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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. To‘g‘ri burchakli uchburchakka ichki va tashqi chizilgan aylanalar radiuslari 2 va 5 ga teng. Katetlarning yig‘indisini toping.

A) 9

B) 14

C) 7

D) 12

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2. Poyezd 3 minutda 7 kilometrmasofani , motosikl 5 minutda 7 kilometrmasofanibosibo`tdi. Mototsikltezligipoyezdtezliginingnecha % initashkilqiladi?

A) 70%

B) 71%

C) 61%

D) 60%

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

A) 6

B) 8

C) 12

D) 10

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4. Hisoblang: (1 − tg5°)(1 − tg20°)2 + 2(2tg20° + tg5°)

A) 1

B) 2

C) 3

D) 4

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5. 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) 1.4

B) √5/2

C) √6/2

D) 4/3

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

A) -2⁹

B) -2¹¹

C) -4⁵

D) -4⁴

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

A) 16

B) 15

C) 14

D) 12

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8. 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) 65

B) 73

C) 55

D) 53

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

A) 6√5 + 1

B) 3√5 + 6

C) 6√5 + 2

D) √5 + 6

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10. Tenglamalar sistemasi nechta yechimga ega?

A) 10

B) 12

C) 6

D) 8

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

A) 1

B) 2

C) 4

D) 3

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12. Soddalashtiring:

A) 2sinα

B) 2

C) cosα

D) 1

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13. Hisoblang: 1 + 2 ∙ 2 + 3 ∙ 2² + 4 ∙ 2³+. … + 100 ∙ 2⁹⁹

A) C

B) D

C) B

D) A

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

A) 30°

B) 120°

C) 90°

D) 60°

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15. Soddalashtiring:

A) cosx

B) 3cosx

C) 3sin2x

D) 3cos2x

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16. (?²+ 4? + 8)² + 3?³ + 14?² + 24? = 0 tenglamaning ildizlari yig’indisini toping.

A) -11

B) -5

C) -7

D) -6

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17. Sektor perimetri 28 sm ga teng, uning yuzi esa 49 sm2. Sektor yoyining uzunligini toping.

A) 7 sm

B) 12 sm

C) 18 sm

D) 14 sm

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18. Soddalashtiring:

A) II

B) IV

C) I

D) III

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19. To‘g‘ri burchakli uchburchakka ichki va tashqi chizilgan aylana radiuslari nisbati 2 : 5 bo‘lsa, uchburchakning katta o‘tkir burchagining kosinusini toping.

A) 0.5

B) 0.2

C) 0.8

D) 0.6

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

A) 3

B) 2

C) 0

D) 1

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21. 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) D

D) A

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22. 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) 175/48

B) 211/47

C) 3

D) 177/48

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

A) 20/3

B) 120/17

C) 60/7

D) 260/17

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