Matematika (28.12.2025)

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;

Hisoblang:

\underbrace{222\ldots222}_{50\text{ ta } 2} : 22

a = 2{,}(14)

;

b = 2{,}1(4)

;

c = 2{,}11(4)

sonlarni kamayish tartibida joylashtiring.

Dehqon bozorga 5 ta kamazda qovun va tarvuz olib keldi. Har bir kamazda qovun yoki tarvuzlar bor. Birinchi kamazda 300 ta, ikkinchisida 315 ta, uchinchisida 330 ta, to'rtinchisida 345 ta va beshinchisida 390 ta tarvuz yoki qovunlar bor. Bozorfurushga har bir kamazdagi tarvuz va qovunlar to'liq sotilmaguncha boshqa kamazdagi tarvuz va qovunlarni sotishni taqiqladi. Bitta kamaz to'liq sotilgandan keyin qovunlar tarvuzlardan 3 marta ko'p qolgani ma'lum bo'ldi. Qancha qovunlar qolgan?

15 ta ko'ylak va 20 ta shimning birgalikdagi narxi 950 000 so'm. 15 ta ko'ylakni

28\%

va 20 ta shimni

20\%

chegirma bilan birgalikdagi sotgan narxi 724 000 so'm bo'ldi. Bitta ko'ylak narxini toping.

Hisoblang:

\frac{10^2 + 11^2 + 12^2 + 13^2 + 14^2}{146}

Hisoblang:

\frac{100^3 - 14^3}{\sqrt{\frac{114^4 + 100^4 + 14^4}{2}}}

Hisoblang:

\sqrt{\frac{666666^2}{12345654321}}

Arifmetik progressiyada

a_3 = a_1 + a_2

a_1 \cdot a_2 \cdot a_3 = 384

tengliklar o'rinli bo'lsa,

S_{10}

ning qiymatini toping.

Geometrik progressiyada

b_{10} = 5

bo'lsa, dastlabki 19 ta hadlari ko'paytmasini toping.

10.

Ifodani soddalashtiring:

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

11.

\left(x^2 + ax + b\right)(x + 3) = x^3 + 4x^2 + 5x + 6

tenglik ayniyat bo'lsa,

a

b

ning qiymatini toping.

12.

Hisoblang:

(1 + \operatorname{tg} 7^\circ)(1 + \operatorname{tg} 8^\circ)(1 + \operatorname{tg} 37^\circ)(1 + \operatorname{tg} 38^\circ)

13.

\sin x \cdot \cos x = \frac{1}{2}\arcsin\left(-\frac{\sqrt{3}}{2}\right)

tenglama nechta ildizga ega?

14.

\frac{2}{3} - \log_x 3 = \log_3 2 \cdot \log_{32} x - \frac{2}{5}\log_{\sqrt{x}} 3

tenglamaning haqiqiy ildizlari ko'paytmasini toping.

15.

14^{\sin x} = (0{,}5)^{\cos x} \cdot 7^{\sin x}

tenglamaning

x \in \left[-4\pi; -\frac{5\pi}{2}\right]

oraliqdagi yechimlari yig'indisini (agar u yagona bo'lsa, shu ildizini) toping.

16.

\frac{x^3}{|x|} + 10x - \frac{11|x|}{x} = 0

tenglama nechta haqiqiy ildizga ega?

17.

Tengsizlikni nechta butun son qanoatlantiradi?

\frac{x^2 - 7|x| + 10}{x^2 - 6x + 9} < 0

18.

x, y

natural sonlar uchun

xy - x + 2y = 5

tenglik o'rinli bo'lsa,

x + y

ning qiymatini toping.

19.

\frac{1}{\left(\sqrt{\frac{2 - x}{x}} - \frac{x + 1}{2x}\right)^2} \ge 0

tengsizlikni nechta butun son qanoatlantiradi?

20.

y = x^2 - 6ax + 10a^2 - a - 6

kvadrat funksiyaning uchining absissasi musbat, ordinatasi manfiy bo'ladigan barcha butun

a

larning yig'indisini toping.

21.

y = \frac{x^2 - 3x - 4}{x^2 + 3x + 4}

funksiyaning qiymatlar sohasini toping.

22.

3(3x + 4)^{20} = C_0 x^{20} + C_1 x^{19} + C_2 x^{18} + \ldots + C_{20}

tenglik o'rinli bo'lsa,

20 \cdot 19C_0 - 19 \cdot 18C_1 + 18 \cdot 17C_2 - \ldots - 3 \cdot 2C_{17} + 2 \cdot 1C_{18}

ning qiymatini toping.

23.

Aniq integralni hisoblang:

\int_0^{\frac{\pi}{4}} \frac{\sin x}{\cos^3 x}\,dx

24.

ABC

uchburchakning tomonlari

AB = 7

;

BC = 13

AC = 15

ga teng. Uchburchakka ichki chizilgan aylanaga va uchburchakning

AB

AC

tomonlariga urinuvchi aylana chizilgan. Shu kichik aylana radiusini toping.

25.

Teng yonli uchburchakning ikkita ichki burchagi yig'indisi uchinchi ichki burchagiga teng. Eng katta tomoniga tushgan medianasi uzunligi 6 ga teng bo'lsa, uchburchakning yuzini toping.

26.

Katta aylana radiusi

O_1A = 6

ga teng. Agar

\angle ABE = 45^\circ

;

\angle CDE = 60^\circ

\angle O_1AC = 15^\circ

ga teng bo'lsa,

O_1

O_2

nuqtalar orasidagi masofani toping.

27.

Muntazam oltiburchakning ichidagi

P

nuqtadan tomonlarigacha bo'lgan eng qisqa masofalarning yig'indisi 18 ga teng. Muntazam oltiburchakning yuzini toping.

28.

ABCD

parallelogrammning

B

o'tmas burchagi

120^\circ

ga teng.

BC

tomondan

E

nuqta olingan.

\left|\overrightarrow{AE} + \overrightarrow{BD} + \overrightarrow{EC}\right| = 16

\left|\overrightarrow{AE} - \overrightarrow{BD} + \overrightarrow{EC}\right| = 10

bo'lsa,

AED

uchburchakning yuzini toping.

29.

ABCD

parallelogramm o'tkir burchagi

60^\circ

ga, diagonallari kvadratlarining nisbati

\left(\frac{AC^2}{BD^2}\right) = \frac{19}{7}

ga teng.

AD > AB

bo'lsa,

\frac{AD}{AB}

ning qiymatini toping.

30.

\alpha

tekislikda yotmaydigan

T

nuqtadan

TH = 4

perpendikulyar va

\alpha

tekislik bilan

30^\circ

burchak tashkil qiluvchi

TA

;

TB

;

TC

og'malar tushirilgan.

\angle AHB = 90^\circ

;

\angle BHC = 60^\circ

;

\angle AHC = 150^\circ

bo'lsa,

THABC

piramida hajmini toping.

31.

A = \{x \in N \mid 4k + 2\}

B = \{x \in N \mid 6k + 2\}

to'plamlar berilgan.

A \cap B

to'plamni toping.

32.

0; 2; 4; 6; 8

raqamlardan barcha uch xonali sonlar tuzildi (raqamlari takrorlanmaydigan) va hosil bo'lgan sonlar kartochkalarga yozildi. Ixtiyoriy tanlangan kartochkadagi sonni 4 ga karrali bo'lish ehtimolini toping.

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

33-35.

Oktaedrga (muntazam sakkizyoqqa) hajmi

8\sqrt{6}\pi

ga teng bo'lgan shar ichki chizildi. Masalani yechishda

\pi = 3

deb hisoblang.

Javob variantlari:

72\sqrt{2}

72\sqrt{3}

36\sqrt{3}

36\sqrt{2}

36.

\begin{cases} x^2 + xy + y^2 = 133 \\ x + \sqrt{xy} + y = 19 \end{cases}

tenglamalar sistemasini yeching.

(x_n, y_n)

juftliklar berilgan sistemaning yechimlari bo'lsa,

\frac{y_n}{x_n}

ning eng kichik qiymatini toping.

b) Tenglamaning ildizlar sonini toping.

37.

Tenglamani yeching:

\sin^3 x \cdot \sin 3x + \cos^3 x \cdot \cos 3x = \cos^3 6x

a) Tenglamaning

[-\pi; \pi]

oraliqda ildizlari sonini toping.

b) Tenglamaning

(0; 2\pi)

oraliqdagi ildizlari yig'indisini toping.

38.

Chizmada

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

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

funksiyalar grafiklari berilgan.

f(x)

funksiya

x = 2024

nuqtada va

g(x)

funksiya esa

x = 0

nuqtada

Ox

o'qiga urinadi. Ikkala grafik ham

O(0; 0)

(2024; 0)

nuqtalardan o'tadi.

\frac{ap + c}{b + q + d}

ning qiymatini toping.

\frac{g(2025)}{f(2025)} + \frac{g(2023)}{f(2023)}

ning qiymatini toping.

39.

f(x)

funksiyaning uchinchi tartibli (

f'''(x)

) hosilasi mavjud. Bunda

f(x) = \left(\frac{x}{1 - 2^x} - \frac{x}{2}\right)\cos^{2025} x \cdot \sin^{2025}\left(\ln\left(x + \sqrt{1 + x^2}\right)\right)

ga teng.

\frac{f'(2025) - f'(-2025)}{f'(2025) + f'(-2025)}

ning qiymatini toping.

\frac{f''(2) + f''(-2)}{f'''(1) + f'''(-1)}

ning qiymatini toping.

40.

f(x) = \sqrt{ax^2 + bx + c} + d

g(x) = kx^2 + px + q

funksiyalar

A(2; 4)

B(6; 4)

nuqtalarda kesishadi.

g(x)

funksiya

Ox

o'qiga chizmadagidek urinadi.

f(x)

grafigi —

AB

ustidagi yarim aylana (eng yuqori nuqtasi

(4; 6)

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

ning qiymatini toping.

b) Bo'yalgan soha yuzini toping.

41.

ABC

uchburchakka ichki chizilgan aylana markazidan o'tuvchi hamda

AC

asosga parallel

EF

kesma berilgan.

E

F

nuqtalar mos ravishda

AB

BC

tomonlari bilan kesishgan nuqtalar.

S_{EBF} = 37\frac{1}{3}

S_{AEFC} = 46\frac{2}{3}

ga teng,

AC = 14

a) Aylana radiusini toping.

AEFC

to'rtburchak perimetrini toping.

42.

ABCD

trapetsiyada

A

B

burchak bissektrisalari

H

nuqtada,

B

C

burchak bissektrisalari

E

nuqtada,

C

D

burchak bissektrisalari

F

nuqtada,

D

A

burchak bissektrisalari

G

nuqtada kesishadi.

AB = 5

;

CD = 9

E

nuqta

AD

tomonda yotadi.

HF

orqali o'tgan to'g'ri chiziq

CD

tomonni

K

nuqtada kesadi.

\frac{CK}{KD}

ning qiymatini toping.

GF : HE = 3 : 7

bo'lsa,

GH : EF

ning qiymatini toping.

43.

ABC

uchburchakka tashqi chizilgan aylana radiusi

2\sqrt{3}

ga teng.

E

nuqta

AB

tomon o'rtasi va

E

nuqtadan

AC

asosga parallel to'g'ri chiziq

BC

tomonni

F

nuqtada, aylanani esa

M

N

nuqtalarda kesadi.

BE = 2

\angle BAC = 60^\circ

ga teng.

OF

ning qiymatini toping.

MN

ning uzunligini toping.

44.

Asosi

ABCDEF

oltiburchakli piramidaning balandligi

4\sqrt{3}

ga teng. Oltiburchakning barcha burchaklari teng va tomonlari

AB = 1

;

BC = 4

;

CD = 2

;

DE = 4

ga teng.

ABCDEF

oltiburchakning perimetrini toping.

b) Piramida hajmini toping.

45.

Kondan yo'lgacha bo'lgan masofa 8 km ga teng. Yuk bilan kondan zavodga dala bo'ylab borishda 8 km/soat tezlik bilan, yo'l bo'ylab borishda esa 10 km/soat tezlik bilan yuradi. Zavoddan kongacha yuksiz yo'l bo'ylab 20 km/soat tezlik bilan, dala bo'ylab esa 12 km/soat tezlik bilan harakatlanadi.

a) Kondan zavodga yuk bilan eng kam qancha vaqt (daqiqa) sarflaydi?

b) Zavoddan konga yuksiz qaytish uchun eng kam qancha vaqt (daqiqa) sarflaydi?

Savol	A	B	C	D	E	F
33. Shar sirti yuzini toping.
34. Oktaedrning to'la sirti yuzini toping.
35. Oktaedrning hajmini toping.