Matematika (01.12.2024) 3-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;

Agar

A

B

natural sonlarining ko'paytmasi

A \cdot B = 3^4 \cdot 5^2 \cdot 7^3 \cdot 11

ga teng bo'lsa,

EKUB(A; B)

ning eng katta qiymatini toping.

Hisoblang:

6 - \frac{31}{30} - \frac{61}{60} - \frac{101}{100} - \frac{151}{150} - \frac{211}{210}

Ikki turli natural sonlarning yig'indisi 36 va kvadratlarining yig'indisi eng kichik bo'lsa, ularning ko'paytmasini toping.

Ish haqi dastlab 10% ga, keyin esa 20% ga oshdi. Ish haqi dastlabgi holatiga nisbatan necha % ga oshgan?

7^{2023}

o'nlar xonasidagi raqamni toping.

Soddalashtiring:

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

Hisoblang:

\frac{(\sqrt{22} - 2)\sqrt{26 + 2\sqrt{88}}}{9}

Hadlari

1, 3, 7, 13, 21 \ldots

shu tartibda ketgan ketma-ketlikni ikkita ketma-ket hadlari ayirmasi, hadlari

2, 4, 6, 8 \ldots

ga teng bo'lgan arifmetik progressiyani tashkil qiladi. Berilgan ketma-ketlikning nechanchi hadi 4033 ga teng?

Agar geometrik progressiya uchun

b_{m+n} = A

b_{m-n} = B

ushbu rekurrent tengliklar berilgan bo'lsa,

b_m

ni toping.

10.

a

soni

1, 4, 7 \ldots

arifmetik progressiyaning biror hadi bo'lib quyidagi ifodaning qiymati 2 ga teng bo'lsa,

a

ning qiymatini toping.

\frac{1}{\sqrt{4} + \sqrt{1}} + \frac{1}{\sqrt{7} + \sqrt{4}} + \ldots + \frac{1}{\sqrt{a + 3} + \sqrt{a}} = 2

11.

Soddalashtiring:

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

12.

P(0; 1)

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

13.

Agar

\sin a + \cos a = \frac{2}{\sqrt{5}}

bo'lsa,

\frac{\sin^4 a}{\cos^2 a} + \frac{\cos^4 a}{\sin^2 a} = ?

14.

Tengsizlikni yeching:

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

15.

Tenglamaning ildizlari ko'paytmasini toping.

\frac{4\log_{16}^2 x + \frac{1}{2}}{\log_{16} x - \frac{1}{4}} \cdot (\log_2 x - 1) - \log_2 x = 8

16.

|2x + 3| = 15

tenglama ildizlari yig'indisini (agar u bitta bo'lsa shu ildizning o'zini) toping.

17.

Tenglama nechta haqiqiy ildizga ega?

(x+1)^{2023} + (x+1)^{2022}(x-2) + (x+1)^{2021}(x-2)^2 + \ldots + (x+1)(x-2)^{2022} + (x-2)^{2023} = 0

18.

Tengsizlikni yeching:

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

19.

Tengsizlikning butun ildizlari yig'indisini toping.

\frac{x + 1}{(x - 2)^2} \cdot \sqrt{6 + 5x - x^2} \ge 0

20.

Agar funksiya

f(x) = \frac{6x - 5}{x + 6}

bo'lsa, ushbu funksiyaga teskari funksiyani 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.

O(12; 5)

nuqta aylana markazi va aylana

Ox

o'qiga urinadi. Aylana radiusini toping?

24.

h(x) = kx + l

f(x) = mx + n

bo'lsa,

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

25.

Agar

ABC

MNC

uchburchaklar to'g'ri burchakli uchburchaklar va

AB = 6

MN = 3

NC = 4

bo'lsa,

x

ning qiymatini 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.

Rombning bir uchi aylana markazida qolgan uchlari esa aylanada yotadi. Agar shu aylanaga tashqi chizilgan muntazam uchburchak yuzi

48\sqrt{3}

bo'lsa, rombning yuzini toping.

28.

Agar rasmda muntazam sakkizburchak va muntazam oltiburchak tasvirlangan bo'lsa,

\alpha

burchakni toping.

29.

Tekislikda yotmagan

M

nuqtadan tekislikning

A

B

nuqtalariga tekislik bilan

30^\circ

60^\circ

burchak hosil qiluvchi

MA

MB

og'malar tushirildi. Agar

M

nuqtadan tekislikkacha eng qisqa masofa

2\sqrt{3}

bo'lsa,

A

B

nuqtalar orasidagi eng qisqa masofani toping.

30.

ABCDA_1B_1C_1D_1

kub berilgan bo'lib,

\overrightarrow{AC} \cdot \overrightarrow{AC_1} = 12

bo'lsa, kub qirrasi uzunligini toping.

31.

0, 1, 2, 4, 8, 9

raqamlari yordamida raqamlari takrorlanmaydigan uch xonali sonlar hosil qilindi. Bular ichidan tanlangan bir sonning toq bo'lishi ehtimolini toping.

32.

Quyida universal to'plam ichida

A

B

to'plamlar berilgan.

(A \cup B)' \cup A

qism to'plamlari sonini toping.

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

33-35.

Quyida to'g'ri parallelepipeddan qirqib olingan karniz tasvirlangan. Chizmadagi ma'lumotlarga asosan (

3-rasm

) (

\pi \approx 3

deb oling)

36.

(x^2 - 2x + k)^2 + 3x(x^2 + 2x + k) = 30x^2

tenglama berilgan. Tenglama yagona ildizga ega bo'ladigan

k

ning qiymati

k_1

bo'lsin. Tenglama ikkita, uchta va to'rtta ildizga ega bo'ladigan

k

ning qiymati eng katta qiymatlari mos ravishda

k_2

k_3

k_4

lar bo'lsin.

a) Tenglama yechimga ega bo'lmaydigan

k

ning eng kichik butun qiymatini toping.

k_1 \cdot k_3(k_2 + k_4)

ning qiymatini toping.

37.

\sqrt{6x - 5 - x^2} \cdot \left(\cos 2x - \frac{1}{2}\right) = 0

tenglamani yeching.

a) Tenglama nechta haqiqiy ildizga ega?

b) Tenglamaning eng kichik musbat haqiqiy ildizini toping.

38.

f(x) + 4f\left(\frac{1}{x}\right) = \frac{1}{x}

funksional tenglama berilgan.

f\left(\frac{1}{2}\right)

ning qiymatini toping.

f(x)

funksiya grafigiga

x_0 = 1

nuqtada o'tkazilgan urinmaning burchak koeffitsiyentini toping.

39.

ABC

uchburchak shakldagi konsert maydonini qurish uchun

ABC

uchburchak shaklidagi qismiga beton quyildi hamda uning parabola shaklidagi qismi taxta bilan qoplandi.

a) Qanday yuzadagi taxta ishlatilgan?

b) Beton bilan qoplangan

ABC

uchburchak shakldagi sahnani yuzini toping.

40.

f(x) = \sqrt{4 - x^2} + 2

funksiya berilgan.

a) Koordinatalar boshidan funksiya grafigigacha bo'lgan eng qisqa masofani toping.

\int_0^2 f(x)\,dx

integralni hisoblang.

41.

To'g'ri burchakli uchburchakda

CM

mediana va

BL

bissektrisa. Bunda

BL = \frac{8}{\sqrt{3}}

BC = 8

ga teng.

a) Kichik katet uzunligini toping.

CM

mediana uzunligini toping.

42.

ABCD

rombning

AB

tomonini

DH

balandlik

AH : HB = 3:2

nisbatda bo'ladi. Rombga

DC

tomoniga

M

nuqtada urinuvchi doira ichki chizilgan.

a) Romb yuzining doira yuziga nisbatini toping (

\pi \approx 3

deb oling).

|CM| : |MD|

nisbatni toping.

43.

Muntazam sakkizburchakka aylana, aylanaga muntazam oltiburchak ichki chizilgan. Sakkizburchak tomoni

a

ga, oltiburchak tomoni

b

ga teng.

\frac{a}{b}

nisbatni toping.

b) Muntazam oltiburchak yuzi

3\sqrt{6} + 3\sqrt{3}

ga teng bo'lsa, sakkizburchak yuzini toping.

44.

Silindrga muntazam oltiburchakli piramida ichki chizilgan. Piramida uchi silindr yuqori asosida, piramida asosi silindr asosida joylashgan. Silindr va piramida yon sirtlarining nisbati

\frac{\sqrt{2}\pi}{3}

ga teng.

a) Silindr va piramida hajmlarining nisbatini toping.

b) Piramida apofemasi asos tekisligi bilan qanday burchak tashkil qiladi?

45.

ABCD

to'g'ri to'rtburchak shaklidagi yerning

A

nuqtasida suv ombori va

C

nuqtasida issiqxona tasvirlangan. Suv omboridan issiqxonaga

A

dan

E

orqali

C

nuqtaga suv olib borish uchun

AE

yo'l bo'yicha soatiga 3 m,

EC

yo'l bo'yicha esa soatiga 5 m tezlikda ariq qazilmoqda. Bunda

AB = 30\ m

BC = 80\ m

a) Agar ariq o'tkazish uchun eng minimal vaqt sarflangan bo'lsa, suv omboridan issiqxonagacha bo'lgan ariqni qazish uchun qancha vaqt sarflangan?

b) Ariq qazish minimal vaqtda bajarilgan bo'lsa,

EC

ariq uzunligini toping.

Savol	A	B	C	D	E	F
33. O'q kesimi yuzini toping. ( $cm^2$ )
34. To'la sirti yuzini toping. ( $dm^2$ )
35. Hajmini toping. ( $cm^3$ )