Matematika (30.12.2025)

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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;

\underbrace{222\ldots222}_{50\text{ ta}} \cdot 11

ni hisoblang.

Hisoblang:

\frac{4 - \frac{1}{\frac{1}{3} + \frac{1}{4}}}{4 + \frac{1}{\frac{1}{3} - \frac{1}{4}}}

Fermer bozorga qovun va tarvuzni 5 ta mashina bilan olib borayotgan bo'lsa va bitta mashinada faqat tarvuz yoki qovun bo'lsa, birinchi mashinada 300, ikkinchi mashinada 315, uchinchisida 330, to'rtinchi mashinada 345, beshinchisida 390 ta mahsulot bor. Bozorkom bitta mashinani sotib bo'lmaguncha qolganlarini sotishga ruxsat bermasdi. Bitta mashina sotib bo'lingandan so'ng qovunlar soni tarvuzlar sonidan 3 marta ko'pligi aniqlandi. Qovunlarning soni nechta?

O'rik tarkibida

92\%

suv bor. O'rik quritilganda qoqining tarkibida

16\%

suv bor. 42 kg o'rik quritilganda necha kg qoqi olinadi?

Ifoda ratsional bo'ladigan

n

ning eng kichik natural qiymatini toping:

\left(\frac{2^{\frac{2}{3}} \cdot 4^{\frac{1}{3}} \cdot 3^2}{8^{\frac{3}{4}} \cdot 3^{\frac{1}{2}} \cdot 9^{\frac{1}{3}}}\right)^n

Hisoblang:

\left(52 + 6\sqrt{43}\right)^{\frac{3}{2}} - \left(52 - 6\sqrt{43}\right)^{\frac{3}{2}}

Hisoblang:

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

O'suvchi arifmetik progressiyada

S_8 + S_7 = 26

S_8 \cdot S_7 = 144

bo'lsa,

S_{15}

ni toping.

x, y, z

sonlari kamayuvchi geometrik progressiyaning ketma-ket hadlari.

x

2y

3z

sonlari arifmetik progressiyaning ketma-ket hadlari bo'lsa, geometrik progressiyaning maxrajini toping.

10.

Agar

a > b > 0 > c

bo'lsa,

\frac{|ab - c|}{ab - c} + \frac{|ac - b|}{ac - b} + \frac{|bc - a|}{bc - a}

ni soddalashtiring.

11.

Ifodani soddalashtiring:

\frac{(x - 2)^4 + (x - 1)^2 - 1}{(x - 2)(x - 3) + 2}

12.

Soddalashtiring:

\cos 2\alpha + \sin 2\alpha \cdot \operatorname{tg}\alpha

13.

Hisoblang:

\cos\frac{2\pi}{7} + \cos\frac{4\pi}{7} + \cos\frac{6\pi}{7}

14.

Tenglamaning nechta haqiqiy ildiziga ega?

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

15.

Hisoblang:

\frac{3}{7}\left(\log_2 32 + 27\log_{\frac{1}{3}} 4\right) + \log_3\frac{81}{\sqrt{7} - \sqrt{5}} + \log_3\sqrt{12 - 2\sqrt{35}}

16.

Tenglama nechta butun yechimga ega?

\frac{x^3}{|x|} + 10x - \frac{11x}{|x|} = 0

17.

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

tenglama nechta haqiqiy yechimga ega?

18.

Tengsizlikni qanoatlantiruvchi butun sonlar nechta?

\left||x + 1| - 3\right| \le 6

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 nollari

x_1

12x_1

bo'lsa, shunday

a

larning yig'indisini toping.

21.

Agar

f(x + 1) = x^2 + 2x + 2

bo'lsa,

f(-3) + f(1)

ning qiymatini toping.

22.

y = \sqrt{\frac{7}{2}x^2 - 7}

funksiya grafigiga o'tkazilgan urinma

y = ax + b

bo'lsin. Agar

y = -\frac{1}{2}x + 3

to'g'ri chiziq grafigi

y = ax + b

funksiya bilan o'zaro perpendikulyar bo'lsa,

a + b

ning qiymatini toping.

23.

Aniq integralni hisoblang:

\int_0^{\frac{\pi}{12}} \left(\sin^4 3x + \cos^4 3x\right) dx

24.

Aylanadagi

A

nuqtadan uzunliklari

\sqrt{2 + \sqrt{3}}

\sqrt{3}

teng bo'lgan

AB

AC

vatarlar o'tkazilgan. Agar

AB

AC

vatarlar orasidagi burchak

45^\circ

ga teng bo'lsa, aylana markazidan

AC

vatargacha bo'lgan masofani toping.

25.

ABC

to'g'ri burchakli uchburchakning gipotenuzasiga tushgan

CM

mediana uzunligi katetlaridan birining uzunligiga teng. Agar

CM = 2

bo'lsa,

ABC

uchburchak yuzini toping.

26.

ABC

to'g'ri burchakli uchburchakka to'g'ri burchagi umumiy bo'lgan

ADEF

kvadrat ichki chizildi. Agar

BE = 10

EC = 15

bo'lsa,

ADEF

kvadratning yuzini toping.

27.

Chizmada

ABCD

— parallelogramm tasvirlangan. Agar

\angle BAD = 60^\circ

\left(\frac{|AC|}{|BD|}\right)^2 = \frac{19}{7}

AD > AB

bo'lsa,

AD : AB

ning qiymatini toping.

28.

Muntazam o'n ikkiburchakning tomoni 2 ga teng bo'lsa, bo'yalgan soha yuzini toping.

29.

ABCD

parallelogrammning

B

o'tmas burchagi

120^\circ

ga teng.

BC

tomondan

E

nuqta olingan. Agar

\left|\overrightarrow{AE} + \overrightarrow{BD} + \overrightarrow{EC}\right| = 16

\left|\overrightarrow{AE} - \overrightarrow{BD} + \overrightarrow{EC}\right| = 10

bo'lsa,

AED

uchburchakning yuzini toping.

30.

Tekislikda yotmagan nuqtadan tekislikka og'ma va perpendikulyar tushirilgan. Og'ma va tekislik orasidagi burchak

\arccos 0{,}28

bo'lsa, og'maning tekislikdagi proyeksiyasi 21 bo'lsa, perpendikulyar uzunligini toping.

31.

Agar

A = \{(x, y) \mid x^2 + y^2 = 9;\ x, y \in R\}

B = \{(x, y) \mid x + y = 3;\ x, y \in R\}

bo'lsa,

A \cap B

ni toping.

32.

Binomial yoyilmaning koeffitsiyentlar yig'indisini toping:

C_{99}^1 + C_{99}^3 + C_{99}^5 + \ldots + C_{99}^{97}

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

33-35.

SABCD

muntazam to'rtburchakli piramidaning yon yoqlari asos tekisligi bilan

30^\circ

li burchak tashkil etadi. Bu piramidaga asosi piramida asosida va yon yoqlariga urinuvchi yarim shar ichki chizilgan. Piramida asosi tomoni bilan yarim shar diametri ayirmasi 1 ga teng (

\pi \approx 3

deb hisoblang).

Javob variantlari:

\frac{\sqrt{2}}{2}

\frac{\sqrt{3}}{4}

\frac{4\sqrt{3}}{9}

\frac{2}{3}

\frac{3}{2}

\frac{1}{2}

36.

Tenglamani yeching:

x^4 - 12x^2 + 16\sqrt{2}x - 12 = 0

a) Tenglama nechta haqiqiy ildizga ega?

b) Tenglamaning haqiqiy ildizlari ko'paytmasini toping.

37.

Tenglama berilgan:

\sin\left(\frac{\pi}{2} - x\right) + \cos 3x = \cos 4x + 1

a) Tenglamaning eng kichik musbat ildizini toping.

[0; \pi]

oraliqdagi ildizlari sonini toping.

38.

Quyidagi

f(x)

g(x)

funksiyalar uchun

f(x) = x^2 + 12x + 30

g(x) = f(f(f(f(f(x)))))

o'rinli bo'lsa,

g(x) = 0

tenglama nechta haqiqiy ildizga ega?

g(-5)

ning qiymatini toping.

39.

f(x) = -x^4 - 2x^3 + 3x^2 + \frac{14x}{3} + 2

funksiya ikkita lokal maksimum, bitta lokal minimum nuqtaga ega.

g(x) = kx + l

funksiya

f(x)

ga ikkita nuqtada urinadi.

k + l

ning qiymatini toping.

f(x)

g(x)

funksiyalarning urinish nuqtalari orasidagi masofani toping.

40.

Rasmda

y = a\cos(kx) + b

funksiya grafigi tasvirlangan.

\frac{a + b}{k}

ning qiymatini toping.

[2\pi; 6\pi]

oraliqda funksiyani

Ox

o'qi atrofida

360^\circ

ga aylantirishdan hosil bo'lgan jism hajmini toping.

41.

ABC

muntazam uchburchak ichidan

P

nuqta olinib, uchburchak uchlari bilan tutashtirildi.

ACP

uchburchakka markazi

O

nuqta bo'lgan aylana ichki chizilgan. Agar

AP = 3

PB = 5

PC = \sqrt{19}

bo'lsa,

ABC

uchburchak perimetrini toping.

P

nuqtadan aylana markazigacha bo'lgan masofani toping.

42.

Markazlari

O_1

O_2

O_3

O_4

bo'lgan teng radiusli aylanalar

ABCD

rombga rasmdagi ko'rinishda ichki chizildi (

O_1

aylana

AB

AD

ga,

O_2

aylana

AD

BD

ga urinadi,

O_1

O_2

aylanalar o'zaro urinadi;

O_3

O_4

— simmetrik). Agar

BD = 12

AD = 10

bo'lsa,

O_1

O_3

aylana markazlari orasidagi masofani toping.

O_1O_2O_3O_4

to'rtburchak yuzini toping.

43.

Aylanaga tomonlari soni juft bo'lgan muntazam ko'pburchak ichki chizilgan. Ko'pburchakning bir uchidan chiqqan eng katta va eng kichik diagonallari orasidagi burchak

60^\circ

ga teng bo'lsa,

a) Ko'pburchakning diagonallari sonini toping.

b) Aylana radiusi 6 ga teng bo'lsa, muntazam ko'pburchak yuzini toping.

44.

ABCD

muntazam tetraedrning ichki sohasiga uchi

AC

tomonning o'rtasida bo'lgan konus chizilgan. Konusning asosi tetraedrning

BC

tomoni o'rtasidan o'tuvchi hamda

BD

AC

qirralariga parallel bo'lgan kesimga ichki chizilgan. Agar konusning yon sirti

\sqrt{3}\pi

ga teng bo'lsa (

\pi \approx 3

deb oling),

a) Konus hajmini toping.

b) Tetraedr hajmini toping.

45.

Uyning tom qismi asosining tomonlari 8 m va 5 m va ular orasidagi burchak

60^\circ

ga teng bo'lgan uchburchakli to'g'ri prizmadan iborat (rasmga qarang). Tomning ikki yon yog'i tunukadan iborat. Uyning pastki qismi parallelepipeddan iborat bo'lib, uning ikki yog'i kvadrat, qolgan ikki yog'i esa to'g'ri to'rtburchakdan iborat.

a) Uyning tom qismini qoplashga necha

m^2

tunuka ishlatilgan?

b) Uyning hajmini toping (

m^3

Savol	A	B	C	D	E	F
33. Yarim shar radiusini toping.
34. Yarim sharning piramida yon yog'iga urinish nuqtasidan piramida asosigacha bo'lgan masofani toping.
35. Piramida hajmini toping.