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Matematika olimpiada Tuman 2022-2023 o'quv yili

11-sinf Matematika olimpiada №4

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Avgust 19, 2024

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11-sinf Matematika olimpiada №4

2022-yil 12-oktyabr:

2022-2023 o'quv yilida tuman bosqichida tushgan savollar!

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1. To‘g‘ri burchakli uchburchakda AC=3, BC=4, CD-balandlik va CE-bissektrisa. DE ni toping.

A) 12/35

B) 15/31

C) 23/35

D) 21

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2. ABCD tetraedrning D uchidagi barcha yassi burchaklari to`g`ri. Shu tetraedrda kub shunday ichki chizilganki, kubning bitta uchi D nuqtada, unga qarama−qarshi uchi esa ABC yoqda yotibdi. Agar DA=5, DB=6 va DC=10 bo`lsa, kub qirrasining uzunligini toping.

A) 15/7

B) 25/12

C) 2√2

D) 2

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3. f(x)=x³/3-x²-35x+2 funksiya uchun f `(x)=0 bo`lsa x ni toping.

A) −7 va −5

B) 5 va 7

C) −7 va 5

D) −5 va 7

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

A) 120°

B) 90°

C) 60°

D) 30°

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5. 80 ni qanday musbat butun songa bo‘lganda, hosil bo‘ladigan qoldiq bo‘linmaning 25%iga teng bo‘ladi?

A) 21

B) 10

C) 19

D) 11

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6. bo’lsa, P(−1) ni toping.

A) 105

B) 115

C) 100

D) 120

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7. Nargizada 2 ta olma va 3 ta nok bor. U 5 kun ketma-ket har kuni singliga bittadan meva beradi. Bu ishni necha usul bilan amalga oshirish mumkin?

A) 6

B) 8

C) 10

D) 12

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8. Soddalashtiring: bunda, |?| < 1.

A) 2a+4

B) 2-2a

C) -2

D) 4-2a

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9. Soddalashtiring:

A) -1

B) −сtg1°

C) −сtg²1°

D) −сtg13°

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10. Hisoblang:

A) D

B) C

C) B

D) A

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11. y = x² − 2x + 3 parabolaga y = 3 to‘g‘ri chiziqqa nisbatan simmetrik bo‘lgan parabola tenglamasini tuzing.

A) y =-x² + 2x - 3

B) y = -x² − 2x + 3

C) y = x² − 2x - 3

D) y =-x² + 2x + 3

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12. Muntazam uchburchak 36 ta yuzi 1 ga teng bo‘lgan kichkina muntazam uchburchaklardan iborat (chizmaga qarang!). ABC uchburchak yuzini toping.

A) 13

B) 10

C) 11

D) 12

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13. ∠CAD = 2∠ACD, AC =13a, BD = 5a va AB = CD bo’lsa, 20⋅cos ∠ACD ni toping.(chizmaga qarang!)

A) 16

B) 19

C) 17

D) 15

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14. Arifmetik progressiyada ?₂ = 12, ?₇ − ?₄ = 9 bo‘lsa, ?₁₉ni toping.

A) 63

B) 61

C) 54

D) 56

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15. Raqamlari yig‘indisi 10 dan kam bo‘lmagan, raqamlari ko‘paytmasi esa 10 dan katta bo‘lmagan uch xonali sonlar nechta?

A) 106

B) 89

C) 96

D) 100

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16. Soatning minut mili 144° ga burilganda, soatning soat mili qanday burchakka burilada?

A) 6°

B) 18°

C) 12°

D) 9°

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17. (3 − cos²x− 2sinx)(lg²y+ 2lgy+ 4) ≤ 3 bo‘lsa, sin²x+ 20y+ 1 ni toping.

A) 2

B) 3

C) 5

D) 4

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18. Tengsizliklar sistemasinining butun yechimlari sonini toping.

A) 7

B) 6

C) 5

D) 4

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19. Agar ifodaning qiymatini toping.

A) 4

B) 0.8

C) 20

D) 0.05

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20. Doskaga ?₁, ?₂, ?₃, . . . , ?₂₀₀ sonlari yozilgan. Ma’lumki, ?₁ = 3, ?₂ = 9. Agar ixtiyoriy n natural son uchun ?_n+2 = ?_n+1 − ?_n tenglik o‘rinli bo‘lsa, ?₂₀₀ ni toping.

A) 3

B) 9

C) 8

D) 6

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21. u_nketma-ketlik quyidagicha berilgan: u₁ = 1, u_n+1 = u_n + 8n. U holda, u₅₀ −u₃₀ ni toping.

A) 6320

B) 6320

C) 5320

D) 6230

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22. BC = BE = CD ,∠DAE = 20°, ∠BCD = 60° ,∠ADE = ? (chizmaga qarang)

A) 20°

B) 30°

C) 40°

D) 50°

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23. a²b⁵ = 6²⁰ tenglikni qanoatlantiruvchi nechta (?; ?) butun sonlar juftligi mavjud?

A) 10

B) 12

C) 8

D) 14

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24. Natural son roppa-rosa 2 ta tub bo‘luvchiga, bu sonning kvadrati esa 45 ta turli natural bo‘luvchiga ega. Berilgan sonning kubi eng ko‘pi bilan nechta natural bo‘luvchiga ega bo‘lishi mumkin?

A) 41

B) 40

C) 91

D) 81

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25. ABC uchburchakda ctgA+ctgB= 3 va AB= 12 bo‘lsa, ABC uchburchak yuzini toping.

A) 216

B) 521

C) 123

D) 265

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26. formula bilan berilgan sonli ketma-ketlikning dastlabki 20 ta hadining o‘rta arifmetigini toping.