matematikaƒ›ათემატიკა9... · 5 sesavali ix klassi matematikis sagnis swavlebis...

maTematika 9

nana jafariZe

maia wilosani

nani wulaia

maswavleblis wigni

mascavleblis cigni_IX_kl.._axali.indd 1 03.07.2012 16:28:30

nana jafariZemaia wilosaninani wulaia

maTematika 9maswavleblis wigni

ydis dizaini: marTa TabukaSvili, naTia kvaracxeliadakabadoneba: maia feiqriSvili

© bakur sulakauris gamomcemloba, 2012

pirveli gamocema, 2012

bakur sulakauris gamomcemlobamisamarTi: daviT aRmaSeneblis 150, Tbilisi 0112 tel.: 291 09 54, 291 11 65elfosta: [email protected]

www.sulakauri.ge

ISBN 978-9941-15-634-2

N. JafaridzeM. TsilosaniN. Tsulaia

MaTh 9Teacher’s Book

© Bakur Sulakauri Publishing, 2012Tbilisi, Georgia


s a r C e v i

Sesavali erovnuli saswavlo gegma wlis bolos misaRwevi Sedegebi da maTi indikatorebiSinaarsisa da miznebis rukamoswavlis Sefasebis sistema.gTavazobT ramdenime gakveTilis sanimuSo scenars

I Tavi 1. funqcia2 amovicnoT wrfivi funqcia3. f : x → x2 funqcia4. kvadratuli gantolebis grafikuli amoxsna5. kvadratuli gantolebis amoxsna7. vietis Teorema8. kvadratuli samwevris daSla mamravlebad

II Tavi 1. toldidi da proporciuli nawilebi samkuTxedSi2. toldidi da proporciuli nawilebi trapeciaSi,nebismier oTxkuTxedSi 3. wesieri mravalkuTxedebi4. wrewiris sigrZe, wris farTobi

III Tavi1. kvadratuli funqcia2. f:x → x2+c funqcia3. f:x → (x–d)2+c funqcia4. y=ax2 funqciis grafiki5. y=ax2+bx+c funqciis grafiki7. parabolis mdebareoba sakoordinato RerZebis mimarT8. kvadratuli utolobis amoxsna9. meore xarisxis orucnobian gantolebaTa sistemis amoxsna12. ricxviTi mimdevroba13. ariTmetikuli progresiis pirveli n wevris jamis formula14. geometriuli progresia15. geometriuli progresiis pirveli n wevris jamis gamosaTvleli formula

IV Tavi1. samkuTxedis msgavseba2. samkuTxedebis msgavsebis I niSani3. samkuTxedebis msgavsebis II niSani4. samkuTxedebis msgavsebis III niSani5. proporciuli monakveTebi msgavs samkuTxedebSi6. msgavsi samkuTxedebis farTobebis Sefardeba


8. msgavsebis meTodi geometriul agebebSi9. heronis formula10. rogor gamovTvaloT samkuTxedis farTobi, r o c amocemulobaSi figurirebs gverdebis da medianebis sigrZeebi11. kuTxis sinusi, kosinusi, tangensi da kotangensi 12. ZiriTadi trigonometriuli igiveobebi13. zogierTi kuTxis sinusis, kosinusis, tangensis da kotangensis mniSvneloba 14. marTkuTxa samkuTxedi 15. samkuTxedis farTobis gamosaTvleli formula ori gverdiTa da

maT Soris mdebare kuTxis sinusiT16. ramdenime saintereso amocana

V Tavi 1. naSTTa klasebi 2. Sedareba4. naturaluri ricxvidan namdvil ricxvamde5. n-uri xarisxis fesvi6. ariTmetikuli fesvis Tvisebebi7. miaxloebiTi gamoTvlebi11. orucnobian utolobaTa sistemis amoxsna

VI Tavi1. veqtoris cneba. toli veqtorebi2. veqtorebis Sekreba 3. veqtorebis sxvaoba 4. veqtorebis gamravleba ricxvze6. sibrtyis dafarva 7. marTobi, daxrili, gegmili. manZili wertilidan sibrtyemde9. prizmis kerZo saxeebi 10. piramida

VII Tavi 1. simravle2. simravleTa sxvaoba 3. albaTobis Teoriis elementebi4. xdomilobaTa jamis albaToba5. xdomilobaTa namravlis albaToba. xisebri diagrama 6. monacemTa warmodgenis xerxebi 7. monacemTa dajgufeba. sixSireTa intervaluri ganawileba8. foTlebiani Reroebis msgavsi diagrama 9. SerCeviTi ricxviTi maxasiaTeblebi

sakontrolo weris nimuSebi


5

Sesavali

IX klasSi maTematikis sagnis swavlebis ZiriTadi mizania mozardSi kvlevis Cvevis, agreTve analitikuri, logikuri, sistemuri da simboluri azrovnebis gamomuSaveba. maTematikis swavlam moswavles unda SesZinos is unar-Cvevebi, romelic mas daexmareba cxovrebiseuli, praqtikuli problemebis gadaWraSi.

erovnuli saswavlo gegmis daniSnulebaa daexmaros saskolo ganaTlebis procesis monawileebs am procesis dagegmvasa da warmarTvaSi.

erovnul saswavlo gegmaSi aRwerilia is savaldebulo moTxovnebi, romelsac unda akmayofilebdes yvela moswavle saswavlo wlis dasrulebis mere. es moTxovnebi TiToeuli mimarTulebisaTvis Sedegebisa da maTi indikatorebis enazea Camoyalibebuli.

Sedegi aris debuleba imis Sesaxeb, Tu ra unda SesZlos moswavlem swavlis mocemuli safexuris dasrulebis Semdeg.

indikatori aris debuleba im codnisa da unar-Cvevebis demonstrirebis Sesaxeb, romelic Camoyalibebulia Sesabamis SedegSi. indikatoris ZiriTadi daniSnulebaa imis warmoCena, miRweulia Tu ara Sedegi. indikatori orientirebulia unar-Cvevebze da Camoyalibebulia aqtivobis enaze.

IX klasis warmodgenili saxelmZRvanelos daniSnulebaa xeli Seuwyos erovnuli saswavlo gegmiT gaTvaliswinebuli unar-Cvevebis gamomuSavebas.

saxelmZRvanelo faravs standartis yvela Sedegs.masalis miwodebis ZiriTadi meToduri orientiria problemuri Txroba. moswavle

aris gakveTilis axsnis aqtiuri monawile.gagacnobT wignis struqturas.TiTqmis yvela paragrafi iwyeba situaciuri amocaniT, maprovocirebeli SekiTxviT

an iseTi amocaniT, romelic moswavlisagan kvlevas moiTxovs da romelic iZleva varaudis gamoTqmis saSualebas. gakveTilis etapebi gamoyofilia aqtivobebiT, riTac xdeba axali masalis aTvisebis Semowmeba da gaRrmaveba. varskvlaviT moniSnulia amocanebi maRali SefasebisaTvis.

maswavleblis sarekomendacio wignSi mocemulia ramodenime gakveTilis scenari, aqtivobebis mizani, daniSnuleba, savaraudo da swori pasuxebi, sakontrolos nimuSebi. mocemulia Sefasebis ZiriTadi mdgenelebi, damxmare literatura maswavleblisaTvis.

agreTve gTavazobT savaraudo saaTobriv ganawilebas. sarezervo saaTebi gvaZlevs saSualebas, rom zogierT gakveTils maswavlebelma meti dro dauTmos, gamoiyenos Tavis Sexedulebisamebr.


6

erovnuli saswavlo gegma

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11

Sinaarsisa da miznebis ruka

Sinaarsi Temis kavSiri miznebTan da SedegebTan

sava

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do

xa

ngr

Zliv

oba

1 2 3

I Tavifunqcia.amovicnoT wrfivi funqcia.f: x→x2 funqcia.kvadratuli gantolebis grafikuli amoxsna.kvadratuli gantolebis amoxsna.jgufuri mecadineoba: amovxsnaT kvadratuli utoloba. vietis Teorema.kvadratuli samwevris daSla mamravlebad.

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. .18 sT

sakontrolo wera №1 1 sT

II Tavitoldidi da proporciuli nawilebi samkuTxedSi.toldidi da proporciuli nawilebi trapeciaSi, nebismier oTxkuTxedSi.wrewiris sigrZe, wris farTobi.

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10 sT



12

1 2 3

III Tavikvadratuli funqcia.f: x → x2+c funqcia.f: x → (x–d)2+c funqcia.f: x → ax2 funqciis grafiki.f: x → ax2+bx+c funqciis grafiki.parabolis mdebareoba sakoordinato RerZebis mimarT.kvadratuli utolobis amoxsna.meore xarisxis orucnobian gantolebaTa sistemis amoxsna.amovxsnaT gantolebaTa sistemavietis Teoremis gamoyenebiT.orucnobian utolobaTa sistemis amoxsna.ricxviTi mimdevroba.ariTmetikuli progresiis pirveli n wevris jamis gamosaTvleli formula.geometriuli progresia.geometriuli progresiis pirveli n wevris jamis gamosaTvleli formula.

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42 sT


IV TavisamkuTxedebis msgavseba.samkuTxedebis msgavsebis I niSani.samkuTxedebis msgavsebis II niSani.samkuTxedebis msgavsebis III niSani.proporciuli monakveTebi msgavs samkuTxedebSi.msgavsi samkuTxedebis farTobebis Sefardeba.namdvil ricxvebze moqmedebebis geometriuli gamosaxva .msgavsebis meTodi geometriul agebebSi.heronis formula.rogor gamovTvaloT samkuTxedis farTobi, roca mocemulobaSi figurirebs gverdebis da medianebis sigrZeebi.kuTxis sinusi, kosinusi, tangensi da kotangensi.ZiriTadi trigonometriuli igiveobebi.zogierTi kuTxis sinusis, kosinusis, tangensisa da kotangensis mniSvneloba.marTkuTxa samkuTxedi.samkuTxedis farTobis gamosaTvleli formula ori gverdiT da maT Soris mdebare kuTxis sinusiT.ramdenime saintereso amocana .

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30 sT



13

1 2 3

V Tavikvadratuli funqcia.naSTTa klasebi.Sedareba.ricxvTa gayofadobis erTi saintereso Sedegi.naturaluri ricxvidan namdvil ricx-vamde.n-uri xarisxis fesvi.ariTmetikuli fesvis Tvisebebi.miaxloebiTi gamoTvlebi.

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12 sT


VI Taviveqtoris cneba. toli veqtorebi.veqtorebis Sekreba.veqtorebis sxvaoba.veqtoris gamravleba ricxvze.sibrtyis gardaqmna. sibrtyis dafarva.

1.prizma.2.marTobi, daxrili, gegmili. manZili wertilidan sibrtyemde.3.prizmis kerZo saxeebi.4.piramida.

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12 sT


VII Tavisimravle.simravleTa sxvaoba.albaTobis Teoriis elementebi.xdomilobaTa jamis albaToba.xdomilobaTa namravlis albaToba. xisebri diagrama.monacemTa warmodgenis xerxebi.monacemTa dajgufeba. sixSireTa intervaluri ganawileba.foTlebiani Reroebis msgavsi diagrama.SerCeviTi ricxviTi maxasiaTeblebi.

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10 sT



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17

gTavazobT ramdenime gakveTilis sanimuSo scenars

I Tavi . §7. vietas Teoremareziume: moswavleebi gaecnobian vietas Teoremas.

aqtivobis mizani:

· moswavleebi SeZleben vietas Teoremis gamoyenebiT zepirad dainaxon dayvanili kvadratuli gantolebis mTeli fesvebi

· SeZleben mocemuli fesvebis mixedviT kvadratuli gantolebis Sedgenas.

savaraudo xangrZlivoba 3 gakveTili:

aqtivobis aRwera:

1. gakveTili iwyeba saSinao davalebis analiziT (5-10wT)

2. maswavlebeli avalebs wyvilebs ifiqron paragrafis dasawyisSi mocemul davale-baze (5wT).

davaleba SeiZleba Sesruldes Semdegnairad: Seadginon sistema (Casvan 02 =++ cbxx

dayvanil gantolebaSi x1 da x2).

=++=++

039024

cbcb

.

SesaZloa daweron 0)3)(2( =−− xx . imis gaTvaliswinebiT, rom fesvebia x1=2 da x2=3.

3. Semdeg xdeba vietas Teoremis damtkiceba(10wT);

4. maswavlebeli avalebs moswavleebs Camoayalibon vietas Teoremis Sebrunebuli Teorema da akonkretebs, rom Sebrunebuli Teoremac samarTliania. (5 wT).

5. maswavlebeli iZaxebs dafasTan moswavles da xsnian paragrafSi mocemul garCeul magaliTebs (10 wT).

sasurvelia yuradReba gamaxvildes imaze, rom dayvanil gantolebas aqvs an orive iracionaluri, an orive mTeli, an ar aqvs amonaxsni.

6. maswavlebeli ajamebs axal masalas da aZlevs davalebas №1, 2, 3, 4, 5.danarCeni ori saaTi moxmardeba Seswavlili masalis gaRrmavebasa da darCenili dava-lebebis ( №6-22) amoxsnas.

III Tavi. §5. funqciis y=ax2+bx+c grafiki

reziume:

· moswavleebi ecnobian cbxaxy ++= 2 funqciis grafiks;

· cbxaxy ++= 2 funqciis gamokvlevis zogad sqemas;

· parabolas wveros koordinatebs.

aqtivobis mizani:

· moswavleebi SeZleben cbxaxy ++= 2 funqciis grafikis agebas

· SeZleben funqciis gamokvlevas.


18

· SeZleben koeficientebis mixedviT daadginon funqciis grafikis mdebareoba sa-koordinato RerZebis mimarT.

savaraudo xangrZlivoba 3 sT.

aqtivobis aRwera: (pirveli gakveTili).1. gakveTili iwyeba saSinao davalebis analiziT (5-10wT);

2. maswavlebeli avalebs moswavleebs ifiqron paragrafis dasawyisSi dasmul probl-emur amocanaze da Semdeg xdeba amoxsnis demonstrireba (10wT).

3. maswavlebeli, winaswar gamzadebuli plakatebiT da funqciebis grafikebis gar-daqmnebiT axdens agebis demonstrirebas (10wT).

4. Semdeg maswavlebeli avalebs moswavleebs ifiqron paragrafSi dasmul individu-alur davalebaze (10wT)

5. maswavlebeli iZaxebs moswavles, romelic moaxdens individualuri davalebis prezentacias. maswavlebeli ajamebs Sedegebs da aZlevs saSinao davalebas #1, 2, 3, 4.

aqtivobis aRwera: (meore gakveTili)

1. gakveTili iwyeba saSinao davalebis analiziT (5wT);

2. maswavlebeli, moswavleebis aqtiuri monawileobiT, gamoiyvans parabolis wveros koordinatebis formulebs (5wT).

3. ganixileba funqciis gamokvlevis zogadi sqema da SemTxvevebisaTvis cal-calke (15wT).

4. maswavlebeli avalebs moswavleebs, ifiqron paragrafSi dasmul individualur SekiTxvaze da Semdeg romelime moswavle axdens davalebis prezentacias (10 wT).

5. Semdeg xdeba paragrafSi ganxiluli amocanebis amoxsna, maswavlebeli ajamebs Sede-gebs da aZlevs saSinao davalebas №5, 6, 7, 8, 9, 10.

darCenili erTi saaTi emsaxureba Seswavlili masalis gameorebas da darCenili sa-varjiSoebis amoxsnas.

III Tavi. §8. amovxsnaT kvadratuli utoloba

aqtivoba: kvadratuli utolobis amoxsna

reziume: moswavleebs xsnian kvadratul utolobas.

aqtivobis aRwera:

vixilavT paragrafSi garCeul utolobebs. SegviZlia dafaze davweroT kidev ramden-

ime utoloba a) (x–2)(x+3)≥0, b) xx

32

0#+- ; g) x2(x+3)>0, d) x2>9.

da SevecadoT moswavleebma msjelobiT amoxsnan TiToeuli maTgani.

amovxsnaT (x–2)(x+3) ≥0 utoloba.

kiTxvebi: rodis aris ori gamosaxulebis namravli arauaryofiTi? jer utoloba

amovxsnaT sistemis daxmarebiT. miviRebT: x≤–3 an x≥2.


19

b) SevecadoT xx

32

0#+- utolobis amoxsnas, ricx-

viTi wrfeebis saSualebiT rogorc es paragraf-

Sia naCvenebi. mivaqcioT yuradReba, rom utoloba

ar aris kvadratuli, magram misi amoxsna xdeba kvadratuli utolobis msgavsad.kiTxva: ra gansxvaveba iqneba (x+3)(x–2) ≤0 utolobis pasuxsa da mocemuli utolobis pasuxs Soris? moswavleebi iwyeben msjelobas:Sefardeba iqneba uaryofiTi, roca TiToeul komponents sxvadasxva niSani aqvs.yuradReba gavamaxviloT aramkacr utolobis niSanze. 3 da 2 ricxvebidan romeli unda ekuTvnodes pasuxs da ratom?miviReT x∈(−3;2].

ganvixiloT g) x2(x+3)>0 utoloba, romelic ar aris kvadratuli, magram misi amoxsna wina magaliTebis gaTvaliswinebiT. moswavleebs ar gauWirdebaT.amovxsnaT utoloba msjelobiT: ori gamosaxulebis namravli dadebiTia, Tu orive dadebiTia an orive arauaryofiTi, magram x2 uaryofiTi ver iqneba, amasTan x=0 utolobas ar akmayofilebs, e. i. x∈(−3;0) (0;∞).

davinaxoT amoxsnis procesi ricxviT RerZebze:

d) vxsniT x2>9 utolobas. sanam mivuTiTebdeT moswavleebs rogor unda daviwyoT utolobis amoxsna, vacadoT maT sakuTari azris gamoTqma. Savaraudoa, zogierTma daweros Semdegi pasuxi: x ≥ ±3. gavaanalizoT es tipiuri Secdoma da gadavideT amoxs-nis swor msvlelobaze. SevcvaloT utoloba Semdegi saxiT: (x–3)(x+3)>0. (15 wT).gTavazobT jgufuri muSaobis Catarebis erT models: klass gavyofT or jgufad da vaZlevT davalebas. maswavlebeli acxadebs TiToeuli amocanis Sesabamis qulas, ur-Cevs moswavleebs, rom jgufi daiyos qvejgufebad da am qvejgufebze ganawildes amo-canebi. TiToeuli qvejgufidan sabolood Tavad unda airCion is moswavle, romelic moaxdens am amocanis prezentacias an gauwevs oponirebas (imis mixedviT, Tu romel jgufs Sexvdeba es amocana saprezentaciod). maswavlebeli aZlevs moswavleebs mo-safiqreblad 15 wuTs.prezentaciamde kenWisyriT dadgindeba à~ da `b~ gundi. gundi irCevs kapitans (ro-melic adevnebs Tvals qvejgufebis muSaobas da mere irCevs amocanas). ̀ a~ gundi irCevs amocanas. `b~ gundidan arCeuli moswavle axdens amocanis prezentacias.SekiTxvis dasmis an SeniSvnis micemis ufleba aqvs mxolod à~ gundidan arCeul opo-nents. Tu `b~ gundma uari Tqva amocanaze, miiRebs nul qulas. à~ gundis warmomad-geneli valdebulia mogvawodos amoxsna. Tu amoxsna iqneba araswori, à~ gundi mi-iRebs minus imden qulas, ramdeni quliTac iyo Sefasebuli es amocana. amis Semdeg maswavlebeli anawilebs qulebs da Seaqvs winaswar gamzadebul cxrilSi.Semdeg `b~ gundi irCevs amocanas da procesi grZeldeba analogiurad. cxadia, Sei-Zleba sagakveTilo drois amowurvamde yvela amocanis amoxsna verc moaswron, mTa-varia, kapitanma sworad amoirCios amocana, romelsac jgufi dasvams, rom moipovo maqsimaluri qula.


20

IV Tavi. §1. samkuTxedebis msgavseba

reziume: moswavleebi gaecnobian msgavsi figurebis ganmartebas. iReben msgavsebis faqtis Caweris da am Canaweridan proporciis amoweris Cvevas.

aqtivobis mizani:• msgavsi figurebis gacnoba.• im faqtis gacnoba, rom erT-erTi gverdis paraleluri wrfe samkuTxedisagan mis-save msgavs samkuTxeds mokveTs.• msgavsebis faqtis Caweris da Canaweridan proporciis amoweris unar-Cvevis gamo-muSaveba da am Cvevis gamoyeneba amocanebis amoxsnisas.• gaiTavison msgavsebis arsi.

savaraudo xangrZlivoba 1 sT.

aqtivobis aRwera:1. gakveTili iwyeba saSinao davalebis analiziT (5 wT).2. Semdeg maswavlebeli avalebs wyvilebs ifiqron paragrafis dasawyisSi mocemul amocanebze (5 wT).3. wyvilebi axdenen namuSevris prezentacias. miuxedavad imisa sworad amoxsnes Tu ara dasmuli amocanebi, mainc xdeba bunebrivi gadasvla axal masalaze (5 wT).4. moswavleebs vTavazobT im faqtis damtkicebas, rom Tu samkuTxedis gverdebi ga-dakveTilia mesame gverdis paraleluri wrfiT, maSin am wrfiT miRebuli samkuTxedi Tavdapirvelis msgavsia. (15 wT).5. Semdeg ixileba paragrafSi mocemuli individualuri SekiTxvebi, rac moswavleebs exmareba ufro Rrmad Cawvdnen msgavsebis arss. sasurvelia maswavlebelma sTxovos moswavleebs, rom Tavad moifiqron msgavsi kiTxvebis Tundac TiTo magaliTi, roca pasuxia _ mcdari da roca pasuxia _ WeSmariti. (5 wT).6. paragrafSi ganxiluli ori amocana emsaxureba aqtivobis gaRrmavebagafarToebas.maswavlebeli ajamebs gakveTils da aZlevs moswavleebs saSinao davalebas. (10 wT)

V Tavi. jgufuri mecadineoba$3. gayofadobis Tvisebebis erTi saintereso Sedegi

reziume: moswavleebi gaecnobian ricxvTa gayofadobis Tvisebebis erT saintereso Sedegs da gamoiyeneben mas geometriuli amocanebis gadawyvetaSi.

aqtivobis mizani:• iswavlian, Tu rogor moZebnon erTi kaTetis mixedviT iseTi marTkuTxa samkuTxedi, romlis gverdebis sigrZeebi naturaluri ricxvebiT gamoisaxeba.• naxaven, rom uamravi naturalurgverdebiani marTkuTxa samkuTxedi arsebobs;• aageben monakveTs, romlis sigrZec iracionaluri ricxviT gamoisaxeba.• algebraSi miRebul codnas da gamoTvlebs gamoiyeneben agebis amocanebSi an ZiebaSi.

savaraudo dro – 1 gakveTili.


21

aqtivobis aRwera:1. maswavlebeli acnobs gayofadobis Tvisebas, Tu rogori ricxvebi da rogor SeiZle-ba warmovadginoT ori ricxvis kvadratebis sxvaobis saxiT da garCeuli amocanebis magaliTze aCvenebs, Tu rogor SeiZleba am faqtis gamoyeneba (10wT).2. yofs klass jgufebad da sTxovs Seasrulon paragrafSi mocemuli davalebebi. ganumartavs maT, rom TiTo amocana fasdeba 4 quliT, da rom romelime amocanaSi Sem-Txvevis gamorCena avtomaturad iwvevs Sesabamisi qulis dakargvas da aZlevs mosamza-deblad dros (15 wT).3. jgufebi abareben maswavlebels namuSevars da saTiTaod axdenen maT mier amorCeu-li amocanis prezentacias. prezentaciis dasrulebis Semdeg danarCen jgufebs aqvT SeniSvnis an SekiTxvis dasmis ufleba, romelSic maswavleblis Sexedulebis mixedviT SeuZliaT moipovon damatebiTi qula (15 wT).4. maswavlebeli ajamebs Sedegebs da acnobs jgufebs (5 wT).

VI Tavi. §4. veqtoris gamravleba ricxvze

reziume: moswavleebi ecnobian veqtoris ricxvze gamravlebis wess, iZenen am moq-medebis Sesrulebis Cvevas.

aqtivobis mizani: veqtoris ricxvze gamravlebis wesis Seswavla da amocanebis amoxs-nisas am wesis gamoyenebis unar-Cvevis gamomuSaveba.

savaraudo xangrZlivoba: 1 gakveTili.

aqtivobis aRwera:

1. gakveTili iwyeba saSinao davalebis analiziT. es saSualebas mogvcems gavimeoroT veqtorebis Sekrebis da gamoklebis wesebi. toli, koleniaruli, TanamimarTuli veqtorebis cnebebi (10wT).

2. moswavleebs vTavazobT raime a veqtori aiRon Sesakrebad oTxjer. jami aRvniSnoTAB -Ti. SevasruloT Sesabamisi naxazi. cxadia AB AB da a veqtorebi TanamimarTulia, amasTan AB veqtoris sigrZe oTxjer metia a veqtoris sigrZeze. igive gavakeToT –a veqtorisTvisac, ris Semdegac vayalibebT veqtoris k ricxvze gamravlebis wess, rogorc uaryofiTi, ise dadebiTi k-sTvis (15wT).

3. vayalibebT veqtoris ricxvze gamravlebis Tvisebebs. Semogvaqvs erTeulovani veq-toris cneba, misi aRniSvna. nebismier veqtors warmovadgenT veqtoris sigrZisa da misi TanamimarTuli erTeulovani veqtoris namravlis saxiT (5wT.).

4. aqtivobis gaRrmaveba-gamtkicebas emsaxureba paragrafis bolos dasmuli individ-ualuri SekiTxvebi. TiToeul am SekiTxvaze pasuxs moswavleebisgan viRebT, vaana-lizebT mas miuxedavad misi sisworisa. TiToeuli moswavlisagan veliT misi varau-dis sajarod gacxadebas da komentirebas (10wT).

5. maswavlebeli ajamebs gakveTils da aZlevs saSinao davalebas (5wT).


22

VII Tavi. §1. simravle

aqtivoba: simravle, moqmedebebi simravleebze _ gaerTianeba, TanakveTa.

reziume: moswavleebi ixseneben simravlis cnebas. ixseneben simravleTa TanakveTas da gaerTianebas.

aqtivobis mizani:

• simravleTa konkretul magaliTebze maTi gaerTianebisa da TanakveTis dasaxeleba, sqematurad maTi Cveneba.• simravluri operaciebis gamoyeneba zogierT amocanis amoxsnis dros;• simravleTa gaerTianeba-TanakveTis elementTa raodenobebis daTvla.

savaraudo xangrZlivoba _ 1 gakveTili.

aqtivobis aRwera:gakveTili daviwyoT imiT, rom davavaloT moswavleebs daasaxelon sxvadasxva Sinaar-sis simravleebis magaliTebi. movTxovoT maT ganmarton simravle _ amis Semdeg ga-vaxsenoT pirveladi cneba da avuxsnaT maT, rom simravle pirveladi cnebaa da moicema mxolod ise, rogorc maT es gakveTilis dasawyisSi gaakeTes _ CamoTvliT, aRweriT. gavaxsenoT simravleTa gaerTianeba da TanakveTa.davsvaT Semdegi SekiTxvebi: daasaxeleT gaerTianeba da TanakveTa: a) Tqveni klasis gogonaTa simravlisa da biWebis simravlis; b) klasis im moswavleTa simravlisa, rom-lebic dadian literaturis wreze da imaTi, romlebic dadian maTematikis wreze; g) aradadebiT da arauaryofiT ricxviTi simravleebis; d) marTkuTxedebis da rombebis simravleebis da a.S. (10 wT)gavaxsenoT simravleTa gaerTianebis da TanakveTis Canaweri, davsvaT klasSi para-grafSi ganxiluli amocana da misi msgavsi amocana; magaliTad: eqskursiaze wasuli 40 bavSvidan 15-s acvia jinsis Sarvali, 17-s _ jinsis qurTuki, 6 bavSvs acvia orive _ jinsis qurTukica da Sarvalic. ramden maTgans ar acvia arc jinsis Sarvali da arc jinsis qurTuki?yovelive amis Semdeg mivceT gaerTianebis da TanakveTis raodenobis dasaTvleli formulebi, romlebic sasurvelia moswavleebma Tavad daweron aRniSnuli amo-canebis amoxsnis Semdeg. amocanebis amoxsna unda moxdes eiler-venis diagramebis gamoyenebiT, rac ufro TvalsaCinos xdis daskvnas. dasaxelebuli amocanebisaTvis magaliTi 1-is ganxilva moswavleebs axsenebs albaTobis elementebs, romelic sa-survelia raime formiT TiTqmis yovel gakveTilze gavakeToT. (10 wT)

aqtivobis ganmtkicebisaTvis davsvaT sakontrolo kiTxvebi:

1. ra aris simravle da rogor moicema is?

2. ras warmoadgens A B simravleebis TanakveTa da gaerTianeba?

3. ra aris carieli simravle da rodis SeiZleba iyos ori simravlis TanakveTa cari-eli? gaerTianeba? moiyvaneT Sesabamisi magaliTebi.

4. sityvierad daaxasiaTeT 3-is jerad da 5-is jerad orniSna ricxvTa simravleebis TanakveTa da gaerTianeba. ramden elements Seicavs TiToeuli?


23

5. daasaxeleT Tqveni klasis im moswavleTa simravle, romlebsac yvela saganSi dadeb-iTi Sefaseba aqvT da simravle im moswavleebisa, romlebic romelime saganSi ver aswreben. ra aris albaToba imisa, rom SemTxveviT dasaxelebuli moswavle ekuTv-nis I simravles, II simravles? romelia meti?

6. ra SemTxvevaSia ori simravlis gaerTianeba an TanakveTa erT-erTi simravlis toli?

VII Tavi. §5. xdomilobaTa namravlis albaToba. xisebri diagrama

reziume: moswavleebi ecnobian Tu ra SemTxvevaSia raime a xdomiloba B-ze damokide-buli da ra SemTxvevaSi damoukidebeli. SeZleben damokidebul da damoukidebel xdomilobaTa namravlis albaTobis gamoTvlas.

aqtivobis mizani:

moswavleebma SeZlon xdomilobaTa namravlis albaTobis gamoTvla sxvadasxva xserx-iT.daeuflon xisebri diagramis agebas da misi saSualebiTac ipovon xdomilobaTa nam-ravlis albaToba.aucilebeli wina codna• ras ewodeba xdomilobaTa jami, namravli.• mocemuli cdis SemTxvevaSi elementaruli cdomilobaTa sivrcis Cawera.• mocemuli xdomilobis xelSemwyob xdomilobaTa CamoTvla.• xdomilobis albaTobis gamoTvla.• xisebri grafis ageba.• araTavsebadi, Tu Tavsebadi xdomilobebis jamis albaTobis gamoTvla.

aqtivobis aRwera:Semogvaqvs erTmaneTze damokidebuli da damoukidebeli xdomilobebis cnebebi. vTxovT moswavleebs daasaxelon damoukidebel (ori monetis agdeba, kamaTlebis gag-oreba...) da damokidebul (urnidan birTvebis amoReba Caubruneblad) xdomilobaTa magaliTebi: a) ras udris imis albaToba, Tu urnidan, romelSic 9 TeTri, 1 wiTeli da 2 lurji birTvia, SemTxveviT amoRebuli birTvi iqneba wiTeli an TeTri; b) ras udris imis albaToba, rom amoRebuli ori birTvidan orive TeTria? problema dasmulia da SeviswavloT es sakiTxi.ra Tqma unda, namravlis albaTobis gamosaTvlelad ufro martivi iqneba, roca ur-naSi ori feris birTvia. ganvixilavT saxelmZRvaneloSi mocemul amocanas (birTvebis raodenoba survilisamebr SegiZliaT SecvaloT). yuradReba mivaqcioT im faqts, Tu ratomaa B xdomiloba _ ̀ meored amoRebuli birTvi wiTelia~ damokidebuli A xdomilo-baze, roca birTvs ukan ar abruneben da ratom iqnebian isini damoukidebelni, roca birTvs ukanve Caabruneben. sasurvelia moswavleebs Tavadve davaTvlevinoT (Sesa-Zloa wyvilebSic) AB xdomilobis xelSemwyob xdomilobaTa raodenoba, aseve elemen-tarul xdomilobaTa sivrce orive SemTxvevaSi. II SemTxvevaSi (ukan Caubruneblad). saWiroa mivaxvedroT, rom am SemTxvevaSi (1; 1), (2; 2); (-3; 3) . . . (9; 9); (10; 10) elementa-ruli xdomilobebi ar arsebobs. amis dasazusteblad SesaZlebelia davsvaT kiTxva:


24

pirvelad amovida burTula #1, SesaZlebelia Tu ara meoredac #1 birTvi amovides? da a.S. mas Semdeg, rac daTvlian P(aB)-s orive SemTxvevaSi SegviZlia davsvaT aseTi kiTx-va: rogoraa SesaZlebeli P(a) da P(B)-s mniSvnelobebisagan miviRoT P(aB)? rasac vim-edovnebT, moswavleebi advilad gaarTmeven Tavs.

amis Semdeg avagebT oretapian xisebr diagramas. I etapi _ erTi burTulis amoReba, romelic an TeTria, an wiTeli. II etapi _ meore burTulis amoReba, romelic Sesabam-isad an TeTria, an wiTeli. am principiT ixazeba diagramac.

I etapze. radgan P(T)=109 , amitom isarze vwerT

109 -s → , xolo P(w)= 1

10, e.i. .

II etapi: Tu pirvelad amovida ̀ T~, maSin urnaSi dar-

Cenilia 9 birTvi, aqedan ki 8 TeTria, amitom

iwereba 108 , xolo , radgan urnaSi 9 birTvidan

mxolod erTia wiTeli. Tu pirvelad amovida `w~, maSin urnaSi darCenili 9 birTvidan

cxrave TeTria, amitom vwerT 99 → , xolo .

analogiurad gavacnobT I diagramasac.mivaqcevinoT yuradReba imaze, rom oretapiani (an meti) cdis Sesabamisi diagramis agebiT TvalsaCinod Cans yoveli elementaruli xdomiloba da advilad SegviZlia TiToeulis albaTobis gamoTvla _ yoveli toti gansazRvravs erT elementarul xdomilobas. aseve SegviZlia rTuli xdomilobis gamoTvlac. daasasruls gamovT-valoT orivejer erTi feri birTvis amoRebis albaToba (T; T) an (w; w) araTavsebadi xdomilobebia, amitom

P((T; T) an (w; w)) = 10081

1001

10082

+ = (dabrunebiT)

P((T; T) an (w; w)) = 108 + 0 =

108 (daubruneblad).

da bolos, vubrundebiT paragrafis dasawyisSi dasmul maprovocirebel SekiTxvas,romelzec moswavleebi sakmaod advilad mixvdebian, rom pasuxi uaryofiTia. vagebT xisebr diagramas. vTvliT albaTobebs im SemTxvevebSi, Tu1. pirvelive gagorebisas movida `6~.2. meore gagorebisas movida `6~ da pirvelze ara.3. pirvelze da meoreze ar movida da mesameze movida `6~ diagramaze es procesi eqvs etapiania.1 SekiTxva _ Tu TamaSi iqneboda sametapiani, maSin yvela elementarul xdomilobaTa(yoveli totis) albaTobaTa jami aris 1. eqvsetapiani cdisas yvela elementarul xdomilobaTa (yoveli totis) albaTobaTa jami tolia 1-is.2) amis Semdeg sasurvelia gakveTilze gairCes paragrafis bolos ganxiluli amocana da #1 amocana savarjiSoebidan.

ukan daubruneblad


25

I Tavi

1. funqcia

reziume:

vixsenebT funqciis cnebas, Caweris formebs, funqciis gansazRvris aresa da mniSvn-elobaTa simravles, funqciis grafiks.

amoxsnebi, miTiTebebi:

6. y=8x+12(15–x).

11. ( )4

3 8 23 12211+ + + -

= .

13. funqcia Rebulobs 11 mTel mniSvnelobas. esenia: ±5; ±4; ±3; ±2; ±1; 0.

14. V=x(x+2)(2x+2).

15. D=(0; h92 ].

16. a) R; b) R\{-1}; g) R; d)2;

33j8 ; e) R\{±

23 }; v) R\{±2}.

18. f=S(h)= h2

(m2).

2. amovicnoT wrfivi funqcia

reziume:

paragrafSi mocemuli an misi msgavsi cxrilis ganxilvis Semdeg moswavlem unda daaf-iqsiros kanonzomiereba _ argumentis erTi da imave ricxviT gazrdiT, funqciis mniSvneloba erTsa da imave (dadebiT an uaryofiT) nazrds Rebulobs. amave dros unda SeamCnios damokidebuleba argumentis nazrdsa da funqciis nazrds Soris (a; ka). xazi gavusvaT, rom es Tviseba mxolod wrfiv funqcias axasiaTebs da piriqiT, am Tvisebis mqone yvela funqcia wrfivia.


1. m=5k+3.

2. S=3+2(t–1). S=2t+1.

6. S=1000+100

1000 5$ n=1000+50n.

11. a) x y

x y

x

y x

x

y

3 2 7 2

10 2 6 2

13 13 2

5 3 2

2

2 2

+ =

- =

=

= -

=

=) )) ;


26

b) x y

x y

yy

xy

y

x

2 3 5

3 2 0

3

23 5

3

2

5 3

5 2

+ =

+ =

- + =

=-

=

=-

Z

[

\

]]

]]

) ) .

3. f : x → x2 funqcia

reziume:

moswavles unda SeeZlos y=x2 parabolas ageba, misi Tvisebebis CamoTvla, imis dadgena,mocemuli wertili mdebareobs Tu ara aRniSnul grafikze.


8. S(x)= x2 – 4

10. 27= m32^ h m=±3

a) M(5; 25) mdebareobs;b) M(±6; 8) ar mdebareobs.

11. a) an=–2an-1b) an=4 · 3n–1;

g) an=(2+ 3 )an-1; a1=2- 3 ;

d) an =an-1+ 2 ; a1=1, an an=1+ 2 ·(n–1).

12. 360°=60 danayofs, e.i. 1 danayofi = 6° da 1°=1/6 danayofs.30°= 30 · 1°=30°. 1/6 danayofi=5 danayofs45°=45·1°=45°. 1/6 danayofi=15/2 danayofs 90°=90·1/6 dan.; 180°=30 dan.b) 1 dan=6°; 5 dan=5 · 6°=30°; 10 dan=10 · 6°=60°; 30 dan=30 · 6°=180°.g) didi isari gadis 60 dan/sT-Si, e. i. V

d=60 dan/sT. V

p=5 dan/sT.

13. saaTis isrebs Soris 5 · 4=20 dan. 1 dan=6° 20 · 1 dan=20 · 6°=120°.

14. 6 sT-ze saaTis isrebs Soris 30 danayofia. saaTis isrebi moZraoben wreze erTimimarTulebiT, e. i. manZili ifareba siCqareebis sxvaobiT. V

d–V

p=55 dan/sT. 6 sT-dan

7-is aT wuTamde gavida 10wT=1/6sT. t=1/6sT. V=55dan/sT. e. i. manZili, romelic daifara10wT-Si _ S1=55/6 dan.isrebs Soris iyo 30 danayofi. es manZili Semcirda 55/6 danayofiT. e. i. isrebs Soris manZili iqneba 30 dan _ 55/6 dan= 125/6 dan=125/6 · 6°=125°.

15. 2 sT-ze isrebs Soris aris 10 danayofi. rac naklebia 90°-ze. e. i. didi isari jer undadaewios patara isars da mere unda gauswros 90°=15 danayofiT, anu erTi mimarTulebiTmoZraobisas unda daifaros 25 danayofi. S=25 dan. V=55 dan/sT. t=25 dan: 55dan/sT=5/11 sT.


27

4. kvadratuli gantolebis grafikuli amoxsna

reziume:

moswavlem unda icodes kvadratuli gantolebis zogadi saxe, misi koeficientebi. grafikulad unda xedavdes, ramdeni amonaxsni SeiZleba hqondes kvadratul gantole-bas, unda SeeZlos wrfivi da kvadratuli funqciebis gadakveTis wertilebis povna.


3. a) naxazze mocemuli y=x2 da y= x52

57

- + funqciaTa grafikebi, e.i. x x52

57

02- + = .

b) y=x2 y=x–2.gantolebaa: x2–x+2=0.

6. �R2=4 R= 2

r.

7. 3 sT-ze isrebs Soris aris 15 danayofi. isrebi rom erT wrfeze iyos, maT Soris undaiyos 0°, an 180°. cxadia, pirvelad iqneba 0°, roca didi isari daeweva pataras. e. i. unda daifaros 15 danayofiT.S=15 dan; V=55 dan/sT. t=15/55=3/11 sT.

8. davabrunoT saaTi 3 sT-ze. isrebs Soris 15 danayofia. 1) gvinda isrebs Soris 30°=5 dan.e.i. unda daifaros 10 danayofiT. S=10 dan, V=55 dan/sT= t=2/11 sT. es dro saWiroa 3 sT-

isTvis, e. i. 4-is 10 wuTisaTvis saWiroa 10 wT-iT naklebi e. i.1/6 sT-iT t1=112

61

661

- = sT.

b) davabrunoT saaTi 3 sT-ze. didi isari unda daewios pataras da gaaswros 15 danay-

ofiT, e. i. unda daifaros S=30 dan. t=5530

116

= sT e. i. dro saZiebelze 1/6-iT metia. e. i.

t1=116

61

6625

- = = sT.

11. aBD samkuTxedi gamodis marTkuTxa 30°-iani kuTxiT, e.i. BD=5 3 .

5. kvadratuli gantolebis amoxsna

reziume:

moswavlem unda icodes kvadratuli gantolebis amoxsnis formula. amoxsnis gareSe(diskriminantiT) unda gansazRvros rodis aqvs gantolebas erTi, ori an arc erTi amonaxsni. unda SeeZlos luwi meore koeficientis SemTxvevaSi Sesabamisi formulis gamoyeneba.


5. a) x(x2+8x+9)=0 x1=0 x2,3=-4± 7 ; g) (x+1)2(1+3x)=0 x=-1 x=31

- ;

z) (5x+3)2(6–8x)=0 x=53

- x=43 .

7. a) 5|x| 2+6|x|+1=0 x∈∅. b) 5|x|2–7|x|+2=0


28

|x|=52 |x|=1

x=52

! x=±1.

10. g) D4

0= 1–(a+1)(a–1)=0 a=± 2

13. a) x2+2kx–3=0 D4

=k2+3>0 nebismieri k-sTvis, e.i. x=-k± k 32+ .

b) 4x–2(k–2)x–2k=0 D4

=(k–2)2+8k=(k+2)2≥0 nebismieri k-sTvis x=k2

21

-

R

T

SSSS

.

19. (x+1)2=3x2 x=2

3 1+.

21. a) n2+(n-1)2; b) n2+(n-1)2=113, n=8.

22. ab

43

= b=3x a=4x c=5x x=4.

23. (a+3)2– a2

2` j =144 a=10.

25. a= hh242

2

+-` j .

27. xx

xx

4 91

73

++ =

++ x=5; saZiebeli wiladia

95 .

28. Cqari x km/sT 400 km

sabargo (x+20) km/sT 400 km

x x400

20400

1-+

= , saidanac x=80 km/sT.

30. x x448

448

4+

+-

= , saidanac x=20 km/sT.

31. x x2

21

2221

221

2821

8-

++

= x=7 km/sT.

32. x

420+–

x 7420

32

++

= x=63 km/sT.

33. ,x x0 5

15 151

-- = x=3 t.


29

35. x2–x· x100

=16 x=80% an x=20%.

36. x x4

64

++

=1 lika — 6 sT, Tako — 12 sT.

37. x x56 6-

+ =1 x=15.

38. gvaqvs ori SemTxveva: I didi isari CamorCeba 18°-iT, maSin dawevamde dasafaria 18° da unda gaaswros 66°-iT, e. i. unda daifaros 84°· 84°=14 dan e. i. S=14 dan t=14/55sT. II didi isari winaa 18°-iT, maSin dasafari rCeba 66°-18°=48°=8 dan t=8/55sT.

39. saaTi aCvenebs 3-is 29 wuTs. vTqvaT isrebs Soris aris x danayofi. gavasworoT saaTi2 sTze. isrebs Soris iqneba 10 danayofi. e. i. 2 saaTidan 3-is 29 wuTamde aris S=(10+x) danayofi.t=29wT=29/60sT. V=55 dan/sT.10+x=29/60·55=x=29/12 ·11–10=199/12 dan=199/12·6°=199°/2=99,5°.

41. (x–a) 1x- +^ h=0

0

x a

x

x

x a

x

0

0

1 0 Q

# #

- =

- + =

=> >) )Tu a>0, gantolebas amonaxsni ar eqneba.

42. m

m

4

64

n

n

5

3

=

=

-

+) gavamravloT m8=28, e.i. m=2. CavsvaT romelimeSi 25-n=22, saidanac n=3.

45. a) aucileblad xazi unda gavusvaT, rom moc. gantolebas rom erTi amonaxsni qondes,an a=0 an D=0.

7. vietas Teorema

reziume:

moswavleebs ganvumartoT, rom vietas Teorema mosaxerxebelia ara marto imisTvis, rom zepirad vipovoT gantolebis fesvebi, aramed imisTvisac, rom SevadginoT gan-toleba fesvebis mixedviT, an SevarCioT parametri ise, rom fesvebi akmayofilebdnen mocemul pirobas.


4. a) 42

21 =+ xx 821 =+ xx 221 =xx

421 =xx 0482 =+− xx

6. a) amovxsnaT kvadratuli gantoleba:

x2–8x+15=0 aseTi wyvilebia (3;5) (5,3).


30

8. am amocanis amoxsnamde sasurvelia gakeTdes zogadi monaxazi ax2+bx+c=0 gantolebi-

saTvis. Tu 0≥D da 0>ac

, fesvebi erTnairniSniania, xolo Tu 0≥D da 0<ac

fesvebi

sxvadasxvaniSniania.

I. orive fesvi dadebiTia, Tu

≥

<

>

0

0

0

Dabac

II. orive fesvi uaryofiTia, Tu

≥

>

>

0

0

0

Dabac

III. fesvebi sxvadasxvaniSniania, Tu 0<ac

, SevniSnoT, rom am SemTxvevaSi 0>D piroba avtomaturad sruldeba

a) 0123 2 =−− xx ; 31

−=ac sxvadasxvaniSniani.

b) 0152 2 =++ xx 0>D 021>=

ac

025>=

ab

orive fesvi uaryofiTia.

g) 0342 =+− xx fesvebi dadebiTia.

11. x2–15x+26=0 x x

x x

15

26

1 2

1 2

+ =

=)

x x x xx x1 1

2615

1 2 1 2

2 1+ =+

= .

12. 15x2–7x–3=0 x x

x x

157

51

1 2

1 2

+ =

=-

Z

[

\

]]

]

−=⋅

=+

5933

5733

21

21

xx

xx. 0

59

572 =−− xx .

14. 32

1 −=xx

21 3xx −=⇒ x2+2px–12=0 Dp

412

2= + , yovelTvis dadebiTia

x x p

x x

x x

2

12

3

1 2

1 2

1 2

+ =-

=-

=-

* x2=±2 p=±2

16. ax2+bx+a=0. 121 ==aaxx e.i. x

x1

21

=

19. jer vaCvenoT rom nebismieri oTxkuTxedis gverdebis Suaw-ertilebiT miRebuli oTxkuTxedi yovelTvis paralelogramia.

MNAC2

= =QP; MQBD2

= =NP.

e.i.


31

cxadia Tu amave dros oTxkuTxedis diagonalebi tolia, maSin MNPQ rombi iqneba.

20. aC=18; aM=MB u.v. aK; KC. gavavloT BN||MD, cxadia, aK=KP=PC, e.i. aK=

318 =6 KC=12.

23. mocemuli gantoleba tolfasia

=+−

=++

07405

2

2

axxaxx

443

21

axx

axx

=

=

00

2 ≥≥

DD

; e.i. miviReT sistema: 4

2

4321axxxx =

a

a

a

a

a

a

4

25 4 0

49 4 0

2

425

449

2 !

$

$

#

#

=

-

-

=Z

[

\

]]

]]

Z

[

\

]]

]]

a=2 an a=-2.

8. kvadratuli samwevris daSla mamravlebad

reziume:

vaCvenoT moswavleebs savarjiSoebi, sadac aucileblad gvWirdeba kvadratuli samw-

evris mamravlebad daSla, magaliTad x x

xx5 6

2 13

252

- +

-+

-= gantolebis amoxsnis tipis

savarjiSo.


8. a) x x

x x

7 10 0

6 0

2

2

!

+ + =

- -) x = –5.

9. y=x+2 aris wrfe, xolo yx

x x4

2 82

=-

- - =x+2, x≠4 wertili (4; 6) amovardnilia.

10. a) x x x

x

2 8 16

4

2

$

- = - +) x=6.

12. kata wT Tagvi

5 5 5

1 151 x=5

x 100 100

13. nebismier sam wertilze yovelTvis gaivleba sibrtye, amdenad albaToba imisa, rom samive wertili xvdeba erT naxevarsferoze aris 1.


32

14. amocanis pirobis Tanaxmad SevadginoT swori Cvenebebis cxrili: I _ yviTeli; mwvane II _ wiTeli; mwvane III _ wiTeli; forToxlisferi.

e.i. ori mwvane da ori wiTeli erTmaneTs unda emTxveodes e.i. a aris _ II, B — III, C — IwiTeli _ 2; forToxlisferi _ 4; yviTeli _ 8; mwvane _ 9.

15. a) 3x2–5xy–2y2=3 xy

3+` j(x–2y)=(3x+y)(x–2y). x= y y y

y6

5 73

2

!= -> .

Seamowme Seni codna:

1) g; 2) b; 3) g; 4) d; 5) a; 6) a; 7) b; 8) a; 9) d; 10) b; 11) b; 12) b; 13) b.

I Tavis damatebiTi savarjiSoebi

9. (x2−5)2+4(x2−5)−5=0 x2−5≡y.

y1=-5; y2=1. x

x

x

x

5 5

5 1

0

6

2

2!

- =-

- =

=

=== .

16. a) 2x2−(3a−1)x+a−1=0D=0, (3a–1)2−8(a−1)=09a2−14a+9=0 a∈∅.

19. B wertilis abscisaa x0 x

x2

2 30

0

2= x0= 3 a(– 3 ;3) B( 3 ;3).

20. 1) x2=kx+b; x2–kx+3k=0; a(3; 0).

y=kx+b; o=3k+b; b=-3k.

D=0 k2–12k=0.

a) k=0; k=12; b) (0; 0) an (6; 36).

32. x x

4202

42021

-+

= x=40km/sT; 2 saaTSi.

43. piTagoras TeoremiT

x= ( ) ( )192 3 256 32 2

$ $+ =960 km.

54. x x

x40

20040

101

-+

= =200


33

56. x2+162=(x+4)2 x=30.

57. x= 125 1172 2- =44.

66. x2–3x+1=0 x x

x x

3

1

1 2

1 2

+ =

=)

a) 5( )x x x x

x x5 515

1 2 1 2

1 2+ =

+= ;

b) ( ) 2

35x x x x

x x x x5 5 5 1 2

2

1 2

1

2

2

2

1

2

2

2+ =+ -

=^ h

;

g) ( )( )

x x x x

x x x x x x5 5 5

1

3

2

3

1

3

2

3

1 2 1

2

2

2

1 2+ =

+ + -.

70. davalagoT 2ax2+2(a2–a–2)x–7=0. moduliT tol da niSniT gansxvavebul fesvebs mivi-RebT, Tu a2–a–2=0 da a>0, e.i. a=2.

71. 7x2–ax+a–7=0 x x

a

x xa

7

77

1 2

1 2

+ =

=-

Z

[

\

]]

].


34

II Tavi

1. toldidi da proporciuli nawilebi samkuTxedSi


1. moc. ΔaBC, D∈aC, aD:DC=2:3 ⇒ SΔaBD:SΔBDC=2:3.

2. moc. ΔaBC, D∈aC, SΔaBD:SΔBDC=1:2 ⇒ aD:DC=1:2.

3. S1:S2=1:3 ⇒ S2=3·S1=9sm2. S3:S1=5:1 ⇒ S3=5·S1=15sm2.

4. ganv. ΔaBN. aM=MB ⇒ SΔaMN=SΔMNB=S.

ganv. ΔaBD. aN=ND ⇒ SΔBND=SΔaNB=2S.

ganv. aBCD paralelogrami. BD diagonalia. e.i. ΔaBD= ΔBCD ⇒ SΔBCD=4S.

5. aM=DC ⇒ SΔaBM=SΔDBC=2sm2.

6. SΔMNB=12sm2 ⇒ 7S=12 ⇒S=712 .

SΔaBC=21S=217

123

$ =36sm2.

pasuxi: SaBC=36sm2.

9. ganv. ΔaCD. aM=MD ⇒ SΔaCM=SΔMCD≡S. ganv. aBCD paralelogrami. aC diagonalia. e.i. SΔaBC=SΔaCD=2S.

SaBCM:SMCD=3S:S=3:1.

10. ganv. ΔaMB, aK=KB ⇒ SΔaKM=SΔKMB=2.ganv. ΔaBC. aM=MC ⇒ SΔBMC=SΔaBM=4.SΔaBC=8sm2.

11. ganv. ΔaBN. aM=MB ⇒ SΔaNM=SΔMBN≡S.ganv. ΔaBD. aN=ND ⇒ SΔaBN=SΔBND=2S.SΔBCD=SΔaBD=4S.SaBCD=8S=16 ⇒ S=2.SΔaMN=S=2sm2.

12. ganv. ∆aNB. MBAM

32

= . SΔaNM=2S da SΔMNB=3S.

ganv. ∆aBC. .NCAN

SS

SS

S S25

25 5

25

2BNC

ABN

BNCBNC& & &= = = =

3

3

33 .

a 5x M N 9x 7x C

B

5S 7S 9S

a

2S S S

M D

B C


35

13. A wertilze gavavloT BC-s paraleluri wrfe. aRvmarToT BC monakveTis SuamarTobi. am ori wrfis gadakveTis wertili ≡K-Ti da SevaerToTB da C wertilebTan saZiebeli samkuTxedia BCK, da ΔBCK-s simetriuliΔBCK1, BC wrfis mimarT.

14. a) b)

15. erT-erTi gverdi gavyoT SefardebiT 1:2:3. dayofis wertilebi SevaerToT mopirda-pire wverosTan.

16. vTqvaT iraklis mier amowerili ricxvebis jamia x, maSin ninos mier amowerili ricxvebis jami iqneba 3x. e.i. orives mier amowerili ricxvebis jamia 4x. (x∈N). cxrilSi Cawerili yvela ricxvis jamia 45. e.i. amowerili ricxvebis jamia (radgan 4-is jeradia) 40 an 36. 40-is SemTxvevaSi amosaweri darCa 5. pirvelis mier amowerili ricxvebis jamia 10=1+2+3+4. ninos mier amowerili ricxvebis jamia 30=6+7+8+9.

36-is SemTxvevaSi 36=4x ⇒ x=9. 3x=27.ar varga, radgan umciresi 4 ricxvis jamia 10.pasuxi : amosaweri darCa 5.

17. namravlis bolo cifri 0-ia, Tu namravli Seicavs Tanamamravlebad 2 da 5-s.a) 5; b) 20;

g) 5, 10, 15, 20, 25, 30 ricxvebi Seicavs jamSi 7 cal 5-ians, radgan 25=5·5. e.i. n=30.d) 5; 10; 15; 20; 25; 30; 35; 40; 45; 50 1 1 1 1 2 1 1 1 1 2n=45-isaTvis gvaqvs 10 nuli. 50-isaTvis ki ukve 12, radgan 50=2·5·5, e.i. 11 nuliT aRniS-nuli namravli ar damTavrdeba.

18. I maRaziaSi fasi gaxda x·0,9·0,9=0,81x. meoreSi _ 0,8x.

2. toldidi da proporciuli nawilebi trapeciaSi,nebismier oTxkuTxedSi


1. C wverodan gavavloT BD diagonalis ||CK monakveTi. ganv. ∆aCK. ∠aCK=∠aOD=90°. aK=a+b=16. aC=CK≡x. aK2=aC2+CK2 ⇒ 162=2x2 ⇒ x2=8·16.

SaBCD=SaKC=2

8x21 12 4

$= ·16=64sm2.

S S S

S S

S

a D

O

b a K

B C


36

2. M wertilze gavavloT fuZeebis marTobuli DK mon-akveTi.

ganv. ΔPBM=ΔMKa ⇒ MP=MK≡h. e.i. am samkuTxedebis M wverodan gavlebuli simaRleebi tolia da udristrapeciis simaRlis naxevars.

SΔBMC=21 ·h ⇒6=

21 ·3·h ⇒ h=4. e.i. SaBCD=

25 3+ ·8=32.

3. aBCD trapeciis blagvi kuTxis wverodan gavavloT BD diagonalis paraleluri CK monakveTi fuZis gagrZelebasTan gadakveTaze. miRebuli ΔaCK aBCD trapeciis toldidia.

S=3·3 3 =9 3 .

4. aCK tolferda marTkuTxa samkuTxedi, romelic aBCD trapeciis toldidia (ix. amocana 3).

aK=BC+aD=12 aC=CK=6 2 .

SΔaCK=216 2 6 2 36$ = .

SaBCD=36sm2.

5. ganv. ∆aCK (ix. amocana 3). aC=15; CK=20; CP=h=12. ∆aCP ⇒ aP=9. ∆CPK ⇒ PK=16.

SaBCD=21 (9+16)·12=25·6=150. SaBCD=150sm2.

6. aO:OC=5:3 ⇒ SaOB≡5S. SBOC=3S. SCOD= SaOB≡5S. 5S·5S=3S·SaOD ⇒ SaOD=

325 S=15. e.i. S=

59 .

SaBCD=13S+15=38,4sm2.

M

a K 5 D

P B C 3

a

60°

3 6

30°

K

C

a 12 K

B

15

a

12

P

20

K

C

a

B3S

5S O 3x

5x5x

D

C


37

7. SaBC: SaCD= :ADBC

2 3= .

8. ∆aBC ⇒ OCAO

SS

BOC

AOB= ;

∆BCD ⇒ OBDO

SS

OBDO

SS

OCAO

OBOD

BOC

COD

BOC

AOB& &= = =m m . r.d.g.

9. 17071 (17071–16961=110<200).

10. ricxvi iyofa 36-ze es ricxvi iyofa 4-ze da 9-ze. 4-ze iyofaricxvi, romlis bolo ori cifriT Sedgenili ricxvi iyofa 4-ze. miviReT ori SemTxveva:a) 13*452; b) 13*456.TiToeulSi ise unda CavsvaT erTi cifri, rom miRebuli ricxvi gaiyos 9-ze.a) 1+3+4+5+2=15. e.i. unda CavsvaT 3. 133452.b) 1+3+4+5+6=19. e.i. unda CavsvaT 8. 138456.

3. wesieri mravalkuTxedebi

reziume:

paragrafSi garCeuli ori amocana sakmarisia imisaTvis, rom moswavleebma SeasrulonparagrafSi mocemuli davalebebi. sasurvelia, am garCeuli amocanebis prezentaciisas maswavlebelma sTxovos moswavleebs zogadad Camoayalibon: a) Tu Caxazulia wrewirSi wesieri n-kuTxedi, rogor Caxazon 2n-kuTxedi. b) Caxazulia wesieri n kuTxedi, rogor Semoxazon wesieri n-kuTxedi, wesieri 2n-kuTxedi.


1. wrewiri a1, a2... a6 wertilebiT gaiyofa 6 tol nawilad. cxadia, mravalkuTxedi iqneba

wesieri.

4. a) gavavloT ori urTierTmarTobuli diametri, yvela Semdeg SemTx-vevaSi gavyoT miRebuli mravalkuTxedis gverdi or tol nawilad

da gavavloT am wertilze radiusi.

8. a) 25%-ian seqtors Seesabameba sruli kuTxis meoTxedi, e.i. 90°;b) 360° — 100% 45° — x x=

36045 100$ 12,5%.

10. .... .... ( ) ( ) ... ( )19

10 20 100 180 19019

10 190 20 180 90 110 10019

1900+ + + + +=

+ + + + + + += =100.

11. a3=2t=2000kg. 2a a

8

33

=` j =250 kg.

a

B

O

D

C


38

12. n-5; n-4; n-3; n-2; n-1; n — jami 5n-15. n; n+1; n+2; n+3; n+4; n+5 — jami 5n+15. sxvaoba 30-is tolia.

4. wrewiris sigrZe, wris farTobi

reziume:

wina paragrafSi Sesrulebuli praqtikuli savarjiSoebi daexmareba moswavleebs imisaRqmaSi, rom wrewirSi Caxazuli an wrewirze Semoxazuli wesieri mravalkuTxedis gverdebis raodenobas rac ufro gavzrdiT, miT ufro axlos iqneba mravalkuTxedis perimetri wrewiris sigrZesTan.


3. radiusis toli qordis mier moWimuli rkalis gradusuli zoma 60°-ia, e.i. rkalis

sigrZea r r360

2 603

$r r= .

5. 2�R=64�. R=32.

6. a) n180

42

$$

rr= , n=90°. b) n

1804

2$

r r= , n=22,5°.

7. ,R

n R n180

180crr

= = .

8. SeiZleba proporciiT 36° — 45 360° — l l=450.

9. wesieri eqvskuTxedis gverdi masze Semoxazuli wrewiris radiusis tolia, a6=R.

2�R=6R+7. R=2 6

7r-

. l=3

7rr-

.

10. a) 60�== R180

60$r , R=180, qordis sigrZe=R=180;

b) 60�== R180

90$r , R=120, qordis sigrZe=R 2 120 2= ;

g) 60�== R180120$r , R=90, qordis sigrZe=R 03 9 3= .

13. a) S=πR2–2R2=R2(π–2); b) a=R 3 S=πR2– a

4

32

=πR2– R

4

3 32

=R2·4

3 3r-c m.

g) πR2– R23

32

==R2·4

3 3r-c m.

Seamowme Seni codna:

1. aD=BD=DE ⇒ ∠aEB=90° ⇒ aE=h=4.


39

SaBC=21 ·8·4=16.

ganv. ∆DEC da ∆aEC. maT aqvT saerTo EC fuZe, xolo ∆DEC-s EC-ze daSvebuli simaRle h2

=2, e.i. SDEC=21 SaEC=8.

swori pasuxia: b.

2. ∆DBC-Si B wverodan DC fuZeze daSvebuli simaRle tolia aD=6.

SBDC=21 ·6·3=9.

swori pasuxia: b.

3. ganv. ∆BaK. S∆BaK=21 aK·Bh=

23 ·Bh

p ⇒S∆BaK+S∆DCh=23 ·(Bh+hC)=

23 ·BC=

23 ·12=18.

ganv. ∆DCh. S∆DCh=21 Dh·hC=

23 ·hC

swori pasuxia: d.

4. SaOB=SCOD=S

S2=3·12 ⇒ S=6.

SaBCD=2·6+3+12=27.

swori pasuxia: a.

5. aD:DC=SaBD:SBCD ⇒ aD:DC=12:9=4:3.

swori pasuxia: g.

6. aK:aD-3:5 ⇒ aK:KD=3:2 ⇒ SaCK=3S da SKCD=2S ⇒ S∆aDC=5S.

BD:DC=2:3 ⇒ SaBD:SaDC=2:3 ⇒ SaBD:5S=2:3 ⇒ SaBD=310 S.

S∆aBC=5S+310 S= S S

315 10

325+

=

p ⇒S∆aBC=325 ·3=25,

S∆BDC=21 aB·DC=

21 ·10·6=30

swori pasuxia: e.

7. BC:5BE ⇒ BE:BC=1:5 ⇒ SBED:SBDC=1:5 ⇒

⇒ S∆BDE=51 S∆BDC

p ⇒S∆BDE=51 ·30=6.

S∆BDC=21 aB·DC=

21 ·10·6=30

swori pasuxia: b.

a

B

S 3 O 12

S

C

D


40

8. S∆aBK+S∆aCK S∆aCK=21 aK·Bh

p ⇒ S∆aBK+S∆aCK=21 aK·BC=

21 ·4·16=32.

S∆aCK=21 aK·Ch

swori pasuxia: d.

9. S=21 SaBCD=30sm2. a). 10. SaKD=

21 SaBCD=20sm2. S=

21 SaKD=10sm2. g).

11. b. 12. b). 13. b).

II Tavis damatebiTi savarjiSoebi

1. DCAD

21

= =SaBD:SBCD=1:2 ⇒ SaBD=S, aBDC=2S. 3S=18; S=6.

SaBD=6sm2; SBCD=12sm2;

3. ganv. ∆aBD. aE:ED=2:3 ⇒ S∆aBE=2S; S∆BED=3S. ganv. ∆BEC. BD=DC ⇒ S∆EDC=3S. ∆DCa ⇒ S∆aEC=2S.

10S=40 ⇒ S=4. S∆CBE=6S=6·4=24sm2.

4. aBCF paralelogramia. S∆aBE=S. cxadia, S∆BEC=S∆aEF=S. S∆aBC=2S.

ganv. ∆aFD. aB:BD=1:2 ⇒ S∆aFB:S∆BFD=1:2 ⇒ SS2

21

BFD

= ⇒ SBFD=4S ⇒S∆BFD=2S∆aBC.

5. Suaxazi MN=4 ⇒ aC=8. MN adgens medianasTan 60°-ian kuTxes. aC||MN, e.i. aC adgens medianasTan 60°-ian kuTxes.

(sasurvelia aseT dros MN ar davitanoT naxazze, radgan naxazis gadatvirTvis SemTxvevaSi garTuldeba amoxsnis procesis danaxva).

∠aDB=60°∠KBD=30° ⇒ KD=

29 ·BK=h ⇒ h2=92–

29 2

` j =92 h49

949

43 9

2

9 32

22 2

&$

- = - = = .

SaBC=21 ·8·

2

9 318 3= .

a D

B

C

a

2S 3S

E 2x

3x

2S

D

3S

C

B

x

B

a F

2x

D

S E

S

S

C

a 60°K D C

B


41

6. 3

4

h

h ab

34b

a

&=

==o ⇒ b=aC=4x ⇒ KC=2x.

a=3x ∆BKC ⇒ a2=hb

2+KC2 ⇒ 9x2=9+4x2 ⇒ x2=59 ⇒ x=

5

3 5 .

a=3x=5

9 5 ; b=4x=5

12 5 .

7. EC:BC=1:3 ⇒ S∆aEC: S∆aBC=1:3 ⇒ 24:SaBC=31 ⇒ SaBC=72. aB=BC=x.

SaBC=21 x2=72 ⇒ x2=144 ⇒ x=12. aB=12.

swori pasuxia: g.

8. ganv. ∆BEC. BD=DC ⇒ S∆BDE=S∆EDC≡S ⇒ S∆BEC=2S.

ganv. ∆aBC. aE=EC ⇒ S∆aBE=S∆BEC=2S ⇒ S∆aBC=4S=36 ⇒ S=9.S∆BDE=xsm2=9sm2 ⇒ x=9.

swori pasuxia: g.

9. aD:aC=5:7 ⇒ aD:DC=5:2 ⇒ S∆aBC:S∆BDC=7:2 ⇒ 42:S∆BDC=7:2 ⇒ S∆BDC=12.

swori pasuxia: b.

10. S∆aBh=21 ah·Bh=

21 ·4·6=12.

∆BDh ⇒ S∆BDE≡S. maSin SDEh=2S. ∆aBh ⇒ SBDh=SaDh=3S. S∆BDE=S=2sm2.

swori pasuxia: a.

11. S∆aBC=21 h·aC.

marTkuTxedis fuZe Tu iqneba aC,

maSin meore gverdi unda iyos h2

.

simaRlis Suawertilze gavavloT aC-s paraleluri wrfe. C da a wertilebidan gavavloT simaRlis paraleluri wrfeebi Suaxazis gagrZelebasTan gadakveTamde. ΔaMK=ΔKBD da ΔBDQ=ΔQPC.

a

a

K C

B

b

Z [ \] ] ] ] ] ] ]

B

D

S x

a

3S

2S

E 2x h C

M

a

K D

Q P

B


42

12. AB≡a aBCD kvadratis BC gverdis nebismieri P wertilidan aRvmarToT marTobi, romlissigrZea a.

∆BMa=∆MKP ∆KPQ=∆QCD.

17. SS5

52

AOD

= . SaOD=12,5S. e.i. 24,5S=98.

S=4. SaOD=50.

19. aE:ED=aB:CD=1:3, e.i. aE=1. SCaE=21 ·aE·CD=

21 ·9=4,5.

a

a

P M

a

Q C

a

D

K

CB

a 5x

2S

O5S

2x

5S

D


43

III Tavi

1. kvadratuli funqcia

reziume:

es gakveTili sasurvelia Catardes integrirebulad fizikis maswavlebelTan erTad, kompiuterul laboratoriaSi. moswavleebs vuCvenoT zarbaznis lulidan horizonta-lurad gasrolili Wurvis, kalaTSi nasroli burTis da sxva traeqtoriebi, romlebic paraboliT aRiwereba. vTxovoT moswavleebs Tavad moifiqron msgavsi magaliTebi.


2. a) S=x2+a2+(a–x)2; b) S=a2–x2–(a–x)2;

3. S S S4A B C D ABCD AA D1 1 1 1 1 1= -

S=a2–2x(a–x).

6. ... ...2 1

1

3 2

1

100 99

11

2 1

1

3 2

1

100 999

++

++ +

+=

-+

-+ +

-= .

2. f:x → x2+c funqcia

reziume:

moswavleebi ukve icnoben y=x2 funqcias, mis grafiks, Tvisebebs. paragrafSi mocemuli cxrilebiT advilad davanaxebT gardaqmnas y=x2 → y=x2+c.


6. vipovoT c, -1=4+c c=-5, e.i. y=x2–5.a) mdebareobs; b) ara; g) ara; d) ara.

8. S=x2–4.

9. naxazidan Cans c=-1.y=kx+b gadis wertilebze (1; 0) (4; 1), saidanac k=

31 ; b=

31

- .

11. S=x2–� (sm2).

13. BK=KD, e.i. ∠BDa=45°. S=25sm2.

14. 74k-s bolo cifria 74-is bolo cifri, e.i. 1. 34k+2-s bolo cifria 32=9.

15. a=10k+7=10k+5+2. naSTia 2.

a

B C

DK


44

3. f:x → (x–d)2+c funqcia

reziume:

sasurvelia movTxovoT, rom y=(x–d)2+c funqciisaTvis d da c parametrebs mianiWon raime mniSvneloba da Tavad Seadginon paragrafSi mocemuli cxrilebis msgavsi cxrilebi. es dagvexmareba imaSi, rom moswavleebma Tavad dainaxon rogor unda moxdes gardaqmna imisTvis, rom y=x2+c funqciis grafikidan miiRon y=(x–d)2+c funqciis grafiki. gakveTili sasurvelia Catardes kompiuterul laboratoriaSi, informatikis maswavlebelTan erTad. moswavleebs vaCvenoT y=x2 parabolis ageba kompiuterSi da misi gardaqmnebi _ paraleluri gadatana x da y RerZebis mimarT.


5. y=x2+bx+ca) S=(-5; 0) y=(x+5)2=x2+10x+25b) S=(-2; 1) y=(x+2)2+1=x2+4x+5.

9. a) f(n)=n(n–1); g) 21·20=420; d) n2–n=380. klasSi 20 moswavlea. 375 suraTi gamoy-enebuli ver iqneba.

11. nomrebis raodenobaa 3·2=6, Sesabamisi funqcia f(n)=n(n–1), 5 cifrisaTvis gveqneba 20 nomeri, 10 cifrisaTvis — 90.

12. S (-3; -2) y=(x+3)2–2=x2+6x+7. aB= 81 9 3 10+ = .

13. y=x2+bx+c.a) M (-3; 0) K (2; 0). b c

b c

9 3 0

4 2 0

- + =

+ + =) b=1 c=–6.

18. gantolebas mTel ricxvebSi amonaxseni ar aqvs, imitom, rom 34x+2-is bolo cifria 9, e.i. marcxena mxare 5-ze ar iyofa.

19. u.s.j. (3; 5; 7)=105 saZiebeli ricxvia 107.

20. simaRle=2r=4, ferdi 4 2 Suaxazis tolia, e.i. S=4·4 2 =16 2 .

21. Sifris gasaRebi cxrilis principia; magaliTad, aso „l“ aris -is svetSi da -is striqonSi, e.i. „l“→ .

4. y=ax2 funqciis grafiki

reziume:

sanam moswavleebs davavalebT paragrafSi mocemul wyvilebisTvis gankuTvnil samuS-aos, SeiZleba SevavsebinoT y=ax2 funqciis mniSvnelobaTa cxrili. mag. a=2 da a = –2-Tvis.


45


7. a) radgan TaRis simaRlea 7m, amitom f(x1)= f(x2)=-7.

901

- x2=-7 ⇒ x2=630 x1=3 70 x2=-3 70 , saZiebeli man-

Zilia 6 70 ≈50.

b) -5=a·(-20)2 a=801

- y=801

- x2.

8. koordinatTa saTave davamTxvioT wylis Wavlis dacemis wertils. miviRebT y=ax2 funqcias. naxatze saTavidan Teonas fexebamde manZili 5 metria, Wavli ki gadmoedineba 1 m simaRlidan. e.i. |x|=1 y=5, aqedan a=5. y=7,2 ⇒ 7,2=x2 ⇒ x2=1,44, e.i. x=1,2.

12. (1; 1) albaToba 361 . (1; 3) (2; 2) (3; 1) albaToba

121 .

13. 23455+2729+15121.23455=...72729=...25121=5 jamis bolo cifria 4.

14. naturaluri ricxvis kvadrati SeiZleba bolovdebodes 0; 1; 4; 5; 6; 9; e.i. 5-ze gayofis naSTi SeiZleba iyos 0; 1; 4.

15. c2–b2=225, 225=3·3·5·5. SevadginoT yvela SesaZlo sistema.

c b

c b

3

75

- =

+ =) c=39 b=36; c b

c b

1

225

- =

+ =) c=113 b=112;

c b

c b

5

45

- =

+ =) c=25 b=20; c b

c b

9

25

- =

+ =) c=17 b=8.

17. a) radgan (-2; ∞) y=2(x–a)2 funqciis zrdadobis Sualedia, e.i. wveros abcisaa x0=-2, e.i. a=-2.b) cxadia, parabolis Stoebi zemoT aris mimarTuli, funqcia (1;3) SualedSi zrdadi rom iyos, wveros x0 koordinatisTvis unda Sesruldes , x0∈(-∞; 1], x0=a, e.i. a≤1.


46

5. y=ax2+bx+c funqciis grafiki

reziume:

es gakveTili SeiZleba Catardes rogorc integrirebuli, informatikis maswavle-belTan erTad. kompiuterSi avagoT y=ax2 funqciis grafiki da CavataroT paragrafSi aRwerili gardaqmnebi.


3. aseTi wrfe ar arsebobs, radgan kvadratuli funqciis gansazRvris area R.

5. b) naxazze mocemuli y=ax2+bx+c funqciis grafiki, romelzec mdebareobs M(-2; -4)wertili; xolo (-3;-2) _ parabolas wveroa. miviReT sistema:

a b c

ab

aac b

4 4 2

23

44

2

- = - +

- =-

-

Z

[

\

]]

]]

a = –2; b = –12; c = –20.

6. a) y=ax2+bx+c parabolas wveroa (1;4) e.i. ab2

1- = da a

ac b4

44

2-

= . parabola gadis F(3;0)

wertilze. e.i. 0=9a+3b+c, sistemis amoxsniT vRebulobT a=-1; b=2; c=3.

10. 16x–16x2=0x=0 x=1

12. a) ∆OMK-s fuZe OK=4; xolo simaRle, M wertilis abscisa, 3-is tolia. e.i. SMOK=6.

g) OM= ( )3 5 342 2+ - = . MK= 9 81 90 3 10+ = = .

13. x x

20015

20032

-+

= , saidanac x=60.

16. y=ax2 (50;10). y x2501 2

= .

17. 1255+214202+381+70=...5+...6+...3+1=...5 iyofa 5-ze.

18. 7k+2=5m+3.5m=7k–1 magaliTad k=3 m=4. 23.

19. rombis diagonalebis naxevrebi aRvniSnoT 3x; 4x, maSin rombis gverdi iqneba 5x=30

x=6; rombis diagonalebia 36 da 48. S2

36 48864

$= = .

20. a) y RerZis mimarT a1(2;3); B1(2;6); C1(4;7); D1(5;1).b) O(0;0) wertilis mimarT a2(2;-3) B2(2;-6) C2(4;-7), D2(5;-1).

= –2


47

21. aseT funqciaTa grafikebs saerTo aqvT abscisaTa RerZTan kveTis wertilebi (fesvebi) da simetriis RerZi (wveros abscisa).

24. a>0 SemTxvevaSi fesvebi x=1wrfis mimarT sxvadasxva mxares, xolo a<0 SemTxvevaSi fesvebi x=1 wrfis mimarT erT mxares arian ganlagebuli.

7. parabolis mdebareoba sakoordinato RerZebis mimarT

reziume:

ukve ganxiluli gardaqmnebis gamoyenebiT advilad davadgenT parabolis sakoordi-nato RerZebis urTierTmdebareobis a, b da c koeficientebze damokidebulebas da parabolis wveros koordinatebs, ris Semdegac ukve SegviZlia SevTavazoT funqciis gamokvlevis zogadi sqema.


3. z) y=(x–2)(x+3).

sasurvelia moswavleebma frCxilebis gaxsnis gareSe aagon funqciis grafiki. x1=-3;

x2=2, wveros abscisa x0= 23 2

21- +

=- .

4. rogorc wina magaliTSi aRvniSneT, gantolebis fesvebi simetriulia parabolas

simetriis RerZis mimarT, amitom simetriis RerZis gantoleba iqneba x25 3

=- + .

e.i. x= –1.

5. radgan funqcias aqvs erTi nuli, is exeba x RerZs x=2 wertilSi wveroTi. e.i. simetriisRerZis gantolebaa x=2.

6. pirobidan gamomdinare, c=2, xolo 4b2

- = , e.i. b = –8.

7. y=ax2+bx+c funqciis x RerZTan kveTis wertilia M(–2;0), meore ki N(x2;0). amave dros,

simetriis RerZis gantolebaa x=3, e.i. x3

22 2=

- +, saidanac x2=8. SevadginoT sistema:

b cb

0 4 2

23

= - +

- =* b = –6; c = –16.

1

a>0 a<0

1 1

y y y

0 0 0x x x

y

0 x


48

8. SevadginoT Sesabamisi sistema:

ab

a b c

c

22

4 2 0

8

- =-

- + =

=-

Z

[

\

]]

]], saidanac

a

b

2

8

=-

=-)

14. 121121+3636 jamis bolo cifria 7.

15. SevadginoT 5-ze gayofis naSTebis cxrili, e.i. n2+n gaiyofa 5-ze Tu n=5k an n=5k+4.

16. MC=9 OC=6 aN=12 aO=8 SaOC=24=31 SaBC, e.i. SaBC=72.

17. x=1 da x=-1 wertilebis mimarT funqciis nulebi ganlagebulia sxvadasxva mxares.

8. kvadratuli utolobis amoxsna

reziume:

ukve Seswavlili Tvisebebis safuZvelze moswavleebs SeuZliaT Tavad Camoayalibon kvadratuli utolobis amoxsnis sqema.


3. g) x

x

x

x

3 0

9 0

3 0

9 0<2 2

2

$ $-

-

-

-) ) ∅; d)

x

x

4 0

0

2

!

$-) (-∞;-2]∪[2;∞);

e) x

x

9 0

5 0

2

2

$-

-) (5;∞); e) x2+5x+9>0 x∈R.

5. a) x x

x

6 8 0

1 0

2

2

$

+ +

-) [1;∞).

7. S=x(6–x)>5; 6x–x2>5, e.i. x2–6x+5<0 x∈(1;5).Tu gverdebi mTeli ricxvebia x=2; 3; 4, marTkuTxedis gverdebia 2; 4 an 3; 3.

8. (18–2x)(24–2x)>24·18–(18–2x)(24–2x).

n n2 n2+n0 0 01 1 22 4 13 4 24 1 0


49

x2–21x+54>0 x

x

3

18

<

>= . miviReT — Tavisufali aris sigane unda iyos 3-ze naklebi.

9. a) 2x2–(a+1)x+3>0 utoloba Sesruldeba cvladis nebismieri mniSvnelobisaTvis, roca D<0 (pirveli koeficienti dadebiTia).

D=(a+1)2–24<0 a∈(-1–2 6 ; -1+2 6 ).

b) a2x2+2ax+1>0. (ax+1)2>0.Tu a=0 utoloba sruldeba nebismieri x-sTvis, Tu a≠0, maSin (ax+1)2 gamosaxuleba nebismieri x-isTvis iRebs arauaryofiT mniSvnelobas, e.i. a=0.

10. aRvniSnoT nakveTis sigane xiT, maSin sigrZe iqneba 205–2x.x(205–2x)≥5000.40≤x≤62,5.

11. SevadginoT amocanis Sesabamisi utoloba.

x x528

528

332

#-

++

- . vRebulobT x2–24x–25≥0. e.i. x≥25 kateris sakuTari siCqare unda

iyos aranakleb 25km/sT.

12. D=9–4q<0 q>49 .

13. a2–8<0.

15. 1) pirobebi sakmarisi ar aris. amocana rom amoixsnas an a unda iyos mocemuli, an mocemuli unda iyos wveros ordinata.2) a) Tu f(-4)>f(1), e.i. funqcia zrdadia da k>0; b) f(0)<0 gvaZlevs mxolod im monacems, rom wrfe ordinatTa RerZs kveTs qveda naxevarsibrtyeSi. piroba sakmarisi ar aris.

18. aB=BK=4, KC=2. e.i. SaBCD=4·6sin30°=12.

9. meore xarisxis orucnobian gantolebaTa sistemis amoxsna

reziume:

moswavles unda SeeZlos konkretul sistemisTvis amoarCios amoxsnis Sesabamisi gza.ipovos fesvebi. unda SeZlos grafikuli amoxsnis Cveneba, fesvebis Cawera.


3. a) x y x y

x y x y

18

6

2 2

2 2

+ + + =

- + - =) x x

y y

2 2 24

2 2 12

2

2

+ =

+ =) x x

y y

12 0

6 0

2

2

+ - =

+ - =)


50

an

an

4 3

3 2

x x

y y

=- =

=- =) (–4;–3) (–4;2) (3;–3) (3;2);

v) x y xy

x y xy

5

1

+ + =

+ - =) x y

xy

2 2 6

2 4

+ =

=) x y

xy

3

2

+ =

=) (2;1) (1;2).

4. x2–4x–5=2x–14; x2–6x+9=0 D=0 e.i. 1 wertili.

5. kx2–5x+1=–kx–3kx2+(k–5)x+4=0D=k2–26k+25a) D=0 k=1; 25; b) D>0 k∈(-∞;0)∪(0;1)∪(25;∞); v) D<0 k∈(1;25).

6. a b

a b

17

1692 2

+ =

+ =) 5;12. 9. x y

xy

170

68

+ =

=) es ricxvebia 136 da 34.

15. ( )a b a b

a b ab

10 6 2

10 5 2

+ = + +

+ = +) , saidanac ab=32. 19. x y

x y

4 41

2 29

+ =

+ =

Z

[

\

]]

]] 16;12.

12. ricxviTi mimdevroba

reziume:

gavixsenoT ukve cnobili ricxviTi mimdevrobebi; naturalur ricxvTa, luw ricxvTa, kent ricxvTa mimdevrobebi. vTxovoT moswavleebs sityvebiT, Sinaarsobrivad, daaxa-siaTon ricxviTi mimdevrobebi.


3. a) an= nn1+

; b) an=n3; g) an=n2+(n+1)2; d) an=n3+1.

e) mimdevrobis wevrebi ganlagebulia parabolaze an=n2–6n+7;

v) an=2n2–16n+33.

6. a) a1=2, an=3an-1; b) a1=3, an=(-1)n-1 an-1; g) a1=-1, a2=1, an=nn-1+an-2.

12. cxadia, progresia zrdadia a17< a23<0 a49> a42>0.


51

13. a1+an-d=31; a1=5.

14. vipovoT mimdevrobis udidesi uaryofiTi wevri an<0; 4- n5

<0 n>20.

a21=4–521

51

=- a20=0 a19=4–519

51

= a19+a21=0.

17. a) an=5–8n an>14.

5–8n>14 8n<-9 mimdevrobis aseTi wevri ar arsebobs.

b) an=n2+4n an>-4.

n2+4n>-4 (n+2)2>0 mimdevrobis yvela wevri akmayofilebs pirobas;

g) an=7n–4 an<17.

7n-4<17 n<3 a1;a2.

d) an= nn

24

++ an<2

nn

24

2<++ ;

nn2

0<-+

yvela wevri.

18. a) an= –4n2+8n+4 kvadratuli funqcia udides mniSvnelobas iRebs n=1 wertilSi, xolo a1= 8.

b) n0 = –3–2 = 32 udidesia a1 = a2 = –5.

20. vnaxoT ramdeni 13-is jeradi ricxvia 100-mde. aseTi wevri 7-ia, maT Soris umciresia a13=5, udidesi a91=23.

22. pirveli wamis Semdeg siCqare aCqarebis toli iqneba, e.i. a1=d. an=a1+d(n–1) formulaSi 45=d+14d; 15d=45 d=3.


52

13. ariTmetikuli progresiis pirveli n wevris jamis formula

reziume:

SesaZlebelia, rom gakveTili daviwyoT davalebiT: ipoveT 1-dan 100-mde ricxvebis jami. SeiZleba davexmaroT SekiTxvebiT 1. risi tolia pirveli da mease wevrebis jami. 2. meore wevri gaizarda 1-iT 99-e ki Semcirda erTiT, risi toli iqneba maTi jami?


1. a) d=5, S22=1705.

4. a) a1=-28 n=9 Sn=0. a2

289 0

n $-+

= , e.i. an=28 d=7.

5. b) a a da d

5 10 40 0

22 3

4 14

1 1

1 $

+ + =+

=* d=3.

7. a1=1 an=135; d=2.

8. 2a1+2d(n–1)=2a1+18d. n=10.

9. a1=10, an=99 n=90.

10. an=7n+3. 111<7n+3<186 16≤n≤26, e.i. S=S26–S15.

11. a1=5 d=-2.

13. a d

a d2

2 34 28

22 5

6 48

1

1

$

$

+=

+=

Z

[

\

]]

]; S16=208.

14. ( ) ( ) ...

x

x x1 2 1157

2

- + - + += davTvaloT 1+2+...+(x–1)

a1=1 d=1 an=x–1 n=x–1. e.i. 1+2+...+(x–1)= ( )x x21-

, miviReT ( )

x

x x

2

1157

2

-= , saidanac x=15.

15. Sn=2n2+3n= ( )nn

nn

nn

24 3

23 4 4 4

27 4 2

$ $ $+

=+ - +

=+ - . a1=3,5 d=4.

17. I sxeuli 4 wamSi — 40 m.

II sxeuli 4 wamSi — 2

14 5 34 58

$$

+= .

darCeba 200–98=102.


53

21. a+b+c=f(1) da radgan D<0, e.i. grafiks aqvs Semdegi sqematuri saxe:

11. orucnobian utolobaTa sistemis amoxsna (ix. gv. 85)

14. geometriuli progresia

reziume:

moswavlem unda icodes geometriuli progresiis ganmarteba — rekurentuli for-mulis dawera. unda SeeZlos zogadi wevris formulisa da progresiis ZiriTadi Tvisebis dawera. dainaxon saerTo da gansxvaveba ariTmetikul da geometriul pro-gresiebs Soris. gaanalizon rodis aris geometriuli progresia zrdadi, klebadi, arc erTi.


2. bb

b q

b qq

k

n

k

n

n k

1

1

1

1

$

$= =-

-

-

.

4. x2=9·10=90, saidanac x=± 90 . a.p.T. x=-3 10 .

6. b1=4 b7=161 . 4·q6=

161 q6=

641 q=

21

! .

4; 2; 1; 21 ;

41 ;

81 ;

161 an 4; -2; 1; -

21 ;

41 ; -

81 ;

161 .

7. a) b q b

b q b

2

3

1 1

1

2

1

- =

- =)

q

q

1

1

32

2-

-= e.i. q=

21 ; b1=-4.

8. b bb b

b413

322 3

1 4

3

++

=

=*

( )q q

q

1

1

413

3

+

+= , saidanac q=4;

41 . Sesabamisad b1=2 an 512.

10. a) bn=5n=5·5n-1. b1=5 q=5.

g) bn=3n+1 davamtkicoT, rom bn-1·bn+1=bn2.

3n·3n+2=32n+2==32(n+1)= 3n 1 2+^ h =bn

2.

11. b) b 18 2 108 25

2

$=


54

b 36 35 = (b1 dadebiTia), e.i. q= 6 .

12. a=b1; b= b1q; c=b1q2. samkuTxedis utoloba b1q

2<b1+b1q.

q2–q–1<0, e.i. q<2

1 5+<2.

13. a b c

a b c

211 1 1

127

+ + =

+ + =*

a b c

abcbc ac ab

21

127

+ + =+ +

=* . gaviTvaliswinoT, rom b2=ac, maSin me-2

gantolebidan b

bc b ab127

3

2+ +

= . b

21127

2 = . b2=36, e.i. b=6.

a c

ac

15

36

+ =

=) , saidanac a da c-s mniSvnelobebia 3 da 12.

16. b1; b2; b4 — ariTmetikuli progresiaa.

b1; b1q; b1q3;

b1+ b1q3=2b1q

q3–2q+1=0;

q3–q–q+1=0

q(q2–1)–(q–1)=0

(q–1)(q2+q–1)=0, saidanac q=1 an q=2

1 5!-.

17. a1; a20; a58 — geometriuli progresiaa.

(a1+19d)2= a1(a1+57d)a1

2+38 a1d+361d2= a12+57 a1d.

19 a1d–361d2=0d(a1–19d)=0 an d=0 an a1=19d.Tu d=o q=1, xolo Tu a1=19d, maSin q=

aa d19

1

1 + =2.

18. n4+4=n4+4n2+4–4n2=(n2+2)2–4n2=(n2–2n+2)(n2+2n+2).

2 2 1n n

n n2 2 1

2

2

- + =

+ + == vRebulobT n=1. e.i. es ricxvia 5.

19. x2+3y2+2x+6y+4=x2+2x+1+3y2+6y+3=(x+1)2+3(y+1)2.Tu x=y=-1 S=0.


55

15. geometriuli progresiis pirveli n wevris jamisgamosaTvleli formula


2. mocemulobiT vpoulobT q=21 da b1= 2

3 saZiebeli jami SeiZleba ase warmovidginoT:

S7–S2=12893 .

7. b b b

b b b

1728

63

1 2 3

1 2 3

=

+ + =) b2

3=1728, e.i. b2=12. miviReT: b b

b b

144

51

1 3

1 3

=

+ =) saZiebeli ricxvebia 3; 12; 48.

8. a) 0xx

11

100

--

= . x=-1. b) 0xx

11

101

--

= . x∈∅.

9. ( ) ( )

q

b q

q

b q

1

1

1

112

1

109

1

100

-

--

-

-=

( )12

q

b q q

1

1

109 100

-

-=

( )12

q

b qq

1

11

9

100

$-

-= S9= q

12100 .

10. a) .... ....

... ...( )

( )

( )2

( )

( )( )

xx

xx

xx

xx

xx

xx

x x xx x x

nx

x x

x

x xn

x

x x

x x

xn

x x

x xn

1 1 12

12

22

1

1 1 12

1

11

1

1 21

2

1

1

1

1

1

1 12

n

n

n

n

n

n

n n

n

n

n

n

n n

22

2

2 22

2

4

4

2

2

2 4 2

2 4 2 2

2 2

2

2 2

2

2 2

2 2

2

2 2

2 2 2

+ + + + + + = + + + + + + + + + =

= + + + + + + + + =-

-+

-

-

+ =

=-

-+

-

-+ =

-

+ -+

+

` c cc

j m mm

b) 9+99+999+...+ ...99 9nS=10–1+100–1+1000–1+...+ ...10 0

nS–1= ( )

n9

10 10 1n-

- .

12. a) 0

x

x x

4 7 0

3 4

2 0>

2

$

$

+

- + 4 jami 0 ar SeiZleba iyos.

b) gansazRvris are x

x

6

3

$

#) carielia.

Seamowme Seni codna

1.b; 2. b; 3. d; 4. g; 5. d; 6. a; 7. d; 8. b; 9. d; 10. d; 11. b; 12. g; 13. g; 14. d; 15. d; 16. g; 17. g; 18. d; 19. a; 20. b.


56

III Tavis damatebiTi savarjiSoebi

5. (1) y=ax2+bx+c wvero F(1;1) c=2.

1ab2

- = ; 1a

ac b4

42

-= ; a=1; b=-2.

(3) y=ax2+bx+c.

ab

ac b a

a b c

23

4 423

16 4 3

2

- =-

-=-

- + =-

Z

[

\

]]

]]

.

6. a) y=x2+bx+c.

b2

- =5, e.i. b=-10.

1=9+30+c c=-38.

7. x2<xx(x–1)<0 x∈(0;1).

9. a) y=x2–x+1 y=x2+x+1x2–x+1=x2+x+1 x=0 y=1 (0;1).

11. a=1y=x2 B(x;x 3 ) x 3 =x2 x= 3 S= ( )x

4

2 33 3

2

= .

12. a) 3x–9=x2–(2k+1)x+8k

x2–2(k+2)x+8k+9=0D4

=(k+2)2–8k–9=k2–4k–5=(k–5)(k+1)

Tu k∈(-∞;-1)∪(5;∞) ori gadakveTis wertiliTu k∈-1;5 erTi gadakveTis wertiliTu k∈(-1;5) arc erTi gadakveTis wertili.

17. Tu a∈[0;2) gantolebas eqneba 4 amonaxsni, xolo y=2 wrfe grafiks kveTs sam wer-tilSi.

18. y=ax2+bx+cc=3CavsvaT fesvebi

a b

a b

3 0

9 3 3 0

- + =

+ + =)

a

b

1

2

=-

=)

e.i.y=-x2+2x+3 MN=y0=4.

aac b a2

4 42

-[]

]]

4aa4 4

23

=-


57

19. y=ax2+bx+c c=4.

a b

ab

4 2 4 0

22

- + =

- =-* b=4; a=1.

e.i. f(x)=x2+4x+4a) f(-3)=1b) c=8

a b

ab

4 2 8 0

22

- + =

- =-* b=8; a=2.

f(x)=2x2+8x+8f(-3)=2

20. y=(m+1)x2–2mx+4

2ab2

- = ( )m

m2 1

22

+= . m=-2.

21. C(2;0) SaOCB=2.

25. a) Sn+3–3Sn+2+3Sn+1–Sn=Sn+3–Sn–3(Sn+2–Sn)=an+1+an+2+an+3–3an+2=an+1+an+3–2an+2=2an+2–2an+2=0.

b) ( )

( )

( ) ( )

( ( )) ( )

2 ( ) 2 ( )

( )( )) (

( ) ( )( ))

SS S

a d m nm n

a d mm

a d nn

a d m n m n

a m d m m a n d n n

a d m n m n

a m n d m n m n

m nm n

22 12

2 12

2 1

2 1

2 1

2 1

m n

m n

1

1 1

1

1

2

1

2

1

1

$

$ $

$

$

-=

+ + -+

+ --

+ -

=+ + - +

+ - - + -=

=+ + - +

- + - + -=

+-

+

26. a) x+5=2 x=-3; b) x–3=2 x=5; g) |x|=2 x=±2.

28. a1=1 a10=9,1 e.i. d=0,1.

32. S=S20–S9.

36. g) ymin(4)=,0 5

1-

=-2 ymin(5)=,0 51 =2.

43. bb

b q

b q

3

9

1

2

1

8

$

$= =q6=27.

bb

b q

b q

6

18

1

5

1

17

$

$= =q12=272.

44. ricxvebi 1; 2; 4 adgenen geometriul progresias romlis q=2. x=a gantolebis fes-

via, amitom yvela fesvi Seadgens geometriul progresias, Tu a=21 ; 1; 2; 4; 8.


58

IV Tavi

1. samkuTxedis msgavseba

reziume:

moswavleebi gaecnobian msgavs samkuTxedebs. msgavsebis faqtis sworad Caweras da amCanaweridan proporciis amoweras. gaecnobian im faqts, rom samkuTxedis fuZis para-leluri monakveTi am samkuTxedidan mokveTs missave msgavs samkuTxeds.


1. a) sakmarisia; b) sakmarisia; g) sakmarisia; d) sakmarisia.

2. amocanis ganxilvamde moswavleebs vaCvenoT, rom msgavsi samkuTxedebis perimetrebi ise Seefardeba erTmaneTs, rogorc Sesabamisi gverdebis sigrZeebi. marTlac, Tu erTi samkuTxedis gverdebia a; b; c da k msgavsebis koeficienti, maSin meore samkuTxedis

gverdebi iqneba ak; bk; ck da ( )

PP

a b c

k a b ck

1 =+ +

+ += , amis Semdeg

PP

153

ABC

MNB =3

3 P31 15

3MBN =

P531

MBN = .

3. :MBAM

MBAB MB

MBAB

MNAC

1 1315

1 4 1=-

= - = - = - = .

5. ABMN

ACCM

53

= = aB=10 R= AB2

=5.

8. miTiTeba: ferdebis gagrZelebebis gadakveTis wertili SevaerToT didi fuZisSuawertilTan da vaCvenoT, rom patara fuZec gadaikveTa mis Sua wertilSi.

9. meore samkuTxedis gverdebis Sefardebaa 6:9:12=2:3:4, e. i. misi gverdebis sigrZeebiSegviZlia aRvniSnoT 2x; 3x da 4x-iT.amocanas aqvs 3 amoxsna:I. 2x=24 ⇒ x=12II. 3x=24 ⇒ x=8III. 4x=24 ⇒ x=6.

10. pirveli velosipedisti B punqtSi Cavida 80:10=8 saaTSi, meore ki 80:20+3=7 saaTSi.a) ufro adre Cavida meore velosipedisti;b) velosipedistebi Sexvdnen orjer.meore velosipedisti gaCerda a-dan 40km-is daSorebiT da icdida 3 saaTi. 40km-is gavlaspirvelma moandoma 4sT. e. i. pirvelad Sexvdnen a-dan 40km-Si. meore velosipedistma moZraoba ganaaxla moZraobis dawyebidan 5 saaTSi. am momentisTvis pirveli mas uswrebda 10km-iT. e. i. meore daeweva 1 saaTSi, anu moZraobis dawyebidan 6 saaTSi, e. i. daeweva a-dan 60km-is manZilze. g) 4 saaTiT adre.


59

2. samkuTxedebis msgavsebis I niSani

reziume:

wina klasSi ukve gavecaniT figuraTa msgavsebas. msgavsi samkuTxedebis Tvisebebs. sasurvelia gakveTili daviwyoT msgavsi figurebis gameorebiT.


1. ∆BDC~∆DaB aD=8sm.

2. DO+OE=14 da ∆aOE~∆BOD OE=6sm da OD=8sm.

3. ∆aDC~∆BCa; CD=3sm da BD=9sm.

4. ∆aBC ∠C=90°; CB=9; ∠B=60° Caxazulia MNBP rombi. ∆MaN~∆CaB, saidanac MN=6sm.

5. diagonalebis gadakveTis O wertilze gavavloT OM||BC M∈aB. ∆BFE~∆MFO, saidanac

BE=a cbc2+

.

7. ∆aBC~∆DCa, saidanac aC=18sm.

10. paralelogramSi a·ha= b·hb, e.i. hb=64dm.

11. SaBCD=aD·DC=65.

SBCD=21 SaBCD=

265 , amave dros SBCD=

21 Bh·EC=13, e.i.

265 = BH

213$ Bh=5.

12. z=xy1 , CavsvaT,

+++

=++

+++

+++

=xyx

yxyxyxyx

S1

1111

111

11

1

111

11=

++++

=++

+++

+xyxxyx

xyxxy

xyxx

13. cxadia 1·2·3·4·5···98·99 bolovdeba 0-iT, xolo 1·3·5·7···97·99 — 5-iT, sxvaobis bolo cifri iqneba 5.

14. ganvixiloT zogadi varianti. SevadaroT ba da

b xa x

1010

++ , ganvixiloT sxvaoba

( ) ( )

( )ba

b xa x

b b xab ax ab bx

b b x

x a b

1010

1010 10

10-

++

=+

+ - -=

+

-. Tu a>b, anu Tu wiladi wesieria, maSin

ba

b xa x

1010

>++ , Tu wiladi arawesieria, maSin a<b da Sesabamisad sxvaoba uaryofiTia. Cven

SemTxvevaSi 6737

677377

> .


60

3. samkuTxedebis msgavsebis II niSani

reziume:

paragrafi iwyeba wyvilebisaTvis gankuTvnili samuSaoTi, ris Semdegac moswavleebs vTxovT am amocanis safuZvelze gamoTqvan varaudi. Semdeg vayalibebT samkuTxedebis msgavsebis II niSans da vamtkicebT. Semdeg aris individualuri SekiTxvebi, 1) mc; 2) mc; 3) mc; 4) W; 5) mc; 6) W. maRal Sefasebas imsaxurebs moswavle, Semdegi amocanebis amoxsniT # 7; 8;10; 12.


4. ∠a saerToa. ABAD

ACAE

32

= = , e.i. msgavsia.

7. SevaerToT B da E wertilebi. SaBE=21 SaBCD=30 SaBE=

21 ·10·Bh, e.i. Bh=6.

8. BC gverdze a wverodan daSvebuli simaRle 3-is tolia, xolo BC=b, e.i. b2

3 $ =18, b=12.

9. 10020=1000010>985010.

10. 4m2–12m+10=42

m m23

492

$- +` j+1=4 m23

12

- +` j . gamosaxuleba umcires mniSvnelobas

iRebs, roca m23

= , umciresi mniSvnelobaa 1.

12. xy xz

xy yz

xz zy

5

10

13

+ =

+ =

+ =

* SevkriboT xy+xz+yz=14. miRebul gantolebas gamovakloT sistemis

TiToeuli gantoleba, miviRebT: yz

xz

xy

9

4

1

=

=

=

* , saidanac x2y2z2=36, e.i. xyz=±6, x=±32 , y=±

23 , z=±6.

4. samkuTxedebis msgavsebis III niSani

reziume:

paragrafi iwyeba wyvilebisaTvis gankuTvnili amocaniT. Semdeg vayalibebT da vamt-kicebT samkuTxedebis msgavsebis III niSans. individualuri SekiTxvebis pasuxebi:1) a) ki; b) ara. 2) a) W; b) W. 3) W; 4) mc; 5) W.maRali SefasebisTvis gankuTvnili amocanebia N2, 3, 11.


2. kvadratis gverdi aRvniSnoT x-iT x x4 12

12=

- , saidanac x=3sm.

3. BF

BF420 12

5+

= , saidanac BF=300sm.

4. ∆aEF~∆CED. CDAF

CEAE

= , saidanac AFnam

= , e.i. ( )

BFn

a m n=

-.


61

8. S DE AB21

21

10 5 2 25 2BDE $ $ $= = = .

10. pirveli ricxvia 36x, meore - 36y, sadac usg(x;y)=1. ⇒=+ 4323636 yx 12=+ yx . Se-varCevT imis gaTvaliswinebiT, rom x da y urTierTmartivia.

11.

=+

=+

3530)(

33 yxyxxy 3

+

=+

=+

359033

33

22

yxxyyx

==+

65

xyyx

125)( 3 =+ yxsistemis amonaxsnebia (2;3) (3;2).

12. 9621

=⋅⋅= BOS AOB e.i. )0;3(−B )6;0(A e.i. k=2, b=6, k+b=8.

13. radgan 2n

sruli kvadratia e.i. erTi Tanamamravli unda iyos 32 , radgan n3

kubia,

meore Tanamamravlia 34 miRebuli ricxvia 23·34.

14. 2

32

3)2(2

32 2223

−+=

−+−

=−

+−n

nnnn

nnn

.

−

−=−

33

11

2n

e.i. n=3; 1; 5; -1, radgan n unda iyos naturaluri, e.i. n=3; 1; 5.

15. unda amovxsnaT sistema a a

a a

a a

3 2 0

5 4 0

0

2

2

2

- + =

- + =

- =

Z

[

\

]]

] a=1.

5. proporciuli monakveTebi msgavs samkuTxedebSi

reziume: arsebiTia, rom moswavleebs kargad esmodeT, rom msgavs samkuTxedebSi yvela Sesabamisi wrfivi elementebis Sefardeba Sesabamisi gverdebis Sefardebis tolia da SeeZloT am faqtis gamoyeneba amocanebis amoxsnis dros.


5. SMNC=S; SFNC=2S; SaFC=4S. SFBC=21 SaFC=2S. e.i. SaBC=6S;

SaBCD=12S=36. miviReT: aB=6, aF=4 da MN=2.

6. 2x+3x=20 x=4.R1=2x=8, R2=12.


62

6. msgavsi samkuTxedebis farTobebis Sefardeba

reziume:

paragrafi iwyeba wyvilebisaTvis gankuTvnili samuSaoTi, ris Semdegac vamtkicebT Teoremas msgavsi samkuTxedebis farTobebis Sefardebis Sesaxeb. Semdeg vixilavT paragrafSi garCeul amocanebs. maRali SefasebisaTvis gankuTvnili amocanebia #5, 12, 14, 15,16.


1. 2110

=p

20=p .

3.

=+

=

6842

48

yxy

x

y=1 an 16 x=32 an 2.

4.

23

23

a

xa

ax −= , saidanac )32(3 −= ax S

kv.)347(3 2 −= a .

5. aC gverdze daSvebuli simaRle 5212 -ia. ABCF ⊥

ACABCF ⋅=⋅5212 , saidanac

524

=CF . CFB∆ ~ NMB∆ .

BNMN

CBCF

= x

x−

=88

524

, saidanac x=3.

8. 122

== ABCDSS .


63

9. 32

2

2

=ABMB ;

32

=ABMB

, saidanac 1

2623

2 +=

−=

AMMB

.

11. mocemuli samkuTxedis farTobia 360. 2

3654

360

=

S s=810.

12. 40=BC CBCMACCK ⋅=⋅ (mkveTebis Tviseba)

e.i. ACBC

MCKC

= e.i. ∆KMC~∆BCa, 2

2

BCKC

SS

ABC

KMC = , saidanac

56

=KMCS

554

5612 =−=AKMBS

13.

−=−−=++

=++

542

1

ccabPba

−=−=+

=

11

2

cba

P p=2.

14. mgzavris siCqare _ x eqskalatoris _ y eqskalatoris sigrZe — S.

=

=+

42

24

xS

yxS

241=

+Sy

Sx , saidanac

561

=Sy

e.i. 56=yS

.

16. 000 31721360 +⋅= . davxazoT wrewiri, gavavloT 1OA radiusi da gadavdoT 0

21 21=∠ OAA . Semdeg imave mimarTulebiT gadavdoT 0

32 21=∠ OAA da a.S. 0

1716 21=∠ OAA

miRebuli 117OAA∠ iqneba 3°-iani. radgan 00 3721 ⋅= , amitom mocemuli kuTxis gverde-

bidan mimdevrobiT gadavdebT 3°-ian kuTxes 6-jer.

17. [x]+{x}=x, e.i. vRebulobT x≥|x|, rac SesaZlebelia, Tu x≥0.


64

8. msgavsebis meTodi geometriul agebebSi


1. avagoT nebismieri samkuTxedi α da β kuTxeebiT, Semdeg a1CB1 samkuTxedSi gavataroT C kuTxis CK1 biseqtrisa,gadavzomoT CK1 sxivze CK biseqtrisis toli monakveTi.K wertilze gavavloT a1B1 wrfis paraleluri wrfe, romelic C kuTxis gverdebs kveTs saZiebel a da B wertilebSi.

2. avagoT saZiebeli aBC samkuTxedis msgavsi a1B1C1 samkuTxedi ori kuTxiT. gavavloTB kuTxidan BK1 simaRle. BK1 sxivze B wertilidan gadavdoT BK simaRlis toli mon-akveTi. K wertilze gavavloT aC||a1C1 B kuTxis gverdebTan gadakveTamde.

3. davxazoT saorientacio naxazi.

ageba: avagoT a kuTxe, mis gverdebze gadavzomoT aB1 da aC1monakveTebi ise, rom aB1:aC1=2:3. B1 da C1 wertilebi Sevaer-ToT. aB1C1 samkuTxedSi gavavloT B1C1 gverdis mediana aM1.aM1 sxivze gadavdoT aM medianis toli monakveTi. M wertil-ze gavataroT B1C1-is paraleluri wrfe a kuTxis gverdebTangadakveTamde B da C wertilebSi saZiebeli samkuTxedia aBC.

4. ixileT amocana 3.

5. marTi kuTxis wverodan mis gverdebze gadavzomoT Ca1 da CB1 monakveTebi ise, romCa1: CB1 iyos mocemuli Sefardeba. gavavloT a1CB1 samkuTxedis CM1 mediana. CM1 sxivzegadavdoT CM monakveTi, romelic mocemuli hipotenuzis naxevris tolia. am wrfis gadakveTis wertilebi kuTxis gverdebTan saZiebeli samkuTxedis wveroebia.

8. SaBC=21 ·7·3=10,5.


65

9. ra ricxvzec ikveceba 81334

++

nn

wiladi, imave ricxvze Seikveceba 34813

++

nn

wiladic.

34813

++

nn

341)34(3

+−++

=n

nn34

13+−

+=nn . maSasadame, saZiebel ricxvze unda ikvecebodes

341+−

nn da maSasadame wiladic

134

−+

nn .

174

17)1(4

1744

134

−+=

−+−

=−+−

=−+

nnn

nn

nn .

analogiurad imave ricxvze unda Seikvecos 7

1−n wiladic. es ukanaskneli ki, cxadia

ikveceba 7-ze, roca n–1=7k anu n=7k+1.

10. 34k-s bolo cifria 1, e.i. 19831984+4=...1+4=....5, e.i. iyofa 5-ze.

11. 2222 )1()0()1( −+−=−+ yxyx aris manZili a(0;1) da B(x;y)

wertilebs Soris, analogiurad 22 )0()1( −+− yx aris

manZili B(x;y) da C(1;0) wertilebs Soris. aB+BC umciresia, roca B∈aC monakveTs anu aB+BC-s umciresi mniSvneloba aC monakveTis sigrZea da tolia 2 .

12. 2342 =− acb ; 2b -is 4-ze gayofis SesaZlo naSTebia 0, 1; 23-is ki 3. 4ac=b2–23, b2–23 ar iyofa 4-ze.

13. Canaweri P(E) niSnavs, rom am paraleluri gadatanisas P wertilSi gadadis E P1 wertilSi.cxadia, EKK1P marTkuTxedia, maSasadame

⇒

=

+++==

71

111111

PPPPPKMKPMP

EMPM

PPMK

3571414 =++=P .

9. heronis formula

reziume:

heronis formulis damtkiceba moeTxoveba mxolod cxrianis da aTianis donis bavSvebs, formulis codna da misis gamoyeneba ki _ yvelas.wyvilebSi micemuli davaleba: p=x+y+z; p–a=z; p–b=x da p–c=y. am tolobebis heronis formulaSi Casmis Sedegad miviRebT dasamtkicebel formulas.


3. aBCD paralelogramSi aB=51; aC=74 da BD=40. O diagonalebis kveTis wertilia. vipovoT heronis formuliT aOB samkuTxedis farTobi da gavamravloT 4-ze.

a

B

C


66

5. SaBC=84, e.i. BF simaRle tolia 8. NP=x NBP da aBC samkuTxedebis

msgavsebidan 8

)12(821

xx −−= ,

1366=x .

6. aBC samkuTxedis BC=4, xolo a wverodan daSvebuli simaRle 5-is tolia, e.i. S=10.

7. 20)22()13( 22 =−−++−=AB

20=AC 40=BC , e.i. 20== ACAB 102=BC .

BC gverdze daSvebuli simaRle 101020 =−=h , e.i. 2

BCh = da ABC∆ marTkuTxa

tolferda samkuTxedia ∠a=90°, e.i. h wertili emTxveva a wertils da 20=BH .

10. rogor gamovTvaloT samkuTxedis farTobi, rocamocemulobaSi figurirebs gverdebis da medianebis sigrZeebi

reziume:

sasurvelia, moswavleebs winaswar davavaloT feradi fanqris da danayofebiani saxa-zavis motana. Semdeg gavaxsenoT, rom Tu samkuTxedis gverdisadmi mopirdapire wvero-dan gavlebuli monakveTi gverds yofs a:b SefardebiT, maSin es monakveTi samkuTxedis farTobs yofs, Sesabamisada a:b SefardebiT.


1. ABCCOA SS61

1= , radgan ABCNOA SS

121

1= , e.i. a1N medianaa da

a1C= a1O, e.i. Ca1:aO=Oa1:aO=1:2.

2. ∆aBC-Si aa1=CC1=15; BB1=18 O medianebis gadakveTis wertilia. ganvixiloT ∆aOC.

aO=OC=32 aa1=10; 6

31

11 == BBBO . e.i. aB1=8; 4868 =⋅=AOCS ; 1443 == AOCABC SS .

3. aB=27; BC=29; BM=26. gavagrZeloT MK=BM da vipovoT SaBK heronis formuliT 270== ABKABC SS .

4. aK=9 aO=6; CM=5 CO=310 .

SaOC=21

6310

$ $ ; SaBC=30.


67

5. SaBC=3Q Qh a

32$

= , e.i. ha

Q6= .

11. kuTxis sinusi, kosinusi, tangensi da kotangensi

reziume:

moswavleebi gaecnobian maxvili kuTxis trigonometriul funqciebs. Sesabamis aRniS-vnebs da kavSirs am funqciebs Soris.


1. meore kaTeti SeiZleba gamovTvaloT x6 =tgα TanafardobiT, e.i. x=8sm piTagoras

TeoremiT hipotenuza 10-is tolia.

3. sina=ABBD ; BD=

211 . 6. sinα=

ACh ; sinβ=

BCh ; sinα=

sinsin

ACBC

314

ba= = .

8. sin3a =

53 ; aK=3x; aO=5x; OK=4x.

SaOC= S3ABC =300, e.i. 300=12x2; x=5; aC=30.

10. a=7k+3; b=7n+4.a) ab=(7k+3)(7n+4) naSTebis namravlia 12. e.i. naSTia 5.b) 3a+4b=3(7k+3)+4(7n+4)=21k+9+28n+167(3k+4n+3)+4 naSTia 4.g) (a+2)(b–3)=(7k+5)(7n+1) naSTia 5.d) a(b+2)+4=ab+2a+4, ab-s Svidze gayofis naSTia 5, 2a-si ki — 6.e.i. jamis 5+6+4=15 naSTia 1.e) (a–1)(b+4)=ab+4a–b–45+12–4–4=9 naSTia 2.

12. ZiriTadi trigonometriuli igiveobebi

reziume:

moswavleebi ecnobian ZiriTad trigonometrul igiveobebs.


1. unda avagoT marTkuTxa samkuTxedi, romlis erTi kaTetis sigrZe meoreze 2-jer metiiqneba. meti sigrZis mqone kaTetis mopirdapire maxvili kuTxe saZiebeli kuTxe iqneba.

3. a) tg2α−sin2αtg2α=tg2α(1−sin2α)=sin2α

6*. aageT α kuTxe, Tu cnobilia, rom a) cosα=74 .

a KC

M N

B

2

a


68

avagoT marTkuTxa samkuTxedi kaTetiT (4x) da hipotenuziT (7x), sadac x sigrZis mon-akveTs nebismierad avarCevT.

8. b) 18; d) 2 · 3 · 3 = 18.

13. zogierTi kuTxis sinusis, kosinusis, tangensis dakotangensis mniSvneloba


3. ∆aBC-Si aC=1 ∠a=30°; ∠C= 45°; B wertilidan davuSvaT aC-ze BK simaRle, aRvniSnoT

igi x-iT. BKC tolferda marTkuTxa samkuTxedia, e.i. KC=x da BC=x 3 ; ∆aBK marTkuTxaa

∠a=30°. miviRebT aK=3 3 ; aB=2x.

aC=x 3 +x; x( 3 +1)=1; x=2

3 1-; aB= 3 –1. BC=

2

6 2-.

4. am kaTetis mopirdapire kuTxisTvis sinα=a

a

2 2

1= , e.i. α=β=45°.

5. rombis diagonalebis Sefardeba misi naxevrebis Sefardebis tolia. amitom Tu rombis

erTi kuTxea α; tg ;2 3

1a=

2a =30°. α=60°.

6. Tu trapeciis maxvili kuTxea α; tg2

1 α=45°.

7. marTkuTxa samkuTxedis hipoteza aRvniSnoT c-Ti.

a) 1) c2=1002+102 ⇒ c=10 101 ; 2) c= 100 25 25 172 2+ = ; 3) c=100 2 .

b) 50%.

8.

9. g1; g2; g3; g4; g5; g6s1; s2; s3; s4; s5; s6.

sss; ssg; sgs, sgg, gss, gsg; gvs; ggg.


69

14. marTkuTxa samkuTxedi

reziume:

moswavleebi gaecnobian h2=a´b´, hc=ab; b2=bć, a2=ać formulebs.


1. gamoviyenoT a2=ca´Tanafardoba.

2. ix. amocana 1.

3. CKB da aCB samkuTxedebis msgavsebidan CKBK

ACCB

= viRebT: BK=18.

aK-s povna h2=a´b´ TanafardobiTac SeiZleba aK=1842

2

=98 (an isev msgavsebiT).

4. diametrze dayrdnobili Caxazuli kuTxe marTia.

5. rombis diagonalebi marTi kuTxiT ikveTebian.

6. rombSi Caxazuli wrewiris diametri simaRlis tolia, e. i. simaRle tolia 4-is. rombissimaRle ki diagonaliT miRebuli tolgverda samkuTxedis simaRlecaa. Tu rombis

gverds aRvniSnavT a-Ti. a

42

3= , e.i. a

3

8 3= .

11. a´=9; b´=16, e.i. c=25.a2=ać ⇒ a2=9·25 ⇒ a=15b2=bc=16·25 ⇒ b=20

r= a b c2

+ - =5.

14. kombinaciis pirveli ricxvis arCevis 20 variantia, meore ricxvis _ 19, mesamesi ki _ 18. kombinaciaTa raodenobaa 20 · 19 · 18 .

15. samkuTxedis farTobis gamosaTvleli formula ori gverdiTa da maT Soris mdebare kuTxis sinusiT

reziume:

moswavleebi gaecnobian samkuTxedis farTobis gamosaTvlel formulas. S=21 sin c,

oTxkuTxedis farTobis gamosaTvlel formulas, diagonalebiTa da maT Soris mdebare kuTxis sinusiT.


70


1. moswavleebma ukve ician tolgverda samkuTxedis farTobis formula a

4

32

c m, kidev

erTxel davanaxoT maT am formulis samarTlianoba, Tu farTobs viTvliT gverdebiTa da maT Soris mdebare kuTxis sinusiT.

3. SaBO· SCOD=

21 ·BO·aO·sinα·

21 ·CO·OD·sinα=

=21 ·aO·OD·sin(180°–α)·

21 ·BO·OC·sin(180°–α)=SBOC· SaOD.

4. ∆aDC-ASi CB medianaa, amitom SBDC= SaBC=2.

6. d1≡x ⇒ d2=2x. rombis farTobi gamoiTvleba formuliT: S=21 d1d2,

e.i. 21 x·2x=12 ⇒ x2=12.

rombis gverdi a x x xx

x2 2

24 4

515

2 22

22

= + = + = =` `j j .

7. vipovoT meore diagonali: S=21 d1d2, e.i. d

212 =96. d=16.

piTagoras TeoremiT vipovoT rombis gverdi a=10.

S∆aBD=21 SaBCD=48,

21 ·10h=48, h=9,6.

8. davweroT sistema: d d

d d2

11

12

1 2

1 2

=

+ =* kosinusebis Teoremis Sedegis Tanaxmad: d1

2+d22=4a2.

aviyvanoT sistemis meore gantoleba kvadratSi:

d12+d2

2+2d1d2=144 d12+d2

2=144–44.

e.i. d12+d2

2=100, 4a2=100, a=5, P=20.

9. S=21 ·10·14. 30=5·14·

21 =35.

10. rombis simaRle h=2r. gverdi a=sin sinh r2{ {

= . farTobi S=a2sinφ=sin

sinsin

r r4 4642

2 2

${

{{

= = .

11. a b2+ ·20=400; a b

2+ =20.

12. aK=2

69 51- =9 KD=2

69 51+ =60.

BK= 41 9 32 502 2

$- = =40

SaBCD=BK·KD=40·60=2400.


71

13. 105.

14. kenti cifrebis raodenobaa 5.a) 53; b) 5·4·3=60.

15. asoebis raodenobaa 33, erTniSna ricxvTa ki _ 10 anu sul 43 simboloa.variantTa raodenobaa 434.

16. ramdenime saintereso amocana


1. aC=aDS=18; sinα=

54 ∆aCM-dan

ACAM

54

= . SemoviRoT aRniSvna aM≡4x, e.i.

aC=5x. CM=3x=MD. ∆CKD-Si CDCK

54

= , e.i. CK= x5

24 =aB, ∆aBC-dan

BC= ( )xx x

55

24572

2

- =` j SaBCD=(BC+aD) CK xx

x2 5

75

1024

18= + =` j , saidanac x=8

5 3 .

CK=524

85

3 3 3$ =

2. BK–r=m=2

5 sinα=53 . moc. Tanaxmad, Tu O Caxazuli wris centria

BO=m Tu sinα=53 cosα=

54 ∆BOM-dan OM=r=mcosα=2 2 ; ∆aBK-Si

BK=m+r=2

9 2 aB=2

15 2 .

aK=aBcosα=6 2 S=21

12 22

9 254$ $ = .

3. ∠aBC=α, e.i. ∠aBK=∠CBK=2a . ganvixiloT ∆BOM tg

2a =

21 .

OM=OK=r BM=2r BO=r 5 , e.i. BK=r 5 1+^ h.

aK=BKtg2a =r 5 1+^ h·

21 = r

2

5 1+, e.i. aC=r 5 1+^ h.

S=21 r 5 1+^ h. r 5 1+^ h=

94

3 5+^ h.

4. trapeciis simaRle aRvniSnoT h-iT. fuZeebi ki _ a da b-Ti, maSin ferdis sigrZea a b2+ . P=2(a+b)=48, e.i. a+b=24. simaRlis gavlebiT miRebuli marTkuTxa samkuTxedidan

gamominareobs, rom .ha b

Sa b

h6

42 2

244 48$ $=

+= =

+= = .


72

5. pirveli lekvis arCevis 5 variantia. meoris _ ukve oTxi. e. i. variantTa raodenobaa5·4=20.

6. a) aseulebis cifris arCevis 5 variantia, aTeulebis _ 4. erTeulebis _ sami, e. i. 5·4·3=60.b) radgan aseulebis cifri ver iqneba 0, amitom aseulebis cifris arCevis variantia 4,aTeulebis _ 4, erTeulebis ki _ 3; 4·4·3=48.

7. a) 5·5·5=125; b) 4·5·5=100.

8. 25 24 23 138002 2$ $ = = 6900.

Seamowme Seni codna

I varianti

1) g; 2) d; 3) a; 4) g; 5) d; 6) b; 7) a; 8) d; 9) b; 10) g.

11) g; 12) a; 13) g; 14) b.

IV Tavis damatebiTi savarjiSoebi

1. samkuTxedis fuZe a=hS2

11

222 11= = , tgα=

11

111= , α=45°. SeiZleba SevaxsenoT moswav-

leebs, Tu mediana Sesabamisi gverdis naxevaria, es samkuTxedi marTkuTxaa da Tu igiamave dros tolferdaa, e.i. fuZesTan mdebare kuTxe 45°-ia.

2*. ∠KaC=∠MBC. sinα=53 e.i. SegviZlia SemovitanoT aRniSvna MC≡3x,

maSin BC=5x , BM =4x, PΔaBC=8x. SaBC=21 ·4x·6x=12x2. 12x2=48. x=2.

r=PS

8 248

3$

= = .

3. radgan trapecia Semoxazulia da aD+BC=aB+CD=4h. SaBCD=2h2.

2h2=72, h=6, r= h2

=3.

4. mocemuli fuZiT da ferdiT vpoulobT pirvel diagonals, misi sigrZea 4, xolo radgan meore diagonali kuTxes Suaze yofs, zeda fuZe mocemuli ferdis tolia.

vpoulobT simaRles diagonaliT Sedgenili marTkuTxa samkuTxedidan h512

= da

S2

3 5512

548

$=+

= .


73

5. rombSi Caxazuli wrewiris diametri rombis simaRlis tolia, e.i. rombis maxvili kuTxe30°-ia.S=a2sin30°=25, a2=50, a=5 2 .

6. aC=3x=9 ⇒ x=3.

S=21 ·9·4x=54sm2.

7. ( )S

c

ab

a b

a b

ab

a b

ab

24

10

48

100

196

48

14

482 2

2

& & &=

=

=

+ =

+ =

=

+ =

=o o o) ) )

vietas Sebrunebuli Teoremis Tanaxmad, a da b aris x2–14x+48=0 gantolebis amonaxsnebi,

anu 8 da 6. r a b c2 2

8 6 10=

+ -=

+ - =2sm.

8. viciT, rom diagonalebi gadakveTis wertiliT iyofa fuZeebis proporciul mon-akveTebad, e.i. 1:3. simaRliT da diagonaliT Seqmnil samkuTxedSi gavlebulia fuZis paraleluri wrfe. e.i. simaRlec gaiyofa 1:3 SefardebiT.

9. msgavsi samkuTxedebis gverdebis sigrZeebia 2x; 5x da 4x. 11x=55, e.i. x=5. gverdebia: 10; 25 da 20.

10. pirvelis gverdebia: 6x; 7x da 8x, meoris ki _ 6y; 7y da 8y. umciresi gverdebis jamia:6x+6y=38. Sefardeba 6x:6y=10:9.

( )

yx

x y

x n

y n

n

910

3 19

10

9

3 19 19

&

$

=

+ =

=

=

=

p p**

n

x

y

31

310

39

3

&

=

=

= =

Z

[

\

]]]

]]]

e.i. pirveli samkuTxedis gverdebia 20; 370 da

380 ; meoris ki — 18; 21 da 24.

11. aM=aK=KC=CN=10; samkuTxedebis msgavsebidan MNABMB

20= .

12. ∆aBC~∆DBF ⇒ 11

6

22x

x BF

2

= ⇒ BF=12.

5

4 C

5x

4x B

a

5x

6x F D

a C

B


74

13. ΔaBD da ΔaCB-Si∠aBD=∠aCB mocemulobis Tanaxmad; ∠a saerToa, e.i. isini msgavsia,

ACAB

ABAD

= , aqedan aD=6,4, e.i. DC=3,6.

14. Tu BM=BN=x, maSin aM=MC=18–x, maSin MN monakveTi tolia 40–(12+2)(18–x)=2x–8.

∆MNB~∆aBC x x18 12

2 8=

- , aqedan vRebulobT x=MB=6.

15. ∆aBC~∆DCa ⇒ x

x28

= ⇒ x=4.

16. wrewiris a wertilidan BD diametrze davuSvaT aK marTobi. ΔaBD marTkuTxaa, aK ki marTi kuTxis wverodan hipotenuzaze daSvebuli marTobi, anu aK2=3·12⇒aK=6 sm.

17. Caxazuli wrewiris centri mdebareobs a kuTxis biseqtri-saze.

e.i.

xx

rr

38 20

=- ⇒ 8r=60–3r ⇒ 11r=60 ⇒ r=

1160 .

18. naxazze ganxilulia SemTxveva:

∆aBD ⇒ ABxx

8 23

= ⇒ aB=12.

∆aBC ⇒ 12 4

ACAB

AC108

5

3

&= = ⇒ aC=15.

analogiurad amoixsneba, Tu Oa=2x da OD=3x. ar akmayofilebs samkuTxedis utolobas.

19. ΔaBC ⇒ 152=aM·aC ⇒152=aM·25 ⇒ aM=9. MC=25–9=16. BN biseqtrisaa, e.i.

BCAB

ANAN

ANAN

25 2015

25&=

-=

-.

75–3aN=4aN ⇒ 7aN=75 ⇒ aN=775 . MN = aN−aM =

775 –9=

712 . pasuxi: 9;

712 da 16.

20. ACAB

BCBE

87

= = .

ΔBDC ~ ΔEFC ⇒EFBD

ECBC

x y

y308

15&= = ⇒ x=16.

a

x

8 D

B 2 C

8x

a

M

3x 3x C

B

a O

8

10 3x 2x D

C

B

15

a

20

M N C

B

15 7

a D F

8y x C

B

E


75

21. saerTo gare mxebs vipoviT formuliT. Semdeg msgavsebidan vipoviT danarCeni mon-akveTebis sigrZeebs.

22. viciT, rom trapeciaSi Caxazuli wrewiris centri-dan ferdi Cans marTi kuTxiT.

ΔKCO ⇒ KC=5, e.i. CP=5. ΔCOD ⇒ OC2=CP·CD ⇒ 102=5·CD ⇒ CD=20.

r2=CP·PD=5·15 ⇒ r =5 3 .

aB+CD=BC+aD ⇒ PaBCD=2(aB+CD)=2 5 3 10 10 3 2+ = +^ ^h h.

23. cosABBD BD

2 28 71

&a

= = ⇒BD=4

∆aBD ⇒ aD2=282–42=32·24 ⇒ aD=16 3 .

∆aBD ⇒ODBO

ADAB

rr

r r16 3

28 47 16 3 4 3

7 4 3

16 3& & &= =

-= - =

-.

24. paralelogramis blagvi kuTxis wverodan daSvebul si-maRleebs Soris kuTxe tolia paralelogramis maxvili kuTxis.

∆aBK ⇒ sinADh

ADAD

54

21

54

851

& &a= = = = .

∆BCM ⇒ BCh

BCBC

54 1

54

452

& &= = = . P⇒ 2· 85

810

2815

415

$+ = =` j .

25. aB2=aD·(16+aD) 152=16aD+aD2. aqedan vipoviT aD-s da gamoviyenebT

ra b c

2=

+ - formulas.

26. a2=a′c formulidan vipoviT BD da Da monakveTebs.h2=a′b′ formulidan R D-s.ΔCDa~ΔMNa. aqedan vipoviT MN-s.

a

60°

30°

O 60

°

P

30° 30° D

B K C

a r D

28

C

B

O 2

a

B

a α

α α

h1

K

h2

M

D

C

15

a D 16

C

B

B

M

D N a

C


76

27. ONAO

OMCO

12

= = SegviZlia SemoviRoT aRniSvnebi ON≡x, OM≡y, maSin

aO=2x, CO=2y. ΔONC-Si x2+4y2=9. ΔaOM-Si y2+4x2=16. SevkriboT es tolobebi, vRebulobT 5x2+5y2=25. e. i. x2+y2=5, magram Tu ganvixi-lavT ΔaOC-s.

aC= 2 2AO OC x y x y4 4 52 2 2 2 2 2+ = + = + = .

29. ,

5

12BCAB AB x

BC xAC x

2 41

125

13& &= ==

==o .

aqedan vipoviT a da C kuTxeebis trigonometriul funqciebis mniSvnelobebs.

30. aBCD trapeciaSi BK da CD simaRleebia. ΔCPD ⇒ CP=2=BK. PD2=42–22=12⇒ PD=2 3 . ΔaBK ⇒ aB=2 2 . BC=KP=7–2 3 –2=5–2 3 . P=2 2 +5–2 3 +4+7=16+2( 2 − 3 ) .

35. radiusebis Sefardeba Sesabamisi qordebis Sefardebis tolia. e.i. 13x+5x=36; x=2 .

36. aB=6 ; aC=12 ; ∠a=120°. gavavloT BM||aK, cxadia

∆aBM tolgverdaa, e.i. aM=MB=6. ∆aKC~∆MBC.

MCAC

MBAK

= . saidanac aK=4sm.

B a

C


77

V Tavi

1. naSTTa klasebi

reziume:

moswavlem unda SeZlos xarisxis bolo cifris povna. SeZlos ricxvis 3-ze, 4-ze, 9-ze, 5-ze ,10-ze gayofis naSTebis povna. mocemul ricxvze gayofiT miRebuli naSTTa klasebis Camowera. SeZlos gamoTvalos Tu romel naSTTa klass ekuTvnis m-ze gayofiT miRebuli romelime ori naSTTa klasidan aRebuli elementebis wrfivi kombinacia (Tu a∈Kp da b∈Kq, maSin romel klass ekuTvnis, sazogadod, na+kb ricxvi),


1. a) yoveli me-7 ricxvi 7-ze gayofisas iZleva imave naSTs. –5+7=2, r=2 .

b) 5; g) 7; d) –2+11=9, r=9; e) –3+8; v) –195=17(–12)+9, r=9;

z) –52=12⋅(−5)+8, r=8; T) n=1(n−1)+1, r=1. i) n2+n+1=n(n+1)+1; r=1.

2. kenti ricxvebi 4k+1 an 4k+3 saxisaa, e.i. r = 1 an 3-s.

3. a) 5+5+7=17, r=8; b) 3+5+7+8+1=24, r=6. g) r=2;

5. a) (1+7+1)–(8+5+2)=9–15=–6, r=–6+11=5; b) (1+4)–(7+9)=–1111, r=0;g) 1030007 → 7+3+1−0=11, r=0; d) 2+7+2–(4+3)=4, r = 4.

7. a) 3333=34k+1 bolovdeba 31=3-iT; b) 52722=524k+2 bolovdeba rac 22=4-iT;g) 222523=(....2)4k+3 bolovdeba 23=8-iT; d) 777251= (...7)4k+3 bolovdeba 73=...3-iT;e) 88888=884k+4 bolovdeba 84=....6-iT.

8. evklides algoriTmidan gamomdinareobs a da b ricxvebis udidesi saerTo gamyofiserTi metad saintereso Tviseba: Tu d=u.s.g. (a;b), maSin SesaZlebelia moiZebnos k, l∈Z ricxvebi, rom Sesruldeba: d=ka+lb.a) k=5 da l=−3-Tvis gveqneba: 5(3n+2)−3(5n−3)=19. d=19, e.i. aseTi saxis ricxvebis saerTo gamyofi an 1-is an 19-is tolia. vipovoT a da b-s iseTi wyvilebi, romelTa u.s.g. 19-is rolia. a=19p, b=19q.

(3n+2=19p da 5n–3=19q) ⇒ n= p qp

q

3

19 2

5

19 3

5

3 1&

-=

+=

+.

p -2 3 8 13 18q -1 2 5 8 11

b=19qb=5n–3 -19 38 95

a=19pa=3n+2 -38 57 152

b) a=8n+3, b=2n–1.


78

(8n+3)–4(2n–1)=7 7d ⇒ d=7.

(a=7p, b=7q) ⇒ (8n+3=7p da 2n–1=7q) ⇒ n= p q

8

7 3

2

7 1-=

+ ⇒p=4q+1. magram q≠2k, k∈Z.

p 1 3 5q 5 13 21

a=8n+3a=7p 35 91 147

b=2n–1b=7q 7 21 35

9. a) –12=3·(–4) –12∈K0; –5=–6+1 –5∈K1. –1+3=2 –1∈K2; 1∈K1; 5=3·1+2 5∈K2 . 19=3·6+1; K1; 24=3·8; K0. b) 1) a=3k; b=3k+2 a+b=3n+2 (a+b)∈K2 .2) a–b=...(–2)=...(–2+3=1). (a–b)∈K1.3) ab-Tvis r=0·2=0 ab∈K0 .4) 5a–3b-Tvis r=5·0–3·2=–6, e.i. r=0 (5a–3b)∈K0 .

10. –8+8=0 ⇒ −8∈K0; –7+8=1⇒ −7∈K1; –6+8=2 ⇒ −6∈K2.–3+8=5 –3∈K5 ; –1+8=7 –1∈K7; 0∈K0 .1∈K1; 4∈K4; 5∈K5; 8∈K0; 18=2·8+2 18∈K2.27=3·8+3 27∈K3.

11. a) (a+b):8= ... (5+4)9:8=1 (1), e.i. r(a+b)=1. (a+b)∈K1.b) ab = ... (4·5=20). 20=2·8+4 r(ab) = 4. ab∈K4.g) a–b = ... (5–4=1).d) b–a = ... (–1), –1+8=7 r(a–b)=7.e) a–3b = ... (5–3·4=–7), –7+8=1, r(a–3b)=1.

12. erTi SexedviT meore diagramaze mogeba ufro swrafad izrdeba, magram Tu davak-virdebiT masStabs (ordinatTa RerZze) vnaxavT, rom orive diagramaze monacemebi erTi da igivea. 2005-2006 wlebSi mogeba iyo I-ze 40–30=10 aTasi lari, II-ze 40–30=10 aTasi lari,e.i. erTnairi.

2. Sedareba

reziume:moswavlem unda SeZlos dainaxos, rom mocemul ricxvze gayofisas miRebul erTsa daimave naSTTa klasSi moTavsebuli ricxvebi, erTmaneTis sadari ricxvebia. SeZlos gamoiyenos SedarebaTa Tvisebebi ricxvebze moqmedebebis (Sekrebis, gamoklebis, gam-ravlebis) Sesrulebisas, zogierT ricxvze gayofadobis dasamtkiceblad.


1. a) 4-ze gayofisas yoveli me-4 ricxvi iZleva erTsa da imave naSTs, e.i. erTmaneTis


79

sadaria moduliT 4.15:4=...(3) ⇒3 ≡ 7 ≡ 11 ≡ 15 ≡ 19(mod 4).b) 15:5 ⇒5 ≡ 10 ≡ 15 ≡ 20 ≡ 25(mod5).g) 15:6 =...(3) ⇒3 ≡ 9 ≡ 15 ≡ 21 ≡ 27(mod 6).d) 5 ≡ 15 ≡ 25 ≡ 35 ≡ 45(mod10).

2. anan–1...a0–(a0+a1+...+an–1+an)=10nan+10n–1an-1+...+10a1+100a0–a0–a1–...–an–1–an=an(10n–1)+an–1(10n–

1–1)+...+a1(10–1).miRebuli jamis TiToeuli Sesakrebi iyofa 3-ze. e.i. jamic gaiyofa 3-ze. amrigad, ricxvi da misive cifrTa jami erTmaneTis sadaria moduliT 3 (radgan maTi sxvaoba iyofa 3-ze).

3.a≡5(mod7) a=7k+5a≡-2(mod3) a=3k+1 -2≡-2+3=1(mod3)a≡9(mod13) a=13k+9a≡1(mod9) a=9k+1a≡-1(mod7) a=7k+6 -1≡-1+7=6(mod7)

4. a) 10n–1= ...99 9 9

n

h?

⇒ 10n≡1(mod9).b) nebismieri ricxvi sadaria misi bolo cifrisa moduliT 10.795=74k+3≡73≡3(mod10), 73=a3.g) 8357≡84k+1≡8(mod10)d) 2355≡24k+3≡23≡8–10=-2(mod10)e) (256≡1(mod5)) ⇒ 256256≡1256≡1(mod5).

5. a) 31–1+28+31+30+31+30+31+31+15≡2+0+3+2+3+2+3+3+1≡5(mod7).(1→ kvira) ⇒ (0 → SabaTi).15 seqtemberi kviraa.b) SabaTi; g) samSabaTi; d) SabaTi.

6. a) 5≡1(mod4) ⇒ 5n≡1(mod4) ⇒ (5n–1)4; b) 8≡2(mod6) ⇒ 8n≡2n(mod6);

g) 15≡7(mod8) ⇒ 15n≡7n(mod8); T) 15≡1(mod7) ⇒ ( )

( )

mod

mod

15 1 7

6 6 7

n

/

/o) ⇒ 15n+6≡7≡0(mod7).

k) 33=27≡1(mod13) ⇒ (33)n≡1(mod13); l) 13≡5(mod8) ⇒ 13n+7·5n≡5n+7·5n=8·5n 8;

m) 7≡-2(mod9) ⇒ 733+233≡(-2)33+233≡0(mod9).

7. a) ganv. a3−a=a(a−1)(a+1)6 nebismieri a∈Z -Tvis; b) a5=a4k +1 ⇒ a5 bolovdeba imave cifriT rac a, amitom 5-ze gayofisas mogvcemen erTsa da imave naSTs.

8. a) 3≡ −2(mod5) ⇒3100≡(−2)100=2100(mod5);b) 112≡21(mod100)114≡212≡41(mod100)115≡41⋅11≡51(mod100)1110≡512=2601≡1(mod100).

9. 12+1≡2≡−1(mod3)


80

22+1≡2≡−1(mod3)

32+1≡1(mod3) .

(12+1)≡(42+1)≡...≡(10002+1)(mod3) → sul 334 Tanamamravli, 12+1≡2–1(mod3).

(22+1)≡(52+1)≡...≡ (9982+1)(mod3) → sul 333 Tanamamravli, 22+1≡2–1(mod3).

(32+1) ≡...≡ (9992+1)(mod3) → sul 333 Tanamamravli, 32+1≡1(mod3).

e.i. [(12+1)(42+1)...(10002+1)] [(22+1)(52+1)...(9982+1)] [(32+1)(62+1)...(9992+1)]≡(−1)334(−1)3331333≡

≡−1(mod3). r.d.g.

11. a) 40°≡400°≡760°≡1120°(mod360).

12. 1+2+...+9=45 459 e.i. p=1.

13. 1+2+3+4=10 103 e.i. p=0.

14. a) 52≡1(mod 24) ⇒520≡110(mod 24) ⇒ 520:24 =...(1).

b) 72≡1(mod 24) ⇒ 520≡1(mod 24) ⇒ 748:24 =...(1).

15. ( )

2( 7)

2( 7)

mod

mod

mod

37 2 7

16

23

/

/

/

p ⇒ 37n+2+16n+1+23n≡2n+2+2n+1+2n=2n ⋅ 7≡0(mod 7).

16. ( )

7 25 ( )

mod

mod

a

b

7 3 3 7

4 3 7

/

/

+

+ =-o ⇒ (7a+3)2n+1+(7b+25)2n+1≡32n+1+(−3)2n+1=32n+1−32n+1≡0(mod7) .

17. (1; 5) da (3; 7).

18. b=1; 2; 3; 4 e.i. oTxi variantia.

19. ( ; ); ( ; ); ( ; ); ( ; )

( ; ); ....................... ( ; )

............................................

( ; ); ....................... ( ; )

1 1 1 2 1 3 1 4

2 1 2 4

8 1 8 4

N

P

OOOOO

⇒ 8·4=32 xdomiloba.

20.

21.3 SevadginoT cxrili

1 2 3 4 5 6 7 8 xdomilobaTaraodenobaa 3.esenia (8;2); (7;3); (6;4).

P=(a+b=10)=323

1 2 3 4 5 6 7 8 92 3 4 5 6 7 8 9 103 4 5 6 7 8 9 10 114 5 6 7 8 9 10 11 12

22. a) 10; b) yvelaze uaresi SemTxvevaa , , ...,110 111 200

916 7 8444 444, e.i. 91 ricxvi; g) magaliTad: 111, ...,

119. — sul 9.


81

3. amocanebi jgufuri muSaobisTvis:

1. a≡ .a a a a a a a1 2 3 4 5 6 7 a7+a5+a3+a1≡a, a6+a4+a2≡b, a+b= a1+...a7=1+2+...+7=28.

a gaiyofa 11-ze, Tu (a–b)11. gveqneba sami SemTxveva a–b=0; 11; 22.

1) a b

a b

28

11

+ =

- =o) ⇒ sistemas naturalur ricxvebSi amonaxseni ara aqvs, e.i. albaToba imisa,

rom miRebuli ricxvi gaiyofa 7-ze aris 0.

2) 28

0

a b

a b

+ =

- =o) ⇒ a=b=14 davTvaloT ramdeni sxvadasxvaniSna SvidniSna ricxvi arsebobs

romlebic Caiwereba 1, 2, . . ., 7 cifrebiT.

adgili 1 2 3 4 5 6 7 sul

raodenoba 7 6 5 4 3 2 1 7!=5040

I adgilze (milionebis TanrigSi) SesaZlebelia davweroT Svidi cifri.

II adgilze _ 6 cifri, radgan erTi ukve dakavda pirvel adgilze.

III adgilze _ 5 cifri da a.S. me-7 adgilze darCeba erTi cifri. sul arsebobs 5040

ricxvi, aqedan a6, a4, a2 cifrTa sameuli, ise, rom a6+a4+a2=14. iqneba {1,6,7}, {2,5,7}, {3,4,7}, {3,5,6}. yoveli simravle gansazRvravs 6 aseT sameuls, magaliTad,

(a6; a4; a2)=(1;6;7)=(1;7;6)=(6;7;1)=(6,1,7)=(7,1,6)=(7,6,1).

.a a a a1 6 71 3 5 7 (1) ricxvSi {a1, a3, a5, a7}={2;3;5;4}. am oTxi ricxvisagan SesaZlebelia mivi-

RoT 24 dalagebuli oTxeuli. e.i. (1) saxis 24 gansxvavebuli ricxvi arsebobs, xolo

im ricxvebis raodenoba, romlis luw adgilebze Cawerilia ricxvebi 1; 6; 7 iqneba 6⋅24. analogiurad danarCeni {a2, a4, a6} saxis sameulebisTvis. amrigad 11-is jeradi iarsebebs 4⋅6⋅24=242 ricxvi.

P=1 2 3 4 5 6 7

24354

2

$ $ $ $ $ $= .

3. a b

a b

a

b

28

22

25

3

+ =

- =

=

=o) ⇒ sami gansxvavebuli naturaluri ricxvis jami ≠ 3.

pasuxi: 354 .

2. davakvirdeT, rom saZiebeli A ricxvi romelime mocemul ricxvze gayofisas iZleva 1-iT nakleb naSTs. e.i. a= u.s.j.(1;2;3;4;5;6;7;8;9;10)–1=2519.

3. vTqvaT didi kubis wibo a-s tolia, maSin aseTi kubis asagebad saWiro iqneba a3 raode-nobis patara kubiki. radgan daaklda erTi rigi, e.i. daaklda a cali kubiki. miviReT n=a3−a=a(a−1)(a+1)6 . r.d.g.


82

Tema: naSTebis ariTmetikagv. 70. wyvilebSi. m ariTmetikaSi 0-s ara aqvs gamyofebi, roca m martivi ricxvia da

eqneba gamyofebi, roca m Sedgenili ricxvia, mag. Tu m=10, maSin 2*5=10≡0(mod10). Tu

m=7, maSin namravli iqneba 0, Tu erT-erTi Tanamamravli nulia.

SemogTavazebT fermas Teoremis damtkicebas:

ganvixiloT a-s jeradi ricxvebi, m1=a, m2=2a, m3=3a,...,mp-1=(p-1)a. am ricxvebidan arc

erTi wyvili ar aris erTmaneTis sadari moduliT p, radgan mk–mn=(k–n)a, xolo (k–n)a ar iyofa p-ze. aseve arc erTi mi, i=1,...,(p-1) ar iyofa p-ze. amitom m1, m2,..., mp-1 ricxvebi

p-ze gayofisas sxvadasxva TanmimdevrobiT mogvcemen naSTebs: 1-s, 2-s, ..., (p–1)-s. amitom

isini raRac gadanacvlebiT sadarni arian 1,2,...,(p–1) ricxvebisa.

e.i. m1m2m3...mp-1≡1·2·3...(p–1)(mod p).1·2·3...(p–1)ap–1≡1·2·3...(p–1)(mod p). ap–1≡1(mod(p)).

4. naturaluri ricxvidan namdvil ricxvamdereziume:moswavleebi ukve icnoben naturalur, mTel, racionalur ricxvTa simrav-leebs, ician iracionaluri ricxvic. Aam paragrafSi moxdeba masalis Tavmoyra.


9. a) 1666,061= 223-e cifri iqneba 6;

b) )428571(,073= periodSi 6 cifria. 223=37·6+1 223-e cifri iqneba 4.

g) )384615(,0135= 223-e cifri iqneba 3. d) 1333,0

152= 223-e cifri iqneba 3.

11. –102–101–...0+...+99= 30300098991001011020

−=+−−−−−−=

18. K(x1;0), L(x2;0), miviReT x2–x1=5x1.

x x

x x

6

7

2 1

2 1

=

+ =) x1=1 x2=6 c= x1· x2=6.

5. n-uri xarisxis fesvi

reziume:

sasurvelia gavixsenoT ariTmetikuli kvadratuli fesvis ganmarteba, Semdeg dava-konkretoT, rom analogiurad SegviZlia ganvmartoT nebismieri luwi xarisxis fesvi da mxolod amis Semdeg ganvumartoT kenti xarisxis fesvi.


83


13. a) a3=27 a=3. e.i. saWiroa 8·3=24 m.

15. 31 a2h=9.

22

2

22

26

−

aaa a3=27 a=3. h=a=3.

21. 12734 bolo cifria 1.17772 bolo cifria 9.29393 bolo cifria 9.namravlis bolo cifria 1.

22. y=ax2+bx+c a(0;2) e.i. c=2.y=ax2+bx+2 B(1;8)8=a+b+2 a+b=6e.i. a+b–c=4.

6. ariTmetikuli fesvis Tvisebebi

reziume: moswavleebs gavaxsenoT ariTmetikuli kvadratuli fesvis Tvisebebi, ris Semdegac gauadvildebaT axali masalis aTviseba.


8. a) ( ) 13132

−=− ; b) ( ) 3101034 4−=− ; g)

( ) 1210

21

10 6

6 6 −=

−.

9. v) 4 44 xaxa −= x≤0; z) 4 54 aaa −−=− .

12. ⇒−+

=−1

4)( 2

nmmnnm =+

−=+ 1

)(4

2nmmnnm

2

2

2

)()(4

−+

=−

−+nmnm

nmnmmn

.

13. SeuZlebelia, radgan ricxvebi, cxadia, unda iyos kenti, magram sul aris 5 kenti cifri 1; 3; 5; 7; 9, xolo 5-iT dabolovebuli ricxvi martivi ar iqneba.

14. EEFF=1000E+100E+10F+F=1100E+11F=11(100E+F) ricxvi 11-is jeradia, xolo orniSna ricxvi, romelic 11-is jeradia gansxvavebuli cifrebiT ar Caiwereba.

15. p=3.

16. a) u.s.g.(2100–1; 2120–1)=u.s.g. (2100–1; 2120–2100)=u.s.g.(2100–1; 220–1)=..........=1.

b) u.s.g. =

ba

11;11 u.s.g..(a;b). e.i. =

60100

111;111 u.s.g.(100;60)=10.

17. SeTanxmdnen, rom pirveli gamomsvleli daasaxelebda im fers, romlis Sesabamisi bumbulebis raodenoba iqneboda kenti (cxadia, TeTri da Savi bumbulebidan, erT-erTi luwi raodenobaa da meore kenti). davuSvaT es Savi bumbulia. danarCenebi gadaxedaven erTmaneTs (pirvelis garda) da Tu vinme xedavs kent raodenoba Sav bumbuls e.i. TviTon aqvs TeTri da Tu xedavs luw raodenoba Sav bumbuls, e.i. TviTon aqvs Savi bumbuli.


84

7. miaxloebiTi gamoTvlebi

reziume:

moswavleebma unda SeZlon mocemuli Tanrigis mixedviT, iracionaluri ricxvis miax-loebiTi mniSvnelobis povna.


13. )35(4

345 22

2 +=+= aaaS .

16. davweroT TiToeuli funqciis Sesabamisi gantoleba: kvadratuli y=ax2+4 (2;–1)

–1=4a+4 45

−=a 445 2 +−= xy .

wrfivi y=kx+b (0;–1); (3;2)y=x–1.Sesabamis sistemas eqneba saxe:

−≥

+−≤

1

445 2

xy

xy

17. D=k2–180≥0 56−≤k 56≥k .

19. =−+

+

+−

−+− 62721

13132

12122

2

2( ) 2 ( )

41

3 2 2 4 2 3 1 6 12 2

- - - + + - =^ h

( ) 32626252622232

41 2

=−+−=−+−=.

20. ABFA

AFAB⇒

=∠

==060

8

aBF samkuTxedi tolgverdaa, tolgverdaa misi mgavsi DEF samkuTxedic e.i. DE=DF=2 aD=6.

SaBCD=aB·aD·sin60°=24 3 .


85

11. orucnobian utolobaTa sistemis amoxsna (III Tavis)

reziume:

gavixsenoT gavlili masala. davsvaT SekiTxvebi:1. sityvierad daaxasiaTeT da Semdeg daStrixeT iseT wertilTa simravle, romelTaT-

visac a) y>5; b) x<1; g)

><

51

yx

.

2. romel sakoordinato meoTxedebSi SeiZleba iyos wertili, romlis abscisa metia5-ze da a.S.


2.e)

+≤

−>

11

2

2

xyxy T)

≥≥+

0122

yyx

i)

−>

<+

xy

yx2

922

3. a) y≥–1; b) y>–x+3; g) y≤x2+4x.

4. d — 45kg — x calip — 30kg — y cali

≤

≤+

2

21003045yx

yx

≥≤+

xyyx

214023

≥

+−≤

xy

xy

2

7023

8. I testirebis qula≡x. x≤10; II testirebis qula≡y. y≤15.

∈≤≤

+−≤

0,1510

18

Nyxyx

xy ∆aBC-Si moTavsebuli yvela wertili, romelTa koordinatebi mTeli ricxvebia, e.i. sul 36 TanamSromlis.


86

9. 20092009.9-iT dabolovebuli ricxvis nebismieri kenti xarisxi 9-iT bolovdeba, luwi _ 1-iT. bolo cifria 9.

10. p kentia.

⇒

+=+=

2313

kpkp

+=

+=+=+

+=+

13)4(312310

11310

kpkkp

kp xolo

+=

+=++=+

231231014312

kpkpkp

.

orive SemTxvevaSi am samidan erT-erTi ar aris martivi e.i. es ricxvebia: 3 — 13 — 17.

12. gavavloT MN mxebi, cxadia MN=2r=3.

MCBNADMN trapeciaSi +

+=++=+NBMCMNCBANDMMNAD

aD+BC+12=25, e.i. aD+BC=13.

Seamowme Seni codna

1) d; 2) g; 3) d; 4) b; 5) d; 6) a; 7) g; 8) a; 9) a; 10) g; 11) a; 12) e; 13) d; 14) a; 15) d; 16) g; 17) d; 18) a; 19) d; 20) a; 21) g; 22) g.

V Tavis damatebiTi savarjiSoebi

1. 11.

2. a) (53+63)(532−53⋅63+632 )116;g) (13+23+33+...+1803+1813+1823) davajgufoT:(13+1823)+(23+1813)+(33+1803)+... TiToeuli Sesakrebi iyofa 183-ze.

3. mocemuli namravli iyofa nebismier ricxvze, romelic 101-ze naklebia k⋅a+b rom iyofodes a-ze, b-c unda iyofodes a-ze.

4. 1·2·3...106+1.

5. a) 2006 nebismier xarisxSi 6-iT bolovdeba.b) 1999-is nebismieri luwi xarisxis bolo cifria 1.g) 2004-is nebismieri luwi xarisxis bolo cifria 6.d) 2002-is xarisxi luwia da 4-is jeradi, e.i. xarisxis bolo cifria 6.

6. a) magaliTad n=2, k=2; b) n=2 k nebismieria.

7. a) 1+2+3+...+ 2006+2007+2008=(1+2008)+(2+2007)+...=2009·1004;b) 1+2+3+...+2005+2006+2007=2008⋅1003+1004.


87

8. 34-ze gayofis SesaZlo gansxvavebuli naSTebia 0, 1, 2, . . . , 34 e.i. 35 naturalur ricxvsSoris aucileblad moiZebneba ori mainc erTi da igive naSTis mqone ricxvi.

11. 222132+255243+236637–359346=...6+...5+...6–...1=...6.

12. ricxvis cxraze gayofis naSTi misi cifrTa jamis cxraze gayofis naSTis tolia.ricxvebi miiReba a) (1;5;6); b) (1;5;7); g) (1;6;7); d) (5;6;7) sameulebis kombinaciebiT. SesaZlonaSTebia a) 1+5+6 naSTi iqneba 3 b) 4, g) 5, d) 0.

13. (0;1;6) — naSTi 1 (0;1;7) — 2; (0;6;7) — 1, (1;6;7) — 2.

14. a) (234+1)7–1 (234+1)-is nebismieri xarisxis naSTi 234-ze gayofisas aris 1. e.i. sxvaobagaiyofa 234-ze. e) 33n–1=(26+1)n–1. T) 289n–9n=(280+9)n–9n.

15. by0+ax0=c. b(y0–ak)+a(x1+bk)=by0+ax0+abk=c.

17. a) XI XII I II III IVdReebi 8 31 31 28 31 23naSTebi 1 3 3 0 3 2

Svidze gayofis naSTi 5 iqneba, xuTSabaTi.

22. 4832

32=

4836

43= . 24.

74

145

143

=+ 25. 3514

52=

3515

73= .

mniSvnelis umciresi mniSDvnelobaa 70.

45. z) 310310

1+=

−.

46. a) 22

122

122

832

83=

−+

+=

−+

+ ;

b) 417117417

)174(

)174(3

3 2

−=+−+

=+−

+ ; g) 352

)52(3 2

−=−

−;

d) =⋅−

+⋅− 10

50103,0

23)23( 101023

10)23()23( 2

=⋅−

⋅+−⋅ ;

e) =−+

++

++

++

887

176

165

154

1

2878675645 −=−−+−+−+−= .


88

VI Tavi

1. veqtoris cneba. toli veqtorebi


3. a) AB CD-. ; AB CD= , e.i. AB CD=- . b) CB DA= .

6. a) WeSmariti; b) WeSmariti; g) mcdaria, es veqtorebi SeiZleba iyos sawinaaRmdegod mimarTuli.

8. a) n

37

+ , n=1, n=7; b) 5n+7–n12 , n=2; 3; 4; 6; 12.

2. veqtorebis Sekreba

1. SeiZleba, roca am veqtorTa sigrZeebi tolia, xolo mimarTulebebi sawinaaRmdego.

3. ,F F 3 6 21 2+ = kg.

4. a) AB A D A B A D A C1 1 1 1 1 1 1 1+ = + = ;

g) DA B B C B C C C B1 1 1 1 1+ = + = ;

e) DB BC DB B C DC1 1 1 1 1+ = + = .

10. SS

OCAO

SS

BOC

ABO

OCD

AOD= =

12. a) 7n+8–n4 , n=±1; ±2; ±4.

b) n

n n nn

n15 5 2 2 1

5 21

12

++ + + +

= + ++

, n=0; –2.

F1F

F1

2

+

F2

a1

a

B

D1

D

C

B1 C1


89

3. veqtorebis sxvaoba


2. a b a b 5+ = - = .

3. a da b veqtorebi urTierTmarTobulia.

4. a) OB OA AB- = AB BC= OC OD DC- =

b) OB OA AB- =

6. b) AD MP EK EP MD AD DM MP PK AM MK AK+ + + - = + + + = + = .

7. a) AD CA DC BC CD CB DC BD DB BD 0+ + + + = + + = + =^ ^ ^h h h .

b) AB AC DC CB DC DB- + = + =^ h .

8. miRebul samkuTxedebs toli fuZeebi da toli simaRleebi aqvT.

9. miRebuli samkuTxedi aris tolferda da marTkuTxa. hipotenuza=2R=20,

S=21 ·10·20=100.

10. SBDC=SaBC=2.

12. a) x=7·(5·6·8)+14+ 1 naSTia 1; b) x=7·(5·6·8)–21+ 6 naSTia 6.

4. veqtorebis gamravleba ricxvze


1. a) AB KB2=- ; b) AK AB2= .

2. a) a bKL KC CL21

21

= + = + ;

aLM21

=- ; bMK21

=- .

b) b aAK AC CK21

= + =- - ;

b aCM CA AM21

21

= + = - .

3. a) kAB CD$= k=–1.

b) kAB CD$= k=2.

g) kOB B D1 1$= k=21

- .


90

5. 2 2 2 2AC B D AO OD AO OD AD BC21 1+ = + = + = =^ h .

6. OM ON OA OB OD OC

OA OB OD OC

DA CB AD BC

21

21

21

21

21

21

- = + - + =

= + - - =

= + =- -

^ ^

^

^

h h

h

h

.

II gza: OM ON NM- =

+

___________________________________

NM NC CB BM

NM ND DA AM

NM NC ND CB DA BM AM2

= + +

= + +

= + + + + +

NC ND 0+ =

BM AM 0+ = , e.i. NM BC AD BC AD21

21

= - - =- +^ ^h h.

9. mezobeli gverdebis Suawertilebis SemaerTebeli monakveTi diagonaliT miRebuli samkuTxedis Suaxazia, diagonalis paraleluria da mis naxevars udris.

10. marTkuTxedis diagonalebi tolia.

11. tolferda trapeciis diagonalebi tolia.

6. sibrtyis dafarva


6. a) 10!+8!=91·n! 8!(9·10+1)=91·n!n=8.

b) ! !! !

15 1415 14

-+ =n,

! ( )

! ( ! )n

14 15 1

14 15 1-

+= , n

78

= .

g) 13!(14·15–n)=105·13! 210–n=105, n=105.

d) ( ) ! ( )

( ) !!

( ) !

n n

nn

n n

1 2

1 12019

$+ +

+ += , n=18.

7. x4+x

xx

1 14

2

2

2

= -c m +2=18.

a

D

O

N

M

C

B


91

7. marTobi, daxrili, gegmili. manZili wertilidan sibrtyemde

reziume:

moswavlem unda icodes, rom nebismieri wertilidan sibrtyisadmi gaivleba erTad-erTi marTobi da uamravi daxrili; unda icodes gegmilis cneba, agreTve is faqti, rom tol daxrilebs toli gegmolebi aqvT da piriqiT. amave dros, or daxrils Soris meti gegmili aqvs meti sigrZis mqone daxrils da es faqtic Sebrunebadia.


1. piTagora: 30 40 502 2+ =

5. aO=1∠aBO=∠aCO=30°, e.i. aB=aC=2. ∆aBC marTkuTxaa, BC=2 2 .

6. ( )20 15 9 162 2 2- - = .

7. aD2=212+252=1066.CD= 1066 225 29- =

8. x2–1=4x2–493x2=48 x=4.

9. prizmis kerZo saxeebi


3. a 3 3= a 3= Ssr

=6a2=18 Skv

=a a a2 22

$ = .

4. a2 2 11 2= ; a2=11 Sgv

=4a2=44.

5. tolgverda samkuTxedia gverdiT a 2 5 2=

S= a

4

3

2

25 32

= .

10. piramidareziume:

ganvmartoT piramida, wesieri piramida, wibo, waxnagi, wvero. mivceT n-kuTxa piramidis ganmarteba da davsvaT kiTxvebi: ramdeni wvero, wibo da waxnagi eqneba n-kuTxa pirami-das? paragrafSi dasmuli SekiTxvebis pasuxebia:1. aucileblad; 2. mcdaria, aqvs oTxi waxnagi; 3. WeSmaritia; 4. mcdaria, nebismieri pi-ramidis gverdiTi waxnagi samkuTxedia; 6. mcdaria.

B

a

O

C

a 21

25

C

B

K

D


92


2. amocanis pirobidan Sf=4sm2, e.i. fuZis gverdia 2.

3. gverdiTi waxnagi tolferdaa, e.i. apoTemis K fuZe DC-s Suawertilia, e.i. OK gverdis naxevaria. aB=2x. x x25 12

2- = , saidanac x=4 an 3.

4. ∆NMK~∆aSC, e.i. S S41

41

8 2MNK ASC $= = = .

5. S AB

4

34 3ABC

2

= = , e.i. aB=4.

SaBC=12. SK21

4 12$ = . SK=6.

8. CB= 3 ∠aCK=60°.

SS

KBAK

12

CKB

ACK = = 0sin

sinCB CKAC CK

360

12

$ $$ $

cc= saidanac aC=2, aB= 7 .

Seamowme Seni codna

1. b. 2. b. 3. b. 4. d. 5. g. 6. g. 7. d. 8. a. 9. g. 10. d. 11. b. 12. g. 13. d.

a

B

D

K

S

C

O


93

VI Tavis damatebiTi savarjiSoebi

1. a) W; b) mc; g) W; d) mc; e) mc; v) mc; z) W.

2. kolinearulia.

3. a) ki, Tu isini TanamimarTulia; b) ki _ nebismier aBC samkuTxedSi: dauSvaT, aB<aC da aB<BC. AB AC CB= + ; g) ki, Tu isini sawinaaRmdegoTaa mimarTuli; d) ki, Tu isini TanamimarTulia; e) ki, Tu isini marTobulia.

4. a) AD BC= da A D B C1 1 1 1= ;

b) CD C D C B BD CD BD BC CD BC BD1 1 1 1 1 1 1 1& &= = + = - + = .

5. a) AB BD DC AD DC AC+ + = + = ;

b) AD CB DC AC CB AB+ + = + = ;

g) AB CD BC DA AB BC CD DA AC CA 0+ + + = + + + = + =^ ^h h .

6. a) m a b2 2 2= + 2 2 2n a b= - 2 2 2 2 2 2 4m n a b a b b- = + - + =

3 3 3m n a b a b a b31

31

31

310

38

- = + + - = - m n a b a b a b101

101

101

1011

109

- - =- - - + =- - .

7. Aa) 3 5a b a b2 1$a b+ = + +^ h ⇒α=3 da 2β+1=5 ⇒ β=2.

b) 2a a a b b 0$ $ $ $a b a b+ - + - = (α+β)a +(2α-β)b =1·a +0·b .

8. m k mDC AB& $= = .

c mQD QC CD= + = - (1)

k m b a mk

b aAB AQ QB1

&$= = + = - = -^ h

miRebuli toloba CavsvaT (1)-Si:

ck

b aQD1

= - -^ h.

9. AD AB BD AB BC21

= + = + .

BE BC CE BC CA21

= + = +

CF CA AF CA AB21

= + = +

AD BE CF AB BC CA23

23

0 0$= + = + + = =^ h .

10. a) ara; b) ki. 11. a) ara. 12. a) ki; b) ki.

a B

D C

Q

F

a

D

E C

B


94

VII Tavi

1. simravle

reziume:

moswavle unda flobdes moqmedebebs simravleebze. SeeZlos Cawera simravlisa, romliselementebsac axasiaTebs raime Tviseba. SeZlos daTvalos ori simravlis gaerTianebaSiSemavali elementebis raodenoba.


1. a) W.

b) qorelidan me-9 klasis me-7 Tavidan (gv. 69) mc.

g) W.

d) mc.

2. a) M={1; 2; 3; 4; 5; 6; 7; 8}; b) M={–4; –3; –2 . . .5; 6};

g) M={–6; –5; –4; –3; –2; –1; 0}; d) |x–2|<3 ⇔–3<x–2<3 ⇔ –1<x<5. M={1; 2; 3; 4}.

3. a) 3x–5y=7 ⇔ y= x5

3 7- x –1 4 9y –2 1 4

(–1; –2); (4; 1); (9; 4);

b) 5x+7y=2 ⇔ x= y

5

7 2- + x –1 –8 –15

y 1 6 11 (–1; 1); (–8; 6); (–15; 11).

4. a) n

nn

51

5+= + .

nn 5+ ∈N, roca

n5 ∈N ⇒ n=1;5.

pasuxi: orelementiani.

b) k

kk

5 65

6-= - ⇒ (6k) ⇒ k=1; –1; 2; –2; 3; –3; 6; –6.

pasuxi: a rvaelementiani simravlea.

g) x

x Q

x

x Qx3

5 15 0

516

> >& ,

d dd Q

-- p* * .

d) ( 3) 0

( 2) 0

x

y

2

2

$

$

-

-o) .

ori arauaryofiTi ricxvis jami nulis tolia, roca TiToeuli udris nuls. e. i. x=3 da y=2. gantolebis amonaxsenia wyvili (3; 2). a simravle erTelementiania.


95

5. a) a) a={x∈N/x=3k–1, k∈N}; b) D={x∈N/x=7k–4};

g) B= k N2

1k d) 3; d) C=

nn

n N1

d+

' 1.

6. a)

g)

b)

d)

e)

z)

v)

T)


96

7. a) SesaZlebelia, rom ormocive TanamSromelma, romelmac icis germaneli icis agreTve inglisuric. am SemTxvevaSi mxolod in-glisuri ecodineba 60-40=20 TanamSromels.

b) SesaZlebelia, rom arc erTma TanamSromelma ar icodes orive ena. maSin inglisuri an germanuli ecodineba 60+40=100 TanamSromels.

25+18–x=35 ⇒ x=8.

43=25+18–x ⇒ x=0.

g) 25=25+18–x ⇒ x=18. d)

9. a) gamoviyenoT rTuli procentis formula:

A 5000 11009

500010

1092

1092

4

2 2

= + = =` j =5940,5. mogeba iqneba 940,5 lari.

b) a=3000100112 8

` j .

11. 1760=a ,1

1005 5 4

+c m .

15. a(8;3). ∆aPQ tolferdaa. SaPQ=36–2·6–8=16.

jgufuri mecadineoba

SesaZlebelia klasi daiyos 4 an 5 jgufad. TiToeulma jgufma imuSaos erTsa da imaveamocanebze.1. vTqvaT, samive enas x bavSvi swavlobs. amis Semdeg Seivseba danarCeni ujrebi ise, rogorc naxazzea.

F E 32 – (5–x) – (9–x) – 25 – (5–x) – (17–x) – –x=18+x 5–x – x=13+x x 9–x 7–x G 30 – (9–x) – (7–x) – x= =14+x

I G

I G

8. a) a 25-x x B

18-x

b) a B

a B a B


97

18+x+(5–x)+(9–x)+x+(13+x)+(7–x)+(14+x)=7066+x=70x=4samive enas 4 bavSvi swavlobs.

3. F K 25 –8+x –x –7+x= 30 – 8+x – x – 1 =10+x 8–x – 10+x=12+x x 7–x 10–x M 40 –7+x –x–10+x= =23+x

10 + x + 12 + x + 23 + x + 7 - x + 8 - x + 10 - x + x = 75, x = 5.sportis samive saxeobaze xuTi bavSvi dadis.

2. simravleTa sxvaoba

reziume:

moswavleebs unda SeeZloT ori simravlis sxvaobis povna da Tu a⊂B, maSin ipovon Asim-ravlis B simravlemde damateba.sasurvelia moswavleebs vaCvenoT Tu rogor gamoviyenebT maTematikaSi simravleTa sxvaobas.amovxsnaT

xx x

1

2

-- =0 (1) gantoleba.

x x

x

0

1 0

2

!

- =

-) , Tu M-iTY aRvniSnavT x2–x=0 gantolebis

amonaxsenTa simravles F-iT ki x–1=0 gantolebis amonaxsenTa simravles, maSin (1) gantolebis amonaxsenTa simravle iqneba M\F. M={0; 1}; F={1}. M\F={0}. (1) gantolebis amonaxsenia x=0.


1. a) B\F={–17; 4}; b) D\F={0; 7}; d) CFL= 243$ ..

2. b) {20; 21; 22;...}; g) Z\Z0– =N; v) CQQ0

+ =Q–.

4. a\B iqneba 0-iT daboloebul anu 10-is jerad ricxvTa simravle.

7. a) a∩B∩C g) a\C d) B\a


98

9. a) (–∞; 5)∩(1;∞)=(1; 5); b) [5; 9]∩[–3; 7]=[5; 7].

(–∞; 5) (1;∞) = R [5; 9] [–3; 7]=[–3; 9].

g) [–4; 3]∩[3; 9]={3}; d) [0; 4)∩[4; ∞)=∅;

[–4; 3] [3; 9]=[–4; 9]; [0; 4) [4; ∞)=[0;∞).

10. a) [4; 8]\[7; 10)=[4; 7); b) [–2; 4]\{0}=[–2; 0) (0; 4].

12. 2) ⇒ (d=2b). 3) da 4) ⇒ b

b

2

3

=

== .

6) ⇒a=2 e. i. b=3. ⇒ d=6. radgan с<d, e.i. c=5.

13. a) 72 km/sT= 7200

00

36

102 =20m/wm. l=20m/wm·8=160m.

b) matarebelma gaiara aC=120+lmat.

·120+l=20·12. l=120m.

g) 60 km/sT=00

00

36

600=

350 m/wm. l= 20

350

-` j·30=100m.

d) pirvelma matarebelma 1 wT-Si gaiara Tavisi da meore matareblis sigrZeTajami 2l=(20–50/3)60=200 ⇒ l=100m.

e) 2l=(20+50/3)6=220.

14. 4R2=52+122 ⇒ R2=4

169 R=6,5.

e) (a∩B)\C v) a\(B C) z) (a\B)∩C

l l

R R

12


99

15. MOKC kvadratia OC= 8 ⇒ CK=r=2.BF=BK=3x aseve aF=aM=2x. rogorc erTi wertilidan gamosuli mxebebi. piTagoras Teoremis Tanaxmad:(2x+2)2+(3x+2)2=25x2 ⇒ x=2. aC=6sm; BC=8sm; aB=10sm.

3. albaTobis Teoriis elementebi

reziume:

moswavlem unda SeZlos gaarCios Tavsebadi da araTavsebadi xdomilobebi aucilebeli,SemTxveviTi xdomiloba. ipovos mocemuli xdomilobis sawinaaRmdego xdomiloba, ori xdomilobis jami, namravli, sxvaoba. SeZlos Sesabamisi magaliTebis moyvana.


1. a) a1 → isari ar gaCerdeba kent nomerze an isari gaCerdeba luw nomerze.

b) a4 → isari gaCerdeba nomerze, romelic ar aris 5-is jeradi.

g) a6 + a7 → isari gaCerdeba yviTel an wiTel ferze.

d) a2 a5 → isari gaCerdeba mwvane feris luw nomerze.

z) a1 da a5; a1 da a6; a1 da a7; a2 da a5; a2 da a6; a2 da a7; a2 da a8; da a.S.

T) a1 da a8; a1 da a2; a5 da a6; a5 da a7; a5 da a8; a6 da a7 da a.S.

i) a5. P(a5)=6/16; P(a6)=4/16.

2. a) Ω={sss; ssb;sbs; bss; sbb; bsb; bbs; bbb};

b) 1) safasuris erTxel mosvla. 2) erTnairad mosalodnelia;

g) P(s.2-jer)=3/8; albaToba imisa, rom safasuri mova 2-jer mainc aris P=4/8=1/2;d) P=4/8.

3. a) a1 → amoRebulia feradi birTvi.b) a1 a2 → amoRebuli birTvi arc TeTria da arc lurji.g) Ω\a3 → amoRebuli birTvi ar aris wiTeli an amoRebuli birTvi an TeTria, an lurjian yviTeli.

4. g) davTvaloT ramdeni ricxvi Seicavs cifr 1-ians. 1-dan 99-mde yoveli aTeulSi 1 ricxvi Seicavs 1-ian erteulebis TanrigSi, magram 10-dan _ 19-is CaTvliT ki 10 ricxvi aTeulebis TanrigSi. e. i. [1 — 99] → 10+9=19 ricxvi [100 → 199] → 100 ricxvi.danarCen aseulebSi _ TiToeuli 19 ricxvi da kidev 1000. e.i. sul 9·19+100+1=272 ricxvi

P(a1)=1000272 . davTvaloT im ricxvebis raodenoba, romlis CanawerSic gvxvdeba cifri

5-iani. 1-dan 999-is CaTvliT iqneba imdenive 5-ianis Semcveli ricxvi, ramdenic iyo

1-ianis Semcveli ricxvi e.i. 271. P(a2)=1000271 .


100

6. a) P(moniSnuli)=x

2000 . b) x

x4000 2000

227 2227$

$& =o =113,5.

e. i. moniSnuli SesaZlebelia iyo 113 an 114 Tevzi. ra Tqma unda, ar aris aucilebeli, rommoniSnuli iyos maincadamainc 113 an 114 Tevzi. es Sedegi SesaZlebelia miRweul iqnas Tu am cdas Zalian bevrjer CavatarebT.

9. TiToeuli xels 8-s arTmevs. 9 mosamarTle xels 9·8=72-s CamoarTmevs. magram rdganori mosamarTlis erTmaneTTan xelis CamorTmeva erT xelis CamorTmevad iTvleba, amitom xelis CamorTmevaTa ricxvi iqneba 9·8/2=36.

10. ∠OBa=90°; aB= OA OB2 2- =12

aB=aK=12.

4. xdomilobaTa jamis albaToba

reziume:

moswavle unda flobdes xdomilobaTa jamis albaTobis gamoTvlas rogorc Tavsebadi,ise araTavsebadi xdomilobebisTvis.


1. a) P(wiTeli)=4718 ; b)

4712 ; g)

472 ;

d) P(ara TeTri)=1– ( ( ) ( ))P A P A472

1= - an asec: P(ara TeTri)=47

12 15 184745+ +

= ;

e) „feradia“ es igivea, rac „araTeTri“;

v) P(mwv)+P(yv)=4712

4715

4727

+ = .

2. SevadginoT dominos qvaze mosul qulaTa jamis Sesabamisi cxrili.

0 0 0 1 0 2 0 3 0 4 0 5 0 6

1 1 1 2 1 3 1 4 1 5 1 6

2 2 2 3 2 4 2 5 2 6

3 3 3 4 3 5 3 6

4 4 4 5 4 6

5 5 5 6

6 6


101

a xdomiloba ganxorcieldeba, Tu qulaTa jami gautoleba 0-s, 3-s; 6-s; 9-s; 12-s. ganx-orcieldeba, Tu qulaTa jami toli iqneba: 0-is, 4-is; 8-is; 12-is. C _ Tu qvaze erTi mainc oriania.

a) Ω={(0; 0); (0:1) ..., (0;6), (1; 1)(1, 2) . . . (1; 6) (2;2) . . ., (6; 6)};b) P(a)=10/28; g) P(B)=8/28; d) P(C)=7/28;e) A da B xdomiloba erTdroulad ganxorcieldeba, Tu qulaTa jami gaiyofa 3-zec da 4-

zec. e. i. (0:0); (6; 6) P(aB)=2/28.v) a da C erTdroulad ganxorcieldeba, Tu qulaTa jami gaiyofa 3-ze da qvaze erTi

mainc 2-iani iqneba. aseTi elementaruli xdomilobebia; (1; 2) (2; 4). e. i. P(aC)=2/28.z) (2; 2), (2; 6) P(BC)=2/28;T) A da B Tavsebadi xdomilobebi:

P(a+B)=P(a)+P(B)–P(aB)=10/28+8/28–2/28=16/28;i) P(a+C)=P(a)+P(C)–P(aC)=10/28+7/28–2/28=15/28;k) P(B+C)=8/28+7/28–2/28=13/28.

3. a1, a2 da a3 wyvil-wyvilad araTavsebadi xdomilobebia. amitom

P(a1+a2+a3)=P(a1)+P(a2)+P(a3)=0,21+0,43+0,25=0,89.

4. SevadginoT cxrili:

1 2 3 4 5 61 2 3 4 5 6 72 3 4 5 6 7 83 4 5 6 7 8 94 5 6 7 8 9 105 6 7 8 9 10 116 7 8 9 10 11 12

a ganxorcieldeba, Tu qulaTa jami iqneba: 4; 8; 12.B ganxorcieldeba, Tu qulaTa jami iqneba: 3; 6; 9; 12.

b) P(a)=369 ; g) P(B)=

3612 ;

d) C-s ganxorcielebas xels uwyobs Semdegi elementaruli xdomilobebi

(2;1), (2; 2) (2; 3) (2; 4) (2; 5) (2; 6) (1; 2) (3; 2) (4; 2) (5; 2) (6; 2). e.i. P(C)=3611 .

e) P(aB)=1/36; v) P(aC)=3/36; z) P(BC)=4/36;

T) P(a+B)=9/36+12/36-1/36=20/36; i) P(a+C)=9/36+11/36-3/36=17/36.

6. x=25·10085 =21,25. 21 lari 25 TeTri.

7. a) 5212 ·100≈23,1; b)

3636 ·100=100%; g)

10020 ·100=20%; d)

52 ·100=40%.


102

8. vTqvaT, SesaZlebelia. a=n2. davuSvaT a-s gamyofebia: , , .... , , ...., , ,n n nnA

nA A

11 22 11 2 344 44 _ yovel

gamyofs eyoleba Tavisi mewyvile, magaliTad n1 da a/n1. magram n-s ar eyoleba mewyvile,radgan a/n = n. e.i. a=n2-is gamyofTa raodenoba kentia. ar SeiZleba n2-s hqondes 2002 gamyofi.

10. OB, Oa, OD, OC biseqtrisebia. amitom ∠aOB=∠COD=90°. aO2+OB2=aB2

+

OC2+OD2=CD2

aO2+OB2+OC2+OD2=aB2+CD2=52.

11. x+2x=90 ⇒ x=30°

MC=MB=Ma ⇒ ∠30° ⇒ BC=21 aB=5

aC= AB BC 5 32 2- = .

5. xdomilobaTa namravlis albaToba. xisebri diagramareziume:

moswavlem unda SeZlos gaarCios mocemuli a da B xdomilobebi erTmaneTze damokide-buli xdomilobebia Tu damoukidebeli. SeZlos, orive SemTxvevaSi, aB xdomilobis mox-denis albaTobis gamoTvla. SeZlos namravlis albaTobis gamoTvla xisebri diagramis saSualebiT.

1. axseniT Sinaarsi „sul 216216 “ winadadebisa. Tu TamaSi Sewydeboda me-3 etapze. maSin 6-is

mosvlis yvela SesaZlo SemTxvevaTa albaTobebis jami 1-is tolia.

P(6) P(6 ) P(6 ) P(6 ) 16 66 66+ + + = .


1. SevadginoT xisebri diagrama.

+ _ burTi Cavardeba kalaTSi. – _ burTi ar Cavardeba kalaTSi. a) 0,7·0,2=0,14; b) 0,3·0,8=0,24; g) Cavardeba natosi da ar Cavardeba ninosi an Cavardeba

ninosi da ar Cavardeba natosi:0,3·0,2+0,7·0,8=0,62.

d) 0,3·0,2+0,7·0,8+0,3·0,8=0,62.

Cavarda 1 burTi Cavarda orive burTi

0,3 0,7

+ – 0,8 0,2 0,8 0,2

+ – + –


103

2. I etapi: amova aso „a“ II etapi: amova aso „n“. III etapi: amova aso „a“.

a) P (ana)=146

134

125

915

$ $ =

b) „a“, „n“ da „a“ asoebisagan SesaZlebelia Sedges: „ana“,

„aan“, „naa“ sityvebi. gveqneba sami xe.

wavikiTxoT me-3 xe: aso „n“ aweria 4 birTvze

14-dan e. i. P(n)=144 .

II etapze aso „a“-s amoRebis albaTobaa 136 , radgan aso

„a“ aweria 6 birTvze, magram ukve urnaSi 13 birTvia.

III etapze urnaSi 12 birTvia da aqedan 5-zea aso „a“.

pirveli totis albaTobaa 146

134

125

915

$ $ = .

meore totis — 146

135

124

915

$ $ = .

mesame totis — 144

136

125

915

$ $ = .

P=915

915

915

9115

+ + = .

3. + _ dazianda – _ ar dazianda samiznis dazianeba nSinavs, rom erTma msrolelma

mainc moaxvedra. P=0,8·0,6+0,8·0,4+0,2·0,6=0,92. an asec: P (dazianda)=1 – P (ar dazianda)=1–0,2·0,4=0,92.

4. a) anis mosazreba mcdaria, radgan

P (ar Cavardeba arc erTi)=0,8·0,7·0,5=0,28. b) 0,2·0,7·0,5+0,8·0,3·0,5+0,8·0,7·0,5=

=0,07+0,12+0,28=0,47.

g) 0,2·0,3·0,5+0,2·0,7·0,5+0,8·0,3·0,5==0,03+0,07+0,12=0,22.

d) 0,2·0,3·0,5=0,03. e) ar Seicvleba.

a

n –

a –

–

I

II

III

146

134

125

127

n

a

a

a

n

a

n

a

III

I

II

146

144

134

135

136

125

124

125

0,8 0,2

+ – 0,6 0,4 0,6 0,4

+ – + –

I

II

0,2 0,8

+ – 0,3 0,7 0,3

+ + 0,5 0,5

– – 0,5 0,5 0,5 0,5

+ – + –

0,5 0,5

+ – + –


104

B..... I gaCerdes a1 seqtorze

umjobesia a boqlomi.

II a2-ze III a3-ze IV a4-ze

eseni damoukidebeli xdomilobebia:

P (a1a2a3a4)=101

91

81

71

50401

$ $ $ =

P (a1a2a3a4)= 61

61

61

61

12961

$ $ $ =

17

a..... 10

1

a1

a2

a3

a4

1 9

18

5. a) SesaZlo elementaruli xdomilobebia:

(8; 3), (8; 5), (8; 7)(6; 3), (6; 5), (6; 7) sabas mogebis Sansia

95 .

(1; 3), (1; 5), (1; 7) giorgis mogebis Sansi 94 .

b) (7; 2), (7; 4), (7; 9)(5; 2), (5; 4), (5; 9) sabas mogebis Sansia

95 .

(3; 2), (3; 4), (3; 9) giorgisa - 94 .

d) (9; 1), (9; 6), (9; 8)(4; 1), (4; 6), (4; 8) sabas mogebis Sansia isev

95 .

(2; 1), (2; 6), (2; 8) giorgisa - 94 .

cxadia, g) SekiTxvisas moswavleTa umravlesoba upasuxebs, rom ufro momgebiania a1 ruleti. amrigad, naCqarevi, dausabuTebeli daskvnis gamotaniT advili SesaZlebelia Secdomebi davuSvaT.

6. albaToba imisa, rom bavSvma aawyos sityva „maTematika“ aris:

1101

93

81

71

61

52

41

31

21

3024001

$ $ $ $ $ $ $ $ $ = .

e.i. praqtikulad SeuZlebelia bavSvma SemTxveviT aawyos sityva.

8. vTqvaT, es cifrebia a1, a2, a3.

2


105

jgufuri mecadineobis proeqti

2. Tu a-dan B-Si Cavida erTi romelime gziT,

maSin B-dan C-Si Casvlis sami varianti aqvs.

e. i. a-dan C-Si SesaZlebelia Cavides 4·3=12sxvadasxva marSrutiT.

3. a-dan C-Si Casasvlelad aris 5·4=20 sxvadasxva marSruti. araa SesaZlebeli.

4. kenti cifrebia 1, 3, 5, 7, 9. * * *a) pirvel adgilze SesaZlebelia CavweroT kenti cifrebidan nebismieri, e. i. gvaqvs 5 varianti.II adgilze ki SesaZlebelia CavweroT darCenili oTxidan nebismieri _ oTxi varianti, mesameze ki _ 3 varianti.

1 2 3 sul b) radgan gameoreba SesaZlebelia, amitom TiToeul adgilze SegviZlia CavweroT mocemuli xuTi cifridan nebismieri, e. i. aris TiToeul adgil-ze gvaqvs xuT-xuTi varianti.

ar meordeba 5 4 3 5·4·3=60

meordeba 5 5 5 5·5·5=125

5. 1 2 3 sul

mesame adgilze unda CavweroT cifri 5 _ erT variantze.ar meordeba 4 3 1 12

meordeba 5 5 1 25

6. P(1)=

21

meorem rom moigos, amisTvis pirvels unda mou-vides safasuri.

P(2)=21 ·

21 =

41

mesamem rom moigos, amisTvis pirvelsac da meoresac unda mouvides safasuri.

P(3)=21 ·

21

21

81

$ = P(4)=21

1614

=` j P(5)=21

3215

=` j .

a C B

I

B B

C

A

B B

C C C C C C C

b s

b s

b s

b s

b s

I

II

III

IV

V

21

21

21

21

21

21

21

21

21

21


106

P(1)+P(2)+P(3)+P(4)+ (5)P21

41

81

161

321

321

1= + + + + + = .

P( )521

3215

= =` j . " "5 — mexuTes mouvida safasuri _ TamaSi dasrulda _ mogebaTa

yvela SesaZlo SemTxvevebis albaTobaTa jami 1-is tolia.

7. cifrebia: 2, 3, 4, 5. 1·2·3· ... ·n=n!

1 2 3 4 sul

a) 4 3 2 1 1·2·3·4=4!=24b) 4 4 4 4 44=256

8. radgan I da II tomi gverd-gverd unda idos, SesaZlebelia isini erT wignad warmov-idginoT. e. i. gvaqvs 4 wigni.

1 2 3 4 sulI II an II I

magram radgan I da II tomis gan-lagebas ori variantia, amitom gveqneba 48 SesaZlo varianti.4 3 2 1 24

9. TiToeul kostumTan SesaZlebelia Sarvlis xuTi varianti, e. i. 3·5=15 variantia.

6. monacemTa warmodgenis xerxebi

reziume:

moswavles unda SeeZlos warmoadginos monacemebi svetovani, wertilovani, wriuli diagramis saxiT. SeeZlos gamoTvalos monacemis sixSire, fardobiTisixSire. aagos sixSireTa poligoni.


1. a) yveli; b) rZe aris %1700300

10017300

$ = ; g) 250450 =1,8-jer.

3. 4. a) ≈63667.

5. a) 100

1000000 22$ =220000; b) 1009 .

8. a) y = kx + b .

,

__________________

k b

k b

k

b

k

3 2

18 4

2 5

8

15 6

&-= +

=- +

=-

=

- =

) ) y=–2,5x+8.


107

b) 5 7 9

2 5

x y

x y

+ =

- + =o) ⇔ (–1;2). k=–2. y=–2x+b.

saZiebeli y=–2x+b wrfe gadis (–1;2) wertilze 2=2+b ⇒ b=0. e.i. y=–2x.

g) y=kx+b gadis (0;5) da (02;0) wertilebze.

9. 1+22009+32009+...+20082009≡ ≡1+22009+32009+...+10042009+(–1004)2009+ +...+(–2)2009+(–1)2009= =1+22009+32009+...+10042009–10042009–...– –32009–22009–12009≡0(mod2009). r.d.g.

1005≡–1004(mod2009)1006≡–1003(mod2009). . . . . . . . . . . . . . . . . . . . . . . . . . .2007≡–2(mod2009)2008≡–1(mod2009)

10. martivi procentia: n wlis Semdeg gamoitans a=15000+n·1003 ·1500.

11. a=15000·1003

15

+` j .

12. a)

b) 3=3·1 a6=3⋅25, a7=3⋅26. 6=3⋅2; 12=3⋅22; 24=3⋅23 Semdegi iqneba a5=3⋅24

g) izrdeba 4-iT, ariTmetikuli progresiaa. –3, 1, 5, 9, 13, 17, 21.d) ariTmetikuli progresiaa.

7. monacemTa dajgufeba. sixSireTa intervaluri ganawileba

reziume:

moswavleebma unda icodnen, rom diskretuli monacemebis didi moculobis erToblio-bisa da uwyveti monacemebisTvis gamoyofen raodenobrivi niSnis cvlilebis intervalis dayofas nawilebad da gamoTvlian dajgufebis intervalebSi moTavsebul monacemTa sixSireebs, mocemuli intervaluri variaciuli mwkrivis (intervaluri ganawilebis) grafikuli gamosaxva xdeba sixSireTa an fardobiT sixSireTa histogramis meSveobiT.


4.

+4 +6 +8 +10 +12 +14 +16

–5, –1, 5, 13, 23, 35, 49, 64.

4

3 8/3 7/3 25/3

1

1

sixSireTa histograma

4 7 10 13 16

12/114

8/114

5/114

1

fardobiT sixSireTa histograma

4 7 10 13 16


108

intervali [1;4) [4;7) [7;10) [10;13) [13;16)

sixSire 5 8 12 7 6

hnk

35

38

312

37

36

nhnk

3 385$ 3 38

5$ 3 38

5$ 3 38

5$ 3 38

5$

Tu fardobiT sixSireTa histogramis agebisas masStabs n-jer, Cvens SemTxvevaSi 38-jergavzrdiT, miRebuli histogramis garegnuli saxe iqneba igive, rac sixSireTa histo-gramis.

5. vTqvaT a=95, xolo b=935. h=84.

6.

sixSireTa intervaluri ganawilebis agebisas intervalebis raodenobisa da siganisgansazRvrisaTvis sayovelTaod miRebuli wesi ar arsebobs. statistikur literatur-aSi rekomendebulia 5-dan 20-mde toli sigrZis intervalis ageba, magram sakiTxi yovel kerZo SemTxvevaSi calkea gadasawyveti.

8. skolis direqtoris arCevis 10 variantia. darCenili 9 kan-

didatidan unda SeirCes ori moadgile. yoveli dawyvildeba

danarCen 8-Tan. iqneba sul 2

9 8$ varianti. sul ki —102

9 8$ =360.

intervali

s i x S i r e T a gamoTvla

sixSire

fard. sixSire

[95;375)

37

[375;655)

15

[655;935]

2

5437

5415

542

intervali

sixSireTa

gamoTvla

sixSire

[95;179)

fard. sixSire

5

[179;263)

16

[263;347)

14

[347;431)

6

[431;515)

6

[515;599)

4

[599;683)

1

[683;767)

1

[767;851)

0

0

[851;935)

1

545

5416

5414

546

546

544

541

541

541

9

1

8

2

7

3

6

4

5


109

9. direqtori I moadg. II moadg sul720 varianti

10 9 8 720

10. a) a

D

0

0>

!) ; b) D>0.

8. foTlebiani Reroebis msgavsi diagrama

reziume:

moswavlem unda SeZlos monacemTa sixSireTa ganawileba grafikulad foTlebiani Re-roebis msgavsi diagramiT gamosaxos. maT unda esmodeT, rom aseTi diagrama ar kargavs individualur monacemebs da amave dros TvalsaCinod aRwers sixSireTa ganawilebis formas.


1. 18 2 5 1 0 819 7 2 8 5 5 820 7 5 2 521 2 2 8 1 222 2

3. 1 98 82 81 80 80 80 75 95 892 70 70 70 68 65 63 61 56 40 40 30 30 25 12 10 10

07 03 03 00 00 00 10 10 50 41 71 823 60 57 41 30 204 50 15 05 605 33 236 48 40 10 0010 08 0612 60

monacemTa ZiriTadi umravlesoba moTavsebulia [2000; 3000) intervalSi.

5. x x

20015

20032

-+

= ; 40 wT=32 sT. x=60.

I → 75 km/sT. II → 60 km/sT.

6. ∠BaC=α ⇒ h=8cosα=1 h=1.

a

B C

D

b 8

α


110

7. h=2r=4,5sm. ΔaBK-dan aB=sinha

=4,5·3=13,5.

ΔCMD-dan CD=sinhb

=4,5·7=31,5.

BC+aD=aB+CD ⇒PaBCD=2(aB+CD)=90. PaBCD=90sm.

8. ∆aBD~∆DCB

9. SerCeviTi ricxviTi maxasiaTeblebi

reziume:

moswavlem unda SeZlos mocemuli ricxviTi monacemebisaTvis ipovos rogorc central-

uri tendenciis sazomebi: SerCevis saSualo, SerCevis mediana da moda, aseve monacemTa

gafantulobis sazomebi: gabnevis diapazoni, SerCevis dispersia, standartuli gadaxra.


3. a: 2 3 4x

105 7 10$ $

=+ + + =4,3. moda aris 4.

x2

4 4=

++

=4; gabnevis diapazonia 10–2=8.

S2= ( , ) ( , ) ( , ) ( , )10

2 4 3 3 4 4 3 5 7 4 3 10 4 32 2 2 2

$ $- + - + - + -=3,89. S= S

2 ≈1,94.

b) standartuli gadaxra ar warmoadgens gafantulobis mdgrad sazoms. is gacilebiT

mgrZnobiarea eqstremaluri dakvirvebebis mimarT, vidre saSualo. mocemul monace-

mebSi (an sxva romelimeSi) SecvaleT, klasSi, erTi monacemi da moswavleebi dainaxaven,

rom standartuli gadaxris reagireba aRmoCndeba ufro Zlieri, vidre saSualosi.

4. Sesabamisi variaciuli mwkrivia:

2,3 3,6 5,5 6,1 6,2 6,8 7,2 7,7 8 8,5 8,8 8,9 9,1 9,6 10,2 10,4 11,2 11,4 11,7 11,8 12,1 12,3 13,514,5 15,3 15,9 16,6 18,5 18,7 19,5.

, ,x

210 2 10 4

=++

=10,3.

7. a∪ B - aris xdomiloba ̀ erT-erTze mainc mova safasuri~. P(a∪

B)=1−P(( g;g))=1–21 ·

21 =

43 .


111

8. a)

9. OM gavagrZeloT: MN=R. MN=R. MNO1R marTkuTxedia.

MK=NO1. ΔONO1-dan O1N= OO ON1

2 2- . O1N= 50 40 10 90

2 2

$- = =30. MK=30sm.

Seamowme Seni codna. I varianti1. g. 2. a. 3. g. 4. g. 5. g. 6. g. 7. a. 8. b. 9. b. 10. g.

VII Tavis damatebiTi savarjiSoebi

1. a) b)

2. a) y= x7

2 5- (–1;–1); (6;1); (13;3).

3. a) a={21; 42; 63}.

5. klasSi aris 3+1+0+11+11+5+1=32 moswavle.

a

C

B a B

C

x y –1 6 13

–113

B

a

a

a

a1

a

C

2a

C1

B1

g) B B1

C1a1a a C

KR

RN

MO1O

r

g.

20–11–1– –5=3

12–1= =11

6–1=5 1

q.

24–11–1– –11=1

12–1=11

17–5–1–11=0 m.


112

6.

9. a) orniSna martivi ricxvebia: 11, 13, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79,

83, 87, 89, 97. P=2622 .

b) 2, 3, 5, 7 → P=264 .

10. yovel aTeulSi Sedis erTi a5 ricxvi → sul 9. magram kidev daemateba 50, 51, 52, 53,

54, 56, 57, 58, 59 _ sul 9. aseTi orniSna ricxvia 18. P=9018 .

11. P=107

96

157

$ = a – pirveli mwvanea. B _ meore mwvanea.

unda gamovTvaloT P(aBB). Tan B damokidebulia

a xdomilobze.

13. P(luwi)=63 . P(3)=

61 . P(luwi, 3)=

63

61

121

$ = .

14. pirveli yuTidan: P(st)=103 . P st` j=

107 .

meore yuTidan: P(st)=156 . P st` j=

159 .

a) P(st; st)=103

156

253

$ = .

b) P st st;` j=107

156

257

$ = .

C 28–5–7–

–3=13

8–3= =5

3 10–3=7

T 30–5–3– –2=20

5–3=2

42–7–3–2=30 Φ

100

x=100–(13+20+30+7+5+2+3)=20.

mwv.

mwv.

107

96


113

g) P(st; st` j=103

159

509

$ = .

17. „+“ standartuli „_“ arastandartuli

a) 10080

9920

9916

$ = ;

b) 10080

9979

495316

$ = ;

g) 10020

9980

9916

$ = ; d) 10020

9919

49519

$ = .

18. bostneulsa da baRCeuls uWiravs 14%. e.i. 100

10000 14$ =1400kv.m.

19. %

0% xx

100 360

6 100360 6$

$&

$c

c=o =216°.

20. %%

% %6010

100350

$ = .

21. variaciuli mwkrivia:

a) 0, 0, 1, 2, 2, 3, 4, 4, 4, 5, 5, 5, 6, 7, 7, 7, 7, 7, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 10, 11, 11, 11, 12, 12, 13.

b) 0 1 2 3 4 5 6 7 8 9 10 1 12 13sixSire 2 1 2 1 3 3 1 5 3 8 1 3 2 1

fard.sixS. 236 136 236 136 336 336 136 536 336 836 136 336 236 136

79 99

80 100

+ 20 99

20 100

80 99

– 19 99

+ – + –

8/36

1/36

1 7 2 8 3 9 4 10 5 11 6 12 13


114

g) 2

7 7x =

++

=7, moda aris 9.

1 2 2 3 4 3 5 3 6 7 5 8 3 9 8 10 11 3 12x

362 13

7$ $ $ $ $ $ $ $

=+ + + + + + + + + + + +

=+

.

(0 7) 2 (1 7) (2 7) 2 (3 7) (4 7) 3 (5 7) 3 (6 7) (7 7)

(8 7) 3 (9 7) 8 (10 7) (11 7) 3 (12 7) ( ),

S36

5

36

2 13 7

36419

11 6

2

2

2 2 2 2 2 2 2

2 2 2 2 2 2

$ $ $ $ $

$ $ $ $.

=- + - + - + - + - + - + - + -

+

+- + - + - + - + - + -

=

S≈3,3.

22. intervali [4;6) [6;8) [8;10) [10,12) [12;14) [14;16]

sixSire 4 7 10 12 9 6

nkh 24

=23,5 5 6 4,5 3

26. 5, 7, 7, 8, 8, 8, 10, 11, 12, 14, 20.

a) 12x =+

=7, moda aris 8.

x11

5 2 7 3 8 10 11 12 14 2010

$ $=

+ + + + + + +=

+

.

(5 10) (7 10) 2 (8 10) 2 (10 10) (11 10) (12 10) (14 10) ( 10),S

3620

11172

15 62

2 2 2 2 2 2 2 2

$ $.=

- + - + - + - + - + - + - + -=

S≈ S 42

. .

6 5 4 3 2 1

sixSireTa histograma

4 8 10 12 14 16 6


115

sakontrolo weris nimuSebi

sakontrolo wera №1

1. funqcia mocemulia y=8x+7 formuliT.a) ipoveT funqciis mniSvneloba, romelic argumentis 0,25 mniSvnelobas Seesabameba.b) ipoveT argumentis mniSvneloba, romelic funqciis (–11) mniSvnelobas Seesabameba.g) ekuTvnis Tu ara funqcias a(12;23); B(1;12); C(–1/4; 5) wertilebi.d) ipoveT argumentis mniSvnelobebi, romelTaTvisac funqcia dadebiTia.

2. rogori unda iyos y=kx+b koeficientebis niSnebi, rom grafikma gaiarosa) I; II; III meoTxedebSi. b) mxolod I da II meoTxedebSi.

3. SeadgineT kvadratuli gantoleba, romlis fesvebi iqneba

a) 3; 6, b) 0;4, g) –2; 2, d) 2 ; 3 .

4. ipoveT 0743 2 =−+ xx gantolebis fesvebis Sebrunebul sidideTa jami.

5. ipoveT p-s yvela mniSvneloba, romlisTvisac 0532 2 =++− ppxx gantolebas aqvs sxvadasxva niSnis fesvebi.


1. aBC samkuTxedis aB gverdze aRebulia M wertili,ise rom aM:MB=2:5, xolo BC gverdze N wertili, ise rom BN:NC=2:3. ipoveT MBN samkuTxedis faTobi, Tu aBC samkuTxedis farTobi 21sm2–is tolia.

2. trapeciis fuZeebi ise Seefardeba erTmaneTs, rogorc 2:3. ipoveT trapeciis far-Tobi, Tu S∆aOD=45sm2, sadac O diagonalebis kveTis wertilia.

3. ipoveT im wrewiris radiusi, romlis sigrZea 98p.

4. aBCD paralelogramis aD gverdze aRebulia M wertili, ise rom, aM:MD=1:3. ipoveT MKD samkuTxedis farTobi, Tu SaBCD=40sm2 da K – BC gverdis nebismieri wertilia.


1. dawereT cxy += 2 funqcia, Tu cnobilia, rom parabolis wveroa a) )81;0( ; b) ( 8;0 − ).

2. K(1;2) wertili mdebareobs y=x2+c parabolaze, mdebareobs Tu ara imave parabolaze wertili )4;3(A ; )6;5(B .

3. ipoveT parabolas wveros koordinatebi: a) 462 +− xx ; b) 743 2 ++− xx .


116

4. b da c parametrebis ra mniSvnelobisaTvis mdebareobs a(2;7) da B(–1;2) wertilebi cbxxy ++= 2 funqciis grafikze.

5. amoxseniT grafikulad utoloba 322 +≥ xx .


1. cbxaxy ++= 2 kvadratuli funqciis Sesaxeb cnobilia, rom (a+b+c)·c<0. Semdegi winadadebidan romelia WeSmariti a) D>0; b) D<0; g) D≥0; d) D=0?

2. Tu xaxxa +=−+ 2)2( 2 gantolebebis amoxsnebi Tanabradaa daSorebuli x=1 wertilidan, risi tolia a?

3. ipoveT k da p, Tu pxkkkxy +−+−= )63( 22 kvadratuli funqcia umcires mniSvnelobas Rebulobs x=2 wertilSi da es mniSvneloba -5-is tolia.

4. naxazze gamosaxulia 542 ++−= xxy funqciis grafiki. ipoveT OCAB + .

5. naxazze gamosaxulia cbxaxy ++= 2 funqciis grafiki ( acbD 42 −= ) maSin:

a) ac>0; b) aD>0; g) cD>0; d) ab>0; e) bD>0.


1. mimdevrobis me-8 da me-9 wevrebi Sesabamisad 8–is da 9–is tolia, am mimdevrobis nebismieri sam momdevno wevris jami ki 20-is tolia. ipoveT am mimdevrobis 145-e wevri.

2. ipoveT mimdevrobis udidesi wevri da misi nomeri.

3. mimdevrobis zogadi wevriaa n5

7n = - . ipoveT am mimdevrobis udidesi uaryofiTi da umciresi dadebiTi wevrebi.

4. geometriuli progresia mocemulia formuliT bn = 5n+2 . ipoveT b1 da b2.


117


1. or tolferda samkuTxeds ferdebs Soris kuTxe toli aqvs. erT-erTis gverdebia 10 sm, 10 sm da 8 sm. meoris perimetria 42 sm. ipoveT meore samkuTxedis ferdi.

2. aBC samkuTxedSi gavlebulia aC gverdis paraleluri wrfe, romelic aB da BC gverdebs kveTs Sesabamisad M da N wertilebSi. ipoveT CN, Tu BN=6sm, BM:aB=3:7.

3. ipoveT marTkuTxa samkuTxedis kaTetebi, Tu hipotenuzaze maTi gegmilebia 9sm da 16sm.

4. rombis diagonali rombis gverdTan adgens kuTxes, romlis sinusia 1/4. ipoveT meorediagonali, Tu rombis gverdia 5sm.

5. aBC samkuTxedSi aB=10sm da BC=8sm. gavlebulia BK biseqtrisa da aM mediana, romelTa kveTis wertilia O. ipoveT: SaOB : SaMC.


1. amoxseniT grafikulad Semdegi gantolebebi:a) x2 = 3x + 2; b) x2= 2x – 4.

2. ipoveT a koeficienti Tu ax2 – 4x+2=0 gantolebas aqvs 2-is toli fesvi.

3. motociklistma M punqtidan N punqtamde manZili 5sT-Si gaiara. ukan dabrunebisasman pirveli 36km imave siCqariT gaiara, xolo gzis darCenili manZili 3km/sT-iT meti siCqariT. ra siCqariT midioda motociklisti, Tu ukan dabrunebaze man 15wT-iT nak-lebi dro daxarja, vidre M-dan N-Si Casvlaze?

4. ipoveT ( ) | |

yx

xx

xx x

8

5100

92

2

22

=-

-+ - +

-- funqciis gansazRvris are.


1. aBC da a1B1C1 samkuTxedebi da sivrcis ori – O da P wertilebi. cnobilia, rom; ;OA OP OA OB OP OB OC OP OC11 1+ + + = + =

daamtkiceT, rom Aa1B1C1 samkuTxedis gverdebi Sesabamisad toli da paraleluria aBCsamkuTxedis gverdebis.

2. vTqvaT a m n= + da –b m n= . gamosaxeT m da n-is saSualebiT veqtorebi:

a) 3 5a b+ ; b) –4a b21 ; g) 20 1

a b10

+ .

3. a da b aranulovani da arakolinearuli veqtorebia, gamoTvaleT k da p ricxvebi, Tua) ka + (2p+ 1)b = 5a – 3b ; b) (k+1)a +(p2 – 2)b = 2a +7b .

4. daamtkiceT, rom OA OC OB OD+ = + , sadac O sivrcis nebismieri wertilia, xoloaBCD oTxkuTxedi paralelogramia.

=


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