Matematika (02.03.2026) 2-smena

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;

2026^{2025}

ifodani 7 ga bo'lgandagi qoldiqni toping.

Quyidagilardan nechtasi ratsional sonlar?

1)

\frac{7}{17}

\frac{22}{17}

2 + \lg 3

\sqrt{\ln e^4} + \sqrt{\sqrt[3]{8} - 1}

\sqrt[3]{1 + \sqrt{2}}

Berilgan ikki xonali sonning o'nlar xonasi birlar xonasidan 3 ga katta. Ikki xonali sonning raqamlari yig'indisining shu ikki xonali songa ko'paytmasi 364 ga teng. Bu sonning raqamlari ko'paytmasini toping.

O'quvchi ikkita kitobni 205000 so'mga sotib oldi. Agar birinchi kitob narxi

15\%

ga kamaytirilsa va ikkinchi kitob narxi

20\%

ga oshirilsa, u holda kitoblarning yangi narxlari o'zaro teng bo'ladi. Birinchi kitobning narxi ikkinchi kitob narxidan necha so'mga qimmat ekanini aniqlang.

Hisoblang.

\left(\left(5^{\frac{7}{2}}\right)^{\frac{4}{7}} + (16^3)^{\frac{1}{12}}\right) \cdot 3^{-3}

\left(2\sqrt{11}; 7\sqrt{5}\right)

oraliqda nechta butun son bor?

Hisoblang.

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

n

ta hadli arifmetik progressiyaning

n

ta hadi o'rta arifmetigi

2n

ga teng bo'lsa, arifmetik progressiyaning o'ninchi hadini toping.

Musbat hadli geometrik progressiyada

b_1 + b_2 + b_3 = 21

\frac{1}{b_1} + \frac{1}{b_2} + \frac{1}{b_3} = \frac{7}{12}

tenglik o'rinli bo'lsa,

b_2

ni toping.

10.

Agar

(a + b + 2024)^2 = 4(a + 2023)(b + 1)

bo'lsa,

a - b

ni toping.

11.

Soddalashtiring:

(x > 8)

\sqrt{x + 4\sqrt{x - 4}} + \sqrt{x - 4\sqrt{x - 4}}

12.

Soddalashtiring.

\frac{1 - \sin^4\alpha - \cos^4\alpha}{\operatorname{tg}^2\alpha}

13.

Tenglama nechta haqiqiy yechimga ega?

\sqrt{4 - x^2} \cdot \cos x = 0

14.

Tenglamaning haqiqiy ildizlari yig'indisini toping.

\left(\sqrt{7 - 4\sqrt{3}}\right)^x + \left(\sqrt{7 + 4\sqrt{3}}\right)^x = 14

15.

Tenglamaning haqiqiy ildizlar yig'indisini toping (agar u bitta bo'lsa, shu ildizini toping).

\log_{3x-7}(9 - 12x + 4x^2) + \log_{2x-3}(6x^2 - 23x + 21) = 4

16.

Tenglama nechta haqiqiy ildizga ega?

x^2 - \left(\pi - \sqrt{10}\right)|x| - \sqrt{10}\pi = 0

17.

Tenglamaning eng katta ildizini toping.

x^4 - 10x^3 + 26x^2 - 10x + 1 = 0

18.

Tengsizlik nechta butun yechimga ega?

\bigl||2x - 1| - x\bigr| \le 7

19.

Tengsizlikni qanoatlantiruvchi butun sonlar nechta?

\left|1 - \sqrt{x}\right| < 1

20.

Agar

f(x) = \begin{cases} 8 + x, & \text{agar } x < 0 \\ 2 - x, & \text{agar } x \ge 0 \end{cases}

bo'lsa,

f\bigl(f(f(-5))\bigr)

ni toping.

21.

Aniq integralni hisoblang.

\int_0^1 (x^3 - x + 2)^{2025} \cdot (3x^2 - 1)\,dx

22.

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

funksiyaning

x_0 = \frac{\pi}{4}

nuqtadagi hosilasini toping.

23.

ABC

uchburchakka aylana tashqi chizilgan. Bunda

\angle CBO = 40^\circ

O

— nuqta aylana markazi bo'lsa,

\alpha

burchakni toping.

24.

f(x, y, z) = \frac{x^2 + 1}{x^2 - 1} - \frac{2y^2 - 1}{2y^2 + 1} + \frac{4z^2 - 1}{4z^2 + 1}

|a| > 1

-1 < b \le 1

-1 \le c < 1

funksiya berilgan. Funksiyaning

x = \frac{\sqrt{a^2 - 1}}{a - 1}

y = \sqrt{\frac{1 - b}{2 + 2b}}

z = \frac{\sqrt{1 - c^2}}{2(c - 1)}

bo'lgandagi qiymatini toping.

25.

ABC

uchburchakning

BE

bissektrisasi

AC

tomonni 4 va 8 ga teng kesmalarga ajratadi. Agar

AB = 6

bo'lsa,

BE

kesma uzunligini toping.

26.

ABC

uchburchakda

AB

BC

tomonlariga mos ravishda

D

E

nuqtalar olingan.

AE

CD

kesmalar

O

nuqtada kesishadi. Kesmalar orasida

BD : AB = 1 : 3

BE : BC = 1 : 2

tengliklar o'rinli. Agar

ABC

uchburchakning yuzi 60 ga teng bo'lsa,

AOC

uchburchak yuzini toping.

27.

ABCD

trapetsiyaga aylana ichki chizilgan. Aylana markazidan

C

D

uchlarigacha bo'lgan masofalar mos ravishda 2 va 4 ga teng.

ABCD

trapetsiyaning yuzini toping.

28.

Muntazam oltiburchakning tomoni 1 ga teng. Uning tomonlari o'rtalari tutashtirilib yana oltiburchak hosil qilindi. Hosil bo'lgan oltiburchakka yana shunday tomonlari tutushtirildi va bu jarayon cheksiz davom etdi. Hosil bo'lgan oltiburchaklar yuzini toping.

29.

\overrightarrow{AB}(6; 2)

\overrightarrow{AC}(4; -12)

vektorlardan tashkil topgan

ABC

uchburchakka tashqi chizilgan aylana radiusini toping.

30.

\alpha

tekislikdan tashqarida joylashgan

A

nuqtadan ushbu tekislikka

AB

AC

og'malar hamda

AH

perpendikulyar tushirilgan. Bu yerda

H

— perpendikulyarning asos nuqtasi bo'lib,

B

C

H

nuqtalar

\alpha

tekislikda yotadi. Agar

AH = 4

\angle ABH = 45^\circ

\angle ACH = 60^\circ

\angle BHC = 60^\circ

bo'lsa,

AHBC

piramidaning hajmini toping.

31.

A = \{1; 2; 3; 4; 5; 6\}

to'plam berilgan. Ushbu to'plamning nechta qism to'plamida 3 yoki 5 dan kamida bittasi qatnashadi?

32.

1000 gacha bo'lgan natural sonlar ichida 11 ga qoldiqsiz bo'linadigan va 2 ga ham 3 ga ham qoldiqsiz bo'linmaydigan sonlar nechta?

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

33-35.

Rasmda hajmlari teng bo'lgan konus, silindr va shar berilgan. Silindr o'q kesimi kvadratdan, konus o'q kesimi muntazam uchburchakdan iborat bo'lsa, (bu yerda

r_{konus}

r_{silindr}

r_{shar}

— asoslarning radiuslari)

Javob variantlari:

\frac{\sqrt[6]{12}}{2}

\frac{\sqrt[6]{3}}{\sqrt[3]{6}}

\frac{\sqrt[3]{18}}{2}

\frac{\sqrt[3]{12}}{6}

\frac{\sqrt[3]{18}}{3}

\frac{\sqrt[3]{6}}{2}

Savol	A	B	C	D	E	F
33. Silindr asosi radiusining konus asosining radiusiga nisbatini $\left(\frac{r_{silindr}}{r_{konus}}\right)$ toping.
34. Shar radiusining konus radiusiga nisbatini $\left(\frac{r_{shar}}{r_{konus}}\right)$ toping.
35. Silindr asosi radiusining shar radiusiga nisbatini $\left(\frac{r_{silindr}}{r_{shar}}\right)$ toping.

36.

Tenglama berilgan.

2(x^2 + x + 1)^2 - 13(x^3 - 1) = 7(x - 1)^2

a) Tenglama nechta haqiqiy ildizga ega?

b) Tenglamaning haqiqiy ildizlari ko'paytmasini toping.

37.

Tenglamani yeching.

(\sin x + \cos x)^4 + (\sin x - \cos x)^4 + \sin 4x = 3

a) Eng kichik musbat ildizini toping.

[-\pi; 2\pi]

oraliqda nechta ildizga ega?

38.

f(x)

funksiya uchun

2f(x) + f(1 - x) = \frac{3x^2 + x + 4}{x(1 - x)}

tenglik o'rinli bo'lsa, (bunda

f^{-1}(x)

—

f(x)

ning teskari funksiyasi)

f(2)

ni toping.

f^{-1}(-2)

ni toping.

39.

f(x) = 2\sqrt{x}

funksiya berilgan.

f(x) = 2\sqrt{x}

ga o'tkazilgan urinma

Ox

o'qi bilan

30^\circ

tashkil qiladi. Urinmaning

Ox

o'qi bilan kesishgan nuqtasidan

y = 2\sqrt{x}

funksiyagacha bo'lgan eng qisqa masofani toping.

f(x) = 2\sqrt{x}

funksiyasidan

y = \frac{x}{2} + 3

to'g'ri chiziqqacha bo'lgan eng qisqa masofani toping.

40.

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

funksiya

Ox

o'qini

(-1; 0)

nuqtada kesib o'tadi va

(2; 0)

nuqtada urinadi.

a + b + c

ning qiymatini toping.

b) Chegaralangan soha yuzini toping.

41.

Rasmda tomonlari 2, 4 va 6 ga teng bo'lgan kvadratlar tasvirlangan.

a) Shtrixlangan soha yuzini toping.

|TK|

kesma uzunligini toping.

42.

ABC

uchburchakning

AC

AB

katetlari mos ravishda 3 va 4 ga teng.

AC

kateti va

BC

gipotenuzasi orqali tashqi

BCFE

ACML

kvadratlar chizilgan. Bunda

O_1

O_2

mos ravishda

ACML

BCFE

kvadratlar diagonallari kesishish nuqtasi.

|MF|

uzunligini toping.

AO_1O_2

uchburchak yuzini toping.

43.

ABCD

trapetsiyaga aylana tashqi chizilgan. Trapetsiyaning kichik asosi 3 ga, katta asosi 13 ga,

B

nuqtadan aylanadagi

E

D

nuqtalargacha bo'lgan masofalar 10 ga teng (bunda

E

D

nuqtalar ustma-ust tushmaydi).

AE

kesma uzunligini toping.

ABCDE

beshburchak yuzini toping.

44.

Yarim shar, silindr va konusdan quyidagi jism yasalgan. Konus yasovchisi 5 cm ga, shar radiusi 3 cm ga va jismning umumiy uzunligi 15 cm ga teng bo'lsa,

a) Jismning to'la sirtini (

cm^2

) toping.

b) Jismning hajmini (

cm^3

) toping.

45.

Quyida osma ko'prik parabola shaklida tasvirlangan. Ko'prikning eng katta ustunlari orasidagi masofa 300 m, qolgan ustunlar orasidagi masofalar 15 m dan va katta ustun balandligi 67 m ga teng. Ko'prik markaziga koordinata boshi ya'ni

O(0; 0)

nuqta joylashtirilgan.

a) Parabola tenglamasini tuzing.

b) 10 ta kichik ustunlar uzunliklari yig'indisini toping.