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Matematika olimpiada Viloyat 2021-2022 o'quv yili

11-sinf Matematika olimpiada №3

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Avgust 19, 2024

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11-sinf Matematika olimpiada №3

2021-2022 o'quv yilida viloyat bosqichida tushgan savollar

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1. Koordinata boshidan o‘tuvchi, simmetriya o‘qi ordinata o‘qiga parallel, cho‘qqisi y=x va y+x-2=0 to‘g‘ri chiziqlar kesishgan nuqtada bo‘lgan parabolaning tenglamasini toping.

A) y=-x²-2x

B) y=-x²+2x

C) y=x²-2x

D) y=-x²+1

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2. Asosi ga teng bo`lgan muntazam to`rtburchakli piramidaning uchidan o`tkazilgan kesma asos tomonini 2:4 nisbatda bo`ladi va asos tekisligi bilan 60^o li burchakhosilqiladi. Piramidahajmini toping.

A) √10

B) 6

C) 12√30

D) √30

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3. Ikki xonali sonni qandaydir natural songa bo‘lganda bo‘linmada 3 va qoldiqda 8 hosil bo‘ladi. Agar bo‘linuvchining raqamlari o‘rni almashtirilsa va bo‘luvchi son o‘zgarishsiz qoldirilsa, u holda bo‘linmada 2 va qoldiqda 5 hosil bo‘ladi. Bo‘linuvchning dastlabki qiymatini toping.

A) 55

B) 65

C) 53

D) 73

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4. Soddalashtiring:

A) II

B) IV

C) I

D) III

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5. bo‘lsa, ?² + ?² ni toping.

A) 20

B) 61/20

C) 65/21

D) 20.2

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6. ABC uchburchakning AC tomonida shunday D nuqta olinganki, bunda ∠ABC=∠BDC, 5AB=4BD. Agar BC=10 bo‘lsa, ACni toping.

A) 8

B) 6

C) 10

D) 12

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7. Hisoblang:

A) 1.5

B) 1

C) 2

D) 0

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8. Funksiyaning eng kichik qiymatini toping. ?(?) = (2? + 6)⁵ − 4(2? + 6)⁴, bunda |? + 3| ≤ 2.

A) -2¹¹

B) -4⁴

C) -4⁵

D) -2⁹

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9. Aylananing ОА, ОВ, ОС radiuslari o‘tkazilgan. Agar bo‘lsa, АОВ burchakni toping.

A) 90°

B) 60°

C) 120°

D) 30°

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10. Agar ? + 4/√?= 17 bo‘lsa, ? + 4√? ni toping.

A) 3

B) 2

C) 1

D) 4

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11. Uchburchakning tomonlari 10, 24 va 26. Aylananing markazi uchburchakning katta tomonida yotadi va ikki kichik tomonlari bu aylanaga urinadi. Aylana radiusini toping.

A) 60/7

B) 260/17

C) 120/17

D) 20/3

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12. Soddalashtiring:

A) 3sin2x

B) 3cosx

C) 3cos2x

D) cosx

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13. To‘g‘ri burchakli uchburchak o‘tkir burchaklari yarimlarining tangenslari ko‘paytmasi 1/6 ga teng bo‘lsa, uchburchak o‘tkir burchaklarining kosinuslari yig‘indisini toping.

A) √5/2

B) 4/3

C) 1.4

D) √6/2

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14. АВCDE beshburchak berilgan, М, N, Р va Q nuqtalar —АВ, ВС, CD va DE tomonlarning o‘rtalari. Agar U va V — МP? va NQ ning o‘rtasi bo‘lsa, АЕ: UV nisbatni toping.

A) 3

B) 6

C) 4

D) 5

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15. Quyidagi sonlar orasidan musbat sonni aniqlang.

A) B

B) A

C) C

D) D

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16. Tenglamalar sistemasi nechta yechimga ega?

A) 12

B) 6

C) 10

D) 8

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17. Agar ?(?) = ?²⁰²¹ + 3?²⁰²⁰ + 3?²⁰¹⁹ − ?²⁰¹⁸ + 1 bo‘lsa, ni toping.

A) 3

B) 0

C) 2

D) 1

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18. 80 ni qanday musbat butun songa bo‘lganda, hosil bo‘ladigan qoldiq bo‘linmaning 25%iga teng bo‘ladi?

A) 11

B) 10

C) 21

D) 19

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19. Hisoblang:

A) 172/249

B) 81/109

C) 85/104

D) 95/243

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20. АВС uchburchak ∠C=60° burchagining bissektrisasi 5√3 ga teng. Agar АС : ВС = 5 :2 bo‘lsa, ВС ni toping.

A) C

B) B

C) A

D) D

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21. Tenglama nechta yechimga ega? Bunda, x ∈ (1; 47)

A) 14

B) 12

C) 16

D) 15

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22. Agar a = √5 + 1 va b = √5 − 1 bo‘lsa, 2a² − ab − a² ifodaning qiymatini toping.

A) 3√5 + 6

B) 6√5 + 1

C) √5 + 6

D) 6√5 + 2

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23. (?²+ 4? + 8)² + 3?³ + 14?² + 24? = 0 tenglamaning ildizlari yig’indisini toping.

A) -11

B) -6

C) -7

D) -5

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24. Aylanaga АВС uchburchak ichki chizilgan. Uchburchakning СС1medianasining davomi aylanani ? nuqtada kesib o‘tadi. ni toping.

A) 1

B) 4

C) 2

D) √5

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25. Teng yonli АВС uchburchakda: АВ = ВС = 25 sm va АС = 14 sm. ВС tomonga AD balandlikning D asosida urinuvchi va АС tomon o‘rtasidan o‘tuvchi aylana radiusini toping.

A) 211/47

B) 175/48

C) 3

D) 177/48