Mavzular, xossalar va misollar

1-misol: y = x² - 4x + 3 funksiyaning uchini toping.

x₀ = -(-4)/(2·1) = 2 y₀ = f(2) = 4 - 8 + 3 = -1 Javob: uchi (2; -1), a=1>0 bo'lgani uchun shoxlari yuqoriga qaragan.

1-misol: y = x² - 5x + 6 funksiyaning nollarini toping.

D = 25 - 24 = 1 x = (5±1)/2 x₁ = 3, x₂ = 2

1-misol: x² - 5x + 6 > 0 tengsizlikni yeching.

Ildizlar: x=2, x=3 (D=1) a=1>0 — ildizlar orasida manfiy, tashqarisida musbat Javob: x < 2 yoki x > 3

1-misol: x² + 2x + 3 > 0 tengsizlikni yeching.

D = 4 - 12 = -8 < 0, a=1>0 Demak ifoda barcha haqiqiy x uchun musbat Javob: x ∈ R

1-misol: sinx = 3/5, x — I chetverakda. cosx va tgx ni toping.

cos²x = 1 - 9/25 = 16/25 cosx = 4/5 (I chetverakda cosx>0) tgx = (3/5)/(4/5) = 3/4

1-misol: sin75° ni toping (75°=45°+30°).

sin75° = sin45°cos30° + cos45°sin30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4

1-misol: sinx=3/5 (I chetverak). sin2x va cos2x ni toping.

cosx = 4/5 sin2x = 2·(3/5)·(4/5) = 24/25 cos2x = 1 - 2·(9/25) = 7/25

1-misol: a₁=3, d=4. a₁₀ va S₁₀ ni toping.

a₁₀ = 3 + 9·4 = 39 S₁₀ = 10/2·(3+39) = 210

1-misol: a₁=2, q=3. a₅ va S₅ ni toping.

a₅ = 2·3⁴ = 162 S₅ = 2·(3⁵-1)/(3-1) = 2·242/2 = 242

1-misol: f(x) = 1/(x-3) ning aniqlanish sohasini toping.

x - 3 ≠ 0 => x ≠ 3 Javob: D(f) = (-∞;3)∪(3;+∞)

1-misol: y = x² funksiyaning qiymatlar sohasini toping.

x² ≥ 0 barcha x uchun Javob: E(f) = [0;+∞)

1-misol: f(x) = x²-3 juft funksiyami?

f(-x) = (-x)²-3 = x²-3 = f(x) Javob: ha, juft funksiya.

1-misol: y=x² ning [0;+∞) dagi monotonligini aniqlang.

x₁=1, x₂=2 (x₁<x₂) f(1)=1, f(2)=4, f(1)<f(2) Javob: [0;+∞) da o'suvchi.

1-misol: y=x² grafigidan y=x²+3 grafigini qanday hosil qilish mumkin?

y=x² grafigini OY o'qi bo'ylab 3 birlik yuqoriga ko'chiramiz.

1-misol: y=2x² grafigi y=x² dan qanday hosil bo'ladi?

Har bir ordinata 2 marta orttiriladi — grafik OY bo'ylab 2 marta cho'ziladi (tikroq bo'ladi).

1-misol: Taksi xizmati 5000 so'm boshlang'ich narx + har km uchun 2000 so'm oladi. 10 km uchun narxni toping.

y = 2000x + 5000 x=10: y = 2000·10+5000 = 25000 so'm

1-misol: Jism otilish balandligi h(t) = -5t²+20t (t — soniya). Eng katta balandlikni toping.

t₀ = -20/(2·(-5)) = 2 h(2) = -5·4+40 = 20 metr

1-misol: {(1;2),(2;3),(3;4)} to'plam funksiyami?

Har bir x qiymatiga faqat bitta y mos keladi (1→2, 2→3, 3→4) Javob: ha, funksiya.

1-misol: Bitta kitob narxi 25000 so'm. x ta kitob uchun umumiy narx y ni ifodalovchi funksiyani yozing.

y = 25000x

1-misol: (x+1)/(x-2) = 3 tenglamani yeching.

x+1 = 3(x-2) x+1 = 3x-6 -2x = -7 => x = 3.5 Tekshirish: x≠2, mos keladi.

1-misol: 1/x + 1/(x+2) = 3/4 tenglamani yeching.

Umumiy maxraj: x(x+2) 4(x+2)+4x = 3x(x+2) 8x+8 = 3x²+6x 3x²-2x-8 = 0, D=4+96=100 x = (2±10)/6 => x₁=2, x₂=-4/3

1-misol: {x+y=5; 1/x+1/y=5/6} sistemani yeching.

y=5-x 1/x + 1/(5-x) = 5/6 5/(x(5-x)) = 5/6 => x(5-x)=6 x²-5x+6=0 => x=2 yoki x=3 Javob: (2;3) va (3;2)

1-misol: {1/x+1/y=3; 1/x-1/y=1} sistemani yeching.

Qo'shamiz: 2/x=4 => x=1/2 Ayiramiz: 2/y=2 => y=1 Javob: x=1/2, y=1

1-misol: (x-3)/(x+2) > 0 tengsizlikni yeching.

Hal etuvchi nuqtalar: x=-2, x=3 x=-3: 6>0 ✓; x=0: -1.5<0 ✗; x=4: 0.17>0 ✓ Javob: x<-2 yoki x>3

1-misol: (x-1)(x+3)/(x-2) ≤ 0 tengsizlikni yeching.

Hal etuvchi nuqtalar: -3, 1, 2 (x≠2) Tekshirish orqali: x≤-3 yoki 1≤x<2 Javob: x∈(-∞;-3]∪[1;2)

1-misol: {x²-4>0; x-1<0} sistemani yeching.

1) x²-4>0 => x<-2 yoki x>2 2) x-1<0 => x<1 Kesishma: x<-2

1-misol: {(x-1)/(x+3)>0; x²-4≤0} sistemani yeching.

1) (x-1)/(x+3)>0 => x<-3 yoki x>1 2) x²-4≤0 => -2≤x≤2 Kesishma: 1<x≤2

1-misol: (x²-4)/(x-2) = 3 tenglamani yeching.

ODZ: x≠2 (x-2)(x+2)/(x-2) = 3 => x+2 = 3 => x=1 (ODZ ga mos)

1-misol: (2x-1)/(x+1) = (x+3)/(x-1) tenglamani yeching.

(2x-1)(x-1) = (x+3)(x+1) 2x²-3x+1 = x²+4x+3 x²-7x-2 = 0, D=49+8=57 x = (7±√57)/2

1-misol: √(x+3) = 2 tenglamani yeching.

ODZ: x≥-3 x+3 = 4 => x=1 Tekshirish: √4=2 ✓

1-misol: √(x+1) = x-1 tenglamani yeching.

ODZ: x≥-1, x≥1 (o'ng tomon ≥0 bo'lishi kerak) x+1=(x-1)² => x²-3x=0 => x=0 yoki x=3 Tekshirish: x=0 — yolg'on ildiz (rad etiladi); x=3 — √4=2=3-1 ✓ Javob: x=3

1-misol: √(2x+1) = √(x+4) tenglamani yeching.

ODZ: x≥-1/2 2x+1 = x+4 => x=3 Tekshirish: √7=√7 ✓

1-misol: √(x+2) + √(x-1) = 3 tenglamani yeching.

√(x+2) = 3-√(x-1) x+2 = 9-6√(x-1)+(x-1) 6√(x-1) = 6 => √(x-1)=1 => x=2 Tekshirish: √4+√1=2+1=3 ✓

1-misol: {√x+√y=5; x+y=13} sistemani yeching.

u=√x, v=√y: u+v=5, u²+v²=13 (u+v)²=25=13+2uv => uv=6 t²-5t+6=0 => t=2 yoki t=3 Javob: (x;y)=(4;9) yoki (9;4)

1-misol: {√(x+y)=3; x-y=2} sistemani yeching.

x+y=9, x-y=2 Qo'shib: 2x=11 => x=5.5, y=3.5

1-misol: 1/(√5-2) ifodani ratsionallashtiring.

1/(√5-2) · (√5+2)/(√5+2) = (√5+2)/(5-4) = √5+2

1-misol: (x-4)/(√x-2) ifodani soddalashtiring (x≥0, x≠4).

x-4 = (√x-2)(√x+2) Demak ifoda = √x+2

1-misol: f(x)=2^x va g(x)=(1/2)^x funksiyalarning monotonligini solishtiring.

f(x): a=2>1 => o'suvchi g(x): a=1/2<1 => kamayuvchi

1-misol: 2³·2⁵ ni hisoblang.

2³·2⁵=2⁸=256

1-misol: log₂8 ni hisoblang.

2^x=8=2³ => x=3

1-misol: log₂12 - log₂3 ni hisoblang.

log₂(12/3) = log₂4 = 2

1-misol: log₄8 ni hisoblang.

log₄8 = log₂8 / log₂4 = 3/2

1-misol: 3^(log₃7) ni hisoblang.

a^(logᵢx)=x formulasiga ko'ra javob: 7

1-misol: 2^(x+1) = 32 tenglamani yeching.

32=2⁵ x+1=5 => x=4

1-misol: 4^x - 5·2^x + 4 = 0 tenglamani yeching.

t=2^x: t²-5t+4=0 => t=1 yoki t=4 2^x=1 => x=0; 2^x=4 => x=2

1-misol: log₂(x+1) = log₂5 tenglamani yeching.

x+1=5 => x=4 ODZ: x>-1, mos keladi

1-misol: log₃(x-1) = 2 tenglamani yeching.

x-1 = 3² = 9 => x=10

1-misol: {2^x·2^y=32; 2^x/2^y=2} sistemani yeching.

2^(x+y)=32=2⁵ => x+y=5 2^(x-y)=2 => x-y=1 Qo'shib: 2x=6 => x=3, y=2

1-misol: {log₂x+log₂y=5; log₂x-log₂y=1} sistemani yeching.

Qo'shib: 2log₂x=6 => log₂x=3 => x=8 Ayirib: 2log₂y=4 => log₂y=2 => y=4

1-misol: 2^x > 8 tengsizlikni yeching.

8=2³, a=2>1 Javob: x > 3

1-misol: (1/2)^x < 8 tengsizlikni yeching.

8=(1/2)^(-3), a=1/2<1 Javob: x > -3

1-misol: log₂(x-1) > log₂3 tengsizlikni yeching.

ODZ: x>1 x-1>3 => x>4

1-misol: log₁⁄₂x > log₁⁄₂4 tengsizlikni yeching.

a=1/2<1 => 0<x<4

1-misol: Bankda 1 000 000 so'm, yillik 10% foiz. 3 yildan keyingi summani toping.

N = 1 000 000·(1.1)³ = 1 000 000·1.331 = 1 331 000 so'm

1-misol: Avtomobil narxi 200 000 000 so'm, har yili 15% qadrsizlanadi. 2 yildan keyingi narxni toping.

N = 200 000 000·(0.85)² = 200 000 000·0.7225 = 144 500 000 so'm

1-misol: log₂(4³) ni hisoblang.

4³=2⁶ log₂(2⁶) = 6

1-misol: log₂32 - log₂4 + log₂1 ni hisoblang.

= 5 - 2 + 0 = 3

1-misol: y=sinx funksiyaning davri va qiymatlar sohasini ko'rsating.

Davri: T=2π Qiymatlar sohasi: E(f)=[-1;1]

1-misol: sin30°, cos30°, tg30° larni toping.

sin30°=1/2 cos30°=√3/2 tg30°=√3/3

1-misol: arcsin(1/2) ni toping.

sin(π/6)=1/2, π/6∈[-π/2;π/2] Javob: arcsin(1/2)=π/6

1-misol: arctg(1) ni toping.

tg(π/4)=1 Javob: arctg(1)=π/4

1-misol: sin²x + cos²x + tgx·ctgx ni soddalashtiring (x≠πn/2).

sin²x+cos²x=1, tgx·ctgx=1 Javob: 1+1=2

1-misol: x=π/3 da 2sinx·cosx ni hisoblang.

sin60°=√3/2, cos60°=1/2 2·(√3/2)·(1/2) = √3/2

1-misol: Kunlik harorat T(t)=15+10sin(πt/12) (t — soat). t=6 dagi haroratni toping.

T(6) = 15+10sin(π/2) = 15+10·1 = 25°

1-misol: Charx aylanishida h(t)=5+5sin(2πt/10). Maksimal va minimal balandlikni toping.

Maksimal: 5+5=10 Minimal: 5-5=0

1-misol: sinx = 1/2 tenglamani yeching.

arcsin(1/2)=π/6 Javob: x=(-1)ⁿ·π/6+πn, n∈Z

1-misol: cosx = 1/2 tenglamani yeching.

arccos(1/2)=π/3 Javob: x=±π/3+2πn, n∈Z

1-misol: tgx = 1 tenglamani yeching.

arctg(1)=π/4 Javob: x=π/4+πn, n∈Z

1-misol: ctgx = 1 tenglamani yeching.

arcctg(1)=π/4 Javob: x=π/4+πn, n∈Z

1-misol: sin2x = sinx tenglamani yeching.

2sinx·cosx - sinx = 0 sinx(2cosx-1) = 0 sinx=0 => x=πn cosx=1/2 => x=±π/3+2πn

1-misol: sinx + sin3x = 0 tenglamani yeching.

2sin2x·cosx = 0 sin2x=0 => x=πn/2 cosx=0 => x=π/2+πn

1-misol: sinx > 1/2 tengsizlikni [0;2π] oralig'ida yeching.

sinx=1/2 nuqtalari: x=π/6 va x=5π/6 Javob: π/6 < x < 5π/6

1-misol: tgx > 1 tengsizlikni yeching.

Javob: π/4+πn < x < π/2+πn, n∈Z

1-misol: Soqqa tashlanganda 4 raqami tushish ehtimolini toping.

n=6 (1 dan 6 gacha), m=1 P = 1/6

1-misol: Soqqa tashlanganda 6 tushmasligi ehtimolini toping.

P(6)=1/6 P(6 emas) = 1 - 1/6 = 5/6

1-misol: 5 ta kitobni tokchaga necha xil usulda joylashtirish mumkin?

P₅=5!=120

1-misol: 5 ta o'quvchidan 2 tasini (tartibsiz) tanlash necha usul bilan mumkin?

C₅² = 5!/(2!·3!) = 10

1-misol: Soqqa tashlashda har bir yoqning tushish ehtimoli tengmi?

Ha, "adolatli" soqqada barcha 6 yoq bir xil imkoniyatga ega Har biri uchun P=1/6

1-misol: Ikki marta tanga tashlanganda ikkalasida ham gerb tushish ehtimolini toping.

P = 1/2 · 1/2 = 1/4

1-misol: "Ikki nuqta orqali faqat bitta to'g'ri chiziq o'tkazish mumkin" — bu aksiomami yoki teoremami?

Bu aksioma — isbotsiz qabul qilinadigan asosiy tasdiq.

1-misol: "Agar uchburchak teng yonli bo'lsa, asosidagi burchaklari teng" teoremaning teskarisini ayting.

Teskari teorema: "Agar uchburchakning asosidagi ikki burchagi teng bo'lsa, u teng yonli"

1-misol: Katetlari 3 va 4 bo'lgan to'g'ri burchakli uchburchakning gipotenuzasini toping.

c = √(3²+4²) = √25 = 5

1-misol: Asosi 8, balandligi 5 bo'lgan uchburchakning yuzini toping.

S = 1/2·8·5 = 20

1-misol: Nechta nuqta (bir to'g'ri chiziqda yotmasa) bitta tekislikni aniqlaydi?

3 ta nuqta

1-misol: To'g'ri chiziqning ikki nuqtasi tekislikka tegishli bo'lsa, chiziqning o'zi tekislikka tegishlimi?

Ha, aksiomaga ko'ra butun to'g'ri chiziq tekislikka tegishli bo'ladi.

1-misol: To'g'ri chiziq tekislikka nisbatan qanday joylashishi mumkin?

1) Tekislikda yotadi 2) Bitta nuqtada kesib o'tadi 3) Tekislikka parallel (umumiy nuqtasi yo'q)

1-misol: Nuqtadan tekislikkacha bo'lgan masofa qanday aniqlanadi?

Nuqtadan tekislikka tushirilgan perpendikulyar kesmaning uzunligi orqali.

1-misol: Kubning yoq, qirra va uch sonini toping.

6 ta yoq, 12 ta qirra, 8 ta uch Tekshirish: 6+8-12=2 ✓

1-misol: Nechta muntazam ko'pyoq (Platon jismi) mavjud?

5 ta: tetraedr, kub, oktaedr, dodekaedr, ikosaedr

1-misol: Kubni tekislikda tasvirlashda old yoq qanday shaklda chiziladi?

Old yoq to'g'ri kvadrat shaklda, orqa qirralar esa siljitilgan (parallelogramm ko'rinishida) chiziladi.

1-misol: Kubning yoyilmasi nechta kvadratdan iborat?

6 ta kvadrat (kubning 6 ta yoqiga mos)

1-misol: Bitta tekislikda yotgan ikki to'g'ri chiziq umumiy nuqtaga ega bo'lmasa, ular qanday deyiladi?

Parallel to'g'ri chiziqlar deyiladi.

1-misol: Ikki kesishuvchi to'g'ri chiziq orasidagi burchak qanday aniqlanadi?

Ular hosil qilgan ikki juft burchakdan kichigi (0°-90° oralig'ida) orqali aniqlanadi.

1-misol: Ayqash to'g'ri chiziqlar qanday to'g'ri chiziqlar deyiladi?

Bitta tekislikda yotmaydigan, kesishmaydigan va parallel bo'lmagan to'g'ri chiziqlar.

1-misol: Ayqash to'g'ri chiziqlar orasidagi masofa qanday aniqlanadi?

Ikkala chiziqqa ham perpendikulyar bo'lgan umumiy kesmaning uzunligi orqali.

1-misol: To'g'ri chiziq va tekislik o'zaro qanday joylashishi mumkin?

1) To'g'ri chiziq tekislikda yotadi 2) Bitta nuqtada kesishadi 3) Parallel (umumiy nuqtaga ega emas)

1-misol: To'g'ri chiziq tekislikka perpendikulyar bo'lishi uchun qanday shart bajarilishi kerak?

Tekislikdagi ikkita kesishuvchi to'g'ri chiziqqa perpendikulyar bo'lishi kerak.

1-misol: Ikki tekislik umumiy nuqtaga ega bo'lmasa, ular qanday deyiladi?

Parallel tekisliklar deyiladi.

1-misol: Ikki kesishuvchi tekislik orasidagi burchak qanday aniqlanadi?

Kesishish chizig'iga perpendikulyar bo'lgan va har bir tekislikda yotuvchi nurlar orasidagi burchak orqali.

1-misol: Parallel proyeksiyalashda to'g'ri chiziqning proyeksiyasi odatda nima bo'ladi?

To'g'ri chiziq bo'ladi (agar u proyeksiyalash yo'nalishiga parallel bo'lmasa).

1-misol: Parallelogrammning parallel proyeksiyasi qanday shakl bo'ladi?

Yana parallelogramm bo'ladi, chunki parallellik va tomon nisbatlari saqlanadi.

1-misol: 2457 + 1386 ni hisoblang.

Birlik: 7+6=13, 3 yozamiz, 1 ni o'tkazamiz O'nlik: 5+8+1=14, 4 yozamiz, 1 ni o'tkazamiz Yuzlik: 4+3+1=8 Minglik: 2+1=3 Javob: 3843

1-misol: -7 + 12 ni hisoblang.

|12| > |-7| 12 - 7 = 5 Javob: natija musbat, ya'ni 5

1-misol: 5234 - 1876 ni hisoblang.

Birlik: 4-6 yetmaydi, 14-6=8, qarzga oldik O'nlik: 2-1(qarz)-7 yetmaydi, 12-1-7=4, yana qarz Yuzlik: 2-1(qarz)-8 yetmaydi, 12-1-8=3, yana qarz Minglik: 5-1(qarz)-1=3 Javob: 3358

1-misol: 6 - 15 ni hisoblang.

6 - 15 = 6 + (-15) |15| > |6| => natija manfiy 15 - 6 = 9 Javob: -9

1-misol: 246 × 7 ni hisoblang.

6×7=42, 2 yozamiz, 4 ni o'tkazamiz 4×7=28, +4=32, 2 yozamiz, 3 ni o'tkazamiz 2×7=14, +3=17 Javob: 1722

1-misol: 23 × 15 ni hisoblang.

23×5=115 23×10=230 115+230=345 Javob: 345

1-misol: klass 144 ÷ 12 ni hisoblang.

12 × 12 = 144 Javob: 12

1-misol: 47 ni 5 ga qoldiq bilan bo'ling.

5 × 9 = 45, 47-45=2 Javob: bo'linma 9, qoldiq 2

1-misol: 240 ning 25% ini toping.

240 × 25 / 100 = 240 × 0,25 = 60 Javob: 60

1-misol: Mahsulot narxi 150 000 so'm, unga 20% chegirma qilindi. Yangi narxni toping.

Chegirma: 150000 × 20/100 = 30000 Yangi narx: 150000 - 30000 = 120000 Javob: 120 000 so'm

1-misol: 2⁵ ni hisoblang.

2⁵ = 2·2·2·2·2 = 32 Javob: 32

1-misol: (-2)³ ni hisoblang.

(-2)³ = (-2)·(-2)·(-2) = 4·(-2) = -8 Javob: -8 (daraja toq bo'lgani uchun natija manfiy)

1-misol: 7 × 8 ni hisoblang (karra jadvalidan).

7 × 8 = 56 Javob: 56

1-misol: 60 × 70 ni karra jadvalidan foydalanib hisoblang.

6 × 7 = 42 60 × 70 = 4200 Javob: 4200

1-misol: 38 614 sonining yuzlar xonasidagi raqamni toping.

O'ngdan: 4 — birlar, 1 — o'nlar, 6 — yuzlar Javob: 6

1-misol: 47 386 sonini yuzlar xonasigacha yaxlitlang.

Yuzlar xonasida 3, keyingi (o'nlar) raqami 8 ≥ 5 Demak 3 bittaga ortadi: 47 400 Javob: 47 400

1-misol: −12; 5; −3; 0 sonlaridan eng kichigini toping.

Son o'qida eng chapda turgan son eng kichik Javob: −12

1-misol: (2x+3)² ni yoying.

(2x+3)² = (2x)² + 2·2x·3 + 3² = 4x² + 12x + 9

1-misol: x³ + 8 ni ko'paytuvchilarga ajrating.

x³ + 8 = x³ + 2³ = (x+2)(x² - 2x + 4)

1-misol: 2x² + 2xy - 3x - 3y ni ko'paytuvchilarga ajrating.

2x² + 2xy - 3x - 3y = 2x(x+y) - 3(x+y) = (x+y)(2x-3)

1-misol: 6x² - x - 1 ni ko'paytuvchilarga ajrating.

D = 1 + 24 = 25, √D = 5 x = (1±5)/12 → x₁=1/2, x₂=-1/3 6x²-x-1 = 6(x-1/2)(x+1/3) = (2x-1)(3x+1)

1-misol: |2x-3| = 5 tenglamani yeching.

2x-3 = 5 → x = 4 2x-3 = -5 → x = -1 Javob: x=4 yoki x=-1

1-misol: |x-2| < 3 tengsizlikni yeching.

-3 < x-2 < 3 -1 < x < 5 Javob: x ∈ (-1; 5)

1-misol: a₁=3, d=4 bo'lsa, a₁₀ ni toping.

a₁₀ = 3 + 9·4 = 3 + 36 = 39

1-misol: a₁=3, d=4 progressiyaning dastlabki 10 ta hadi yig'indisini toping.

a₁₀=39 (yuqoridagi misoldan) S₁₀ = 10·(3+39)/2 = 10·42/2 = 210

1-misol: b₁=2, q=3 bo'lsa, b₅ ni toping.

b₅ = 2·3⁴ = 2·81 = 162

1-misol: b₁=8, q=1/2 bo'lgan cheksiz kamayuvchi progressiyaning yig'indisini toping.

S = b₁/(1-q) = 8/(1-1/2) = 8/(1/2) = 16

1-misol: x:15 = 4:5 tenglamadan x ni toping.

5x = 15·4 5x = 60 x = 12

1-misol: 250 ning 12% i nechaga teng?

250·12/100 = 250·0.12 = 30

1-misol: a=4, b=16 sonlarining arifmetik va geometrik o'rtalarini toping.

A = (4+16)/2 = 10 G = √(4·16) = √64 = 8

1-misol: a=4, b=16 sonlarining garmonik o'rtasini toping.

H = 2·4·16/(4+16) = 128/20 = 6.4 (Tekshirish: A=10 ≥ G=8 ≥ H=6.4 — munosabat bajarilmoqda)

1-misol: |2 3; 1 4| determinantni hisoblang.

|2 3; 1 4| = 2·4 - 3·1 = 8 - 3 = 5

1-misol: 2x+y=5, x-y=1 sistemasini Kramer usulida yeching.

D = |2 1;1 -1| = 2·(-1)-1·1 = -3 Dₓ = |5 1;1 -1| = 5·(-1)-1·1 = -6 Dₛ = |2 5;1 1| = 2·1-1·5 = -3 x = Dₓ/D = -6/-3 = 2, y = Dₛ/D = -3/-3 = 1 Javob: (2; 1)

1-misol: lim (x→2) (x²-4)/(x-2) ni toping.

(x²-4)/(x-2) = (x-2)(x+2)/(x-2) = x+2 (x≠2) lim (x→2) (x+2) = 4

1-misol: lim (x→0) sin(3x)/x ni toping.

sin(3x)/x = 3· sin(3x)/(3x) lim (x→0) 3·sin(3x)/(3x) = 3·1 = 3

1-misol: y = x³ - 3x² + 2 funksiyaning hosilasini va y'(1) qiymatini toping.

y' = 3x² - 6x y'(1) = 3·1 - 6·1 = 3 - 6 = -3

1-misol: y = x² funksiyaning x₀=1 nuqtadagi urinma tenglamasini tuzing.

f(1) = 1, f'(x) = 2x, f'(1) = 2 y = 1 + 2(x-1) = 2x - 1

1-misol: 3564 soni 2, 3, 4 va 9 ga bo'linadimi, tekshiring.

Oxirgi raqam 4 (juft) → 2 ga bo'linadi Raqamlar yig'indisi 3+5+6+4=18, 18 ⋮ 3 va ⋮ 9 → 3 va 9 ga bo'linadi Oxirgi ikki xona 64, 64 ⋮ 4 → 4 ga bo'linadi Javob: 3564 soni 2, 3, 4 va 9 ning barchasiga bo'linadi

1-misol: 36 va 48 sonlarining EKUB va EKUKini toping.

36 = 2²·3², 48 = 2⁴·3 EKUB = 2²·3 = 12 EKUK = 2⁴·3² = 144 (tekshirish: 12·144 = 1728 = 36·48)

1-misol: √50 ni soddalashtiring.

√50 = √(25·2) = √25·√2 = 5√2

1-misol: 8²⁄³ ni hisoblang.

8²⁄³ = (³√8)² = 2² = 4

1-misol: x²-9x+20=0 tenglama ildizlarining yig'indisi va ko'paytmasini (tenglamani yechmasdan) toping.

a=1, b=-9, c=20 x₁+x₂ = -b/a = 9 x₁·x₂ = c/a = 20

1-misol: ildizlari yig'indisi 5 va ko'paytmasi 6 bo'lgan kvadrat tenglama tuzing.

S=5, P=6 x² - 5x + 6 = 0

1-misol: √(x+3) = x-3 tenglamani yeching.

x+3 = (x-3)² = x²-6x+9 x²-7x+6=0 → x=1 yoki x=6 Shart: x-3≥0, ya'ni x≥3 — x=1 bu shartga mos kelmaydi (chet ildiz) Tekshirish (x=6): √9=3, 6-3=3 ✓ Javob: x=6

1-misol: √(2x-1) = √(x+4) tenglamani yeching.

2x-1 = x+4 x = 5 Tekshirish: 2·5-1=9≥0 ✓, √9=3, √(5+4)=√9=3 ✓ Javob: x=5

1-misol: y = 1/(x-3) funksiyaning aniqlanish sohasini toping.

Maxraj nolga teng bo'lmasligi kerak: x-3≠0 → x≠3 D(y) = R \ {3}

1-misol: y = √(x-4) funksiyaning aniqlanish sohasini toping.

x-4 ≥ 0 → x ≥ 4 D(y) = [4; +∞)

1-misol: P(x)=x³-2x²-x+2 uchun P(1) ni hisoblang va xulosa chiqaring.

P(1) = 1 - 2 - 1 + 2 = 0 P(1)=0 bo'lgani uchun (x-1) — P(x) ning ko'paytuvchisi

1-misol: P(x)=x³-2x²-x+2 ni Gorner sxemasi bilan (x-1) ga bo'ling.

Koeffitsientlar: 1, -2, -1, 2; a=1 1 | -2+1·1=-1 | -1+1·(-1)=-2 | 2+1·(-2)=0 (qoldiq) Bo'linma: x²-x-2, qoldiq: 0 P(x) = (x-1)(x²-x-2) = (x-1)(x-2)(x+1)

1-misol: log₂8 + log₂4 ni hisoblang.

log₂8 = 3 (chunki 2³=8), log₂4 = 2 (chunki 2²=4) log₂8 + log₂4 = 3 + 2 = 5

1-misol: log₃x = 4 tenglamani yeching.

x = 3⁴ = 81

1-misol: sinx=4/5, x — I chorak bo'lsa, cosx ni toping.

cos²x = 1 - sin²x = 1 - 16/25 = 9/25 I chorakda cosx>0 → cosx = 3/5

1-misol: sin150° ni keltirish formulasi bilan hisoblang.

sin150° = sin(180°-30°) = sin30° = 1/2

1-misol: sinx = 1/2 tenglamani yeching.

arcsin(1/2) = π/6 x = (-1)ⁿ·π/6 + πn, n∈Z

1-misol: cosx = √2/2 tenglamani yeching.

arccos(√2/2) = π/4 x = ±π/4 + 2πn, n∈Z

1-misol: ∫x⁴ dx ni toping.

∫x⁴ dx = x⁵/5 + C

1-misol: ∫(3x²-2x+1)dx ni toping.

∫(3x²-2x+1)dx = x³ - x² + x + C

1-misol: Uchburchakning ikki burchagi 50° va 70°. Uchinchisini toping.

180° − (50° + 70°) = 60°

1-misol: Asosi 10, balandligi 6 bo'lgan uchburchak yuzini toping.

S = ½·10·6 = 30

1-misol: Katetlari 5 va 12 bo'lsa, gipotenuzasini toping.

c² = 25 + 144 = 169, c = 13

1-misol: Qo'shni tomonlari 7 va 5 bo'lsa, perimetrini toping.

P = 2(7 + 5) = 24

1-misol: Rombning diagonallari 6 va 8. Tomonini toping.

a² = 3² + 4² = 25, a = 5

1-misol: Asoslari 12 va 8, balandligi 5 bo'lgan trapetsiya yuzi?

S = (12 + 8)/2·5 = 50

1-misol: Radiusi 5 bo'lgan aylana uzunligi va doira yuzini toping.

C = 10π, S = 25π

1-misol: Markaziy burchak 80° bo'lsa, shu yoyga tiralgan ichki burchak?

80° : 2 = 40°

1-misol: Radiusi 3, markazdan nuqtagacha 5. Urinma kesmasi?

l² = 25 − 9 = 16, l = 4

1-misol: Muntazam oltiburchak ichki burchagi?

(6−2)·180°/6 = 120°

1-misol: Tomoni 4 bo'lgan muntazam oltiburchak tashqi aylana radiusi?

R = a = 4

1-misol: Asosi 3×4 to'g'ri to'rtburchak, balandligi 5 bo'lgan prizma hajmi?

V = 12·5 = 60

1-misol: O'lchamlari 2, 3, 6 bo'lgan parallelepiped diagonali?

d² = 4 + 9 + 36 = 49, d = 7

1-misol: Asosi 6×6 kvadrat, H = 5 bo'lgan piramida hajmi?

V = ⅓·36·5 = 60

1-misol: Muntazam to'rtburchakli piramida: asos tomoni 6, H = 4. Apofemasi?

r = 3, l² = 16 + 9 = 25, l = 5

1-misol: Radiusi 3 bo'lgan teng yonli silindr balandligi?

H = 2r = 6

1-misol: r = 3, H = 5. Hajmini toping.

V = π·9·5 = 45π

1-misol: r = 6, H = 8. Yasovchisini toping.

l² = 36 + 64 = 100, l = 10

1-misol: r = 3, H = 4. Hajmini toping.

V = ⅓π·9·4 = 12π

1-misol: R = 13, d = 5. Kesim radiusi?

r² = 169 − 25 = 144, r = 12

1-misol: R = 3. Hajmini toping.

V = (4/3)π·27 = 36π

1-misol: Katetlari 6 va 8 bo'lgan uchburchak: R va r ni toping.

c = 10, R = 5, r = (6+8−10)/2 = 2

1-misol: Ichki chizilgan to'rtburchakning bir burchagi 80°. Qarama-qarshi burchagi?

180° − 80° = 100°

1-misol: Qirrasi 8 bo'lgan kubga ichki va tashqi sharlar radiuslari?

r = 4, R = 4√3

1-misol: Konus: asos radiusi 3, H = 4. Yasovchi va ichki shar radiusi?

l = 5; r = S/p = 12/8 = 1,5

1-misol: A(1; 2) va B(4; 6) orasidagi masofa va o'rta nuqta.

d = √(9 + 16) = 5; o'rta (2,5; 4)

1-misol: y = 2x − 3 chiziqda x = 4 bo'lsa, y?

y = 8 − 3 = 5

1-misol: a(3; 4), b(1; −2): a + b va |a| ni toping.

a + b = (4; 2); |a| = 5

1-misol: a(2; 3), b(−3; 2) ni tekshiring.

a·b = −6 + 6 = 0 → perpendikulyar

1-misol: A(1; 2; 3), B(3; 6; 7) orasidagi masofa?

d = √(4 + 16 + 16) = 6

1-misol: a(1; 2; 2), b(2; −1; 0): |a| va a·b.

|a| = 3; a·b = 2 − 2 + 0 = 0

1-misol: a = 6, A = 30°. R ni toping.

2R = 6 : ½ = 12, R = 6

1-misol: b = 3, c = 5, A = 60°. a² ni toping.

a² = 9 + 25 − 15 = 19

1-misol: k = 3, kichigining yuzi 4. Kattasining yuzi?

4·9 = 36

1-misol: k = 2, kichik hajm 5. Katta hajm?

5·8 = 40

1-misol: h = 6, p = 8. Og'ma?

l = √(36 + 64) = 10

1-misol: Kubning ikki qo'shni yog'i orasidagi burchak?

90°

1-misol: R = 10, d = 6. Kesim radiusi?

r = √(100 − 36) = 8

1-misol: Kub qirrasi 4. Diagonal kesim yuzi?

4·4√2 = 16√2

1-misol: 7!/5! ni hisoblang.

7·6 = 42

1-misol: 3 xil ko'ylak va 4 xil shimdan necha xil to'plam tuzish mumkin?

3·4 = 12

1-misol: 8 kishidan 3 kishilik guruh necha usulda tanlanadi?

C(8,3) = 8·7·6/6 = 56

1-misol: 10 o'quvchidan sardor va o'rinbosarni necha usulda tanlash mumkin?

Tartib muhim: A(10,2) = 90

1-misol: O'yin kubigi tashlandi. 5 dan katta ochko tushish ehtimolligi?

Qulay: 6 — 1 ta; P = 1/6

1-misol: Tanga ketma-ket 3 marta tashlanganda hammasida gerb tushish ehtimolligi?

(1/2)³ = 1/8

1-misol: A = {1,2,3,4}, B = {3,4,5}. A∪B, A∩B, A\B ni toping.

A∪B = {1,2,3,4,5}; A∩B = {3,4}; A\B = {1,2}

1-misol: Sinfda 30 o'quvchi: 18 tasi ingliz, 15 tasi rus tilini o'rganadi, 8 tasi ikkalasini. Hech birini o'rganmaydiganlar?

Kamida bittasi 18 + 15 − 8 = 25; 30 − 25 = 5

1-misol: (x + 1)⁴ da x² oldidagi koeffitsiyent?

C(4,2) = 6