Ýokary matematika boýunça meseleler we gönükmeler

576 Pages·7.858 MB·Uzbek

by  Körpäýew Kakajan, Ataýew Alladurdy.

TürkmenisTanyň PrezidenTi
GurbanGuly berdimuhamedow

TürkmenisTanyň dÖwleT TuGrasy
TürkmenisTanyň dÖwleT baÝdaGy
TürkmenisTanyň
dÖwleT senasy
Janym gurban saňa, erkana ýurdum,
Mert pederleň ruhy bardyr köňülde.
Bitarap, garaşsyz topragyň nurdur,
Baýdagyň belentdir dünýäň öňünde.
Gaýtalama:
Halkyň guran Baky beýik binasy,
Berkarar döwletim, jigerim-janym.
Başlaryň täji sen, diller senasy,
Dünýä dursun, sen dur, Türkmenistanym!
Gardaşdyr tireler, amandyr iller,
Owal-ahyr birdir biziň ganymyz.
Harasatlar almaz, syndyrmaz siller,
Nesiller döş gerip gorar şanymyz.
Gaýtalama:
Halkyň guran Baky beýik binasy,
Berkarar döwletim, jigerim-janym.
Başlaryň täji sen, diller senasy,
Dünýä dursun, sen dur, Türkmenistanym!
kakajan körpäýew, alladurdy ataýew,
allaberdi aşyrow, orazmämmet annaorazow
Ýokary maTemaTika
boÝunÇa meseleler
we GÖnükmeler
Ýokary okuw mekdepleri üçin okuw gollanmasy
Türkmenistanyň Bilim ministrligi
tarapyndan hödürlenildi
Aşgabat
“Ylym” neşirýaty
2012
UOK 510:378
K 76
körpäýew k. we başg.
K76 Ýokary matematika boýunça meseleler we gönükmeler. Ýokary okuw
mekdepleri üçin okuw gollanmasy. – A.: Ylym, 2012. – 576 sah.
TDKP № 222 KBK 22.11 ýa 73
© Körpäýew K. we başg., 2012
© «Ylym» neşirýaty, 2012
I ÁÜËÞÌ
ÒÅÊÈÇËÈÊÄÅ ÀÍÀËÈÒÈÊ ÃÅÎÌÅÒÐÈÉÀ
¸ 1. Òåêèçëèêäå ãüíþáóð÷ëû äåêàðò êîîðäèíàòàëàð sistemasy
Òåêèçëèêäå àëíàí åðêèí íîêàäûÿ îðíónû êåñãèòëåìúãå ìþìêèí÷èëèê
áåðéúí sistema êîîðäèíàòàëàð sistemasy äèéèëéúð. Øîë sisteìaëàðûÿ èÿ
éüíåêåéè ãüíþáóð÷ëû äåêàðò êîîðäèíàòàëàð sistemasydyr.
Ãîé, òåêèçëèêäå èêè ñàíû üçàðà ïåðïåíäèêóëéàð óãðóêäûðûëàí ãüíè
÷ûçûêëàð áåðëåí áîëñóí. Áó ãüíè ÷ûçûêëàðûÿ áèðèíè êåñå (ãîðèçîíòàë),
áåéëåêèñèíè äèê (âåðòèêàë) éåðëåøäèðåëèÿ. Îëàðûÿ êåñèøìå O íîêàäûíà
êîîðäèíàòàëàð áàøëàíãûùû äèéèëéúð. Êåñå ãüíè ÷ûçûãà àáñèññàëàð îêû (Ox
îêy), äèê ãüíè ÷ûçûãà áîëñà îðäèíàòàëàð îêû (Oy îêy) äèéèëéúð. Àáñèññàëàð
îêóíûÿ položitel ugry àäàò÷à êîîðäèíàòàëàðûÿ áàøëàíãûùûíäàí ñàã òàðàïà,
otrisatel óãðû áîëñà ÷åï òàðàïà óãðóêäûðûëàíäûð. Îðäèíàòàëàð îêóíûÿ
položitel óãðû êîîðäèíàòàëàðûÿ áàøëàíãûùûíäàí éîêàðûê, otrisatel óãðû
áîëñà àøàê óãðóêäûðûëàíäûð.
Ýðêèí M íîêàäûÿ òåêèçëèêäúêè îðíóíû îíóÿ àáñèññàëàð âå îðäèíàòàëàð
îêëàðûíà áîëàí ïðîéýêñèéàëàðû àðêàëû êåñãèòëåìåê áîëàð. A íîêàäûÿ
àáñèññàñû 2, îðäèíàòàñû 3 äèéèï éàçìàãûÿ äåðåãèíå A(2,3) ãüðíþøäå
áåëãèëåíéúð, îêàëàíäà áîëñà A íîêàäûÿ êîîðäèíàòàëàðû 2 âå 3 äèéèï
îêàëéàð. Óìóìàí åðêèí M íîêàäûÿ àáñèññàñû x, îðäèíàòàñû y áîëñà, îë
øåéëå éàçûëéàð: M(x, y).
Åãåð A(x , y ) we B(x , y ) òåêèçëèãèÿ åðêèí íîêàòëàðû áîëñà, îíäà
1 1 2 2
îëàðûÿ àðàñûíäàêû d óçàêëûê àøàêäàêû ôîðìóëà áèëåí êåñãèòëåíéúð:
d = (x −x )2 +(у − у )2 . (1)
2 1 2 1
( ) ( ) ( )
Åãåð A x , y , B x , y we C x ,y íîêàòëàð þ÷áóð÷ëóãûÿ äåïåëåðè
1 1 2 2 3 3
áîëñà, îíäà îíóÿ ìåéäàíû àøàêäàêû ôîðìóëà áèëåí õàñàïëàíûëéàð:
1
S = x (y − y )+x (y − y )+x (y − y ) =
2 1 2 3 2 3 1 3 1 2
1 1 x −x y − y
= (x −x )(y − y )−(x −x )(y − y ) =+ 2 1 2 1 (2)
2 2 1 3 1 3 1 2 1 2 x −x y − y
3 1 3 1

1. Åãåð A(6,0), B(5,2), C(0,3), D(−7,1) we E(−4,−6) íîêàòëàð áúø-
áóð÷ëóãûÿ äåïåëåðè áîëñà, îë áúøáóð÷ëóãû ãóðìàëû.
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2. Ýãåð A(5,3), B(2,−1), C(−1,4) íîêàòëàð þ÷áóð÷ëóãûÿ äåïåëåðè
áîëñà, îíóÿ òàðàïëàðûíûÿ óçûíëûêëàðûíû òàïìàëû.
×üçþëèøè. Èêè íîêàäûÿ àðàsûíäàêû óçàêëûãû tàïìàãûÿ (1) ôîðìóëàñûíû
óëàíûï àëàðûñ:
АВ = (5−2)2 +(3+1)2 = 9+16 =5;
BC = (2+1)2 +(−1−4)2 = 9+25 = 34;
АC = (5+1)2 +(3−4)2 = 36+1= 37. 

3. ÄåïåëåðèA(−2,−4), B(2,8) we C(10,2) íîêàòëàð áîëàí
þ÷áóð÷ëóãûÿ ìåéäàíûíû hàñàïëàìàëû.
×üçþëèøè. (2) ôîðìóëàíû óëàíûï àëàðûñ:
1 1
S = (2+2)(2+4)−(10+2)(8+4) = 24−144 =60kw.bir.

2 2
4. Μ (7,4) we M (3,−5) íîêàòëàðûÿ àðàsûíäàêû óçàêëûãû òàïìàëû.
1 2
5. Àøàêäàêû íîêàòëàðûÿ àðàsûíäàêû óçàêëûãû òàïìàëy:
1) A(2,3) we B(−10,2);
( ) ( )
2) C 2,−7 we D 2 2,0 ;
3) A (3,−4) we B (6,−8);
1 1
4) A (10,0) we B (2,−6);
2 2
5) A(−11,−4) we B (1,−9);
3 3
6) A (8,−4) we B (−2,1).
4 4
6. Äåïåëåðè Î (0, 0), À (1, 1), Á (2, –2) íîêàòëàð áîëàí þ÷áóð÷ëóãûÿ
ãüíüáóð÷ëû þ÷áóð÷ëóêäûãûíû ñóáóò ýòìåëè.
7. Äåïåëåðè À (1, 5), Á (2, 7) âý C (4, 11) íîêàòëàð áîëàí þ÷áóð÷ëóãûÿ
ìåéäàíûíû õàñàïëàìàëû.
8. À (–1, 2), Á (5, 6), C (1, 3) íîêàòëàð þ÷áóð÷ëóãûÿ äåïåëåðè áîëñà, C
äåïåäåí èíäåðèëåí áåéèêëèãèÿ óçûíëûãûíû õàñàïëàìàëû.
9. Äåïåëåðè À (–3, 3), Á (1, 3), C (1, –1) íîêàòëàð áîëàí þ÷áóð÷ëóãûÿ
ãüíþáóð÷ëû þ÷áóð÷ëóêäûãûíû ãüðêåçìåëè.
×üçþëèøè. ÀÁ, ÀC âå ÁC êåñèìëåðèÿ óçûíëûêëàðûíû òàïàëûÿ:
АВ = (1+3)2 +(3−3)2 =4; АС = (1+3)2 +(−1−3)2 =4 2 ;

ВС = (1−1)2 +(−1−3)2 =4.
Îíäà АВ2 + ВС2 = АС2 äåÿëèãèÿ éåðèíå éåòéúíëèãè þ÷èí Ïèôàãîðûÿ
òåîðåìàñûíûÿ ýñàñûíäà þ÷áóð÷ëóê ãüíþáóð÷ëûäûð.
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¸ 2. Êåñèìè áåðëåí ãàòíàøûêäà áüëìåê
M (x ,y ) we M (x ,y ) íîêàòëàðû áèðëåøäèðéúí êåñèìèÿ þñòþíäå
1 1 1 2 2 2
М М
éàòéàí âå 1 =λ äåÿëèãè êàíàãàòëàíäûðéàí M íîêàäûÿ
ММ
2
êîîðäèíàòàëàðû àøàêäàêû ôîðìóëàëàð áèëåí êåñãèòëåíéúð:
x +λx y +λy
x = 1 2 , y = 1 2 . (1)
1+λ 1+λ
Ýãåð M(x,y) íîêàò M M êåñèìè äåÿ èêè áüëåãå áüëéúí áîëñà,
1 2
îíäà λ=1 we (1) formula
x +x y + y
x = 1 2 , y = 1 2 (2)
2 2

gürnþøi alar.
10a.  Äåïåëåðè A(5,3) âå B(2,−1) íîêàòëàðäà áîëàí ABC
þ÷áóð÷ëóãûÿ ìåäèàíàëàðû M(2,2) íîêàòäà êåñèøéúí áîëñàlar îíóÿ
C(x,y)
þ÷þíùè äåïåñèíèÿ êîîðäèíàòàëàðûíû òàïìàëû (1-nji a ÷yzgy).
×üçþëþøè. (2) ôîðìóëàíû óëàíûï, AB êåñèìè äåÿ èêè áüëåãå áüëéúí
N íîêàäûÿ x y êîîðäèíàòàëàðûíû òàïàëûÿ:
0, 0
5+2 7 3−1
x = = , y = =1. 
0 2 2 0 2
Þ÷áóð÷ëóãûÿ þ÷þíùè C äåïåñè AB êåñèìè äåÿ èêè áüëåãå áüëéúí N
íîêàò áèëåí, ìåäèàíàëàðûÿ êåñèøìå M íîêàäûíäàí ãå÷éúí MN ãüíþäå
éàòéàíäûð. Ìåäèàíànûÿ õúñèéåòèíå ãüðú MN êåñèìèÿ óçûíëûãû NC
1
ìåäèàíàíûÿ óçûíëûãûía äåÿäèð. Øîíóÿ þ÷èí MN êåñèìèÿ äàøûíäà
3
NC 3
éàòéàí C íîêàò, êåñèìè λ= =− ãàòíàøûêäà áüëéúð. Îíäà (1) âå
CM 2
7 3
(2) ôîðìóëàëàðäà x = , y =1, x =2, y =2, λ=− ãîéóï, C
1 2 1 2 2 2
íîêàäûÿ x,y êîîðäèíàòàëàðûíû òàïàðûñ.:
7 ⎛ 3⎞ ⎛ 3⎞
+⎜− ⎟⋅2 1+⎜− ⎟⋅2
2 ⎝ 2⎠ ⎝ 2⎠
x = = −1, y = = 4.
⎛ 3⎞ ⎛ 3⎞
1+⎜− ⎟ 1+⎜− ⎟
⎝ 2⎠ ⎝ 2⎠
Øóíëóêäà, C(−1,4).

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1-nji a ÷yzgy
10 b. M (7,5) âå M (−2,3) íîêàòëàðû áèðëåøäèðéúí êåñèìäå M
1 2 1
íîêàäà, M íîêàòäàí 3 ýññå ãîëàé éåðëåøåí M(x,y) íîêàäû òàïìàëû.
2
М М 1 1
×üçþëèøè. Ìåñåëúíèÿ øåðòèíå ãüðú 1 = , îíäà λ= . (1)
ММ 3 3
2
ôîðìóëàëàðû óëàíûï àëéàðûñ:
1
( )
7+ −2
x +λx 3
x = 1 2 = =4,75;
1+λ 1
1+
3
1
5+ ⋅3
y +λy 3
y= 1 2 = =4,5.
1+λ 1
1+
3
Äèéìåê, M(4,75, 4,5) bolar.
11. Äåïåëåðè A(−5,3), B(−2,−1), C(5,3) íîêàòëàðäà áîëàí þ÷áóð÷ëóãûÿ
A äåïåñèíèÿ è÷êè áóð÷óíûÿ áèññåêòðèñàñûíûÿ óçûíëûãûíû òàïìàëû (1-nji
á çyzgy).
×üçþëèøè. Ãîé, A áóð÷óÿ áèññåêòðèñàñû AK áîëñóí. Áåëëè áîëøû éàëû
BK AB
K íîêàò BC êåñèìè = =λ áîëàí ãàòíàøûêäà áüëéúð.
KC AC
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