Matematika (23.11.2025)

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;

2! + 4! + 6! + \ldots + 2024!

yig'indining oxirgi raqamini toping.

Hisoblang:

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

Uchta poliz ekinlari tarvuz, qovun va karamlarning narxlari donasiga 50 000 so'm, 10 000 so'm va 1000 so'mdan narxlangan. Jami 500 000 so'mlik xarid orqali 100 ta mahsulot olingan bo'lsa, karamlar soni nechta?

Yog'liligi

36\%

bo'lgan yangi sog'ilgan sutdan yog' olib tashlangandan keyin yog'liligi

4\%

li sut qoladi. 18 litr

4\%

li sut olish uchun qancha yog'i olinmagan sut kerak?

n

m

sonlar uchun

4n + m = 3

bo'lsa,

3^m \cdot 81^n

ni hisoblang.

Hisoblang:

\sqrt{3{,}25 + \sqrt{3}} + \sqrt{7 - 4\sqrt{3}}

\sqrt[3]{9 + 4\sqrt{5}} - \sqrt[3]{7 - 5\sqrt{2}} + \sqrt[3]{9 - 4\sqrt{5}} - \sqrt[3]{7 + 5\sqrt{2}}

ni hisoblang.

\{a_n\}

arifmetik progressiya hadlari

a_1 = p

;

a_2 = k

;

a_3 = p^2

\{b_n\}

geometrik progressiya hadlari

b_1 = p

b_2 = p^2

b_3 = k

bo'lsa, arifmetik progressiyaning dastlabki 10 ta hadi yig'indisini toping (bunda

p \ne k

Kamayuvchi geometrik progressiya uchun

b_1^2 + b_2^2 + b_3^2 = 84

b_1 + b_2 + b_3 = 14

bo'lsa, geometrik progressiyaning dastlabki oltita hadi yig'indisini toping.

10.

Agar

b < c < 0 < a

bo'lsa,

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

ni soddalashtiring.

11.

Agar

a + b = x

ab = y

bo'lsa,

ab^3 + a^3b + a^2 + b^2 + 5ab + 2a^2b^2

x

y

orqali ifodalang.

12.

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

13.

\sqrt{5}\sin x + \sin 2x = 0

tenglamaning

\left[\frac{\pi}{2}; \frac{5\pi}{2}\right]

oraliqdagi ildizlari yig'indisini toping.

14.

\frac{3 + a + 2^{x+2}}{x + 5} = 0

tenglama

a

ning qanday qiymatlarida yechimga ega bo'ladi?

15.

Tenglamani yeching:

\frac{2\ln 2 - \ln 5 + \ln(2x + 3)}{\ln x + \ln(x + 3)} = 1

16.

Tenglama nechta haqiqiy ildizga ega?

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

17.

Tenglama nechta ildizga ega?

\left(x^2 - x - 1\right)^3 + \left(x^2 - 3x - 2\right)^3 = \left(2x^2 - 4x - 3\right)^3

18.

Tengsizlikning butun yechimlari yig'indisini toping.

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

19.

Tenglamaning haqiqiy ildizlari ko'paytmasini toping.

\sqrt[6]{x^2 + 3x} + \sqrt[3]{x^2 + 3x} = 6

20.

f(x) = ax^3 + bx^2 + cx + d

funksiya grafigi tasvirlangan. Funksiya

(2; 0)

nuqtada

Ox

o'qiga urinadi.

Ox

o'qini

(-3; 0)

Oy

o'qini esa

(0; 4)

nuqtalarda kesib o'tadi.

a + b + c + d

ni toping.

21.

Aniq integralni hisoblang:

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

22.

f(x) = \cos\left(kx^n\right) \cdot \lg\sqrt[n]{x + \sqrt{1 + x^2}}

funksiya berilgan (

n

— juft natural son). Agar

f'(2023) = 2024

bo'lsa,

f'(-2023) + 2024

ni hisoblang.

23.

Bo'yalgan soha yuzini toping (kichik aylana katta aylanaga ichkaridan urinadi; kichik doiraning vatar ustidagi qismi yuzi 9 ga teng).

24.

f(x) = \frac{x}{4} + \frac{4}{x}

funksiyaning

[-6; -2]

oraliqdagi eng katta qiymatini toping.

25.

Muntazam uchburchakning ichidagi ixtiyoriy nuqtadan tomonlariga tushirilgan perpendikulyarlar 3, 4 va 5 ga teng bo'lsa, muntazam uchburchakning yuzini toping.

26.

Rasmda uchta kvadrat tasvirlangan. Bo'yalgan soha yuzini toping.

27.

ABC

uchburchakda

A

uchidan

BC

tomonga

AD

bissektrisa o'tkazilgan. Bunda

BD : DC = 1 : 2

AD = DC

bo'lsa,

\alpha

burchakni (

\angle B

) toping.

28.

Muntazam sakkizburchak tomonlarini davom ettirishdan hosil bo'lgan

\alpha

\beta

burchaklar yig'indisini toping.

29.

\vec{a}, \vec{b}, \vec{c}

birlik vektorlar va

\angle(\vec{a}, \vec{b}) = \angle(\vec{b}, \vec{c}) = \angle(\vec{a}, \vec{c}) = 60^\circ

bo'lsa,

\vec{a}

\vec{b} + \vec{c}

vektorlar orasidagi burchak kosinusini hisoblang.

30.

Muntazam oltiburchakli piramidaning yon qirrasi asos tekisligi bilan

45^\circ

tashkil qiladi, yon qirrasi

4\sqrt{6}

ga teng bo'lsa, piramida hajmini toping.

31.

A

B

bo'sh bo'lmagan to'plamlar:

A = \{6

ga karrali ikki xonali sonlar

\}

B = \{

yozuvida kamida bitta 6 raqami bo'lgan ikki xonali sonlar

\}

A \cap B

to'plamning elementlari soni nechta?

32.

Hasan va Husan markazdan 6 ta teng sektorga bo'lingan aylana yo'li bo'ylab tasodifiy bir yo'nalishni tanlab, 1 soat 5 km/h tezlikda yuradi. Shuning uchun ikkalasi ham markazdan 5 km radiusli aylana bo'ylab joylashadi. 1 soat harakatlangandan so'ng orasidagi masofa 6 km dan kam bo'lmaslik ehtimolligi nechaga teng?

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

33-35.

Silindr va kesik konusdan iborat banka tasvirlangan bo'lib, banka hajmi

1000\pi\ cm^3

, silindr balandligi konus balandligidan 3 marta katta bo'lib, kesik konus kichik asosining diametri 8 cm bo'lsa (bankaning umumiy balandligi 30 cm).

Javob variantlari:

4,5

7,5

Savol	A	B	C	D	E	F
33. Kesik konus balandligi ( $h_1$ ) ni toping.
34. Silindr asosining radiusini toping.
35. Banka ichidagi sut (rasmga qarang) hajmi $432\pi\ cm^3$ bo'lsa, sut mavjud bo'lgan balandlik ( $h_2$ ) ni toping.

36.

Tenglamani yeching:

x^4 + kx^3 + x^2 + 4kx + 16 = 0

a) Tenglama

k = 1

bo'lganda nechta haqiqiy ildizga ega bo'ladi?

b) Tenglama yagona ildizga ega bo'ladigan

k

ning qiymatlari nechta?

37.

Tenglamani yeching:

\operatorname{tg} 3x + \operatorname{ctg}\frac{3x}{2} = 4\cos^2\frac{3x}{2}

a) Tenglama

[0; \pi]

oraliqda nechta yechimga ega?

b) Tenglamaning eng katta manfiy va eng kichik musbat ildizlari yig'indisini toping.

38.

f(x) = -|x - a| + b

g(x) = |x - c| + d

funksiya grafiklari berilgan.

f(x)

g(x)

funksiya grafiklari

A(2; 5)

B(8; 3)

nuqtalarda kesishadi. Rasmdagi ma'lumotlar asosida,

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

ni toping.

C

D

nuqtalar orasidagi masofani toping.

39.

Rasmda

\varphi(x)

funksiya grafigi tasvirlangan bo'lib, rasmdagi ma'lumotlar asosida

\frac{x\varphi(x)}{\varphi'(x)} < 0

tengsizlik

(-8; 5)

oraliqda nechta butun yechimga ega bo'ladi?

\begin{cases} \varphi(x) > 0 \\ \varphi'(x) > 0 \end{cases}

tengsizliklar sistemasi

(-8; 5)

oraliqda nechta butun yechimga ega?

40.

Rasmda

f(x) = x^3 + ax^2 + bx + c

g(x) = x^3 + px^2 + qx + d

funksiya grafiklari tasvirlangan. Berilgan ma'lumotlar asosida

\frac{a + b + c}{p + q + d}

ni toping.

b) Bo'yalgan soha yuzini toping.

41.

ABC

uchburchakda

B

uchidan

BP

— bissektrisa,

BH

— balandlik,

BM

— mediana o'tkazilgan.

L

— nuqta ichki chizilgan aylananing

AC

ga urinish nuqtasi (rasmga qarang).

AB = 4

BC = 6

bo'lsa,

LM

ni toping.

AB = 6

BC = 10

AC = 12

bo'lsa,

HL

ni toping.

42.

ABCD

kvadrat berilgan. Kvadrat ichidagi

E

nuqta uchun

DE = 1

CE = 2

BE = 3

\angle DEC

ni toping.

b) Bo'yalgan soha (

BED

uchburchak) yuzini toping.

43.

ABCDE

beshburchakka tashqi aylana chizilgan (rasmga qarang).

AB = BC = CD = DE = 4

\angle ADB = 30^\circ

F

nuqta

AD

BE

diagonallar kesishish nuqtasi bo'lsa,

BCDF

to'rtburchak yuzini toping.

AEF

uchburchak yuzini toping.

44.

Qalam konus va silindrsimon shakllardan iborat bo'lib, qalamning uzunligi 160 mm bo'lib,

x

ning uzunligi

y

ning uzunligidan 130 mm ga uzun va diametri 10 mm bo'lsa,

a) Qalamning konussimon uchi hajmini (

mm^3

) toping.

b) Qalamning bir qismi bo'lgan silindr hajmini (

mm^3

) toping.

45.

Uyning tomi tunuka va taxta bilan yopilgan. Tomning ikki yon tomoni to'g'ri to'rtburchakli qismi tunuka bilan, to'g'ri burchakli uchburchak shaklidagi ikki tomoni taxta bilan yopilgan. Bunda

AD = 3{,}2

DC = 1{,}8

CE = 12

m ga teng.

a) Tunuka bilan yopilgan qismlar yuzlarining yig'indisini (

m^2

) toping.

b) Agar tunukaning 1

m^2

i 180 000 so'm, taxtaning 1

m^2

i 80 000 so'm bo'lsa, uyning tomini yopish uchun qancha so'm mablag' sarflangan?