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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. a,b-to‘g‘ri burchakli uchburchakning katetlari va h-gipotenuzaga tushirilgan balandligi bo‘lsa, yig‘indini toping.

A) 4

B) 1

C) 2

D) 3

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2. Agar lg5=a va lg3=b bo`lsa, log₃₀⁡8 ni toping.

A) 3(1−a)

B) (3(1-a))/(1+b)

C) 1+b

D) (1−a)(1+b)

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3. Agar 2^a=27, 3^c=16 bo’lsa a ∙ ? ning qiymatini toping.

A) 11

B) 10

C) 12

D) 13

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

A) -2⁹

B) -4⁵

C) -2¹¹

D) -4⁴

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

A) 4

B) 3

C) 1

D) 2

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6. (3 − cos²x− 2sinx)(lg²y+ 2lgy+ 4) ≤ 3 bo‘lsa, sin²x+ 20y+ 1 ni toping.

A) 4

B) 5

C) 2

D) 3

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

A) 10

B) 12

C) 13

D) 11

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

A) 9°

B) 6°

C) 18°

D) 12°

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

A) 106

B) 100

C) 96

D) 89

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10. 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) 37/3

C) 35/3

D) 85/6

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11. ? soniga teskari bo‘lgan son ? ning 9% ini tashkil qiladi. ? ni toping. (?u yerda, ? > 0)

A) 13/3

B) 10/3

C) 16/3

D) 8/3

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12. Soddalashtiring:

A) C

B) B

C) D

D) A

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13. Agar ifodaning qiymatini toping.

A) 20

B) 4

C) 0.8

D) 0.05

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

A) 55

B) 60

C) 50

D) 45

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

A) 20°

B) 50°

C) 40°

D) 30°

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17. Agar bo‘lsa sin²α ni toping.

A) D

B) A

C) B

D) C

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

A) 3

B) 4

C) 2

D) 6

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

A) 63

B) 54

C) 61

D) 56

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

A) 15

B) 17

C) 16

D) 19

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

A) 20

B) 16

C) 15

D) 18

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22. bo’lsa, P(−1) ni toping.

A) 120

B) 105

C) 115

D) 100

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

A) -2

B) 2a+4

C) 2-2a

D) 4-2a

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

A) 8

B) 14

C) 10

D) 12

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25. 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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26. Soddalashtiring:

A) -1

B) −сtg13°

C) −сtg1°

D) −сtg²1°

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27. Tenglamani yeching:

A) ∅

B) 0

C) 10/21

D) 29/462

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28. 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) 91

B) 41

C) 81

D) 40

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

A) 123

B) 265

C) 521

D) 216

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30. Tengsizliklar sistemasinining butun yechimlari sonini toping.