1 pdf-istvis 1-10.pmd - Ganatleba

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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

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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: `oqros 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

maTematika X maswavleblis wigni

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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.

maTematika X maswavleblis wigni

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

maTematikuri ganaTleba, aseTia:

6 maTematika X maswavleblis wigni

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.

maTematika X maswavleblis wigni

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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.

8 maTematika X maswavleblis wigni

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

savaldebulo sagani “maTematika”

maTematikis erTiani kursi

VII – IX klasebi (sabazo skola)

savaldebulo sagani “maTematika”

maTematikis erTiani kursi

X klasi (saSualo skola)

savaldebulo sagani “maTematika”

maTematikis erTiani kursi

→ →

dawyebiTi skola

sabazo skola

saSualo skola

XI – XII klasebi

savaldebulo sagani “maTematika”

maTematikis zogadi kursi






maTematika X maswavleblis wigni

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

10 maTematika X maswavleblis wigni

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

Sedegi TvalsaCinoa, Tu moswavle:

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

Sedegi TvalsaCinoa, Tu moswavle:

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

Sedegi TvalsaCinoa, Tu moswavle:

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.).

maTematika X maswavleblis wigni

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

Sedegi TvalsaCinoa, Tu moswavle:

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.

12 maTematika X maswavleblis wigni

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

Sedegi TvalsaCinoa, Tu moswavle:

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

Sedegi TvalsaCinoa, Tu moswavle:

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.

Sedegi TvalsaCinoa, Tu moswavle:

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.

maTematika X maswavleblis wigni

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.

14 maTematika X maswavleblis wigni

mimarTuleba: geometria da sivrcis aRqma

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

formulirebis xerxebs

Sedegi TvalsaCinoa, Tu moswavle:

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

Sedegi TvalsaCinoa, Tu moswavle:

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

Sedegi TvalsaCinoa, Tu moswavle:

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

Sedegi TvalsaCinoa, Tu moswavle:

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

Sedegi TvalsaCinoa, Tu moswavle:

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

Sedegi TvalsaCinoa, Tu moswavle:

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

Sedegi TvalsaCinoa, Tu moswavle:

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

Sedegi TvalsaCinoa, Tu moswavle:

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

maTematika X maswavleblis wigni

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

sakontrolo samuSao #2

sarezervo dro

II Tavi _ informaciebi da ricxvebi

10 ricxviTi sistemebi

11 informaciis kodireba

12 modularuli ariTmetika

13 saidumlo damwerlobebi

sakontrolo samuSao #3

sarezervo dro

III Tavi _ orcvladiani gantolebebi da utolobebi

14 orcvladiani gantoleba

15 orcvladian gatolebaTa sistema

16 amocanebi orcvladian gantolebaTa sistemaze

sakontrolo samuSao #4

16 maTematika X maswavleblis wigni

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

sakontrolo samuSao #6

24 ricxviTi mimdevrobebi

25 fibonaCis ricxvebi da oqros kveTa

26 Zalian didi da Zalian mcire

27 ras gvamcnoben monacemebi

sakontrolo samuSao #7

sarezervo dro

V Tavi _ figuraTa gardaqmnebi

28 figuraTa gardaqmnis magaliTebi

29 gadaadgileba

30 kongruentuli figurebi

sakontrolo samuSao #8

31 msgavsebis gardaqmna

sarezervo dro

III trimestri _ 50 sT.

msgavsebis gardaqmna

32 grafTa Teoriis elementebi

sakontrolo samuSao #9

VI Tavi _ xdomilobaTa albaToba

33 elementarul xdomilobaTa sivrce

34 Sedgenili xdomiloba

35 SemTxveviTi xdomilobis albaToba

sakontrolo samuSao #10

36 geometriuli albaToba

37 statistikuri albaToba

38 teselacia

sakontrolo samuSao #11

sarezervo dro

VII Tavi _ saswavlo wlis Semajamebeli samuSaoebi

39 raSi gvWirdeba funqcia?

40 pirobiTi winadadebebi

41 msjelobaTa saxeebi

sakontrolo samuSao #12

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

maTematika X maswavleblis wigni

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~,

`asaxva~, 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).

18 maTematika X maswavleblis wigni

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.

maTematika X maswavleblis wigni

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.

20 maTematika X maswavleblis wigni

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.

maTematika X maswavleblis wigni

21


22 maTematika X maswavleblis wigni

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.

maTematika X maswavleblis wigni

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.

24 maTematika X maswavleblis wigni

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 dro: 45 wT.

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.

maTematika X maswavleblis wigni

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.

26 maTematika X maswavleblis wigni

§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 `algebruli~ 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

maTematika X maswavleblis wigni

K

8

6

27


q. abulaZe

b. barbaqaZe

g. gabunia

t. iaSvili

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

M K

4

5

6

7

8

9

28 maTematika X maswavleblis wigni

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.

gakveTilis dro: 45 wT.

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.

maTematika X maswavleblis wigni

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.

30 maTematika X maswavleblis wigni


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: `oqros kveTa _ harmoniuli proporcia~ *

reziume

moswavleebi ecnobian cnebas `oqros 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. `oqros kveTa~ maTematikaSi: amocanis dasma, misi algebruli da geometriuli amonaxsni

2. `oqros 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 `ilia WavWavaZis~ erovnuli saswavlo gegmebisa da saxelmZRvaneloebis jgufis

mier.

maTematika X maswavleblis wigni

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

1 xelis CamorTmeva

A

C

B

D

4 amxanagi

6 xelis CamorTmeva

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.

maTematika X maswavleblis wigni

33


* SemoTavazebulia proeqt `ilia WavWavaZis~ erovnuli saswavlo gegmebisa da saxelmZRvaneloebis jgufis

mier.

34 maTematika X maswavleblis wigni

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 `axlosaa” ermaneTTan monacemTa

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

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

`axlosaa” 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 _

maTematika X maswavleblis wigni

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.

maTematika X maswavleblis wigni

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.

40 maTematika X maswavleblis wigni

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

sakontrolo samuSao #1

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

sakontrolo samuSao #2

II varianti

ipoveT Semdegi ricxvebis da ricxviTi gamosaxulebebis mniSvnelobebi

measedamde sizustiT:

daalageT xarisxebi zrdadobis mixedviT:

sul: 20 qula

damatebiTi amocana

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

maTematika X maswavleblis wigni

2

3

3

4

2

3

3

41


42 maTematika X maswavleblis wigni

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?


damatebiTi amocana

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.

maTematika X maswavleblis wigni

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)

damatebiTi amocana

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.

44 maTematika X maswavleblis wigni

sakontrolo samuSao #5

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.

sakontrolo samuSao # 6

# 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.

maTematika X maswavleblis wigni

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.

46 maTematika X maswavleblis wigni

sakontrolo samuSao # 7

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 )

damatebiTi amocana

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:

sakontrolo samuSao # 8

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

damatebiTi amocana

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).

maTematika X maswavleblis wigni

2

2

2

2

2

2

2

2

2

2

47


damatebiTi amocana

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

48 maTematika X maswavleblis wigni

sakontrolo samuSao # 9

# 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


damatebiTi amocana

sakontrolo samuSao # 10

# I varianti II varianti qulebi

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

kamaTelze dajda `5~}

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

kamaTelze dajda `3~}

B = {ori kamaTlis gagorebisas erTerT

kamaTelze dajda `4~}

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 (`amogdebuli~ wertiliT) sworad ageba _ 1 quliT

(`amogdebuli~ wertilis daufiqsirebloba isjeba am 1 qulis daklebiT).

maTematika X maswavleblis wigni

5

4

3

3

sul 15 qula

49


50 maTematika X maswavleblis wigni

sakontrolo samuSao # 11

# I varianti II varianti qulebi

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.

sakontrolo samuSao # 12

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

# I varianti II varianti qulebi

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

# I varianti II varianti qulebi

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.

maTematika X maswavleblis wigni

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.

52 maTematika X maswavleblis wigni

zogierTi sakontrolo samuSaos amoxsnis nimuSi:

[ −1;

3]

( − ∞;

−3,

5)

∪ ( − 0,

5;

+∞)

sakontrolo samuSao #2

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


damatebiTi amocana.

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. (I varianti)

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)

4. (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.

damatebiTi amocana.

x

2

2

2

= −

13

23

=

13

sakontrolo samuSao #4

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

4. (II varianti)


x

⎧xy

+ 600 = ( y + 5)(

x + 10)


⎩xy

= 1200



x


⎪y

⎪⎩

= −

= −

3) (II varianti)

23

13

28

13

= 9

maTematika X maswavleblis wigni

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


2. (I varianti)

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

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x

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AB wertilze gamavali wrfis gantolebaa

y = − x −1

⎧y

≥ −x

−1


y ≤ −x

−1


pasuxi: ⎪x

≤ 1


⎩x

≥ −3

sakontrolo samuSao #5

y


4

1

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O 1

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2. (II varianti)

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.

3. (II varianti)

a)

4

⎧2x

− y < 4


⎩x

− 3y

≥ 6

4. (II varianti)

y


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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 < B = 90 ABCD 0 , e. i. roca AB = x, BC = 16 – x

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

maTematika X maswavleblis wigni

55


⎛ a + b + c + d ⎞

miviRebT St. < ⎜ ⎟

⎝ 4 ⎠

⎛1000


= ⎜ ⎟

⎝ 4 ⎠

= S kv.

iolad SeiZleba Sedardes kvadratis da rombis farTobebic:

marTkuTxedis Sedareba kvadratTan.

250 – x

56 maTematika X maswavleblis wigni

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.

maTematika X maswavleblis wigni

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

58 maTematika X maswavleblis wigni

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 `ar 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

maTematika X maswavleblis wigni

=

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;

3 0 ← ganayofi 3, naSTi 0. 3 gavyoT 2-ze;

1 1 ← ganayofi 1, naSTi 1. 1 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

60 maTematika X maswavleblis wigni

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

maTematika X maswavleblis wigni

61


12.10. samkuTxedebis msgavsebiT:

15

62 maTematika X maswavleblis wigni

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

maTematika X maswavleblis wigni

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

64 maTematika X maswavleblis wigni

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.

⎪⎧


⎪⎩

maTematika X maswavleblis wigni

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

66 maTematika X maswavleblis wigni

gia salome lali suliko biZina

b) b) roca (–∞; –2], maSin x + 2 – x > 2

2 > 2


y ↑

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O

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– 2

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pasuxi: gia, salome, lali,

biZina, suliko (SesaZlebelia,

rom biZina, lali da suliko

erTi simaRlisanic

iyvnen).


x


+ 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 ↑

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x

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O

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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

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– 4

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x

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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

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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

maTematika X maswavleblis wigni

O

– 4

2

4

4

3 3

a ∈ (− ; )

5 5

67


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17.8. (5-quliani)

g)

y


d)

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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)

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– 2

2 ⎧(

x −1)

+ ( y + 2)


⎩2x

− y −1

= 0

68 maTematika X maswavleblis wigni

O

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O

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– 2

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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

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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

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18. 3. (3 quliani) d) y ↑

(4 quliani) b)


x

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O

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→ x

– 4

O

– 4

2

4


x

3 3

a ∈ (− ; )

5 5

y↑

O

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→ x

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(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 ↑

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§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

maTematika X maswavleblis wigni

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

70 maTematika X maswavleblis wigni

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

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A

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maTematika X maswavleblis wigni

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 −

72 maTematika X maswavleblis wigni

(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

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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

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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

maTematika X maswavleblis wigni

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 =

74 maTematika X maswavleblis wigni


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

maTematika X maswavleblis wigni

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

76 maTematika X maswavleblis wigni

, 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

maTematika X maswavleblis wigni

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+

...

78 maTematika X maswavleblis wigni

§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.

maTematika X maswavleblis wigni

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

80 maTematika X maswavleblis wigni

+ – +

-

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

maTematika X maswavleblis wigni


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

82 maTematika X maswavleblis wigni

⎧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

maTematika X maswavleblis wigni

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)

84 maTematika X maswavleblis wigni

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

maTematika X maswavleblis wigni

85


y↑

(– x; y) (x; y)

O


x

86 maTematika X maswavleblis wigni

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

32.7. (3-quliani) (4-quliani) (5-quliani)

1

5

7

6

3

2

1

3

1

§32

y ↑

(– x; y) (x 0; y)

y = – x

A 1 (–1,5; 1,5)

32.8. (3-quliani) (4-quliani) (5-quliani)

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

maTematika X maswavleblis wigni

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

88 maTematika X maswavleblis wigni

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.

maTematika X maswavleblis wigni

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)

90 maTematika X maswavleblis wigni

§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

`eqvsiani~ SeiZleba movides 91 SemTxvevaSi, aqedan

`eqvsiani~ 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

maTematika X maswavleblis wigni

91


36.1. (3-quliani) d.

(4-quliani) b.

(5-quliani) b.

36.2. (3-quliani) P = .

92 maTematika X maswavleblis wigni

§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


/ 3

=


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

maTematika X maswavleblis wigni

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) =

94 maTematika X maswavleblis wigni

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.

maTematika X maswavleblis wigni

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

96 maTematika X maswavleblis wigni

≈ 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 = .

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