isotlng: ( ) + = () ( ) ( ) ( ) ( ) ( ) + = + + = = = = Xuddi shung o xshsh, ( ) formulni hm isotlsh mumkin = + (4) () v (4) formullr, mos rvishd, yig indining kui v yirmning kui formullri de tldi () v (4) formullr hm qisq ko pytirish formullri hisolndi Mshqlr Quyidgi mshqlrd ikkihdning kvdrtini ko phd shklid tsvirlng (65 7): 65 ) ( c + d) ) ( + x ) 5) ( y + ) ) ( x y) 4) ( x + ) 6) ( + m) 66 ) ( m ) ) ( ) m 5) + 7 ) ( x ) 4) ( ) y 6 6) + 7

113 67 ) ( q + p) ) ( x + y) ) ( 6 4 ) 4) ( z t) 5 68 ) ( + ) ) ( + ) ) ( x + n ) 4) ( x + y ) 69 ) m ) ) 5 x y 4) ) ( + ) x y ) x 0, 0, ) ( ) 7 ( ) 4 0,4 0,5 c 4) = formulg qndy geometrik m no er olsiz? Nuqtlr o rnig mos so zlrni qo ying: Qirrsining uzunligi v o lgn ysymiz O lchmlri ½ ½ v ½ ½ o lgn ysymiz Ulrni shundy txlymizki, hosil o ldi 7 ) ( 4 5 ) ) ( x + xy ) ) ( ) 4) ( xy + y ) 0, 5 4 0,5 Qisq ko pytirish formullridn foydlni, mllrni jring (7 75): 7 ) ( 90 ) ) ( ) 40+ ) 0 4) ) 999 ) 00 ) 5 4) 9 75 ) 7 ) 57 ) 997 4) 00 Ifodni soddlshtiring (76 77): 76 ) ( ) + ( + ) x y x y ) ( + ) ( ) ) ( x + y) ( x y ) 4) ( + ) + ( ) 77 ) ( + ) + ( ) ) ( ) + ( ) 8 Alger, 7- sinf 4 5

114 4 ) ( x ) ( x + ) 4) ( + x) + ( x) Tenglmni yeching (78 79): 5 78 ) 6x ( 4x 5) = 5 ) x( x ) ( x ) ) 64x ( 8x ) = 87 4) ( x ) ( x ) 79 ) ( x ) ( x ) = ) ( y ) ( y ) ( y ) ) ( x ) ( x ) ( x ) 4) ( y ) ( y ) ( y ) 0 + = = = 7 80 Ifodning qiymtini toping: ) ( ) ( ) = 0 + = ,und = 6 y 5 4 y 4 y,und y = 7 5m m 5m 6 m,und m = 0, ) ( ) ( ) ) ( ) ( ) 4) ( ) ( )( ) 4x 7x + 5x 5x +,und x = x ni shundy irhdg lmshtiringki, ntijd tenglik jrilsin: x 4 = ) ( ) ) ( ) ) ( ) 6 x + 7c = c+ 49 c + x = x = ) ( ) 8 Ifodni ikkihdning kvdrti shklid tsvirlng: 4 ) 0+ 5 ) k + k + ) x x 4) p,6p+ 0,64 8 x ni shundy irhdg lmshtiringki, ntijd ikkihdning kvdrti hosil o lsin:

115 ) x ) 6 x + 49 ) p 0,5 p+ x 4) 6 + x 84 ning qndy qiymtlrid ifodni ikkihdning kvdrti ko rinishid yozish mumkin: ) ( ) ( ) ) ( ) ( ) x 5 + 4x + + x 7x + 0 5x 8 + x? 85 Isot qiling: ) ( ) = ( ) 4) ( ) = ( ) ) ( ) = ( + ) 5) ( ) ) ( ) ( + ) = ( + ) 6) ( ) + = = + - Kvdrtlr yirmsi formulsi y ni Ikki son yig indisini ulrning yirmsig ko pytirmiz: ( )( ) + = + =, ( + )( ) = () ( ) ( ) = + () Ikki son kvdrtlrining yirmsi shu sonlr yirmsi iln ulr yig indisining ko pytmsig teng () v () tenglikd, istlgn sonlr yoki lgerik ifodlrdir, msln: ( ) ( ) ) nm+ k nm k = nm 9 k ( )( ) 4 4 )4 5 = ( + ) = ( + )( + + ) )

116 6 () formulning geometrik tlqini () formul hm qisq ko pytirish formulsi deyildi Uni hisolshlrni soddlshtirish uchun qo llnildi Msln: ( )( ) )6 57= = 600 9= 59 ( )( ) ) 98 0 = = 00 = = 9996 () tenglik kvdrtlr yirmsi formulsi deyildi U ko phdlrni ko pytuvchilrg jrtishd qo llnildi Msln: ) 9 = = + ( ) ( ) ( ) ( ) ( )( ) 4 ) 4 0,64c = 0,8c = 0,8c + 0,8 c ( ) = ( ) ( + ) ) 4) B C H E F A G D M S ABCD = S AEFG = S GFEBCD = S EBHL S GFEBCD = S EBHL =( )(+) ( ) ( c) ( c)( c) = ( + c)( + c) + = = () formuldn foydlni, ko pytirishni jring (86 94): 86 ) ( c+ d)( c d) ) ( + c)( c ) ) ( p+ q)( p q) 4) ( m n)( m n) L Mshqlr +

117 87 ) ( x + 5)( x 5 ) ) ( )( + ) ) ( + )( ) 4) ( + x)( x ) ) ( + )( ) ) ( y + 6x) ( 6 x y) ) ( c+ d)( c d) 4) ( m n)( n+ m) 89 ) 4d + 4 d ) y x y + x ) + 4) m+ n m n 90 ) ( c + d ) ( c d ) ) ( x 4 y ) ( y + x 4 ) ) ( + ) ( ) 4) ( m n ) ( m + n ) 9 ) ( + 4 ) ( 4 ) ) ( t + p 4 ) ( p 4 t ) ) ( m 4 5n )( 5n + m 4 ) 4) ( )( + ) 4 4 0, 0,5 0,5 0,, 0,, 0, 9 ) + ) 0,5q + p 0,5 q p ) x y x + y 4),5c +,5 c 9 ) ( xy 4xy ) ( x y + 4 xy ) ) ( + xy ) ( xy ) ) ( 5 + ) ( 5 ) 4) ( 4xy) ( + 4 xy ) ) ( + x) ( x) ( 9 + x ) ) ( 4x + y ) ( x + y) ( x y) ) ( x + ) ( x + ) ( x ) 4) ( ) ( + ) ( + ) 9 4 Qisq ko pytirish formullridn foydlni, hisolng (95 96): 95 ) 48 5 ) 68 7 ) 4 7 4) ) 7 ) 44 6 ) )

118 97 Soddlshtiring: ) ( c ) ( c+ )( c) ) ( + ) ( + )( ) ) ( x + y)( x y) + ( x + y) 4) ( )( + ) ( ) 5) ( )( + ) + + 6) ( )( ) Ifodning qiymtini toping: ) ( ) ( )( ) ) ( ) ( )( ) 4m m+ + m m+,und m =,4 x + 4 0x x x,und x = 0, k 7 k + 5 k 5 k k,und k = ) ( )( ) ( ) ( )( ) 4) ( ) ( )( ) ( )( ) ,und = 5 99 Tenglmni yeching: ) ( x ) ( x )( x ) ) ( x ) ( x )( x) = = 49 ) x + x 9x 8= 0 4) y y 4y + = Kvdrtning ikki qrm-qrshi tomonining hr iri 8sm g uzytirildi, qolgn ikki tomoni es shunch qisqrtirildi Shklning yuzi qndy o zgrdi? 40 Hisolng: 4 5 0,8 5 0, ,5 8

119 - Ko phdni ko pytuvchilrg jrtishning ir nech usulini qo llsh Ko phdni ko pytuvchilrg jrtishd zn ir ems, lki ir nech usullr qo llnildi Misollr keltirmiz: ) ko phdni ko pytuvchilrg jrting: ( ) ( ) ( ) = = + Bu yerd ikkit usuldn foydlnilgn: umumiy ko pytuvchini qvsdn tshqrig chiqrish v kvdrtlr yirmsi formulsini qo llsh ) ( +) 4 ko phdni ko pytuvchilrg jrting: ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) + 4 = + = = = = = ( ) ( ) = + Bu yerd qo shiluvchilr umumiy ko pytuvchig eg emsligi sli, vvl kvdrtlr yirmsi formulsidn foydlnildi, so ngr yig indi v yirm kvdrtlrining formullridn foydlnildi Yn ir misol yechi ko rylik: ) 4x y 4x y ( 4x y ) ( 4x y) + + = + + = ( x y)( x y) ( x y) ( x y)( x y ) = = + + Birhdlr umumiy ko pytuvchig eg o lmgni v iror formulni qo llsh mumkin o lmgni uchun, u yerd vvl guruhlsh usulidn foydlnildi, so ngr es kvdrtlr yirmsi formulsi qo llnildi Ko ri chiqilgn u misollr ko phdni ko pytuvchilrg jrtishg doir topshiriqlrni jrishd quyidgi trtig rioy qilish foydli eknligini ko rstdi: ) umumiy ko pytuvchini (gr u or o ls) qvsdn tshqrig chiqrish 9

120 0 ) ko phdni qisq ko pytirish formullri o yich ko pytuvchilrg jrtishg urini ko rish ) gr oldingi usullr mqsdg oli kelms, guruhlsh usulini qo llshg hrkt qilish Msl Tenglikni isotlng: ( ) ( ) + = + + () Tenglikning o ng tomonidgi qvslrni ochmiz: ( ) ( ) + + = = + Tenglikning o ng tomoni chp tomonig tengligi keli chiqdi, y ni () tenglik isot qilindi Xuddi shu ki tenglikning to g riligi isotlndi ( ) ( ) = + + () () v () tengliklr, mos rvishd, kulr yig indisi v kulr yirmsi formullri de tldi Bu formullr hm ko phdni ko pytuvchilrg jrtishd qo llnildi Msln: ( ) ( ) ( ) ( ) ( )( ) )7+ = + = ) x 8xy = x x 8y = x x y = x x y x + xy + 4 y 40Hisolng: ) 47 7 ) ) 50,7 50,6 4) 9,4 9, 40 (Og zki) Ko pytuvchilrg jrting: Mshqlr ) 6 x ) 5 ) y 4)

121 404 ) ( + ) = + 4 ) ( ) = 4 9 tengliklr hqid nim dey olsiz? ) ulr qysi v lrd to g ri-yu, qysilrid noto g ri? ) ixtiyoriy v lr uchun ulrni to g ri o ldign qilishni uddsidn chiqsizmi? Ko pytuvchilrg jrting (405 46): 405 ) 5x 9 ) 4 9 ) 64y 6 x 4) ) cd 9 ) 6 ) 4 9 4) 6x 5 y ) y x ) 0, ) 4) 0,09x 6 y ) 6xy ) xy 6 ) ) ) 4 4 ) ) 6 4) 8 40 ) ( + ) c ) ( ) ) ( m n) k 4) ( ) + 9 x y 4 y 4 ) ( + ) ( c) ) ( + ) ( + ) ) ( + ) ( + c) 4) ( ) ( + ) 4 ) 9 6+ ) ) + c+ c 4) 8 8 x + x 4 ) 9x + 4x + 6 ) 6m + mn+ n ) ) ) x + xy + y ) 4c 4 + c ) p pq + q 4) ) ) ) )

122 46 ) ) ) ) 47 Ifodning son qiymtini toping: ) 5m 0mn+ 5 n,und m = 4, n = 4 ) 6m + mn+ 6 n,und m = 56, n = 44 ) 6 + 4,und = 4, = ) 64 8,und = 6, = Tenglmni yeching: ) x 6 = 0 ) 4x + 4x + = 0 ) x = 0 4) 5 0x + x = Hisolng: ) ) 48 8 ) ) Tushiri qoldirilgn shundy uchhdni topingki, tenglik jrilsin: + = + x y = x y ) x y ( x y) ( ) ) ( ) ( ) ) ( x + y) = ( x + y) () 4) ( x y) = ( x y) ( ) 4 Ko pytuvchilrg jrting: ) x y ) x + 7 5) n 64 7) p ) + c d 4) 7 6) + 8) 5 Ko pytuvchilrg jrting (4 44): 64 5 y ) 5 + 4) 64 y ) ) + 7 4) ) 7m 8 ) 4 ) 8

123 9 44 ) ) 6 6 ) 6 x 79 4) 6 64 y Ifodni qisq ko pytirish formullridn foydlni, ikkihd shklid yozing (45 46): 45 ) ( z + 5) ( z 5z + 5 ) ) ( x + y) ( 4x 6xy + 9 y ) ) ( y + ) ( y y + 4 ) 4) ( c d) ( c + cd + d ) 46 ) ( 0 )( ) ) ( 5) ( ) ) m n m + mn+ n ) x y x + xy + y Ko pytuvchilrg jrting: ) ( ) ( ) ) ( + ) + ( + ) ) ( ) ( ) 4) ( ) + ( ) 48 Hisolng: ) ) ,98 7,98,0+,0 7,98 +,0 49 Qvslr ichig shundy hdlr yozingki, hosil o lgn ifod x ning rch qiymtlrid hm o zgrms o lsin: ) ( x ) + ( x + ) ( ) ) ( x ) ( x ) ( + ) Tenglmni yeching: ) ( x ) ( x x ) x( x ) ( x ) ) ( x ) ( x x ) x( x ) ( x ) ) ( x ) ( x + x + ) x( x ) = 4) ( x ) ( x x ) x( x ) = = = 7

124 Ko pytuvchilrg jrting (4 44): 4 ) ) y y ) mn mn 4) 4 4 ) 4 6 xy xy ) 8 7 xy 4 ) 4 7cd 6 c 4) 4 ) ) 8p 6p+ 8 ) m + n 4 mn 5) ) 5 4 5x + 0xy + 5 y 6) mn+ 4mn+ mn 4 44 ) c + d ) cd 6 c 5) 7x 56 xy ) 54x 6 4) 6) Hisolng: 9,7 8, + 8 8,6 46 )Agr n toq son o ls, (n+) ifodning 8 g ) ixtiyoriy nturl son nd n +n +n ifodning 6 g o linishini isotlng Ko pytuvchilrg jrting (47 48): 47 ) ( ) + + c ) ) ( x xy + y ) 4) + ( x xy y ) 4 48 ) + + ) x y x + y 5) m 5 m + m ) 4) x x x + 6) soni g o linishini isotlng 4 x x x n istlgn utun son o lgnd (7n ) (n 7) ifodning qiymti 5 g o linishini 9 g o linishini isot qiling 44 Tenglmni yeching: ) ( x ) ( x + x + 9) ( x 7) = x ) x ( x x ) ( x ) x( x ) ( x ) = 0

125 44 Motorli qyiqning oqim o yich tezligi 8 km/sot, oqimg qrshi tezligi es 4 km/sot Dryo oqimining tezligini v qyiqning turg un suvdgi tezligini toping O zingizni tekshiri ko ring! Ifodni stndrt ko phd ko rinishid tsvirlng: ( ) + ( )( + ) + 6 Ko pytuvchilrg jrting: ) xy y ) 6 8 ) x 6 x 4) x 0x + 5 5) ( x ) + yx ( ) 6) 4+ Ko phdni ko pytuvchilrg jrting v uning =, = o lgndgi son qiymtini toping: + 9 I V o g d o i r m s h q l r Ko pytuvchilrg jrting (44 447): 44 ) 6 ( + ) + ( + ) ) ( ) + ( ) ) 4 ( x y) + ( x y ) 4) ( ) ( ) 444 ) ( x+ y)( x y) + ( x+ y ) ) ( ) ( + )( ) ) ( + ) ( + ) x y x x y 4) ( ) ( ) 445 ) ( y + z)( x + x) + ( y z)( x + x) ) ( y z)( x x) + ( y z)( x + x) ) ( 6x ) + 7x( 6x ) 4y( 6x ) ) x( x y) y( x y) ( x y)

126 446 ) c c ) 0x + 0xy + 5x + 5 y ) 5x + 4xy 0y 4 x 4) 48xz + xy 5yz 0 y 447 ) 6 5c 0c + c ) 6mnk + 5mk 4nk 5 mn ) 8c+ 5c 0cx + 8 x 4) 4x 5c + 40c+ 9 cx 448 Ifodni soddlshtiring: ) ( x ) ( x ) + 7 ) ( x + ) ( x ) 7 x ) y ( y ) + ( y ) ( y + ) 4) ( y + ) ( y ) + ( y ) 0 y Ikkit ketm-ket nturl son kvdrtlri yirmsining moduli toq son o lishini isotlng 450 Ksrni qisqrtiring: ) 5 7 ) ) 8 7 4) x v y ning istlgn qiymtlrid tenglik to g ri o lishini ¹8 isotlng: ( + )( ) = ( )( + ) x y x y x y x y ) Oildgi 6 t qizning hr irining ksi or Shu oild necht frznd or? ) Muhmmdjonning klri qnch o ls, oplri hm shunch Ktt opsining uklri soni singillri sonidn mrt ko p Shu oild necht o g il, necht qiz or?

127 IV og doir sinov mshqlri testlr Umumiy ko pytuvchini qvsdn tshqrig chiqring: 4 0 A) 6 (4 5 ) B) 6 (4 5 ) C) 6 (4 5 ) D) 6 (4 5 ) Ko pytuvchilrg jrting: 5( ) + ( ) ( ) A) ( )( + ) B) ( )( 8) C) ( )(8 ) D) ( )( + 8) Ko pytuvchilrg jrting: 4 x ( y) + 4z + 7( y x z) A) ( x y + z)(4 7) B)(y x z)(7 +4) C) ( x y z)(4 7) D) ( x y + z)(4+ 7) 4 Hisolng: 6,9 6,9,7 6,9, A) 69 B),69 C)6,9 D),69 5 Ko pytuvchilrg jrting: x + x y y A) ( + )( x + y) B) ( )( x + y) C) ( )( x y) D) ( + )( x y) 6 Ko pytuvchilrg jrting: 7 (5 ) 0+ 6 A) (5+ )(7 ) B) ( 5 )(7+ ) C) (5 )(7 ) D) (5 )(7 + ) 7 Tenglmni yeching: (x + ) (x 4) = A) 4 B) C) 5 D 4 8 Ko pytuvchilrg jrting: 8 7 A) ( )( + ) B) (+ ) ( ) C) ( ) ( ) D) ( )( ) 9 Hisolng: ( ):( ) A) 6 B)00 C)600 D)

128 Trixiy m lumotlr Al-Koshiyning Arifmetik kliti srid ikkihdni ixtiyoriy nturl drjg ko trish qoidlri erilgn Turli lgerik formullrni isotlshd, tenglmlrni yechishd geometrik mulohzlrdn foydlnish qdimgi Xitoy, Yunoniston, Hindiston, O rt Osiyo mtemtiklri srlrid uchrydi Ulr ( + ) = + +, ( ) = +, =( )½ ½(+)(yoki ( ) = ( ) + ( ) )ki yniytlrni geometrik usuld isotlgnlr Msln, = ( )( + ) for- mulni isotlshg shundy yondshilgn: tomoni g teng kvdrtdn tomoni g teng kvdrtni qirqi olins, qolgn shklning yuzi: ( ) + ( ) = ( )( + ) g, yoki riir, ( ) + ( ) g teng o lishi -rsmdn rvshn ko rini turidi Demk, = ( )( + ) formul to g ri To g ri urchkli uchurchkning tomonlrini utun (yoki rtsionl) sonlrd ifodlsh uchun Xitoy mtemtiklri miloddn vvlgi irinchi ming yillrdyoq tenglikdn foydlngnlr 8 - rsm p q p + q + ( pq) =

129 ALGEBRAIK KASRLAR Algerik ksr Ksrlrni qisqrtirish msl Kterning turg un suvdgi tezligi sotig kitrg, dryo oqimining tezligi sotig kilometrg teng ning dryo oqimi o yich hrkt tezligi uning dryo ig qrshi hrkt tezligidn nech mrt ortiq? Kterning dryo oqimi o yich tezligi sotig ( + ) kitrg teng oqimg qrshi tezligi sotig ( ) kilometrg Shuning uchun dryo oqimi o yich hrkt tezligi g qrshi hrkt tezligidn + ortiq o ldi + ifod lgerik ksr deyildi Bu ksrning surti +, ji es mumn, surt v mxrji lgerik ifodlr o lgn ksr rik ksr deyildi lgerik ksrlrg doir yn ir nech misollr keltirmiz: x( + c) x + y c y( c) gr lgerik ksrg kiruvchi hrflr o rnig iror sonlr ls, u hold zrur hisolshlr jrilgndn keyin shu rik ksrning son qiymti hosil o ldi sln, =0, = 8 o lgnd + lgerik ksrning iymti = = 9 g teng o ldi 0 8