Matematika (27.12.2025)

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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;

88, 95, 96

a

sonlarining o'rta arifmetigi 94 ga teng bo'lsa,

a

ning qiymatini toping.

Quyidagilardan nechtasi irratsional son:

1)

\pi + \sqrt{10}

\frac{22}{7}

\sqrt[3]{2 - \sqrt{5}} + \sqrt[3]{2 + \sqrt{5}}

3{,}2(525)

\sqrt{4 - 2\sqrt{3}} + \sqrt{4 + 2\sqrt{3}}

Bir ishni bajarishda Anvar Sherzodga qaraganda 5 kun ko'p ishlaydi, Jasur esa Sherzodga qaraganda 4 kun kam ishlaydi. Agar uchchalasi butun ishni 3 kunda bajargan bo'lsa, Sherzod bir o'zi ishlasa ishni necha kunda bajaradi?

Mahsulotning dastlabki narxi 400 000 so'm edi. Avval mahsulot narxi

30\%

ga kamaytirildi, shundan so'ng hosil bo'lgan narx

10\%

ga oshirildi. Mahsulotning oxirgi narxini toping.

Hisoblang:

\frac{1^3 + 2^3 + \ldots + 9^3}{2025}

Hisoblang:

\sqrt{\frac{11^4 + 100^4 + 111^4}{2}}

Agar

\left(2x^3 + x^2 + 1\right)^5 \cdot \left(x^2 - 2\right)^3

ko'paytmani soddalashtirgandan keyin

x^2

li hadining oldidagi koeffitsiyentini toping.

Arifmetik progressiyada

a_2 + a_4 = 16

a_1 \cdot a_5 = 28

bo'lsa,

a_2

ni toping.

O'suvchi geometrik progressiyada

b_1 + b_2 + b_3 = 91

b_3 - b_2 = 42

bo'lsa,

S_2

ni toping.

10.

Agar

x + \frac{1}{x} = 6

bo'lsa,

\frac{x^4 + 1}{4x^2}

ning qiymatini toping.

11.

Soddalashtiring:

\sqrt[6]{2x\left(7 + 3\sqrt{5}\right)} \cdot \sqrt[3]{\sqrt{5x} - 3\sqrt{x}}

12.

Soddalashtiring:

\frac{2\sin 20^\circ + \sin 40^\circ}{2\sin 20^\circ - \sin 40^\circ}

13.

\begin{cases} x + y = \frac{\pi}{3} \\ \sin x \cdot \sin y = \frac{1}{4} \end{cases}

tenglamalar sistemasi

[-2\pi; 2\pi]

oraliqda nechta ildizga ega?

14.

8 - 2^x \cdot x + 2^{3-x} = x

tenglamaning nechta haqiqiy ildizi bor?

15.

Tenglamaning ildizlari yig'indisini toping.

\log_2^2 x + (x - 1)\log_2 x = 6 - 2x

16.

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

tenglamalar sistemasi berilgan bo'lsa,

x + y

ning qiymatini toping.

17.

x^3 - (x + 1)^2 = 2001

tenglama nechta haqiqiy ildizga ega?

18.

Tengsizlikni qanoatlantiruvchi butun sonlar nechta?

x^2 - 8x - \frac{3x - 12}{|x - 4|} + 18 \le 0

19.

Tengsizlikni nechta butun son qanoatlantiradi?

\sqrt{x - 8\sqrt{8 + x} + 24} - \frac{16 + x}{2\sqrt{8 + x} - 10} < 0

20.

f(x) = 2x^2 + (a - 10)x + 6

funksiya nollaridan biri

x_1

12x_1

bo'lsa, shunday

a

larning yig'indisini toping.

21.

Agar

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

bo'lsa,

f^{-1}(2)

ning qiymatini toping. (

f^{-1}(x)

—

f(x)

funksiyaning teskari funksiyasi).

22.

f(x) = \sqrt{\frac{7}{2}x^2 - 7}

funksiyaning grafigiga o'tkazilgan urinma

y_1 = ax + b

bo'lsin. Agar bu urinma bilan

y_2 = -\frac{1}{2}x + \frac{3}{4}

to'g'ri chiziqning grafiklari o'zaro perpendikulyar bo'lsa,

a + b

ning qiymatini toping.

23.

Aniq integralni hisoblang:

\int_0^1 \frac{6x^2}{\left(4 - x^3\right)^2}\,dx

24.

Aylanada o'zaro teng va perpendikulyar vatarlar o'tkazilgan bo'lib, ular aylanadagi

A

nuqtada umumiy nuqtaga ega. Agar aylana markazidan vatarlargacha bo'lgan eng qisqa masofalar 8 sm bo'lsa, vatar uzunligini toping.

25.

Rasmdagi ma'lumotlar asosida

x

ning uzunligini toping.

26.

ABC

uchburchakning

BC

tomonidan

D

nuqta olingan.

AD

kesma

ABC

uchburchakni ikkita

ABD

ADC

uchburchaklarga ajratadi.

ABD

ADC

uchburchaklarga ichki chizilgan aylanalar

AD

kesmaga mos ravishda

E

F

nuqtalarda urinadi. Agar

AB = 5

AC = 7

BD = 5

DC = 3

bo'lsa,

EF

kesma uzunligini toping.

27.

ABCD

trapetsiyaning

AB

CD

asoslari mos ravishda 4 va 12 ga teng. Trapetsiyaning

AB

CD

asoslariga parallel

EF

kesma o'tkazilgan.

EF

kesma

BD

diagonal bilan

O

nuqtada kesishadi. Agar

EOD

BOF

uchburchaklar yuzlari teng bo'lsa,

EF

kesma uzunligini 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.

EH

ning uzunligini toping.

29.

Koordinatalar sistemasida

A(-5; 4)

B(14; 5)

C(3; -3)

nuqtalar berilgan. Shu nuqtalarni tutashtirib

ABC

uchburchak hosil qilingan. Koordinata boshidan

ABC

uchburchakning og'irlik markazigacha bo'lgan masofani toping.

30.

\alpha

\beta

tekisliklar o'zaro parallel.

\alpha

tekislikda

A, B

nuqta 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.

Agar

A = \{a; b; c; d\}

B = \{a; b; m; n; k\}

bo'lsa,

A \cup B

ning qism to'plamlar sonini toping.

32.

Barcha raqamlari toq raqamlardan iborat bo'lgan uch xonali sonlar

n

ta, barcha raqamlari juft raqamlardan iborat bo'lgan to'rt xonali sonlar

m

ta bo'lsa,

\frac{n}{m}

ning qiymatini toping.

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

33-35.

ABCDA_1B_1C_1D_1

kubga markazi

O

nuqtada bo'lgan shar ichki chizilgan bo'lib, shar kubning uchta

ABCD

AA_1B_1B

BB_1C_1C

yoqlariga va

DD_1

qirrasiga urinadi. Kubning qirrasi uzunligi

\sqrt{2}

ga teng.

Javob variantlari:

\sqrt{2} - 1

2\sqrt{2} - 2

4\sqrt{2} - 4

2 - \sqrt{2}

2\sqrt{2} - 1

4 - 2\sqrt{2}

36.

\begin{cases} x^2 + y^2 = 4 - 2ay - a^2 \\ y - x^2 = 0 \end{cases}

tenglamalar sistemasi berilgan.

a) Tenglama yagona ildizga ega bo'ladigan

a

ning butun qiymatlari yig'indisini (agar u yagona bo'lsa, shu sonni) toping.

b) Tenglama ikkita har xil ildizga ega bo'ladigan

a

ning eng kichik qiymatini toping.

37.

Tenglamani yeching:

\cos x + \cos 2x + \cos 3x = \sin x + \sin 2x + \sin 3x

a) Tenglamaning eng kichik musbat yechimini toping.

[-\pi; \pi]

oraliqda tenglama nechta ildizga ega?

38.

f(x) = (x + a)^2 + b

x \in [-a; \infty)

berilgan.

g(x)

funksiya

f(x)

funksiyaning o'suvchi tarmog'iga teskari,

f(x)

g(x)

funksiyalar

(0; 3)

nuqtada kesishadi.

a + b

ning qiymatini toping.

f(x)

g(x)

funksiyalar nechta nuqtada kesishadi?

39.

Rasmda

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

parabola uchining koordinatasi

B(2; -3)

berilgan. Parabola

Oy

o'qini

(0; -2)

nuqtada kesib o'tadi.

\frac{b \cdot c}{a}

ning qiymatini toping.

A(4; 2)

nuqtadan parabolagacha bo'lgan eng qisqa masofani toping.

40.

Chizmada

y = ax^2 + bx + c

y = -|x + m| + n

funksiya grafiklari tasvirlangan.

\frac{a + b - c}{n - m}

ning qiymatini toping.

b) Bo'yalgan soha yuzini toping.

41.

ABC

uchburchakka ichki aylana chizilgan. Bunda

D

nuqta aylanaga urinish nuqtasi va

AB = 7

DC = 3

\angle C = 60^\circ

ga teng.

a) Aylana radiusini toping.

ABC

uchburchakning yuzini toping.

42.

ABCD

to'g'ri to'rtburchakning

AD

BC

tomonlaridan mos ravishda

E

F

nuqtalar olingan.

AE

BF

chiziqlar

M

nuqtada,

CF

DE

chiziqlar esa

N

nuqtada kesishadi.

AMF

BME

CEN

DNF

uchburchaklarning yuzalari mos ravishda 32, 8, 27, 3 ga teng bo'lsa,

a) Bo'yalgan soha yuzini toping.

ABCD

to'g'ri to'rtburchak yuzini toping.

43.

ACDE

to'g'ri to'rtburchak ichida muntazam oltiburchak berilgan.

AB_1B_6

uchburchakning yuzi 7 ga,

EB_6B_5B_4

to'rtburchakning yuzi 9 ga teng bo'lsa,

a) Bo'yalgan soha yuzini toping.

ACDE

to'g'ri to'rtburchakning yuzini toping.

44.

Asosi trapetsiyadan iborat piramida berilgan. Trapetsiyaning kichik asosi yon tomonlariga teng. Katta asosi 12 sm va o'tmas burchagi

120^\circ

ga teng. Piramidaning barcha yon qirralari asos tekisligi bilan

60^\circ

li burchak tashkil qiladi.

a) Piramidaning to'la sirti yuzini toping.

b) Piramidaning hajmini toping.

45.

Uzunligi 12 sm lik sim berilgan, shu simdan yuzasi eng katta shakl yasaldi. Masalani yechishda

\pi \approx 3

deb olinsin.

a) Shakl yuzini toping.

b) Shu shaklga uchburchak ichki chizildi, uchburchak uchlari shaklni uzunliklari 3, 4 va 5 bo'lgan qismlarga ajratdi. Uchburchak yuzasini toping.

Savol	A	B	C	D	E	F
33. Sharning radiusini toping.
34. $O$ nuqtadan $AA_1D_1D$ yoqqacha bo'lgan eng qisqa masofani toping.
35. $O$ nuqtadan $D$ nuqtagacha bo'lgan masofani toping.