Matematika (23.02.2025) 2-variant

45 ta savol · saqlash uchun kiring

Matematika formulalar varaqasi / Лист формул по математике

Tenglamalar / Уравнения

ax^{2}+bx+c=0; \quad x_{1:2}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}; \quad \begin{cases}x_{1}+x_{2}=-\frac{b}{a}\\ x_{1}\cdot x_{2}=\frac{c}{a}\end{cases}

|f(x)|+|g(x)|=|f(x)+g(x)|\Rightarrow f(x)\cdot g(x)\ge0;

\frac{Ax}{ax^{2}\pm bx-c}\pm\frac{Bx}{ax^{2}\pm dx+c}=k \quad \Rightarrow \quad \frac{A}{ax-\frac{c}{x}\pm b}\pm\frac{B}{ax+\frac{c}{x}\pm d}=k;

(ax^{2}\pm bx-c)(ax^{2}\pm dx+c)=Ax^{2} \quad \Rightarrow \quad (ax-\frac{c}{x}\pm b)(ax+\frac{c}{x}\pm d)=A;

ax+\frac{c}{x}=z

o'zgaruvchi kiritib yechiladi / решается c помошью замены

(x-a)(x-b)(x-c)(x-d) = A \quad \Rightarrow \quad a+b=c+d

bo'lsa,

t=x^{2}-(a+b)x

o'zgaruvchi kiritib yechiladi / решается c помошью замены

Qisqa ko'paytirish formulalari / Формулы сокращённого умножения

a^{2}-b^{2}=(a-b)(a+b)

(a\pm b)^{2}=(a^{2}\pm2ab+b^{2})

(a\pm b)^{3}=a^{3}\pm3a^{2}b+3ab^{2}\pm b^{3}

a^{3}\pm b^{3}=(a\pm b)(a^{2}\mp ab+b^{2})

a^{n}-b^{n}=(a-b)(a^{n-1}+a^{n-2}b+...+ab^{n-2}+b^{n-1})

a^{2n}\pm b^{2n}=(a\pm b)(a^{2n-1}\mp a^{2n-2}b+...+ab^{2n-2}\mp b^{2n-1})

a^{2n+1}\pm b^{2n+1}=(a\pm b)(a^{2n}\mp a^{2n-1}b+...+ab^{2n-1}\mp b^{2n})

(a+b)^{n}=a^{n}+na^{n-1}b+\frac{n(n-1)}{2!}a^{n-2}b^{2}+...

...+\frac{n(n-1)(n-2)\cdot...\cdot(n-(k-1))}{k!}+...+nab^{n-1}+b^{n}

Arifmetik progressiya / Арифметическая прогрессия

a_{n}=a_{n-1}+d=a_{1}+(n-1)d;

a_{n}=\frac{a_{n-k}+a_{n+k}}{2} \quad (1\le k\le n-1);

S_{n}=\frac{a_{1}+a_{n}}{2}\cdot n=\frac{2a_{1}+d(n-1)}{2}\cdot n

Geometrik progressiya / Геометрическая прогрессия

b_{n}=b_{1}\cdot q^{n-1}; \quad b_{n}^{2}=b_{n-m}\cdot b_{n+m} \quad (1\le m\le n-1);

S_{n}=\frac{b_{1}(1-q^{n})}{1-q} \quad (|q|>1);

S_{n}=\frac{b_{1}}{1-q} \quad (|q|<1);

Logarifmning xossalari / Свойства логарифмов

log_{a}1=0; \quad log_{a}(x_{1}\cdot x_{2})=log_{a}|x_{1}|+log_{a}|x_{2}|;

log_{a}\frac{x_{1}}{x_{2}}=log_{a}|x_{1}|-log_{a}|x_{2}|; \quad log_{a}b=\frac{log_{c}b}{log_{c}a};

log_{a^{q}}x^{p}=\frac{p}{q}\cdot log_{a}x; \quad log_{a}b=\frac{1}{log_{b}a};

lg10^{n}=n; \quad lne^{n}=n;

Daraja va ildiz / Степень и корень

a^{n}\cdot a^{m}=a^{m+n}; \quad a^{n}:a^{m}=a^{m-n}; \quad (a^{m})^{n}=a^{m\cdot n};

(a\cdot b)^{n}=a^{n}\cdot b^{n}; \quad (\frac{a}{b})^{n}=\frac{a^{n}}{b^{n}}; \quad a^{-n}=\frac{1}{a^{n}};

(\frac{a^{n}}{b^{m}})^{-k}=(\frac{b^{m}}{a^{n}})^{k}; \quad (-a)^{2n}=a^{2n}; \quad (-a)^{2n+1}=-a^{2n+1};

\sqrt[n]{a^{k}}=a^{\frac{k}{n}}; \quad \sqrt[n]{a^{k}\cdot b^{m}}=\sqrt[n]{a^{k}}\cdot\sqrt[n]{b^{m}};

\sqrt[n]{\frac{a^{k}}{b^{m}}}=\frac{\sqrt[n]{a^{k}}}{\sqrt[n]{b^{m}}}; \quad (\sqrt[n]{a^{m}})^{k}=\sqrt[n]{a^{mk}}; \quad \sqrt[n]{a^{n+k}}=a\sqrt[n]{a^{k}};

a^{-\frac{k}{n}}=\frac{1}{\sqrt[n]{a^{k}}}; \quad \sqrt[n]{\sqrt[m]{a^{k}}}=\sqrt[nm]{a^{k}};

\sqrt[n]{a^{k}}\cdot\sqrt[m]{b^{p}}=\sqrt[nm]{a^{km}\cdot b^{pn}}; \quad \sqrt[n]{a^{n}}=|a| \quad (n=2k);

\sqrt[2n]{(a-b)^{2n}}=\begin{cases}a-b, & a\ge b\\ b-a, & b\ge a\end{cases}

Urinma va normal tenglamasi / Уравнение касательной и нормали

y=f(x_{0})+f^{\prime}(x_{0})(x-x_{0});

y=f(x_{0})-\frac{1}{f^{\prime}(x_{0})}(x-x_{0});

Hosilaga doir formulalar va xossalari / Таблица производных и свойства

(x^{a})^{\prime}=ax^{a-1}; \quad (a^{x})^{\prime}=a^{x}ln a; \quad (e^{x})^{\prime}=e^{x};

(log_{a}x)^{\prime}=\frac{1}{x ln a}; \quad (ln x)^{\prime}=\frac{1}{x}; \quad (sin x)^{\prime}=cos x;

(cos x)^{\prime}=-sin x; \quad (tg x)^{\prime}=\frac{1}{cos^{2}x}; \quad (ctg x)^{\prime}=-\frac{1}{sin^{2}x};

(u\pm v)^{\prime}=u^{\prime}\pm v^{\prime}; \quad (k\cdot u)^{\prime}=k\cdot u^{\prime};

(u\cdot v)^{\prime}=u^{\prime}\cdot v+u\cdot v^{\prime}; \quad (\frac{u}{v})^{\prime}=\frac{u^{\prime}\cdot v-u\cdot v^{\prime}}{v^{2}};

Integralga doir formulalar va xossalari / Таблица и свойства интегралов

\int x^{n}dx=\frac{x^{n+1}}{n+1}+C; \quad \int\frac{dx}{x}=ln|x|+C;

\int e^{x}dx=e^{x}+C; \quad \int a^{x}dx=\frac{a^{x}}{lna}+C;

\int sinx dx=-cosx+C; \quad \int cosx dx=sinx+C;

\int k\cdot f(x)dx=k\cdot\int f(x)dx;

\int(f(x)+g(x))dx=\int f(x)dx+\int g(x)dx;

\int_{a}^{b}f(x)dx=F(b)-F(a); \quad \int_{a}^{b}f(x)dx=-\int_{b}^{a}f(x)dx;

\int_{a}^{b}f(x)dx=\int_{a}^{c}f(x)dx+\int_{c}^{b}f(x)dx \quad (a\le c\le b);

O'rta arifmetik va o'rta geometrik / Среднее арифметическое и среднее геометрическое

A=\frac{a_{1}+a_{2}+...+a_{n}}{n}; \quad G=\sqrt[n]{a_{1}\cdot a_{2}\cdot...\cdot a_{n}}

Ba'zi trigonometrik formulalar / Основные формулы тригонометрии

Asosiy trigonometrik ayniyatlar / Основные тригонометрические свойства:

sin^{2}\alpha+cos^{2}\alpha=1; \quad tg\alpha=\frac{sin\alpha}{cos\alpha}; \quad ctg\alpha=\frac{cos\alpha}{sin\alpha};

tg^{2}\alpha+1=\frac{1}{cos^{2}\alpha}; \quad ctg^{2}\alpha+1=\frac{1}{sin^{2}\alpha};

cos(\alpha\pm\beta)=cos\alpha\cdot cos\beta\mp sin\alpha\cdot sin\beta;

sin(\alpha\pm\beta)=sin\alpha\cdot cos\beta\pm cos\alpha\cdot sin\beta;

tg(\alpha\pm\beta)=\frac{tg\alpha\pm tg\beta}{1\mp tg\alpha\cdot tg\beta}; \quad ctg(\alpha\pm\beta)=\frac{ctg\alpha\cdot ctg\beta\mp1}{ctg\alpha\pm ctg\beta};

sin\alpha+sin\beta=2sin\frac{\alpha+\beta}{2}cos\frac{\alpha-\beta}{2};

sin\alpha-sin\beta=2sin\frac{\alpha-\beta}{2}cos\frac{\alpha+\beta}{2};

cos\alpha+cos\beta=2cos\frac{\alpha+\beta}{2}cos\frac{\alpha-\beta}{2};

cos\alpha-cos\beta=-2sin\frac{\alpha+\beta}{2}sin\frac{\alpha-\beta}{2};

Trigonometrik funksiyaning ba'zi burchaklardagi qiymatlari / Значения тригонометрической функции под некоторыми углами

$\alpha$	0	$\frac{\pi}{6}$	$\frac{\pi}{4}$	$\frac{\pi}{3}$	$\frac{\pi}{2}$
$\sin\alpha$	0	$\frac{1}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{3}}{2}$	1
$\cos\alpha$	1	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{1}{2}$	0

To'g'ri burchakli uchburchak uchun formulalar / Формулы для прямоугольного треугольника

a^2 + b^2 = c^2;

a = c \cdot \sin\alpha

b = c \cdot \cos\alpha;

a = b \cdot \text{tg}\alpha

Ixtiyoriy uchburchak uchun formulalar / Формулы для произвольного треугольника

S = \frac{1}{2}ab\sin\gamma

S = \sqrt{p(p - a)(p - b)(p - c)}

p = \frac{1}{2}(a + b + c); \quad S = \frac{1}{2}h_c \cdot c

\frac{\sin\alpha}{a} = \frac{\sin\beta}{b} = \frac{\sin\gamma}{c}

a^2 = b^2 + c^2 - 2bc\cos\alpha

Uchburchak uchun formulalar (umumiy) / Формулы для треугольника (общие)

S=\frac{1}{2}ab sin\gamma; \quad S=\sqrt{p(p-a)(p-b)(p-c)};

p=\frac{1}{2}(a+b+c); \quad S=\frac{1}{2}h_{c}\cdot c;

\frac{a}{sin\alpha}=\frac{b}{sin\beta}=\frac{c}{sin\gamma}=2R; \quad a^{2}=b^{2}+c^{2}-2bc cos\alpha;

To'g'ri burchakli uchburchak / прямоугольный треугольник:

a^{2}+b^{2}=c^{2}; \quad a=c\cdot sin\alpha; \quad b=c\cdot cos\alpha; \quad a=b\cdot tg\alpha;

Uchburchakka ichki va tashqi chizilgan aylana radiuslarini hisoblash formulasi /
Формула вычисления радиусов окружности, вписанной внутрь и вне треугольника

r=\frac{2S}{a+b+c}; \quad R=\frac{abc}{4S};

To'rtburchakning yuzlari uchun formulalar / Формулы граней четырехугольника

Parallelogramm / параллелограмм:

S=a\cdot h_{a}; \quad S=a\cdot b\cdot sin\alpha;

Romb / ромб:

S=\frac{1}{2}\cdot d_{1}\cdot d_{2}; \quad S=a^{2}\cdot sin\alpha;

Trapetsiya / трапеция:

S=\frac{a+b}{2}\cdot h;

Aylana va doira Doiraviy sektor va segment yuzi / Круговой сектор и сегментная грань

l=2\pi R; \quad S=\pi R^{2};

S_{sektor}=\frac{\pi R^{2}}{360}\cdot\alpha; \quad S_{segment}=\frac{R^{2}}{2}(\alpha-sin\alpha);

Vektorlar / Векторы

|\vec{a}|=\sqrt{x^{2}+y^{2}+z^{2}}

vektorning uzunligi / длина вектора

\vec{a}\cdot\vec{b}=|\vec{a}|\cdot|\vec{b}|\cdot cos\varphi

skalyar ko'paytmasi / скалярное умножение

\vec{a}\cdot\vec{b}=x_{1}\cdot x_{2}+y_{1}\cdot y_{2}

vektorlarning skalyar ko'paytmasi / скалярное произведение векторов

\vec{a}\cdot\lambda=\vec{c}(x\cdot\lambda;y\cdot\lambda;z\cdot\lambda)

Stereometriya formulalari / Формулы стереометрии

Prizma / призма | Piramida / пирамида:

V=S\cdot H; \quad V=\frac{1}{3}S\cdot H;

Kesik piramida / усеченная пирамида:

V=\frac{1}{3}h(S_{1}+\sqrt{S_{1}\cdot S_{2}}+S_{2});

Silindr / цилиндр:

S_{yon}=2\pi RH; \quad V=\pi R^{2}H;

Konus / конус:

S_{yon}=\pi R l; \quad V=\frac{1}{3}\pi R^{2}H;

Kesik konus / усеченный конус:

V=\frac{1}{3}\pi h(R_{1}^{2}+R_{1}\cdot R_{2}+R_{2}^{2});

Shar / шар:

S=4\pi R^{2}; \quad V=\frac{4}{3}\pi R^{3};

Shar segmentining hajmi / объем шар сегмента:

V=\pi H^{2}(R-\frac{H}{3});

Shar segmentining sirtining yuzi / площадь поверхности шарового сегмента:

S_{sirt}=2\pi RH;

Agar

a, b

raqamlar uchun

5a + \frac{5}{11}b = 45

tenglik o'rinli bo'lsa,

2a + b

ni toping.

Agar

x = 1 + 4 + 7 + \ldots + 301

y = 2 + 8 + 14 + \ldots + 602

bo'lsa,

\frac{y + 2022x}{x}

ni hisoblang.

Basseynda 4 ta quvur bor. Birinchi va ikkinchi quvur basseyni to'ldiriladi. Uchinchi va to'rtinchi quvur basseynni bo'shatadi. 4 ta quvur birdaniga ochilsa basseyn 5 soatda to'ladi. Faqat birinchi, ikkinchi va uchinchi quvurlar ochilsa 3 soatda to'ladi. Faqat birinchi, uchinchi va to'rtinchi quvurlar ochilsa 30 soatda to'ladi. Faqat birinchi va uchinchi quvurlar ochilsa basseyn necha soatda to'ladi?

Meva tarkibida 80% suv bor. Quritilgandan keyin

25\ kg

ga kamayib, suvi 60% bo'lib qoldi. Dastlab meva necha

kg

bo'lgan?

Hisoblang:

\frac{3^9 \cdot 3^{-6} \cdot 5^4 + 625 \cdot 27 \cdot \left(\frac{1}{5}\right)^{-1}}{2 \cdot 3^2 \cdot 5^2 - 4 \cdot 9 \cdot \left(\frac{2}{5}\right)^{-2}}

Agar

a = 7 - \sqrt{3}

ga teng bo'lsa,

a^2 - 14a + 51

ning qiymatini toping.

Ifodani soddalashtiring:

\frac{\left(\sqrt{3} - \sqrt[4]{45}\right)^2}{1 - 2\sqrt[4]{5} + \sqrt{5}}

Arifmetik progressiyada

a_4 + a_7 + a_9 + a_{11} + a_{14} = 45

bo'lsa, dastlabki o'n yettita hadi yig'indini toping.

Geometrik progressiyada

S_4 = 80

S_8 = 6560

ga teng bo'lsa,

S_6

ni toping.

10.

Agar

|c| < |b| < |a|

a < 0

b > 0

c < 0

bo'lsa,

\frac{a|b - c|}{|a|} + \frac{b|c - a|}{|b|} + \frac{c|a - b|}{|c|}

ni hisoblang.

11.

Ifodani soddalashtiring:

\frac{a^4 - 64}{a^2 + 8} - \frac{a^6 - 512}{a^4 + 8a^2 + 64}

12.

Ifodani soddalashtiring:

\frac{\sin\alpha}{\operatorname{tg}\left(\frac{\pi}{4} - \frac{\alpha}{2}\right) \cdot (1 + \sin\alpha)}

13.

Quyidagi tenglamaning

\left[\frac{\pi}{2}; \frac{17\pi}{2}\right]

oralig'ida nechta yechimga ega:

\cos 2x - 1 = 5\sin x + 2

14.

Tengsizlikni yeching:

x^2 \cdot 3^x + 9 > 3^x + 9 \cdot x^2

15.

\log_{0,5}\log_3\log_2\frac{x^2 + 8x}{x - 5} \le 0

tengsizlikning eng kichik butun yechimini toping.

16.

Quyidagi tenglamalar sistemasi nechta yechimga ega?

\begin{cases} x^2 = 4x + 5y \\ y^2 = 5x + 4y \end{cases}

17.

Agar

x_1

x_2

sonlari

2x^2 + 9x + 5 = 0

tenglamaning ildizlari bo'lsa,

\frac{1}{x_1^2} + \frac{1}{x_2^2}

ni hisoblang.

18.

Tengsizlikning natural yechimlari yig'indisini toping.

245x - x^6 > 0

19.

Tengsizlikning butun yechimlari yig'indisini toping:

(x^2 - 5x + 10)(x^2 - 5x + 12) \le 48

20.

Rasmdagi ma'lumotlar asosida

a, b, c

larni taqqoslang.

21.

Aniq integralni hisoblang:

\int_2^4 \frac{3}{5 - 4x - x^2}\,dx

22.

Agar

f(x) = \frac{2^{2x}}{\sqrt{2 - 2^{2x}}}

ga teng bo'lsa,

f'(0)

ni toping.

23.

Ikkita aylana tashqi ravishda urinadi. Ularning diametrlari

10\ dm

18\ dm

ga teng bo'lsa, aylanalar markazlari orasidagi masofani toping.

24.

Nechta natural son

f(x) = 6\sin x - 4\cos x

funksiyaning qiymatlar sohasiga tegishli?

25.

Teng yonli uchburchakning uchidagi burchagi

30^\circ

ga teng. Asosidan olingan

E

nuqtadan yon tomonlarigacha bo'lgan eng qisqa masofalar 3 va 9 ga teng. Uchburchak yuzini toping.

26.

ABC

to'g'ri burchakli uchburchak gipotenuzasi uzunligi 15 ga,

\sin\angle ACB = 0{,}6

ga teng bo'lsa, uchburchak yuzini toping.

27.

Ikkita bir xil kvadrat ustama-ust qo'yilib biri ikkinchisi ustida

45^\circ

burildi. Rasmga qarang. Bo'yalgan soha yuzining

ABCD

kvadrat yuziga nisbatini toping.

28.

Muntazam ko'pburchakning tomoni

2\sqrt{3}

ga va ichki chizilgan aylana radiusi 3 ga teng bo'lsa, ko'pburchak yuzini toping.

29.

Rasmda qirrasi uzunligi 3 ga teng bo'lgan

ABCDA_1B_1C_1D_1

kub tasvirlangan. Shunga ko'ra

\left(\overrightarrow{AD} + \overrightarrow{DC}\right) \cdot \left(\overrightarrow{CC_1} + \overrightarrow{C_1D_1}\right)

ni hisoblang.

30.

ABCD

to'g'ri to'rtburchak tekisligiga

S

nuqtadan

SA

perpendikulyar tushirildi.

SB = 6

SC = 9

SD = 7

bo'lsa,

SA

ni toping.

31.

A = \{24

ning natural bo'luvchilari

\}

B = \{16

ning natural bo'luvchilari

\}

C = \{20

ning natural bo'luvchilari

\}

bo'lsa,

n(A) + n(B) - n(C)

ni hisoblang. (Bu yerda

n(A)

—

A

to'plamning elementlar soni)

32.

3, 5, 7

raqamlaridan foydalanib

10^5

dan kichik nechta natural son tuzish mumkin?

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

33-35.

Prizma va silindr hajmlari teng. Prizmaning asosi trapetsiyadan iborat. Trapetsiyaning kichik asosi trapetsiyaning balandligiga teng, kichik asosi katta asosidan 2 marta kichik. (

\pi = 3

deb oling)

Javob variantlari:

\sqrt{3}

\sqrt{2}

Savol	A	B	C	D	E	F
33. Prizmaning balandligi bilan silindrning balandligi teng. Trapetsiyaning kichik asosi silindrning radiusidan necha marta katta.
34. Trapetsiyaning kichik asosi silindrning radiusidan 2 marta katta. Silindrning balandligi prizmaning balandligiga nisbatini toping.
35. Trapetsiyaning kichik asosi silindrning radiusidan 2 marta katta. Prizma asosining yuzi 6 ga teng. Silindrning asosini yuzini toping.

36.

Tenglamani yeching:

\begin{cases} \frac{4}{x + y} + \frac{4}{x - y} = 3 \\ (x + y)^2 + (x - y)^2 = 20 \end{cases}

a) Tenglamalar sistemasining nechta juft

(x; y)

yechimi bor?

b) Agar tenglamaning yechimlari

(x_1; y_1), (x_2; y_2), \ldots (x_n; y_n)

bo'lsa,

x_1 + x_2 + \ldots + x_n

ni toping.

37.

Tenglamani yeching:

\left|\cos\left(\frac{2\pi}{3}\sin\frac{2x}{3}\right) + 1\right| = \frac{1}{2}

a) Tenglamaning

[-\pi; 4\pi]

oraliqdagi ildizlar sonini toping.

b) Tenglamaning

[-\pi; 4\pi]

oraliqdagi ildizlari yig'indisini toping.

38.

f(x)

funksiya uchun

(x + 1)f(x) + f\left(\frac{1}{x}\right) = \frac{4x + 2}{x + 1}

bo'lsin.

f(2)

ni toping.

f^2(3)

ni toping.

39.

Rasmda uchlari

y_1

y_2

funksiyalarda bo'lgan eng katta yuzali

ABCD

to'g'ri to'rtburchak tasvirlangan.

ABCD

to'g'ri to'rtburchak yuzasini toping.

AC

diagonal uzunligini toping.

40.

Rasmda

y_1

— to'g'ri chiziq,

y_2

— parabola va

x = 1

simmetriya o'qi berilgan bo'lsa, rasmdagi ma'lumotlar asosida,

x + y

ni hisoblang.

b) Bo'yalgan soha yuzasini toping.

41.

ABC

uchburchakning

A

uchidan

AH

balandlik tushirilgan.

ABH

AHC

uchburchaklarga ichki aylanalar chizilgan bo'lib, bunda

M

O

nuqta

ABH

uchburchakka ichki chizilgan aylana mos ravishda

BH

AH

tomonga urinuvchi nuqtalar,

T

N

nuqtalar

AHC

uchburchakka ichki chizilgan aylananing mos ravishda

AH

HC

tomonlariga urinuvchi nuqtalari. Agar

AB = 13

AC = 15

BC = 14

a) Shtrixlangan yuzini toping.

K

nuqtadan

AH

gacha bo'lgan eng qisqa masofani toping.

42.

ABCD

to'rtburchakka aylana tashqi chizilgan.

AC

BD

diagonallar

N

nuqtada kesishadi.

N

nuqtadan

AB, BC, CD, AD

tomonlarga perpendikulyarlar tushirilgan. Ularning qiymatlari mos ravishda

4, \sqrt{5}, \frac{8}{\sqrt{21}}, 8\sqrt{\frac{5}{21}}

ga teng.

\frac{AN}{NC}

ni toping.

b) Agar

AC = 2\sqrt{21}

bo'lsa,

BD

ni toping.

43.

Rasmda

ABCDEFGH

qavariq sakkiz burchak berilgan. Agar

AB = BC = CD = DE = EF = FG = \sqrt{3}

bo'lsin.

a) Agar

MN

kesma sakkizburchak yuzini teng ikkiga bo'lsa,

MN

kesma ajratgan shakllar perimetrlar musbat ayirmasini toping.

|P_{ABCDEFMN} - P_{MGHN}|

b) Agar

FM = \sqrt{2}

bo'lsa,

MN

kesma uzunligini toping.

44.

Sharga eng katta hajmli silindr ichki chizilgan.

R

— shar radiusi,

r

— silindr asosi radiusi.

\frac{r}{R}

nisbatni toping.

b) Shar hajmining silindr hajmiga nisbatini toping.

45.

Anvar 24000000 so'm pulini

A

B

bankga omonatga qo'yadi.

A

bankdan 1 yilda oddiy foizda 16% foyda qiladi,

B

bankdan 1 yilda oddiy foizda 18% foyda qiladi.

a) Agar

A

B

banklardan 1 yilda 4090000 so'm foyda qilgan bo'lsa,

A

bankga qancha pul qo'ygan?

b) Agar

A

bankdan bir yilda 2240000 so'm foyda qilgan bo'lsa,

B

bankdan qancha foyda ko'radi?