Matematika (23.02.2025) 1-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, b, c, d, f

turli raqamlar bo'lib,

\overline{aaaa} - \overline{bbb} + \overline{cc} + d + f = 5000

bo'lsa,

a \cdot b - d \cdot f + c

ning qiymatini toping.

Hisoblang:

\left(4{,}(6) \cdot 2\frac{4}{7} - 3\right) \cdot 1{,}0(5) - \frac{1}{0{,}4}

Basseynda 4 ta quvur bor. Birinchi va ikkinchi quvur basseyni to'ldiriladi. Uchinchi va to'rtinchi quvur basseynni bo'shatadi. 4 ta quvur birdaniga ochilsa basseyn 5 soatda to'ladi. Faqat birinchi, ikkinchi va uchinchi quvurlar ochilsa 3 soatda to'ladi. Faqat birinchi, uchinchi va to'rtinchi quvurlar ochilsa 30 soatda to'ladi. Faqat birinchi va uchinchi quvurlar ochilsa basseyn necha soatda to'ladi?

Ko'ylakning narxi 100000 so'm edi. Dastlab ko'ylak narxi 10% ga oshirildi, keyin 15% ga arzonlashtirildi. Ko'ylakning hozirgi narxini toping.

Agar

x^2 = 2048^{24}

\sqrt[3]{y} = 512^5

bo'lsa,

\frac{x}{y}

ni toping.

Soddalashtiring:

\frac{(\sqrt{11} - \sqrt{5})(\sqrt{15} + \sqrt{33} - \sqrt{22} - \sqrt{10})}{\sqrt{108} - \sqrt{72}}

Kasr maxrajini irratsionallikdan qutqaring.

\frac{1}{\sqrt{2} + \sqrt{3} - \sqrt{5}}

Arifmetik progressiyada

a_{14} = 20

S_{28} = 600

ga teng bo'lsa,

a_{28}

ni toping.

Kamayuvchi geometrik progressiyada dastlabki 3 ta hadi yig'indisi 14 ga, ularning kvadratlar yig'indisi 84 ga teng bo'lsa, progressiyaning dastlabki 6 ta hadi yig'indisini toping.

10.

Agar

|c| < |b| < |a|

a < 0

b > 0

c < 0

bo'lsa,

\frac{a|b - c|}{|a|} + \frac{b|c - a|}{|b|} + \frac{c|a - b|}{|c|}

ni hisoblang.

11.

Ifodani soddalashtiring:

\frac{-4 \cdot 4^{x-1} \cdot 3^{2x+1} - 12^{x+1} \cdot 3^{x+2}}{36^x \cdot 8 + 9^x \cdot 2^{2x+1}}

12.

P(0; 1)

nuqtani 3 radianga burish natijasida hosil bo'lgan nuqta qaysi chorakda yotadi?

13.

Ifodani soddalashtiring:

$

\frac{\sin^2 2\alpha \cdot \operatorname{tg}^2 2\alpha}{1 - \operatorname{ctg}^2 2\alpha} + \frac{\cos^2 2\alpha \cdot \operatorname{ctg}^2 2\alpha}{1 - \operatorname{tg}^2 2\alpha} + 2

14.

Tengsizlikni yeching:

3^x - 2^{x+2} + 2^{x-3} + 2^{x-1} - 3^{x-1} \ge 0

15.

Ifodaning qiymatini toping.

2023^{\log_{2023}^2 2024} - 2024^{\log_{2023} 2024} - \log_2 \log_2 \underbrace{\sqrt{\sqrt{\ldots\sqrt{2}}}}_{2023\ ta}

16.

Tenglama nechta haqiqiy ildizga ega?

\frac{(x + 2)^2 - 9}{x + 7} \cdot (x + 7) = 16

17.

Tenglamaning haqiqiy ildizlari ko'paytmasini toping.

(x - 3)^2 + 3x = \sqrt{x^2 - 3x + 7} + 22

18.

2x^2 + 2\sqrt{2}x + 1 \le 0

tengsizlik nechta yechimga ega?

19.

Tengsizlikni qanoatlantiruvchi butun sonlar yig'indisini toping.

\frac{x^2 - 5x + 64}{x^2 - 10x + 24} \le \frac{10}{4 - x}

20.

y = x^2 - (a^2 - 2a - 3)x + 6

funksiya nollari

x = 2

x = 3

ga teng bo'ladigan

a

ning qiymatlarini toping.

21.

Aniq integralni hisoblang:

\int_2^4 \frac{3}{5 - 4x - x^2}\,dx

22.

Agar

f(x) = \left|\log_{\cos x}\sin x - \ln\pi\right|

bo'lsa,

f'\left(\frac{\pi}{4}\right)

ni toping.

23.

Tashqi burchagi

45^\circ

ga teng bo'lgan muntazam ko'pburchakka aylana tashqi chizilgan. Aylana radiusi 2 ga teng bo'lsa, ko'pburchak yuzini toping.

24.

Nechta natural son

f(x) = 6\sin x - 4\cos x

funksiyaning qiymatlar sohasiga tegishli?

25.

ABC

to'g'ri burchakli uchburchak gipotenuzasi uzunligi 15 ga,

\sin\angle ACB = 0{,}6

ga teng bo'lsa, uchburchak yuzini toping.

26.

ABC

uchburchakka ichki chizilgan aylana uchburchakning

AB

BC

AC

tomonlariga mos ravishda

F, D

E

nuqtalarda urinadi. Agar

|AB| = 9

|BC| = 7

|AC| = 10

ga teng bo'lsa,

AE

kesma uzunligini toping.

27.

ABCD

kvadrat tomoni uzunligi 6 ga teng.

AF = 2

CE = 3

\angle FMA = \beta

\angle CNE = \alpha

bo'lsa,

\alpha + \beta

ni toping.

28.

ABC

to'g'ri burchakli uchburchakka

ADEF

kvadrat shunday ichki chizilganki, ular uchun to'g'ri burchak umumiy va

A

burchagi to'g'ri. Agar

|BE| = 20

|EC| = 30

ga teng bo'lsa, bo'yalgan soha yuzini toping.

29.

O'zaro

120^\circ

li burchak tashkil qiluvchi ikki yoqli tekislik berilgan.

A, B, C

D

nuqtalar tekisliklarga tegishli. Agar

|AC| = 7\sqrt{3}

|AB| = 12

|DC| = 4\sqrt{3}

\angle BAC = 30^\circ

\angle ACD = 60^\circ

bo'lsa,

B

D

nuqtalar orasidagi masofani toping.

30.

Rasmda

x^2 + y^2 = 36

aylana va

y = kx + b

to'g'ri chiziq grafiklari tasvirlangan.

B

nuqtadan

OX

o'qigacha bo'lgan eng qisqa masofani toping.

31.

Quyidagi universal to'plam ichida

A

B

to'plamlar berilgan.

(A \cup B)' \cup A

qism to'plamlari sonini toping.

32.

Barcha raqamlari toq bo'lgan, uchga karrali uch xonali sonlar nechta?

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

33-35.

Chodirning tomi konusdan, pastki qismi silindrdan iborat. Konus va silindr asoslari ustma-ust tushadi va balandliklari bir xil. Chodirning balandligi 6 ga teng bo'lsa,

Javob variantlari:

64\pi

68\pi

54\pi

56\pi

60\pi

72\pi

Savol	A	B	C	D	E	F
33. Chodirning tom qismini uchun qoplash uchun $20\pi$ ( $m^2$ ) gazlama ketgan bo'lsa, chodirning hajmini toping.
34. Konusning hajmi $81\pi$ ( $m^3$ ) bo'lsa, silindr yon sirtini toping.
35. Konus yasovchisi konus asosidan $\frac{\sqrt{17}}{4}$ marta katta bo'lsa, silindr yon sirtini toping.

36.

x^4 - 4x^3 - 4x^2 + 16x - 8 = 0

tenglama berilgan.

a) Tenglamaning haqiqiy ildizlari nechta?

b) Tenglamaning haqiqiy ildizlari

x_1, x_2, \ldots, x_n

bo'lsa,

|x_1| + |x_2| + \ldots + |x_n|

ni toping.

37.

Agar

\begin{cases} \operatorname{tg} x + \operatorname{tg}\frac{y}{2} = \frac{4}{\sqrt{3}} \\ \operatorname{ctg} x + \operatorname{ctg}\frac{y}{2} = \frac{4}{\sqrt{3}} \end{cases}

bo'lsa,

\operatorname{tg} y

ning eng kichik qiymatini toping.

x

ning eng kichik musbat qiymatini toping.

38.

f(x) + \frac{x - 1}{2x + 1} \cdot f\left(\frac{1}{x}\right) = \frac{3}{x - 1}

funksiya berilgan bo'lsa,

f(2)

ning qiymatini toping.

f(x) = 0

tenglamaning ildizlari yig'indisini (agar tenglama ildizi bitta bo'lsa, shu ildizini) toping.

39.

Rasmda

y = f(x)

grafigi tasvirlangan.

f(x)

funksiyaning

(-6; 6)

oraliqda nechta lokal maksimum nuqtasi bor?

f'(x) = 0

bo'lganda

(-6; 6)

oraliqda ildizlar sonini toping.

40.

y_1 = \frac{(x + 2)^3}{16}

shu funksiyaga

x_0 = 2

nuqtada urinma o'tkazilgan. Bu to'g'ri chiziq

y_2 = kx + b

bo'lsa,

k \cdot b^2

ni toping.

b) Shtrixlangan soha yuzasini toping.

41.

ABC

to'g'ri burchakli uchburchakda

BH

balandligi uni ikkita

ABH

BCH

to'g'ri burchakli uchburchaklarga ajratadi.

ABH

BCH

uchburchaklarga ichki chizilgan aylana radiuslari mos ravishda 3 va 4 ga teng bo'lsa,

BH

balandlikni toping.

ABC

uchburchakning yuzini toping.

42.

Tomonlari 6, 6, 8, 8 ga teng bo'lgan qavariq to'rtburchak berilgan va ichki va tashqi aylanalar chizilgan bo'lsa,

a) To'rtburchakning kichik diagonalini toping.

b) Ichki va tashqi chizilgan aylanalar markazlari orasidagi masofani toping.

43.

ABCDEF

qavariq oltiburchak berilgan. Bunda

BD = FE

\angle A = \angle E = 90^\circ

ga teng.

a) Agar oltiburchakning barcha tomonlari 2 ga teng bo'lsa,

BFD

uchburchak yuzini toping.

b) Agar oltiburchakning barcha tomonlari

\frac{10}{\sqrt{4 + \sqrt{7} + \sqrt{3}}}

ga teng bo'lsa, oltiburchak yuzasini toping.

44.

Sharga konus ichki chizilgan. Konus yon sirti

18\sqrt{2 + \sqrt{3}}\,\pi

ga teng. Konus yasovchisi va asos tekisligi orasidagi burchak

75^\circ

ga teng bo'lsa,

a) Konus asosining radiusini toping.

b) Shar hajmini toping (

\pi \approx 3

deb olinsin).

45.

Uy hayvonlari uchun yarim silindr shaklidagi temir listdan bo'yi

200\ cm

, diametri

40\ cm

suv idishi yasalgan bo'lsa, (

\pi \approx 3

deb olinsin)

a) Suv idishini chiqindi chiqmaydigan qilib tayyorlash uchun qancha (

dm^2

) temir list ishlatadi.

b) Suv idishiga qancha litr suv ketadi.