Matematika (28.02.2026) 1-smena

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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;

\underbrace{666\ldots66}_{50\text{ ta}} + \underbrace{555\ldots55}_{50\text{ ta}}

ni hisoblang.

Hisoblang.

\left(2{,}28 - 4\frac{7}{20}\right) : 2\frac{1}{4} - \left(\frac{1}{2} - 1\frac{7}{8}\right)

Oralaridagi masofa 275 km bo'lgan

A

B

shaharlaridan bir vaqtning o'zida bir-biriga qarab motosiklchi va velosipedchi yo'lga chiqdi. Velosipedchi yo'lda 15 daqiqa dam olib motosiklchi tomon yo'lni davom ettirdi. Motosiklchi esa soatiga 50 km tezlik bilan, velosipedchi esa soatiga 20 km tezlik bilan harakatlandi. Ular uchrashgan paytda motosiklchining bosib o'tgan masofasini (km) toping.

To'g'ri burchakli uchburchakning gipotenuzasi

3\sqrt{5}

ga teng. Agar uning bir kateti

133\frac{1}{3}\%

ga, ikkinchi kateti esa

16\frac{2}{3}\%

ga oshirilsa, hosil bo'lgan katetlar yig'indisi 14 ga teng bo'ladi. Uchburchakning yuzini toping.

Hisoblang.

\left(\frac{2}{3}\right)^{-3} \cdot (1{,}875)^{-1}

Hisoblang.

\sqrt{\frac{666666^2}{12345654321}}

Hisoblang.

\sqrt[6]{7 - \sqrt{40}} \cdot \sqrt[3]{\sqrt{5} + \sqrt{2}} \cdot \sqrt[3]{9}

Arifmetik progressiyada

a_3 = a_1 + a_2

a_1 \cdot a_2 \cdot a_3 = 384

bo'lsa,

S_{10}

ni toping.

Kamayuvchi geometrik progressiyada

b_1 + b_2 + b_3 = 19

b_1^2 + b_2^2 + b_3^2 = 133

bo'lsa, dastlabki beshta hadi yig'indisini hisoblang.

10.

Agar

\frac{x}{y} = 4

y \ne -\frac{3}{5}

bo'lsa,

\frac{x + y + 3}{x + 6y + 6}

ning qiymatini toping.

11.

Soddalashtiring.

\frac{\left(\frac{1}{x} + \frac{1}{y} + \frac{1}{xy}\right)(x + y - 1)}{\frac{1}{x^2} + \frac{1}{y^2} + \frac{2}{xy} - \frac{1}{x^2y^2}}

12.

Agar

\cos\alpha = \frac{3}{5}

\left(0 < \alpha < \frac{\pi}{2}\right)

a = \sin\alpha

b = \operatorname{tg}\alpha

c = \operatorname{ctg}\alpha

bo'lsa,

a

b

c

larni o'sish tartibida joylashtiring.

13.

Agar

\cos\alpha + \sqrt{3}\cos\beta = 0

2\alpha + \beta = 180^\circ

bo'lsa,

|\alpha - \beta|

ning qiymatini toping.

14.

Tenglamaning haqiqiy ildizlari yig'indisini hisoblang.

4^{x^2 - x} - 17 \cdot 2^{x^2 - x + 2} + 256 = 0

15.

Tenglamaning haqiqiy ildizlar ko'paytmasini toping. (Agar u bitta bo'lsa, shu ildizini toping.)

\frac{2}{3} - \log_x 3 = \log_3 2 \cdot \log_{32} x - \frac{2}{5}\log_{\sqrt{x}} 3

16.

Tenglama nechta butun yechimga ega?

x^4 - \frac{2027!}{2026! + 2025!}x^2 + 2025 = 0

17.

Tenglama nechta butun yechimga ega?

x^4 - \sqrt{|x| - 45} = 2025^2

18.

Tengsizlik nechta butun yechimga ega?

\frac{x^2 - 7|x| + 10}{x^2 - 6x + 9} < 0

19.

Tengsizlikni qanoatlantiruvchi butun sonlar nechta?

\sqrt{x^2 - 4x + 4} - 5 \cdot \sqrt{2 - x} - 6 < 0

20.

Rasmda

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

funksiya grafigi tasvirlangan. Parabola uchining koordinatalari

(3; 5)

Oy

o'qini

\left(0; 3\frac{1}{5}\right)

nuqtada kesib o'tadi.

f(18)

ning qiymatini toping.

21.

Integralni hisoblang.

\int_{-1}^{1} \frac{1}{x^4 + x^2}\,dx

22.

Agar

3(3x + 4)^{20} = C_0x^{20} + C_1x^{19} + C_2x^{18} + \ldots + C_{19}x + C_{20}

tenglik o'rinli bo'lsa,

20 \cdot 19 \cdot C_0 - 19 \cdot 18 \cdot C_1 + 18 \cdot 17 \cdot C_2 - \ldots - 3 \cdot 2 \cdot C_{17} + C_{18}

ni hisoblang.

23.

Aylananing vatari 8 ga teng, u

90^\circ

li yoyni tortib tursa, aylana radiusini toping.

24.

Quyidagi grafiklardan nechtasi funksiya bo'ladi?

25.

ABC

to'g'ri burchakli uchburchakning

B

to'g'ri burchak bissektrisasi

AC

gipotenuzani

\sqrt{5}

2\sqrt{5}

kesmalarga ajratsa,

ABC

to'g'ri burchakli uchburchak yuzini toping.

26.

ABC

uchburchakning

A

B

uchlaridan

AD

BE

bissektrisalar tushirildi.

BE

AD

bissektrisalar

K

nuqtada kesishadi. Agar

AB = 6

AE = 4

EC = 8

bo'lsa,

KDCE

to'rtburchak yuzini toping.

27.

ABCD

trapetsiyaning

BC

AD

asoslari mos ravishda 7 va 27 ga,

AB

CD

yon tomonlari esa mos ravishda 12 va 16 ga teng bo'lsa,

ABCD

trapetsiya yuzini toping.

28.

Muntazam

n

burchakning eng kichik diagonallari orasidagi burchak

120^\circ

ga teng bo'lsa, muntazam

n

burchakning nechta tomoni bor?

29.

ABC

uchburchak dekart koordinatalar sistemasiga joylashtirildi. Uchburchak uchlarining koordinatalari

A(1; 3)

B(5; 1)

C(4; 4)

ga teng bo'lsa, bunda

AL

bissektrisa uzunligini toping.

30.

Muntazam to'rtburchakli piramidaning to'la sirtining yuzi asosining yuzidan 3 marta katta bo'lsa, yon yog'i bilan asos tekisligi orasidagi burchak tangensini toping.

31.

A = \{60\text{ ning natural bo'luvchilari}\}

B = \{48\text{ ning natural bo'luvchilari}\}

bo'lsa,

A \cap B

ning qism to'plamlar sonini toping.

32.

Raqamlar yig'indisi 7 ga teng bo'lgan uch xonali sonlar qog'ozchalarga yozilgan. Bulardan tanlangan son juft son bo'lish ehtimolligini toping.

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

33-35.

Rasmda konus ichiga eng katta hajmli silindr ichki chizilgan.

Javob variantlari:

\frac{1}{3}

\frac{2}{3}

\frac{3}{4}

\frac{4}{9}

\frac{2}{9}

36.

Tenglamalar sistemasini yeching.

\begin{cases} 26x^2 + 42xy + 17y^2 = 10 \\ 10x^2 + 18xy + 8y^2 = 6 \end{cases}

a) Tenglamalar sistemasi yechimi

(x_1; y_1)

(x_2; y_2)

, ...,

(x_n; y_n)

bo'lsa, tenglamalar sistemasi nechta yechimga ega?

x_n + y_n

ning eng katta qiymatini toping.

37.

Tenglamani yeching.

\sin^3 x + \sin^3 2x + \sin^3 3x = (\sin x + \sin 2x + \sin 3x)^3

a) Tenglamaning eng kichik musbat ildizini toping.

b) Tenglama

[-\pi; \pi]

oraliqda nechta ildizga ega?

38.

f(x) = \frac{ax + b}{cx + d}

funksiya

Ox

o'qini

(3; 0)

nuqtada kesib o'tadi. Agar

f(x)

funksiyaning qiymatlar sohasi

E(y) = (-\infty; 2) \cup (2; \infty)

f(x) = f^{-1}(x)

bo'lsa (

f^{-1}(x)

—

f(x)

funksiyaning teskari funksiyasi),

f(x)

ning aniqlanish sohasiga tegishli bo'lmagan sonni toping.

f^{-1}(6)

ning qiymatini toping.

39.

f(x) = \frac{x^2}{8} - \frac{3x}{2}

kvadrat funksiya va

x^2 - 18x + y^2 - 12y + 97 = 0

aylana tenglamasi berilgan.

a) Aylana markazi va parabola uchi orasidagi masofani

d_1

toping.

b) Paraboladan aylanagacha eng qisqa masofa

d_2

ni toping.

40.

f(x) = -2x^2 + 4x

g(x) = x^2 + px + q

funksiyalar kesishishidan hosil bo'lgan soha yuzi

\frac{32}{27}

ga teng. Agar

x = \frac{4}{3}

to'g'ri chiziq chegaralangan sohani teng ikkiga ajratsa,

f(x)

g(x)

funksiyalar kesishish nuqtalari absissalari yig'indisini toping.

p \cdot q

ning qiymatini toping.

41.

ABC

uchburchakka aylana ichki chizilgan.

ABC

uchburchakning

A

uchidan

AD

bissektrisa o'tkazildi.

BC

tomoniga parallel va aylana markazidan o'tuvchi

EF

kesma o'tkazilgan. Agar

BC = 30

EB = 8

FC = 12

bo'lsa,

DC

kesma uzunligini toping.

AB

AC

tomonlari ayirmasi modulini toping.

42.

ABCD

kvadrat ichidan

P

nuqta olingan. Agar

AP = 1

BP = 5

PC = 7

bo'lsa,

PD

kesma uzunligini toping.

b) Kvadratning yuzini toping.

43.

Rasmda

ADFK

parallelogram va

EG : GC = 1 : 2

DE : EF = 2 : 3

bo'lsin.

\frac{EG}{BC}

ni toping.

b) Agar

S_{ABC} = 242

bo'lsa,

S_{ADFK}

ni toping.

44.

AB

diametrli radiusi 3 ga teng bo'lgan sfera berilgan.

C

D

nuqtalar sferada joylashgan.

DC

qirrasi o'rtasidan

E

nuqta olindi. Sferaga eng katta hajmli

ABCD

piramida ichki chizilgan bo'lsa,

a) Piramida hajmini toping.

AE

chiziq va asos tekisligi orasidagi burchak sinusini toping.

45.

Silindr shaklidagi yog'och hajmi

V = 162\pi\ cm^3

va yon sirti

S = 72\pi\ cm^2

ga teng. Usta uni yo'nib shar shakliga keltirdi. Sharning hajmi eng katta bo'lsin. (

\pi \approx 3

deb olinsin.)

a) Kesib olingan shar sirtining yuzini toping.

b) Silindrdan eng katta shar kesib olinganda chiqindi hajmini toping.

Savol	A	B	C	D	E	F
33. Silindr asosining radiusini konus asosi radiusiga nisbatini toping.
34. Silindr balandligini konus balandligiga nisbatini toping.
35. Silindr hajmini konus hajmiga nisbatini toping.