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

A) 9

B) 12

C) 7

D) 14

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2. Ifodani soddalashtiring:

A) √(a²-1)

B) 1+√(a+1)

C) 0

D) 1−√(a-1)

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3. y=6sin2x+sin12x funksiyaning hosilasini toping.

A) 24cos7xcos5x

B) −24sin7xcos5x

C) 24sin7xsin5x

D) 24sin5xcos7x

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4. Quyidagi sonlar orasidan musbat sonni aniqlang.

A) D

B) B

C) A

D) C

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

A) A

B) B

C) C

D) D

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

A) 12

B) 10

C) 11

D) 13

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7. Agar bo‘lsa sin²α ni toping.

A) A

B) D

C) C

D) B

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

A) 25/6

B) 20/3

C) 23/6

D) 22/3

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9. 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) 9

B) 8

C) 3

D) 6

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10. Tenglamani yeching:

A) 0

B) 29/462

C) ∅

D) 10/21

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

A) 216

B) 265

C) 521

D) 123

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12. 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) √3

C) 2√3

D) √5

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13. Agar ifodaning qiymatini toping.

A) 4

B) 20

C) 0.05

D) 0.8

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

A) 12

B) 14

C) 8

D) 10

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15. 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) 83/6

B) 35/3

C) 37/3

D) 85/6

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

A) 20°

B) 50°

C) 30°

D) 40°

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

A) 5320

B) 6320

C) 6320

D) 6230

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

A) 55

B) 45

C) 50

D) 60

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

A) 96

B) 89

C) 106

D) 100

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21. bo’lsa, P(−1) ni toping.

A) 115

B) 120

C) 105

D) 100

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

A) 4-2a

B) 2-2a

C) -2

D) 2a+4

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23. Uchlari A(2; −3), B(6; −1), C(6; 4) va D(2; 2) nuqtalarda bo‘lgan ABCD to‘rtburchak yuzini toping.

A) 16

B) 20

C) 18

D) 15

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24. Tengsizliklar sistemasinining butun yechimlari sonini toping.

A) 5

B) 6

C) 4

D) 7

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25. ?(3) ∙ (? − 2) + ?(? − 1) = 3? bo‘lsa, ?(1) ni toping.

A) 2

B) 6

C) 4

D) 3

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26. (3 − cos²x− 2sinx)(lg²y+ 2lgy+ 4) ≤ 3 bo‘lsa, sin²x+ 20y+ 1 ni toping.

A) 3

B) 5

C) 2

D) 4

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27. 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) 81

B) 40

C) 91

D) 41

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28. Soddalashtiring: