1.
Agar m=243612203m = 24^3 \cdot 6^{12} \cdot 20^3 bo'lsa, m22n\frac{m^2}{2^n} kasrning qiymati natural son bo'ladigan nn ning eng katta qiymatini toping.
2.
Agar x[12;2]x \in \left[\frac{1}{2}; 2\right] bo'lsa, 3x1xx2\bigl||3x - 1| - |x|\bigr| - |x - 2| ni soddalashtiring.
3.
Ikkita kenguru orasidagi masofa 30 metr (katta kenguru kichigini quvmoqda). Katta kenguru bir sakrashda 2 metr, kichigi 1 metr sakraydi va katta kenguru ikki marta sakraganda kichigi uch marta sakraydi. Katta kenguru kichigini quvib yetganda katta kenguru qancha masofa o'tgan bo'ladi?
4.
Uch nafar ishchi birgalikda 4 080 000 so'm ish haqi oldi. Birinchi ishchining olgan pulining ikkinchi ishchining olgan puliga nisbati 712:747\frac{1}{2} : \frac{7}{4}, uchinchi ishchi esa birinchi ishchi olgan pulining 4313%43\frac{1}{3}\% qismini oladi. Shunda uchinchi ishchi qancha so'm olganini aniqlang.
5.
Eng katta sonni toping.
6.
Hisoblang.

2(51)(3+5)35\sqrt{2}(\sqrt{5} - 1)(3 + \sqrt{5})\sqrt{3 - \sqrt{5}}
7.
Soddalashtiring.

125x65543\sqrt[3]{\frac{125 \cdot x^{\frac{6}{5}}}{54}}
8.
Arifmetik progressiyada a5=5a_5 = 5 va a7a10a_7 \cdot a_{10} eng kichik bo'lsa, dd ni toping.
9.
{an}\{a_n\} arifmetik progressiya va {bn}\{b_n\} geometrik progressiya berilgan. Agar b1+b2+b3=16b_1 + b_2 + b_3 = 16, b1=a2b_1 = a_2, b2=a4b_2 = a_4, b3=a6b_3 = a_6 va a1+a2+a3++an=176a_1 + a_2 + a_3 + \ldots + a_n = 176 bo'lsa, nn ni toping.
10.
Soddalashtiring.

a3b3(2a+b)23a(a+b)+(a2+b)(b2+a)ab(ab+1)a2ab+b2\frac{a^3 - b^3}{(2a+b)^2 - 3a(a+b)} + \frac{(a^2+b)(b^2+a) - ab(ab+1)}{a^2 - ab + b^2}
11.
Agar a3+a1=0a^3 + a - 1 = 0 bo'lsa, a4+a3+a2+9a5a2a+6\frac{a^4 + a^3 + a^2 + 9}{a^5 - a^2 - a + 6} ni toping.
12.
Hisoblang.

3sin103cos103cos70\frac{3\sin 10^\circ - \sqrt{3}\cos 10^\circ}{3\cos 70^\circ}
13.
α\alpha, β\beta va γ\gamma uchburchakning ichki burchaklari. Agar sinαsinβsinγ=12\sin\alpha \sin\beta \sin\gamma = \frac{1}{2} bo'lsa, sin2α+sin2β+sin2γ\sin 2\alpha + \sin 2\beta + \sin 2\gamma ni toping.
14.
Tenglama nechta haqiqiy ildizga ega?

7x11x77x121x=1\frac{7^x - 11^x}{\sqrt{77^x - 121^x}} = 1
15.
Tengsizlikni nechta butun son qanoatlantiradi?

log12(2x512)log4x1\log_{\frac{1}{2}}\left(2x - 5\frac{1}{2}\right) \ge \log_{4-x} 1
16.
x23x+a2=0x^2 - 3x + a^2 = 0 tenglama haqiqiy ildizga ega bo'ladigan aa ning nechta butun qiymati mavjud?
17.
Tenglamaning haqiqiy ildizlari yig'indisini (agar u bitta bo'lsa, shu ildizini) toping.

(x+3)236(x3)23=9x23\sqrt[3]{(x+3)^2} - 6\sqrt[3]{(x-3)^2} = \sqrt[3]{9 - x^2}
18.
Tengsizlikni qanoatlantiradigan butun yechimlari yig'indisini toping.

x2+3x+46x27x+1085x\frac{x^2 + 3x + 46}{x^2 - 7x + 10} \le \frac{8}{5 - x}
19.
Tengsizlikni yeching.

x11+12x>0\sqrt{x - 11} + \sqrt{12 - x} > 0
20.
Funksiyaning aniqlanish sohasiga tegishli bo'lgan butun sonlar yig'indisini toping.

f(x)=x+36xx210x+25f(x) = \frac{\sqrt{x + 3} - \sqrt{6 - x}}{x^2 - 10x + 25}
21.
Aniqmas integralni hisoblang.

dxx3+1\int \frac{dx}{x^3 + 1}
22.
Agar f(x)=x+x2+x3++x2024x+x2+x3++x1012f(x) = \frac{x + x^2 + x^3 + \ldots + x^{2024}}{x + x^2 + x^3 + \ldots + x^{1012}} bo'lsa, f(1)f'(1) ni toping.
23.
Radiusi uzunligi 1+21 + \sqrt{2} ga teng bo'lgan chorak aylanaga O1O_1 markazli aylana ichki chizilgan. O1O_1 markazli aylana radiusini toping.
Savol
24.
f(x)=x2ax+3f(x) = x^2 - ax + 3 va g(x)=2x1g(x) = 2x - 1 funksiya grafiklari bir nuqtada kesishadigan aa ning barcha qiymatlari yig'indisini toping. (Agar u bitta bo'lsa, shu aa ni toping.)
25.
Uchburchakning ikkita ichki burchaklari 6464^\circ va 7676^\circ ga teng. Uchburchakning bissektrisalari kesishishidan hosil bo'lgan eng kichik burchakni toping.
26.
ABCABC uchburchakda AB=8AB = 8, BC=13BC = 13 va AC=15AC = 15. Agar LL nuqta uchburchakning bissektrisalar kesishish, MM nuqta medianalar kesishgan nuqtasi bo'lsa, LMLM kesma uzunligini toping.
Savol
27.
Rasmda berilgan ma'lumotlar asosida, kvadratning yuzasini toping.
Savol
28.
Aylanaga muntazam oltiburchak ichki va tashqi chizilgan. Tashqi chizilgan muntazam oltiburchak yuzining ichki chizilgan muntazam oltiburchak yuziga nisbatini toping.
29.
AB\overrightarrow{AB} vektorning boshi A(6;4;2)A(6; 4; -2) nuqta, OxyOxy tekisligida oxiri BB nuqta bo'lsa, hamda CD(4;3;1)\overrightarrow{CD}(4; -3; 1) vektor AB\overrightarrow{AB} vektorga kollinearligi aniq bo'lsa, AB\overrightarrow{AB} vektor koordinatalari yig'indisini toping.
30.
Silindrning o'q kesim diagonali asos radiusidan 4 marta katta. Yon sirti yuzi 123π12\sqrt{3}\pi bo'lsa, silindr hajmini toping.
31.
Ushbu to'plam ko'rinishini aniqlang.
Savol
32.
Raqamlar yig'indisi 9 ga teng bo'lgan nechta uch xonali son bor?

Topshiriqlar (33-35) va javob variantlari (A-F) ni o'zaro moslashtiring.

33-35.
Konus asosiga kvadrat ichki chizilgan. Kvadrat tomoni 4 cm ga teng. Kvadratni tomonini ikki uchi tutashtirilib uchburchak kesim hosil qilindi. Shu kesimda ikki yon qirralari orasidagi burchak 6060^\circ ga teng.
Savol

Javob variantlari:

A)
16216\sqrt{2}
B)
828\sqrt{2}
C)
222\sqrt{2}
D)
424\sqrt{2}
E)
434\sqrt{3}
F)
232\sqrt{3}
SavolABCDEF
33.
Konus balandligini toping (cm)(cm).
34.
Kesim ABSABS yuzasini toping (cm2)(cm^2).
35.
Konus hajmini toping (cm3)(cm^3). (π3\pi \approx 3 deb oling.)
36.
Tenglamalar sistemasini yeching:

{x4+y4=17(xy)xy=2(xy)\begin{cases} x^4 + y^4 = 17(x - y) \\ xy = 2(x - y) \end{cases}
a) Tenglamalar sistemasi (x1;y1)(x_1; y_1), (x2;y2)(x_2; y_2), ..., (xn;yn)(x_n; y_n) yechimlari bo'lsa, nechta yechimga ega?
b) xn+ynx_n + y_n ning eng kichik qiymatini toping.
37.
Tenglamani yeching.

cos3xsin3x+sin3xcos3x=34\cos 3x \cdot \sin^3 x + \sin 3x \cdot \cos^3 x = \frac{3}{4}
a) Tenglamaning eng kichik musbat ildizini toping.
b) Tenglama [2π;2π][-2\pi; 2\pi] oraliqda nechta ildizga ega?
38.
f(x)=cos(2x)+cosx2f(x) = \cos(\sqrt{2}x) + \cos\frac{x}{\sqrt{2}} funksiya berilgan.
a) Funksiyaning eng kichik musbat davrini toping.
b) Funksiyaning eng kichik qiymatini toping.
39.
Rasmda (1;13)(-1; 13) oraliqdagi y=f(x)y = f'(x) funksiya grafigi tasvirlangan. (Bunda f(x)f'(x)f(x)f(x) funksiyaning birinchi tartibli hosilasi.) Funksiya grafigi 1×11 \times 1 o'lchamdagi katakchalarda tasvirlangan.
Savol
a) y=f(x)y = f(x) funksiyaning nechta ekstremum nuqtasi bor?
b) y=f(x)y = f(x) funksiya [5;11][5; 11] oraliqdagi qaysi nuqtada eng katta qiymatga erishadi?
40.
f(x)=x24x+5f(x) = x^2 - 4x + 5 funksiya berilgan. Rasmda g(x)=x+1g(x) = x + 1 va h(x)=x+5h(x) = x + 5 to'g'ri chiziqlari ham tasvirlangan.
Savol
a) f(x)f(x) va g(x)g(x) funksiyalar bilan chegaralangan soha (S1)(S_1) ni toping.
b) f(x)f(x), h(x)h(x) va g(x)g(x) funksiyalar bilan chegaralangan soha (S2)(S_2) ni toping.
41.
ABCABC uchburchakning ichidan PP nuqta olingan bo'lib, BP=3BP = \sqrt{3}, PC=1PC = 1. APAP kesma BACBAC burchak bissektrisa. Agar BAC=30\angle BAC = 30^\circ va BPC=120\angle BPC = 120^\circ bo'lsa,
Savol
a) sinβ\sin\beta ni toping.
b) ACAC tomon uzunligini toping.
42.
ABCDABCD to'g'ri to'rtburchakning ichidan EE nuqta olingan. Agar AE=1AE = 1, ED=4ED = 4, EB=2EB = \sqrt{2} va BC=3ABBC = 3AB berilgan bo'lsa,
Savol
a) cosEAD\cos\angle EAD ni toping.
b) ABCDABCD to'g'ri to'rtburchak yuzini toping.
43.
Muntazam o'n ikkiburchakning tomoni 62\sqrt{6} - \sqrt{2} ga teng. Muntazam o'n ikkiburchakka aylana tashqi chizilgan.
a) Aylana radiusini toping.
b) O'n ikkiburchak yuzini toping.
44.
Asosi rombdan iborat piramida berilgan. Rombning kichik burchagi 3030^\circ va asosiga ichki chizilgan doira radiusi 3\sqrt{3} ga teng. Piramida yon yoqlari asos tekisligi bilan 6060^\circ burchak hosil qiladi.
a) Piramida to'la sirtini toping.
b) Piramida hajmini toping.
45.
Beshta ishchi bir ishni bajarmoqda. Birinchi, ikkinchi va uchinchi ishchilar birgalikda ishni 7,5 soatda, birinchi, uchinchi va beshinchi ishchilar 5 soatda, birinchi, uchinchi va to'rtinchi ishchilar 6 soatda, ikkinchi, to'rtinchi va beshinchi ishchilar 4 soatda birgalikda ishni tugatishdi.
a) Ikkinchi ishchi yolg'iz o'zi ishni necha soatda tugatadi?
b) Beshta ishchi birgalikda ishni necha soatda tugatadi?