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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. tenglama nechta ildizga ega?

A) 2

B) 3

C) 1

D) 4

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2. f(x)=70cosxcos6x uchun boshlang`ich funksiya toping.

A) 7sin5x+5sin7x+C

B) −7cos5x−5cos7x+c

C) 7cos5x−5cos7x+c

D) 7sin5x−5sin7x+c

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3. a+b=−1 a+c=6 va b+c=1 bo`lsa, a² (3b+3c+2a)+b² (3a+3c+2b)+c² (3a+3b+2c) ning son qiymatini toping.

A) 216

B) 27

C) 45

D) 100

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

A) 2

B) 3

C) 1

D) 4

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

A) -2⁹

B) -2¹¹

C) -4⁴

D) -4⁵

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

A) 16

B) 19

C) 15

D) 17

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7. 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) 40

B) 81

C) 41

D) 91

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8. Tengsizliklar sistemasinining butun yechimlari sonini toping.

A) 7

B) 5

C) 4

D) 6

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9. To‘g‘ri burchakli uchburchakning bir burchagi 60° ga teng. Bu burchakdan chiqarilgan bissektrisaning uzunligi 2 m. Uchburchakning gipotenuzasi uzunligini toping.

A) √5 +1

B) 2√3

C) √5

D) √3

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

A) 54

B) 56

C) 63

D) 61

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11. O‘tkir burchakli uchburchakning ikki tomon uzunliklari ayirmasi 8, bu tomonlarning uchinchi tomondagi proyeksiyalari mos ravishda 8 va 20 ga teng. Berilgan uchburchakka tashqi chizilgan aylana radiusini toping.

A) 35/3

B) 37/3

C) 85/6

D) 83/6

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12. Soddalashtiring:

A) D

B) A

C) B

D) C

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

A) 6°

B) 9°

C) 12°

D) 18°

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14. Hisoblang:

A) A

B) B

C) C

D) D

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

A) 25/6

B) 20/3

C) 22/3

D) 23/6

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16. 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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17. Agar ifodaning qiymatini toping.

A) 0.8

B) 20

C) 0.05

D) 4

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18. ?(3) ∙ (? − 2) + ?(? − 1) = 3? bo‘lsa, ?(1) ni toping.

A) 3

B) 2

C) 6

D) 4

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

A) 14

B) 12

C) 8

D) 10

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

A) 5

B) 2

C) 4

D) 3

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21. Soddalashtiring:

A) −сtg13°

B) −сtg1°

C) -1

D) −сtg²1°

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

A) 6320

B) 6320

C) 6230

D) 5320

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23. ABCD - kvadrat, AC-diagonal. Agar AE=BE+CE ( a + b = c ) bo‘lsa, ∠AEB burchakni toping(chizmaga qarang!).

A) 45

B) 55

C) 60

D) 50

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24. 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.