Matematika (29.12.2025)

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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;

156 sonining natural bo'luvchilari yig'indisini toping.

Hisoblang:

\frac{2^4 + 2^2 + 1}{2^7 - 2} + \frac{3^4 + 3^2 + 1}{3^7 - 3} + \frac{4^4 + 4^2 + 1}{4^7 - 4}

Usta bir ishni 4 kunda, shogirdi esa 12 kunda bajaradi. Usta va shogird birgalikda ishning bir qismini 1 kunda bajardi. Qolgan ishni esa shogird tugatdi. Jami ish necha kunda bajarilgan?

Palto narxi 1 200 000 edi. Birinchi marta narxini

40\%

ga oshirildi, keyin

10\%

ga kamaytirildi. Paltoning hozirgi narxini toping.

1^3 + 2^3 + \ldots + 59^3

ifoda quyidagi qaysi songa qoldiqsiz bo'linadi?

Hisoblang:

\sqrt{4^9 + 6^{10} + 9^{10}}

Agar

x = \frac{9 - \sqrt{80}}{2}

bo'lsa,

\sqrt{x^2 + 18x + 81} + \sqrt{x^2 + 8\sqrt{5}x + 80}

ni hisoblang.

Arifmetik progressiyada

a_{20} + a_{17} = 35

a_{21} \cdot a_{16} = 150

bo'lsa,

a_2

ni toping.

Arifmetik progressiya hadlari

(a_n)

va geometrik progressiya hadlari

(b_n)

. Agar

b_1 + b_2 + b_3 = 65

b_2 > b_1

b_1 - a_1 = 1

b_2 - a_2 = 8

b_3 - a_3 = 35

b_1 + b_2 + \ldots + b_n = 200

bo'lsa,

n

ning qiymatini toping.

10.

Ifodani soddalashtiring:

\frac{a^{-6} + a^{-4} + a^{-2}}{a^6 + a^4 + a^2}

11.

Agar

a + b = x

ab = y

bo'lsa,

ab^3 + a^3b + a^2 + b^2 + 5ab + 2a^2b^2

x

y

orqali ifodalang.

12.

Ifodani soddalashtiring:

\frac{2\cos^2\alpha}{\operatorname{tg} 2\alpha + \sin 2\alpha}

13.

Tengsizlikni yeching:

\sqrt{2\sin x} \le 1

14.

Tenglamaning haqiqiy ildizlari ko'paytmasini toping:

\left(\sqrt[4]{17 - 12\sqrt{2}}\right)^x + \left(\sqrt[4]{17 + 12\sqrt{2}}\right)^x = 6

15.

Tenglama nechta haqiqiy ildizga ega:

\frac{2\ln 2 - \ln 5 + \ln(2x + 3)}{\ln x + \ln(x + 3)} = 1

16.

\begin{cases} 3x - y = 16 \\ x + 2y = 17 \end{cases}

tenglamalar sistemasi berilgan bo'lsa,

x + y

ning qiymatini toping.

17.

x^4 - 3x^2 + 6x - 3 = 0

tenglama nechta haqiqiy ildizga ega?

18.

Tengsizlikning nechta butun yechimi bor:

\frac{7}{x^2 - 5x + 6} + 1 \le \frac{9}{3 - x}

19.

Nechta butun son tengsizlik yechimi bo'la olmaydi:

\sqrt{x^2 - 1} \ge x - 2

20.

f(x) = x + \frac{1}{x}

funksiya berilgan. Quyidagilardan qaysi biri to'g'ri?

21.

Agar

f(x) = \frac{3}{\sqrt{x + 1}}

bo'lsa,

f^{-1}(2)

ning qiymatini toping. (

f^{-1}(x)

—

f(x)

funksiyaning teskari funksiyasi).

22.

f(x) = \cos\left(kx^n\right) \cdot \lg\sqrt{x + \sqrt{1 + x^2}}

funksiya berilgan (

n

— juft natural son). Agar

f'(2023) = 2024

bo'lsa,

f'(-2023) + 2024

ni hisoblang.

23.

Aniq integralni hisoblang:

\int_0^{\frac{\pi}{2}} (x - 1)\cos x\,dx

24.

Rasmdagi ma'lumotlar asosida

x

ning qiymatini toping.

25.

Rasmda tomoni

\sqrt{3}

ga teng bo'lgan kvadrat berilgan. Kvadratning bir tomoni muntazam uchburchakning balandligiga teng bo'lsa, muntazam uchburchakning yuzasini toping.

26.

ABC

uchburchakning yuzi 96 ga teng. Uchburchakning

AL

BD

CG'

medianalaridan mos ravishda

F

E

K

nuqtalar olingan va

AF = FL

BD = 3BE

CG' = 4CK

bo'lsa,

FEK

uchburchakning yuzini toping.

27.

ABCD

to'g'ri to'rtburchak berilgan.

AB = 9

BC = 11

bo'lsa, bo'yalgan soha

EFGH

to'rtburchakning yuzini toping.

28.

ABCDE

muntazam beshburchak berilgan.

DE

tomoni davom ettirilib

O

nuqta olindi.

DO

kesma uzunligi 8 ga teng.

O, D, E

nuqtalar bitta to'g'ri chiziqda yotadi. Bunda

G

nuqta

AB

tomonning o'rtasi bo'lib,

OG

kesma

EA

tomonni

F

nuqtada kesib o'tadi.

\angle EDF = 54^\circ

H

nuqta

BC

tomonning o'rtasi.

EM

tomonning uzunligini toping.

29.

\vec{a}, \vec{b}, \vec{c}

birlik vektorlarning ixtiyoriy ikkitasining orasidagi burchak

60^\circ

ga teng.

\vec{a}

\vec{b} + \vec{c}

vektorlar orasidagi burchak kosinusini toping.

30.

\alpha

\beta

tekisliklar o'zaro parallel.

\alpha

tekislikda

A, B

nuqtalar olingan.

A

B

nuqtalardan

\beta

tekisligiga

AC

BD

og'malar tushirilgan. Agar

AC = 37

BD = 125

AC

og'ma proyeksiyasi 12 ga teng bo'lsa,

BD

ning

\beta

tekislikdagi proyeksiyasini toping.

31.

Quyidagilardan nechtasi bo'sh to'plam? (

I

— irratsional sonlar,

N

— natural sonlar,

Z

— butun sonlar,

R

— haqiqiy sonlar)

1.

\{x \mid x^2 - 2x = 2024, x \in N\}

;
2.

\{x \mid x^2 + 19x = x!, x \in N\}

;
3.

\{x \mid x^2 - 3\sqrt{2} - 2\sqrt{3} = 0, x \in I\}

;
4.

\{x \mid 2024x^2 + 2023x - 1 = 0, x \in Z\}

;
5.

\{x \mid |x| + 10 - \pi = 0, x \in R\}

32.

Aylana shaklidagi istirohat bog'ida Hasan va Husan markazdan 6 ta teng sektorga bo'lingan aylana yo'li bo'ylab tasodifiy bir yo'nalishni tanlab, 1 soat 5 km/soat tezlikda yuradi. Shuning uchun ikkalasi ham markazdan 5 km radiusli aylana bo'ylab joylashadi. 1 soat harakatlangandan so'ng orasidagi masofa 6 km dan kam bo'lmaslik ehtimolligi nechaga teng?

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

33-35.

Uchburchakning tomonlari 9, 10 va 17 ga teng. Bu uchburchakning eng katta balandligi atrofida

360^\circ

aylantirishdan jism hosil bo'ldi. (

\pi = 3

deb olinsin).

Javob variantlari:

600

945

512

567

1462

1512

36.

\sqrt{1 - x^2} = 4x^3 - 3x

tenglama berilgan.

a) Tenglama nechta haqiqiy yechimga ega?

b) Tenglamaning haqiqiy ildizlari ko'paytmasini toping.

37.

Tenglamani yeching:

5\sin^2 x - \cos^2 x + 8\cos x = |\cos x| - 1

a) Tenglama

\left[-\frac{5\pi}{4}; \frac{5\pi}{4}\right]

oraliqda nechta yechimga ega?

b) Tenglamaning

[0; 6\pi]

kesmadagi yechimlari yig'indisini toping.

38.

f(x) = ax^2 + bx + c

juft funksiya

y = 4x - 4

to'g'ri chiziqqa bitta nuqtada urinadi va shu urinish nuqtasi

Ox

o'qida yotadi.

\frac{a + b}{c}

ning qiymatini toping.

f(-4)

ning qiymatini toping.

39.

Rasmda

y_1(x)

yarim aylana,

y_2(x)

y_3(x)

to'g'ri chiziqlar tasvirlangan.

y_2(x)

y_3(x)

to'g'ri chiziqlar o'zaro perpendikulyar.

y_1(x)

yarim aylana

(0; 0)

(x_1; 0)

nuqtalardan o'tadi.

y_2(x) = 2\sqrt{2}x - 6\sqrt{2}

to'g'ri chiziq yarim aylana markazidan o'tadi.

y_3(x)

to'g'ri chiziq

Ox

o'qini

(x_2; 0)

nuqtada kesib o'tadi va

y_1(x)

yarim aylanaga urinadi. Bunda

O

aylana markazi.

x_1

ni toping.

x_2

ni toping.

40.

f(x) = a\cos(kx + \varphi) + b

funksiya grafigi berilgan.

\frac{|a| + |b|}{|k|}

ni toping.

b) Bo'yalgan soha yuzini toping (

\pi = 3

deb olinsin).

41.

ABC

uchburchakning

B

uchidan

AC

asosiga

BP

BE

kesmalar o'tkazilgan bo'lib, bu kesmalar

GL

o'rta chizig'ini mos ravishda

H

F

nuqtalarda kesib o'tadi.

BPE

uchburchakning yuzi esa 104 ga teng. Agar

\frac{|AP| + |EC|}{AC} = \frac{2}{3}

\frac{S_{BEC}}{S_{ABP}} = \frac{4}{9}

tengliklar o'rinli bo'lsa:

HGB

uchburchakning yuzini toping.

HLCP

to'rtburchakning yuzini toping.

42.

Aylanaga

ABCD

to'rtburchak tashqi chizilgan.

BC

tomonning o'rtasidan

E

nuqta,

AD

tomonning o'rtasidan esa

F

nuqta olingan.

EF

to'g'ri chiziq hamda

AC

BD

diagonallar

P

nuqtada kesishadi va

PF = 3EP

. Agar

AB = 6

AC = 2\sqrt{13}

bo'lsa:

\cos(\angle ADC)

ni toping.

b) Aylana radiusini toping.

43.

Muntazam oltiburchak ichidan

K

nuqta olingan va

B, D, F

uchlarigacha bo'lgan masofalar mos ravishda 11, 10, 5 ga teng bo'lsa,

a) Muntazam oltiburchak tomonini toping.

AK

kesma uzunligini toping.

44.

Asosining tomonlari 3 va 4 ga teng bo'lgan muntazam uchburchakli kesik piramidaga shar ichki chizilgan.

a) Shar radiusini toping.

b) Kesik piramida hajmini toping.

45.

Tomoni 6 m ga teng bo'lgan

ABCD

kvadrat berilgan. Kvadrat ichidan olingan ixtiyoriy

O

nuqtadan

AB

tomongacha bo'lgan eng qisqa masofa 2 m ga,

BC

tomongacha bo'lgan eng qisqa masofa 3 m ga teng.

O

nuqta orqali o'tkazilgan to'g'ri chiziq kvadratdan qanday eng katta yuzali (

m^2

) uchburchak ajratadi?

O

nuqta orqali to'g'ri chiziq o'tkazilganda hosil bo'ladigan shakllarning yuzalari

S_1

S_2

(

m^2

) ga teng bo'lsa,

|S_1 - S_2|

ning eng katta qiymatini (

sm^2

) toping.

Savol	A	B	C	D	E	F
33. Hosil bo'lgan jismning asosini (doiraviy halqasini) yuzini toping.
34. Hosil bo'lgan jismning to'la sirtidan asosi yuzining ayirmasini toping.
35. Jism hajmini toping.