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Matematika olimpiada Viloyat 2021-2022 o'quv yili

10-sinf Matematika olimpiada №3

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Avgust 19, 2024

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10-sinf Matematika olimpiada №3

2021-2022 o'quv yili viloyat bosqichida tushgan savollar.

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1. a ning qanday qiymatlarida $4x^{2}-4(a-1)x+1=0$ kvadrat tenglamaning ildizlari yig‘indisi manfiy bo‘ladi?

A) a<1

B) a<0

C) 0<a<2

D) 0<a<1

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2. 2x²-2ax+1=0 kvadrat tenglama ildizlari uchun

tengliko`rinlibo`lsa, a ni toping.

A) 7

B) -1

C) 1

D) -3

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3. Agar ni toping.

A) 0.8

B) 0.81

C) 0.96

D) 0.9

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4. Agar ? > 0, ? + ?² = 7,25; ?² − ? = 2 va ?² = √(? − 1) ∙ √(2 − ?) bo‘lsa, ?(√(? − 1) + √(2 − ?)) ning qiymatini toping.

A) 7

B) 4

C) 5

D) 6

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5. Uchburchak tomonlarining uzunliklari berilgan tenglamaning ildizlariga mos keladi. ?³ − 24?² + 183? − 440 = 0. Uchburchakning yuzini hisoblang.

A) 35

B) 2√21

C) 4√21

D) √21

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6. Agar

A) 3π/2

B) π

C) 0

D) π/2

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7. a,b,c−ABC uchburchakning tomonlari. Agar a⁴ + b⁴ + c⁴ + 32 = 2(a²b² + b²c² + a²c²) bo‘lsa, ABC uchburchak yuzini toping.

A) √3

B) 1

C) √2

D) 2√2

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8. ABCD to‘rtburchakda АВ = CD = 9 va bu to‘rtburchakka radiusi 4 ga teng aylana ichki chizilgan. ABCD to‘rtburchak yuzini toping.

A) 36

B) 72

C) 81

D) 144

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9. Agar bo‘lsa, ?(0) − ?(5) ayirmani toping.

A) -27

B) -24

C) -26

D) -25

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10. ABCD to‘rtburchakda А va В burchaklar- to‘g‘ri, tg∠D = 3/4 va ВС = AD/2 = АВ + 2 bo‘lsa, АС ni toping.

A) 15

B) 9

C) 10

D) 8

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11. ABC uchburchakning AC tomonida D nuqta olingan, bunda ∠ABC = ∠BDC. Agar AD = 10, CD = 8 bo‘lsa, BC ni toping.

A) 12

B) 10

C) 15

D) 9

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12. Tenglama nechta yechimga ega?

A) 4

B) 3

C) 2

D) 5

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13. Agar cos ∠A = 1/5 va sin∠B = 1/2 bo‘lsa, АВС uchburchakning mos ravishda А va В uchlaridan tushirilgan balandliklar nisbatini toping.

A) 5√6/24

B) 2/5

C) 5√2/5

D) 5√3/12

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14. Ayirmasi noldan farqli bo‘lgan arifmetik progressiyaning 4-hadidan boshlab 14- hadigacha bo‘lgan hadlar yig‘indisi 77 ga teng. Progressiyaning 7 ga teng bo‘lgan had nomerini toping.

A) 10

B) 11

C) 9

D) 8

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15. Agar ?₁ = 6? − 6 va у₂//?₁ hamda у₂ to‘g‘ri chiziq ?(6; 6) nuqtadan o‘tsa, ?₂ ni toping.

A) 6? − 24

B) 6? − 30

C) −6? + 6

D) −6? + 42

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16. tenglama nechta yechimga ega, agar x ∈ (0; 50)?

A) 16

B) 15

C) 14

D) 17

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17. ABC – gipotenuzasi AB bo‘lgan to‘g‘ri burchakli uchburchak. Gipotenuzaning ikki tomon davomida AB to‘g‘ri chiziqda AK = AC va BM = BC shartlar bilan kesmalar ajratilgan. KCM burchakni toping.

A) 90°

B) 120°

C) 150°

D) 135°

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18. ?(? − 1) = 2?(5? + 4) va ?(2? − 1) = 4? + 4 bo‘lsa, ?(?) ni toping

A) 20x+48

B) x+25

C) 0

D) 20x+50

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19. ?/? kasr (?, ? −natural sonlar)— qisqarmas kasr va (7?+6?)/(3?+2?) kasr esa qisqaradi. Ushbu kasr qanday songa qisqaradi?

A) 5

B) 3

C) 8

D) 2

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20. bo‘lsa ? ning qiymatini toping.

A) 12

B) 37

C) 25

D) 40

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21. Tenglamani yeching: 3^x+3 + 8·3^x+2 = 33

A) 1

B) -1

C) -2

D) 0

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22. ?(?) = ?⁵ − 7?⁴ + 3?³ − ? + 2 ko‘phadni ?² + ? ga bo‘lgandagi qoldiqni aniqlang.

A) 4? + 2

B) 8? + 2

C) 6? + 2

D) 10? + 2

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23. ?² + |? − 3| ≤ |?2 + ?| − 3 tengsizlikni yeching.

A) x>3

B) 3 ≤ x

C) x ≤ 3

D) x<0

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24. Tenglamalar sistemasi nechta yechimga ega?

A) 8

B) 10

C) 12

D) 6

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25. Qavariq to‘rtburchakning diagonallari 3 va 4 ga teng. Agar qarama qarshi tomonlarining o‘rtalarini tutashtirishdan hosil bo‘lgan kesmalar uzunliklari o‘zaro teng bo‘lsa, qavariq to‘rtburchakning yuzini toping.

A) 7

B) 8

C) 6

D) 5