Matematika (19.02.2023)

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;

Natural

a

b

sonlar uchun

a + b = 111

tenglik bajarilsa,

ab - 1

ayirmaning eng katta qiymatini toping.

Hisoblang.

\frac{1 - 3 + 5 - 7 + 9 - 11 + \ldots + 2021 - 2023}{1 - 2 + 3 - 4 + 5 - 6 + \ldots + 2021 - 2022}

Ikki ishchi bir ishning

\frac{2}{3}

qismini 4 kunda bajara oladi. Agar ular alohida ishlasa, birinchi ishchi ishni ikkinchisidan 5 kun oldinroq bajarib tugatadi. Yolg'iz ishlaganda birinchi ishchi ishni necha kunda tugata oladi?

500 kg ruda, tarkibida 12,5 % temir bo'lgan 200 kg ruda ajratib olindi. Qolgan rudada temir ulushi dastlabkidan 20% ortiq bo'ldi. Dastlab ruda tarkibida qancha (kg) temir bo'lgan?

Hisoblang.

\frac{10^{55} + 10^{54} + 10^{50}}{10^{49} + 10^{53} + 10^{54}}

Hisoblang.

\sqrt{\frac{2022 \cdot 2024 + 1}{2024 \cdot 2030 + 9} \cdot \frac{2026 \cdot 2028 + 1}{2020 \cdot 2026 + 9} - 1 + \sqrt{25}}

Agar

a < a^3 < a^2

b^3 < b < b^2

c^3 < c^2 < c

bo'lsa,

\frac{ab|b - c - 1|}{|ab|} + \frac{bc|a + c + 1|}{|bc|} + \frac{ac|a + 1 + b|}{|ac|}

ifodani soddalashtiring.

Arifmetik progressiyada

a_5 + a_{n-4} = 24

S_n = 96

bo'lsa,

n

ni toping.

Birinchi hadi 5 dan katta bo'lgan geometrik progressiyada

b_1 + b_2 + b_3 = 21

b_3 + b_4 + b_5 = 84

bo'lsa, uning dastlabki oltita hadining yig'indisini toping.

10.

Agar

a - b = 4

bo'lsa,

\frac{a^2 - b^2 + 3a - 3b}{a^2 - b^2 + 6a + 9}

ni hisoblang.

11.

Ifodani soddalashtiring.

\frac{a^2 + b^2}{a + b} : \left(\frac{a^2 + b^2}{ab} + \frac{b^2}{a^2 - ab} - \frac{a^2}{ab + b^2}\right)

12.

Soddalashtiring.

2\operatorname{tg} 53^\circ \left(\frac{1}{\sin 106^\circ} + \frac{1}{\operatorname{tg} 106^\circ}\right)

13.

Soddalashtiring.

\frac{1}{\cos x \cos 2x} + \frac{1}{\cos 2x \cos 3x} + \frac{1}{\cos 3x \cos 4x} + \ldots + \frac{1}{\cos 2022x \cos 2023x}

14.

Tengsizlikning

[2; 2023]

oraliqda nechta natural yechimi bor?

2^x + 3^x + 4^x + 5^x > 54

15.

Tengsizlikning butun yechimlari nechta?

3^{\log_3 \sqrt{x-6}} > 3^{\log_9 (x-4)}

16.

Tenglama ildizlari

x

y

bo'lsa,

x \cdot y

ni hisoblang.

|x + y - 12| + (x - y - 2)^2 = 0

17.

Tenglamaning haqiqiy ildizlari sonini toping.

x^2 + \frac{4x^2}{(5x + 2)^2} = \frac{9}{5}

18.

Tengsizlikning

[1; 2023]

kesmada nechta natural yechimga ega?

\frac{\pi x - \sqrt{10}}{\pi - \sqrt{10}} < 0

19.

Tengsizlikning

[-4; 5]

oraliqdagi butun yechimlari yig'indisini toping.

(2x + 3)(x^2 + 3x) - \frac{16(2x + 3)}{x^2 + 3x} \ge 0

20.

Tenglamani yeching.

x + \frac{x}{1+2} + \frac{x}{1+2+3} + \ldots + \frac{x}{1+2+3+\ldots+7} = 7

21.

y = ax^2 + bx + c

funksiya uchun

f'(1) = 0

f(2) - f'(2) = 1

\int_0^1 f(x)\,dx = \frac{2}{3}

bo'lsa,

b - a - c

toping.

22.

y = \cos^n x \cdot \cos nx

bo'lsa,

y'(x)

ni toping.

23.

Aylanada

A, B, C, D

nuqtalar olingan.

O

— aylana markazi,

A

D

nuqtalar bilan bir to'g'ri chiziqda yotadi. Agar

\angle ABC = 30^\circ

OD = 3

bo'lsa,

CD

ni toping.

24.

Quyidagi funksiyalardan nechtasi toq?

1)

y = 2^x + 2^{-x}

y = x\sqrt{1 + x^2}

y = 1 + \sin 2x

y = \log_2\left(x + \sqrt{1 + x^2}\right)

25.

ABC

uchburchakda

AD

BN

CM

medianalar kesishgan nuqta

G

nuqta bo'lsin. Agar

ABC

uchburchak yuzasi 48 ga teng bo'lsa,

GDN

uchburchak yuzasini toping.

26.

A(-10; -3)

B(-4; 5)

C(5; 5)

D(11; -3)

nuqtalardan o'tuvchi aylana radiusini toping.

27.

Agar aholining elektr energiyasiga talabi har yili 2,5 % ortsa, necha yilda 9 marta ortadi?

28.

Tomoni 4 ga teng bo'lgan

ABCDEF

muntazam oltiburchak uchta tomonining o'rtalarini tutashtirilib muntazam uchburchak yasalgan. Bo'yalgan soha yuzini toping.

29.

Tekis burchagi

60^\circ

bo'lgan ikki yoqli burchakning qirrasida

A

B

nuqtalar olingan. Ulardan ikki yoqli burchakning turli yoqlarida

A_1, B_1

nuqtalardan burchakning qirrasiga perpendikulyar qilib

AA_1 = 16

BB_1 = 13

. Agar

A_1B_1 = 17

bo'lsa,

AB

ni toping.

30.

ABCDA_1B_1C_1D_1

kubda

\overrightarrow{BC} + \overrightarrow{CD}

\overrightarrow{D_1A_1} + \overrightarrow{A_1B}

vektorlar orasidagi burchak kosinusini toping.

31.

A

to'plamning elementlar soni 9 ta,

B

to'plamning elementlar soni 7 ta,

U

universal elementlar soni 16.

n(A \cup B)'

eng ko'pi bilan nechaga teng? (

A'

—

A

to'plamning universal to'plamgacha to'ldiruvchisi)

32.

10 kishidan 1 ta boshliq, 2 ta yordamchi, 2 ta mutaxassisni necha xil usulda tanlanadi.

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

33-35.

Gipotenuzasi

4\sqrt{2}

bo'lgan to'g'ri burchakli uchburchakning to'g'ri burchagi uchidagi bissektrisasi uni ikkita teng yonli uchburchakka ajratadi. Berilgan uchburchak gipotenuza atrofida

360^\circ

ga aylantirildi. (

\pi \approx 3

)

Javob variantlari:

16\sqrt{2}

32\sqrt{2}

48\sqrt{2}

30\sqrt{2}

15\sqrt{2}

36.

Tenglamani yeching.

(x^2 - x - 1)^3 + (x^2 - 3x + 2)^3 = (2x^2 - 4x + 1)^3

a) Tenglamaning nechta haqiqiy ildizi bor?

b) Tenglamaning nechta musbat ildizi bor?

37.

Tenglamani yeching.

\frac{\sin 2^x}{\sin 2^{x-2} \cdot \cos 2^{x-2}} = 2\sqrt{3}

a) Tenglamaning

(0; 5)

oraliqda nechta ildizi bor?

b) Tenglamaning

(0; 5)

oraliqdagi ildizlaridan eng kattasi va eng kichigining ildizlarining musbat ayirmasini toping.

38.

y = g'(x)

funksiya grafigi parabola bo'lib,

(2; 7), (4; 3)

(6; -9)

nuqtalardan o'tishi ma'lum.

y = g(x)

funksiya grafigining

x_0 = -2

nuqtasiga o'tkazilgan urinmaning burchak koeffitsientini toping.

b) Funksiyaning statsionar nuqtalar yig'indisini toping.

39.

Koordinatalar tekisligida

y_1 = \sqrt{3}x

y_2 = \frac{x}{\sqrt{3}}

funksiyalarning grafiklari yasalgan. Birinchi chorakda bu to'g'ri chiziqlar orasida

A(a; b)

nuqta olingan va

a^2 + b^2 = 144

AL

AK

kesmalar mos ravishda

A

nuqtadan

y_1

y_2

to'g'ri chiziqlargacha bo'lgan eng yaqin masofalar.

M

N

nuqtalar

A

nuqtaga mos ravishda

K

L

nuqtalarga simmetrik bo'lgan nuqtalar.

KL

ni toping.

NM

ni toping.

40.

Grafigi

OY

o'qiga nisbatan simmetrik bo'lgan

y = n\sqrt[3]{x^2 + m} + d

funksiya grafigi

(0; 4)

(1; 3)

nuqtalardan o'tadi.

n + m + d

ni toping.

b) Funksiya grafigi va abssissalar o'qi chegaralangan soha yuzini toping.

41.

O'tkir burchakli

ABC

uchburchakning

BH

AD

balandliklari mos ravishda

F

E

nuqtalarni

\angle EFC = \angle BEC = 90^\circ

bo'ladigan qilib olingan. Bunda

AH = 5

HC = 3

BD = 2

CE

kesma uzunligini toping.

b) Agar

EF = \sqrt{48 - 24\sqrt{3}}

bo'lsa,

CEF

uchburchakning yuzini toping.

42.

ABCD

kvadrat

BD

diagonalining

D

uchi davomida

E

nuqta olingan va

DE = 3

a) Kvadratning diagonali 18 bo'lsa,

CE

ni toping.

b) Agar

CE = \sqrt{29}

bo'lsa, kvadratning yuzini toping.

43.

Burchaklari teng bo'lgan

ABCDEF

oltiburchakda

AB = 1

BC = 4

AF = 2

EF = 3

bo'lsa,

CD

tomoni toping.

ABCDEF

oltiburchakning yuzasini toping.

44.

Qirrasi

6\sqrt{2}

bo'lgan muntazam tetraedrga shar ichki chizilgan.

a) Muntazam tetraedrning hajmini toping.

b) Sharning to'la sirtini toping.

45.

Binoning hajmi 96 m³. Balandligi va eni bir xil

a

, uzunligi

b

bo'lgan binoning bir devori uy devoriga yopishgan. Binoning uy devori bilan umumiy bo'lmagan qirralari armaturadan, uchta yon devorini shisha oynadan va tomini tunukadan tiklagan.

a) Agar bino uchun armaturani eng kam miqdorda sarflagan bo'lsa, jami qancha armatura sarflagan (m)?

b) Agar 1 m armatura 1 \

, 1 m² shisha oyna 10 \

va 1 m² tunuka 5 \

tursa, binoni tiklash uchun jami qancha (\

) sarflagan?

Savol	A	B	C	D	E	F
33. Hosil bo'lgan jismning to'la sirtini toping.
34. Hosil bo'lgan jismning hajmini toping.
35. Hosil bo'lgan jismga ichki chizilgan shar hajmini toping.