Milliy sertifikat — Matematika (04.12.2022)

45 ta savol · natijani saqlash uchun tizimga kiring

Matematika fanidan asosiy formulalar / Основные формулы математики

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;

a, b, c

− turli raqamlar bo'lsa,

100a + 10b + c

ning eng katta qiymatini toping.

\left(2022 - \frac{1}{2022}\right) : \frac{2021}{2022} \cdot \frac{1}{2023}

ni hisoblang.

2 va 3 ga bo'linmaydigan barcha ikki xonali natural sonlar yig'indisini toping.

A

shahardan

B

shaharga ikkita mashina yo'lga chiqdi. Tezliklari

v_1

v_2

(

v_1 > v_2

). Birinchi mashina

B

ga borib shu zahoti qaytib ikkinchisi bilan uchrashdi. Ikkinchi mashina bosib o'tgan masofa

AB

masofaning necha foizini tashkil etadi?

3 ta quvur berilgan. 1-quvur basseynni 5 soatda to'ldiradi, 2-quvur 3 soatda to'ldiradi, 3-quvur 2 soatda bo'shatadi. 3 ta quvur bir vaqtda necha soatda basseynni to'ldiradi?

Hisoblang.

\sqrt[3]{\frac{12}{25} \cdot \sqrt{\frac{244}{15 \cdot (38^2 - 23^2)}}}

Ishorasi almashinuvchi geometrik progressiyada

b_1 = a - 5

;

b_2 = a + 4

;

b_3 = 5a + 8

bo'lsa,

b_4

ni toping.

\frac{1}{(x+y)^2} \cdot \left(\frac{1}{x^2} + \frac{1}{y^2}\right) + \frac{2}{(x+y)^3} \cdot \left(\frac{1}{x} + \frac{1}{y}\right)

ni soddalashtiring.

2x^2 - 5x + 4 = 0

tenglama nechta haqiqiy ildizga ega?

2\sin\frac{7\pi}{6} + \cos^2\frac{7\pi}{4}

ni hisoblang.

a > 5

bo'lsa,

\sqrt{(3 - a)^2} - \sqrt{(a - 5)^2}

ni hisoblang.

\frac{a}{b} = 4, \frac{b}{c} = 10

bo'lsa,

\frac{a^2+b^2+c^2}{ac} + \frac{a}{c}

ni hisoblang.

\frac{2\cos\alpha + \cos 3\alpha + \cos 5\alpha}{\cos 3\alpha + \sin\alpha \cdot \sin 2\alpha}

ni soddalashtiring.

3 \cdot 4^x + 2 \cdot 25^x = 5 \cdot 10^x

tenglama nechta haqiqiy ildizga ega?

Toq funksiyani toping.

x^2 + 4x + 1 = 2\sqrt{x^2 + 4x + 4}

tenglamaning haqiqiy ildizlar yig'indisini toping.

||2x - 1| - 7| \le 5

tengsizlikni qanoatlantiradigan butun yechimlari nechta?

\frac{\sqrt{4-x^2}}{x+1} \ge 0

tengsizlikni yeching.

32^3 \cdot 8^{12} \cdot 125^{17}

necha xonali son?

\log_{\cos 2x}(\sin 2x) \le 1

tengsizlikni yeching.

f(g^{-1}(g(x)) + 1) = x^2 + 5x + 6

berilgan.

g^{-1}(x)

−

g(x)

ning teskari funksiyasi.

f(-1)

ni toping.

f(x) = \sin^2 5x - |\cos 2x + x|

funksiya berilgan.

f'\left(-\frac{\pi}{6}\right)

ni toping.

Chizmadagi ma'lumotlardan foydalanib

x

ni toping. (

AB \parallel EF

). Chizmada

ABC

uchburchak,

AC

tomonida

F

BC

tomonida

E

nuqta.

AB=16, AC=20, EF=12, EC=x

Tekislikka ikkita

AB

BC

og'ma va

BH

perpendikulyar tushirilgan.

BH = HC

AB = 2HC

\angle AHC = 90^\circ

bo'lsa,

\cos\angle BAC

ni toping.

Tomoni 4 ga teng kvadratga doira ichki chizilgan. Doiraning yuzini toping.

ABC

uchburchakda

\angle BAC = 90^\circ

\angle BCA = 30^\circ

BD

bissektrisa

2\sqrt{2}

ga teng bo'lsa,

ABC

uchburchak yuzini toping.

Diagonallar soni tomonlar sonidan 6 marta ko'p bo'lgan ko'pburchakning ichki burchaklar yig'indisini toping.

Rombning katta diagonali

10\sqrt{4 + 2\sqrt{2}}

ga teng, o'tkir burchagi

45^\circ

bo'lsa, diagonallari kesishish nuqtasidan tomonlarigacha bo'lgan eng qisqa masofalar yig'indisini toping.

\int_{-3}^{4} 2x^2|2x|\, dx

ni hisoblang.

ABC

uchburchakning yuzi 60 ga teng.

\vec{BE} = \vec{AB}

bo'lsa,

AEC

uchburchakning yuzini toping.

(A \cap B) \cup (A \cup B)'

ning elementlar sonini toping. (Eyler-Venn:

A

da faqat o'zida 1, 4, 9, 10, 11, 12; kesishmada 3, 5, 7, 8;

B

da faqat o'zida 8, 14, 17.)

Merganning nishonga tekkizish ehtimolligi 0,8. 5 marta o'q uzilgan. Dastlabki 3 o'q tegib, keyingi 2 o'q tegmasligi ehtimolligini toping.

33-35

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

Muntazam oltiburchakli piramidaga shar ichki chizilgan. Shar ichiga eng katta hajmli konus ichki chizilgan. Piramida asosining tomoni 6 ga, yon yog'i asos tekisligi bilan

60^\circ

tashkil etadi. (

\pi \approx 3

Savol	A) $162\sqrt{3}$	B) 108	C) 32	D) 81	E) $81\sqrt{3}$	F) 27
33. Piramida hajmini toping.
34. Shar hajmini toping.
35. Konus hajmini toping.

\frac{6x-k}{x-1} = 3 - \frac{2x+k}{x+1}

tenglama berilgan.

a) Tenglama bitta ildizga ega bo'ladigan

k

ning eng katta qiymatini toping.

b) Tenglama bitta ildizga ega bo'ladigan

k

ning barcha qiymatlari yig'indisini toping.

3(\log_2 \sin x)^2 + \log_2(1 - \cos 2x) = 2

tenglamani yeching.

a) Tenglamaning eng kichik musbat ildizini toping.

b) Tenglamaning

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

kesmada nechta ildizi bor?

g(x) = kx + b

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

funksiya grafiklari berilgan. Grafikdan foydalanib hisoblang. (Grafikda

f(x)

uchi

(0; 4)

x

o'qini

-2

+2

da kesadi.)

f'(2)

ni hisoblang.

f'\left(\frac{1}{a}\right) - 5f'\left(\frac{1}{b}\right) + 2f'\left(\frac{1}{c}\right)

ni hisoblang.

f(x) = x^4 - 2x^3 + 2x^2 - 2x + 1

bo'lsa,

f(1 + \sqrt{2}) + f(1 - \sqrt{2})

ni hisoblang.

f(x) = 0

nechta turli haqiqiy ildizga ega?

Rasmda

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

parabola va

g(x) = kx + l

to'g'ri chiziq grafiklari tasvirlangan.

(-2; 0)

(x_0; y_0)

nuqtalar ularning kesishgan nuqtalari bo'lsa,

x_0 + y_0

ni hisoblang.

b) Bo'yalgan sohaning yuzini toping.

ABC

uchburchakda

AB = 7, BC = 10, AC = 15

. Uchburchakka ikkita aylana shunday ichki chizilganki,

R

radiusli aylana uchburchak tomonlariga,

r

radiusli aylana esa

AC, BC

R

radiusli aylanaga urinadi.

R = ?

r = ?

Yuzi 42 ga teng bo'lgan

ABCD

trapetsiya.

BC

yon tomonidan

K

L

nuqtalar olingan.

\frac{CL}{LK} = \frac{3}{2}; \frac{KB}{LK} = 2

\frac{DC}{AB} = \frac{5}{9}

bo'lsa,

AKB

uchburchakning yuzini toping.

AKLD

yuzini toping.

Rasmda aylananing

\frac{1}{4}

qismi tasvirlangan.

AC = AB = R

(aylana radiusi),

CD = 1, BD = 3\sqrt{2}

bo'lsa,

\sin\angle CAD

ni toping.

CAD

uchburchakning yuzini toping.

Silindrga oktaedr ichki chizilgan. Oktaedrning hajmi 36 ga teng.

a) Oktaedrning qirrasini toping.

b) Silindr hajmini toping.

Tomonlari 2 va 10 ga teng to'g'ri to'rtburchakli metaldan asoslari ikkita doira va yon sirti qirqib olinib silindr yasaldi. (

\pi \approx 3

a) Eng kam chiqindi bilan silindr yasalsa, uning hajmini toping.

b) Silindr yasashdan hosil bo'lgan chiqindi metalning necha foizini tashkil etadi?