Matematika (01.12.2024) 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;

a \cdot b = 3^4 \cdot 5^2 \cdot 7^3 \cdot 11

bo'lsa,

EKUB(a; b)

ning eng katta qiymatini toping.

Hisoblang:

\frac{1}{30} + \frac{1}{60} + \frac{1}{100} + \frac{1}{150} + \frac{1}{210}

A

B

shahardan ikki turist

C

shaharga yo'lga chiqdi. Agar

A

B

shahardagi turistlar bir biriga qarab yursa

t_1

vaqtda uchrashadi. Ikki turist

C

shaharga

t_2

vaqtda birga yetib boradi. Turistlarning tezliklari ko'paytmasini toping.

Bir oilaning oylik maoshi 15000000. Uning 60 % i oilaviy harajatlarga sarflanadi. Qolgan pulning 30 % ini bankga omonatga qo'yadi. Oila bankga necha so'm pul qo'yadi?

8 \cdot 40^{45} \cdot 36^2 \cdot 625^{25} \cdot 216

necha xonali son?

Hisoblang:

\frac{1}{2\sqrt{6} + 5} - 5 + 2\sqrt{6}

Hisoblang:

\sqrt[3]{54 + 30\sqrt{3}} + \sqrt[3]{54 - 30\sqrt{3}}

Kamayuvchi arifmetik progressiyada

a_1 + a_3 + a_5 = 15

a_2^2 + a_3^2 + a_4^2 = 83

bo'lsa,

a_{13} = ?

6 ta haddan iborat bo'lgan geometrik progressiyada dastlabki uchta hadi yig'indisi 26 ga, oxirgi uchta hadi yig'indisi 702 ga teng bo'lsa, maxrajini toping.

10.

7^a = 6

6^b = 12

bo'lsa,

\log_{81} 98\sqrt{3}

a

b

orqali ifodalang.

11.

\left(\frac{8a^3 + b^3}{4a^2 - b^2} + b\right) : \frac{a^2}{2a - b}

ifodani soddalashtiring.

12.

\operatorname{tg}\frac{\pi}{4} + \sqrt{3}\sin\frac{7\pi}{3}

ni hisoblang.

13.

2\sin^2\left(x - \frac{\pi}{3}\right) - 5\sin\left(x - \frac{\pi}{3}\right) + 2 = 0

tenglama

[-\pi; 2\pi]

nechta yechimga ega?

14.

\frac{3^x - 3^{x-2}}{3^x - 2^x} > 1 + \left(\frac{2}{3}\right)^x

tengsizlikni yeching.

15.

\lg^2(x + 1) = \lg(x - 1) \cdot \lg(x + 1) + 2\lg^2(x - 1)

haqiqiy ildizlar ko'paytmasini toping.

16.

|x + 2| + |x + 5| = 9 + |x - 4|

tenglama nechta haqiqiy ildizga ega?

17.

a

ning qanday qiymatida

\sqrt{25 - x^2} = a

tenglama haqiqiy ildizga ega?

18.

-5\sqrt[4]{x} + \sqrt{x} + 6 \ge 0

tengsizlikni yeching.

19.

\left(2x + 1 - \frac{6}{x}\right)\left(\frac{28}{x} - 7 + \left(\sqrt{-3 - 2x}\right)^2\right) \ge 0

tengsizlikning butun yechimlari yig'indisini toping.

20.

y = x^2 - \sqrt{9 + x^2}

funksiyaning eng kichik butun qiymatini toping.

21.

f(x) = |x^2 + 3x + 2|

g(x) = x + 2

funksiyalar bilan chegaralangan sohalar yuzini toping.

22.

f(x) = 3^x \log_3(3x)

bo'lsa,

f'(1)

ni toping.

23.

Rasmda

O

markazli aylana bilan

ABCD

kvadrat tasvirlangan. Agar

K

L

nuqtalar aylana va kvadrat kesishgan nuqtalar va

OD = 4

DL = 2

BC = 6

bo'lsa,

KL

kichik yoyning uzunligini toping.

24.

h(x) = kx + l

f(x) = mx + n

bo'lsa,

\frac{k + l}{m + n} = ?

25.

ABC

uchburchakda

AB = 5

BC = 4

AC = 3

ga teng.

C

burchak uchidan

AB

tomonga

CH

balandlik va

CL

bissektrisa tushirilgan bo'lsa,

HL

ni toping.

26.

ABC

to'g'ri burchakli uchburchakka aylana shunday chizilganki, aylana

AB

katetga urinib shu katet qarshisidagi burchakdan o'tadi. Aylana markazi

O

nuqta

BC

gipotenuzada joylashgan, aylana gipotenuzani

E

nuqtada kesib o'tadi. Agar

BE = 1

OC = 4

\angle BCD = \angle ACD

bo'lsa,

O

nuqtadan

DC

tomongacha bo'lgan masofani toping.

27.

ABCD

to'g'ri burchakli trapetsiya kichik asosi katta yon tomoniga teng, balandligi 4 ga teng va katta diagonali

4\sqrt{5}

ga teng bo'lsa, trapetsiya yuzini toping.

28.

Tomonlari soni

n

juft son bo'lgan muntazam ko'pburchakning bir uchidan chiqqan eng katta va eng kichik diagonallari orasidagi burchak

45^\circ

ga teng. Muntazam ko'pburchak ichki burchaklar yig'indisini toping.

29.

Tekislikda yotmagan

A

nuqta orqali tekislikka

AH

balandlik va

AM

og'ma tushirilgan. Agar og'ma va tekislik orasidagi burchak

37^\circ

AH = 6

bo'lsa,

H

uchidan

AM

og'magacha bo'lgan eng qisqa masofani toping. (

\cos 37^\circ = 0{,}8

deb oling)

30.

ABC

uchburchak yuzasi 60 ga teng.

\overrightarrow{BE} = -\frac{1}{2}\overrightarrow{BA}

bo'lsa,

ACE

uchburchak yuzini toping.

31.

Bo'sh to'plamlar sonini toping.

R

— haqiqiy sonlar to'plami,

I

— irratsional sonlar to'plami.

1)

\{x \mid x^2 - \sqrt{10} + \pi = 0; x \in R\}

\{x \mid x^2 + 1 = 0; x \in R\}

\{x \mid |x| = -3; x \in R\}

\{x \mid x^2 + 3x + 2 = 0; x \in I\}

32.

To'rt xonali sonlar ichidan raqamlari toq sonlardan iborat bo'lgan va 5 ga karrali son bo'lish ehtimolligini toping.

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

33-35.

O(0; 0; 0)

;

A(0; 0; 3)

B(\sqrt{2}; \sqrt{2}; 0)

bo'lib,

D

C

nuqtalar mos ravishda

OX

OY

o'qlarida joylashgan.

OB

kesma

OCD

uchburchakning balandligi va medianasi bo'ladi. (

\pi \approx 3

deb hisoblang)

Javob variantlari:

Savol	A	B	C	D	E	F
33. $AOCD$ piramida hajmini toping.
34. $AOB$ uchburchakning $AO$ tomoni atrofida $360^\circ$ ga aylantirishdan hosil bo'lgan jism hajmini toping.
35. $OCD$ uchburchakning $CD$ tomon atrofida $360^\circ$ ga aylantirishdan hosil bo'lgan jism hajmini toping.

36.

a, b, c

sonlar

x^3 - 9x^2 + 11x - 1 = 0

tenglamaning haqiqiy ildizlari.

t = \sqrt{a} + \sqrt{b} + \sqrt{c}

bo'lsa,

a + b + c + abc + ab + bc + ac

ni toping.

t^4 - 18t^2 - 8t

ni hisoblang.

37.

Quyidagi tenglama berilgan:

\sqrt{7 + 8\cos^2\frac{\pi x}{18}} = 4\sin^2\frac{\pi x}{18}

[0; 18]

oraliqdagi yechimlar yig'indisini toping.

[20; 30]

oraliqdagi ildizlari sonini toping.

38.

Agar

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

berilgan bo'lsa, (

f^{-1}(x)

—

f(x)

funksiyaga teskari funksiya)

f^{-1}(2)

ni hisoblang.

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

tenglamaning ildizi bo'la olmaydigan nechta butun bor?

39.

To'g'ri burchakli trapetsiyaning ichki sohasiga eng katta yuzali to'g'ri to'rtburchak chizilgan. To'g'ri to'rtburchakning bir tomoni trapetsiyaning katta asosida yotadi. Trapetsiyaning asoslari 6 va 16 ga teng. Trapetsiya balandligi 15 ga teng.

a) To'g'ri to'rtburchak yuzini toping.

b) Trapetsiya yuzidan to'g'ri to'rtburchak yuzalar ayirmasining modulini toping.

40.

f(x)

funksiya uchun

f''(x) = 12x^2 + 6x

f'(1) = 9

f(1) = 5

tengliklar o'rinli bo'lsa,

f(0)

ni hisoblang.

f(2024) = 2

f(2023) = 3

bo'lsa,

\int_{2023}^{2024}(f(x) + xf'(x))\,dx

ni hisoblang.

41.

ABC

to'g'ri burchakli uchburchakning

AC

BC

katetlarida mos ravishda

D

E

nuqtalar shunday tanlab olinganki

AD = BC

CD = BE

tengliklar o'rinli bo'lsa,

\angle AOD

burchakni toping.

BC = 10

AE = 2\sqrt{58}

bo'lsa,

S_{ABC}

ni hisoblang.

42.

ABCD

parallelogramning diagonallarida olingan

A_1

B_1

C_1

D_1

E

nuqtalar uchun,

A_1E \parallel AD

C_1E \parallel CD

bo'lib,

AC : A_1C_1 = 5:3

BD : B_1D_1 = 3:2

nisbatlar o'rinli bo'lsa,

\frac{S_{A_1C_1E}}{S_{ACD}}

ni toping.

\frac{S_{A_1B_1C_1D_1}}{S_{ABCD}}

ni toping.

43.

Rasmda

ABCD

kvadrat va markazi

B

nuqtada va radiusi

r

ga teng aylana berilgan. (

\pi \approx 3

deb oling)

a) Kvadrat ichidagi bo'yalgan sohaning bo'yalmagan soha yuziga nisbatini toping.

b) Agar

r = 2

bo'lsa, bo'yalgan soha yuzini toping.

44.

Sharga shar hajmining

\frac{1}{4}

qismini tashkil qiluvchi konus ichki chizilgan. (

\pi \approx 3

deb oling)

H_{konus} = \sqrt[3]{2 + \sqrt{5}}

bo'lsa, shar hajmini toping. (

H_{konus} > R_{shar}

)

R_{shar} = H_{konus} = \sqrt{3 + 2\sqrt{2}}

bo'lsa, konusga ichki chizilgan sfera sirtining yuzini toping.

45.

Sirk karkasining eng baland nuqtasidan yergacha bo'lgan masofa 8 ga va kengligi 20 ga teng bo'lgan parabola shaklida. Karkas qulab ketmasligi uchun balandligi 6 ga teng nuqtasida yerga tros tortilgan.

\operatorname{tg}\alpha

ni toping.

OD

ni toping.