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Matematika olimpiada Tuman 2019-2020 o'quv yili

11-sinf Matematika olimpiada №2

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Avgust 19, 2024

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11-sinf Matematika olimpiada №2

2019-yil tuman bosqichida tushgan savollar

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1. ABC uchburchakka aylana tashqi chizilgan. Uchburchakning AM medianasi aylana bilan K nuqtada kesishguncha davom ettirilgan. Agar AM=18,MK=8,BK=10 bo‘lsa, AC tomonini toping.

A) 12

B) 15

C) 16

D) 14

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

A) 23/35

B) 15/31

C) 12/35

D) 21

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3. Asosi ga teng bo`lgan muntazam to`rtburchakli piramidaning uchidan o`tkazilgan kesma asos tomonini 2:4 nisbatda bo`ladi va asos tekisligi bilan 60^o li burchakhosilqiladi. Piramidahajmini toping.

A) 6

B) 12√30

C) √30

D) √10

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4. funksiyaning boshlang`ich funksiyasini toping.

A) III

B) II

C) IV

D) I

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5. ni hisoblang.

A) 13

B) -13

C) 12

D) -12

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6. Poyezd 3 minutda 7 kilometrmasofani , motosikl 5 minutda 7 kilometrmasofanibosibo`tdi. Mototsikltezligipoyezdtezliginingnecha % initashkilqiladi?

A) 71%

B) 70%

C) 60%

D) 61%

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7. 3cos2x-3 sin2x=0 trigonometrik tenglamani yeching.

A) π/12+πk/2; k∈Z)

B) π/6+πk; k∈Z

C) π/12+πk; k∈Z

D) π/12+2πk; k∈Z

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8. 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) 2

B) 25/12

C) 2√2

D) 15/7

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9. Tengsizlikni yeching.

A) (2/3;2)

B) (4/3;3)

C) (1/2;∞)

D) (5/3;2)

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10. |x²+2x-8|=3a tenglama a ning qanday qiymatlarida 3 ta haqiqiy yechimga ega bo`ladi.

A) a>3, a=0

B) 1

C) (0;3)

D) a=3

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11. GH kesmani O nuqta, G nuqtadan boshlab hisoblaganda, 5 : 7 kabi, P nuqtaesa 5:11 kabi nisbatda bo’ladi. O va P nuqtalar orasidagi masofa 30 sm bo’lsa, GH kesmaning uzunligini toping.

A) 234

B) 72

C) 288

D) 18

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12. Perimetri 4 ga o`tkir burchagi 30^oga va shu burchak qarshisidagi tomoni √3 ga teng bo`lgan uchburchakka ichki chizlgan aylana radiusini toping.

A) (√3-1)/2

B) 4√3+7

C) (1+√3)/2

D) 7-4√3

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13. Ifodani soddalashtiring:

A) 1−√(a-1)

B) 0

C) 1+√(a+1)

D) √(a²-1)

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14. 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) 27

B) 100

C) 216

D) 45

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15. funksiyaning aniqlanish sohasini toping.

A) (2/3;∞)

B) (-∞;2/3)∪(2/3;∞)

C) (-∞;4/3)

D) (-∞;4/3)∪(4/3;∞)

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16. Bitta tekislikda yotuvchi n ta turli tog`ri chiziq o`tkazilgan. Ularning ixtiyoriy ikkitasi parallel emas va ixtiyoriy uchtasi bitta nuqtadan o`tmaydi ,ya`ni bitta nuqtada kesishmaydi. Bu to`g`ri chiziqlar tekislikni 1276 qismga ajratadi. n ni aniqlang?

A) 49

B) 50

C) 52

D) 51

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17. Agar f(g(x))= − x+13 bo`lsa, f(x) va g(x) to`g`ri chiziqlar orasidagi burchakni toping .

A) 0

B) π/3

C) π/2

D) π/4

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18. Tenglamani yeching: |x²-3x-2|=x²+3|x+2|+4

A) [2;∞)

B) [−2;0]

C) [−2;∞)

D) (−∞;−2]

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

A) 11

B) 13

C) 10

D) 12

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20. Radiusi 6 ga teng bo`lgan uchta aylana o`zaro tashqi urinishidan hosil bo`lgan egri chiziqli uchburchakka ichki chizilgan aylana radiusini toping.

A) 4√3-6

B) 1

C) 2√2−1

D) 6-2√3

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

A) 1+b

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

C) 3(1−a)

D) (1−a)(1+b)

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22. ifodani soddalshtiring

A) 1

B) 2cosα

C) cosα

D) 2sinα

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23. Tenglamani yeching:

A) πk; π−arcctg4+πk; k∈Z

B) π −arcctg4+πk; k∈Z

C) arccos0,8+2πk; k∈Z

D) πk; −arcctg4+πk; k∈Z

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24. Bir savdogar 1 mеtrini 1200 tiyindan olgan matosini yuvib, kuritgandan so`ng 1800 tiyindan sotyapti. Mato yuvilib kuritilganidan so`ng 20 foizi qisqardi. Bularga ko`ra savdogar nеcha foiz foyda ko`radi?

A) 20

B) 18

C) 15

D) 10

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

A) 7sin5x−5sin7x+c

B) −7cos5x−5cos7x+c

C) 7cos5x−5cos7x+c

D) 7sin5x+5sin7x+C