Matematika (25.05.2025) 1-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;

42^n

n \in N

ning qanday eng kichik qiymatida

49^5 \cdot 12^3

soniga qoldiqsiz bo'linadi?

Hisoblang:

\sqrt{\frac{2026!}{2025! + 2024!}} - \sqrt{\frac{10!}{9! + 8!}}

1 soatda Anvar 20 ta detal, Sardor 21 ta detal ishlab chiqardi. Sardor 10 daqiqa oldin ishni tugatgan bo'lsa, har biri nechtadan detall yasagan?

Zavodda 4020 ta avtomobil 1 oyda ishlab chiqarildi. Bu bir oylik rejaning 120% ni tashkil etadi. Zavoda rejadan tashqari nechta avtomobil ishlab chiqarilgan?

Hisoblang:

\frac{2^{98} + 1}{2^{49} + 2^{25} + 1}

Sonlarni kamayish tartibida yozing.

a = \sqrt[6]{26}

b = \sqrt[3]{5}

c = \sqrt{3}

Agar

2025^3 - 2023^3 = a

bo'lsa,

\sqrt{\frac{a - 2}{6}}

ni toping.

Agar arifmetik progressiyada

a_1 + a_m = 90

S_m = 270

ga teng bo'lsa,

m

ning qiymatini toping.

Arifmetik va geometrik progressiya hadlari uchun

a_1 = b_1 = 1

a_9 = b_9

a_1 + a_2 + a_3 + \ldots + a_9 = 369

bo'lsa,

b_7

ni toping.

10.

Agar

\left(\frac{1}{125}\right)^{a^2 + 4ab} = \left(\sqrt[3]{625}\right)^{3a^2 - 10ab}

tenglik o'rinli bo'lsa,

\frac{a}{b}

ni toping.

11.

Agar

a + b + c = 0

a^2 + b^2 + c^2 = 1

bo'lsa,

a^4 + b^4 + c^4

ning qiymatini toping.

12.

Hisoblang:

\arcsin\left(-\frac{\sqrt{3}}{2}\right) + \operatorname{arctg}\left(\frac{\sqrt{3}}{3}\right)

13.

Agar

\sin\alpha - \cos\alpha = \frac{1}{\sqrt{5}}

\alpha \in \left(0; \frac{\pi}{2}\right)

ga teng bo'lsa,

\sin 2\alpha - \cos 2\alpha

ning qiymatini toping.

14.

Tenglama nechta yechimga ega?

222^{3x} + 222^x = 222

15.

Tenglama nechta haqiqiy ildizga ega.

\left(\frac{1}{4}\right)^{\frac{\log_{12}\log_8 x}{\log_{12} 0{,}25}} = \log_{\frac{1}{8}} 3

16.

|x - 4| - \sqrt{4x + 17} = -3

tenglamaning haqiqiy ildizlari yig'indisini toping.

17.

Tenglamaning haqiqiy ildizlari yig'indisini toping (agar u bitta bo'lsa o'sha ildizini toping).

x\sqrt{x} - 4x + 3\sqrt{x} = 4x - 16\sqrt{x} + 12

18.

[-20; 20]

kesmada tengsizlik nechta butun yechimga ega?

\sqrt{x^2 - 10x + 16} < 2x - 4

19.

\sqrt{x^2 + 9} < 5

nechta natural son tengsizlikni qanoatlantiradi?

20.

Agar

g(x) = 6x - 5

g(f(x)) = x^2

ga teng bo'lsa,

f(2)

ni toping.

21.

Aniq integralni hisoblang:

\int_{\frac{\pi}{2}}^{\frac{2\pi}{3}} \frac{1}{1 + \sin x + \cos x}\,dx

22.

Agar

f(x)

g(x)

funksiyalar uchun

g(x) = \frac{x + 1}{x}

g(f(x)) = \frac{x - 1}{x - 2}

bo'lsa,

f(2023)

ni toping.

23.

Rasmda aylana va to'g'ri to'rtburchak tasvirlangan. Aylana bilan chegaralangan soha yuzining to'g'ri to'rtburchak yuziga nisbatini toping.

24.

f(x) = \frac{\cos^2 x + 6\cos x + 9}{\sin^2 x}

funksiyaning

\left[-\frac{\pi}{3}; \frac{3\pi}{2}\right]

oraliqdagi eng kichik qiymatini toping.

25.

ABC

uchburchakda

BC = 8

AB = 13

\angle AOB = 150^\circ

ga teng bo'lsa,

ABC

uchburchak yuzini toping.

26.

ABCD

parallelogramning

A

B

burchak bissektrisalari

F

nuqtada kesishadi. Agar parallelogram yuzi 48 ga teng bo'lsa,

AFD

uchburchak yuzini toping.

27.

Muntazam

ABC

uchburchak ichidan

F

nuqta olingan. Bunda

F

nuqtadan

AC

BC

tomonlargacha bo'lgan eng qisqa masofalar teng.

F

nuqtadan

AB

tomongacha bo'lgan eng qisqa masofa 1 ga teng. Agar

AB = 4\sqrt{3}

ga teng bo'lsa,

F

nuqtadan

AC

tomongacha bo'lgan eng qisqa masofani toping.

28.

Rasmda muntazam sakkizburchak tasvirlangan.

\frac{S_2}{S_1}

nisbatni toping.

29.

Rasmda muntazam oltiburchak tasvirlangan.

E

nuqtaning koordinatalar ko'paytmasini toping.

30.

Uchburchakning tomonlari 10, 17 va 21 ga teng. Uchburchakning katta burchagi uchidan uchburchak tekisligiga perpendikulyar o'tkazilgan bo'lib, uning uzunligi 15 ga teng. Bu perpendikulyarning tekislik bilan kesishmagan uchidan uchburchakning katta tomonigacha bo'lgan masofani aniqlang.

31.

Rasmdagi ma'lumotlar asosida

(A \cup B) \cap C

ning qism to'plamlar sonini toping.

32.

Seyf maxfiy 6 kod bilan ochiladi. Kodlar raqamlardan iborat. Eng ko'pi bilan seyfni necha xil usulda ochish mumkin?

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

33-35.

Rasmda qirralari 2 ga teng bo'lgan muntazam oltiburchakli prizma tasvirlangan.

36.

Tenglamalar sistemasi berilgan bo'lsin

\begin{cases} x^2 + y^2 = 36 \\ |x| + y = a \end{cases}

a) Tenglamalar sistemasi 3 ta yechimga ega bo'ladigan

a

lar yig'indisini toping (agar bitta bo'lsa o'zini toping).

b) Tenglamalar sistemasi 4 ta yechimga ega bo'ladigan

a

ning barcha butun qiymatlari yig'indisini toping (agar bitta bo'lsa o'zini toping).

37.

Tenglamaning

\frac{0{,}5 - \cos 2x}{\cos 2x - \sin x - 1} = \cos 2x + \sin x - 1

a) Tenglamaning eng kichik musbat va eng katta manfiy yechimlari yig'indisi toping.

b) Tenglama

[-\pi; \pi]

oraliqda nechta yechimga ega.

38.

Rasmda berilgan grafikda

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

g(x) = k\sqrt{x + l}

berilgan.

f(x)

funksiyasi

Ox

(m; 0)

(5; 0)

nuqtalarida kesib o'tgan va parabola uchi

(2; -4{,}5)

bo'lsin.

f(x)

bilan

g(x)

kesishish

(0; n)

(m; 0)

nuqtalarida bo'lsin.

f(-2)

ni toping.

\frac{a - b - l - c}{k - m - n}

ni toping.

39.

Agar

f(x)

funksiyada

f(0) = 1

f'(0) = 2

f''(x) = 4f'(x) - 3f(x) + 1

lar berilgan.

f''(x)

f^{IV}(x)

lar

f(x)

funksiyaning mos ravishda ikkinchi va to'rtinchi darajali hosilalari.

f''(0)

ni toping.

f^{IV}(0)

ni toping.

40.

Rasmda berilgan ma'lumotlardan foydalanib,

B

nuqta koordinatalari yig'indisini toping.

b) Bo'yalgan soha yuzini ya'ni uchburchak yuzini toping.

41.

ABC

muntazam uchburchak bo'lsa

OC = \frac{14}{\sqrt{3}}

PB = 10

AP = 6

ga teng bo'lsa,

a) Bo'yalgan soha yuzini toping.

PC

ni uzunligini toping.

42.

Tomonlari 13, 13, 15, 15 qavariq to'rtburchakka aylana ichki chizilgan. Diagonallardan biri diagonallar kesishish nuqtasida

5:9

nisbatda bo'linadi.

a) To'rtburchakning yuzini toping.

b) Diagonallari kesishish nuqtasidan aylana markazigacha bo'lgan masofani toping.

43.

Rasmda muntazam

ABCDEF

oltiburchak berilgan.

a) Muntazam oltiburchakning tomoni 4 ga teng bo'lsa,

PDC

uchburchak yuzini toping.

EPF

uchburchakning yuzini

ABCP

to'rtburchak yuziga nisbatini toping.

44.

Muntazam oltiburchakli kesik piramidaning ichida kub joylashgan. Kubning yuqori asosi kesik piramidaning yuqori asosida, kubning pastki asosi piramidaning pastki asosida joylashgan. Kesik piramidaning kichik asosi yuzi katta asosi yuzidan 4 marta kichik.

a) Kesik piramidaning kichik asosining tomon

1 + \sqrt{3}

ga teng bo'lsa, kub hajmini toping.

b) Kubning to'la sirti

24 - 12\sqrt{3}

ga teng bo'lsa, kesik piramidaning katta asosini tomonini toping.

45.

Maktab o'quvchisi uyidan maktabgacha yo'l kartasi ko'rsatilgan bo'lsin. Maktabdan daryogacha bo'lgan masofa

AB = 30\ km

, daryodagi ko'prik uzunligi

DE = 20\ km

ga teng (ko'prikning qalinligi hisobga olinmasin) va daryodan uygacha masofa

FC = 50\ km

bo'lsin. Agar maktabdan uygacha eng qisqa masofa

AC = 20\sqrt{34}\ km

bo'lsa,

BD

ni toping?

AD

DE

EC

oraliqlar yig'indisini toping.

Savol	A	B	C	D	E	F
33. $ACD_1F_1$ to'rtburchak yuzini toping.
34. $B_1CDE_1$ to'rtburchak yuzini toping.
35. $ABE_1$ nuqtalardan kesib o'tuvchi soha yuzini toping.