1 pdf-istvis 1-10.pmd - Ganatleba

T. beitriSvili 

g. berikelaSvili 

d. kapanaZe 

q. manjgalaZe 

maTematika 10 

maswavleblis wigni 

saqarTvelos sapatriarqo 

da fondi `bavSvebi Cveni momavalia~ 

gilocavT axali saswavlo wlis dawyebas! 

maTematika X maswavleblis wigni 

1

sarCevi 

1 Sesavali .................................................................................................................................................................. 4 

2 „maTematika – X klasis“ mokle Sinaarsi ......................................................................................................... 4 

3 erovnuli saswavlo gegma ...................................................................................................................................... 5 

4 Sinaarsisa da miznebis ruka ................................................................................................................................ 16 

5 swavlebis interaqtiuli meTodebi .................................................................................................................. 18 

6 gakveTilis dagegmva (zogadi principebi) ...................................................................................................... 20 

7 saSinao davalebis prezentacia ......................................................................................................................... 21 

8 gakveTilebis scenarebi ....................................................................................................................................... 22 

9 aqtivoba: òqros kveTa _ harmoniuli proporcia~ ..................................................................................... 31 

10 kenigsbergis Svidi xidi ...................................................................................................................................... 33 

11 aqtivoba “saSualo xelfasi”............................................................................................................................. 34 

12 moswavleTa trimestruli da wliuri Sefasebebi ...................................................................................... 36 

13 sakontrolo samuSaoebi.................................................................................................................................... 38 

14 zogierTi sakontrolo samuSaos amoxsnis nimuSi .................................................................................... 50 

15 zogierTi savarjiSos amoxsna ......................................................................................................................... 53 


3

4 maTematika X maswavleblis wigni 

Sesavali 

daaxloebiT 30 wlis win maTematika xuTi damoukidebeli sagnis saxiT iswavleboda: „ariTmetika“, 

„algebra“, planimetria“, „stereometria“ da „trigonometria“. saatestacio da umaRles 

saswavleblebSi misaRebi gamocdebic, ZiriTadad, ori maTematikuri nawilisgan Sedgeboda. 

didi xania, rac umaRles saswavleblebSi misaReb gamocdas „maTematika“ hqvia. es bunebrivicaa, 

radgan am erTiani sagnis xelovnuri diferencireba xSirad problemebsac qmnida, magaliTad, 

erTi da igive cneba algebraSi da geometriaSi erTmaneTisgan gansxvavebuli SinaarsiT iswavleboda. 

Tu 2005 wlis ivlisSi Catarebul pirvel erovnul gamocdebsac gavixsenebT (gadavxedavT 

zogadi unarebis maTematikur nawilsa da maTematikis sagamocdo sakiTxebs), advilad davrwmundebiT, 

rom moswavlis maTematikuri ganaTlebisadmi moTxovnebi radikalurad Seicvala. 

axali saxelmwifo saganmanaTleblo standartebis Sesabamisad Sedgenili saxelmZRvanelo 

„maTematika – X klasi“ algebrisa da geometriis ubralo integracia ar aris. masSi gaerTianda 

maTematikis oTxi ZiriTadi mimarTuleba: „ricxvebi da maTze moqmedebebi“, „kanonzomierebani 

da algebra“, „geometria da sivrcis aRqma“, „statistika da albaToba“. magram aqac am oTxi mimarTulebis 

ubralo gaerTianebaze ar aris saubari: wina planze wamoweulia is Sedegebi da maTi 

indikatorebi, romelTa realizaciasac unda mieZRvnas swavleba-swavlis procesi. icvleba am 

procesSi monawileTa aqtivobebi: TiToeuli moswavle maqsimalurad erTveba muSaobaSi, xolo 

maswavlebeli am muSaobis organizatoris, wamyvanis (zogjer msajis) funqciiT Semoifargleba. 

qvemoT Tqven gaecnobiT: 

• Cven mier Sedgenili me-10 klasis maTematikis axali saxelmZRvanelos mokle Sinaarss; 

• maTematikis me-10 klasis kursis Sesabamis Sinaarsisa da miznebis rukas; 

• zogierTi gakveTilis scenars; 

• sakontrolo samuSaoebis variantebsa da moswavleTa namuSevrebis gasworeba-Sefasebis kriteriumebs. 

SeniSvna: saswavlo-Tematur gegmaSi yovel sakontrolo samuSaoze 2 saaTia gamoyofili. es, 

ZiriTadad, imis surviliTaa nakarnaxevi, rom SeZlebisdagvarad orsaaTiani sakontrolo samu- 

Saoebic CavataroT, riTac mowavleebs SedarebiT grZelvadiani muSaobis unari gamoumuSavdebaT 

(erovnul gamocdebze maT xom 2-3 astronomiuli saaTis ganmavlobaSi uwyveti muSaoba mouxdebaT). 

es SesaZlebelia, saswavlo wlis dasawyisSive mogvardes 5 saaTiani kvireuli datviTvis 3- 

4 dReze ganawilebiT, Tumca arc maswavlebelTaTvis kargad cnobili „sxva gaveTilis sesxeba“ 

ikrZaleba. maTematikis maswavleblebma, survilis mixedviT, me-2 saaTi SeiZleba gamoiyenon sakontrolo 

samuSaos Sejameba-analizisaTvis (romelsac zogjer erTi gakveTilic ki ar yofnis). 

„maTematika – X klasis“ mokle Sinaarsi 

moswavleze orientirebuli swavlebis principebi, romelic saganmanaTleblo standartebSia 

Camoyalibebuli, upirveles yovlisa, saxelmZRvanelom da Sesabamisma meTodikam unda ganaxorcielos. 

am moTxovnidan gamomdinare, pirvel rigSi moswavlis asakobriv Taviseburebebs, interesebsa 

da SesaZleblobebs viTvaliswinebT: 

– saxelmZRvanelos Svidi Tavidan TiToeuli iwyeba sam svetad warmodgenili rubrikebiT: „gavixsenebT“, 

„viswavliT“, „SevZlebT“, rac moswavles warmodgenas Seuqmnis momaval samuSaoebze; 

– yovel paragrafSi gverdis marcxena svetSi warmodgenilia wina klasebSi Seswavlili is 

ZiriTadi sakiTxebi (gansazRvrebebi, Tvisebebi, formulebi), romelTa aqve gaxseneba axali masalis 

kargad gaazrebis winapirobaa. amiT moswavles „ukan dasaxevi gza“ aRar rCeba da maSinac ki, 

roca raime mizeziT gakveTilebs gaacdens, sruli SesaZlebloba eqmneba, damoukideblad Caataros 

gamotovebuli samuSaoebi; 

– miuxedavad imisa, rom ZiriTadi cnebebi, gansazRvrebebi, Tvisebebi, formulebi Tu daskvnebi 

gansakuTrebuladaa gamoyofili, teqsti moswavles imasac mianiSnebs, rom dazepirebas 

moeridos; 

– kiTxvebisa da savarjiSoebis sistema imgvaradaa diferencirebuli, rom specialuri aRniSvnebiT 

sam svetad warmodgenili masalebi 3, 4 da 5 quliani horizontuli aRmavlobis garda, 

sirTulis vertikalur aRmavlobasac (zemodan qvemoT) iTvaliswinebs. amgvari sistema aradiskriminaciulia, 

radgan maswavlebeli mxolod davalebis nomris miTiTebiT Semoifargleba, moswavle 

ki Tavad airCevs davalebas Tavisi SesaZleblobisa Tu interesis Sesabamisad; 

– „kiTxvebi diskusiisaTvis“, „maTematikuri Sejibri“, „davalebebi jgufuri mecadineobisaTvis“, 

„damoukidebeli saklaso samuSao“, „es Tqvenc SegiZliaT“ da sxva rubrikebi imaze miu-

TiTebs, rom maswavlebeli xSirad damkvirveblis, msajis an wamyvanis funqciiT Semoifargleba 

– danarCens klasi asrulebs. 

saxelmZRvanelo Sedgeba 38 paragafisagan, romelic 7 TavSia gadanawilebuli miaxloebiTi 

Tematikis mixedviT. aq umTavres principad saganmanaTleblo standartSi warmodgenili mimar- 

Tulebebisa da Sedegebis logikuri Tanmimdevrulobis dacva aris aRiarebuli. pirvelive paragrafebidan 

algebruli da geometriuli sakiTxebi, simravleTa Teoriisa Tu statistikis 

elementebi erTmaneTis TanmimdevrobiTa da logikuri urTierTkavSiriT gadmoicema. 

qvemoT warmodgenilia erovnuli saswavlo gegma Sinaarsisa da miznebis ruka, romelic mokled 

dagvanaxvebs saxelmZRvanelos erovnul saswavlo gegmasTan Sesabamisobas, Sinaarss, Sedegebisa da 

indikatorebis Tanmimdevrul ganawilebas drosa da periodebSi. SevniSnavT, rom TiToeul Temaze 

miTiTebulia saaTebis minimaluri raodenoba, romelic maswavlebels SeuZlia Secvalos. 

erovnuli saswavlo gegma 

zogadsaganmanaTlebo skolaSi maTematikis swavlebis koncefcia 

saskolo maTematikuri ganaTlebis roli da miznebi 

Tanamedrove epoqaSi maTematika farTod gamoiyeneba adamianis saqmianobis yvela sferoSi: 

mecnierebasa da teqnologiebSi, medicinaSi, ekonomikaSi, garemos dacvasa da aRdgena-keTilmowyobaSi, 

socialur gadawyvetilebaTa miRebaSi. aRsaniSnavia agreTve maTematikis gansakuTrebuli 

roli kacobriobis ganviTarebaSi da Tanamedrove civilizaciis CamoyalibebaSi. sainformacio 

da gamoTvliTi teqnologiebis ganviTareba, sivrce-drois struqturis ukeT gaazreba, 

bunebaSi arsebuli mravali kanonzomierebis aRmoCena da aRwera naTlad warmoaCens maTematikis 

samecniero da kulturul Rirebulebas. rac gansakuTrebiT mniSvnelovania, maTematika 

xels uwyobs adamianis gonebrivi SesaZleblobebis ganviTarebas. igi iZleva efeqtiani, lakoniuri 

da araorazrovani komunikaciis saSualebas. maTematikis gamoyenebiT SesaZlebelia rTuli 

situaciis TvalsaCino warmoCena, movlenebis axsna da maTi Sedegebis ganWvreta. maTematikaSi 

Seqmnili abstraqtuli sistemebi da Teoriuli modelebi gamoiyeneba kanonzomierebebis Sesaswavlad, 

situaciis gasaanalizeblad da problemebis gadasaWrelad. 

problemis gadaWrisas aucilebelia mis arsSi wvdoma, adekvaturi maTematikuri aparatis 

SerCeva, xolo aseTis ar arsebobis SemTxvevaSi - misi SemuSaveba; Sesaswavli procesis gaazrebuli 

modelis Seqmna, miRebuli modelis saSualebiT saWiro daskvnebis gakeTeba da Semdeg maTi gamoyenebiTi 

interpretacia. praqtikuli Tu samecniero problemebi, Tavis mxriv maTematikas amaragebs 

mniSvnelovani da saintereso amocanebiT. Sesabamisad, swavlebisas ZiriTadi yuradReba 

unda mieqces maTematikuri meTodebis gamoyenebas garemomcveli samyaros Semecnebisas, socialur-ekonomikuri 

Tu teqnikuri procesebis marTvisas, sayofacxovrebo Tu mecnieruli problemebis 

gadaWrisas da maTematikuri codnis, rogorc logikurad gamarTuli sistemis Camoyalibebas 

da gadacemas. garda amisa, maTematikis swavlebisas, ZiriTadi fokusis gadatana (rogorc 

praqtikuli aseve mecnieruli xasiaTis) problemebis gadaWraze, aZlierebs moswavleTa enTuziazms 

da aRZravs interess maTematikisadmi. 

aqedan gamomdinare, zogadsaganmanaTleblo skolaSi maTematikis swavlebis miznebia moswavleTaTvis: 

• azrovnebis unaris ganviTareba; 

• deduqciuri da induqciuri msjelobis, SexedulebaTa dasabuTebis, movlenebisa da faqtebis 

analizis unaris ganviTareba; 

• maTematikis rogorc samyaros aRwerisa da mecnierebis universaluri enis aTviseba; 

• maTematikis rogorc zogadsakacobrio kulturis Semadgeneli nawilis gacnobiereba; 

• swavlis Semdgomi etapisaTvis an profesiuli saqmianobisaTvis momzadeba. 

• cxovrebiseuli amocanebis gadasawyvetad saWiro codnis gadacema da am codnis gamoyenebis 

unaris ganviTareba. 

ZiriTadi unar-Cvevebi, romelTa gamomuSavebasac xels uwyobs maTematikis 

saskolo kursi 

maTematikis codna niSnavs maTematikuri cnebebisa da procedurebis flobas, maTi gamoyenebis 

unars realuri problemebis gadaWrisas; agreTve komunikaciis im saSualebebis flobas, 

romlebic saWiroa informaciis misaRebad da gadasacemad maTematikuri enisa da saSualebebis 

gamoyenebiT. 


5

is ZiriTadi unar-Cvevebi, romelTa Camoyalibebasac emsaxureba problemebis gadaWraze orientirebuli 

maTematikuri ganaTleba, aseTia: 


msjeloba-dasabuTeba 

• varaudis gamoTqma da misi kvleva kerZo magaliTebze 

• sawyisi monacemebis gadarCeva da organizeba (maT Soris aqsiomebis, ukve cnobili faqtebis), 

arsebiTi Tvisebebisa da monacemebsi gamoyofa 

• damtkicebis, dasabuTebis meTodis SerCeva (mag. sawinaaRmdegos daSvebis meTodis gamoyeneba, 

evristikuli meTodis gamoyeneba dasabuTebisas) 

• sxvadasxva tipis gamonaTqvamis adekvaturi gamoyeneba; magaliTad: pirobiTi (“Tu ... maSin”) da 

raodenobrivi xasiaTis, daSvebis, gansazRvrebis, Teoremis, hipoTezis, SemTxvevaTa CamonaTvalis 

• arCeuli strategiis vargisianobisa da misi gamoyenebis sazRvrebis ganxilva 

• msjelobis xazis ganviTareba, alternatiuri gzis moZebna, miRebuli gadawyvetilebis sisworisa 

da efeqtianobis dasabuTeba; ganzogadoebiT an deduqciT miRebuli daskvnebis axsna da 

dasabuTeba 

• Teoremebis-debulebebis daskvnis analizi erTi an ramdenime pirobis, SezRudvis SesutebamoxniT 

• gamonaklisi SemTxvevebis aRniSvna da maTi ganzogadoebis aramarTebulobis dasabuTeba kontrmagaliTis 

moZebniT 

komunikacia 

• terminologiis, maTematikuri aRniSvnebisa da simboloebis koreqtuli gamoyeneba 

• informaciis warmodgenis xerxebisa da meTodebis floba, gamoyeneba; sxvadasxva gziT warmodgenili 

informaciis interpretacia, masze msjeloba, erTmaneTTan dakavSireba 

• sxvisi naazrevis gageba da gaanalizeba 

• informaciis miRebisa da gadacemis Sesaferisi saSualebebis SerCeva auditoriisa da sakiTxis 

gaTvaliswinebiT 

• informaciis gadacemisas sakiTxis arsis (mag. obieqtis arsebiTi Tvisebebis) warmoCena 

gamoyeneba, modelireba 

• figurebis da obieqtebis zomebis, agreTve maT Soris manZilebis, masis, temperaturis da drois 

gasazomad gzebisa da meTodebis povna da gamoyeneba; procesis an realuri viTarebis modelirebisaTvis 

saWiro monacemebis SerCeva da mopoveba 

• Cveul garemoSi (yoveldRiur cxovrebaSi) maTematikuri obieqtebisa da procesebis SemCneva 

da maTi Tvisebebis gamoyeneba modelis agebisas, praqtikuli (yofiTi) amocanebis gadaWrisas 

• mocemuli modelis elementebis interpretireba, im realobis konteqstSi, romelsac igi aRwers 

da piriqiT – realuri viTarebis dakvirvebis Sedegad miRebuli monacemebis interpretireba 

Sesabamisi modelis enaze 

• mocemuli modelis gaanalizeba da Sefaseba, kerZod, misi moqmedebis arealisa da modelis 

adekvaturobis dadgena; SesaZlo alternativebis ganxilva da Sedareba 

problemebis gadaWra 

• amocanis Sinaarsis aRqma, amocanis monacemebisa da saZiebeli sidideebis gaazreba-gamijvna 

• problemis gansazRvra da misi Camoyalibeba, maT Soris arastandartul viTarebaSi (mag. rodesac 

problemis gadasaWrelad saWiro maTematikuri procedura calsaxad araa gansazRvruli) 

• kopleqsuri (rTuli) problemebis safexurebad, martiv amocanebad dayofa da etapobrivad 

gadaWra (amoxsna), maT Soris standartuli midgomebisa da procedurebis gamoyenebiT 

• problemis gadasaWrelad saWiro strategiebisa da resursebis SerCeva, maTi gamoyeneba da 

efeqtianobis monitoringi 

• ukve cnobili faqtebisa da strategiebis SerCeva da erTmaTeTTan dakavSireba maRali sirTulis 

problemebis gadasaWrelad

• miRebuli Sedegis kritikuli Sefaseba konteqstis gaTvaliswinebiT da zRvruli SemTxvevebis 

kvleva 

• problemis gadaWrisas adekvaturi damxmare teqnikuri saSualebebisa da teqnologiebis Ser- 

Ceva da maTi gamoyeneba 

damokidebuleba 

• TanamSromloba jgufuri samuSaoebis Sesrulebisas; koreqtuloba maswavlebelTan da megobrebTan 

mimarTebaSi 

• samuSaos organizebisa da dagegmvis xerxebisa da meTodebis floba 

• maTematikis adgilisa da mniSvnelobis Sefaseba sxvadasxva disciplinebSi, biznesSi, xelovnebaSi 

da adamianis moRvaweobis sxvadasxva sferoebSi 

• informaciuli teqnologiebis gamoyenebisas eTikur/socialuri xasiaTis problemebis gacnobiereba 

da eTikuri normebis dacva. 

maTematikis erovnuli saswavlo gegmis struqtura 

CamoTvlili unar-Cvevebis Camoyalibeba da ganviTareba SesaZlebelia mxolod Sesabamisi Sinaarsis 

(cnebebis, debulebebisa da procedurebis) gamoyenebiT. 

erovnuli saswavlo gegma maTematikaSi pirobiTad dayofilia oTx ZiriTad mimarTulebad: 

ricxvebi da moqmedebebi; geometria da sivrcis aRqma; monacemTa analizi, statistika da albaToba; 

kanonzomierebebi da algebra. 

es mimarTulebebi mWidro urTierTkavSirSia da moicavs im codnas da unar-Cvevebs, romelsac 

moswavle unda daeuflos zogadsaganmanaTleblo skolaSi swavlis ganmavlobaSi. saswavlo gegmis 

dayofa mimarTulebebad ar niSnavs kursis analogiur dayofas, igi mxolod warmoaCens Sesaswavli 

masalis speqtrs da saSualebas iZleva mieTiTos, Tu raze unda gamaxvildes meti yurad- 

Reba swavlebis ama Tu im safexurze. 

1. ricxvebi da moqmedebebi: 

• ricxvebi, maTi gamoyenebebi da ricxvis warmodgenis saSualebebi 

• moqmedebebi ricxvebze da ricxviTi Tanafardobebi 

• raodenobaTa Sefaseba da miaxloeba 

• sidideebi, zomis erTeulebi da ricxvebis sxva gamoyenebebi 

2. geometria da sivrcis aRqma: 

• geometriuli obieqtebi: maTi Tvisebebi, urTierTmimarTeba da konstruireba 

• zoma da gazomvis saSualebebi 

• gardaqmnebi da figuraTa simetriuloba 

• koordinatebi da maTi gamoyeneba geometriaSi 

3. monacemTa analizi, albaToba da statistika: 

• monacemTa wyaroebi da monacemTa mopovebis saSualebebi 

• monacemTa mowesrigebis xerxebi da monacemTa warmodgenis saSualebebi 

• monacemTa Semajamebeli ricxviTi maxasiaTeblebi 

• albaTuri modelebi 

• SerCeviTi meTodi da SerCevis ricxviTi maxasiaTeblebi 

4. kanonzomierebebi da algebra: 

• simravleebi, asaxvebi, funqciebi da maTi gamoyeneba 

• diskretuli maTematikis elementebi da maTi gamoyeneba 

• algoriTmebi da rekursia 

• algebruli operaciebi da maTi Tvisebebi 

mimarTulebebis miznebi swavlebis safexurebis mixedviT 

saswavlo kursi zogadsaganmanaTleblo skolaSi dayofilia sam safexurad: dawyebiTi skola 

(I – VI klasebi), sabazo skola (VII – IX klasebi) da saSualo skola (X – XII klasebi). maTematikis 

saswavlo kursis agebis principi unda iTvaliswinebdes am dayofas da TiToeul safexurze maTematikis 

swavlebas unda hqondes mkafiod Camoyalibebuli miznebi. 


7

icxvebi da moqmedebebi 

am mimarTulebis ZiriTadi miznebia `ricxvis SegrZnebis” ganviTareba, Tvlis principebis 

aTviseba, ariTmetikuli moqmedebebisa da maTi Tvisebebis Seswavla, gamoTvlis xerxebis aTviseba 

da Sedegebis Sefaseba; Caweris poziciuri sistemebis Seswavla, maTi urTierTSedareba da gamoyeneba 

ariTmetikuli moqmedebebis Sesrulebisas da praqtikuli amocanebis gadaWrisas; ricxviTi 

sistemebis Seswavla. 

dawyebiTi skola. am safexurze unda moxdes ricxvebTan dakavSirebuli problemebis gadaWrisas 

ariTmetikuli moqmedebebis da maTi adekvaturad gamoyenebis unaris Camoyalibeba; ariTmetikuli 

moqmedebebis Tvisebebisa da maT Soris kavSirebis danaxva; ariTmetikuli moqmedebebis 

Sedegisa da ricxviTi gamosaxulebis mniSvnelobis Sefasebis unaris ganviTareba. 

garda amisa, moswavles unda Camouyalibdes aTobiTi poziciuri sistemis srulyofili gageba 

da mravalniSna ricxvebze moqmedebebis Sesrulebisas misi gamoyenebis unari; wiladis sxvadasxva 

aspeqtis (rogorc mTelis nawili, erTobliobis nawili, pozicia ricxviT RerZze da gayofis 

Sedegi) gageba. 

sabazo skola. am safexurze moswavlem unda gaiRrmavos Tavisi codna mTel ricxvebTan, wiladebTan, 

aTwiladebTan da procentebTan dakavSirebiT ise, rom safexuris dasrulebis Semdeg 

iyenebdes wiladebis ekvivalentobas, aTwiladebs, proporcias da procentebs amocanebis amoxsnisas 

da realur viTarebaSi. ricxvis cnebis gageba unda gafarTovdes racionalur ricxvebamde. 

mas unda SeeZlos ricxviT RerZze racionaluri ricxvis miaxloebiTi poziciis miTiTeba da 

unda Seeqmnas sawyisi warmodgenebi iracionalur ricxvebze. 

saSualo skola. ricxvebze ariTmetikuli moqmedebebis Sesrulebis unari da maTi Tvisebebis 

codna/gamoyeneba unda gaxdes algebruli struqturebisa da kanonzomierebebis ukeT gaazrebis 

safuZveli. am safexurze, moswavles unda SeeZlos rogorc ricxviTi sistemis, aseve ariTmetikuli 

operaciis cnebis gafarToeba, mag. veqtorebsa da matricebze. garda amisa, unda moxdes 

mTel ricxvTa sistemis ufro Rrmad Seswavla ricxvTa Teoriis elementebis gamoyenebiT. 

kanonzomierebebi da algebra 

am mimarTulebis ZiriTadi mizania moswavles Camouyalibdes kanonzomierebebis, algebruli 

mimarTebebisa da funqciuri damokidebulebebis amocnobis; agreTve, maTi saSualebiT movlenebis 

modelirebisa da problemebis gadaWris unari. 

dawyebiTi skola. am safexurze mimarTulebis mizania martivi kanonzomierebebisa da sidideebs 

Soris damokidebulebis amocnobis unaris ganviTareba, ariTmetikuli operaciebis Tvisebebis 

da asoiTi aRniSvnebis gamoyenebis Seswavla. 

sabazo skola. am safexurze mimarTulebis mizania sidideebs Soris damokidebulebebTan dakavSirebuli 

cnebebisa da procedurebis Seswavla, agreTve maTi gamosaxvis sxvadasxva xerxis 

erTmaneTTan dakavSirebisa da Sedarebis unaris ganviTareba; problemis gadaWrisas asoiTi gamosaxulebis 

gamoyenebis, maT Soris gantolebis Sedgenisa da amoxsnis unaris ganviTareba; sawyisi 

warmodgenebis Seqmna simravlur cnebebsa da operaciebze. 

saSualo skola. am safexuris mizania funqciaTa ojaxebis, maTi Sedarebisa kvlevis meTodebis 

Seswavla; sxvadasxva konteqstSi arsebuli damokidebulebis gamosaxvisas iteraciuli da rekurentuli 

formebis gamoyenebis unaris ganviTareba; struqturis aRwerisa da Seswavlisas 

diskretuli maTematikis aparatis gamoyenebis unaris ganviTareba. 


geometria da sivrcis aRqma 

am mimarTulebis ZiriTadi mizania geometriuli obieqtebisa da maTi Tvisebebis, gazomvebis, 

geometriuli gardaqmnebisa da geometriaSi algebruli meTodebis gamoyenebis Seswavla. 

dawyebiTi skola. am safexurze mimarTulebis mizania geometriuli obieqtebis urTierTganlagebis 

aRwerisa da demonstrirebis, geometriul obieqtTa komponentebis amocnobisa da 

maTi urTierTmimarTebis aRweris, atributebis mixedviT figuraTa dajgufebis, sityvieri aRwerilobis 

mixedviT figuris amocnobisa da misi modelis Seqmnis unaris ganviTareba. 

sabazo skola. am safexurze mimarTulebis mizania geometriul obieqtTa Seswavlisas, geometriul 

obieqtTa Soris mimarTebebis dadgenisas da geometriul obieqtTa klasifikaciisas, gazomvis, 

Sedarebisa da geometriuli gardaqmnebis gamoyenebis unaris ganviTareba. garemoSi orientirebisas 

koordinatebis gamoyenebis da arapirdapiri gziT obieqtTa zomebis dadgenis Seswavla; 

induqciuri/deduqciuri msjelobisa da varaudis gamoTqma-Semowmebis unaris ganviTareba.

saSualo skola. aRniSnul safexurze unda moxdes deduqciuri/induqciuri msjelobisa da 

geometriuli kvlevis Sedegebis ganzogadebis unaris ganmtkiceba. koordinatebis, trigonometriis 

da gardaqmnebis gamoyeneba praqtikuli geometriuli problemebis gadasaWrelad da 

am xerxebidan yvelaze efeqtianis SerCevis unaris ganviTareba. 

monacemTa analizi, albaToba da statistika 

zogadsaganmanaTleblo skolaSi statistikuri cnebebisa da aparatis Semotanis mizania moswavleTa 

intuiciuri warmodgenebis mowesrigeba, struqturebad Camoyalibeba da maTi albaTurstatistikuri 

intuiciisa da azrovnebis ganviTareba. 

dawyebiTi skola. am safexurze mimarTulebis swavlebis mizania moswavleebi gaecnon aRweriTi 

statistikis elementebs – Tvisebriv da diskretul raodenobriv monacemTa Segrovebis, 

mowesrigebis, warmodgenisa da interpretaciis saSualebebs. 

sabazo skola. am safexurze mimarTulebis swavlebis mizania moswavleebi daeuflon aRweriTi 

statistikis ZiriTad cnebebsa da meTodebs, raTa maTi saSualebiT gaerkvnen monacemTa TaviseburebebSi 

da SeZlon varaudis gamoTqma monacemebze dayrdnobiT. garda amisa, swavlebis mizania 

moswavleebi gaecnon albaTobis Teoriis sawyisebs da gaacnobieron gansxvaveba deterministul 

da SemTxveviTobis Semcvel viTarebebs Soris. 

saSualo skola. am safexurze mimarTulebis swavlebis mizania moswavleebs SeeqmnaT sistematizebuli 

warmodgenebi albaTobis Teoriisa da statistikis Sesaxeb, raTa gaakeTon an/da Seafason 

daskvnebi gaurkvevlobis Semcvel viTarebaSi, amoicnon SemTxveviTobis roli ama Tu im 

wamowyebaSi da moaxdinon misi kvantifikacia gadawyvetilebis misaRebad. 

maTematikis saskolo kursis organizacia 

zogadsaganmanaTleblo skolaSi swavlis savaldebuli kursi moicavs pirvel cxra klass, xolo 

mecxre klasis Semdeg moswavleTa nawilma SeiZleba aRar gaagrZelos swavla zogadsaganmanaTleblo 

skolaSi. zogadsaganmanaTleblo skolis yovel safexurze maTematika Seiswavleba rogorc 

savaldebulo sagani. meaTe klasSi moswavleebi miiReben iseT ganaTlebas romelic maT xels 

Seuwyobs ukeT gaerkvnen TavianT midrekilebebsa da interesebSi, ris Sedegadac XI – XII klasebSi 

maT unda gaakeTon arCevani maTematikis gaRrmavebul da zogad kurss Soris (ix. diagrama). 

XI – XII klasebi 

savaldebulo sagani “maTematika” 

maTematikis gaRrmavebuli kursi 

I – VI klasebi 


maTematikis erTiani kursi 

VII – IX klasebi (sabazo skola) 



X klasi (saSualo skola) 



→ → 

dawyebiTi skola 

sabazo skola 

saSualo skola 

XI – XII klasebi 


maTematikis zogadi kursi 

→ 

→ 

→ 

→ 

→ 


9

erovnuli saswavlo gegmis daniSnuleba da misi struqtura 

erovnuli saswavlo gegmis daniSnulebaa daexmaros saskolo ganaTlebis procesis 

monawileebs (maswavlebelebs, moswavleTa mSoblebs, saxelmZRvaneloebis avtorebs da 

ganaTlebis procesis marTvis specialistebs), am procesis dagegmvasa da warmarTvaSi. 

erovnul saswavlo gegmaSi aRwerilia is savaldebulo moTxovnebi, romelsac unda 

akmayofilebdes yoveli moswavle TiToeuli klasis dasrulebis Semdeg. es moTxovnebi, 

TiToeuli mimarTulebisaTvis, Camoyalibebulia Sedegebisa da maTi indikatorebis enaze. 

Sedegi aris debuleba imis Sesaxeb Tu ra unda SeeZlos moswavles 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. SedegTan dakavSirebuli indikatorebis erToblioba faravs Sedegs da amave 

dros TiToeuli maTgani warmoaCens Sedegs romelime kuTxiT. is Tu ramdenadaa esa Tu is Sedegi 

miRweuli, ganisazRvreba Sesabamisi indikatorebis im raodenobiT romelsac moswavle 

akmayofilebs. Sedegi iTvleba miRweulad Tu moswavle akmayofilebs am Sedegis Sesabamisi 

indikatorebis umetesobas. 

TiToeuli safexuris Sesabamisi Sedegebisa da maTi indikatorebis erTobliobas Tan erTvis 

Sinaarsi, romelic aris saswavlo masalis im sakiTxTa CamonaTvali romlis gamoyenebiTac 

SesaZlebelia Camoyalibebuli Sedegebis miRweva swavlebis mocemul safexurze. dokumentSi 

warmodgenili Sinaarsi mxolod sarekomendacio xasiaTisaa da aqedan gamomdinare igi ar 

ganixileba, rogorc am safexurisaTvis gankuTvnili savaldebulo saswavlo masala. igulisxmeba 

rom erTi da igive Sedegis miRweva (e.i. im unarebis ganviTareba, romlebis am SedegSia aRwerili) 

SesaZlebelia gansxvavebul saswavlo masalaze dayrdnobiTac. 

wlis bolos misaRwevi Sedegebi: 

maT. X 

ricxvebi da 

moqmedebebi 

1. ganasxvavebs namdvil 

ricxvTa qvesistemebs 

2. akavSirebs sxvadasxva 

poziciur sistemebs 

/ namdvil 

ricxvTa qvesimravleebs 

erTmaneTTan 

3. asrulebs namdvil 

ricxvebze moqmedebebs 

da afasebs maT 

Sedegs 

4. iyenebs msjelobadasabuTebissxvadasxva 

xerxs 

5. 

wyvets praqtikuli 

saqmianobidan momdinare 

problemebs 


kanonzomierebebi 

da algebra 

6. ikvlevs funqciis 

Tvisebebs da iyenebs 

maT sidideebs Sorisdamokidebulebis 

Sesaswavlad 

7. iyenebs gantolebaTa 

da utolobaTa 

sistemebs modelirebissaSualebiT 

problemis 

gadaWrisas 

8. iyenebs diskretuli 

maTematikis 

e l e m e n t e b s 

problemis modelirebisa 

da analizisTvis. 

mimarTuleba: 

geometria da 

sivrcis aRqma 

9. flobs da iyenebs 

geometriul figuraTa 

warmodgenisa 

da debulebaTa 

formulirebis xerxebs 

10. poulobs obieqt- 

Ta zomebsa da 

obieqtTa Soris 

manZilebs 

11. asabuTebs geometriuli 

debulebebis 

marTebulobas 

12. ikvlevs figuraTa 

geometriul gardaqmnebs 

sibrtyeze 

da iyenebs maT geometriuliproblemebis 

gadaWrisas 

monacemTa analizi, 

albaToba da 

statistika 

13. moipovebs dasmuli 

amocanis amosaxsnelad 

saWiro 

Tvisebriv da raodenobrivmebsmonace- 

14. awesrigebs da warmoadgens 

Tvisebriv 

da raodenobriv monacemebs 

dasmuli 

amocanis amosaxsnelad 

xelsayreli 

formiT 

15. aRwers SemTxvevi- 

Tobas albaTuri modelebisbiTsaSuale- 

16. iyenebs statistikur 

da albaTur 

cnebebs/Tvalsazrisebs 

yoveldRiur 

viTarebebSi

wlis bolos misaRwevi Sedegebi da maTi indikatorebi 

mimarTuleba: ricxvebi da moqmedebebi 

maT. X1. ganasxvavebs namdvil ricxvTa qvesisistemebs 

Sedegi TvalsaCinoa, Tu moswavle: 

a) ganasxvavebs racionalur da iracionalur ricxvebs, rogorc periodul da araperiodul aTwiladebs; 

axdens iracionaluri ricxvis racionaluri mimdevrobiT miaxloebis demonstrirebas 

modelis gamoyenebiT. 

b) mocemuli sizustiT amrgvalebs namdvil ricxvebs; ganasxvavebs usasrulo perioduli aTwiladis 

SemoklebiT CawerasDdamrgvalebisgan. 

g) asaxelebs mocemul or namdvil ricxvs Soris moTavsebul racionalur ricxvs. (mag. asaxelebs 

0.6(5) –sa da 0.66 –s Soris moTavsebul racionalur ricxvs). 

d) axdens namdvili ricxvis aTobiTi poziciuri sistemiT Caweris interpretacias da/an mis 

demonstrirebas modelze (mag. axdens 1-ze naklebi dadebiTi namdvili ricxvis miaxloebas 

[0,1] monakveTis Tanmimdevruli danawilebiT). 

maT. X.2. akavSirebs sxvadasxva poziciur sistemebs / namdvil ricxvTa qvesimravleebs 

erTmaneTTan 


a) adarebs sxvadasxva poziciur sistemebs erTmaneTs; msjelobs TiToeulis upiratesobaze ricxvebis 

Cawerisas. (mag. aTobiTi poziciuri sistema, romauli, egvipturi – sadac aTis xarisxebisTvis 

Sesabamisi ricxviTi saxelebi/ieroglifebi arsebobda). 

b) moyavs informaciis cifruli kodirebis/teqnologiebis magaliTebi; akavSirebs ricxvis 

sxvadasxva poziciur sistemaSi Caweras erTmaneTTan (mag. orobiT poziciur sistemaSi Caweril 

ricxvs wers aTobiT poziciur sistemaSi). 

g) akavSirebs namdvil ricxvTa qvesimravleebs erTmaneTTan simravleTa Teoriis enis gamoyenebiT 

(qvesimravle, simravleTa TanakveTa, gaerTianeba, sxvaoba, damateba; am mimarTebebis 

venis diagramis gamoyenebiT gamosaxva). 

d) sxvadasxva formiT wers namdvil ricxvebs (mag. periodul aTwilads Cawers wiladis saxiT); 

adarebs da alagebs sxvadasxva formiT Caweril namdvil ricxvebs (aTwiladi, wiladi; erTi da 

igive mTelis nawili da procenti; ricxvis standartuli forma, aTobiTi da orobiTi poziciuri 

sistema; ricxvis xarisxi da iracionaluri gamosaxuleba). 

maT. X.3. asrulebs namdvil ricxvebze moqmedebebs da afasebs maT Sedegs 


a) amartivebs namdvil ricxvebze moqmedebebis (agreTve modulis) Semcvel gamosaxulebas moqmedebaTa 

Tvisebebis, Tanmimdevrobisa da maT Soris kavSiris gamoyenebiT. 

b) axdens wilad-maCvenebliani xarisxis cnebis interpretacias da misi Tvisebebis demonstrirebas; 

adarebs da alagebs erTi da igive fuZis mqone xarisxebs. 

g) amocanis konteqstis gaTvaliswinebiT irCevs ra ufro mizanSewonilia moqmedebaTa Sedegis 

Sefaseba, Tu misi zusti mniSvnelobis povna; iyenebs Sefasebas namdvili ricxvebze Sesrulebuli 

gamoTvlebis Sedegis adeqvaturobis Sesamowmeblad. 

d) erTi ariTmetikuli moqmedebis Semcvel gamosaxulebaSi amrgvalebs wevrebs (namdvili ricxvebs) 

da poulobs moqmedebebaTa Sedegis miaxloebiT mniSvnelobas; msjelobs damrgvalebiT 

gamowveul gansxvavebaze. 

e) moyavs fardobiTi azriT ~Zalian didi” da “Zalian mcire~ sidideTa magaliTebi (mag. sinaTlis 

weli, eleqtronis masa); axdens ~usasrulod mcires/didis~ cnebis interpretirebas zRvruli 

procesebis saSualebiT. 

maT. X.4. iyenebs msjeloba-dasabuTebis sxvadasxva xerxs 


a) asabuTebs martiv debulebas ricxvebis Tvisebebis an ricxviTi kanonzomierebebis Sesaxeb; 

Sesabamis SemTxvevaSi axdens hipoTezis uaryofas kontrmagaliTiT (mag. WeSmaritia Tu mcdari: 

nebismieri ori kenti ricxvis namravli kentia; nebismieri luwi da kenti ricxvebis sxvaoba 

luwia da a.S.). 


11

) msjelobis nimuSebSi amoicnobs deduqcias, ganzogadebas da analogias; iyenebs maT mTel ricxvebs 

Soris damokidebulebebis dasadgenad (mag. romeli cifri dgas ricxvis 23455 erTeulebis TanrigSi?). 

g) amocanebis amoxsnisas iyenebs ricxviT simravleebs Soris damokidebulebis gamosaxvis zogierT 

xerxs (mag. venis diagramebs). 

d) iyenebs “winaaRmdegis daSvebis” meTods ricxvebis Sesaxeb martivi debulebebis damtkicebisas. 

maT. X.5. wyvets praqtikuli saqmianobidan momdinare problemebs 


a) asrulebs gamoTvlebs da adarebs or martivad/rTulad daricxul saprocento ganakveTs, 

sxvadasxvagvar fasdaklebas, dabegvras; msjelobs maT Soris Soris gansxvavebaze. 

b) msjelobs teqnologiebis gamoyenebasTan dakavSirebiT wamoWrili eTikuri/socialuri xasiaTis 

problemebis Sesaxeb (prezentaciis magaliTi: sxvadasxvagvari informacia internetSi; 

sainformacio teqnologiebis/programuli uzrunvelyofis momxmareblis ufleba/movaleobebi, 

momsaxure mxaris ufleba/movaleobebi). 

g) msjelobs informaciis Teoriisa da ricxvTa Teoriis praqtikul mxareze, maT rolze/gavlenaze 

Zvel/Tanamedrove sazogadoebaSi (jgufuri samuSaos nimuSi: teqsturi informaciis kodireba 

/ dekodireba romelime xerxiT; prezentaciis magaliTi: oqros kveTa arqiteqturasa da xelovnebaSi; 

an fibonaCis mimdevroba da bunebrivi procesebis modelireba/simulireba; anbanis 

wanacvlebiT daSifvris magaliTebi istoriidan - iulius keisaris Sifri: 5-asoTi wanacvlebuli 

anbani; mag. meore msoflio omis droindeli germanuli daSifvris manqana ~enigma~). 

d) iyenebs kuTxis zomis erTeulebs Soris kavSirebs wrewirze mobrunebasTan da/an brunvis 

Sedegad gadaadgilebasTan dakavSirebuli amocanebis amoxsnisas (mag. lilvTan dakavSirebuli 

amocanebi). 

SeniSvna: me-2 da me-3 indikatorebidan erTi mainc savaldebuloa. 


Sinaarsi 

1. namdvil ricxvTa qvesimravleebi (racionalur da iracionalur ricxvTa simravleebi) 

• iracionaluri ricxvis racionaluri ricxvebis mimdevrobiT miaxloeba 

2. aTobiTisgan gansxvavebuli ricxviTi sistemebi 

• aTobiTisgan gansxvavebul sistemaSi ricxvebis Caweris praqtikuli magaliTebi (mag. 

orobiT sistemaSi) 

• kavSirebi sxvadasxva poziciur sistemebs Soris (mag. aTobiTi poziciuri sistemaSi 

mocemuli ricxvis warmodgena orobiT sistemaSi da piriqiT). 

3. aTobiT sistemaSi mocemuli ricxvis Cawera standartuli formiT; standartuli formiT 

mocemuli ricxvis Cawera aTobiT poziciur sistemaSi. 

4. sxvadasxva saxiT mocemuli ricxvebis Sedareba/dalageba. 

5. ariTmetikuli moqmedebebi namdvil ricxvebze 

6. namdvili ricxvebis damrgvaleba da ariTmetikuli moqmedebebis Sedegis Sefaseba. 

7. racionalur-maCvenebliani xarisxi da misi Tvisebebi 

8. modularuli (naSTebis) ariTmetikis elementebi (gacnobis wesiT, `bolo cifris ariTmetika~). 

9. zomis erTeulebi: kuTxis radianuli zoma. 

mimarTuleba: kanonzomierebebi da algebra 

maT. X.6. ikvlevs funqciis Tvisebebs da iyenebs maT sidideebs Soris damokidebulebis 

Sesaswavlad 


a) sidideebs Soris damokidebulebis aRmweri funqciisaTvis (maT Soris realur viTarebaSi) 

k 

asaxelebs funqciis tips (wrfivi, modulis Semcveli, kvadratuli, f ( x ) = ) am funqciis gamo- 

x 

saxvis xerxisagan damoukideblad.

) sidideebs Soris damokidebulebis aRmweri funqciisaTvis, maT Soris realur viTarebaSi, 

poulobs funqciis nulebs, funqciis maqsimums/minimums, zrdadoba/klebadobisa da niSanmudmivobis 

Sualedebs; axdens am monacemebis interpretacias realuri viTarebis konteqstSi. 

g) cvlis funqciis parametrebs da axdens am cvlilebis Sedegis interpretirebas im procesis 

konteqstSi, romelic am funqciiT aRiwereba (mag. manZilis droze damokidebulebis aRmwer 

funqciaSi S ( t) 

= v ⋅ t + S ra gavlenas axdens siCqaris cvlileba ganvlil manZilze?). 

0 

d) adarebs or funqcias, romlebic realur process gamosaxavs (poulobs im simravles sadac 

erTi funqcia metia/naklebia meore funqciaze, tolia meore funqciis) da axdens Sedarebis 

Sedegis interpretirebas konteqstTan mimarTebaSi. 

maT. X.7. iyenebs gantolebaTa da utolobaTa sistemebs modelirebis saSualebiT problemis 

gadaWrisas 


a) teqsturi amocanis amosaxsnelad adgens da xsnis orucnobian gantolebaTa sistemas, romelSic 

erTi gantoleba wrfivia, xolo meoris xarisxi ar aRemateba ors; axdens amonaxsnis interpretacias 

amocanis konteqstis gaTvaliswinebiT. 

b) irCevs da iyenebs gantolebaTa/utolobaTa sistemis (ucnobebisa da gantolebebis/utolobebis 

raodenoba ar aRemateba 2-s) amoxsnis xerxs (mag. Casmis, Sekrebis), grafikulad gamosaxavs 

amonaxsns da axdens amonaxsnis simravlur interpretacias. 

g) wrfivi utolobis an/da ori wrfivi utolobis Semcveli sistemis saSualebiT gamosaxavs amocanis 

pirobaSi mocemul SezRudvebs (mag. firmam sareklamo kompaniaze unda daxarjos araumetes 

2000 larisa. maT dagegmili aqvT gamoaqveynon aranakleb 10 sareklamo gancxadebisa. 

dasvenebis dReebSi sareklamo gancxadebis Rirebulebaa 20 lari, xolo kviris danarCen dReebSi 

10 lari.). 

maT. X.8. iyenebs diskretuli maTematikis elementebs problemis modelirebisa da 

analizisTvis. 


a) iyenebs xisebr diagramebs an/da grafebs variantebis dasaTvlelad, gegmis/ganrigis Sesadgenad, 

optimizaciis sasruli amocanebis amosaxsnelad (algoriTmebis gareSe) (mag. or obieqts 

Soris mciresi manZilis moZebna). 

b) realuri procesebis diskretuli modelebiT aRwerisas iyenebs rekursias (mag., mosaxleobis 

raodenobis yovelwliuri mudmivi procentuli zrda); ganavrcobs rekurentuli wesiT 

mocemul mimdevrobas. 

g) adekvaturad iyenebs simravlur terminebs da cnebebs (mag., funqciis gansazRvris are da 

mniSvnelobaTa simravle) da operaciebs sasrul simravleebze (TanakveTa, gaerTianeba, 

sxvaoba, damateba), maT Soris realuri viTarebis modelirebisas an aRwerisas. 

Sinaarsi 

1. wrfivi, modulis Semcveli, kvadratuli da funqciebi. 

2. simravlis cneba; operaciebi sasrul simravleebze: TanakveTa, gaerTianeba, simravlis damateba; 

venis diagramebi. 

3. funqciis gansazRvris are da mniSvnelobaTa simravle. 

4. funqciis zrdadoba/klebadobisa da niSanmudmivobis Sualedebi. 

5. funqciis nulebi da maqsimumebis/minimumebis wertilebi da Sesabamisi mniSvnelobebi. 

6. orucnobian gantolebaTa sistema, romelSic erTi gantoleba wrfivia xolo meoris xarisxi 

ar aRemateba ors. 

7. orucnobian wrfiv utolobaTa sistema. 


13

8. grafebi (aramkacrad: wirebiT SeerTebuli wertilebi sibrtyeze), maTi zogierTi saxeoba 

da Tviseba gacnobis wesiT: orientirebuli/araorientirebuli, ciklebi, grafis ori wveros 

SemaerTebeli gzebi. 

9. ricxviTi mimdevrobis mocemis rekurentuli xerxi. 


mimarTuleba: geometria da sivrcis aRqma 

maT. X.9. flobs da iyenebs geometriul figuraTa warmodgenisa da debulebaTa 

formulirebis xerxebs 


a) aRwers geometriul obieqtebs da maT grafikul gamosaxulebebs Sesabamisi terminologiis 

gamoyenebiT. 

b) iyenebs maTematikur simboloebs geometriuli debulebebisa da faqtebis gadmocemisas; sworad 

iyenebs terminebs: “yvela”, “arcerTi”, “zogierTi”, “yoveli”, “nebismieri”, “arsebobs” 

da “TiToeuli”. 

g) msjeloba-dasabuTebisas iyenebs mocemuli pirobiTi winadadebis/debulebis Sebrunebul, 

mopirdapire da Sebrunebulis mopirdapire winadadebas/debulebebs. 

maT. X.10. poulobs obieqtTa zomebsa da obieqtTa Soris manZilebs 


b) obieqtTa zomebisa da obieqtTa Soris manZilebis dasadgenad (maT Soris realur viTarebaSi) 

iyenebs figuraTa (mravalkuTxedebis, wreebis/wrewirebis) msgavsebas an/da damokidebulebebs 

figuris elementebis zomebs Soris (magaliTad im sagnis simaRlis gazomva, romlis fuZe 

miudgomelia, miudgomel wertilamde manZilis gamoTvla). 

g) poulobs brtyeli figuris farTobs da iyenebs mas optimizaciis zogierTi problemis gadasaWrelad 

(maT Soris realur viTarebaSi). 

d) iyenebs koordinatebs sibrtyeze geometriuli figuris zomebis dasadgenad. 

maT. X.11. asabuTebs geometriuli debulebebis marTebulobas 


a) deduqciuri da induqciuri msjelobis nimuSSi aRadgens gamotovebul safexurs/safexurebs. 

b) iyenebs algebrul gardaqmnebs, tolobisa da utolobebis Tvisebebs geometriul debuleba- 

Ta dasabuTebisas. 

g) iyenebs koordinatebs geometriuli obieqtis Tvisebebis dasadgenad da dasabuTebisTvis. 

maT. X.12. ikvlevs figuraTa geometriul gardaqmnebs sibrtyeze da iyenebs maT 

geometriuli problemebis gadaWrisas 


a) axdens geometriul gardaqmnebs sibrtyeze da martiv SemTxvevebSi iyenebs maT figuraTa tolobis 

dasadgenad. 

b) iyenebs koordinatebs geometriuli gardaqmnebis (paraleluri gadatana, RerZuli/centruli 

simetria) Sesrulebisa da gamosaxvisaTvis. 

g) msjelobs da akeTebs daskvnas erTi da igive tipis geometriuli gardaqmnebis (paraleluri 

gadatana, mobrunebebi erTi da igive centris garSemo, RerZuli simetriebi paraleluri 

RerZebis mimarT, saerTo centris mqone homoTetiebi) kompoziciebis Sesaxeb. 

d) figuris da/an geometriuli gardaqmnebis Tvisebebis mixedviT msjelobs mocemuli figurebiT 

sibrtyis dafarvis SesaZleblobis Sesaxeb; Sesabamis SemTxvevaSi axdens sibrtyis (lokalurad) 

dafarvis demonstrirebas. 

Sinaarsi 

1. figuraTa msgavseba da msgavsebis niSnebi. 

2. or wertils Soris manZilis formula koordinatebSi.

3. geometriuli gardaqmnebi sibrtyeze: RerZuli simetria, mobruneba, homoTetia, paraleluri 

gadatana; geometriuli gardaqmnebis kompoziciebi. 

4. mravalwaxnagebi da maTi niSan-Tvisebebi. 

mimarTuleba: monacemTa analizi, albaToba da statistika 

maT. X.13. moipovebs dasmuli amocanis amosaxsnelad saWiro Tvisebriv da raodenobriv 

monacemebs 


a) iyenebs monacemTa Segrovebis xerxebs (dakvirveba, gazomva, miTiTebul respondentTa jgufis 

gamokiTxva mza anketiT/kiTxvariT). 

b) atarebs statistikur (maT Soris, SemTxveviT) eqsperiments da agrovebs monacemebs. 

g) ikvlevs da iyenebs monacemTa sxvadasxva istoriul da Tanamedrove wyaroebs (mag., sainformacio 

cnobari, interneti, katalogi da sxva). 

maT. X.14. awesrigebs da warmoadgens Tvisebriv da raodenobriv monacemebs dasmuli 

amocanis amosaxsnelad xelsayreli formiT 


a) irCevs Tvisebriv da raodenobriv (daujgufebel) monacemTa warmodgenis Sesaferis 

grafikul formas, asabuTebs Tavis arCevans da qmnis cxrils/diagramas. 

b) agebs sxvadasxva diagramebs erTi da igive Tvisebrivi an raodenobrivi monacemebisTvis da 

msjelobs, Tu monacemTa ramdenad mniSvnelovan aspeqtebs warmoaCens TiToeuli da ra upiratesoba 

gaaCnia TiToeuls. 

g) axdens monacemTa dajgufebas/dalagebas, msjelobs dajgufebis/dalagebis principze. 

maT. X.15. aRwers SemTxveviTobas albaTuri modelebis saSualebiT 


a) aRwers SemTxveviTi eqsperimentis elementarul xdomilobaTa sivrces, iTvlis xdomilobaTa 

albaTobebs variantebis daTvlis xerxebis gamoyenebiT (mag., xisebri diagramis saSualebiT). 

b) atarebs eqsperiments SemTxveviTobis warmomqmneli romelime mowyobilobiT da afasebs xdomilobis 

albaTobas eqsperimentuli monacemebis safuZvelze (fardobiTi sixSiris saSualebiT), 

msjelobs gansxvavebaze Teoriul (mosalodnel) Sedegsa da empiriul (eqsperimentul) 

Sedegs Soris. 

g) mocemuli sasruli albaTuri sivrcisaTvis aRwers SemTxveviTobis warmomqmnel mowyobilobas, 

romlis albaTur modelsac warmoadgens es sivrce, asabuTebs mowyobilobis dizains. 

maT. X.16. iyenebs statistikur da albaTur cnebebs/Tvalsazrisebs yoveldRiur 

viTarebebSi 


a) ganixilavs im statistikur viTarebebs, romelTa gamocdilebac gaaCnia (mag. mosaxleobis 

aRwera, arCevnebi, sazogadoebrivi azris gamokiTxva), iyenebs gamoqveynebul faqtebs/monacemebs 

da msjelobs mocemuli problemis Sesaxeb (mag. ekologiuri sakiTxebis Sesaxeb). 

b) msjelobs albaTuri modelebis gamoyenebis Sesaxeb dazRvevaSi, sociologiur kvlevebSi, 

demografiaSi. 

g) mohyavs albaTur-statistikuri modelebis gamoyenebis magaliTebi bunebismetyvelebaSi da 

medicinaSi (mag., mikro da makro nawilakebis fizika, genetika), xsnis movlenebs SemTxveviTobis 

meqanizmis moqmedebis saSualebiT. 

Sinaarsi 

1. monacemTa wyaroebi da monacemTa mopovebis xerxebi mecnierebaSi (sabunebismetyvelo, 

humanitaruli, socialuri, teqnikuri mecnierebebi), warmoebaSi, marTvaSi, ekonomikaSi, 

ganaTlebaSi, sportSi, medicinaSi, momsaxurebasa da soflis meurneobaSi: 

• dakvirveba, eqsperimenti, mza kiTxvariT gamokiTxva 

2. monacemTa klasifikacia da organizacia: 

a) Tvisebrivi da raodenobrivi monacemebi 


15

) monacemTa dalageba zrdadoba-klebadobiT an leqsikografiuli meTodiT 

3. monacemTa mowesrigebuli erTobliobebis raodenobrivi da Tvisebrivi niSnebi: 

a) monacemTa raodenoba, pozicia da Tanmimdevroba erTobliobaSi 

b) monacemTa sixSire da fardobiTi sixSire 

4. monacemTa warmodgenis saSualebani Tvisebrivi da raodenobrivi (maT Soris dajgufebuli 

monacemebisTvis): 

a) sia, cxrili, piqtograma 

b) diagramis nairsaxeobani (wertilovani, meseruli, xazovani, svetovani, wriuli) 

5. Semajamebeli ricxviTi maxasiaTeblebi Tvisebrivi da daujgufebeli raodenobrivi 

monacemebisTvis: 

a) centraluri tendenciis sazomebi (saSualo, moda, mediana) 

b) monacemTa gafantulobis sazomebi (gabnevis diapazoni, saSualo kvadratuli gadaxra) 

6. albaToba: 

a) SemTxveviTi eqsperimenti, elementarul xdomilobaTa sivrce (sasruli sivrcis 

SemTxveva) 

b) SemTxveviTobis warmomqmneli mowyobilobebi (moneta, kamaTeli, ruleti, urna) 

g) xdomilobis albaToba, albaTobebis gamoTvla variantebis daTvlis xerxebis 

gamoyenebiT 

7. fardobiT sixSiresa da albaTobas Soris kavSiri 

§ paragrafis saxelwodeba 

I trimestri _ 70 sT. 

I Tavi _ kanonzomierebani 

1 kanonzomierebani da maTematika 

2 n-uri xarisxis fesvi 

3 mudmivi Sefardebebi 

4 π ricxvi iracionaluria 

5 wrisa da wrewiris nawilebis gazomva 

sakontrolo samuSao #1 

6 nebismieri kuTxis sinusi, kosinusi da tangensi 

7 monacemebi. maTi mopoveba, warmodgena da analizi 

8 simravle. operaciebi simravleebze 

9 racionalurmaCvenebliani xarisxi 


sarezervo dro 

II Tavi _ informaciebi da ricxvebi 

10 ricxviTi sistemebi 

11 informaciis kodireba 

12 modularuli ariTmetika 

13 saidumlo damwerlobebi 


sarezervo dro 

III Tavi _ orcvladiani gantolebebi da utolobebi 

14 orcvladiani gantoleba 

15 orcvladian gatolebaTa sistema 

16 amocanebi orcvladian gantolebaTa sistemaze 



Sinaarsisa da miznebis ruka 

Sedegebi da indikatorebi 

X. 1b. X4b, X10a. 

X. 1 ab. gd; X 3 a, g. 

X.10 a. X.11a 

X1a X9b 

X10 b X9a 

X.12bg 

X.13 ab. X.14abg 

X.2g, X.4g, X9b 

X.1g, X.2gd, X.3ab, X4d 

X.1gd, X.2 abd 

X.2b, X5 bg 

X.4 ab 

X.2b 

X.7b 

X.7ab 

X.7ab, 

X.7ab, X.8a 

saaT. raod. 

32 

2 

4 

2 

2 

3 

2 

3 

3 

3 

4 

2 

2 

14 

3 

2 

2 

3 

2 

2 

24 

2 

4 

4 

2

17 orcvladiani utoloba da misi grafiki 

18 orcvladian utolobaTa sistema 

19 amocanebi orcvladian gantolebTa da utolobaTa 

sistemebze 

sakontrolo saumuSao #5 

sarezervo dro 

II trimestri _ 55 sT. 

IV Tavi _ funqcia 

20 asaxva 

21 ricxviTi funqciebi 

22 funqciaTa Tvisebebi 

23 aqilevsi da ku 


24 ricxviTi mimdevrobebi 

25 fibonaCis ricxvebi da oqros kveTa 

26 Zalian didi da Zalian mcire 

27 ras gvamcnoben monacemebi 


sarezervo dro 

V Tavi _ figuraTa gardaqmnebi 

28 figuraTa gardaqmnis magaliTebi 

29 gadaadgileba 

30 kongruentuli figurebi 


31 msgavsebis gardaqmna 

sarezervo dro 

III trimestri _ 50 sT. 

msgavsebis gardaqmna 

32 grafTa Teoriis elementebi 


VI Tavi _ xdomilobaTa albaToba 

33 elementarul xdomilobaTa sivrce 

34 Sedgenili xdomiloba 

35 SemTxveviTi xdomilobis albaToba 


36 geometriuli albaToba 

37 statistikuri albaToba 

38 teselacia 


sarezervo dro 

VII Tavi _ saswavlo wlis Semajamebeli samuSaoebi 

39 raSi gvWirdeba funqcia? 

40 pirobiTi winadadebebi 

41 msjelobaTa saxeebi 


saswavlo wlis Semajamebeli samuSaoebi (testebze 

muSaoba) 

sarezervo dro 

X.7bg, X.8a, X.9ab 

X.7bg, X.10g 

X.7bg, X.11bg 

X.8ag 

X.6ab, X.8g 

X.6abgd, X.8abg 

X.3de, X.4a 

X.1bgd, X.5a, X8ab 

X.5g, X8b 

X.3e. 

X.13 abg, X.14abg 

X.12a 

X.11g, X12 abg 

X.9abg, X.12 abg 

X.10a, X.11abg 

X.10a, X.11abg 

X.8a 

X.15a, X.16a 

X.15a, X.16a 

X.15bm, X.16bm 

X.15 abg, X.16ab 

X.16abg 

X.12ad 

X.6bgd, X.10bg 

X.9bg, X.11a 

X.9bg 


2 

3 

3 

2 

2 

38 

5 

4 

5 

5 

2 

4 

4 

3 

2 

2 

2 

18 

3 

4 

4 

2 

3 

2 

45 

2 

3 

2 

22 

2 

2 

4 

2 

3 

2 

3 

2 

2 

16 

3 

2 

3 

2 

4 

2 

sul: 170 saaTi 

17

amdenime SeniSvna 

maswavleblebi miCveuli iyvnen ganaTlebis saministrodan miRebuli programebis, Tematuri 

gegmebisa da sxvadasxva instruqciebis mixedviT muSaobas. imis miuxedavadac ki, rom am dokumentebidan 

mxolod programis Sesruleba iyo savaldebulo, mbrZaneblur-administraciuli 

reJimi aSkarad zRudavda maswavleblis yovelgvar TviTSemoqmedebas. amJamad maswavleblis mu- 

Saobis xarisxis ganmsazRvreli mxolod is Sedegebia, romelTac erovnuli saswavlo gegma iTvaliswinebs. 

zemoT mocemuli `Sinaarsisa da miznebis ruka~ erovnuli saswavlo gegmis Sedegebisa 

da maTi indikatorebis ganawilebas warmogvidgens saxelmZRvanelos paragrafebisa da savaraudo 

drois mixedviT. amitom TiToeul maswavlebels SeuZlia masSi cvlilebebis Setana 

muSaobis TaviseburebaTa gaTvaliswinebiT. unda gaviTvaliswinoT, rom: 

_ `Sinaarsisa da miznebis rukaSi~ zogierTi Sedegi da indikatori meordeba; zogierTi maTgani 

ki (magaliTad, X.3) bunebrivad Sedis TiTqmis yvela TemaSi; 

_ zogierT paragrafs miTiTebuli Sedegebisa da indikatoris mxolod nawili Seesatyviseba 

da isini momdevno TavebSi dasruldeba (amis Sesaxeb maswavleblis am wignSi ufro dawvrilebiT 

vimsjelebT); 

_ Cveni saxelmZRvanelo faravs saswavlo gegmis yvela punqts da gaTvaliswinebulze meti 

sakiTxisagan Sedgeba. aseTebia, magaliTad, sakiTxebi: `kuTxis sinusi, kosinusi da tangensi~, 

àsaxva~, romlebic XI klasis gegmaSicaa Setanili, magram Cveni koncefciis mixedviT, maTi Seswavla 

X klasidan unda iwyebodes. zogierTi Tema am saxelmZRvaneloSi ufro dawvrilebiTaa 

warmodgenili, vidre moiTxoveboda da es ZiriTadad 5-quliani amocanebis saxiTaa realizebuli 

(rac bunebrivad maT arasavaldebulod gamocxadebasac gulisxmobs). 


swavlebis interaqtiuli meTodebi 

interaqcionizmi aris mimarTuleba Tanamedrove socialur fsiqologiaSi, romelic damyarebulia 

amerikeli fsiqologis j. midis koncefciaze. 

interaqtivizmSi (urTierTqmedebaSi) igulisxmeba pirovnebaTaSorisi komunikacia, romlis 

mTavar Taviseburebebad miCneulia adamianis unari `miiRos sxvisi roli~, warmoadginos, Tu 

rogor aRiqvams mas partniori an jgufi, amis Sesabamisad moaxdinos situaciis interpretacia 

da aagos sakuTari mosazreba. 

interaqtiuli meTodebiT swavleba Zireulad gansxvavdeba swavlebis tradiciuli meTodisagan. 

tradiciul swavlebisas gakveTilze centraluri figura maswavlebelia. igi Sesaswavl 

masalas mza saxiT awvdis (gadascems) moswavles, romelic codnis pasiur mimRebis rolSi imyofeba. 

interaqtiuli meTodikiT swavlebis dros TviT moswavle imyofeba Sesaswavli movlenebis 

centrSi. igi Tavis codnasa da gamocdilebaze dayrdnobiT sxva moswavleebTan erTad aanalizebs, 

afasebs, ayalibebs mosazrebebs, eufleba unar-Cvevebs, uyalibdeba ganwyoba-damokidebulebebi 

faqtebisa da movlenebisadmi. 

interaqtiuli meTodebia: 

I. rolebis gaTamaSeba. 

am procesisaTvis erTi an meti moswavle asrulebs sxva funqcias, romelic misi uSualo funqciisagan 

gansxvavdeba. misi mizania adamianur urTierTobebSi realuri an warmosaxviTi problemebis 

gadaWra. 

germaneli fsiqologi biuleri Tvlida, rom TamaSi aris saqmianoba, romlis ganxorcielebas 

safuZvlad funqciuri siamovnebis miReba udevs. 

rolebis gaTamaSeba gakveTilze gaTamaSebuli mcire dadgmaa. 

Cvens SemTxvevaSi SeiZleba erTi moswavle (oponenti) kiTxvebs aZlevdes meores (momxsenebels) 

da mis pasuxebs oponirebas uwevdes. afiqsirebdes Secdomebs da aZlevdes axal kiTxvebs. 

aqve SeiZleba mesame moswavle (recenzenti) akvirdebodes da maTi kamaTis (TamaSis) damTavrebis 

Semdeg akeTebdes daskvnebs, rogori kiTxvebi daisva, rogori pasuxebi gaeca, sad dauSves Secdoma, 

ra SeiZleboda ukeTesi yofiliyo da sxva. 

maswavlebeli am SemTxvevaSi mxolod `wamyvanis~ rolSi gamodis, romelic aregulirebs procesebs 

da bolos swor daskvnebs akeTebs da adarebs mas moswavleTa Sexedulebebs. Tavidan ki 

moswavleebs acnobs TamaSis wesebs. 

II. saklaso diskusia – saswavlo programis an saswavlo meTodis specialuri saxea, romelic 

vizualur aRqmazea dafuZnebuli. misi mniSvnelovani mxare isaa, rom stimuls aZlevs moswavleebs, 

gamoiCinon iniciativa da warmoadginon Tavisi azri, Tundac mcdari.

diskusiis dros xdeba: 

– monawileTa Soris informaciis gacvla; 

– erTi da imave sakiTxis gadawyvetisadmi sxvadasxvagvari midgomis Zieba; 

– sxvadasxva Tvalsazrisis (xSirad erTmaneTis gamomricxavis) Tanaarseboba; 

– saerTo azris an gadawyvetilebis misaRebad jgufuri SeTanxmebis Zieba. 

arsebobs diskusiis Semdegi formebi: 

• `mrgvali magida~ – moswavleTa mcire jgufi (4-5 moswavle) axdens azrTa urTierTgacvlas. 

• `paneluri diskusia~ – diskusias uZRveba jgufis mier winaswar daniSnuli lideri. 

• `forumi~ – mTeli klasi axdens azrTa da ideaTa urTierTgacvlas. 

• `simpoziumi~ – monawileni gamodian informaciiT, romelic winaswar aqvT momzadebuli. 

• `debatebi~ – kamaTi, agebuli monawileTa winaswar dagegmil gamosvlaze. kamaTSi monawileobs 

jgufis TiTo warmomadgeneli rigrigobiT. 

• `kolokviumi~ – azrTa gacvla-gamocvla maswavlebels an mowveul specialistsa da auditorias 

Soris. 

maswavlebeli aqac `wamyvanis~ rolSi gamodis, romelic aregulirebs procesebs. Tavidan 

gegmavs diskusiis formebs, bolos ki ajamebs da gamohyavs swori daskvna. 

III gonebrivi ieriSi erT-erTi interaqtiuli meTodia, romelic amerikelma fsiqologma 

j. osbornma SeimuSava. misi mizania problemis gadaWra mTeli jgufis monawileobiT, ideaTa 

Tavisufali gamoTqmis gziT. 

maswavlebeli winaswar arCevs problemas da SekiTxvis saxiT mkafiod Camoayalibebs mas. amis 

Semdeg iwyeba gonebrivi ieriSis pirveli etapi, romelsac ideaTa generacia (dagrovebis) etapi 

ewodeba. am dros daculi unda iyos Semdegi wesebi: 

1) dauSvebelia moswavlis mier gamoTqmuli azris kritika, kamaTi an Sefaseba. 

2) moswavleebi ideebs gamoTqvamen nebayoflobiT da ara maswavleblis survilisamebr. 

3) TiTo moswavlem SeiZleba gamoTqvas erTi an ramdenime mosazreba. SeiZleba iyos sxvisi 

mosazrebis msgavsi. 

4) yvela idea unda daiweros dafaze, yvelaze miuRebelic ki. 

5) dro SeiZleba ganisazRvros winaswar an maswavlebelma Sewyvitos garkveuli masalis dagrovebisTanave. 

ideaTa Sefasebis etapi. 

1) xdeba gamoTqmuli mosazrebebis mimoxilva. 

2) Tu romelime mosazrebebi msgavsia, xdeba maTi ganzogadeba-gaerTianeba. 

3) moswavleebma daalagon ideebi, romelic, maTi azriT, yvelaze Rirebulia da es dalageba 

moxdes mniSvnelobebis mixedviT (yvelaze mniSvnelovani idea iwereba pirvelad). Tu gamovyofT 

cxra ideas mas `briliantis TamaSs~ eZaxian. 

maswavlebeli rogorc yovelTvis `wamyvanis~ rols asrulebs, aregulirebs procesebs. 

SesaZlebelia gonebrivi ieriSis dros WeSmaritebas ver mivaRwioT, aq maswavlebelma unda 

gaakeTos swori daskvna da Seadaros igi moswavleTa ideebs, gamoyos kazusebi (Secdomebi) da 

gaaanalizos. 

IV. moderacia – meTodi moicavs problemis gadaWras, romelic dafuZnebulia diskusiis gziT 

TanamSromlobis damyarebaze rogorc jgufSi, ise plenarul sesiebze, sadac gamoyenebulia 

TvalsaCino masala. meTodi zrdis urTierTobis `xarisxs~ da gamoiyeneba garkveuli saxis saswavlo 

amocanebisaTvis. 

V. jgufuri muSaoba. 

gakveTilze jgufuri muSaobis organizaciis ZiriTadi Taviseburebebia 

1) klasi iyofa ramdenime jgufad, romlebic irCeven liderebs (jgufebis Sedgenisas, kargi 

iqneba, Tu gaviTvaliswinebT gardneris `mravalmxriv inteleqts~. sasurvelia, jgufSi sxvadasxva 

inteleqtis mqone moswavleebi iyvnen. 

2) TiToeul jgufs eZleva konkretuli davaleba (davaleba yvela jgufisaTvis SeiZleba iyos 

erTi an sxvadasxva). 

3) jgufSi davaleba moswavleebze iseTnairad nawildeba (lideris mier) da sruldeba, rom 

SesaZlebelia jgufis TiToeuli wevris individualuri wvlilis Setana. 

4) jgufebi unda muSaobdnen SeTanxmebulad, ukonfliqtod, ar unda iTrgunebodes arc erTi 

bavSvi. 

5) samuSao unda Sesruldes maswavleblis mier winaswar gansazRvrul droSi. 

6) jgufi muSaobs erT magidasTan. 

7) jgufebSi muSaobis dasrulebis Semdeg yoveli jgufidan rigrigobiT gamodian moswavleebi 

da prezentacias ukeTeben Sesrulebul samuSaoebs. 


19

Tu klass gavyofT sam jgufad, vTqvaT, A, B da C jgufebs mivcemT davalebas (davalebebs), 

Semdgars ramdenime punqtisagan, maSin SeiZleba gavmarToT maTematikuri Widili. A jgufi (oponenti) 

iZaxebs B jgufs, (momxsenebels) davalebis romelime punqtze. C jgufi recenzentia. dafasTan 

gamodis TiTo moswavle TiTo jgufidan. B jgufis romelime moswavle (momxsenebeli) 

pasuxs scems dasmul kiTxvebs. mas ekamaTeba A jgufis romelime moswavle (oponenti), xolo C 

jgufis romelime moswavle (recenzenti) usmens da Semdeg ajamebs maT kamaTs (Widili gadadis 

rolebis gaTamaSebaSi). bolos maswavlebeli ajamebs Sedegebs (rogorc Jiuri) da uwers qulebs 

winaswari SeTaxmebis safuZvelze. Semdeg B jgufi iZaxebs C-s, C ki A-s da a. S. dafasTan erTi 

moswavlis orjer an metjer gamosvla rekomendebuli ar aris. Tu romelime jgufma gamoZaxebaze 

uari Tqva, maSin kiTxvas TviTon kiTxvis damsmeli upasuxebs da is iqneba momxsenebeli, oponenti 

ki – uaris mTqmeli jgufi. 

VI proeqti (grZelvadiani davaleba) – es aris moswavleTa mier winaswar dagegmili da damoukideblad 

Sesrulebuli samuSao, romelic maT SeuZliaT warmoadginon gamokvlevebis, TvalsaCinoebis, 

moxsenebis, leqciis an sxva saxiT. 

proeqtis Temas irCevs maswavlebeli. moswavleebi gegmaven TavianT moqmedebebs da proeqtis 

Sesrulebis xangrZlivobas. 

proeqtis meTodiT muSaoba exmareba moswavleebs: 

– daamyaron uSualo kavSiri gare samyarosTan, faqtebTan da movlenebTan; 

– dagegmon sakuTari moqmedebebi da dro; 

– moiZion masalebi da informaciebi; 

– akontrolon sakuTari swavlebis Sedegebi da sxva. 

maswavlebeli garkveuli drois Semdeg moswavleebs warmoadgeninebs Tav-TavianT proeqtebs 

da afasebs maT. 

gaxsovdeT! 

(gardneris `mravalmxrivi inteleqtidan~ gamomdinare): 

zogierTi moswavle kargad akeTebs proeqtebs, zogi kargia rolebis gaTamaSebis dros, zogi 

– jgufuri muSaobis dros, zogs gonebrivi ieriSi itacebs. zogi lideria, zogs prezentacia 

exerxeba, zogic weriT ukeT amJRavnebs Tavis codnas. 

yoveli moswavle Tavisi SesaZleblobis mixedviT unda Sefasdes. 

amitom kargad SearCieT Sefasebis komponentebi da maTi procentuli wili. 

erTi da igive Tema paralelur klasebSi sxvadasxvanairad SeiZleba gaSuqdes, radgan iq sxvadasxva 

SesaZleblobis moswavleebi swavloben. 


gakveTilis dagegmva (zogadi principebi) 

winaswar daugegmav gakveTils, ragind mcodne da gamocdili maswavlebeli atarebdes mas, 

yovelTvis axlavs xarvezebi. amitom gakveTilis gegmas calke adgili aqvs gamoyofili maswavleblis 

potrfolioSi. mas ZiriTadad mokle Canawerebis saxe aqvs da pedagogi gakveTilis msvlelobisas 

iyenebs. magram gegmis struqtura da Sinaarsi sxvadasxva faqtorebzea damokidebuli: 

saswavlo wlis pirveli gakveTilia Tu bolo; sakontrolo weraa, misi wina gakveTili Tu misi 

momdevno; gvaqvs teqnikuri da TvalsaCino saSualebebiT aRWurvili maTematikis kabineti Tu 

ara da sxva. 

nebismieri gakveTilis gegmas viwyebT Temisa da gakveTilis tipis miTiTebiT (axali masalis 

axsna, ganmtkiceba, Temis (Tavis) Sejameba, damoukidebeli da sakontrolo samuSao, saklaso Tu 

gasvliTi praqtikuli samuSao da sxva); Temisa da im erT an or (dawyvilebuli gakveTilis SemTxvevaSi) 

saaTSi misaRwevi Sedegebis CamonaTvaliT (raSic moviSveliebT erovnul saswavlo gegmasa 

da Sinaarsisa da miznebis rukas). vuTiTebT moswavlis saxelmZRvanelos Sesabamisi gverdis (an 

gverdebis) nomers, klasSi gasarCevi da saSinao davalebad misacemi savarjiSoebis nomrebs (jgufuri, 

damoukidebeli an sakontrolo samuSaoebis dagegmvisas vuTiTebT maT mdebareobas an 

iqve vwerT Sesabamis variantebs). 

imisaTvis, rom klasis TiToeuli moswavle saSinao davalebisa Tu proeqtis saprezentaciod 

trimestrSi 2-3-jer mainc iyos gamoZaxebuli, mizanSewonilia, gegmaSi mivuTiToT momaval gakveTilze 

prezentatori moswavleebis rigiTi nomrebi klasis moswavleTa siidan. 

gegmaSi mivuTiTebT TvalsaCinoebis, teqnikuri da sxva damxmare saSualebebis nusxas. 

SeniSvna: imdenad kargad unda gvqondes moazrebuli klasSi gansaxilavi da saSinao davalebad 

micebuli savarjiSoebi, rom: 

_ klasSi amocanebis amoxsna Cven ki ar daviwyoT, aramed moswavleebs mivagnebinoT bunebrivlogikuri 

kiTxvebis dasma-pasuxebi;

_ SegveZlos Sedegebis (warmatebisa Tu warumateblobis) prognozireba da proeqtisa Tu 

saSinao davalebis prezentaciisas moswavlis namuSevris swrafi Sefaseba (amgvar muSaobaSi dagvexmareba 

moswavlis saxelmZRvaneloSi mocemul savarjiSoTa pasuxebi da am wignSi arsebul 

amocanaTa amoxsnis nimuSebi). 

am wignSi gavecnobiT ramdenime sanimuSo gakveTilis gegmasa da scenars, moswavlis wignis 

TiToeuli struqturuli elementis Sesabamis detalur komentarebsa da maswavleblis warmatebuli 

muSaobisaTvis aucilebel sxva masalebs. 

aq warmodgenili masalebi (Sinaarsisa da miznebis ruka, gakveTilis dagegmva, ramdenime gakveTilis 

gegma da scenari, sakontrolo samuSaoebi da maTi Sefasebis sqemebi, amocanebis amoxsnis 

nimuSebi da sxva) da moswavlis saxelmZRvaneloc isea agebuli, rom zogierTi gakveTilis dagegmva 

arc ki saWiroebs detalur komentarebs (wardgenis fazebis miTiTebas). es zrdis maswavleblis 

Semoqmedebis ares. 

saSinao davalebis prezentacia 

saSinao davalebis micemis win klass aucileblad unda ganvumartoT, rom davalebad micemuli 

TiToeuli nomeri sami sxvadasxva donis sakiTxisagan Sedgeba. 3, 4 da 5-quliani sakiTxebidan maT 

erT-erTi unda Seasrulon, magram Sefasebac Sesabamisi iqneba. amasTan ar aris rekomendebuli 

saSinao davalebad 2-3 nomerze metis micema. 

gakveTils daviwyebT saSinao davalebis gamokiTxviT. SevniSnavT, rom TiToeulma moswavlem 

davlebad micemuli sami nomridan ori amocana unda Seasrulos. aq moswavles „ukandasaxevi gza“ 

_ sizarmacis „ver SevZeli“ pasuxiT dafarva _ moWrili aqvs: Tu ver amoxseni 5-quliani amocana, 

warmoadgine 4-quliani an 3-quliani mainc. moswavles kidev erTxel unda avuxsnaT Sefasebis 

kriteriumebic (sasurvelia, qvemoT mocemuli kriteriumebi klasSi gamovakraT). 

magram saSinao davalebis amgvari Semowmeba mxolod rekomendaciis saxes atarebs da misi 

dacva savaldebulo ar aris. arsebobs saSinao davalebis Semowmebis sxva formebic (magaliTad, 

zepirad vikiTxavT, TiToeul sakiTxze ra pasuxebi miiRes moswavleebma da Secdomebis sajaro 

gasworebis gziT amovwuravT davalebis Semowmebasac. maswavlebelma individualurad SeiZleba 

SearCios davalebis Semowmebis forma misi Sinarsis, mniSvnelobisa da klasis momzadebis donis 

Sesabamisad. 

1) davalebis saprezentaciod gamoZaxebuli moswavle maswavlebels warudgens saSinao davalebebis 

rveuls, romelSic miniSnebuli iqneba ori amocana (magaliTad, #3-is 5-quliani da #5is 

4-quliani); 

2) Tu moswavlis namuSevarSi amocanis SinaarsSi garkvevis mcdeloba mainc SeiniSneba, igi 

Sefasdeba sul mcire 1 quliT mainc. amitom weriT namuSevarSi aseTi moswavle moipovebs aranakleb 

2 qulisa; 

3) araumetes 6 qulis mompovebel moswavles klasi da maswavlebeli dausvams 4 kiTxvas (Ziri- 

Tadad mimdinare Temaze). TiToeul sworad pasuxgacemul kiTxvaze prezentors TiTo qula miemateba. 

Tu weriTi namuSevari 7 qulas imsaxurebs, am moswavles dausvamen 3 damatebiT kiTxvas da 

a. S. Tu moswavlem ori 5-quliani amocana amoxsna sworad, maswavlebels ufleba aqvs, 10 quliT 

Seafasos misi codna zedmeti kiTxvebis gareSe. 

maswavlebels ufleba aqvs Sefasebis obieqturobaSi dasarwmuneblad gamokiTxuli moswavlisagan 

moiTxovos orive (an erTi mainc) nomris sajaro prezentacia klasis winaSe. 


21


gakveTilebis scenarebi 

maTematikis pirveli gakveTili 

kanonzomierebani da maTematika“ swored Zveli Semoqmedi pedagogebis gamocdilebis gamo- 

Zaxilia. garemoSi arsebul procesebze dakvirveba, kanonzomierebaTa aRmoCena da maTematikuri 

modelireba, romelic maTematikis swavlebis erT-erTi mniSvnelovani mizania, moswavleTaTvis 

ukve kargad cnobili magaliTebis gaxsenebiTa da maTze msjelobiT unda daiwyos. klasSi maswavleblis 

mier kargad organizebuli diskusia, romelSic mas am paragrafis dasawyisSi mocmuli 

suraTebi da magaliTebi daexmareba, moswavleebs Zaldautaneblad CarTavs muSaobaSi. aq mniSvnelovania 

„TamaSis“ iseTi wesebis dacva, rogorebicaa: urTierTmosmenis kultura, sakuTari 

naazrevis sxartad gadmocema (zogierTi maswavlebeli, calkeuli moswavlis monologebisagan 

Tavdacvis mizniT, iyenebs 1-3 wuTian qviSis saaTebs), diskusiebSi zogierTi moswavlis „iZulebiTi“ 

CarTva _ rogoria Seni azri? Sen ras ityvi? – kiTxvebiT. aq maswavlebeli, rogorc es 

araerTxel aRvniSneT, mxolod kiTxvebis dasmiTa da mokle komentarebiT Semofarglavs Tavis 

funqciebs. samive magaliTi problemis saxiT daismeba da maT gadawyvetaze muSaobisas maswavlebeli 

maqsimalurad pasiuri unda iyos im gagebiT, rom Tavad ar daiwyos dafaze muSaoba, Zveleburi 

„axsna“ da mxolod kiTxvebiTa da komentarebiT „aiZulos“ moswavleebi Tavadve, saxelmZRvanelos 

daxmarebiT, gaerkvnen situaciaSi da damoukideblad Camoayalibon daskvnebi. 

„magaliTi 2“-is ganxilvisas moswavleebi dafaze akeTeben Sesabamis naxazs. 

A 

maswavlebeli klass saxelmZRvaneloSi mocemul am naxazze gamosaxul oTx marTkuTxa samkuTxedze 

asoebis daweras sTavazobs da, svams kiTxvebs: 

– romel wertilSia naTura? (B wertilSi); 

– ris tolia damkvirveblis simaRle? (B C = B C = h); 

1 1 2 2 

– ra manZilze Catarda dakvirvebebi? (C C = B D da CC = DB ); 

1 1 2 2 

– ra unda davamtkicoT? (rom b>a) 

– ratomaa WeSmariti ∠BAC=∠BB D=β da ∠BA C=∠BB D=α tolobebi? (isini paralelur gverde- 

2 1 1 

biani kuTxebia); 

h BD h BD 

– h, a da b sidideebis CasarTavad romeli tolobebi gamoviyenoT? (tgα= = , tgβ = = ) 

a B1D 

b B D 

BD BD 

BD BD 

– SegviZlia Tu ara da Sefardebebis Sedareba? (radgan B D , 

1 2 

B D B D 

B D B D 

1 

2 

B 2 

) A2 )) 

b C2 h 

) )) 

h h 

e. i. > , saidanac α > β). 

a b 

aqac da, bunebrivia, me-3 magaliTis ganxilvis procesSic, klasis mzaobis donis Sesabamisad 

SesaZlebelia am tipis mimaniSnebeli kiTxvebis raodenobis sagrZnoblad Semcireba. 

saSinao davalebis micemis win klass aucileblad unda SevaxsenoT, rom davalebad micemuli 

TiToeuli nomeri sami sxvadasxva donis sakiTxisagan Sedgeba. 3, 4 da 5-quliani sakiTxebidan maT 

erT-erTi unda warmoadginon. 

a 

h 

C 1 

α 

B 

D 

C 

1 

2 

2

§6. nebismieri kuTxis sinusi, kosinusi da tangensi 

gakveTilis Tema: nebismieri kuTxis sinusis, kosinusisa da tangensis gansazRvra. 

gakveTilis tipi: axali masalis axsna. 

gakveTilis dro: 45 wT. 

gakveTilis mizani: wina gakveTilze Catarebuli sakontrolo weris analizi da maxvili kuTxis 

sinusis, kosinusisa da tangensis ganmartebis safuZvelze azrovnebis logikuri msjelobis gziT 

nebismieri kuTxis sinusis, kosinusisa da tangensis ganmartebebis Camoyalibeba. 

muSaobis forma: individualuri, wyvilebi. 

muSaobis meTodebi: gonebrivi ieriSi, leqciuri. 

TvalsaCinoeba: plakatebi. 

misaRwevi Sedegebi: moswavlem unda SeZlos, maxvili kuTxis sinusis, kosinusisa da tangensis 

gansazRvris safuZvelze logikurad Camoayalibos nebismieri kuTxis sinusis, tangensisa da 

kotangensis gansazRvra da 90 0 , 180 0 , 270 0 , 360 0 -ze maTi mniSvnelobebis povna. 

gakveTilis msvleloba (leqciuri nawili): maswavlebeli moswavleebs acnobs sakontrolo 

weris Sedegebs, axarisxebs mas siZnelis mixedviT, gamohyavs miRweuli Sedegebis procenti. ake- 

Tebs daskvnas warmatebebisa Tu warumateblobis Sesaxeb, saubrobs mizezebze da moswavleebis 

daxmarebiT aanalizebs Secdomebs (Tu arsebobs). 15 wT. 

maswavlebeli mimarTavs gonebriv ieriSs. svams kiTxvebs: ra aris marTkuTxa samkuTxedi? ra 

kavSirSia erTmaneTTan kaTetebi da hipotenuza? ras ewodeba maxvili kuTxis sinusi, kosinusi 

da tangensi? yvela pasuxi iwereba dafaze da keTdeba swori daskvna. 5 wT. 

leqciuri nawili. maswavlebeli dafaze xazavs (an plakatze miuTiTebs) erTeulovan wrewirs 

da masze svams p 0 wertils. moswavleebs aZlevs informacias p 0 wertilis koordinatebze da abrunebs 

mas wrewirze saaTis isris moZraobis sawinaaRmdegod. aqve akeTebs daskvnas: α kuTxis 

Sesabamis yovel p α wertils wrewirze Seesabameba erTaderTi p α (x α , y α ) wertili. 5 wT 

davaleba (wyvilebSi samuSaod). maswavlebeli moswavleTa wyvilebs aZlevs davalebas, sakoordinato 

sibrtyeze daxazon wrewiri, romlis centri koordinatTa saTaveSia. α kuTxis konkretuli 

mniSvnelobebisaTvis moZebnon masze Sesabamisi p α wertilebi (maswavlebels SeuZlia, 

α-s mniSvnelobebi misces wignidan an TviTon SearCios). maswavlebeli CamovliT amowmebs Sesrulebul 

samuSaos da adgilze akeTebs komentarebs. 5 wT. 

individualuri davaleba. ipoveT p α -s koordinatebi, roca α=0 0 , 45 0 , 90 0 . ra kavSirSia aRniSnuli 

wertilis koordinatebi sinα-sa da cosα-s mniSvnelobebTan amave kuTxeebisaTvis? (saWiroebis 

SemTxvevaSi maswavlebeli miTiTebebs iZleva). maswavlebeli amowmebs yvela moswavlis namuSevars 

da saboloo daskvnas wers dafaze: p 0 = (1; 0); p 45 = ( ; ); p 90 = (0; 1). aRniSnuli wertilis 

koordinatebi Sesabamisad tolia cosα-sa da sinα-s mniSvnelobebisa amave kuTxeebze: cosα = x α da 

sinα = y α. mciredi miTiTebis Semdeg moswavleebi TviTon ayalibeben kuTxis sinusis, kosinusisa 

da tangensis gansazRvrebebs da adgenen maT mniSvnelobebs 90 0 , 180 0 , 270 0 -ze; Rebuloben 

cos 2 α + sin 2 α = 1 igiveobas. 9 wT. 

saSinao davalebis micema: savarjiSo ## 2; 3. 1 wT. 

§7. monacemebi da xdomilobebi 

Tema: monacemebi da xdomilobebi. 

gakveTilis mizani: moswavleebma SeZlon gakveTilze miRweuli Tu miuRweveli Sedegebis statistikurad 

warmodgena da misi ganzogadebis safuZvelze monacemebisa da xdomilobebis daxasiaTeba. 

gakveTilis tipi: ganmazogadebeli 

muSaobis forma: individualuri, wyvilebSi, rigebSi 

gamoyenebuli meTodebi: gonebrivi ieriSi, jgufuri, leqciuri 

gakveTilis dro: 45 wT 

gamoyenebuli masala: baraTebi, plakatebi, slaidebi da sxva. 

gakveTilis msvleloba (gaTvaliswinebuli unda iyos is, rom §7-is win aris paragrafi nebismieri 

kuTxis sinusis, kosinusisa da tangensis Sesaxeb. moswavleebs swored am axal gakveTilze 

moeTxoveba misi codna da bunebrivia maT aqvT davaleba micemuli): 

– fiqsirdeba gakveTilze msxdom moswavleTa raodenoba, vTqvaT 36. 


23

– fiqsirdeba, ramdens aqvs davaleba da ramdens ara. vTqvaT, 6 moswavles davaleba ara aqvs. 

– fiqsirdeba, ramdens aqvs srulyofilad amoxsnili da ramdens ara. vTqvaT, srulyofilad 

amoxsnili aqvs 21 moswavles. 

es monacemebi SeiZleba dafaze gadavitanoT Semdegnairad: 

moswavleTa raodenoba 

36 

21 

– varigebT baraTebs, romlebzec moswavleebs vawerinebT sinusis, kosinusis, tangensis gansazRvrebebs 

da kidev sxva kiTxvebis pasuxebs. swori da araswori pasuxebis raodenobas vwerT 

dafaze. vTqvaT samma moswavlem pasuxi ver gasca ver erT kiTxvas. 10-ma moswavlem sruli pasuxi 

gasca yvela kiTxvas, 9 moswavlem ver dawera tangensis gansazRvreba. miRebuli statistika 

dafaze maswavlebelma ase warmoadgina: 

maswavlebeli svams sxva kiTxvebsac, romelic gakveTils Seexeba da yovel monacems afiqsirebs 

dafaze. amave dros fasdebian moswavleebi. vTqvaT, daiwera Semdegi qulebi: 10; 9; 8; 8; 6. esec 

dafiqsirdeba dafaze. 20 wT. 

maswavlebeli gadadis dafaze dafiqsirebuli masalis analizze. svams kiTxvebs: 

1) ramden moswavles ar hqonda davaleba. 

2) ramdenma ar icis tangensis ganmarteba da sxva. 

ixazeba sxva diagramac (SeiZleba wignidan) da amis Sesaxebac ismeba kiTxvebi. 

yuradReba maxvildeba miRebul niSnebze da daismis kiTxva. 

1) ra saSualo niSani daiwera klasSi? pasuxi = 7. 

2) romeli niSani daiwera ufro xSirad? pasuxi 8. 

3) davalagoT miRebuli niSnebi zrdadobis mixedviT. 6; 8; 8; 9; 10. romeli niSania SuaTana? 

pasuxi 8. 

4) ra iqneba kidura wevrebis (niSnebis) sxvaoba? pasuxi 18 − 6 = 4. 10 wT. 

leqciuri nawili. maswavlebeli amcnobs moswavleebs, rom nebismieri monacemebi SeiZleba 

warmovadginoT svetovani diagramiT, wriuli diagramiT, xazovani diagramiT. aCvenebs maT slaidebiT, 

TvalsaCinoebebze da sxva. yveba mis gamoyenebaze, vTqvaT, reitingebis dros da sxva. 


O 

↑ 

9 

6 

dawera sinusis da 

kosinusis gans. 

veraferi 

upasuxa 

kl. mosw. 

raod. 

daval. 

sruly. 

daval. 

arasruly. 

14 mosw. 

10 mosw. 

3 mosw. 

9 mosw. 

daval. 

ar aqvs 

sruli 

pasuxi 

ver dawera 

tang. gan. 

→

maswavlebels Semoaqvs gansazRvrebebi: 

– nebismieri monacemebis saSualo ariTmetikuls davarqvaT monacemTa saSualo. e. i. miRebuli 

niSnebis saSualo aris `7~. 

– im monacems, romelic xSirad meordeba, davarqvaT monacemTa moda. e. i. miRebuli niSnebis 

moda aris `8~. 

– Tu monacemebs davalagebT zrdadobis mixedviT, maSin mediana davarqvaT: 

a) Sua monacems, Tu monacemTa ricxvi kentia; 

b) ori Sua monacemis saSualo ariTmetikuls, Tu monacemTa raodenoba luwia. 

Cvens SemTxvevaSi medianaa `8~. 

– Tu kidura wevrebis sxvaobas ganvixilavT, mas davarqvaT diapazoni. 

Cvens SemTxvevaSi diapazonia `4~. 5 wT. 

moswavleebs wyvilebSi da Semdeg rigebSi vaZlevT davalebas saxelmZRvanelofan, vTqvaT, 

N4-s. mas dafasTanac aanalizebs romelime moswavle. 9 wT. 

davaleba. saxelmZRvanelodan. vTqvaT, ## 5; 6; 7. 

§15. orcvladian gantolebaTa sistema 

gakveTilis Tema: orcvladian gantolebaTa sistema, romlis erTi gantoleba arawrfivia. 

gakveTilis tipi: kombinirebuli (davalebis Semowmeba, gamokiTxva, maT Soris ganvlili masalis 

Semowmeba, axali masalis axsna, magaliTebis amoxsna, davalebis micema): 


gakveTilis mizani: orcvladian gantolebaTa sistemis amoxsnis sxvadasxva xerxis Seswavla, 

sadac erTi gantoleba arawrfivia. 

muSaobis forma: individualuri, wyvilebSi. 

muSaobis meTodi: roluri, gonebrivi ieriSi, leqciuri, diskusia, moderacia. 

resursi: plakatebi, baraTebi. 

misaRwevi Sedegi: moswavlem Zvel codnaze dayrdnobiT unda SeZlos logikuri gziT gadavides 

axalze da amoxsnas orcvladiani gantoleba (sadac erTi gantoleba arawrfivia) sxvadasxva 

xerxiT. 

gakveTilis msvleloba (viTvaliswinebT, rom moswavleebs micemuli aqvT davaleba orcvladian 

gantolebaze. maT unda icodnen wina gakveTilidan ganmartebebi, orcvladiani gantolebis 

grafikuli warmodgena da sxva): 

maswavlebels dafasTan gamohyavs ori moswavle, erTs aZlevs momxseneblis rols (mopasuxe), 

meores _ oponentis rols (kiTxvebis damsmeli). oponents ufleba aqvs, dausvas kiTxvebi mxolod 

orcvladiani gantolebis irgvliv da amoaxsnevinos magaliTebi (SeiZleba saSinao davalebidanac 

ki). dafasTan mimdinareobs kamaTi, klasi recenzentis rolSi gamodis, maswavlebeli aregulirebs 

process da amowmebs saSinao davalebis Sesrulebas. aqve winaswar gamzadebul baraTs (baraTze 

weria ramdenime kiTxva an magaliTi) aZlevs wyvilebs Sesasruleblad. 

Tu momxsenebelma kiTxvas ver upasuxa, maSin oponenti TviTonve pasuxobs, Tu ara da klasi. 

rodesac kiTxvebi amoiwureba, SesaZlebelia klasma dausvas kiTxvebi rogorc momxsenebels ise 

oponents. kiTxvebma mTlianad unda amowuros gakveTilis masala. Tu moswavleebma ver amowures 

kiTxvebi, maSin maswavlebeli akeTebs amas gonebrivi ieriSis gziT. bolos maswavlebeli kidev 

erTxel ajamebs kiTxvebs da xazgasmiT gamoyofs im nawils, romelic saWiroa axali gakveTilisaTvis. 

kiTxvebi aucileblad unda moicavdes: 

gansazRvrebebs: orcvladiani gantolebis, misi amonaxsnisa da amoxsnis xerxebis. gantolebis 

grafikulad warmodgenis. pirveli xarisxis orcvladiani gantolebisa da misi grafikis agebis 

xerxs. kvadratul funqciasa da mis grafiks. wrewiris gantolebasa da mis grafiks (yovelive es 

maswavlebels plakatis saxiT unda hqondes warmodgenili). 

SeniSvna. dafasTan mdgomi moswavleebis Secvla sxva moswavleebiT dasaSvebia, garkveuli kiTxva-pasuxis 

Semdeg. fasdeba yvela imis mixedviT, Tu rogori pasuxebi gaeca, rogori kiTxvebi 

daisva, klasSi vin iaqtiura, baraTebiT micemuli davaleba rogor Sesrulda da sxva. 

dafasTan moswavlis gamoZaxebis, baraTebis gziT davalebis micemisa da sxva dros gaTvaliswinebuli 

unda iqnes moswavlis inteleqti, anu unda SeecadoT, moswavles komfortuli garemo 

SeuqmnaT. 15-17 wT. 


25

maswavlebeli svams kiTxvebs da gonebrivi ieriSis gziT midis daskvnamde. mag.: rogor vipovoT 

ori orcvladiani gantolebis saerTo amonaxsni. garkveuli pasuxebis Semdeg gamoikveTeba, rom 

unda amoixsnas TiToeuli maTgani da vipovoT amonaxsnTa TanakveTa. gamocdileba gviCvenebs, 

rom moswavleTa ZiriTadi pasuxebi aris is, rom maT aerTianeben sistemaSi da Semdeg Casmis, 

Sekrebis an garkveuli xerxiT xsnian mas. 

ra Tqma unda, aseTi pasuxi kargia. 5-7 wT. 

maswavlebeli iZleva individualur davalebas: moswavleebma Casmis xerxiT amoxsnan iseTi 

gantolebaTa sistema, sadac erTi gantoleba arawrfivia. bunebrivia, moswavleebi analogiisa 

da logikuri azrovnebis gziT davalebas Tavs arTmeven, Tu, ra Tqma unda, maT ician wina gakve- 

Tili. maswavlebeli Camovlis gziT amowmebs namuSevrebs da iqve iZleva SeniSvnebs. Tu es saWiroa 

(aseTi magaliTebi moyvanilia wignSi. SesaZlebelia misi micema an maswavlebeli TviTon akeTebs 

mas), davalebis Sesrulebis Semdeg romelime moswavle gamodis dafasTan da Tavis namuSevars 

ukeTebs prezentacias (anu ubralod amoxsnis dafaze). imave moswavles an sxvas maswavlebeli 

dafaze samuSaod (klasi exmareba) aZlevs imave an sxva magaliTs grafikuli gziT amosaxsnelad. 

10 wT. 

leqciuri nawili. maswavlebeli kidev erTxel aanalizebs orcvladian gantolebas, mis amonaxsns, 

grafiks, orcvladian gantolebaTa sistemas (ori gantoleba wrfivia), misi amoxsnis xerxebs 

(Casma, grafiki, Sekreba) da adgens, rom imave wesiT SesaZlebelia amoixsnas orcvladian 

gantolebaTa sistema, sadac erTi gantoleba arawrfivia. 5 wT. 

diskusiuri nawili. maswavlebeli dafaze wers orcvladian gantolebaTa sistemas, sadac erTi 

gantoleba kvadratulia, meore wrewiris (ix. gakveTilis magaliTi 3 an TviTon adgens maswavlebeli). 

kiTxva: ramdeni amonaxsni aqvs gantolebaTa sistemas? 

diskusia mimdinareobs maswavlebelsa da klass Soris. 

Tu Casmis meTods mivmarTavT, saqme gaZneldeba. kargi iqneba, amovxsnaT grafikulad igi da 

amoixsneba kidec. 9 wT. 

SeniSvna. SesaZlebelia Sefasdnen axali moswavleebic. moswavleTa Sefaseba unda moxdes winaswar 

SemoRebuli komponentebis mixedviT. vTqvaT, jgufuri muSaoba, individualuri muSaoba, 

davalebis prezentacia, aqtiuroba. Cvens SemTxvevaSi davalebis prezentaciisTvis unda Sefasdes 

oponenti da momxsenebeli. baraTebis pasuxisaTvis unda Sefasdes individualuri muSaobis 

komponentiT. aqtiurobisaTvis SesaZlebelia klasSi kidev sxvebic Sefasdnen. 

davaleba. savarjiSo NN 2; a) g) z) an sxva. 


§20. asaxvebi 

erTi simravlis meore simravleze asaxvis cneba da Tvisebebi calke sakiTxad standartis 

arc SedegebSia dafiqsirebuli da arc SinaarsSi. amitom erTi SexedviT SeiZleba zedmetad 

mogveCvenos am didi moculobis paragrafis saxelmZRvaneloSi CarTva. Tavis droze maTematikis 

calkeul sagnebad diferencirebam bolos am sagnebs Soris garkveuli bzarebic ki gaaCina 

(magaliTad, amocanis amoxsnis àlgebruli~ da `geometriuli~ xerxebi). dRes ki, roca 

maTematikuri dargebis erT ansamblSi gaerTianebis amocanis gadawyvetaa saWiro, Tu am sagnebs 

Soris arsebuli bunebrivi kavSirebi aqtiurad ar iqneba gamoyenebuli, SeiZleba uTavbolo 

narevi, an, ufro uaresi, paradoqsuli situaciebi miviRoT. magaliTad, SeiZleba algebrul 

nawilSi ori simravlis tolobisa da geometriul nawilSi wertilTa simravleebis tolobis 

cnebebi winaaRmdegobaSi aRmoCndes. 

erTi simravlis meore simravleze asaxvis es zogadi cneba yovelgvari xelovnuri elferis 

gareSe bunebrivad akavSirebs ricxviTi funqciebisa da figuraTa gardaqmnis cnebebs da, amasTan, 

Tavidan agvacilebs am sakiTxebis logikuri kavSirebis gareSe ganxilvis saSiSroebas. 

gakveTilis scenari 

meore trimestris pirveli gakveTili iwyeba sadiskusio amocaniT, romlis Sinaarsis gacnobis 

pirvel cdas 2-3 wuTi daeTmoba. Semdeg, amocanis Sinaarsis ukeTesad gaazrebisaTvis, 

moswavleebi dafaze da rveulebSi akeTeben aseTi tipis sqemas amocanis teqstis xelmeored, 

nawil-nawil wakiTxvis paralelurad (moswavleTa Seyovnebis SemTxvevaSi maswavlebeli mxolod 

mokle kiTxvebs svams). 

amocanaSi dasmuli kiTxvebi bunebrivad iwvevs msjelobas: `X aRniSnavs im moswavleTa 

k 

raodenobas, romelTac k qula miiRes~.

x = 1 1 

x = 1 2 

x = 1 3 

x = 1 4 

x = 5 5 

x > 5 6 

x > x > x > x > 2 

6 7 8 9 

x = 2 10 

x ≥ 3 9 

x ≥ 4 8 

x ≥ 5 7 

x ≥ 6 6 

_____________________ 

x + x + x + x ≥ 18 

6 7 8 9 

radgan x + x + x + x + x + x = 11, 

1 2 3 4 5 10 

amitom x + x + x + x = 30 – 11 = 19 da zemo utolobebis gamo 

6 7 8 9 

x + x + x + da x ricxvebis jami 18-ze 1-iT meti rom iyos, aucilebelia Sesruldes tolobebi: x 6 7 8 9 6 

= 7, x = 5, x = 4 da x = 3. 

7 8 9 

am amocanis amoxsnis amgvarad dasrulebis Semdeg moswavleebi gaixseneben Sesabamisobisa da 

asaxvis cnebebs da am paragrafis dasawyisSi mocemul suraTze (`vis ra moswons?~) fanqriT 

gakeTebuli isrebiT mianiSneben or sasrul simravles Soris `vis ra moswons~ principiT 

gansazRvrul asaxvaze. amis Semdeg maswavlebeli svams kiTxvebs: 

_ sadiskusio amocanis amoxsnisas romeli simravleebi ganvixileT? (pasuxad SeiZleba araswori 

mosazrebebic gamoiTqvas, magram Sesabamisi kiTxvebiT maswavlebels klasi mihyavs amocanis 

amoxsnis procesSi dafaze mocemuli CanawerebiT dafiqsirebuli Semdegi simravleebis 

amocnobamde: A (meaTeklaselTa 30-elementiani simravle), B = {1; 2; ...; 10} qulebis simravle da 

C = {1; 2; 3; 4; 5; 7}. 

_ romeli simravlis romel simravleze asaxvaa aq gadmocemuli? (moswavleebma SeiZleba A 

simravlis B simravleze asaxvac daasaxelon, magram aq maswavlebeli, Tu am mosazrebis uaryofa 

klasma ver SeZlo, svams damatebiT kiTxvebs: _ viciT A simravlis yvela elementi? mocemuli 

yofila A simravle?). moswavleebi unda mivaxvedroT, rom mxolod amoxsnis Semdeg dadginda 

B → C asaxva: am asaxvas moswavleebi dafaze aRweren isrebiT: 

B 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

C 

1 

5 

7 

4 

3 

2 

siis mixedviT gamodian moswavleebi da avseben cxrils. 

amiT azusteben K simravles (SeiZleba, rom am klasSi 4-ze 

naklebi niSani aravis gamouvida da 10-ianic veravin 

daimsaxura, amitom aseT SemTxvevaSi K = {4; 5; 6; 7; 8; 9}. 

dafaze moswavleebi waSlian cxrilis marjvena svets, 

svetis saxiT Caweren dalagebul K simravles da isrebiT 

warmoadgenen M → K asaxvas. 

maswavleblebs ar vurCevT zemoT aRwerili procesis 

xelovnurad daCqarebas. imis Sesamowmeblad, ramdenad 

kargad gaiazra klasma asaxvis cneba, maswavlebeli svams 

kiTxvebs: 

_ cnobilia Tu ara Cveni klasis moswavleTa M 

simravle? 

_ gvaxsovs Tu ara pirvel trimestrSi maTematikaSi 

miRebuli saboloo niSnebi? 

maswavlebeli klass sTavazobs aseT davalebas: 

dafasTan 

moswavleTa 

moswavleTa 

simravle 

simravle 

simravle 

M 

qeTino abulaZe 

besik barbaqaZe 

niSnebis niSnebis 

niSnebis 

simravle 

simravle 


K 

8 

6 

27

q. abulaZe 

b. barbaqaZe 

g. gabunia 

t. iaSvili 

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

M K 

4 

5 

6 

7 

8 

9 


darCenil droSi moswavleebi Seiswavlian asaxvis 

gansazRvris mniSvnelobaTa simravlisa da asaxvis 

cnebebs, gaarCeven 1-l magaliTs. 

saSinao saSinao davalebad davalebad mivcemT 20.1, 20.5 da 20.6 amocanebs. 

gunduri maTematikuri Widili 

Widilis wesebi 

WidilSi monawileobs sami gundi, romlebic irCeven liderebs. gundebs qmnis maswavlebeli. 

sasurvelia, gundebi Tanabari inteleqtualuri potenciis iyos. gundebs SeiZleba davarqvaT 

gonebamaxviluri saxelebi. kenWisyris Sedegad unda davadginoT pirveli gundi (A), meore gundi 

(B) da mesame gundi (C). 

maswavlebeli gundebs davalebas wina dRes an ramdenime dRiT adre aZlevs, SesaZlebelia, 

imave dResac. mizanSewonilia, SerCeul iqnes imdeni amocana, ramdeni moswavlecaa TiToeul 

gundSi. amocanis amoxsnis saidumlo daculi unda iyos gundis wevrebis mier. 

Widilis dawyebis win gundebi ikribebian da lideris mier TiToeul moswavleze, amocanis 

amoxsnis codnis mixedviT, nawildeba TiToeuli amocana. 

maswavlebeli yoveli amocanis amoxsnas afasebs qulebiT 3; 5 an sxva quliT. 

TamaSi iwyeba Semdegnairad: A gundis lideri B gunds uxmobs romelime amocanis amosaxsnelad. 

B gunds SeuZlia daTanxmdes an ar daTanxmdes am gamoZaxebas. Tu uari ganacxada, mas qula akldeba 

da rolebi Seicvleba: pasuxs gascems kiTxvis damsmeli. Semdeg B gundi eZaxis C-s da a. S. anu 

gamoZaxeba xdeba Semdegi principiT: 

A → B → C → B → A → C → A (erTi wre). SeiZleba TamaSi gaagrZeloT an SewyvitoT. 

rodesac A → B (anu A iZaxebs B-s), maSin momxsenebeli aris B gundis is warmomadgeneli, romelic 

am konkretul amocanazea mimagrebuli. oponenti aris B gundis is warmomadgeneli, romelic 

mimagrebulia am amocanaze, xolo C gundis is warmomadgeneli, romelic mimagrebulia amave 

amocanaze, aris recenzeti. 

momxsenebeli iwyebs amocanis amoxsnas. oponents SeuZlia, nebismier dros Seawyvetinos (Tu 

surs) da dausvas axali SekiTxva (SekiTxva uSualod am sakiTxs unda exebodes), anu `muSaobs~ 

misTvis qulebis warTmevaze. oponenti kamaTis dasrulebis Semdeg akeTebs daskvnas, Tu ra Secdomebi 

iyo daSvebuli, ra SeiZleboda ukeTesad Tqmuliyo da sxv. SeiZleba amiT qulebi moipovos. 

qulebi ganawilebuli unda iyos amocanaze gamoyofili qulidan. 

yoveli moswavle mxolod erTxel gamodis. amocanebi ar meordeba. disciplinis darRvevaze 

SeiZleba dawesdes sajarimo qulebi. 

maTematikuri Widili SeiZleba Catardes nebismier dros, Tumca Cven ufro mizanSewonilad 

migvaCnia, igi romelime Tavis damTavrebis Semdeg, sakontrolo weris win Catardes. 

gTavazobT maTematikuri Widilis konkretul nimuSs. 

Tema. III Tavi _ gantolebebi da utolobebi. 

mosamzadebeli samuSao: 

maswavlebeli klass yofs jgufebad. jgufebi irCeven liderebs. kenWisyriT dgindeba A, B da 

C gundebi. 

wina dRiT maswavlebeli klass aZlevs amocanebs saxelmZRvanelos III Tavidan. maswavlebeli 

amocanebs Tavisi Sexedulebisamebr arCevs, vTqvaT, 14.3 xuT quliani 

14.8 (b) oTxqulianidan. 14.11 samqulianidan. 15.1(a) xuTqulianidan. 15.3 samqulianidan. 15.8 

xuTqulianidan. 15.11 xuTqulianidan. 16.5 samqulianidan 16.9 xuTqulianidan 17.3 oTxqulianidan. 

17.10 samqulianidan. 18.1 xuTqulianidan. 18.6 oTxqulianidan. 19.2 samqulianidan. 19.10. oTxqulianidan, 

III Tavis damatebiTi savarjiSoebidan ## 7, oTxqulianidan 12. 

savarjiSoebi isea SerCeuli, rom yvela sxvadasxva tipis da xasiaTisaa. 

TamaSis dro: sasurvelia 90 wT. an gakveTilebis Semdeg.

TamaSis msvleloba: moswavleebi ikribebian jgufebis mixedviT, sasurvelia, sxvadasxva oTaxebSi. 

Semdeg isini erT oTaxSi moiyrian Tavs da jgufebis mixedviT sxdebian ‘mrgval magidasTan~. 

sasurvelia, TamaSs eswrebodnen mSoblebi da maswavleblebi. 

TamaSis dawyebas auwyebs maswavlebeli. A gundis lideri dgeba da gundTan SeTanxmebiT B 

gunds iZaxebs. sTavazobs mas, magaliTad, savarjiSo 14.3-s xuTqulianidan. B gundis warmomadgeneli 

iwyebs amocanis amoxsnas. A gundis warmomadgeneli ekamaTeba. C gundis warmomadgeneli ki 

usmens. 

B gundis warmomadgeneli – orniSna ricxvi Caiwereba ase: 10x + y... 

A gundis warmomadgeneli – ra aris x da y? 

B – x aris aTeulSi aRniSnuli cifri, y ki erTeulebis aRmniSvneli cifri. 

A – gTxovT, ganmartoT, ra aris ricxvi da ra aris cifri? 

B – (SeiZleba ganmartos, SeiZleba vera). 

10x 

+ y 

igi agrZelebs: Tu mocemul orniSna ricxvs gavyofT cifrTa x + y jamze, igi ase Caiwereba . 

x + y 

10x 

+ y 13 

ganayofi Tu aris 5 da naSTi 13, igi warmodgeba 

= 5 + saxiT. 

x + y x + y 

A-s SeiZleba gauCndes kiTxva, SeiZleba ara. magaliTad, SeiZleba gaCndes kiTxva: 

ratom Caiwereba aseTi saxiT ganayofi da naSTi? 

B-m SeiZleba upasuxos, SeiZleba vera. 

B – (agrZelebs amoxsnas) 10x + y = 5(x + y) + 13. sabolood miiRo: 5x – 4y = 13 

A – ra hqvia Tqven mier miRebul gantolebas? 

B – pasuxobs an vera. Semdeg igi agrZelebs orcvladiani gantolebebis amonaxsnebis wyvilebis 

dadgenas, roca x da y cifrebia. cxadia, x = 5 da y = 3 an x = 9 da y = 8,anu miiReba aseTi orniSna 

ricxvebi: 53 da 98. 

SeiZleba daisvas sxva SekiTxvebic. 

rodesac diskusia damTavrdeba, C jgufis warmomadgeneli akeTebs analizs (an ver akeTebs): 

ra iyo swori da ra – ara. ra Secdomebi iqna daSvebuli da ra unda mieRoT pasuxad. 

bolos maswavlebeli akeTebs daskvnas da anawilebs (Tu momxsenebeli raimeSi CaiWra) qulebs. 

SeiZleba momxsenebelma miiRos maqsimaluri qula, sxvebma – vera. SeiZleba momxsenebelma da 

oponentma gainawilon qulebi da recenzentma veraferi miiRos, SeiZleba qulebi samiveze ganawildes. 

ase gagrZeldeba Semdegi gamoZaxebebi. pirveli wris Semdeg SeiZleba Catardes liderebis 

Sejibri, ris Semdeg sabolood gamovlindeba gamarjvebuli gundi. 

niSnebi SeiZleba daeweroT prezentaciaSi im moswavleebs, romlebmac dafasTan Tavisi amocanis 

amoxsna warmoadgines, aseve aqtivobebSi oponents da recenzents, gamarjvebuli gundis 

wevrebs gundur komponentSi da sxva. 

proeqtis (grzelvadiani davaleba) ganxilva 

Tema – oTxkuTxedi 

davaleba – moswavleebma unda moipovon wina klasis saxelmZRvaneloebidan oTxkuTxedis 

ganmarteba, misi klasifikacia gverdebisa da kuTxeebis mixedviT. paralelogramis, marTkuTxedis, 

rombis, kvadratis ganmartebebi, maTi Tvisebebis Seswavla da maTi dayofa msgavsebebisa 

da ganmasxvavebeli niSnebis mixedviT. vTqvaT, kvadratis gansazRvra oTxkuTxedze, paralelogramze 

da rombze dayrdnobiT. 

mizani: moswavleebisTvis sxvadasxva saxelmZRvaneloebidan informaciis mopovebisa da maTi 

damuSavebis safuZvelze codnisa da Ziebis unar-Cvevebis gaRrmaveba. 

muSaobis forma: individualuri, jgufuri. 

gamoyenebuli meTodebi: diskusia, prezentacia; gonebrivi ieriSi. 

proeqtis vada: maqsimum erTi kvira. 


gamoyenebuli masala: wina klasebis saxelmZRvaneloebi, plakatebi. 

gakveTilis msvleloba: gakveTili iwyeba gonebrivi ieriSis I etapiT, anu mopovebuli masalis 

romelime moswavlis mier dafaze gadataniT. kerZod: 

1) oTxkuTxedi ewodeba ... 

2) oTxkuTxedSi SeiZleba mopirdapir gverdebi iyos: a) paraleleri, b) SeiZleba ori iyos 

paraleluri, ori ara; g) SeiZleba paraleluri gverdebi ar iyos da ixazeba. 


29

a b g 

3) a figuras vuwodoT paralelogrami; b figuras _ trapecia, xolo g-s _ ubralod oTxkuTxedi. 

4) paralelograms aqvs Tvisebebi: mopirdapire gverdebi da kuTxeebi tolia, diagonalebi 

gadakveTis wertiliT Suaze iyofa. 

5) paralelogrami SeiZle davaxarisxoT kuTxeebisa da gverdebis mixedviT: 

a) Tu paralelogramSi yvela kuTxe tolia, aseT figuras 

marTkuTxedi ewodeba. 

b) Tu paralelogramSi yvela gverdi tolia, aseT figuras 

rombi ewodeba. 

g) Tu paralelogramSi yvela kuTxe da yvela gverdi tolia, 

kvadrati ewodeba. 

SeniSvna. Tu romelime moswavles sxva informacia aqvs, masac dafaze vwerT. 

6) cxadia, radgan marTkuTxedi, rombi da kvadrati pirvel rigSi paralelogramia, amitom 

maT paralelogramis Tvisebebi aqvT da garda amisa sakuTari axali Tvisebac, riTac isini 

erTmaneTisgan gansxvavdeba. kerZod, 

marTkuTxedi – diagonalebi tolia. 

rombi – diagonalebi marTobulia da warmoadgens kuTxis biseqtrisebs. 

kvadrati – mas marTkuTxedisa da rombis yvela Tviseba gaaCnia. 

7) kvadratis gansazRvra oTxkuTxedidan SesaZlebelia Semdegnairad: 

oTxkuTxeds, romlis mopirdapire gverdebi paraleluria, kuTxeebi da gverdebi ki tolia, 

kvadrati ewodeba. 

kvadratis gansazRvra marTkuTxedidan SesaZlebelia Semdegnairad: 

marTkuTxeds, romlis gverdebi tolia, kvadrati ewodeba. 

aseve SesaZlebelia igi ganisazRvros rombidan. 25 wT. 

yvela sxva Sexeduleba an sxva azri iwereba dafaze da gonebrivi ieriSis I I etapis mixedviT 

vadgenT: 

← 

← 

← 

← 

gamovyoT paralelogramis, marTkuTxedis, rombisa da kvadratis saerTo Tvisebebi. 

es aris paralelogramis Tvisebebi. 

ganmasxvavebeli Tvisebebi: 

marTkuTxedisaTvis – diagonalebi tolia. 


← 

kvadrati 

← 

oTxkuTxedi 

← 

paralelogrami trapecia 

marTkuTxedi rombi

ombisaTvis – diagonalebi marTobulia da warmoadgens kuTxis biseqtrisebs. 

kvadratisaTvis – radgan kvadrati marTkuTxedicaa da rombic, amitom mas yvela maTi Tviseba 

aqvs, anu kvadratSi Tavs iyris yvela Tviseba. 15 wT. 

SeniSvna. SesaZlebelia dafasTan icvlebodnen moswavleebi da TavianT moZiebul ganmartebebs 

da Tvisebebs werdnen. kargi iqneba, moswavle iqve Tu miuTiTebs, romeli wyarodanaa igi mopovebuli. 

Sefaseba: 1-3 qula, Tu moswavles daxazuli aqvs oTxkuTxedi da romelime misi kerZo saxe. 

4-5 qula. Tu moswavles aqvs raime naxazi da aRniSnuli aqvs romelime Tviseba. 

6-7 qula. Tu moswavle erkveva figuraTa TvisebebSi da ganmartebaSi. 

8-9 qula. Tu moswavle ayalibebs ganmartebebs, Tvisebebs, magram ver axdens maT klasifikacias 

da ver adgens maT Soris kavSirs. 

10 qula. Tu moswavlem unaklod Seasrula davaleba. 5 wT. 

SeniSvna. proeqti SeiZleba micemul iqnas saxelmZRvanelodan an maswavleblis mier Seuswavleli 

masalidan. 

§25. aqtivoba: òqros kveTa _ harmoniuli proporcia~ * 

reziume 

moswavleebi ecnobian cnebas òqros kveTa”, swavloben monakveTis gayofas oqros kveTis 

proporciiT, ifarToeben Tvalsawiers, ecnobian maTematikis gamoyenebis nimuSebs xelovnebasa 

da bunebaSi. 

aqtivobis mizani 

• moswavlis SemecnebiTi interesis xelSewyoba, misi Tvalsawieris gafarToeba 

• garemomcveli samyaros silamazisa da harmoniis “kanonebis” gacnoba da maTze msjeloba. 

gakveTilis gegma: 

1. òqros kveTa~ maTematikaSi: amocanis dasma, misi algebruli da geometriuli amonaxsni 

2. òqros kveTa~ bunebasa da xelovnebaSi 

savaraudo dro: 1 gakveTili 

masala: didi zomis qaRaldi (formati), plakatebi adamianis, mcenarisa da cxovelis gamosaxulebiT 

(gvaTxovebs biologiis maswavlebeli), saxazavi, fanqari, fargali, Suasaukuneebis arqiteqturis 

an mxatvrobis nimuSebis reproduqciebi. 

aqtivobis aRwera: 

1. maswavlebeli Sesaval sityvaSi saubrobs `mudmiv~ problemaze: eqvemdebareba Tu ara silamaze 

da harmonia raime maTematikur kanonebs da SesaZlebelia Tu ara maTi `gazomva~ raime algebruli 

xerxebiT (SesaZlebelia mokle istoriuli eqskursi Zvel saberZneTsa da aRorZinebis 

epoqaSi) 

2. Semdeg maswavlebeli svams amocanas _ rogor gavyoT AB monakveTi C wertiliT ise, rom AB 

: AC = AC : BC. 

3. klasi iyofa or jgufad. pirveli jgufi eZebs dasmuli amocanis algebrul amonaxsns, xolo 

meore xsnis mas geometriulad (agebis amocanis sirTulis gamo, sjobs maswavlebelma warmoadginos 

agebis gza da moswavleebs mosTxovos misi samarTlianobis dasabuTeba). 

4. orive jgufi warmoadgens TavianT namuSevars da maswavleblis mier xdeba gaweuli samuSaos 

Sejameba. maswavlebeli gaacnobs moswavleebs agreTve oqros samkuTxedisa da oqros marTkuTxedis 

cnebebs. 

5. amis Semdeg, pirvel jgufs miecema plakati adamianis sxeulis gamosaxulebiT da eZleva 

davaleba, daadginos saxis da xelis mtevnis ra nawilebia oqros proporciebSi (vfiqrob, jgufSi 

gamoCndeba moxalise, romelic gariskavs daadginos TviTon ramdenad jdeba oqros kveTis proporciebSi). 

* SemoTavazebulia proeqt ìlia WavWavaZis~ erovnuli saswavlo gegmebisa da saxelmZRvaneloebis jgufis 

mier. 


31

6. meore jgufis davalebaa mcenarisa da cxovelis nawilebs Soris proporciebis dadgena 

7. Seajamebs ra jgufebis muSaobis Sedegebs, maswavlebeli 

aCvenebs moswavleebs Sua saukuneebis romelime xuroTmoZRvruli 

Zeglis reproduqcias (mag. parizis RvTismSoblis taZari. 

am reproduqciis naxva SesaZlebelia Tundac qarTul sabWoTa 

enciklopediaSi) da uCvenebs moswavleebs romeli nawilebia 

oqros kveTis proporciebSi. 

8. Semdeg imarTeba mcire diskusia garemomcveli samyaros silamazisa da harmoniis `kanonebis~ 

da Cveni maTdami damokidebulebis Sesaxeb. (igulisxmeba, rom maswavlebels aqvs saRi azri da 

iumoris grZnoba) 

9. maswavlebeli moswavleebs aZlevs saSinao davalebas: 

• moiZion masalebi oqros kveTis Sesaxeb (mag. qarTuli sabWoTa enciklopediis me-6 tomi, a. 

benduqiZis `maTematika-seriozuli da saxaliso~ da a.S.). 

• daadginos, eqvemdebareba Tu ara oqros kveTis princips romelime qarTuli xuroTmoZRvruli 

Zegli misi reproduqciis mixedviT. 

32 maTematika X maswavleblis wigni

marsi 

iupiteri 

§32. kenigsbergis Svidi xidi 

gansaxilveli Tema skolebisaTvis aratradiciulia. mis Seswavlas maTematikuris garda zogadsaganmanaTleblo 

mniSvnelobac aqvs. mravali maTematikuri amocana gamartivdeba, Tu moxerxda 

maTTvis grafebis gamoyeneba. monacemTa warmodgena grafis saxiT maT TvalsaCinoebas 

aniWebs. grafebiT Cven sakmaod xSirad vsargeblobT, magaliTad, rkinigzis an metropolitenis 

sqema warmoadgens grafs, sadac sadgurebi grafis wveroebia, gadasarbenebi ki _ grafis wiboebi. 

mravalwaxnagebis (kubi, piramida da a. S.) wveroebic da wiboebic grafs qmnian. 

samotivacio (kenigsbergis Svidi xidis) amocanis formulirebis Semdeg moswavleebs mivceT 

4-5 wuTi, eZebon amoxsna. pasuxs nu vetyviT, is Semdegi gakveTilebisaTvis SemovinaxoT. 

Tu grafis wveroTa raime wyvili dakavSirebulia ori, sami da a. S. wiboTi, vityviT, rom gvaqvs 

orjeradi, samjeradi da a. S. wibo. 

vTxovoT moswavleebs, daxazon grafebi jeradi wiboebiT. 

wiboTa erTobliobaze mag. 2-Si vambobT, rom is ar aris simravle. 

saqme isaa, rom maTematikaSi simravles mxolod iseTi elementebis erTobliobas uwodeben, 

romlebic erTmaneTisgan ganirCeva. vTqvaT, erToblioba `kalami, fanqari, fanqari~ ar aris sami 

elementisagan Sedgenili simravle, magram Tu raime ganmasxvavebul niSans mianiWebT, magaliTad 

`kalami, wiTeli fanqari, mwvane fanqari~, ukve samelementiani simravle gveqneba. 

araa aucilebeli, grafis wvero naxazze mxolod wertiliT gamoisaxebodes. sxvagvar gamosaxvas 

iyeneben, Tu saWiro xdeba wverosTan raime informaciis miwera. 

amocana 1-is Sesabamisi grafi aseTia. rogorc Cans, 

dedamiwidan marsze ver movxvdebiT. 

am tipis (arabmul) grafebs Cven aRar ganvixilavT. 

saturni 

urani 

neptuni 

amocana 2-is Sesabamisi grafi aseTi saxis SeiZleba iyos. 

wveroTa ricxvia 6, wiboTa ricxvia 10. 

3 amxanagi 

3 xelis CamorTmeva 

dedamiwa 

plutoni merkuri 

E 

F 

venera 

aqve SevniSnoT, rom, marTalia, naxazze {AC} da {BE} wiboebi gadajvaredinda, magram maT saerTo 

wertili ar aqvT. 

SevTavazoT moswavleebs amocana: `n raodenobis amxanagi Sexvedrisas xels arTmevs erTmaneTs. 

ris tolia xelis CamorTmevaTa ricxvi?~ 

konkretuli SemTxvevebisaTvis (2, 3, 4, ... amxanagi) Seadginon saTanado grafebi, sadac wveroebs 

amxanagebi warmoadgenen, wiboebi ki gamosaxaven xelis CamorTmevas. gaakeTon daskvna. msjeloba ase 

SeiZleba warimarTos: 

2 amxanagi 


A 

C 

B 

D 

4 amxanagi 


n( 

n −1) 

zogadi kanonzomiereba aseTia: xelis CamorTmevaTa ricxvi = . 

2 

msjeloba asec SeiZleba warmarTuliyo: TiToeulma xeli CamoarTva danarCen (n - 1)-s. sul 

gveqneboda n • (n - 1), magram aseTi daTvlisas TiToeuli xelis CamorTmeva orjeraa naangariSebi, e. 

i. saWiroa misi 2-ze gayofa. 


33

* SemoTavazebulia proeqt ìlia WavWavaZis~ erovnuli saswavlo gegmebisa da saxelmZRvaneloebis jgufis 

mier. 


aqtivoba “saSualo xelfasi” 

reziume 

moswavleebi cxrilis saxiT warmodgenili monacemebis safuZvelze aanalizeben sawarmoSi TanamSromlebsa 

da direqtors Soris xelfasis momatebis Taobaze warmoqmnil winaaRmdegobas. amisaTvis 

isini iTvlian xelfasebis diapazons, saSualos, modas da medianas. Sesrulebuli gamoTvlebis 

safuZvelze isini ayalibeben sakuTari pozicias da asabuTeben mis marTebulobas Ser- 

Ceuli (sakuTari poziciisTvis Sesaferisi) Semajamebeli ricxviTi maxasiaTeblis saSualebiT. 

aqtivobis mizani 

• moswavleebi gaerkvnen tipuri monacemebis aRwerisas centraluri tendenciis sxvadasxva 

sazomebis SesaZleblobebSi 

• moswavleebi gaiwafon gadawyvetilebis miRebisas statistikuri mosazrebebis gaTvaliswinebaSi 

• moswavleebma ganiviTaron statistikuri xasiaTis argumentebis saSualebiT poziciis 

dasabuTebis unari 

aucilebeli wina codna 

• Semajamebeli ricxviTi maxasiaTeblebi – diapazoni, saSualo, moda, mediana 

• cxrilis wakiTxva 

saWiro masalebi 

• situaciuri amocanis teqsti 

• saweri qaRaldi da kalami 

• kalkulatori 

• kompiuteri (ar aris savaldebulo) 

aqtivobis aRwera 

moswavleTa codnis aqtivacia 

sTxoveT moswavleebs ganixilon SemTxvevebi, rodesac monacemTa diapazoni sakmaod mcirea/ 

didia da sami-oTxi monacemis SemTxvevaSi daakvirdnen: 1) ramdenad àxlosaa” ermaneTTan monacemTa 

saSualo, moda da mediana; 2) centraluri tendenciis romelime sazomis ricxviTi mniSvneloba 

xom ar emTxveva romelime konkretul monacems; 3) saSualo kidura wevrebTan ufro 

àxlosaa” Tu medianasTan. 

SeekiTxeT moswavleebs, maTi azriT romeli sazomi ufro `grZnobieria” kidura wevrebis 

sididis mkveTri cvlilebebisadmi? ratom? 

gamokvleva 

sTxoveT moswavleebs yuradRebiT waikiTxon situaciuri amocanis teqsti da cxrilis dasamuSaveblad 

erTmaneTs dausvan kiTxvebi; SesaZlebelia isini daiyon jgufebad da gaiTamaSon 

amocanaSi miTiTebul TanamSromelTa CamonaTvalis mixedviT rolebi sakuTari mosazrebebis 

mkafio formulirebiT da cxrilidan moxmobili monacemebis warmoCeniT. 

SeekiTxeT moswavleebs – amocanis pirobis mixedviT rogor iyeneben sawarmos TanamSromlebi 

cxrilSi moyvanil informaciasa da Semajamebel ricxviT maxasiaTebleblebs sakuTari interesebis 

warmosaCenad da dasacavad? 

sTxoveT moswavleebs gamoTqvan varaudi saSualos, modas da medianis mniSvnelobebis Sesaxeb 

im SemTxvevaSi, Tuki mcire xelfasis mqone 24 TanamSromlis xelfasi 15 000 lari gaxdeba. 

Semdeg sTxoveT moswavleebs gamoTvalon kalkulatoris an kompiteris saSualebiT centraluri 

tendenciis sazomebi gazrdili xelfasebis SemTxvevaSi da Seadaron sakuTari varaudi 

gamoTvlebis Sedegebs. 

dausviT moswavleebs Semdegi SekiTxvebi: 1) centraluri tendenciis romeli sazomis ricxviTi 

mniSvneloba ar Seicvala? 2) centraluri tendenciis romeli sazomis ricxviTi mniSvneloba 

Seicvala? 3) Tundac erTi xelfasi rom SegecvalaT, centraluri tendenciis romeli sazomis 

mniSvneloba Seicvleboda aucileblad? (saSualo) 4) mxolod erTi-ori xelfasi rom Se-

gecvalaT, savaraudod centraluri tendenciis romeli sazomis mniSvneloba ar Seicvleboda? 

(savaraudod, moda) 5) mxolod erTi-ori xelfasi rom SegecvalaT, ramdenad savaraudoa, rom 

medianis mniSvneloba Seicvleboda? (sazogadod, es damokidebulia medianis mdebareobaze - Tu 

mediana ganTavsebulia mravali toli xelfasis SuaSi, maSin igi ar Seicvleboda, xolo Tu igi 

erTmaneTisgan mkveTrad gansxvavebuli xelfasebis maxloblobaSia, maSin igi albaT Seicvleboda) 

situaciuri amocanis bolo kiTxva sadiskusioa. roluri TamaSis SemTxvevaSi mmarTvelebi 

upiratesobas mianiWebdnen saSualos, gaerTianebis Tavmjdomare – medianas, xolo dabalxelfasiani 

TanamSromlebi – modas. waaxaliseT moswavleTa alternatiuli xedvebi da poziciebi 

ise, rom bunebrivi moeCvenoT amocanebi, romelTac ramdenime amonaxsni/pasuxi aqvs da kvlevisas 

eZebon gansxvavebuli gzebi da dasabuTebebi. 

gaRrmaveba-gafarToeba 

sTxoveT moswavleebs iwinaswarmetyvelon, Tu direqtoris xelfasis rogori nazrdi gamoiwvevda 

saSualos gazrdas 1000 lariT da Seamowmon sakuTari varaudi gamoTvlebiT kalkulatorze 

an kompiuterze. 

SeekiTxeT moswavleebs gaizrdeboda, Semcirdeboda Tu igive darCeboda saSualo xelfasi 

im SemTxvevaSi, Tuki direqtori daiqiravebda kidev erT inJenersa da ostats. sTxoveT moswavleebs 

daasabuTon varaudi. mxolod amis Semdeg CaatarebineT saWiro gamoTvlebi. 

aqtivobis ganxilva/Sefaseba 

naTelia, rom mocemuli aqtivoba xels uwyobs ara marto specifikuri – statistikuri codnis 

dauflebas da statistikuri alRos Camoyalibebas, aramed zogadi inteleqtualuri unarianobis 

ganviTarebasac – esaa problemaTa gadaWris unari, kritikuli azrovnebis unari, gadawyvetilebis 

miRebis unari, msjelobisa da dasabuTebis unari, saswavlo-kvleviTi unarebi. ramdenime 

am tipis sayofacxovrebo konteqstis mqone amocanis gadaWris Sedegad moswavleebs uviTardebaT 

imis garkvevis unaric, Tu rodis romeli sazomis gamoyeneba sjobia monacemTa simravlis 

warmosadgenad da romeli sazomi ufro “mdgradia” cvlilebebis mimarT. 

situaciuri amocana 

sawarmos TanamSromelTa profesiuli gaerTianebis Tavmjdomare molaparakebas awarmoebda 

sawarmos direqtorTan TanamSromelTa xelfasebis Taobaze. misi azriT cxovreba gaZvirda, saqonelze 

fasebi gaizarda da sawarmos TanamSromlebi met fuls saWiroeben sayofacxovrebo 

danaxarjebisTvis, gaerTianebis wevrebidan ki arc erTi ar iRebs weliwadSi 18 000 larze mets. 

sawarmos direqtori daeTanxma mas imaSi, rom marTlac yvelaferi gaZvirda, maT Soris, nedli 

masalis fasic; Sesabamisad, sawarmos mogeba Semcirda. garda amisa, sawarmoSi saSualo xelfasi 

xom weliwadSi 22 000 lars aRemateba. amitom mas verc warmoudgenia aseT viTarebaSi xelfasebis 

momateba. 

im saRamosve gaerTianebis Tavmjdomarem Caatara gaerTianebis wevrTa kreba. gamyidvelma 

sityva iTxova. “Cven, gamyidvelebi mxolod 10 000 lars viRebT weliwadSi, gaerTianebis wevrTa 

umravlesoba ki 15 000 lars. Cven gvsurs, rom Cveni xelfasebi am donemde mainc gaizardos”. 

gaerTianebis Tavmjdomarem gadawyvita, rom yuradRebiT Seeswavla sruli informacia sawarmos 

TanamSromelTa xelfasebis Sesaxeb. saxelfaso ganyofilebaSi mas misces Semdegi cxrili: 

samuSaos 

dasaxeleba 

TanamSromelTa 

raodenoba 

xelfasi 

larebSi 

gaerTianebis 

wevroba 

direqtori 1 250 000 ara 

moadgile 2 130 000 ara 

inJineri 3 55 000 ara 

ostati 12 18 000 diax 

muSa 30 15 000 diax 

buRalteri 3 13 500 diax 

mdivani 6 12 000 diax 

gamyidveli 10 10 000 diax 

mcveli 5 8 000 diax 

sul 72 1 593 500 _ 


35

TanamSromelTa gaerTianebis Tavmjdomarem gamoTvala sawarmos TanamSromelTa saSualo 

xelfasi da miiRo daaxloebiT 22 131, 94 lari. “direqtori marTalia,- gaifiqra man,- magram 

xelfasebis saSualo maRalia sawarmos xelmZRvanelTa maRali xelfasebis xarjze. xelfasebis 

saSualo ar iZleva swor warmodgenas TanamSromelTa tipuri xelfasis Sesaxeb”. 

Semdeg man gaifiqra: `gamyidvelic marTalia. ocdaaTi muSidan TiToeuls 15 000 lari aqvs. 

esaa xelfasebis moda. miuxedavad amisa, gaerTianebis danarCeni 36 wevridan, vinc ar iRebs 15 000 

lars, 24 iRebs amaze naklebs”. 

bolos gaerTianebis Tavmjdomarem gadawyvita gaerkvia risi toli iyo WeSmaritad Sua xelfasi 

da sawarmos xelfasebi daalaga zrdadobis mixedviT. aRmoCnda, rom Sua anu medianuri xelfasi 

36-e da 37-e xelfasebs Sorisaa da raki orive es xelfasi 15 000 laria, amitom medianac 15 000 

laria. 

kiTxva: rogor Seicvleboda xelfasebis mediana, moda da saSualo, Tuki yvelaze mcirexelfasiani 

24 TanamSromlis xelfasi 15 000 laramde gaizrdeboda? Tqven ra pozicias dauWerdiT 

mxars xelfasebTan dakavSirebiT? 

moswavleTa trimestruli da wliuri Sefasebebi 

ukve gavecaniT moswavleTa codnisa da unarebis Sefasebis kriteriumebs jgufur mecadineobebze, 

proeqtisa Tu saSinao davalebis prezetnaciisas, damoukidebel-saklaso da sakontrolo 

samuSaoebze. (gavixsenoT: damoukidebeli saklaso Tu sakontrolo samuSaoebis TiToeuli 

sakiTxi Sefasebulia qulebiT. Tu am qulaTa jamia M, xolo moswavlem samuSaoSi daagrova 

K qula, maSin moswavles ewereba 10K : M gamosaxulebis ricxviTi mniSvnelobis erTeulebis sizustiT 

damrgvalebuli ricxvi. cxadia, Tu sagangebo SemTxvevaSi es ricxvi 0-ia, moswavles ewereba 

Sefaseba ‘1~). rogorc vxedavT, es kriteriumi martivi, gasagebi, obieqturi, gamWvirvale da samarTliania. 

gamocdilebamac aCvena, rom moswavleebisaTvisac gasagebi da misaRebia igi. bunebrivia, 

rom Semajamebeli Sefasebis dadgenis kriteiumsac igive xuTi (xazgasmuli) moTxovna 

wavuyenoT. erTwlianma gamocdilebam aCvena (da es winaswar advilad prognozirebadac iyo), 

rom portfolioSi moswavleTa trimestruli Sefasebis ramdenme parametriani da sxvadasxva 

koeficiantiani kriteriumebi ver akmayofilebda zemo xuTi moTxovnidan zogierTs (maswavlebelTa 

umravlesobisaTvis is dResac gaugebaria). 

magram es sakiTxi arc ise martivi gadasawyvetia, rogorc SeiZleba mogveCvenos. magaliTad, 

saSualos gamoangariSebiT kriteriumi martivi, gasagebi da gamWvirvalea, magram is obieqturi 

da samarTliani ver iqneba, radgan adeqvaturad ver asaxavs moswavlis codnisa da unarebis 

aTvisebis xarisxs (magaliTad, xSiria situaciebi, roca moswavlis mimdinare Sefasebebi maRalia, 

xolo sakontrolo werebis Sefasebebi _ dabali, an piriqiT). sakamaTo ar unda iyos is faqti, 

rom gansakuTrebuli mniSvneloba eniWeba moswavleTa momzadebas sapasuxismgeblo wesebisaTvis 

(gansakuTrebiT X-XI I klasebSi), anu Temis, trimestrul Tu wliur Semajamebel individualur 

samuSaoebsa da Sesabamis Sefasebebs. magram warmoudgenelia Sefasebis iseTi mniSvnelovani 

komponentis gauTvaliswinebloba, rogoricaa, magaliTad, `klasSi CarTuloba~. imisaTvis, rom 

SemecnebiTma miznebma da amocanebma ar daCrdilos aRmzrdelobiTi amocanebi, amitom maTi 

ganxorcieleba erTdroulad unda mimdinareobdes. amasTan, moswavle mxolod maswavlebels ki 

ar warudgens Tavis Teoriul Tu praqtikul namuSevars, aramed Tavisi TanaklaselebisTvis 

gasagebi formiTa da eniT gadmoscems davalebisa Tu kiTxvis pasuxs. aq Semfaseblis funqcia 

klassac eZleva. amitom prezetnaciisas (es saSinao davalebaa, proeqtia Tu jgufuri muSaoba) 

moswavlem: 

_ koreqtulad unda gamoiyenos terminologia, aRniSvnebi da simboloebi; 

_ gasagebad gadmosces sakiTxis arsi; 

_ igi koreqtuli unda iyos maswavlebelTan da TanaklaselebTan mimarTebaSi, SeZlos sxvaTa 

naazrevis moTminebiT mosmena da gaanalizeba; 

_ unda aqtiuri iyos jgufuri davalebebisa Tu proeqtebis Sesrulebisas. 

amas garda, `klasSi CarTuloba~ gulisxmobs prezentatorisadmi konkretuli moswavlis mier 

dasmuli kiTxvebis raodenobisa da Sinaarsis xarisxs (SekiTxvis koreqtuloba da TemasTan Sesabamisoba). 

amgvari aqtiurobis SemTxvevaSi maswavlebeli da klasi iRebs gadawyvetilebas am moswavlis 

mier sarezervo quliT dasaCuqrebis Sesaxeb (magaliTad, grafaSi erTi varskvlavis aRniSvniT 

SegviZlia davimaxsovroT). rodesac es moswavle prezentatori iqneba, gavixsenebT am 

`varskvlavs~ da moswavlis Sefasebas 1 erTeuliT gavzrdiT (es sagrZnoblad gazrdis moswavlis 

motivacias aqtiurobisadmi). 

36 maTematika X maswavleblis wigni

tradiciulad Temis Semajamebeli individualuri samuSaos (testirebis, sakontrolo samu- 

Saoebis) Sefasebebi JurnalSi wiTeli feris cifrebis saxiT Segvqonda (Tumca aq fers mniSvneloba 

ara aqvs), radgan yvela moswavlis erTdrouli Sefasebebi isedac gamokveTilia). cal-calke 

amovwerT prezetnaciebSi miRebul niSnebsa da Semajamebel samuSaoebze miRebul Sefasebebs 

TanmimdevrobiT (trimestris dawyebidan dasasrulamde). 

Tu SefasebaTa es ori mimdevroba miaxloebiT zrdadi an klebadi ar aris, viyenebT saSualoTa 

gamoTvlis wess, vipoviT pirveli mimdevrobis S 1 ariTmetikul saSualos da meore mimdevrobis 

S 2 ariTmetikul saSualos meaTedamde sizustiT. maSin 0,4S 1 + 0,6 _ S 2 gamosaxulebis ricxviTi 

mniSvnelobis erTeulamde sizustiT damrgvalebuli ricxvi iqneba moswavlis trimestruli Sefaseba. 

magaliTad, Tu salome gamyreliZem trimestrSi miiRo Sefasebebi: 

7, 8, 8, 6, 7, 8, 9, 7, 8; 6, 5, 8, 6, 7, 

maSin S 1 = (7+8+8+6+7+8+9+7+8) : 9 ≈ 7,6 da S 2 = (6+5+8+6+7) : 5 ≈ 6,4 

xolo 0,4S 1 + 0,6 ⋅ S 2 ≈ 7 

magram, Tu Sefasebebi miaxloebiT progresirebadia: 

3; 4; 4; 5; 7; 8; 9; 8 

3; 4; 7; 8; 9; 8, 

(S 1 = 48 : 8 = 6; S 2 = 39: 6 = 6,5, xolo 0,4S 1 + 0,6S 2 ≈ 6) 

aseTi moswavle rom aSkarad 8-ianis Rirsia (miT umetes maTematikaSi!) _ es sakamaToc ar unda 

iyos. amitom nu viqnebiT pedanturi da `samarTals~ raRac formulebiT nu davamaxinjebT. arc 

aSkarad regresirebadi (anu ukuRma warmodgenili zemo Sefasebebi) SemTxveva davtovoT formulebis 

amara. ra Tqma unda, aman didi safiqrali unda gagviCinos, veZioT warumateblobis mizezebi 

(pirvel rigSi Cvens muSaobaSi!), magram aq 6-ianis dawera da TviTkmayofileba danaSaulis 

tolfasqmedebad unda aRviqvaT. 

isic ar unda gagvikvirdes, rom moswavleebic aqtiurad Caebmebian trimestruli Sefasebis 

dadgenis procesSi (maT xom saswavlo wlis dasawyisidan miiRes `grZelvadiani davaleba~ 

ramdenime saganSi Tavis Sefasebebis Segrovebis, warmodgenisa da ZiriTadi parametrebis mixedviT 

informaciis analizis Sesaxeb). 

martivi ar aris arc wliuri Sefasebis dadgena. aq, Tu am sami Sefasebis diapazoni 1-s ar aRemateba, 

martivia saSualos gamoyeneba (da, 5; 5; 6-is SemTxvevaSi, SegviZlia 6-ianis gamoyvanac), 

magram Tu diapazoni 1-ze metia (magaliTad, 6, 7, 9 aris an piriqiT), maSin samarTliani iqneba 

saSualoze 1-iT meti an 1-iT naklebi Sefasebis gamoyvana. 


37

damatebiTi amocana 

amoxseniT gantoleba: 

3 x − 15x 

+ 20 

3 2 

= x _ 2 

Sefasebis kriteriumi: 

1. a) zusti mniSvnelobis Cawera fasdeba 1 quliT; 

miaxloebiTi mniSvnelobis sworad Cawera – 1 quliT; 

b) perioduli aTwiladebis Cveulebriv wiladebad warmodgena – 1 quliT; 

zusti mniSvnelobis gamoTvla – 1 quliT; 

miaxloebiTi mniSvnelobis sworad gamoTvla – 2 quliT. 

2. mixvedra imisa, rom 2,221 (3,551) akmayofilebs pirobas fasdeba 1 quliT; 

imis gaxseneba, rom 0,001010010001… iracionaluri ricxvia da amitom 2,22101001… (3,55101001…) 

iracionaluri ricxvia – 2 quliT. 

3. imis Cawera, rom 5 3 = 125 (5 4 = 625), xolo 6 3 > 147 (4 4 < 600) – 1 quliT; 

m = 6 (m = 4) pasuxis miReba – 1 quliT. 


sakontrolo samuSaoebi 

I Tavis Sesabamisi masalebis aTvisebisa da swavlebi miznebis miRwevis xarisxis dasadgenad 

vatarebT erT Sualedur da erT Semajamebel sakontrolo samuSaos, romelsac win uswrebs „mo- 

TelviTi“ damoukidebeli saklaso samuSaoebi. qvemoT mocemulia #1 da #2 sakontrolo 

samuSaoebis variantebi da moswavleTa weriTi namuSevrebis Sefasebis kriteriumebi. 

# 

1. 

a) 3 _ 

I varianti 

b) 2,(6) _ 3,(81) + 

2. 

3. 

4. 

5. 

a) 

b) 

= 3 

x 6 + x 3 _6 = 0 


II varianti 

ipoveT Semdeg gamosaxulebaTa ricxviTi mniSvnelobebi measedamde sizustiT: 

2 _ 

3,(4) _ 2,(27) _ 

dawereT erTi racionaluri da erTi iracionaluri ricxvi, romelic mdebareobs 

a da b ricxvebs Soris, Tu 

a = 2,22 da b = 2,(2) a = 3,55 da b = 3,(5) 

ipoveT iseTi mTeli m ricxvi, romlisTvisac 

m _ 1 < 3 147 < m m < 4 600 < m +1 

ipoveT wesieri n-kuTxedis im nawilis farTobi, romelic 

mdebareobs masSi Caxazuli wrewiris gareT, Tu 

n = 6 da r = 10sm. n = 4 da r = 6sm. 

amoxseniT gantolebebi: 

3 x − 5 = _3 

x8 + 15x4 _ 16 = 0 

qula 

2 

4 

3 

2 

4 

2 

3 

sul: 20 qula

5 

7 10 − 

4. Sesabamisi naxazis warmodgena fasdeba 1 quliT; 

imis dafiqsireba (mixvedra), rom S = 6S 1 (S = 4S 1 ) – 1 quliT; 

π 

(S1 = r2 π 

= tipis Canaweri – 1 quliT. 

4 

3 

2 S = r – 1 

3 

r 

6 

2 

r 2 

pasuxi: S = 100(2 3 − π ) sm2 (S = 36 (4 – π ) sm2 ) – 1 quliT. 

5. a) imis dafiqsireba, rom gantolebis orive mxare me-4 xarisxSia (kubSia) asayvani fasdeba 1 

quliT; umartivesi gantolebis amoxsna da pasuxis dafiqsireba – 1 quliT. 

b) gantolebaSi x 3 = a (x 4 = a) tipis aRniSvna – 1 quliT; miRebuli kvadratuli gantolebis 

sworad amoxsna – 1 quliT; sawyisi kuburi (me-4 xarisxis) gantolebebis amoxsna – 1 quliT. 

sasurvelia, rom nebismeiri sakontrolo samuSaosTvis gaviTvaliswinoT damatebiTi sakiTxebi. 

isini, maTi raodenobisa da sirTulis miuxedavad yovelTvis mxolod 2 quliT fasdeba. 

es saSualebas gvaZlevs, moswavle klasSi „davakavoT“ da davexmaroT Tavis ukeTesad warmo- 

CenaSi (magaliTad 20-quliani 5-ianis nacvlad 22-quliani 5-ianis miReba an ZiriTad samuSaoSi 

„SemTxveviT“ dakarguli qulebis anazRaureba). 

# 

1. 

a) _5 

b) 2 . 

2. 

3. 

4. 

5. 

I varianti 


II varianti 

ipoveT Semdegi ricxvebis da ricxviTi gamosaxulebebis mniSvnelobebi 

measedamde sizustiT: 

daalageT xarisxebi zrdadobis mixedviT: 

sul: 20 qula 


daasaxeleT erTi racionaluri da erTi iracionaulri ricxvi, romelic mdebareobs 6,28-sa da 

2π-s Soris (2 qula) 

– 3 

3 . 

10 -2 ; - ; ; ; ; ; 

amoxseniT Semdegi gantolebebi: 

= x – 2 = x – 1 

y4 + 7y = 0 y5 – 6y2 = 0 

amoxseniT utolobaTa sistema: 

ipoveT α, Tuα 0 -iani seqtoris farTobi da wris radiusia Sesabamisad: 

12πsm 2 da 6sm 5πsm 2 da 10sm 

qula 


2 

3 

3 

4 

2 

3 

3 

41


sakontrolo samuSao # 3 (2 sT.) 

# I varianti 

II varianti 

1. 

a) 

b) 

g) 

2. 

3. 

4. 

5. 

vTqvaT m=n+10, sadac n Tqveni rigiTi nomeria saklaso JurnalSi. 

CawereT m ricxvi orobiT poziciur sistemaSi; 

gamoTvaleT 27(mod m); 

romeli cifriT bolovdeba 2 m+2000 -is mniSvneloba? 

vTqvaT, megobrisagan miiReT orgverdiani werili, 25 striqoniT TiToeulze. 

striqonSi simboloebis raodenoba. 

40 60 

gamoTvaleT kilobaitebSi miRebuli informaciis sidide. 

daadgineT gamosaxulebis niSani 

sin2000 · cos800 sin900 · cos300 O 1 da O 2 tolradiusiani wrewirebis centrebia. gamoTvaleT 

marTkuTxedis gaferadebuli nawilis farTobi mTelebis 

sizustiT m 2 -ebSi, Tu am marTkuTxedis sigrZe da siganea 

7 m da 4 m 9 m da 6 m 

cilindruli formis ori qvabi toli simaRlisaa. amasTan, pirveli maTganis 

fuZis diametri k-jer metia meorisaze. ramdenjer meti wyali Caeteva pirvelSi, 

Tu 

k = 4 k = 2 

(cilindris moculoba tolia fuZis farTobis X simaRleze namravlis) 

O 1 

O 2 

qulebi 

2 

2 

3 

2 

3 

4 

4 

sul 20 qula 

damatebiTi amocana: 100 gramiani gafcqvnili forToxali 99% wyals Seicavs. aorTqlebis 

Sedegad wylis Semcveloba masSi 98% gaxda. ramdens iwonis forToxali axla?


sakontrolo samuSao # 4 

# I varianti II varianti 

qulebi 

1. Semdegi ricxvTa wyvilebidan: 

2 

(0; -25), (1; -1) (-3; 5), (3; 11) (1; 1), (2; 5), (-2; 3), (-1; 1) 

2. 

3. 

4. 

a) 

b) 

romeli ar aris amonaxsni orcvladiani gantolebisa: 

3x 2 –2y=5 2x 2 –3y+1=0 

amoxseniT Semdegi gantolebaTa sistemebi: 

mocemulia wrewiri, romlis centria (-1; 0) wertili, xolo radiusia 3 

erTeuli. dawereT am wrewiris mxebis gantoleba, romelic gadis: 

(-4; 0) wertilze 

(-1; -3) wertilze 

Tu markuTxedis formis miwis nakveTis sigrZesa da siganes Sesabamisad gavzrdiT: 

10 metriTa da 5 metriT, 5 metriTa da 10 metriT, 

farTobi gaizrdeba: 

550 m 2 -iT 600 m 2 -iT 

ipoveT am marTkuTxedis sigrZe da sigane, Tu misi farTobia 0,12 heqtaria. 

bankSi erTi wlis win Setanili Tanxis mogebam 2000 lari Seadgina, amitom Tamarma es fulic 

Tavis angariSze datova bankSi da iangariSa, rom erTi wlis Semdeg igi sul 24200 lars gamoitanda. 

ra Tanxa Seitana bankSi Tamarma erTi wlis win? (2 qula) 

1. davalebis Sinaarsidan gamomdinare, 2 qulis damsaxurebisaTvis sakmarisia, rom moswavlem 

sworad daafiqsiros pasuxi: (-3; 5) an (2; 5) _ II variantSi). aq mas damatebiTi msjelobis an 

dasabuTebis warmodgena ar moeTxoveba, Tumca aseTi msjeloba, cxadia, mas Secdomad ar 

CaeTvleba. amitom, Tu moswavlem aseTi msjeloba SecdomebiT Cawera, an 2-3 wyvilisTvis mainc 

aCvena, rom isini mocemul gantolebas akmayofilebs, mxolod 1 qula daiwereba. 

2. (a) da (b) sistemebisaTvis Sefasebis sqema erTnairia: erT-erTi cvladis meore cvladiT 

gamosaxva fasdeba 1 quliT; miRebuli mniSvnelobis (Tundac SecdomiT gamosaxulis) pirvel 

gantolebaSi Casma da kvadratuli gantolebis miReba _ 1 quliT; kvadratul gantolebis sworad 

amoxsna _ 1 quliT; sistemis TiToeuli amonaxsnisTvis moswavles TiTqo qula dewereba. 

3. wrwewiris gantolebis sworad Cawera fasdeba 1 quliT; imis aRmoCena (dasabuTeba), rom 

mocemul wrewirze mdebareobs _ 1 quliT (es SesaZlebelia mxolod grafikulad aCvenos; wrfis 

gantolebis sworad Cawera _ 1 quliT. 


5 

5 

3 

5 

sul 20 qula 

43

4. I gantolebis Cawera fasdeba 1 quliT, II gantolebis Cawera _ 1 quliT, gantolebaTa sistemis 

amoxsna _ 2 quliT, pasuxis dafiqsireba _ 1 quliT. 

amgvarad, ZiriTadi samuSao 20-quliania, amitom moswavlis namuSevris Sefasebisas 

portfolioSi vafiqsirebT mis mier dagrovil qulaTa jamis naxevars (cxadia, Tu igi 0 ar aris!) 

rac Seexeba 2-qulian damatebiT amocanas, is im moswavleebisTvis aris gaTvaliswinebuli, 

romlebic adre daamTavreben ZiriTad samuSaos. cxadia, 21 an 22 qulis dagrovebis SemTxvevaSi 

moswavles ver SevafasebT 11 quliT, magram klasSi gamocxaddeba aseTi sasiamovno precedentis 

Sesaxeb. 

# 

1. 

2. 

3. 

4. 

b) 


mocemulia wrewiri, romlis centria (-1; 1) wertili, xolo radiusia 2 erTeuli. dawereT 

utolobaTa sistema, romlis grafikia romelime iseTi AB segmenti, sadac AB=900 sul: 20 qula 

. (2 qula) 

am sakontrolo samuSaos sakiTxebis qulobrivi Sefasebis sqema aseTia: 

1. TiToeuli sworad moZebnili amonaxsnisaTvis vwerT TiTo qulas. 

2. aRniSvnebis sworad SemoReba da ⎨ 

⎩ ⎧5 

≤ x ≤ 6 ⎛4 

< x < 5⎞ 

⎜ ⎟ tipis utolobaTa sistemis Cawera 

7 ≤ x ≤ 8 ⎝6 

< x < 7⎠ 

(met-naklebobis simkacres yuradRebas nu mivaqcevT) fasdeba 2 quliT. 

⎧25 

≤ 5x 

≤ 30 

⎨ 

tipis Canaweri _ 2 quliT 

⎩35 

≤ 5y 

≤ 40 

60 ≤ 5x + 5y ≤ 70 _ am tipis Canaweri da pasuxSi `60-dan 70 tonamde~ dafiqsireba _ 1 quliT. 



I varianti II varianti 

daasaxeleT romelime ori amonaxsni Semdegi utolobisa: 

3x 2 – 5xy < 4 4x 2 – 3xy > 3 

samSeneblo masalebis gadasazidad gamoiyenes sxvadasxva simZlavris ori 

TviTmcleli avtomanqana. maTgan pirvelze 

5-dan 6 tonamde 4-dan 5 tonamde 

xolo meoreze 

7-dan 8 tonamde 6-dan 7 tonamde 

tvirTi eteoda. gadazidvebs am ori manqaniT erTdroulad axorcielebdnen, 

amitom am samSeneblo masalebis gadazidvas sul 

5 reisi 6 reisi 

dasWirda. ramdeni tona samSeneblo masala SeiZleba gadaezidaT aseT pirobebSi? 

amoxseniT grafikulad Semdegi utolobaTa sistemebi: 

a) 

dawereT utolobaTa sistema, romlis grafikia AB monakveTi, Tu cnobilia, 

rom 

A(–3; 2) da B(1; –2) A(1; –2) da B(–1; 3) 

2 

5 

4 

5 

4

3. a) TiToeuli wrfis sworad ageba TiTo quliT Sefasdeba, TiToeuli utolobis amonaxsnTa 

simravlis sworad daStrixvis Sedegad swori pasuxis miReba (an mxolod pasuxis anu amonaxsnTa 

simravlis sworad daStrixva) _ 2 quliT. Tu moswavlem wyvetili wrfis nacvlad wre warmoadgina, 

mas 1 qula akldeba. 

b) tolobaTa sistemaSi mocemuli TiToeuli wrfis sworad ageba fasdeba 3 quliT, xolo 

sistemis amonaxsnTa simravlis sworad warmodgena _ 2 quliT. 


# I I varianti 

varianti 

II II varianti varianti 

qulebi qulebi 

qulebi 

1. 

2. 

3. 

mocemulia f : A → B asaxva, romelSic 

A = {mtkvari, volga, enguri, 

nilosi, doni} 

B = {xmelTaSua, Savi, kaspiis} 

simravleebi Sesabamisad mdinareebi 

da zRvebia. 

A = {kievi, ankara, romi, madridi, 

deli} 

B = {espaneTi, indoeTi, 

TurqeTi, ukraina, egvipte} 

Sesabamisad, dedaqalaqebisa da qveynebis 

simravleebia, xolo f asaxva A 

simravlis nebismier elements Seusabamebs 

im zRvas, romelSic is Caedineba, 

qveyanas, romlis dedaqalaqicaa. 

aRwereT es asaxva cxriliT da daadgineT, aris Tu ara f urTierTcalsaxa? 

ipoveT y = f(x) funqciis: a) gansazRvris are da mniSvnelobaTa simravle; 

b) nulebi; g) niSan-mudmivobis Sualedebi; d) zrdadobisa da klebadobis 

Sualedebi: e) Seqceuli funqcia, Tu 

x + 3 

x − 2 

f ( x) 

= f ( x) 

= 

x −1 

x + 1 

v) aageT am funqciis grafiki. 

mocemulia f : D → E funqcia, romelSic 

D = [-1; 3] 

f(x) = 2x – x 2 

D = [-1; 4] 

f(x) = 3x – x 2 

ipoveT: zrdadobisa da klebadobis Sualedebi, udidesi da umciresi 

mniSvnelobebi. 

sul 20 qula. 

damatebiTi damatebiTi amocana 

amocana 

ipoveT im piramidis zedapiris farTobi, romlis fuZea 8 sm da 12 sm sigrZis gverdebis mqone 

marTkuTxedi, romlis gverdiTi wibos sigrZea 10 sm. 

sakontrolo sakontrolo namuSevrebis namuSevrebis Sefasebis Sefasebis sqema sqema aseTia: 

aseTia: 

1. funqciis cxrilis saxiT sworad warmodgena 2 quliT fasdeba, xolo urTierTcalsaxobis 

Sesaxeb swori pasuxi _ 2 quliT. 

2. gansazRvris aris sworad dadgena fasdeba 1 quliT, mniSvnelobaTa simravlisa _ 2 quliT, 

zrdadoba-klebadobis Sualedebis warmodgena _ 2 quliT, nulebis dadgena _ 2 quliT, Seqceuli 

funqciis dadgena _ 3 quliT, funqciis grafikis ageba _ 2 quliT. 


4 

12 

4 

45

3. klebadobis Sualedebis dadgena _ TiTo quliT, udidesi da umciresi mniSvnelobebis povna 

_ TiTo quliT. 

rogorc yovelTvis, am sakontrolo samuSaosTvisac viTvaliswinebT orqulian damatebiT 

samuSaos (ar dagvaviwydes, rom es 2 qula sirTulis Sesabamisi ar aris. realurad, zogierTi 

aseTi amocana 4 da meti quliTac SeiZleboda Sefasebuliyo). 

# 

1. 

2. 

3. 

4. 

5. 

6. 



I I varianti varianti varianti 

II II varianti varianti 

qulebi 

qulebi 

gamoTvaleT usasrulo klebadi geometriuli progresiis wevrTa jami, 

romelSic 

b = 1, 

b = 2, 1 

1 

q = - 0,2 

q = – 0, 1 

ipoveT (a ) ariTmetikuli progresiis pirveli 10 wevris jami, Tu 

n 

a + a = 29 

5 2 a + a = 30 

2 6 

a + a = 35 

3 6 a + a = 46 

3 9 

ipoveT (b ) geometriuli progresiis pirveli wevri da mniSvneli, Tu 

n 

b b = 324 

2 4 b b = 144 

2 4 

b + b = 20 

1 3 b + b = 15 

1 3 

Semdegi mimdevrobebisaTvis dawereT rekurentuli da cxadi formulebi 

5; 8; 11; 14; .... 

3; 7; 11; 15; ... 

mTel ricxvamde sizustiT moZebneT im oqros marTkuTxedis didi gverdi, 

romlis mcire gevrdis sigrZea 

15 erTeuli 

18 erTeuli 

a) 8 ⋅ 1020 – 5 ⋅ 1019 SeasruleT naCvenebi moqmedebebi. Sedegebi warmoadgineT standartuli 

(samecniero) formiT 

a) 5 ⋅ 1017 + 4 ⋅ 1016 b) (5 ⋅ 1017 ) ⋅ (4 ⋅ 1016 ) 

g) (4 ⋅ 1016 ) : (5 ⋅ 1017 ) 


b) (8 ⋅ 1020 ) ⋅ (5 ⋅ 1019 ) 

g) (5 ⋅ 1019 ) : (8 ⋅ 1020 ) 

3 

4 

5 

3 

2 

1 

1 

1 

sul 20 qula 

5 7 

mocemul (a ) mimdevrobis pirveli wevris jamis gamosaTvleli formulaa S = 

n n 

3 

2 + 

. 

aris Tu ara es mimdevroba ariTmetikuli an geometriuli progresia? pasuxi daasabuTeT. 

(2 qula) 

n −

# 

1. 

2. 

3. 

I I varianti varianti varianti 

II II varianti 

varianti 

a) mocemuli xuTkuTxedis gardasaqmnelad 

mis kongruentul xuTkuTxedad: 


a) aageT Δ ABC, romlis wveroebis koordinatebia: 

A (-1; -2), B (2; 2) da C (2; -2); A (-3; 4), B (3; -4) da C (3; 4); 

b) aageT S (Δ ABC) = Δ A B C o 1 1 1; 

g) dawereT A , B da C wertilebis koordinatebi, Tu 

2 2 2 

Δ A B C = S (Δ ABC) da S (Δ ABC) = Δ A B C da 

2 2 2 x y 2 2 2 

x _ abscisaTa RerZia. y _ ordinatTa RerZia. 

d) ramdeni kvadratuli erTeulia A B C -is farTobi? 

2 2 2 

mocemulia wrfe, romlis gantolebaa 

y = -2x + 4. y = 3x - 6. 

a) dawereT am funqciis Seqceuli funqciis gantoleba; 

b) dawereT mocemuli wrfis simetriuli wrfis gantoleba, Tu S simetriis 

l 

l RerZis gantolebaa y = x; 

90 

g) dawereT im wrfis gantoleba, romelic miiReba mocemuli wrfis Ro mobrunebis Sedegad; 

d) romel wrfed gardaiqmneba mocemuli wrfe ƒ (-1; 1) gardaqmniT? 

α S , S ƒ (a; b) da R saxis gadaadgilebebidan SearCieT isini, romelTa kompo- 

o l o 

zicia nebismier SemTxvevaSi sakmarisia 

mocemuli SvidkuTxedis gardasaqmnelad 

mis kongruentul 

SvidkuTxedad; 

b) daasaxeleT am kompoziciebis kidev ori gansxvavebuli varianti. 

qulebi 

sul 20 qula 


cal-calke aRwereT S A , S l da ƒ (a; b) gadaadgilebebi (anu ipoveT: A centris koordinatebi, l wrfis 

gantoleba da a da b ricxvebi), romlebic (x-2) 2 + (y + 1) 2 = 5 wrewirs gardaqmnis (x + 3) 2 + (y - 1) 2 = 5 

wrewirad. (2 qula) 

rogorc vxedavT, am sakontrolo samuSaos Sefasebis sqema sakmaod gaamartiva 2-qulian sakiTxebad 

dayofam. es saSualebas gvaZlevs, zogierT SemTxvevaSi 2-quliani sakiTxis Sesruleba 1 quliTac 

SevafasoT. jamSi: a) 10-baliani Sefasebisas dagrovili qulebi 2-ze gaiyofa da erTeulebamde 

sizustiT damrgvaldeba; b) 5-quliani (anu tradiciuli) Sefasebisas ki _ 4-ze gaiyofa da erTeulebamde 

sizustiT damrgvaldeba. 

moswavleTa damatebiTi muSaobis wasaxaliseblad `damatebiTi amocanis~ amoxsniT damsaxurebuli 

1 an 2 qula moswavlis mier ZiriTad davalebaSi dagrovil qulebs emateba (ase SesaZlebelia, rom 

moswavlem 21 an 22-quliani 10-iani miiRos). 


2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

47


daxazeT 5 -wveroiani grafi, romlis wveroTa xarisxebia 1, 2, 2, 3, 4. (2 qula) 



# I varianti II varianti qulebi 

1. 

2. 

3 a) 

b) 

g) 

4. 

5. 

6. 

(O; R) wrewiris AB qordas diametri yofs or monakveTad, romelTa sigrZeebia 

3 sm da 4 sm, xolo R = 4 sm. 

4 sm da 5 sm, xolo R = 6 sm. 

ipoveT am qordis gadamkveTi diametris monakveTebis sigrZeebi. 

ABC samkuTxedSi BD biseqtrisaa. ipoveT AD, Tu 

AB = 6 sm, BC = 8 sm da AC = 7 sm. AB = 9 sm, BC = 12 sm da AC = 14 sm. 

aageT romelime wrewiri da masSi Caxazuli wesieri Δ ABC; 

2 3 aageT H (Δ ABC); aageT HB (Δ ABC); 

A 

ipoveT miRebuli samkuTxedisa da mocemuli samkuTxedis farTobebis Sefardeba. 

aTeulebamde damrgvalebuli ramdeni lari unda gamoeweros uwyisSi 

TanamSromels, rom 12%-iani saSemosavlo gadasaxadis daqviTvis Semdeg 

man xelze miiRos 

400-dan 420 laramde Tanxa. 580-dan 600 laramde Tanxa. 

ramdeni gansxvavebuli samniSna ricxvi SeiZleba Caiweros 

0, 1, 2 cifrebis gamoyenebiT? 

A qalaqidan B qalaqamde TviTmfrinaviT mgzavroba 370 lari jdeba, A-dan 

C-mde _ 550 lari, A-dan D-mde _ 300 lari, C-dan B-mde _170 lari, C-dan D-mde 

_ 260 lari. SearCieT A-dan C-mde misasvleli yvelaze iaffasiani marSruti. 

3 

3 

2 

2 

2 

3 

2 

3 

sul 20 qula




1. 

2. 

3. 

4. 

ipoveT A xdomilobis xelSemwyob 

elementarul xdomilobaTa ricxvi. 

cda: isvrian patara da didi zomis kama- 

Tels. 

A = {qulaTa jamia 3 an 4} 

ipoveT A da B xdomilobaTa namravli 

da gaerTianeba. 

A = {ori kamaTlis gagorebisas erTerT 

kamaTelze dajda `1~} 

B = {ori kamaTlis gagorebisas erTerT 


urnaSi 5 wiTeli da 6 Savi burTulaa. 

amoRebuli burTula wiTeli aRmoCnda, 

burTula isev Cades urnaSi. ras udris 

albaToba imisa, rom meored amoRebuli 

burTula isev wiTeli iqneba? 

aageT y=f(x) funqciis grafiki, Tu 

2x 8x 

f(X) = 

x 4 

2 

− 

− 

`jokeris~ TamaSisas (romelSic 36 `kartidan~ 2 `jokeria~) or moTamaSes daurigda 9-9 `karti~. 

ipoveT albaToba imisa, rom: 

a) orive jokeri erT-erTi moTamaSis xelSia; 

b) jokerebi ganawilda, romeliRac or moTamaSes Soris. 

(2 qula) 

am sakontrolo samuSaos Sefasebis sqema aseTia: 

1. a) A xdomilobis Semadgeneli yvela elementaruli SemTxveviTi xdomilobebis ricxvis 

dadgena fasdeba 2 quliT; 

b) A-s xelSemwyob xdomilobaTa ricxvis dadgena _ 2 quliT; 

g) A xdomilobis albaTobis dadgena _ 1 quliT. 

SeniSvna: (a) da (b) punqtebSi mcire uzustobebis gamo vaklebT TiTo qulas. 

2. a) A-s albaTobis gamoTvla fasdeba 1 quliT; 

b) B-s albaTobis gamoTvla _ 1 quliT; 

g) A∪ B-s albaTobis gamoTvla _ 1 quliT; 

d) A∪ B -s albaTobis gamoTvla _ 1 quliT. 

3. msjelobis TiToeuli safexuris warmodgena fasdeba 1quliT. 

ipoveT A xdomilobis xelSemwyob 

elementarul xdomilobaTa ricxvi. 

cda: isvrian wiTel da lurj kamaTels. 

A = {qulaTa jamia 2 an 5} 

ipoveT A da B xdomilobaTa namravli 

da gaerTianeba. 

A = {ori kamaTlis gagorebisas erTerT 


B = {ori kamaTlis gagorebisas erTerT 


urnaSi 5 wiTeli da 6 Savi burTulaa. 

amoRebuli burTula Savi aRmoCnda. igi 

urnaSi ar daabrunes. ras udris albaToba 

imisa, rom meored amoRebuli burTula 

isev Savi iqneba? 

aageT y=f(x) funqciis grafiki, Tu 

f(X) = 

x 2x 

0, 

5x 

2 + 

4. a) D(f)-is sworad dadgena fasdeba 1 quliT; 

b) wiladis Sekvecis sworad ganxorcieleba _ 1 quliT; 

g) miRebuli wrfivi funqciis grafikis (àmogdebuli~ wertiliT) sworad ageba _ 1 quliT 

(àmogdebuli~ wertilis daufiqsirebloba isjeba am 1 qulis daklebiT). 


5 

4 

3 

3 

sul 15 qula 

49




1. 

2. 

3. 

4. 

demografebi amtkiceben, rom biWis 

dabadebis albaToba daaxloebiT 0,51 

tolia. 10 000 mSobiarobidan ramdenjeraa 

mosalodneli biWis gaCena? 

ujraSi 50 erTnairi detalia, romel- 

Tagan 5 SeRebilia. ras udris albaToba 

imisa, rom alalbedze amoRebuli detali 

ar iqneba SeRebili? 

Tofidan srolisas moxvedraTa fardobiTi 

sixSire 0,85-is toli aRmoCnda. 

gamoTvaleT moxvedraTa ricxvi, Tu 

sul 120 gasrola dafiqsirda. 

daamtkiceT, rom ar arsebobs naxevrad 

wesieri teselacia, romlis saxelia 

4.5.6. 


demografebi amtkiceben, rom tyupebis 

dabadebis albaToba daaxloebiT 

0,012 tolia. 20 000 mSobiarobidan ramdenjeraa 

mosalodneli tyupebis ga- 

Cena? 

klasSi 32 moswavlea, romelTagan 17 

gogonaa. ras udris albaToba imisa, rom 

dafasTan biWs gamoiZaxeben? 

mSvildidan srolisas moxvedraTa 

fardobiTi sixSire 0,75-is toli aRmoCnda. 

gamoTvaleT moxvedraTa ricxvi, Tu 

sul 200 gasrola dafiqsirda. 

daamtkiceT, rom ar arsebobs naxevrad 

wesieri teselacia, romlis saxelia 

5.6.7. 

2 

3 

2 

3 

sul 10 qula 


1. 

a) 

b) 

g) 

2. 

a) 

b) 

sqematuri gafikebis gamoyenebiT amoxseniT Semdegi gantolebebi da 

utolobebi 

x 3 + x 2 + 4 = 0 x 3 − x 2 − 18 = 0 

x2 6 10 

2 − 1 = x + 1 = 

x 

x 

x 3 ≥ 12 − x 2 3 x + x + 2 ≥ 0 

daadgineT, ramdeni amonaxsni aqvT Semdeg gatolebebs: 

3 x + 1 + x = 2 x3 + x2 = 4 

x3 5 

= 

x 

6 

3 x + 1 = 

x 

3 

3 

3 

3 

2

gagrZeleba 


3. 

a) 

b) 

g) 

d) 

4. 

a) 

b) 

d) 

d) 

5. 

qvemoT CamoTvlili winadadebebidan romelia WeSmariti? 

zogierTi ori kenti ricxvis sxvaobac 

kenti ricxvia 

zogierTi ori martivi ricxvis namravli 

martivi ricxvia 

martivi ricxvebis sxvaoba SeiZleba 

martivi ricxvi aRmoCndes 

nebismieri ori martivi ricxvis jami 

Sedgenili ricxvia 

qvemoT CamoTvlili winadadebebidan romelia mcdari? 

samkuTxedis ori gverdis namravlis 

naxevari mis farTobs ar aRemateba 

nebismeri prizmis wveroebis raodenoba 

luwi ricxvia 

zogierTi samkuTxedi tolferdac 

aris da marTkuTxac 

nebismieri samkuTxedis samive simaRle 

erT wertilSi ikveTeba 

dawereT Semdegi debulebis Sebrunebuli winadadeba: 

Tu samkuTxedis ori gverdis sigrZe- 

Ta kvardatebis jami mesame gverdis sigrZis 

kvadratebis tolia, maSin es samkuTxedi 

marTkuTxaa 

zogierTi ori luwi ricxvis namravli 

kenti ricxvia 

zogierTi ori luwi ricxvis ganayofi 

kenti ricxvia 

martivi ricxvebis sxvaoba yovel- 

Tvis luwi ricxvia 

yvela martivi ricxvi kentia 

paralelogramis ori gverdi namravli 

mis farTobs ar aRemateba 

nebismieri piramidis wveroebis raodenoba 

luwi ricxvia 

zogierTi samkuTxedis sami simaRle 

erT wertilSi ar ikveTeba 

nebismieri samkuTxedsis samive biseqtrisa 

erT wertilSi ikveTeba 

Tu trapeciis diagonalebi kongruentulia, 

maSin is tolferdaa 

rogorc SesavalSive aRvniSneT, saswavlo wlis bolo 4-5 gakveTili daeTmoba Semajamebel samu- 

Saoebs. amaSi maswavlebels did daxmarebas gauwevs saxelmZRvanelos boloSi mocemuli 8 testi. 

maTgan me-7 da me-8 testi 3-saaTiania (iseve, rogor cerovnul gmocdebze). moswavleebs, romelTac 

surT Tavi gamoscadon aseTi tipis samuSasaTvis mzaobaSi, SeuZliaT Sin Seasrulon erT-erTi am 

testebidan. maswavlebels SeuZlia, magaliTad, me-8 testi or nawilad gayos: pirveli nawili klasSi 

daawerinos, xolo meore _ saSinao davalebad misces. aq mTavari gamovlenil xarvezebze muSaobaa. 

ver daveTanxmebiT zogierTi maTematikosis mosazrebas imis Taobaze, rom xarvezebze muSaoba saswavlo 

wlis bolo nawilSi aRar unda davisaxoT miznad. migvaCnia,rom xarvezebze muSaoba mudmivi 

procesi unda iyos _ saswavlo wlis dasawyisidan bolo zaramde. 

cxadia, yvela maswavleblis ocnebaa maTi moswavleebis uxarvezebo namuSevrebis xilva. Cvenc 

wlis bolos aseTi Sedegebis miRwevas gisurvebT. 


2 

2 

2 

sul 20 qula 

51

3. a) = x − 2 

x 3 − 26 = (x − 2) 3 = x 3 − 6x 2 + 12x − 8 

6x 2 − 12x − 18 = 0 

x 2 − 2x − 3 = 0 

x = − 1; x = 3 

pasuxi: x = − 1; x = 3; 

4. (I varianti) 

roca y < 1 ⇒ − (y −1) ≤ 2 ⇒ y ≥ −1 

amonaxsnia [−1; 1) 

roca y ≥ 1 ⇒ y − 1 ≤ 2; ⇒ y ≤ 3 

amonaxsnia [1; 3] 

⏐y − 1⏐≤ 2 amonaxsnia [−1; 3] 

roca y < −2 ⇒ − (y +2) >1,5; y < −3,5 

roca y ≥ −2 ⇒ y + 2 >1,5; y > −0,5 

⏐y + 2⏐ > 1,5 amonaxsnia (−∞; − 3,5) ∪ (−0,5, +∞) 

⎪⎧ 

y −1 

≤ 2 

e. i. ⎨ 

⎪⎩ y + 2 > 1, 

5 

pasuxi ( −0,5; 3] 

⇒ 

⎧ 

⎨ 

⎩ 

π 6 α 

5. radgan 

360 

2 

= 12π 

e. i. α = 12 . 10 

α =120 

pasuxi: α = 120. 


zogierTi sakontrolo samuSaos amoxsnis nimuSi: 

[ −1; 

3] 

( − ∞; 

−3, 

5) 

∪ ( − 0, 

5; 

+∞) 


damatebiTi amocana. 

aviRoT π ≈ 3,1415 maSin 2π ≈ 6,28318 

cxadia 6,28 da 6,28318 Soris moTavsebulia 6, 281 da sxva 

π 

6,28


100 g wyalSi iqneba 1% forToxali anu 1 g. wylis aorTqlebis Sedegad forToxali iqneba 2%. 

vTqvaT miviReT x g. cxadia masSi iqneba 2% forToxali anu 1 g. e. i. = 1 ⇒ x = 50 g. 

pasuxi: 50 g. 


2 ⎧( 

x + 2 ) + ( y −1) 

⎨ 

⎩2x 

+ 3y 

= 5 

⎧13 

+ 28x 

+ 4 = 0 

⎪ 

⎨ 5 − 2x 

⎪y 

= 

⎩ 3 

2 

x 

2 

= 4 

x1 

⎧ 2 5 − 2x 

⎪( 

x + 2 ) + ( −1) 

3 

⎨ 

⎪ 5 − 2x 

y = 

⎪⎩ 

3 

⎨ 

⎩ ⎧ = −2 

y1 

= 3 

2 23 

pasuxi: {(_2; 3) (_ ; )} 

13 13 

3) (I varianti) 


⎧xy 

+ 550 = ( y + 11)( 

x + 5) 

⎨ 

⎩xy 

= 1200 

⎧ 

⎪ 

x 

⎨ 

⎪y 

⎪⎩ 

aSkaraa, rom (_4; 0) ekuTvnis 

(x +1) 2 + y 2 = 9 wrewirs. 

marTlac, (_4+1) 2 + 0 2 = 9 + 0 = 9 

saZiebeli mxebis gantolebaa 

x = _4. 


x 

2 

2 

2 

= − 

13 

23 

= 

13 


x = _4 

− 4 

2 

= 4 

pasuxi: x = 20 da y = 60 

vTqvaT erTi wlis win Seitana x lari mogeba iqneba 

y 

y + 10 

x + 5 

2. (II varianti) 

⎧ 2 3x 

+ 1 

2 

2 ⎪ 

( x −1) 

+ ( y + ) 

⎧( 

x −1) 

+ ( y + 1) 

= 9 

2 

⎨ 

⎨ 

⎩3x 

+ 2y 

= −1 

⎪ 3x 

+ 1 

⎪ 

y = 

⎩ 2 

⎧13 

+ 10x 

− 23 = 0 

⎪ 

⎨ 3x 

+ 1 

⎪y 

= 

⎩ 2 

2 

x 

x1 

⎨ 

⎩ ⎧ = 1 

y1 

= 2 

23 28 

pasuxi: {(1; 2) (_ ; _ )} 

13 13 

y 

↑ 

1 

O 1 

y = _3 


→ 

x 

⎧xy 

+ 600 = ( y + 5)( 

x + 10) 

⎨ 

⎩xy 

= 1200 

⎧ 

⎪ 

x 

⎨ 

⎪y 

⎪⎩ 

= − 

= − 

3) (II varianti) 

23 

13 

28 

13 

= 9 


2 

2 

(_1; _3) ekuTvnis 

(x +1) 2 + y 2 = 9 wrewirs. 

saZiebeli mxebis gantolebaa 

y = _3. 

pasuxi: x = 30 da y = 40 

200000 

%. Tu bankSi aqvs x + 2000 lari, Semdegi 

x 

200000 

( x + 2000) 

erTi wlis Semdeg igive procents moimatebs anu eqneba x + 2000 + 

x = 24200 

100 

x = 20000 lari 

x 

y 

y + 5 

2 

x + 10 

53


vTqvaT pirvel avtomanqanaze eteva xt, 

xolo meoreze yt. maSin 

5 ≤ x < 6 da 7 ≤ y < 8 

erTi reisiT gadava 12 ≤ x+y 2x 

− 4 

⎪ 

⎨ 1 

⎪y 

≤ x − 2 

⎩ 3 

12345678901234567 

12345678901234567 

12345678901234567 

12345678901234567 

12345678901234567 

12345678901234567 

12345678901234567 

12345678901234567 

12345678901234567 

12345678901234567 

→ 

x 

12345678901234567 

12345678901234567 

12345678901234567 

12345678901234567 

12345678901234567 

AB wertilze gamavali wrfis gantolebaa 

y = − x −1 

⎧y 

≥ −x 

−1 

⎪ 

y ≤ −x 

−1 

⎨ 

pasuxi: ⎪x 

≤ 1 

⎪ 

⎩x 

≥ −3 


y 

↑ 

4 

1 

1234567890 

1234567890 

1234567890 

O 1 

1234567890 

1234567890 

1234567890 

1234567890 

1234567890 

1234567890 

1234567890 

1234567890 

1234567890 

1234567890 


vTqvaT pirvel avtomanqanaze eteva x t, xolo 

meoreze – y t. maSin 

4 ≤ x < 5 da 6 ≤ y < 7 

erTi reisiT gadaitanen 10 ≤ x+y < 12 

eqvsi reisiT ki 60 ≤ 6(x+y) < 72 

pasuxi: eqvsi reisiT SeiZleba gadaitanon 

60-dan 72 tonamde tvirTi. 


a) 

4 

⎧2x 

− y < 4 

⎨ 

⎩x 

− 3y 

≥ 6 


y 

↑ 

1234567890123456 

1234567890123456 

1234567890123456 

1234567890123456 

1234567890123456 

1234567890123456 

1234567890123456 

1234567890123456 

1234567890123456 

1234567890123456 

1234567890123456 

1234567890123456 

1234567890123456 

→ 

x 

⎧y 

> 2x 

− 4 

⎪ 

⎨ 1 

⎪y 

≤ x − 2 

⎩ 3 

1 

O 1 

→ 

x 

AB wertilze gamavali wrfis gantolebaa 

5 1 

y = − x + 

2 2 

⎧ 5 1 

⎪ 

y ≥ − x + 

2 2 

⎪ 

⎪ 5 1 

⎨y 

≤ − x + 

pasuxi: ⎪ 

2 2 

⎪x 

≤ −1 

⎪ 

⎩x 

≤ 1

2) ubralo dakvirvebiTac advilad SevamCnevT, rom isrebi saaTSi 2-jer Seadgenen erTmaneTTan 

90 0 -ian kuTxes. maTematikuradac, a n = 

3) moswavleebi advilad SeamCneven, rom isrebi 60 0 -ian kuTxes saaTSi orjer adgenen, e. i. dRe-Ra- 

meSi 24-jer. radgan 2 saaTze isrebi 60 0 -ian kuTxes Seadgenen, momdevno aseTi momenti 21 wuTis 

(6a – 0,5a – 60 = 60, a = 21 ) Semdeg dgeba, kidev erTi aseTi momenti ki TiTqmis 1 saaTis Semdeg dadgeba. 

amrigad, es procesi araperiodulia. 

1.3. 24 km/sT = 400 m/wT, S = vt formulidan t = S/v 1000 : 400 = 2,5 (wT). 

10 br = 20 m, e. i. t = 20 : 400 = wT = 3 (wm) da a. S. 

zogierTi savarjiSos amoxsna 

1.1. Sesabamisad saaTis didi da patara isrebisaTvis viyenebT proporciebs: 

60 wT : 3600 = a wT. : xo , 60 wT : 300 = a wT : yo anu x = 6a, y = 0,5a 

a) 6 sT. 20 wT. 

1800 + 0,50 ⋅ 20 _ 60 ⋅ 20 = 1900 _ 1200 = 700 b) 6 sT. 30 wT. 

1800 + 0,50 ⋅ 30 _ 60 ⋅ 30 = 150 . 

1.2. radgan isarTa mobrunebis kuTxeebs Soris sxvaoba: 

0 

90 ( 2n 

−1) 

180 

180 540 900 

= ( 2n 

−1), 

a1 

= , a2 

= , a3 

= . 

5, 

5 11 

11 11 11 

360 

amgvarad, es momenti dRe-RameSi 48-jer dgeba da periodia wT. 

11 

1.5. es amocanebi emyareba samkuTxedis utolobasa da farTobis formulebs, radgan 

S = 0,5 ⋅ 6 ⋅ 10 sm2 ⋅ sinα = 30 sinα sm2 < 36, amitom pasuxia (g) Tu ΔABC-s SevavsebT ABCD parelelogramamde, 

maSin S = AB ⋅ BC ⋅ sin ∠B, erTi mxriv, namravli udidesia, roca 

da maSin S = x (16 – x) = –x2 + 16x. aq moswavleebs unda axsovdeT y = – x2 + 16x funqciis udidesi mniSvnelobis 

povna (am sakiTxebs Cven funqciebze msjelobisas mivubrundebiT). x = 8, e. i. AC = 8 sm. 

1.6. (3-quliani) SevadaroT trapeciis farTobi kvadratis farTobs: 

§1 

1) kenti 180 0 -ia, amitom 6a – 0,5a = 180(2n-1), e. i. a = , radgan Ramis 12 saaTis Semdeg pirveli 

aseTi momenti dgeba a 1 = wuTze (n = 1), me-n momenti ki a n = wuTis Semdeg, xolo a n+1 – a n 

= = , > 0. amitom, 24 saaTSi 24-jer dadgeba aseTi momenti (es pasuxi 

720 

moswavleebma dakvirvebis Sedegad mainc unda miiRon). am procesis periodia wT. 

11 

c 

h 

a 

b 

d 

St. = 

gamoviyenoT axla utoloba 

, anu 


55

⎛ a + b + c + d ⎞ 

miviRebT St. < ⎜ ⎟ 

⎝ 4 ⎠ 

⎛1000 

⎞ 

= ⎜ ⎟ 

⎝ 4 ⎠ 

= S kv. 

iolad SeiZleba Sedardes kvadratis da rombis farTobebic: 

marTkuTxedis Sedareba kvadratTan. 

250 – x 


2 

2 

S r = 250 . h < 250 . 250 = Skv. 

S m = (250 – x)(250 + x) = 250 2 – x 2 < 250 2 = Skv. 

amrigad, pasuxia b) 

1. 10. radgan α aris naxazze gaCenili blagvkuTxa samkuTxedis gare kuTxis zoma, amitom α > β. Tu 

naTura H simaRlezea, Cven erT SemTxvevaSi vdgavarT H simaRlis fuZidan a 1 manZilze, meore 

SemTxvevaSi ki _ b 1 manZilze. amitom tgα = , tgβ = da radgan b + b 1 > a + a 1 , gveqneba 

tgα > tgβ. meore mxriv, tgα = da tgβ = , e. i. > , saidanac b > a. 

2. 1. a) ; ; 

b) = ; 

g) = = ; 

2.2. a) = = = ; 

b) = 7 + + 2 + 7 _ = 14 + 2 . 6 = 26. 

§2 

2.4. radgan am kaTetis mopirdapire kuTxea 30 0 , amitom hipotenuzis sigrZea 6 sm ⋅ 2 = 12 sm, meore 

kaTetisa ki = 6 sm da p = (18 + 6 ) sm. 

2) 

250 

h 

250 

250 + x 

B C 

A D 

60 sm : 4 = 15 sm, e. i. AB = BC = 15 sm, h = 12 sm, 

sin ∠A = 12/5 = 0,8, AM = = 9 sm. 

tgα = BM : AM = 12 : 9 = 4 / 3

2.5. a) 

= – 2, 

x2 – 3 = – 8 

x2 = –5 

pasuxi: ∅ 

2. 9. 

b) y6 + 7y3 – 8 = 0 

3 y = – 8 

1 

3 y = 1 2 

y = –2 1 

y = 1 2 

pasuxi: {-2; 1}. 

g) 2 + 3 – 5 = 0 

= y 

2y 2 + 3y – 5 = 0 

y = 

y 1 = – y 2 = 1 

≠ – = 1 

pasuxi: {1}. 

x = 1. 

2.11. a) (4-quliani) – 5 7 = − 25⋅ 

7 = − 175; 

−14 

< − 175 < −13, 

e. i. m = – 14. 

a) (5-quliani) m – 47 < 2 < m – 46, magram 2 = < 5, e. i. m – 46 = 5, m = 51. 

b) (4-quliani) m + 11 < 2 < m + 12. magram 2 = 44 < 7, e. i. m + 12 = 7, saidanac m = – 5. 

b) (5-quliani) Tu m < 47 – 2 < m +1, maSin m – 47 < – 2 < m – 46, e. i. 46 – m < < 47 – m. 

magram 4 < < 5, saidanac 47 – m = 5, anu m = 42. 

2.13. aseTi α da β uamravia. vTqvaT, α = +1, β = –1, maSin α – β = 2; 

α 

= 

β 

+ 1 

= 

2 −1 

( 2 + 1) 

) = 3 + 2 2; 

2 2 

= – 3, 

x 2 – 12x + 27 = 0 

x 1 = 3, 

x 2 = 9. 

pasuxi: {3; 9} 

2.8. radgan 3 < < 4, amitom n = 3. 

α − β 2 

= = 2. 

αβ 1 

g) 

| 2 – | = 3 

2 – = –3; 

2 – = +3 

= 5; x = 125; 

= –1; x = –1. 

pasuxi: {–1; 125}. 

2) radgan 5 5 = 25 ⋅ 25 ⋅ 5 = 3125, xolo 4 5 = 16 ⋅ 16 4 = 24, amitom 4 < < 5 e. i. n = 4. 

3) radgan _ 3125 < _ 3000 < _ 1024, amitom _5 < < _4 e. i. n = – 4. 

§3 

| 3 + | =2 

3 + ≠ –2; 

3 + = 2 

≠ –1 

pasuxi: ∅. 

3.2. Tu sin∠A = 0,6 = 3 /5 , maSin avagebT marTkuTxa samkuTxeds, romlis erTi kaTetia 3 erTeuli, xolo 

hipotenuza – 5 erTeuli. Tu sin∠A = 2 /5 , maSin avagebT iseTi tolgverda samkuTxedis simaRles, 

romlis gverdi 4 erTeulia (maSin h = 2 erT.). Semdeg avagebT marTkuTxa samkuTxeds, romlis 

kaTeti da hipotenuzaa 2 da 5 erTeuli. 


57

3.3. (5-quliani) radgan didi kuTxis pirdapir didi gverdi Zevs, amitom saZiebeli A kuTxe mdebareobs 

AB = 12 sm da AC = 13 sm gverdebs Soris. 

heronis formulis Tanaxmad: 

B 

A 

12 7 

13 


S = = 4 . 2 . 3 (sm) 

e. i. . 13 . h = 24 da h = 

sinA = : 12 = 

pasuxi: sinA = 4 /13 . 

3.6. ucnobi manZilebis gazomva sxvadasxva gzebiT SegviZlia: 

1) tbaze: a) napirze SevarCioT iseTi C wertili, romlisTvisac ∠ACB = 90 0 (amas sarebis gamoyenebiT 

movaxerxebT) da SegveZleba AC da CB manZilebis gazomva, maSin AB = ; 

b) SegviZlia napirze SevarCioT iseTi C wertili, rom ∠ABC = 90 0 SegveZleba BC manZilisa da sin 

∠ACB-s gamoTvla. Tu AB = x, BC = a da sin ∠ ACB = m, maSin x : AC = m, da AC = da x 2 + a 2 = , saidanac 

vipoviT saZiebel x manZils. 

2) mdinaris SemTxvevaSic (es VI I klasis saxelmZRvaneloSicaa aRwerili) erT-erTi variantia napirze 

iseTi C wertilis moZebna, romlisTvisac ∠ BAC = 900 . gavzomavT AC manZils da gamovTvliT tg ∠ACB 

= m ricxvs, saidanac AB : AC = m da AB = m ⋅ AC. 

4.1. (4-quliani) radgan , α = πRα /180, amitom πR ⋅ 120 : 180 = 10 π tolobidan R =15 sm, e. i. d = 30 sm. 

(5-quliani) miTiTeba: ixileT 3.3 (5-quliani) 

4.5. radgan 3,1415 < π < 3,1416, 3,1428 < 3 < 3,1429 

h 

3,1408 < 3 < 3,1409, amitom 3,1428 _ 3,1415 < 3 _ π < 3,1429 _ 3,1415 da 

§4 

3,1415 _ 3,1409 < π _ 3 < 3,1416 _ 0,0012 < 3 _ π < 0,0014; 0,0006 < π _ 3 < 0,0008 

am utolobebidan cxadia, rom 3 ufro axlosaa π ricxvTan. 

radgan = , amitom mocemuli winadadeba WeSmaritia. 

§5 

5.1. (1) S = [(2R) 1 2 –πR2 ] : 4 = 0,25 (4 – π) ⋅ R2 , radgan R = 4 sm, amitom S = 4(4 –π) sm 1 2 . 

(2) S 2 = [πR 2 – (R ) 2 ] : 4. 

5.5. moswavleebi daadgenen eileris formulas: m + k – n = 2. 

C

1+ 

§6 

6.1. (1) naxazidan davaskvniT, rom samkuTxedi tolferdaa, erTi ferdis meoreze gegmili 

= 0,5 2 + 3 

2 

( 2 − 3) 

= 8 − 4 3 = 2 2 − 3 

-is tolia, e. i. fuZis ferdze gegmili (2 – )-ia. amitom tg15 0 = 2 – fuZis sigrZea 

. 

, xolo sin750 1 

2 − 3 

= = 0, 

5⋅ 

= 0, 

5 2 − 3 ( 2 + 3) 

2 

2 − 

3 

2 − 

AD : 15 = cos∠ A, amitom AD = 15 ⋅ 0,6 = 9 

15 

DC = 14 – 9 = 5; DB = 2 2 

15 − 9 = 12 

BC = 2 2 

9 + 12 = 15 

A 

D 

14 

C sin ∠ C = DB : BC = 0,8; cos ∠ C = 0,6 

(rogorc vxedavT, saxelmZRvaneloSi mocemul miTiTebas àr davujereT~. cxadia, im SemTxvevaSic 

aseTive procesi dagvWirdeba). 

§7 

7.1. (4-quliani) mediana 4-is tolia, saSualo ki daaxloebiT 4,3-ia. magram yvelaze xSirad marikam 

miiRo `5~ da Tu am faqtiT visargeblebT, agreTve niSnebis tendenciiT, maSin maswavleblis gadawyvetileba 

samarTliania. 

§8 

8.1. (4-quliani) miTiTeba: amocanas ori amonaxsni aqvs. erT-erTi maTgania 

(5-quliani) 

B 

123 

12 

12 

123 

12 

123 

123 

12 

123 

R U R U 

12 

12 

⇒ 

radgan 22 + 35 = 57, amitom 9 moswavlem (57_48) icis orive ena. 

8.2. (1) radgan |m| < 25, amitom – 25 < m < 25 da B = {–48; –46; ...; 48}. 

aqedan A∩B = {2; 4; ...; 48}, A∪B = {–48; –46; ...}; A\B = {50; 52; ...}; B\A = {–48; –46: ...; 0}. 

B simravlis elementebis raodenobaa 49, xolo A∩Bsimravlisa _ 24. 

(2) radgan |M| ≤ 50, amitom B = {–50; –49; ...; 50}; A = {...; –1; 0}. aqedan A∩B = {–50; –49; ...; 0} 

A∪B = {...; –1; 0; 1; ...50}, A\B = {...; –52; –51}, B\A = {1; 2; ...;50}. 

9.6. z) mricxveli da mniSvneli gavamravloT 3 2 3 ( 3 + 3 + 1) 

T) mricxveli da mniSvneli gavamravloT 3 3 ( 5 − 2) 

§9 

§10 

-ze 

10.1. 1001 2 = 1 ⋅ 2 3 + 1 ⋅ 2 0 ; 236 16 = 2 ⋅ 16 2 + 3 ⋅ 16 1 + 6 ⋅ 16 0 

∪ 

123 

10101 2 = 1 ⋅ 2 4 + 1 2 2 + 1 ⋅ 2 0 = 16 + 4 + 1 = 21 10 ; 

-ze 

3 

E 


= 

F A 

B 

O 

D 

C 

59

10.2. B5 16 = B ⋅ 16 1 + 5 ⋅ 16 0 = 11 ⋅ 16 + 5 = 176 + 5 = 181 10 

123F 16 = 1 ⋅ 16 3 + 2 ⋅ 16 2 + 3 ⋅ 16 1 + 15 ⋅ 16 0 = 4096 + 512 + 48 + 240 = 4896 10 

10.3. 1100 ⋅ 13 = (1 ⋅ 2 2 10 3 + 1 ⋅ 22 ) ⋅ 13 = 12 ⋅ 13 = 156; 

11 + 11A + 12 = (1 ⋅ 2 2 16 10 1 + 1 ⋅ 20 ) + (1 ⋅ 162 + 1 ⋅ 161 + 10 ⋅ 160 ) + 12 = 297; 

10.4. (am magaliTebSi mivaqcioT yuradReba, rom 10 aris orobiTi sistemis ricxvi) 

1⋅ 10 5 +1 ⋅ 10 4 = 110000 2 ; 1 ⋅ 10 6 + 1 ⋅ 10 4 + 1 ⋅ 10 2 = 1010100 2 ; 

1 ⋅ 10 7 + 1 ⋅ 10 3 + 1 ⋅ 10 0 = 10001001 2 

10.5. 24 10 : 2 24 gavyoT 2-ze 

12 0 ← ganayofi 12; naSTi 0. 12 gavyoT 2-ze; 

6 0 ← ganayofi 6, naSTi 0. 6 gavyoT 2-ze; 



0 1 ← ganayofi 0, naSTi 1. 

pasuxi: 24 = 11000. 

10.6 CCXXII = C + C + X + X + I + I =100 + 100 + 10 + 10 + 1 + 1 = 222; 

CXCIV = C + (C – X) + (V – I) = 100 + (100 – 10) + (5 – 1) = 194; 

MCMXIX = M + (M – C) + X + (X – 1) = 100 + (1000 –100) + 10+ (10 – 1) = ... 

10.7 111 = CXI; 555 = DLV; 999 = CMXCIX 

10.10. 2 3 = 2 1 ⋅ 2 2 ; 2 5 = 2 3 ⋅ 2; 2 9 = 2 ⋅ 2 8 da a. S. 

10.14 ΔB OA ⇒ x 1 2 + (2y) 2 = 1,52 ΔBOA ⇒ (2x) 1 2 + y2 = 22 A 

1,5 

C 

B 1 

O 

y 

A 1 

2y 

14 


x 

§11 

5x2 + 5y2 = 6,25; 

x2 + y2 = 1,25; 

ΔAOB ⇒ AB2 = (2x) 2 + (2y) 2 = 4(x2 + y2 ) = 5 

AB = 

11.2. 1 -s mosdevs 10 ; 2 2 1 -s mosdevs 2 ; 8 8 F -s mosdevs 10 ; 

16 16 

101 -s mosdevs 110 ; 2 2 7 -s mosdevs 10 ; 8 8 1F-s mosdevs 20 . 

16 16 

11.4. (3-quliani) radgan jami 5-niSna gamovida, B = 1 

I I svetidan ⇒ C = 2 an C = 3 

A B C D 

+ 

bolo svetidan ⇒ C luwia, e. i. C = 2 

A B C D 

igive bolo svetidan ⇒ D = 6 an D =1; 

B D C E C 

D = 1 ar gamodgeba, radgan ukve gvaqvs B = 1; 

e. i. D = 6 

magaliTi aseT saxes miiRebs: 

A 1 2 6 

+ 

A 1 2 6 

⇒ E = 4; A = 3 

1 6 2E 2 

(5-quliani) Sesabamisobas aseTi TanmimdevrobiT vpoulobT: 

F = 1, A = 9, N = 8, L =0, C = 5, B = 4, D = 3, M =7. 

2x 

B 

+

11.6. (4-quliani) 

x aseuli, y aTeuli, z erTeuli 

z kentia 

x + y + z = 15 

x – y = y – z 

x > y + z 

(5-quliani) 9567 + 1085 = 10652 

gv. 73-ze (daamtkiceT damoukideblad): 

vaCvenoT, rom Tu a = b (mod m) da c = d (mod m), maSin a + c = (b + d)(mod m) 

marTlac, a = q m + r , b = q m + r (a da b–s toli naSTebi aqvT) 

1 1 2 1 

c = q m + r , d = q m + r (c da d–s toli naSTebi aqvT) 

3 2 4 2 

(a + c) – (b – d) = (q + q ) m + (r + r ) – (q + q ) m – (r + r ) = (q + q – q – q ) m. 

1 3 1 2 1 4 1 2 1 3 2 4 

aqedan Cans, rom (a + c) – (b + d) unaSTod iyofa m-ze. r.d.g. 

§12 

axla vaCvenoT, rom igive pirobebSi ac = bd (mod m), e. i. saWiroa vaCvenoT, rom ac – bd unaSTod 

iyofa m–ze. 

ac – bd = (q m + r )(q m + r ) – (q m + r )(q m + r ) = ... = [q (q m + r ) + q r – q (q m + r ) – q r ] ⋅ m. r.d.g. 

1 1 3 2 2 1 4 2 1 3 2 3 1 2 4 2 4 1 

Tu ac = bd(mod m) gamosaxulebaSi aviRebT c = a, d = b ⇒ a 2 = b 2 (mod m) da a. S. a k = b k (mod m) 

12.1. 80 = 7 ⋅ 11 + 3, amitom 80 (mod 11) = 3 ⇒ 200 _ 80 (mod 11) = 197 

11 = 0 ⋅ 20 + 11, amitom 11 (mod 20) = 11 ⇒ 9 ⋅ 11 (mod 20) = 99 

12.2. 103 (mod 17) = 1, 42 (mod 17) = 8 amitom (103 ⋅ 42) (mod 17) = 8 WeSmaritia 

17 97 unaSTod iyofa 17-ze. amitom 17 97 (mod 17) = 0 WeSmaritia 

12.4. (3-quliani) 9 50 = 9 2⋅25 = 8125 ⇒ bolovdeba 1-iT. 

12.8. 

z kentia 

(2y –z) + y + z = 15 

x = 2y – z 

x > y + z 

y = 5 

(4-quliani) cxadia, 2 4 = 6 (mod 10) ⇒ (2 4 ) 50 = 6 50 = 6 (mod 10) 

(radgan 6-is nebismieri xarisxi bolovdeba 6-ianiT); 

2 2000 = 6 (mod 10) , 2 5 = 2 (mod 10) ⇒ 2 2005 = 6 ⋅ 2 (mod 10) 

amrigad, 2 2005 -is gayofiT 10-ze naSTi rCeba 2; 

pasuxi: es ricxvi bolovdeba 2-iT. 

(5-quliani) 9-is kenti xarisxebi bolovdeba 9-ianiT. ⇒ 9 2005 = 9 (mod 10) 

viciT, rom 2 2005 = 2 (mod 10) ⇒ (9 ⋅ 2) 2005 = 9 ⋅ 2 = 8 (mod 10) 

ricxvi bolovdeba 8-ianiT. 

12 3 

y = 4 y (radgan marcxena mxareSi igulisxmeba, rom y arauaryofiTia). 

12 4 

y = 3 y (marcxena mxare nebismieria y-saTvis arauaryofiTia). 

4 10 8 10 

4 

4 

5 4 

x = x = x = x ⋅ x = 

12.9. moswavleebma es sityvebi SeiZleba Secvalon martivi naxatebiT. 

x 

4 

x 

z kentia 

x = 10 – z 

x > 5 + z 

⇒ x = 9; z = 1 


61

12.10. samkuTxedebis msgavsebiT: 

15 


x 

↑ 

1,8 

↓ 

§13 

13.4. (3-quliani) – 50 (mod 7) = ? 

– 50 7q + r, 0 ≤ r < 7 ⇒ 0 ≤ – 50 – 7q < 7; 

– 57 < 7q ≤ – 50; q = –8; amrigad, r = – 7q – 50 = 6 

e. i. –50(mod 7) = 6; 

(4-quliani) –50 = – 6 ⋅ 9 + 4 ⇒ –50 (mod 9) = 4; 

(5-quliani) –50 = –5 ⋅ 11 + 5 ⇒ –50 (mod 11) = 5. 

13. 8. qvelmoqmedebajj. 

I I Tavis damatebiTi savarjiSoebi 

4. 7100 = 72⋅50 = 4950 ; 9-iT daboloebuli ricxvis luwi xarisxi bolovdeba 1-iT. 

5. 

S =1– . 

gaferad. 

6. a = b (mod m) piroba niSnavs, rom (a – b) unaSTod iyofa m-ze. 

cxadia (a + c) – (b + c) = a – b -c iyofa unaSTod m-ze. e. i. a + c = (b + c) (mod m). 

aseve, ac – bc = (a – b)c tolobis marjvena mxare iyofa unaSTod m-ze, gaiyofa 

marcxena mxarec, e. i. ac = bc (mod m). 

7. 

A 

11. 

A 

B 

M 

12. (14 – 6) ⋅ 3 = 24. 

E 

B 

K 

O 

x 

2 x 

N 

C 

17. yvelaze metad mainc kiSis saocari nadiroba ancvifrebda xalxs. 

↑ 

↓ 

2,5 

15 = 

x 

x = 10,8 

2, 

5 

centrebis SeerTebiT miRebuli kvadratis gverdia 1, farTobia 1. 

1, 

8 

igi Sedgeba S gaferad. + 4 ⋅ = 1; S gaferad. + = 1; 

C 

heronis formuliT S ABC = 84; ⇒ BK = 8; 

ΔABC ~ ΔMBN ⇒ 

AC 

= 

MN 

BK 

BE 

Tu kvadratis gverds aRvniSnavT x-iT, 

21 8 168 

maSin = ; x = 

x 8 − x 29 

ΔAOD ~ ΔBOC ⇒ 

= 1; = 4 ⇒ (BC + AD) x = 6; S ABCD = = 9 

D 

S 

S 

AOD 

BOC 

= 4 ⇒ 

msgavsebis koeficienti = 2. 

maSasadame, ΔAOD-s simaRle 2-jer metia ΔBOC-s 

Sesabamis simaRleze.

14.3. (4 quliani) Tu am ricxvSi x aTeuli da y erTeulia, miviRebT sistemas: 

⇒ pasuxi: 62; 83. 

§14 

( y + 1) 

⎧x 

= 2 

⎨ 

⎩x 

+ y > 6 

(5 quliani) Tu am ricxvSi aTeuli da erTeulia, miviRebT sistemas: ⎨ 

⎩ ⎧5x 

− 4y 

= 13 

x + y > 13 

SerCevis gziT aqedan miiReba x=9, y=8. pasuxi: 98. 

14.4. (3 quliani) im nawilebis raodenoba, romelTa mniSvnelia 2 aris 1; romelTa mniSvnelicaa 3 _ 

iqneba 2 da a. S. romelTa mniSvnelia 9 _ iqneba 8 wiladi. sul 1+2+ ... +8=36 wiladi. 

(4 quliani) saZiebeli wiladebis raodenobaa 9+10+ ... +28=370 

(5 quliani) saZiebeli wiladebis raodenobaa 10+12+ ... +98=2430 

14.7. (4 quliani) SevitanoT mocemuli wertilebis koordinatebi y=kx+b wrfis gantolebaSi. mii- 

⎧ 1 

⎪k 

= ; 

⎧− 

k + b = 2; 

3 

⎨ ⇒ ⎨ 

+ 8 

Reba sistema ⎩2k 

+ b = 3. 

⎪ 2 e.i. = 

b = 2 . 3 

⎪⎩ 

3 

x 

y . 

14.8. (5 quliani) |x−2| + |y+3| = 5; roca x = 0, gveqneba |y+3| = 3 ⇒ y = 0 an y = −6; e.i. OY RerZTan gadakveTis 

wertilebia (0; 0) da (0; −6). roca y = 0, gveqneba |x−2| = 2 ⇒ x = 0 an x = 4. amrigad, OX RerZi ikveTeba 

(0; 0) da (4; 0) wertilebSi. pasuxi: (0; 0), (0; −6), (4; 0). 

14.10. (4 quliani) b) wrewiri centriT wertilSi (2; −1), 5-is toli radiusiT. 

(5 quliani) a) gantoleba ase gadaiwereba: (x−2y) 2 = 0 ⇒ x − 2y = 0. am gantolebis grafikia wrfe. 

b) (x – 3) 2 + (y – 4) 2 = 25. grafikia 5-is toli radiusiani wrewiri, centriT (3; 4) wertilSi. 

g) (x – 3y) (x + 3y) = 0. am gantolebis grafikia (x – 3y) = 0 da (x + 3y) = 0 wrfeebis erToblioba. 

y↑ 

− 6 3 

d) 2y 

= ; y = − 

x − 2 

x − 2 

14.11. (3 quliani) gadavweroT gantolebebi ase: 

paralelobisaTvis saWiroa sakuTxo koeficientebis toloba, e.i. 

→ 

O 2 

x 

3 

a 7 

= x − 2, 

y = − x + 

2 

6 6 

y . 

3 

2 

a 

= − ⇒ a = −9 

6 


63

(5 quliani) gadavweroT gantolebebi ase: 

12345678901234 

12345678901234 

12345678901234 

123 

14.14. (3-quliani) (4-quliani) (5-quliani) 

12345678901234 

12345678901234 

12345678901234 

12345678901234 

12345678901234 

12345678901234 

12345678901234 

12345678901234 


3 

1 3 

y = x − 2, 

y = − x + . wrfeTa marTobulobisaT- 

2 

a a 

vis saWiroa, rom sakuTxo koeficientebis namravli −1-is toli iyos, e.i. 

14.12. miTiTeba: visargebloT trigonometriuli kompasiT. 

14.13. (3 quliani) 

6 ⋅8 

S = ⋅sin150° 

= 12 

2 

(4 quliani) B 

ΔABK ⇒ 

A 

§15 

x = 

2 2 

15.2 (5-quliani) T) ⎧ y + xy = 6; 

⎧xy( 

x + y) 

= 

⎨ 

⇒ ⎨ 

xy + ( x + y) 

pasuxi: (2; 1), (1; 2). 

⎩xy 

+ x + y = 5. 

⎩ 

1 

S = ⋅ AC ⋅ x = 144 

2 

6; 

= 5. 

3 ⎛ 1 ⎞ 

⎜− 

⎟ = −1⇒ 

a = 

2 ⎝ a ⎠ 

( 3− 

3) 

x aRvniS. x+y = t da xy = v ⇒ 

t = 3, v = 2 

⇒ 1 1 davubrundeT aRniSvnebs: 

t = 2, v = 3 

2 2 

( x + y) 

2 2 

2 

⎪⎧ 

x + y + x + y = 18; 

⎪⎧ 

x + y + − 2xy 

= 18; 

k) miTiTeba ⎨ 

⇒ 

2 2 ⎨ 2 

⎪⎩ xy + x + y = 19. 

⎪⎩ ( x + y) 

− xy = 19. 

l) miTiTeba: ⎪⎩ 

24 −x 

A 

K 

x 

B 

⎪ 

⎧ 

2 2 

x + y − y − 4x 

= 5 

⎨ 

3 2 2 

y y − 4x 

= 0. 

x 

3 

2 

1234567 

1234567 

1234567 

1234567 

1234567 

1234567 

aRvniS. x+y = t da xy = v ⇒ 

. meore gantolebidan Cans, rom an y = 0 (rac pirveli gan- 

tolebisaTvis ar gamodgeba), an y2 − 4x2 = 0. e.i. gveqneba ⎨ 

⎩ ⎧ − 4 = 0 

+ = 5 

2 2 

y x 

. 

x y 

C 

1234567 

1234567 

1234567 

123 

123 

1234567 

A B A B 

123 

123 

123 

⇒ 

⇒ ∅ 

x 1 = 2, y 1 = 1 

x 2 = 2, y 2 = 3

15.3. (3 quliani) 

3 

e.i. a = − . 

2 

15. 4. (3-quliani) ori wrfivi gantolebis sistemas erTaderTi amonaxsni aqvs, Tu ucnobebis 

Sesabamisi koeficientebi proporciuli araa: 3 : 2 ≠ − a : 3 ⇒ a ≠ _ 9/2 

(5-quliani) miTiTeba: meore gantolebidan amoxsnili y CavsvaT pirvelSi; gavutoloT O-s mi- 

Rebuli kvadratuli gantolebis diskriminanti. 

15.5. (3 quliani) parabolis zogad gantolebas y = ax 2 + bx + c unda akmayofilebdes samive wertilis 

koordinatebi, e.i. x = 0 da y = 2, x = 1 da y = 0, x = −1 da y = 1 wyvilebi. Casmis Semdeg miiReba sistema: 

⎧c 

= 2 

⎪ 

⎨a 

+ b + c = 0 ⇒ a = − 

⎪ 

⎩a 

− b + c = 1 

3 

, 

2 

I da I I gantolebebidan Cans, rom (8 − a) 2 = a 2 , e.i. a = 4, amitom sistema martivdeba: 

16 + ( b + 1) 

2 ( b − 3) 

= 

15. 15. 15. 6. 6. (5-quliani) (5-quliani) SevitanoT mocemuli wertilebis koordinatebi (x – a) 2 + (y – b) 2 = R2 wrewiris zogad 

gantolebaSi: 

(2 –a) 2 + (6 – b) 2 = R2 (–6 –a) 2 + (–2 –b) 2 = R2 (2 – a) 2 + (–10 –b) 2 = R2 8 (–4 – 2a) + 8 (4 – 2b) = 0 

16 (-4 -2b) = 0 

am sidideebis gaTvaliswinebiT I gantolebidan ⇒ R = 8 

pasuxi: pasuxi: (x – 2) 2 + (y + 2) 2 a = 2 

b = –2 

= 64 

15.8. (5 quliani) miTiTeba: ΔAOB-s gverdebia R, R da R , amitom iolad davadgenT, rom 

∠AOB = 60 0 , e.i. ∠AOB = 120 0 . 

1 

3 

b = − , c = 2 . pasuxi: 2 

2 

2 2 

2 

x x 

y = − − + 

(4 quliani) miTiTeba: wrewiris gantolebas (x − a) 2 + (y − b) 2 = R2 unda akmayofilebdes samive 

wertilis koordinatebi, e.i. x = 8 da y = −1, x = 0 da y = −1, x = 4 da y = 3 wyvilebi. miiReba sistema 

⎧ 

⎪ 

⎨ 

⎪ 

⎩ 

2 

2 

( 8 − a) 

+ ( b + 1) 

2 

2 

a + ( b + 1) 

= R 

2 ( 4 − a) 

+ ( b − 3) 

2 

2 

= R 

2 

= R 

⎧4x 

− 3y 

= 7 

⎨ 

⎩2x 

+ ay = 5 

2 

⎧4x 

− 3y 

= 7 

⎨ 

− 2 ⎩− 

4x 

− 2ay 

= −10 

amonaxsni ar aqvs, roca 3+2a=0, 

− 

( 3 + 2a) 

y = −3. 

gamovakloT I gantolebas I I, 

Semdeg I gantolebas gamovakloT I I I. 

miiReba 

15.9. (3 quliani) 3 + < m < 4 + ; radgan 4 < < 5, amitom cxadia m = 8. 

(4 quliani) 6 − < m < 7 − . cxadia, rom 5 < < 6, amitom 0 < m < 2, 

e.i. m = 1. 

⎪⎧ 

⎨ 

⎪⎩ 


2 

R 

= R 

2 

2 

65

16.9. (5-quliani). im wrfis gantoleba, romelic gadis A da B wertilebze, aris y = x – 1; am 

wrfisa da mocemuli wrewiris (x + 2) 2 + (y – 3) 2 = 36 TanakveTa iqneba M(4;3) da N (–2; –3) wertilebi. 

MN qordis sigrZea 6 , radgan radiusi aris 6. amitom ∠MON = 90 0 (0 wrewiris centria). AB wrfiT 

miRebuli erTi segmentis farTobia 9π-18, xolo meore seqtorisa 27π+18. maTi Sefardebaa 

1 2 

x 

− y > 2 

1 

y < − 2 2 

x 


gia salome lali suliko biZina 

b) b) roca (–∞; –2], maSin x + 2 – x > 2 

2 > 2 

∅ 

y ↑ 

123456789012345678901 

123456789012345678901 

123456789012345678901 123456789012345678901 

123456789012345678901 

123456789012345678901 

123456789012345678901 123456789012345678901 

123456789012345678901 

123456789012345678901 

123456789012345678901 123456789012345678901 

123456789012345678901 

123456789012345678901 

123456789012345678901 123456789012345678901 

123456789012345678901 

123456789012345678901 

123456789012345678901 123456789012345678901 

123456789012345678901 

123456789012345678901 

123456789012345678901 123456789012345678901 

123456789012345678901 

123456789012345678901 

123456789012345678901 123456789012345678901 

O 

123456789012345678901 

123456789012345678901 

123456789012345678901 123456789012345678901 

123456789012345678901 

123456789012345678901 

123456789012345678901 123456789012345678901 

– 2 

123456789012345678901 

123456789012345678901 

123456789012345678901 123456789012345678901 

123456789012345678901 

123456789012345678901 

123456789012345678901 123456789012345678901 

123456789012345678901 

123456789012345678901 

123456789012345678901 123456789012345678901 

pasuxi: gia, salome, lali, 

biZina, suliko (SesaZlebelia, 

rom biZina, lali da suliko 

erTi simaRlisanic 

iyvnen). 

→ 

x 

3π 

+ 2 

. 

π − 2 

16.10. (5-quliani). vTqvaT, marto me-3 miliT avzi ivseba x saaTSi, maSin marto pirveli miliT 

aivseba 4x sT-Si, xolo marto me-2 miliT _ (x + 4) sT-Si. 1 sT-Si TiToeuli mili aavsebs Sesabamisad 

naw., naw. da nawils. amitom . miviRebT 2x 2 – 19x – 60 = 0. gantolebas: 

x = = . aqedan x = 12, radgan = 0,5, amitom gveqneba pasuxi: mesame mili saaTSi 

aavsebs avzis naxevars. 

§17. amocana 1. 

amocana 2. vTqvaT, 1 kg kartofili Rirs x TeTri, xolo 1 kg xaxvi – y TeTri, maSin 

30 ≤ x ≤ 60 da 40 ≤ y ≤ 80. aqedan, 180 ≤ 6x ≤ 360 da 160 ≤ 4y ≤ 320, saidanac vRebulobT: 340 ≤ 6x + 4y ≤ 680 

radgan 1 lari mgzavrobisTvis iyo saWiro, xolo dedas xurda fuli ar hqonda, 

500 ≤ 6x + 4y + 100 < 800 (8 lars ar miscemda deda, radgan igi `zedmets arasodes iZleoda~) 

e. i. dedas erTi orlarianis garda TemosTvis SeiZleba mieca 3, 4 an 5 erTlariani). 

17.1. ; > 0 ; e. i. > 0 

pasuxi: x ∈ (− ; 0) ∪ (1; ∞) 

17.2. ⏐x + 2⏐− x > 2 

a) a) roca x ∈ (-∞; –2], maSin − x – 2 – x > 2 

– 2 x > 4 

x < –2 

(–∞; –2] 

pasuxi: pasuxi: (-∞; -2)

⎡⎧y 

≥ −x 

y ≥ −x 

⎢⎨ 

⎢⎩x 

+ y < 4 y < −x 

+ 4 

⎢⎧y 

≤ −x 

y ≤ −x 

⎢⎨ 

⎢⎣ 

⎩− 

x − y < 4 y > −x 

− 4 

17.4. (4-quliani). a) 

a) 

(4-quliani). b) 

b) 

(x – 2) 2 + (y + 1) 2 > 3 

R = , centri: (2; -1) 

(4-quliani). d) d) 

d) 

3 

x − < 

y 

y ↑ 

12345678901234567890123456 

12345678901234567890123456 

12345678901234567890123456 

12345678901234567890123456 

12345678901234567890123456 

→ 

x 

12345678901234567890123456 

12345678901234567890123456 

O 

12345678901234567890123456 

12345678901234567890123456 

12345678901234567890123456 

12345678901234567890123456 

12345678901234567890123456 

12345678901234567890123456 

12345678901234567890123456 

12345678901234567890123456 

12345678901234567890123456 

12345678901234567890123456 

1234567890123456789012345678901212345678901 

1234567890123456789012345678901212345678901 

1234567890123456789012345678901212345678901 

1234567890123456789012345678901212345678901 

1234567890123456789012345678901212345678901 

1234567890123456789012345678901212345678901 

1234567890123456789012345678901212345678901 

1234567890123456789012345678901212345678901 

1234567890123456789012345678901212345678901 

1234567890123456789012345678901212345678901 

1234567890123456789012345678901212345678901 

1234567890123456789012345678901212345678901 

1234567890123456789012345678901212345678901 

1234567890123456789012345678901212345678901 

1234567890123456789012345678901212345678901 

1234567890123456789012345678901212345678901 

1234567890123456789012345678901212345678901 

1234567890123456789012345678901212345678901 

1234567890123456789012345678901212345678901 

1234567890123456789012345678901212345678901 

1234567890123456789012345678901212345678901 

1234567890123456789012 

1234567890123456789012345678901212345678901 

1234567890123456789012 

1234567890123456789012 

1234567890123456789012 

1234567890123456789012 

1234567890123456789012 

1234567890123456789012 

1234567890123456789012 

1234567890123456789012 

1234567890123456789012 

1234567890123456789012 

1234567890123456789012 

1234567890123456789012 

1234567890123456789012 

1234567890123456789012 

1234567890123456789012 

1234567890123456789012 

1234567890123456789012 

1234567890123456789012 

1234567890123456789012 

1234567890123456789012 

2 

(5-quliani). b) 

b) 

(x + 2) 2 + (y – 1) 2 ≤ 0 

misi amonaxsnia erTaderTi wertili: 

(-2; 1), radgan (x + 2) 2 + (y – 1 ) 2 ≥ 0 

(5-quliani) d) 

d) 

|x + y| < 4 

y ↑ 

O 

y ↑ 

→ 

x 

12345678901234567890123456 

12345678901234567890123456 

12345678901234567890123456 

12345678901234567890123456 

12345678901234567890123456 

12345678901234567890123456 

– 4 

12345678901234567890123456 

12345678901234567890123456 

→ 

x 

12345678901234567890123456 

12345678901234567890123456 

12345678901234567890123456 

12345678901234567890123456 

12345678901234567890123456 

12345678901234567890123456 

12345678901234567890123456 

12345678901234567890123456 

12345678901234567890123456 

17.14. 17.14. (3-quliani) (3-quliani) 

⇒ x2 + ax + 1 = 0; D = a2 – 4 < 0; a∈ (–2; 2) 

(4-quliani) 

(4-quliani) 

5x 2 + 2x + (2 – a 2 ) = 0. 

(5-quliani) 

(5-quliani) 

y ↑ 

O 2 

1234567890123456789012 

3 

x 

< + 

y 

2 ⎧( 

x −1) 

+ ( y + 2) 

⎨ 

⎩2x 

− y −1 

= 0 

2 

= a 

2 

⎧y 

= 2x 

−1 

⎨ 2 

⎩( 

x −1) 

+ ( 2x 

+ 1) 

D 9 

2 2 2 =1 – 5 (2 – a ) < 0; –9 + 5a < 0; a < 

4 

5 

⎧3 

⎪ − y = 1 

⎨ x 

⎪ 2 

⎩x 

+ ( y + 1) 

2 

= a 

2 

⎧ 3 

⎪y 

= −1 

x 

⎨ 

⎪ 2 9 

x + = a 

⎪ 2 

⎩ x 

x 4 – a 2 x 2 + 9 = 0; D = a 4 – 36 < 0; a 2 < 6; a ∈ (– ; ) 

2 

→ x 

2 

2 

= a 


O 

– 4 

2 

4 

4 

3 3 

a ∈ (− ; ) 

5 5 

67

12345678901234567890123 

17.8. (5-quliani) 

g) 

y 

↑ 

d) 

12345678901234567890123 

12345678901234567890123 

12345678901234567890123 

12345678901234567890123 

12345678901234567890123 

12345678901234567890123 

12345678901234567890123 

12345678901234567890123 

12345678901234567890123 

12345678901234567890123 

12345678901234567890123 

12345678901234567890123 

12345678901234567890123 

12345678901234567890123 

12345678901234567890123 

12345678901234567890123 

12345678901234567890123 

12345678901234567890123 

12345678901234567890123 

12345678901234567890123 

17.10. (3-quliani) {(x; y) | x ≥ 0, y ∈ (− ∞; + ∞)} ∪ {(x; 0) | x ∈ (− ∞; + ∞)} 

(4-quliani) {(x; y) | x ∈ (− ∞; 0) ∪ (0; + ∞), y > 0} 

y 

(5-quliani) ↑ 

F figura 1234567890123456 

2 

naxazze daStrixuli kvadratia. 

17.14. (3-quliani) ⇒ x 2 + ax + 1 = 0; D = a 2 – 4 < 0; a∈ (–2; 2) 

(4-quliani) 

12345678901234567890123 

– 2 

2 ⎧( 

x −1) 

+ ( y + 2) 

⎨ 

⎩2x 

− y −1 

= 0 


O 

1234567890123456 

1234567890123456 

1234567890123456 

1234567890123456 

1234567890123456 

1234567890123456 

O 

1234567890123456 

1234567890123456 

1234567890123456 

1234567890123456 

1234567890123456 

1234567890123456 

1234567890123456 

– 2 

1234567890123456 

1234567890123456 

→ 

x 

2 

2 

= a 

2 

⎧y 

= 2x 

−1 

⎨ 2 

⎩( 

x −1) 

+ ( 2x 

+ 1) 

2 

= a 

5x2 + 2x + (2 – a2 ) = 0. 

D 

2 =1 – 5 (2 – a ) < 0; 

4 

2 –9 + 5a < 0; 

9 

2 a < 

5 

(5-quliani) 

x 

⎧ 3 

⎧3 

⎪y 

= −1 

⎪ − y = 1 

x 

⎨ x 

⎨ 

⎪ 2 

2 2 ⎪ 2 9 2 

⎩x 

+ ( y + 1) 

= a x + = a 

⎪ 2 

⎩ x 

4 – a2x2 + 9 = 0; D = a4 – 36 < 0; a2 < 6; a ∈ (– ; ) 

§18 

12345678901234567890123 

12345678901234567890123 

12345678901234567890123 

12345678901234567890123 

12345678901234567890123 

12345678901234567890123 

12345678901234567890123 

12345678901234567890123 

12345678901234567890123 

12345678901234567890123 

12345678901234567890123 

12345678901234567890123 

12345678901234567890123 

12345678901234567890123 

12345678901234567890123 

12345678901234567890123 

12345678901234567890123 

12345678901234567890123 

12345678901234567890123 

12345678901234567890123 

12345678901234567890123 

12345678901234567890123 

misi gverdi = 2 2 

farTobia 8. 

y ↑ 

4 

17.11. (5-quliani) a) mTeli sibrtye 

b) mTeli sibrtye, koordinatTa saTavis gamoklebiT 

g) mxolod koordinatTa saTave 

123456789012345678901 

18. 3. (3 quliani) d) y ↑ 

(4 quliani) b) 

→ 

x 

123456789012345678901 

123456789012345678901 

123456789012345678901 

123456789012345678901 

123456789012345678901 

123456789012345678901 

123456789012345678901 

123456789012345678901 

123456789012345678901 

O 

12345678901234567890 

123456789012345678901 

12345678901234567890 

12345678901234567890 

12345678901234567890 

12345678901234567890 

12345678901234567890 

12345678901234567890 

12345678901234567890 

12345678901234567890 

12345678901234567890 

12345678901234567890 

→ x 

– 4 

O 

– 4 

2 

4 

→ 

x 

3 3 

a ∈ (− ; ) 

5 5 

y↑ 

O 

123456789012 123456789012 

123456789012 

123456789012 

123456789012 

123456789012 

123456789012 

123456789012 

123456789012 

123456789012 

123456789012 

123456789012 

123456789012 

123456789012 

123456789012 

123456789012 

123456789012 

123456789012 

123456789012 

123456789012 

→ x 

123456789012 

123456789012

(5 quliani) d) y 2 ≤ x 2 ; (y – x) (y + x) ≤ 0. 

jer avagoT y = x da y = –x wrfeebi. isini sibrtyes oTx nawilad yofen. 

imis Sesamowmeblad, Tu romeli maTgani unda davStrixoT, am areebSi 

viRebT Siga wertilebs. 

magaliTYad, (0; 5) da (0; –5) wertilebis koordinatebi ar akmayofileben. 

y 2 ≤ x 2 utolobas; (5; 0) da 

(-5; 0) wertilebis koordinatebi 

ki akmayofileben, 

amitom davStrixavT 

Sesabamis areebs. 

y ↑ 

123456789 

123456789 

123456789 123456789 

123456789 

123456789 

123456789 123456789 

123456789 

123456789 

123456789 123456789 

123456789 

123456789 

123456789 123456789 

123456789 

123456789 

123456789 123456789 

123456789 

123456789 

123456789 123456789 

123456789 

123456789 

123456789 123456789 

123456789 

123456789 

123456789 123456789 

123456789 123456789 

§19 

19.1. (3-quliani) vTqvaT limonaTis fasia lari, bananis _ b lari, amocanis pirobis Tanaxmad 

a) ⇒ 2 – 2b > 0 e. i. > b 

b) Tu b ≥ 0,5, maSin 3 + 5 ⋅ 0,5 ≤ 7 ⇒ ≤ 1,5 

O 

axla, Tu gaviTvaliswinebiT 

|x| ≥ 2 utolobasac, 

miviRebT sistemis 

saZiebel grafiks. 

(4-quliani) a) zemoT miRebuli sistemidan ⇒ 8 + 8 b > 14 e. i. + b > 1,75. amitom 1,4, larad ver 

SeZlebda, sistemis II gantolebidan ⇒ 3 

7 

+ 3b < 7, + b < < 2,4 

3 

b) + b ≤ 2 utolobis ZaliT sistemis II tolobidan 

7 = 3( + b) + 2b ≤ 3 ⋅ 2 + 2b, b ≤ 2b ≥ 1, b ≥ 0,5. 

pasuxi: bananis umciresi fasia 0,5 lari. 

⇒ 

7 − 3l 

(5-quliani) a) sistemis II gantolebidan ⇒ b = , amitom 4 

5 

+2b ≤ 7-dan miviRebT 

7 − 3l 

2l 

+ 7 3 + 7 

4 + ≤ 7, ≤ + b ≤ + − ≤ = 2 . e. i. SeeZlo. 

5 5 5 

→ x 

3 

16 ⋅ + 21 

b) 5 + 3b = 5 + 21− 

9l 

16l 

+ 21 

= ≤ 

2 

= 9 = 7 + 2 e. i. sakmarisia 2 laris damateba. 

5 5 5 

19.10. Tu marTkuTxa samkuTxedis kaTetebia a, b, hipotenuza ki c, maSin a 3 + b 3 < a 2 c + b 2 c = (a 2 + 

b 2 ) ⋅ c=c 3 

20.1. (5-quliani) mocemulia ocdaxuTive (3+10+7+5=25) moswavlis Sedegi 

(25-1) ⋅ 2 = 48, (25 - 2) ⋅ 2 = 46, (25-3) ⋅ 2= 44 da (25-6) ⋅ 2 = 38. amitom B = {48; 46; 44; 38}. 

§20 

y↑ 

→ x 


O 

123456 

123456 

123456 

123456 

123456 

123456 

123456 

123456 

123456 

123456 

123456 

123456 

123456 

123456 

123456 

123456 

123456 

20.4. miTiTeba: f : [-1;1] → [1;5] SesaZlo funqciebidan ganvixilavT: 

1. wrfiv y = kx + b tipis funqcias. aq intervalTa boloebis mixedviT vwerT Sesabamis sistemas: 

⎧− 

k + b = 12b 

= 6, 

b = 3 

⎨ 

⎩k 

+ b = 5 k = 2 

y = 2x + 3 

y 

↑ 

O 

123456 

123456 

123456 

123456 

123456 

123456 

→ x 

123456 

123456 

123456 

123456 

123456 

123456 

123456 

123456 

123456 

123456 

123456 

69

2. 

⎧ k 

k 

⎪ = 1 

y = 1 

tipis funqcias. am SemTxvevaSi −1 

+ b b = k + 

x + b 

⎨ 

⎪ k k = 5b 

+ 5 

= 5 

⎪⎩ 

1+ 

b 

20.10. (4-quliani) radgan rkinis detalis masa moculobis proporciulia, xolo simaRle igive rCeba, 

amitom 2-jer naklebi moculobis pirobebSi wesieri eqvskuTxedis farTobi orjer meti iqneba 

gamoCarxuli cilindris fuZis wris farTobze. 2πR2 = 628 mm2 , R2 ≈ 100mm2 ⎧x 

> y 

sistemis II, IV utolobebidan ⇒ 10x > 350, x > 35. 

⎪ 

maSin ⎪xy 

< 1300 

II, III-dan x(x − 10) < 1300 ⇒ x ≤ 41. 

⎨ 

⎪y 

+ 10 > x 

Semowmebebi gviCvenebs, rom x SeiZleba iyos 36; 37; 38; 39; 40; 41. 

⎪ 

⎩x( 

y + 10) 

> 1650 

, R ≈ 10mm, d ≈ 20mm. 

(5-quliani) gaviTvaliswinoT, rom masa moculobis proporciulia. pirobis Tanaxmad 

πR2 12 , 5 

2 ⋅ h = πR ⋅ H. 

100 

r : R = h : H, 

125 

3 3 e. i. r : R = saidanac 

1000 

r : R = 5 : 10 = 1:2. R = 2r, H = 2h. aqedan, SX : 40 = 1 : 2, amitom SX = 20 sm. 

20.11. (4-quliani) 

⎧h 

= AD 

S = AC ⋅ BD, sadac [BD] ⊥ [AC]. BD = h saidanac S = 10h 

ABCD ABCD ⎨ 

⎩h 

: DC = tg30 


A 

B C 

45 o 

30 o 

D 

10 

30 o 

D 

⎧ 

3 5 

⎨4b 

= −6, 

b = − , k = − 

⎩ 

2 2 

20.6. (4-quliani) moswavleebi iyeneben gamoricxvis meTods (B ⊂ A mcdaria; A ∩ B = (–2; 5) mcdaria, 

radgan 5∈ A∩ B da A = B tolobac mcdaria. amitom WeSmaritia (a)), an iyeneben pirdapiri amoxsnis 

meTods 

((_3; 5] ∪ [-2; 6) = (-3; 6); e. i. WeSmaritia (a)). 

(5-quliani) amocanis amoxsnisas moswavlem unda daweros x2 – 3x ≤ 0 utolobisa da x – 1 > 0 

utolobis amonaxsnTa simravleebia: A = [0; 3]; B = (1; ∞). 

amitom A ∪ B = [0; ∞), xolo A \ B = [0; 1] ( [0; 3] simravles `CamovacliT~ 1-ze met ricxvebs). 

20. 7. (5-quliani) (O; 1) wrewiris gantolebaa x2 + y2 = 1. erTmaneTs SevusabamoT 

centridan gamosul sxivze mdebare wrewirTa wertilebi (rogorc es naxazzea 

naCvenebi). (x1 ; y1 y 

↑ 

(x ; y ) 1 1 

(x; y) 

→ 

) (x ; 7 ) wertili 3-jer ufro Sorsaa centridan, vidre (x; y). amitom 

O 1 3 x 

1 1 

x = x1/3 , y = y1/3 . am tolobebis gaTvaliswinebiT (O; 1) wrewiris gantolebidan 

2 2 3 

miviRebT x + y1 = 9. e. i. (O; 3) wrewiris aRwerili asaxva warmoadgens Ho 1 

homoTetias. 

20.9. (4-quliani) vTqvaT, 1 kg kartofili Rirs x TeTri, xolo 1 nayini _ y TeTri. nika daxarjavda 

(100x+y) TeTrs, rac 5 larsa da 40 TeTrs anu 540 TeTrs Seadgens, natom ki daxarja 470 TeTri: 

7x+80+40=470. aqedan x = 50 10 · 50 + y = 540, e. i. y = 40 

pasuxi: nikas uyidia 40-TeTriani nayini. 

(5-quliani) vTqvaT, mocemuli marTkuTxedis sigrZea x m, xolo sigane _ y m. 

o 

h 

= 

10 − h 

1 

3

3h = 10 − h, 

10 

h = , h = 5( 

3 + 1 

3 -1), S = 50 ( 3 - 1) sm2 . 

kalkulatoris an cxrilis gamoyenebiT gamoviTvliT: 3 -1 ≈ 0,732, e. i. S ABCD ≈ 36,6 sm 2 . 

(5-quliani) vTqvaT, ukve agebulia iseTi ABCD trapecia, romelSic BC = a, AD = b, AB = c 

da CD = d 

a 

b 

c 

d 

(viyenebT informaciis erTad Tavmoyris princips). Tu [BM] | | [CD], maSin BM = d, xolo AM = b–a. 

amgvarad, jer avagebT ΔABM-s sami gverdiT: AB=c, BM = d da AM = b – a, Semdeg, AM sxivze 

avagebT D wertils: MD = a. 

Tu B da D wertilebidan, rogorc centrebidan, SemovxazavT Sesabamisad a da d radiusian 

rkalebs, gadakveTis wertili iqneba C da BCDM _ paralelogrami (mopirdapire gverdebi wyvilwyvilad 

kongruentulia). e. i. ABCD iqneba trapecia, romelSic 

AB = c, BC = a, CD = d da AD = b – a + a = b. 

21.2 a) (5-quliani) y = funqciisaTvis x 2 – x – 6 = 0, roca x 1 = – 2 da x 2 = 3 

e. i. D(y) = R \ {–2; 3} (an D = (–∞; –2) ∪ (–2; 3) ∪ (3; ∞). 

§21 

mniSvnelobaTa simravlis mosaZebnad viyenebT nacnob meTods: vadgenT t-s ra mniSvnelo- 

bebisaTvis aqvs amonaxsni = t gantolebas. aq, cxadia, t ≠ 0, tx 2 – tx – 6t +1 = 0 

D = t2 + 4t (6t-1) ≥ 0 25t2 – 4t ≥ 0 t(25t – 4) ≥ 0 e. i. t∈(–∞;0) ∪ [0,16; ∞) 

e. i. f funqciis mniSvnelobaTa simravlea (–∞; 0) ∪ [0,16; ∞) 

b) y = x + funqciis gansazRvris area D = (–∞; 0) ∪ (0; ∞). roca x < 0, maSin x – = a gantolebas aqvs 

amonaxsni a-s nebismieri mniSvnelobisaTvis: x 2 – ax – 1 = 0, D = a 2 + 4 > 0 e. i. E = R. 

21.4 (5-quliani) a) AC wrfis gantolebaa y = 0; 

k = 

⎧− 

3k 

+ b = 0 

⎨ 

⎩0k 

+ b = 2 

b = 

2 

3 

2, 

2 

e.i. y = x + 2. 

3 

A 

c 

B a C 

+ 4 

21.3 (5-quliani) a) y = 2x – 4 aqedan = 

2 

y 

x , e. i. mocemuli funqciis Seqceuli funqciaa 

y = 0,5x + 2. 

3 3 

b) y = + 1 , x ≠ 0 yx = x + 3, x ( y – 1) = 3 x = 

x 

y −1 

e. i. Seqceuli funqciis formulaa 

b-a 

b) vTqvaT, AB wrfis gantolebaa y = kx + b, maSin 

d 

M 

3 

y = , x ≠ 1 

x −1 

a 

d 

D 

1234567890123456 

1234567890123456 

1234567890123456 

1234567890123456 

1234567890123456 

1234567890123456 

1234567890123456 

1234567890123456 

A 

1234567890123456 

1234567890123456 

1234567890123456 

1234567890123456 


B 

C 

71

g) vTqvaT, BC wrfis gantolebaa y = kx + b , maSin 

ΔABC-s Sesabamisi utolobaTa sistemaa 

21.5. (5-quliani) radgan wuTebis isari yovel wuTSi 6 0 -ian rkals Semowers, xolo saaTebis isari 

_ 0,5 0 -ian rkals, amitom saZiebeli asaxvis formula iqneba y = 5,5 0 t, Tu D = {1; 2; ....; 60}, maSin 

E = {5,5; 11; 16,5; ....; 330} 

am funqciis grafikia ⎨ 

⎩ ⎧y 

= 5, 

5t 

sistemiT gansazRvrul sxivze mdebare 60 wertilisagan Sedgenili 

t > 0 

simravle. 

21.6 (3-quliani) 

wrfe gadis (0;1) da (1;0) 

wertilebze. amitom 

misi gantolebaa 

y = -x + 1, e. i. 

f(x) = - x +1 

21.9. (5-quliani) a) x − 2 = 2x 

− 3 

x ≤ 1, 

5 

⎧x 

≤ 1, 

5 x ≤ 1, 

5 

⎨ 

2 

⎩x 

− 2 = 3 − 2x 

3x 

= 5 x = 1 

3 

2 

x − 3x 

b) ≥ x 

x − 2 

∅ 

2 ≤ 

x 

x − 


(4-quliani) 

y = ax 2 + bx + c gadis (2;3), (- 

1,0) da (1;0) wertilebze. 

amitom 

0 

⎧y 

≥ 0 

⎪ 

2 

⎨y 

≤ x + 2 

⎪ 3 

⎪⎩ 

y ≤ −2x 

+ 2 

4a + 2b + c = 3 

a – b + c = 0 

a + b + c = 0 

f (x) = x 2 – 1 

, x ∈ [0; 2) 

⎧0k 

+ b = 2 b = 2 

⎨ 

⎩k 

+ b = 0 k = −2, 

e. i. y = −2x 

+ 2 

⎧x 

> 1, 

5 x > 1, 

5 

⎨ 

∅ 

⎩x 

− 2 = 2x 

− 3 x = 1 

21.11. (3-quliani) a) arc xy aris mudmivi, arc x/y, amitom arc ukuproporciulobaa, arc 

pirdapirproporciuloba. 

b) 1/6 = 2/12 = 3/18. pirdapirproporciulobaa. 

(4-quliani) a) 4 ⋅ 10 = 8 ⋅ 5 = 2 ⋅ 20; ukuproporciulobaa 

b) arc erTi maTgani 

(5-quliani) a) 2/4 = 3/6 = 10/20; pirdapirproporciulobaa 

b) 8 ⋅ 22 = 11 ⋅ 16 = 44 ⋅ 4; ukuproporciulobaa. 

21.12. (3-quliani) araTanamkveT simravleTa gaerTianebis simZlavre (elementTa raodenoba) 

tolia Semadgenel simravleTa simZlavreebis jamisa. 

12 

123 

123 

12 

R U R U 

123 

123 

123 

123 

⇒ 

∪ 

12 

12 

12 

12 

45 + (53 – x) = 80; x = 18. 

(4-quliani) radgan 98 studentidan samive ena arc erTma ar icis, amitom 

45 + 56 + 30 = 131 jamSi `zedmeti~ 33 (98-Tan SedarebiT) ori enis mcodneTa xarjze 

miviReT. pasuxi: 33. 

a = 1 

b = 0 

c = – a 

(5-quliani) 

wrewiris centria (0;0), 

xolo R = 2. e. i. x 2 + y 2 = 4. 

radgan y ≥ 0, amitom 

2 

y = 4 − x 

f ( x) 

= 4 − x 

2 

pasuxi: ∅ 

123 

123 

123 R U 

123 

123 

123 

123 

G 

12 

123 

12 

e. i.

R U 

12345678 

12345678 

12345678 

12345678 

12345678 

12345678 

G 

(5-quliani) radgan (45+56+30) – 100 = 31, amitom ori ena mainc icis 31-ma studentma. 

radgan 23+6+8=37 da 37-31=6, amitom samive ena icis 6-ma studentma. 

21.13. (3-quliani) g(3) = –2 ⋅ 3 + 1 = –5; f(g(3)) = f(–5) = – 10 

(4-quliani) g(3) = –5; f(g(3)) = f(–4) = – 8 

(5-quliani) (fog)(x) = f(g(x)) = 2 ⋅ g(x) =–-4x + 2 

22.1. b) (5-quliani) = x −1 

− 2, 

y −1 

= 2, 

§22 

x e. i. x – 1 = – 2, x 1 = 1, an x – 1 = 2, x 2 = 3 

e. i. am funqciis nulebia –1 da 3. 

g) y = x2 + 3x + 3 funqcias nulebi ar aqvs, radgan x2 + 3x + 3 = 0 gantolebisaTvis D = 9 – 12 < 0 . 

− 3 

> 

x + 1 

22.2. (4-quliani) b) 0 roca x + 1 < 0 , e. i. x < –1 

(–∞; –1) SualedSi funqcia dadebiTia, (–1; ∞) SualedSi _ uaryofiTia. 

(5-quliani) b) 

6 

< 0 

x − 3 

roca x < 3, e. i. x ∈ (–3;3) amitom 

(-3;3) SualedSi funqcia uaryofiTia; (–∞;3) ∪ (3;∞) SualedSi _ dadebiTia. 

(5-quliani) d) 

12345678 

12 

12345678 

12345678 

12345678 

12345678 

12 

12345678 

12 

12345678 

12345678 

12 

12 

12345678 

12345678 ⇒ 

R 

∪ 

U 

⇒ 

R ∪ ∪ G 

G 

12 

−1 

< 0 

x −1 

roca x −1 

> 12 e. i. x – 1 < – 12 an x – 1 > 12 

(–∞; –11) ∪ (13; ∞) SualedSi funqcia uaryofiTia. x =1-ze funqcia ar aris gansazRvruli. 

es funqcia dadebiTia (–11;1) ∪ (1;13) SualedSi. 

22.3 22.3. 22.3 savarjiSoebis amoxsnisas SesaZloa visargebloT sqematurad agebuli funqciebis grafikebiT. 

(4-quliani) b) 

y = 

rogorc naxazidan Cans, funqciis udidesi mniSvneloba [–1; 4] SualedSi aris f (–1) = 6, umciresia 

6 

f (4) = 3 

7 

– 3 

y↑ 

O 

→ 

x 

y = 3 + 

– 3 

y↑ 

O 


U 

3 

→ 

x 

73

(5-quliani) a) f(x) = x 2 – 2x – 3 funqciis grafikis wveros abscisaa x o = 1, romelic ekuTvnis [_1;4] 

Sualeds, e. i. 

f-is umciresi mniSvnelobaa f(1) = 1 – 2 – 3 = – 4. 

radgan f(-1) = 1+2–3=0 da f(4) = 16 – 8 – 3 = 5, amitom f-is udidesi mniSvnelobaa 5. 

g) f(x) = |2x _ 3| _ 1 funqciaSi |2x _ 3|≥0 da roca x = 1,5 maSin 2x – 3 = 0. radgan 1,5 ∈ [–1;4], amitom f-is umciresi 

mniSvnelobaa _1. 

f(–1) = |_2 _ 3| – 1 = 4, f(4) = |8 _ 3|_ 1 = 4 e. i. am funqciis udidesi mniSvnelobaa 4. 

g) 

y ↑ 

O 3/ 2 

⎛ 3 ⎞ 

naxazidan Cans, rom funqciis umciresi mniSvnelobaa f ⎜ ⎟ = _1, udidesi ki mosalodnelia 

⎝ 2 ⎠ 

miiRweodes [-1; 4] Sualedis boloebze. f (-1) = 4, f(4) = 4. e. i. umciresia 4. 

22.5 22.5. 22.5 (5-quliani) b) 

y ↑ 

y = |2x -3| 

rogorc grafikidan Cans, y zrdadia (-∞; 1) da (1; +∞) Sualedebze. 

g) y = – 2x2 4 

– 4x + 7 parabolas wveros abscisaa x o = − = −1 

. radgan am funqciia grafikis 

4 

Stoebi qveviTaa mimarTuli (–2 < 0), amitom es funqcia zrdadia (–∞; –1] SualedSi, klebadia (1; ∞) 

SualedSi. 

d) y =⎥ 3 – 2x⎥ funqciis umciresi mniSvnelobaa 0, romelic miiRweva, x o = 1,5 wertilze. amitom is 

klebadia (– ∞; 1,5) SualedSi, zrdadia (1,5; ∞) SualedSi. 

22.6. (4-quliani) b) 

y = 1 – 

O 1 

D(y) = (– ∞; –1) ∪ (–1; ∞) 

y = 


→ 

x 

(1; 0) da (0; –1) RerZebTan gadakveTis 

wertilebia. 

zrdadia (–∞; –1) da (–1; ∞) SualedebSi. 

→ 

x 

y↑ 

O 

–1 

y↑ 

O 

3/ 2 

1 

–1 

y↑ 

1 

O 1 

1 

→ 

x 

y = 1 – 

→ 

x 

→ 

x

(5-quliani) a) 

y = ⎮ ⎮ 

D(y) = (–∞;1) ∪ (1; ∞) grafiki gadis (0; –3) 

wertilze. 

y > 0 

22.7. (4-quliani) roca ΔABC tolferdaa, maSin a = 

m = h . vTqvaT, b > c. Tu [AD] ⊥ (BC), maSin h < a a a a da ha < ma . 

did daxrils didi gegmili Seesabameba, amitom c < b 

utolobis Tanaxmad BD < DC, xolo [BC]-s Suawertili 

(K) Zevs [DC]-ze: BK = KC = m . gavixsenoT biseqtrisis 

a 

Tvisebac: Tu AM biseqtrisaa, maSin c : b = BM : MC. 

aqedan BM < BK da CM > CK. amitom DM < DK da AM < AK, 

e. i. h a ≤ a ≤ m a . B 

c 

h a 

A 

y ↑ 

O 1 

a 

3 

m a 

D M K 

(5-quliani) mocemulia: P = 1m. gavixsenoT: Tu r aris wesier n-kuTxedSi Caxazuli 

n n 

0 

180 

wrewiris radiusi, maSin Pn = n ⋅ 2rn 

8tg 

, xolo Sn n 

1 

= Pn 

⋅ rn 

. 

2 

1 

radgan P = 1 amitom n rn = , da 

1 1 

S 0 4 − S3 

= − > 0, 

e . i. S > S 4 3 

180 

4 ⋅ 4 ⋅1 

4 ⋅ 3 ⋅ 3 

2ntg 

n 

1 

S 6 − S4 

= 

4 ⋅6 

⋅ 

1 1 1 

− = − > 0, 

S > S . 6 4 

3 4 ⋅ 4 8 3 8 ⋅ 2 

. 

3 

22.9. asagebia uban-uban gansazRvruli funqciis grafikebi 

a) 

y ↑ 

b) 

y↑ 

g) ↑ 

y 

3 

3 

O 2 

–1 

→ 

x 

2 

O 

– 

–1 

1/ 4 

1 2 3 

→ 

x 

22.10. (5-quliani) 20 m/wm = 1200 m : 60 wm. = 1200 m/wT = 1,2 km/wT. 

h = 1200 · 3 m = 3600 m., (3600 : 8) (m/wT) = 450 m/wT. 

h 

↑ 

m 

3600 

2400 

1200 

wT 

→ t 

3 23 31 

O 1 


b 

→ 

x 

⎧1200t, 

roca 0 ≤ t ≤ 3 

⎪ 

h( 

t) 

= ⎨3600, 

roca 3 < t ≤ 23 

⎪ 

⎩3600 

− 450t, 

roca 23 < t ≤ 31 

C 

→ 

x 

75

22.12. (O; 6) wrewirSi Caxazul ABCD trapeciaSi BC = 6. MN = 8, piTagoras 

TeoremiT OM= 3 3 , amitom ON=8 – 3 3 . 

amgvarad, mikrokalkulatoris daxmarebiT gamoviTvliT, rom AB ≈ 83mm. AD ≈ 10,6sm = 106mm 


, S ABCD ≈ 6640 mm 2 

22.13. (5-quliani) a) f(x) = 5 – 3x, 5 – 3x = y 3x = 5 – y 

b) 

5 − y 

5 − x 

x = e. i. g( 

x ) = 

3 

3 

6 

6 

f ( x ) = 1 − 

1 − = y, 

x + 2 

x + 2 

x ≠ – 2 

2y 

+ 4 

x = 

1 − y 

2x 

+ 4 

g( 

x ) = , x ≠ 1 

1 − x 

22.14. (3-quliani) a) y = x 2 ar gaaCnia, radgan, magaliTad, y(2) = 4 da y(-2) = 4. is tol mniSvnelobebs 

iRebs sxvadasxva x-saTvis. 

(4-quliani) a) y = 800 – x gaaCnia. 

(5-quliani) a) y = 3x2 7 ± 1 

– 7x + 4 ar gaaCnia, radgan, magaliTad, x = , 

6 

4 

x = mniSvnelobebisaTvis y(x1 ) = y(x ). 

2 2 

3 

23.1. mocemulis wina ricxvi ganvsazRvroT am ricxvisagan 1-is gamoklebiT. 

§23 

23.2. y / x = 2 ; y da x ver iqneba mTelebi, radgan 2 iracionaluri 

ricxvia. 

2 −1 

23.5. (3-quliani) a) = ; 

5 

x 

y 

aqedan: 

maSasadame, 

20 3 

1 

12 3 

100 16 

1 

FF 16 

5 + 1 

= 

2 

y 

x 

, 

30 4 

1 

23 4 

A10 16 

1 

A0F 16 

5y 

+ 1 

( y) 

= , 

2 

−1 

f anu 

100 8 

1 

77 8 

5A 16 

1 

59 16 

5x 

+ 1 

( x) 

= 

2 

−1 

f . 

10 16 

1 

F 16 

110 3 

1 

102 3 

–1 

↑ 

y 

1 

O 

–1 

anu, roca x 1 = 1 da 

1 

→ 

x

(5-quliani) a)f(x)=-x2 , x≥0, CavweroT es funqcia gantolebis saxiT y = 

– x2 ; amovxsnaT x-is mimarT: 

x = − y , x = − − y unda SevarCioT pirveli maTgani, radgan x≥0. 

−1 

−1 

vwerT f ( y) 

= − y , y ≤ 0 anu f ( x) 

= − x , x ≤ 0. 

−1 

⋅ x 

23.6. (3-quliani) f(g(x)) = 11g + 1 = 11 11 

+ 1 = x g(f(x)) = 

amrigad, f(g(x)) = g(f(x)) = x, e. i. f da g urTierTSeqceulebia. 

f −1 

11x 

+ 1− 

1 

= = x 

11 11 

1 

1 

1 

(5-quliani) f ( x) 

= , x > 1; 

CavweroT gantolebis saxiT: y = , x > 1; 

⇒ x = + 1, 

y > 0. 

x −1 

x −1 

y 

−1 

1 

−1 

1 

amitom, f ( y) 

= + 1, 

y > 0 , anu f ( x) 

= + 1, 

x > 0. 

y 

x 

23.7. (5-quliani) Tu P(3;2) wertili kx+5y=5k wrfeze Zevs, maSin k ⋅ 3 + 5 ⋅ 2 = 5k ⇒ k =5. 

23.8. (4-quliani) 

x n 

9 2 

( ) , 

5 − n 

9/(5-n) 2 

= n ∈ N, usasrulod mcire iqneba, Tu nebismieri ε>0-saTvis 

3 

+ 5 

ε 

4 − n 

(5-quliani) , n∈N; x 

n 

23.10. 111 2 =2 2 +2 1 +2 0 =7 10 ; 777 8 =7⋅8 2 +7⋅8 1 +7⋅8 0 =511 10 ; FFF 16 =15⋅16 2 +15⋅16 1 +15⋅16 0 =4095 10 

23.11. 22 4 ; 23 4 ; 30 4 ; 31 4 ; 32 4 . 14 8 ; 15 8 ; 16 8 ; 17 8 ; 20 8 . 17 9 ; 18 9 ; 20 9 ; 21 9 ; 22 9 . 

23.13. maT Soris aviRoT ricxvi . e. i. 5< < 6 da racionaluria. 

SeiZleba asec: da SevarCevT ricxvs . 

23.16. samive varianti erTnairad ixsneba: SuaxaziT moWrili samkuTxedis farTobi mTliani samkuTxedis 

farTobis meoTxedia. amitom, S PQR = S ABC : 16 = 4 

24.3. (3-quliani) Tu 3; x; 7 geometriul progresias Seadgens, maSin 

(4-quliani) a 7 = a 5 ⋅ q 2 ; 192 = 48q 2 ; q 2 = 4; q 1 = 2, q 2 = -2. a 5 = a 1 q 4 , amitom 48 = a 1 ⋅ 16; a 1 = 3. 

§24 

pasuxi: a n = 3 ⋅ 2 n-1 an a n = 3 ⋅ (-2) n-1 


77

24.10. (3-quliani) a) (f ο g)(5) = f(g(5)) = f(25) = 2 ⋅ 25 + 3 = 53; 

b) (g ο f)(5) = f(g(5)) = g(13) = 13 2 = 169. 

24.11. (4-quliani) f(x) = x 2 – x, g(x) = 3x+1 ϕ(x) =(g ο f) (x) = f(g(x)) = g 2 – g = (3x+1) 2 – (3x+1) 

ϕ(x) = 9x 2 + 3x; ψ(x) = (g ο f)(x) = g(f(x)) = 3 ⋅ f(x) + 1 = 3(x 2 -x)+1 ψ(x) = 3x 2 – 3x + 1 

24.13. (3-quliani) a n+1 = a n + 3, a 1 = 5, 

(4-quliani) a n+1 = a n + 3 n , a 1 = 1, 

(5-quliani) a n+1 = a n + n + 1, a 1 = 1. 

24.14. (g ο f) (–3) = g(f(–3)) = g(2) = –3 da a. S. 

25.1. (3-quliani) aRvniSnoT x = , maSin x2 = 1 + 1 + 1+ 

... 

e. i. x2 1+ 

5 

= 1 + x ⇒ x = = ϕ 

2 

1 

(4-quliani) aRvniSnoT x = 1 + 

1 

1+ 

1+ 

... 


§25 

1 

maSin x = 1 + ⇒ x = ϕ 

x 

25.6. sawyis mniSvnelobebad nebismieri ricxvebis aRebis SemTxvevaSi Fn/F , n = 2, 3, ... mimdevrobis 

n-1 

wevrebi kvlav oqros Sefardebas uaxlovdeba. 

25.7. wriul diagramaze 3600-s Seesabameba 100%, 900-s _ 25%. frenburTis Sesabamisi seqtoris kuTxe 

daaxloebiT 900-is 3 /4-ia, amitom Sesabamisi procentuli Sefaseba unda iyos g). 

25.8. (5-quliani) wrfivi damokidebulebi gantolebaa y = kx + b. 

radgan (0; 4) wertili am wrfes ekuTvnis, amitom gveqneba b=4 e. i. gvaqvs y = kx + 4. 

(-1; 6) wertilic wrfeze Zevs, amitom 6 = –k+4, e. i. k = –2; y = –2x+4; CavsvaT aq (–7; a) wertilis 

koordinatebi: a = –2 ⋅ (–7) + 4 = 18. 

25.11. a) (fof)(–2) = f(f(–2) = f(0) = –1; b) (f -1 og) (2) = f -1 (g(2)) = f -1 (0) = –2; 

g) (g of -1 ) (2) = g(f -1 (2)) = g(1) = –2; d) (fof -1 )(2) = f(f -1 )(2)) = f(1) = 2. 

25.14. (gof) (–3) = g(f(–3)) = g(1) = –1; (gof) (–2) = g(f(–2)) = g(–1) = 3 da a. S. 

25.16. (3-quliani) a 1 =2, d=7; a n = a 1 +d(n–1); 275 = 2+7(n-1); 7n = 280; n = 40; e. i. 275 progresiis wevria. 

375 = 2+7(n–1); 7n=380, amonaxsni araa mTeli. e. i. 375 araa progresiis wevri. 

(4-quliani) a 1 = 385, d = 378–385 = –7; a n = a 1 + d(n–1) = 385–7(n–1)>0; 7n

(5-quliani) a 2 =50, a 8 = 8, an a 2 = 8, a 8 = 50 pirvel SemTxvevaSi a 1 = 57, d = –7 ⇒ S 10 = 255. 

meore SemTxvevaSi a 1 = 1, d = 7 ⇒ S 10 = 325. 

25. 17. vTqvaT a n = 1/6. amocanis pirobiT S n – 

n ( 1− 

q ) 

a 

1− 

q 

1 = 

31 

6 

aqedan 

gantolebaTa sistema 

a1 = 

1− 

q 

16 

3 

1 ⎛13 

⎞ 31 

= 30 ⋅⎜ 

− Sn 

⎟ ⇒ Sn 

= . 

6 ⎝ 6 ⎠ 6 

tolobis gaTvaliswinebiT miviRebT qn = 1 /32 . amrigad, gvaqvs 

a 1 q n-1 = 1 /6 ; a 1 qn-1 = 1 /16 ⇒ q = 1 – q, q = 1 /2 . 

a 

1− 

q 

1 = 

16 

3 

§26 

26.1. 50 ⋅ 294 = 100 ⋅ 293 = 102 ⋅ (210 ) 9 ⋅ 8 ≈ 102 (103 ) 9 ⋅ 10 = 1030 ; 

2109 = (210 ) 11 : 2 ≈ (103 ) 11 : 2 = 1032 ⋅ 10 : 2 = 5 ⋅ 1032 . 

aseT miaxloebebs did sizustesTan pretenzia ar aqvT da adgenen, 10-is mimarT romeli rigisaa 

ricxvi (e. i. 10-is romeli xarisxi gvaqvs). 

ufro zusti gaTvlebiT, magaliTad, gveqneboda 2109 ≈ 6,5 ⋅ 1032 ; 

237 = (210 ) 3 ⋅ 27 ≈ (103 ) 3 ⋅ 128 ≈ 109 ⋅ 1,3 ⋅ 102 = 1,3 ⋅ 1011 ; 

299 = (210 ) 10 : 2 = (103 ) 10 : 2 = 1030 : 2 = 5 ⋅ 1029 ; 

ufro zusti gaTvlebiT unda yofiliyo 6,3 ⋅ 10 29 . 

n = 5 

215 ⋅ 2 39 = 215 ⋅ (2 10 ) 4 : 2 ≈ 1,1 ⋅ 10 2 ⋅ (10 3 ) 4 = 1,1 ⋅ 10 14 ; 222 ⋅ 2 222 = 888 ⋅ (2 10 ) 22 ≈ 10 3 ⋅ (10 3 ) 22 = 10 69 ; 

26.2. a) 10 51 = (10 3 ) 17 ≈ (2 10 ) 17 = 2 170 ; 32 ⋅ 10 30 = 2 5 ⋅ (10 3 ) 10 = 2 5 ⋅ (2 10 ) = 2 105 ; 

130 ⋅ 10 15 ≈ 128 ⋅ (10 3 ) 5 ≈ 2 7 (2 10 ) 5 = 2 57 ; 

b) 10 80 = (10 3 ) 26 ⋅ 100 ≈ (2 10 ) 26 ⋅ 2 6 = 2 266 . 65 ⋅ 10 50 ≈ 2 2 ⋅ (2 10 ) 17 = 2 172 . 

128 ⋅ 10 37 = 2 7 ⋅ (10 3 ) 12 ⋅ 10 ≈ 2 7 ⋅ (2 10 ) 12 ⋅ 2 3 = 2 130 ; 

65 ⋅ 10 50 = 65 ⋅ (10 3 ) 17 : 10 = 6,5 ⋅ (10 3 ) 17 . radgan 10 3 ≈ 2 10 metobiT, amitom aviRoT 6,5 ≈ 2 2 an naklebobiT. 

§27 

27.2. (pirveli 5-quliani) I maRaziis monacemebidan erT-erTi (200 000) mkveTrad gansxvavdeba sxvebisgan. 

amitom mediana ufro kargad aRwers tipur Semosavals. I I maRaziis monacemebis `tipuri~ 

SeiZleba iyos rogorc saSualo, ise mediana. 

IV Tavis damatebiTi savarjiSoebi 

2. `turniris erT-erT Cempions…~ ⇒ Cempioni erTi ar yofila; 

`bolo adgilze aRmoCnda erTi…~ ⇒ 2, 5 qula daagrova erTma; 

turnirze 28 TamaSi gaimarTa ⇒ qulebis jamia 28. 

amocanis gaanalizebis Semdeg miiReba: 4 + 4 + 4 + 4 3,5 + 3 + 3 + 2,5 = 28, 

4,5 + 4,5 + 4,5 + 3 + 3 + 3 + 3 2,5 = 28, 

5 + 5 + 3,5 + 3 + 3 + 3 3 + 2,5 = 28. 

pasuxi: Cempions SeiZleba daegrovebina 4 qula, 4,5 qula an 5 qula. 


79

4. b, e, T: am funqciebs nulebi ar aqvs. 

5. g) |x – 4| – 2 > 0; |x – 4| > 2 ⇒ x – 4 > 2 an x – 4 < –2; 

x > 6 an x < 2 

pasuxi: y > 0, roca x∈ (–∞; 2) ∪ (6; + ∞); y < 0, roca x∈ (2; 6) 

5 5 

5 2 

v) amitom 3 x + 1 > 5; 

x + 1 > ⇒ x + 1 > an x + 1 < − ; x > an 

3 3 

3 3 

pasuxi: y > 0, roca x∈ (– ∞; – ) ∪ ( ; + ∞) y < 0, roca x∈ (– ; – 1) ∪ (– 1; ). 

6 − 2x 

− 2 > 0; 

x + 3 x + 3 

z) > 0 ⇒ 

pasuxi: y > 0, roca x∈ (–3; 0), y < 0, roca x∈ (– ∞; – 3) ∪ (0; + ∞); 

6. d) funqciis gansazRvris area (– ∞; 1) ∪ (1; + ∞). 

vTqvaT x , x ∈(– ∞; 1) da x < x , maSin |x – 1| = 1– x , 

1 2 1 2 1 1 

|x 2 –1| = 1 – x 2 da y(x 1 ) – y(x 2 ) = 

y(x ) – y(x ) < 0, e. i. (–∞; 1) SualedSi funqcia klebadia. 

1 2 

analogiurad, Tu x , x ∈(1; + ∞) da x < x , gveqneba 

1 2 1 2 

e. i. funqcia zrdadia (1; + ∞) SualedSi. 

7. miTiTeba: TvalsaCinoebisTvis avagoT sqematuri grafikebi. 

a) umciresi mniSvneloba -7; udidesi 8. 

b) umciresi mniSvneloba -6; udidesi -2 

g) umciresi mniSvneloba 2; udidesi 7. 

d) umciresi mniSvneloba -10; udidesi 18 

e) umciresi mniSvneloba -9; udidesi 2 


+ – + 

- 

8 

3 

– + – 

-1 

-3 0 

3(x – x ) 2 1 

(1 – x ) (1 – x ) 

1 2 

8 

x < − 

3 

9. B C 

a) vxazavT mocemulradiusian wrewirs. 

b) nebismieri A wertilidan fargliT movniSnavT B da C 

wertilebs (cnobili sigrZeebis gamoyenebiT). 

A 

D 

g) radgan diagonalebi tolia, B-dan fargliT SegviZlia 

D wertilis moniSvnac. 

> 0 

2 

3

10. vTqvaT, ΔABC-Si AC udidesi gverdia. 

B 

a) jer avagebT MNKE kvadrats, romlis ME 

gverdi AC gverdze Zevs, xolo N∈[AB]; 

N1 K1 b) avagoT AK sxivisa da BC gverdis gadakveTis 

K wertili; K E ⊥ AC, E ∈[AC]; (K N) || (AC), N ∈ [AB]; 

1 1 1 1 1 1 

N K 

(N M ) ⊥ (AC), M ∈ [AC]. radgan AN : AN = AK : AK = 

1 1 1 1 1 

N K : NK = AM : AM = AE : AE (samkuTxedebis 

1 1 1 1 

msgavsebis Tvisebis mixedviT), amitom advilad 

A M M 1 E E1 C 

k davaskvniT, rom HA homoTetia, sadac K = AK1 : AK, MNKE kvadrats asaxavs saWiro (mocemulobidan) 

Tvisebebis mqone M N K E kvadratze. 

1 1 1 1 

12. miTiTeba: wrewirebi ikveTebian (– 3 /4 ; – 1 + 5 /4 ) da (– 3 /4 ; – 1 – 5 /4 ) wertilebSi. 

13. miTiTeba: grafikebi ikveTeba (– 1; 0) da ( 1; 2) wertilebSi. 

saZiebelia didi segmentis farTobi. 

15. 2 2 2 b q = 6; (b + b + b +…) ⋅ 8 = b + b2 + b3 + …, 1 1 2 3 1 

(b1 + b q + b q 1 1 2 8b1/(1 – q) 

2 2 2 2 4 +…) 8 = (b + b1 q + b1 q + …), 

1 

= b 2 

1 /(1 – q2 ) ⇒ q = 1 /2 ; 

2x 

+ 1 t + 1 

16. aRvniSnoT = t ⇒ x = , t ≠ 2. am tolobaTa gaTvaliswinebiT miviRebT: 

x −1 

t − 2 

1 2 

⎛ t + ⎞ 

f(t) = ⎜ ⎟ 

⎝ t − 2 ⎠ 

t + 1 

⎛ x + ⎞ 

+ , t ≠ 2 anu f(x) = ⎜ ⎟ 

t − 2 

⎝ x − 2 ⎠ 

17. Tu d 1 = 3 ⇒ a 1 = 7, S 6 = 87. Tu d 2 = –3 ⇒ a 1 = 19, S 6 = 69. 

18. S m+1 – S m = (m + 1) 2 – 5(m + 1) – m 2 + 5m = 2m-4 ⇒ a m+1 = 2m–4; 

maSin a = 2(m–1) – 4 = 2m–6; 

m 

a – a = 2, e. i. progresia ariTmetikulia. a = 2 ⋅ 5 – 6 = 4 

m+1 m 5 

19. S = 102 + 108 + … + 996 = 6 ⋅ (17 + 18 + … + 166); 

1 2 

t + 1 

+ , x ≠ 2 

t − 2 

frCxilebis SigniT 166 – 17 + 1 = 150 Sesakrebia. amitom ariTmetikuli progresiis wevrTa jamis 

formulis gamoyenebiT S = 3 . (17 + 166) . 150 = 82350 

20. 

21. 

⎧a 

⎪ 

a 

⎨ 

⎪a 

⎪⎩ 

a 

13 

3 

18 

7 

= 3 

⎧a 

⎨ 

8 

= 2 + 

⎩a 

a 

7 

13 

18 

= 3a 

3 

= 2a 

4 

⎪⎧ 

b1q( 

1+ 

q ) = 34 

⎨ 

⇒ q = 

2 4 

⎪⎩ b1q 

( 1+ 

q ) = 68 

7 

⎧a 

⎨ 

+ 8⎩a 

2, 

b 

1 

1 

1 

+ 12d 

= 3( 

a1 

+ 2d) 

⇒ a1 

= 12 27, 

d = 49 

. 

+ 17d 

= 2( 

a + 6d) 

+ 8 

= 1. 

1 

↑ 

y 

2 

–1 O 1 


→ 

x 

81

22. vTqvaT, marTkuTxedis gverdebia a da b, xolo ϕ = 1,618 oqros Sefardebaa. amocanis 

pirobis Tanaxmad, 

⎧2( 

a + b) 

= 2 

⎨ 

⎩b 

/ a = ϕ 


⎧a 

+ b = 1 1 1 

⎨ ⇒ a = , b = . 

⎩b 

= aϕ 

ϕ + 1 ϕ 

§28 

28.2. amoxsnisas gaviTvaliswinoT, rom K, M, H simravleebis simZlavre 3-is tolia. G, T-s simZlavreebi 

ki SeiZleba iyos: 1 (tolgverda samkuTxedisTvis), 2 (tolferda samkuTxedisTvis), 3 

(sxvadasxvagverda samkuTxedisTvis). magaliTad: 

1 

1 

28.5. (3-quliani) 

A 

5 5 

(4-quliani) 

A 

1 

5 

2 

B 

(5-quliani) 

X Y 

→ 

B 

C 

Y 1 

A 

28.6. (5-quliani) 

X 

→ 

→ 

samkuTxedisTvis G = {60 0 }, T = {5} 

samkuTxedisTvis G = {45 0 ; 90 0 }, T = {1; } 

samkuTxedisTvis G = {30 0 ; 60 0 ; 90 0 }, T = {1; ; 2} 

X 1 

C 

X Y 

→ 

→ 

B 

Y 

C 

C 

→ D 

A Y 

B 

X 

→ 

C 

D 

f(A) = C, f(B) = B, 

f(x) = Y, 

sadac nebismieri x-sTvis (XY) || (AC). 

f : [AB] → [CD] iseTia, rom 

f(A) = D, f(B) = C da 

f(x)= Y, (xy) || BC, 

x aris [AB]-s Sigawertili. 

f(A) = D, f(B) = C, f(O) = O; 

f(x) = Y, sadac x∈[AO] da (xy) || (AD); 

f(x 1 ) = Y 1 , sadac x 1 ∈ [OB] da (x 1 y 1 ) || (BC) . 

meore asaxva SeiZleba analogiurad ganisazRvros: 

[AO]-si [OD]-ze da [BO]-si [OC]-ze. 

vTqvaT, mocemulia raime wrewiris rkali AB (Cvens SemTxvevaSi 

ADB) . [CD] aris [AB]-s marTobuli diametri. 

fc : AB → [AB] asaxva, romlisTvisac fc (A) = A, fc (B) = B da fc (X) = 

Y, sadac nebismieri X∈AB wertilisaTvis Y = [CX] ∩ [AB] ur- 

TierTcalsaxaa. viciT, rom aseve urTierTcalsaxad SeiZleba 

erTi monakveTis meoreze asaxva.

urTierTcalsaxa asaxvebis kompoziciac urTierTcalsaxa iqneba. maSasadame, AB da 

nebismier MN monakveTs Soris SeiZleba urTierTcalsaxa Sesabamisobis damyareba. 

28.8. (4-quliani) 

29.3. (5-quliani) b) 

mniSvnelobaTa simravlea E = (–∞; 1]. am simravlis y = 1 

elementze aisaxeba R-is erTaderTi elementi (x = 3) 

gansaxilvel funqcias Seqceuli ar gaaCnia, radgan is 

tol mniSvnelobebs Rebulobs argumentis 

gansxvavebuli mniSvnelobebisaTvis. mag., f(2) = f(4) = 0 

aqve SeiZleba gavixsenoT horizontaluri wrfis 

testi. 

§29 

vTqvaT, (AA 1 )⊥(BD), maSin S BD (ΔABD) = ΔA 1 BD. 

vTqvaT, aris [A 1 C]-s SuamarTobuli wrfe. maSin 

S (ΔA 1 BD) = ΔABC. 

e. i. f = S οS BD kompozicia ΔABD-s asaxavs ΔABC-ze. 

29.5. (4-quliani) vTqvaT, ∠ABC = ∠MNK, aris [BN]-is SuamarTYobuli wrfe. 

1 

A 

maSin S (∠ABC) = ∠ A NC , 1 1 

1 

∠A NC = ∠MNK. 

1 1 

Tu 2 aris ∠A NM-is Suaze gamyofi wrfe, maSin 

1 

S (∠A NC ) = ∠MNK. 

1 1 

2 

amgvarad, S 

2 o S kompozicia ∠ABC-s asaxavs 

1 

∠MNK-ze. 

(5-quliani) 4-quliani amocanis amoxsnisas davrwmundiT, rom ori RerZuli simetria sakmarisia 

erTi kuTxis mis kongruentul kuTxeze asasaxavad. Tu ΔABC H ≅ ΔMNK, sadac ∠A = ∠M, maSin ori 

RerZuli simetriis f kompoziciiT f(∠A) = ∠M (da SesaZloa f(A) = N, maSin f(ΔABC) = ΔMNK). 

A 

B 

B 

C 

A 

A 

C 1 

y ↑ 

1 

O 

A 1 

1 

2 

N 

A 1 

B C 

C 

M 1 

B 

3 

D 

A 

K 

O 

→ 

x 

vTqvaT, f(ΔABC) = ΔM 1 AN 1 . 

Tu O = [M 1 N 1 ] ) ∩ [MN], xolo L aris 

∠NON -isa da ∠M OM-is biseqtrisebis ga- 

1 1 

erTianeba, maSin S (N ) = N da S (M ) = M (dam- 

l 1 l 1 

tkicebebi martivia da moswavleebi damoukideblad 

Caatareben!). amgvarad, sami 

RerZuli simetriis kompoziciiT ΔABC 

aisaxeba mis kongruentul ΔMNK-ze. 

29. 6. (3-quliani) T (2;3) (x; y) = (x +2; y +3). amitom, magaliTad, T (2;3) (A) = T (2;3) (0; –3) = (2; 0) 

(4-quliani) T (a;b) (2; –1) = (1;1) ⇒ a + 2 = 1 da b – 1 = 1 e. i. a = –1, b = 2. T (–1; 2) (0;2) = (–1;4) 

M 

N 1 

M 


83

(5 quliani) T (1;2) = (a +1; b +2) wertili imave 

(a,b) 

sxivzea. amitom b + 2 = 2(a + 1), b = 2a. Tu piTagoras 

Teoremasac gamoviyenebT gveqneba sistema 

a2 + b2 = 5 

⇒ a = 1, b = 2. 

b = 2a 

29. 7. (3 quliani) 


avagoT ABCD marTkuTxedi. 

AB paraleluria 0y RerZisa, amitom 

paraleluri gadatanaa T (0;5) ; 

A 1 = T (0;5) (–3; –1) = (–3; 4) da a. S. 

(4-quliani) radgan [AC] diagonalis sigrZe 5 erTeulia (SeamowmeT!), amitom aRniSnuli paraleluri 

gadataniT A gadadis C-Si. e. i. T (a,b) (–3; –1) = (1; 2) ⇒ a = 4; b = 3. 

gveqneba B 1 = T (4; 3) (–3; 2) = (1; 5) da a. S. 

(5-quliani) radgan BO monakveTis sigrZe swored erTeulia, amitom paraleluri gadataniT 

B aisaxeba O-ze, e. i. T (a; b) (–3; 2) = (0; 0) ⇒ a = 3, b = –2. gveqneba A 1 = T (3;-2) (–3; –1) = (0; –3) da a. S. 

29. 8. (3-quliani) T (x;y) = (x ;y) sadac x = x , y + 5 = y . CavsvaT wrfis gantolebaSi x = x , y = y – 5: 

(0;5) 1 1 1 1 1 

y – 5 = 2x – 4, an, rac igivea, y = 2x +1. 

1 1 

(4-quliani) T (x;y) = (x ; y ) sadac x – 5 = x , y = y CavsvaT wrfis gantolebaSi x = 5 + x , y = y . 

(–5;0) 1 1 1 1 1 1 

y = 2 (5 + x ) – 4 an, rac igivea, y = 2x + 6. 

1 1 

5-quliani 

B(–3; 2) 

y↑ 

O 

B(–3; –1) B(1; –1) 

y↑ 

O 

– 4 

1 3 

y 0 

M 

5 

1 

x 0 

N 

B(1; 2) 

→ 

x 

vTqvaT, [OM) ⊥ , sadac aris y = 2x – 4 wrfis grafiki, 

xolo 1 || , 1 ∩ [OM) = N da MN = 5, radgan 

| y 0 | = 

. OM = 2 . 4, xolo 

( OM 5) 

4 ⋅ + 

OM 

20 

= 4 + 

OM 

4 

OM = , e. i. yo = – (4 + 5 ). 

5 

OM ON 

= , amitom 

4 yo 

amgvarad, 1 wrfis gantolebaa y = 2x – (4+5 ). 

29.13. (3-quliani) 

1800-iani mobrunebiT wrewiris O (2; 2) centri aisaxeba O (–2; –2)-ze, radiusis sidide ki igive rCeba. 

1 2 

amitom mobrunebiT miRebuli wrewiris gantolebaa (x +2) 2 + (y + 2) 2 = 4; O O = 4 . 

1 2 

y ↑ 

(4-quliani) 90 0 -iani mobrunebiT O 1 (2; 2) → O 2 (–2; 2); O 1 O 2 = 4 . 

→ x 

2 

O 

1 

a 

b 

→ 

x

(5-quliani) 450-iani mobrunebisas O (2; 2) → O (0; 2 1 2 ) amitom asaxviT miRebuli wrewiris gantolebaa 

x2 + (y – 2 ) 2 = 4; O O = 16 – 8 1 2 . 

29.15. (3-quliani) ujredebian furcelze Sesrulebuli naxaziT moswavleebi darwmundebian, rom 

x = y wrfis mimarT simetria S a : (x; y) → (y; x), e. i. abscisa da ordinati adgilebs icvlis. amitom 

y = 2x +3 → x = 2y + 3. 

(5-quliani) vTqvaT, f da g Seqceuli funqciebia da 

(x o ; y o ) Zevs y = f(x)-is grafikze, e. i. y o = f(x o ). maSin x o =g(y o ), 

e. i. (y o ; x o ) Zevs y = g(x)-is grafikze. magram (x o ; y o ) da (y o ; x o ) 

simetriuli wertilebia y = x biseqtris mimarT. amrigad, 

Tu romelime wertili Zevs y = g(x)-is grafikze, maSin 

misi simetriuli wertili Zevs y = f(x)-is grafikze da piriqiT. 

29. 16. radgan mobrunebis centri adgilze rCeba, amitom A warmoadgens mocemuli wrfisa da 

misi anasaxis gadakveTis wertils. 

(3 quliani) y = 2x da y = – x/ 2 + 5 ⇒ A (2; 4) 

α = 900 iolad dgindeba naxazidan , an Semdegi wesis gamoyenebiT: y = k x + b da y = k x + b wrfeebi 

1 1 2 2 

marTobulia, Tu k ⋅ k = – 1 

1 2 

29.23. (3-quliani) uaryofiTi ver iqneba. 

(4-quliani) Tu yvela monacemi tolia, igive iqneba maTi saSualoc. aseT SemTxvevaSi saSualosgan 

gadaxra = 0 da standartuli gadaxrac = 0. 

(5-quliani) Tu standartuli gadaxraa S < 1, maSin S 2 < S, e. i. standartuli gadaxra meti iqneba 

dispersiaze. 

30. 9. mocemuli wrewiris centria A (3; –1) wertili. 

a) koordinatTa saTavis mimarT A-s simetriulia A (–3; 1), amitom saZiebeli gantolebaa 

1 

(x +3) 2 + (y – 1) 2 = 25 

§30 

b) Tu A 2 (x; y) aris A(3; –1)-is simetriuli (–3; 2) wertilis mimarT, maSin es ukanaskneli iqneba [AA 2 ] 

x + 3 y −1 

monakveTis Sua wertili. amitom = −3, 

= 2 ⇒ x = – 9, y = 5. saZiebeli wrewiris gantolebaa 

2 2 

(x + 9) 2 + (y – 5) 2 = 25 

x 

g) mocemuli wrewiris centri A(3; –1) Zevs y = 3x wrfis marTobul y = – wrfeze (SeasruleT 

3 

naxazi). amitom A-s anasaxi iqneba A 3 (–3; 1) wertili. 

30.12. (3 quliani) AC wrfis mimarT simetria ΔABC-s asaxavs mis kongruentul ΔAB 1 C-ze. amitom 

S ABC B 1 = 2 ⋅ S ABC = 2 ⋅ ⋅ 6 ⋅ 8 sin30 0 = 24 

30. 15. (3-quliani) ordinatTa RerZis, aseve misi paraleluri nebismieri wrfis mimarT simetria 

wertilis ordinats ar Secvlis. 

y ↑ 

y 0 

x 0 

O 

x 0 

(x 0; y 0 ) 

(y 0; x 0 ) 

y 0 

→ 

x 


85

y↑ 

(– x; y) (x; y) 

O 

→ 

x 


vTqvaT, S 2 (–x; y) = x o ; y) 

−x 

+ x0 

maSin , = 3 e. i. x = 6 + x 

0 2 

amrigad, S 

2 ο S (x; y) = (x + 6; y), e. i. 

1 

aRniSnuli kompozicia aris T (6;0) 

paraleluri gadatana x RerZis 

mimarTulebiT 6 erTeulze. 

(5-quliani) moswavleebi am kompoziciis moZebnas mxolod 

naxazis gamoyenebiT Tu SeZleben. moswavleebi am kompoziciis 

moZebnas mxolod naxazis gamoyenebiT Tu SeZleben. gavavloT 

1 da 2 wrfeebi da maTi marTobuli y = – x wrfec. 

y = – x ⇒ A(1,5; –1,5) 

y = x – 3 

cxadia, S 1 (1,5; –1,5) = A 1 (–1,5; 1,5). Tu axla am wertils avsaxavT 

S 2 simetriiT A 2 (x 0 ; y 0 ) wertilze, maSin A iqneba [A 1 A 2 ] monakve- 

⎧ xo 

+ ( −1, 

5) 

⎪ = 1, 

5 

Tis Suawertili: 

2 

⎨ 

⎪ yo 

+ 1, 

5 

= −1, 

5 

⎪⎩ 

2 

T (3;-3) paraleluri gadatana. 

⇒ x = 4,5 da y = –4,5. amrigad, S o S (1,5; –1,5) = (4,5; –4,5) yofila 

0 0 2 1 

30.17. (4-quliani) radgan orive parabolis wveros abscisa x 0 = 1 erTi da igivea, amitom k = 4. amasTan, 

meore funqcia nuldeba x = 1-ze ⇒ c = –2. pirveli funqciis grafiki gadis (0;1)-ze, ⇒ b = 1 


1 

5 

7 

6 

3 

2 

1 

3 

1 

§32 

y ↑ 

(– x; y) (x 0; y) 

y = – x 

A 1 (–1,5; 1,5) 


A 

D 

B 

E 

5+1+2+4=12 

C 

A 

D 

B 

E 

2 

7 

C 

1+2+1+3=7 

3 

G 

4 

F 

A 

B 

6 

O 

y↑ 

A 

O 

10 

F 

3 

E 

y = x 

→ 

x 

y = x – 3 

A(1,5; –1,5) 

B 

8+2+5+4+3=22 

6 4 

C 

D 

E 

→ x 

C 

D

32.9. 

32.10. a) 

b) 

a) 

200 

A 150 

B 

C 

3 

400 

210 

3 5 

3 5 3 5 

3 5 3 5 3 5 3 5 

a 

240 

D 

marSruti: AD _ 400 dolari. 

ABD _ 150 + 240 = 390 dolari. 

ACD _ 200 + 210 = 410 dolari. 

pasuxi: ABD. 

rogorc am grafidan Cans, cifri 3-iT (Cvens 

amocanaSi) iwyeba rva oTxniSna ricxvi, amdenive 

iqneba 5-iT dawyebuli. 

pasuxi: 16 ricxvi. 

a b c 

a b c a b c a b c 

a b c a b c a b c a bc a b c a b c a b c a b c a bc 

vTqvaT, gansxvavebuli cifrebia a, b, c. rogorc grafidan Cans, a-Ti iwyeba 27 ricxvi. Tu b-Ti da 

c-Ti dawyebulebsac mivaTvliT, sul gveqneba 3 . 27=81 ricxvi. 

32.11. 

B 

D 

C 

A 

kenti wveroebia B da D. 

arsebobs eileris Semovla, romelic 

erT-erTi maTganidan unda iwyebodes. 

32.12. A B 

A B 

C D E 

F 

figuris gareTa F are A, B, C, E areebs unda ukavSirdebodes or-ori wiboTi, D-s ki _ erTi wiboTi 

da a. S., rogorc es grafzea naCvenebi. grafis oTxi wvero (F, A, B, D) kentia, amitom saZiebeli marSruti 

marTlac ar arsebobs. 

C 

D 

F 


E 

87

32.13. (3-quliani) amocanisaTvis Sedgenili grafi ase gamoiyureba. masze 

mxolod ori wveroa kenti (C da D), Semovlac erT-erTi maTganidan iwyeba. 

(4-quliani) 

(5-quliani) kubis wiboebis sigrZeTa jamia 12 . 10 = 120 sm. aseTive sigrZis mavTulisgan karkasi 

gaWris gareSe ar damzaddeba, radgan grafze ar arsebobs eileris Semovla. kubis wveroebi da 

wiboebi qmnian grafs, romlis yvela wvero kentia. 

7. Seadare 30.15 (3 quliani) 

8. a + a + a + a + a = 40 

1 3 5 7 9 

a + a + a + a + a = 60 

2 4 6 8 10 


4 

1 

2 

V Tavis damatebiTi savarjiSoebi 

⇒ (a 2 – a 9 ) + ... (a 10 – a a ) = 20 

5d = 20; d = 4 

10. miTiTeba: grafis wveroTa xarisxebis jami orjer metia wiboTa ricxvze (ix. savarjiSo 32.3, 

3-quliani). 

§33 

33.1. (3-quliani) A 1 _ SemTxveviTi, A 2 _ aucilebeli, A 3 _ SeuZlebeli, A 4 _ SemTxveviTi, A 5 _ SeuZlebeli. 

( 5-quliani) eileris amocana. A 1 _ SemTxveviTi, A 2 _ SemTxveviTi, A 3 _ SemTxveviTi, A 4 _ SeuZlebeli. 

33.2. (3-quliani) unda iyidoT 901 bileTi. 

(4-quliani) quliani) B _ SeuZlebeli, B _ SeuZlebeli, B _ SemTxveviTi. 

1 2 3 

SeiZleba vkiTxoT, am amocanaSi romeli xdomiloba iqneba aucilebeli. aucilebeli xdomiloba _ 

amoRebuli burTulebidan mxolod erTi feris ar iqneba, erTi burTula mainc gansxvavebuli ferisa 

iqneba. 

(5-quliani) cda 1. ori monetis agdebisas yvela SesaZlo varianti: bs, sb, ss, bb. 

cda 2. ori SesaZlo Sedegi: samizneSi moxvedra, samizneSi armoxvedra. 

cda 3. isari gaCerdeba wiTel seqtorze, isari gaCerdeba Sav seqtorze, isari 

gaCerdeba TeTr seqtorze. 

2. kamaTlis gagorebisas yvela SesaZlo 36 variantia: 

pirveli kamaTeli 

1 2 3 4 5 6 

1 11 12 13 14 15 16 

2 21 22 23 24 25 26 

3 31 32 33 34 35 36 

4 41 42 43 44 45 46 

5 51 52 53 54 55 56 

6 61 62 63 64 65 66 

6 

B 

C 

D 

meore kamaTeli 

A

33.4. (4-quliani) by = a(4b – x) tolobaSi marjvena mxare iyofa a-ze. ⇒ a-ze iyofa marcxena mxarec da 

radgan b urTierTmartivia a-sTan, amitom y iyofa: a-ze. y = ay 1. am tolobis gaTvaliswinebiT, mocemuli 

gantolebidan ⇒ ax + bay 1 = 4 ab ⇒ x + by 1 = 4b; x = b (4 – y 1 ). aq marjvena mxare iyofa b-ze, amitom 

marcxena mxarec unda iyofodes: x = bx 1 ⇒ bx 1 = b(4 – y 1 ), x 1 = 4 – y 1 . 

y 1 -ma SeiZleba miiRos mniSvnelobebi 1, 2, 3 ; Sesabamisad x 1 –is mniSvnelobebi iqneba 3, 2, 1. pasuxi 

miiReba Zvel cvladebze dabrunebiT: (3a; b), (2a; 2b), (a; 3b) 

(5-quliani) (y – x) (y2 + xy + x2 ) = 13 ⋅ 7 tolobis marcxena mxareSi Tanamamravlebi SeiZleba iyos 1 

da 91, –1 da –91 (oTxi varianti) an 13 da 7, –13 da –7 (kidev oTxi varianti) miRebuli sistemebidan 

mxolod orsa aqvs mTeli amonaxsnebi: 

y – x = 7 

y2 + xy + x2 sistemis amonaxsnebia wyvilebi (-3; 4), (-4; 3) 

= 13 

y – x = 1 

y2 + xy + x2 = 91 

sistemis amonaxsneba wyvilebi (-6; -5), (5; 6) 

§34 

34.1. (4-quliani) A da B _ araTavsebadi xdomilobebia. 

1 1 

A da B _ Tavsebadi xdomilobebia. 

2 2 

A da B _ Tavsebadia. 

3 3 

A = {TSSS an TTSS}, B = {SSST an SSSS} 

3 3 

(5-quliani) karadidan unda amoiRon 11 cali fexsacmeli. miTiTeba: unda ganvixiloT yvelaze 

cudi situacia, roca viRebT sul sxvadasxva zomas. me-11 cali aucileblad romelime calis wyvili 

aRmoCndeba. 

34.2. (3-quliani) a) A xdomilobas aqvs 9 xelSemwyobi 

A = { (1,4), (4, 1), (2, 3), (3, 2), (1, 5) (5, 1) (2, 4) (4, 2) (3, 3}. 

b) A xdomilobas aqvs 5 xelSemwyobi (18-dan). 

g) 36 elementaruli xdomiloba {(11) (12)... (65) (66)}. 

(4-quliani) A 1 -s aqvs 12 xelSemwyobi, A 2 -s _ xelSemwyob ,A 3 -s _16 xelSemwyobi. 

A = {sami kamaTlis gagorebisas jami udris 5} = {(113), (131) (311), (212), (221), (122)}. 

(5-quliani) A, C, E, G, H _ SemTxveviTi, F _ SeuZlebeli, B, D _ aucilebeli. 

34.3. (4-quliani) a) A = {4T, 3T1S, 2T2S, 1T3S}. 

1 

B = {4S, 3S1T, 2S2T, 1S3T}. 

1 

A ∩B = {3S1T, 2S2T, 1S1T}. 

1 1 

A ∪B = {4T, 4S, 3S1T, 2S2T, 1S1T}. 

1 1 

b) A = {1T3S, 2T2S, 3T1S, 4T}. 

2 

B = {1S3T, 2S2T, 3S1T, 4S}. 

2 

A ∩B = {1T3S, 2T2S, 3T1S}. 

2 2 

A ∪B = {1T3S, 2T2S, 3T1S, 4T, 4S}. 

2 2 

g) A = {2S 2T} 

3 

B = SeuZlebeli xdomiloba, radganac Tu 4 burTulaSi 1 Savi burTulaa, 2 TeTri unda iyos, 

3 

magram 1S + 2T = 3 ≠ 4. 

A ∩ B = Ø. 

3 3 

A ∪ B = A 3 3 3. 


89

4 

1. 

10 

(4-quliani) quliani) a) Â ={monetis agdebisas mova `safasuri~}. 

1 

b) Â = {kamaTlis gagorebisas kenti ricxvi dajda}. 

2 

g) Â = {urnidan amoRebuli burTula ar aris TeTri}. 

3 

d) Â = {axalSobili gogonaa}. 

4 

35.1. (3-quliani) 


§35 

damoukidebeli saklaso samuSao #10-is pasuxebi da miTiTebebi 

I varianti 

98 

2. 

100 

3. albaToba imisa, rom monetis erTi agdebiT 

mova safasuri, -is tolia. albaToba imisa, rom 

safasuri meore agdebaze mova, anu ganxorci- 

eldeba xdomiloba {bs}, udris P{bs} = ; 

P{bbs}= . 

5 

= 

50 

1 

10 

(4-quliani) cifr `5~-s Seicavs 18 orniSna ricxvi: 15, 25, 35, 45, 50, 51, 52, 53, 54, 55, 56, 57, 

(5-quliani) IV, I I , I, I I I. 

35.2. (3-quliani) P (4 an 5, an 6) = 

82 

58, 59, 65, 75, 85, 95. P = . 

100 

3 1 

= . 

6 2 

(4-quliani) P (11, 22, 33, 44, 55, 66) = 

(5-quliani) avirCevdiT I rulets. 

6 1 

= . 

36 6 

4 

1. 

16 

3 8 

35.3. (3-quliani) a) ; b) P(Savi ar iqneba) = . 

15 

15 

I I varianti 

2. P(erTi gasroliT mizans ver moaxvedreben) = 

= 0,3. 

1 

3. P (6) = . ori gagorebiT rom boloSi dajdes 

6 

6, unda ganxorcieldes romelime xelSemwyobi 

5 

xdomiloba: (16), (26), (36), (46), (56). P = . 

36 

sami kamaTlis gagorebis 216 SesaZlo Sedegidan 

èqvsiani~ SeiZleba movides 91 SemTxvevaSi, aqedan 

èqvsiani~ boloSi dajdeba 25 SemTxvevaSi: 

(116, 126, 136, 146, 156, 216, 316, 416, 516, 226, 

236, 246, 256, 326, 426, 526, 336, 346, 356, 346, 365, 

25 

446, 456, 546, 556). P = . 

216

(4-quliani) P(A) = P (61, 62, 63, 64, 65, 66) = 

(5-quliani) A waxnags. 

1 

35.4. (3-quliani) a) P (4) = . 

6 

1 

P(B) = P (15, 25, 35, 45, 55, 65) = . 

6 

6 1 

= . 

36 6 

11 

P(C) = P (11, 12, 13, 14, 15, 16, 21, 31, 41, 51, 61) = . 

36 

11 

P(A∪B) = P (61, 62, 63, 64, 65, 66, 15, 25, 35, 45, 55) = . 

36 

b) P (mova aranakleb 3-isa) = P (3, 4, 5, 6) = 

g) P (mova luwi) = 

3 1 

= . 

6 2 

4 2 

= . 

6 3 

(4-quliani) 

1 

a) P (3,3,) = ; 

36 

5 

b) P (jami 8) = P (2,6), (62), (35), (53) (4,4)) = ; 

36 

g) P(qulaTa jami > 8-ze) = P (jami 10, jami 11, jami 12) = 

1 

(5-quliani) a) 

216 

35.5. (3-quliani) P (tuzi) = 

(4-quliani) P (jvriani) = 

10 

= P ((3,6) (63) (45) (54), (46) (64) (55) (56) (65) (66)) = . 

36 

4 1 

= ; 

36 9 

9 1 

= . 

36 4 

1 

b) 

216 

1 

(5-quliani) P = . 

36 

35.6. TiTo waxnagi eqneba SeRebili (100 – 36) . 6 = 384 kubiks; or-ori waxnagi aRmouCndeba im kubikebs, 

romlebic didi kubis patara wiboebs esazRvreba. maTi raodenoba iqneba 12 . 8 = 96. sami waxnagi 

SeRebili eqneba kubis wveroebTan mdebare kubikebs, e. i. sul 8 cals. 

384 

(3-quliani) P (erTi waxnagi SeRebili) = . 

1000 

96 

(4-quliani) P (ori waxnagi SeRebili) = . 

1000 

8 

(5-quliani) P (sami waxnagi SeRebili) = . 

1000 


91

36.1. (3-quliani) d. 

(4-quliani) b. 

(5-quliani) b. 

36.2. (3-quliani) P = . 


§36 

(4-quliani) manZili 40 km-sa da 70 km-s Soris, anu 30 km, unda daiyos sam tol nawilad. radgan 

brigada 52-e km-ze imyofeba, ajobebs moZraoba 70 km-saken, radgan am ubanze dazianebis albaToba 

iqneba. 

(5-quliani) P = 

1 

36.3. (3-quliani) P = . 

2 

(4-quliani) P = 1. 

1 

(5-quliani) P = . 

16 

2π 

/ 3 

= 

2π 

1 

3 

. 

36.4. (3-quliani) amoRebuli TeTri burTula urnaSi ar daabrunes, urnaSi 10 burTulaa, iqidan 4 

4 

TeTria, amitom P = . 

10 

(4-quliani) urnaSi darCa 9 burTula P (Savi) = 

(5-quliani) P (9-ianSi an aTianSi) = 0,1 + 0,4 = 0,5. 

36.7. (3-quliani) P = 0. 

(4-quliani) 

2 

P = 

3 

P = 0. 

3 

37.1. (3-quliani) P (arastandartuli) = . 

80 

§37 

6 2 

= . 

9 3 

(4-quliani) `1~-is mosvlis albaTobas SevafasebdiT P ≈ 0,167. 

224 + 160 

(5-quliani) a) P (230000 km mets) = = 0, 

768 . 

500 

b) P = 1. 

37.2. (3-quliani) ar SeiZleba. 

(4-quliani) ori kamaTlis 10-jer gagorebisas qulaTa jami arc erTxel ar aRmoCnda 12-is 

toli, magram es ar gvaZlevs saSualebas vamtkicoT, rom jami 12-is mosvlis albaToba udris 0-s. 

Zalian cota iyo Catarebuli cda (mxolod 10). jamis SesaZlo mniSvneloba ki 11-ia (2, 3, 4, 5, 6, 7,

8, 9, 10, 11, 12). amitom sruliad mosalodnelia rom 10-jer gagorebisas romelime erTi ganxorcielda. 

3 

37.3. (3-quliani) _ daWerili markirebuli Tevzis fardobiTi sixSire. 

36 

48 

≈ markirebuli Tevzis daWeris albaToba, sadac N auzSi Tevzis raodenobaa. N ≈ 

N 

48⋅ 

36 

= 576 . 

3 

8 4 

(4-quliani) P (wunis) = 0,05 fn (wunis) = = = 0,16, rac gacilebiT metia 0,05-ze, amitom 

50 25 

aucileblad saWiroa sawarmo procesSi Careva da xelsawyoebis gamorTva. 

(5-quliani) ... kamaTlis gagorebis Sedegebs aRwers cxrili 1. asanTis kolofis agdebis Sedegebs 

cxrili 2. 

1359671 

37.4. (3-quliani) biWis dabadebis albaToba P ≈ 

≈ 0, 51 

1359671+ 

1285086 

(4-quliani) P (gogos dabadeba) = 1 _ 0.515 = 0, 475 

(5-quliani) firma A 

foTlebiani Reroebis diagrama 

8 5, 6 

9 7, 8, 7, 2 

10 2, 1, 7, 6, 2, 2 

11 3, 6, 2, 6 

12 0 

moda = 10,2. 

mediana = 10,2. 

diapazoni = 12 _ 8,5 = 3,5. 

saSualo = 10,7. 

7 

P (10,0mm ≤ sisqe ≤ 11,0) = . 

18 

firma B 

foTlebiani Reroebis diagrama 

6 4 

7 7 

8 2 

9 4, 5, 9, 7 

10 2, 5, 6 

11 5, 5 

12 6, 2 

13 0, 2, 5 

14 9 

moda = 11,5 

mediana = 10,5 

diapazoni = 14,9 _ 6,4 = 8,5 


93

saSualo = 10,8 

3 

P (10,0 mm ≤ sisqe ≤ 11,0) = . 

18 

airCevdiT firma A-s minebs. monacemebis gabneva naklebia (d = 3,5). albaToba, rom minis sisqe 

7 

10,0 mm-sa da 11 mm-s Sorisaa, metia ( ). 

18 

37.5. (3-quliani) a) P (lambaqSi kenWis moxvedris albaToba) = 


2 

π 5 

π 25 

2 

25π 

= = 

625π 

1 

b) daaxloebiT 25 kenWi unda CaagdoT, rom prizi moigoT 4P = 25 ⋅ = 1 

25 

g) 250 kaci 1250 kenWs Caagdebs. unda velodoT rom miaxloebiT 1256 kenWi moxvdeba 

lambaqSi anu unda iyos momzadebuli ara nakleb 56 prizisa. 

38.1. (3-quliani) Tu m = 6 gantoleba ase Caiwereba: (n-3) (k-3) = 9, 6 ≤ n ≤ k; 

3 ≤ n – 3 ≤ k – 3 → n = 6, k = 6. pasuxi (6; 6; 6). 

(4-quliani) Tu m = 4 maSin (n-4) (k-4) = 16, 4 ≤ n ≤ k; 0 < n – 4 ≤ k – 4; 

n – 4 = 1, k – 4 = 16, e. i. n = 5, k = 20; 

n – 4 = 2, k – 4 = 8, e. i. n = 6, k = 12; 

n – 4 = 4, k – 4 = 4, e. i. n = 8, k = 8. 

pasuxi: (4; 5; 20), (4; 6; 12), (4; 8; 8). 

(5-quliani) Tu m = 5 maSin (3n – 10) (3k – 10) = 100, 5 ≤ n ≤ k; 15 ≤ 3n ≤ 3k; 

5 ≤ 3n – 10 ≤ 3k – 10; 

pasuxi: (5; 5; 10) 

38.2. (3-quliani) 4 ⋅ 6 ⋅ 12 

(4-quliani) 3 ⋅ 3 ⋅ 3 ⋅ 4 ⋅ 4 

(5-quliani) 3 ⋅ 3 ⋅ 4 ⋅ 3 ⋅ 4 

38.3. m ≤ n ≤ k da , romelTa gaTvaliswinebiT gantolebidan gveqneba , e. i. m ≤ 6. 

§38 

38.4. vTqvaT, kvanZTan Tavmoyrilia k cali wesieri mravalkuTxedi, Sesabamisad α 1 ,α 2 ,...,α k -s 

toli kuTxeebiT. maSin α 1 + α 2 + ...+ α k = 360 0 . magram TiToeuli am kuTxis sidide ≥ 60 0 . amitom k ⋅ 60 0 

≤ 360 0 da k ≤ 6. 

38.5. (3-quliani) n = 4 

(4-quliani) 3 

(5-quliani) 6 

1 

25 

.

38.6. (5-quliani) vTqvaT, aRniSnuli teselacia Sedgenilia n cali samkuTxediTa da m cali 

eqvskuTxediT. maSin n ⋅ 60 0 + m ⋅ 120 0 = 360 0 , n + 2m = 6 → n =4 , m = 1 da n = 2, m = 2. 

pasuxi: 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 6, 3 ⋅ 6 ⋅ 3 ⋅ 6 

baa 

1. a) wiTeli burTulas amoRebis albaTobaa 

VI Tavis damatebiTi savarjiSoebi 

3 21 

= , amitom urnaSi aranakleb 10 wiTeli burTulaa. 

5 35 

b) urnaSi aranakleb 21 TeTri burTulaa. 

2 10 

= , xolo TeTri burTulas amoRebis albaTo- 

7 35 

4 

g) urnaSi 35 _ (10 + 21) = 4 Savi burTula, Savi burTulas amoRebis albaTobaa . 

35 

1 

2. mease agdebaze safasuris mosvlis albaTobaa . yovel cdaSi borjRalos mosvlis albaToba 

2 

ar aris damokidebuli imaze, ra dajda wina cdaSi, Tu moneta simetriulia. radganac 99 cdaSi 

borjRalos da safasuris TiTqmis erTnairi sixSire dafiqsirda, SeiZleba CavTvaloT, rom 

moneta simetriulia. 

312 

3. A kandidati miiRebs xmebis miaxloebiT ⋅100% = 52%. B kandidati miiRebs xmebis miax- 

600 

loebiT 

240 

⋅ 100% = 40%. amomrCevelTa (100 _ 92)% = 8% ar miiRebs monawileobas arCevnebSi. 

600 

4. samniSna ricxvi, romelic iyofa 2-ze: 354, 534. 

samniSna ricxvi, romelic iyofa 3-ze: 345, 435, 534, 345, 453, 543. 

samniSna ricxvi, romelic iyofa 6-ze: 354, 534. 

samniSna ricxvi, romelic iyofa 5-ze: 345, 435. 

5. iyo gaTamaSebuli 15 partia 

12 13 14 15 16 

23 24 25 26 

34 35 36 

45 46 

56 

6. sityva `guli~-s anagramebi SeiZleba miviRoT xisebri diagramis gamoyenebiT. 

anagramebi: guli, guil, glui, gliu, giul, gilu, ugli, ugil, ulgi, ulig, uigl, uilg, lgui, 

lgiu, lugi, luig, liug, ligu, igul, iglu, iugl, iulg, ilgu. 

7. gamoviyenoT xisebri diagrama 

gzebi I1, I 2, I I 1, I I 2, I I I 1, I I I 2. A qalaqidan C qalaqamde B qalaqis gavliT arsebobs 6 sxvadasxva 

marSruti. 

8. unda iyidoT 18 bileTi. 


95

9. ganvixiloT yvelaze `cudi~ situacia, rodesac CanTidan sul TeTr burTulebs viRebT. maSin 

minimum 6 burTula unda amoviRoT, rom maT Soris erTi TeTri da erTi wiTeli burTula iyos. 

11. monakveTi AB davyoT oTx tol nawilad. 

SemTxveviT arCeuli wertili ufro axlos iqneba M-Tan, vidre A-sTan an B-sTan, Tu moxvdeba 

CD monakveTze P = 

12. P = 

CD 1 

= . 

AB 2 

daStrixuliarisfarTobi 

mTeli farTobi 

2 2 

πr − 2r 

π − 2 

13. P = = 2 

≈ 0,36. 

πr 

π 

14. a) P = 

2, 

25π 

2. 

25π⋅ 

2 

= 

3⋅14⋅14⋅ 

sin 60 3⋅196⋅ 

3 


≈ 0,014. 

b) 200 ⋅ 0,014 ≈ 2,8. Tu wertils 200-jer airCeven, mosalodnelia, rom is 3-jer moxvdeba daStrixul 

areSi. 

g) albaToba oTxjer gaizrdeba: P ≈ 0,056. 

15. Tqven saxlSi ar iyaviT 17 00 sT-dan 17 10 sT-mde, anu 10 wuTi. 

P (amxanagis zars ascdiT) = = 

16. = 

19. P = 

20. a) P = = = 0,912. 

23. P = .

1 pdf-istvis 1-10.pmd - Ganatleba

Create successful ePaper yourself

Delete template?

Save as template?