Matematika (12.10.2024) 2-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;

Natural

a

b

sonlar uchun

a \cdot b = 24

tenglik bajarilsa,

a + 2b - 1

ning eng katta qiymatini toping.

Hisoblang:

81^{0,75} \cdot 32^{-\frac{2}{5}} - 27^{0,(3)} \cdot 16^{-0,5} + 256^{\frac{1}{2}}

A

shahardan

B

shaharga 1-avtomobil o'zgarmas 120 km/h tezlik bilan, 10 daqiqadan so'ng 2-avtomobil 150 km/h tezlik bilan yo'lga chiqdi. Ular

B

shaharga bir vaqtda yetib keldi.

A

B

shaharlar orasidagi masofani toping.

To'g'ri to'rtburchakning eni 25 % ga, bo'yi 20 % ga ortgan bo'lsa, yuzi necha foizga ortgan?

Ifodani soddalashtiring.

\frac{77^{2m}}{7^{m-1} \cdot 11^m \cdot 7^m \cdot 11^{m+1}}

Hisoblang.

\left(\frac{\sqrt{6} + 2\sqrt{3} - \sqrt{2} - 2}{\sqrt{6} + \sqrt{3} - \sqrt{2} - 1}\right)^2

Hisoblang:

\frac{3}{2\sqrt[3]{2} + 2\sqrt[3]{4}} - \frac{3}{2\sqrt[3]{2} - 2\sqrt[3]{4}} + \sqrt[3]{4}

Arifmetik progressiyada

a_{11} = 40

a_{31} = -20

bo'lsa,

\frac{a_{101}}{a_{91}}

ni toping.

Geometrik progressiyaning dastlabki uchta hadi

\log_2 x - 2

;

\log_2(2x)

;

\log_2(4x) + 5

ga teng bo'lsa,

x

ni toping.

10.

Agar

\frac{x}{y} = 2023

bo'lsa,

\frac{x^2 - xy}{x \cdot y}

ning qiymatini toping.

11.

x = 3

y = \sqrt{3}

bo'lsa,

\left(x^3 + xy^2 + \frac{xy(x^2 + y^2)}{x - y}\right) \cdot \left(\frac{xy}{x + y} - x\right) : \frac{x^2(x^2 + y^2)}{x^2 - y^2}

12.

Ifodani soddalashtiring.

\frac{(\sin x - \cos x)^2}{\sin^2\left(\frac{\pi}{4} - x\right)}

13.

Tenglama

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

oraliqda nechta yechimga ega?

\cos 3x - \sin 2x = 0

14.

Tengsizlikni nechta butun son qanoatlantiradi?

7^{3x} + 1 \le 7^{x-1} \cdot (55 \cdot 7^x - 41)

15.

Tenglamaning ildizlar ko'paytmasini toping.

\log_2^2 x + (x - 1)\log_2 x + 2x = 6

16.

Tengsizlikni yeching:

\frac{25x - 5}{|x - 1|} > 0

17.

7x^2 - 6x + 1 = 0

tenglamaning ildizlari

x_1

x_2

bo'lsa, ildizlari

\frac{1}{x_1^2}

\frac{1}{x_2^2}

bo'lgan keltirilgan kvadrat tenglamaning koeffitsiyentlari yig'indisini toping.

18.

Tengsizlikni qanoatlantiruvchi natural sonlar ko'paytmasini toping.

\frac{x^2 - 6x + 8}{x - 1} - \frac{x - 4}{x^2 - 3x + 2} \le 0

19.

(x + 1)(x + 4)(x - 2)(x + 7) = 19

tenglamaning ildizlar yig'indisini toping.

20.

Quyidagi funksiyalardan nechtasi juft funksiya?

y_1 = \sin^2 x + \sqrt[3]{|x|}

;

y_2 = \cos^2 x + \sqrt[3]{x^2}

;

y_3 = \operatorname{tg}^3 x + \sqrt{|x|}

;

y_4 = \operatorname{ctg}^4 x + \sqrt[4]{|x|^3}

21.

Integralni hisoblang:

\int \frac{2x^2 + 1}{x^4 + x^2}\,dx

22.

y = |x^2 - 4x + 3|

funksiyaning

[2; 4]

oraliqda eng katta va eng kichik qiymatlari yig'indisini toping.

23.

Markazi

(4; 3)

nuqtada bo'lgan aylana

Oy

o'qiga urinadi. Shu aylana radiusini toping.

24.

Agar

|x - 5| \le 3

bo'lsa,

9\left(\frac{1}{2}x - 2\right)^2 - \left(\frac{1}{2}x - 2\right)^4

ifodaning eng katta qiymatini toping.

25.

x

ning qabul qilishi mumkin bo'lgan eng katta natural qiymatini toping.

26.

CF = 6

CG = 7

GE = 5

ED = 6

bo'lsa,

FG

kesma uzunligini toping.

27.

EFLM

FGNL

MNKA

ABCD

— kvadratlar berilgan. Agar

EM = 5

bo'lsa,

FNC

uchburchak yuzini toping.

28.

ABCDEF

oltiburchakning yuzi 36 ga teng. Bo'yalgan

BDF

uchburchakning yuzini toping.

29.

ABC

uchburchakning yuzi 60 ga teng.

\overrightarrow{BF} = \frac{1}{2}\overrightarrow{AB}

\overrightarrow{BE} = \frac{1}{2}\overrightarrow{BC}

bo'lsa,

BEF

uchburchak yuzini toping.

30.

Tekislikka ikkita og'ma va perpendikulyar tushirilgan. Agar

AD = 4

\angle ABD = 45^\circ

\angle BAC = 30^\circ

\angle ACD = 30^\circ

bo'lsa, tekislikdagi og'ma uchlari orasidagi masofa (

BC

) ni toping.

31.

A = \{a, b, c, d, n, l\}

B = \{a, b, k, d, m, n\}

bo'lsa,

A \cap B

ning bo'sh bo'lmagan qism to'plamlar sonini toping.

32.

Raqamlari 6 dan kichik bo'lmagan uch xonali 3 ga bo'linadigan natural sonlar nechta?

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

33-35.

Rasmda usti ochiq silindr va yarim shar tasvirlangan. (

\pi \approx 3

)

Javob variantlari:

Savol	A	B	C	D	E	F
33. Yarim shar hajmi silindr hajmiga teng. $AB = 20\ cm$ bo'lsa, shar radiusini toping.
34. $R = AB = 2\ dm$ bo'lsa, jismning (idishning) to'la sirtining yuzini toping.
35. $R = AB = 2\ dm$ bo'lsa, berilgan idishga necha litr suv ketadi?

36.

3x^2 - 4(3a - 2)x + a^2 + 2a = 0

kvadrat tenglama berilgan.

a) Tenglama bitta ildizga ega bo'ladigan

a

ning barcha qiymatlarining yig'indisini toping.

b) Yechimga ega bo'lmaydigan

a

ning butun qiymatlari nechta?

37.

\sin x - \cos x + \sin 3x = \cos 3x + \cos 5x - \sin 5x

tenglamani yeching.

a) Tenglamaning

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

oraliqda yechimlari nechta?

b) Tenglamaning

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

kesmadagi barcha yechimlari yig'indisini toping.

38.

f(x) = x^3 + ax + b

(a \ne b)

funksiyaga

x_0 = a

x_0 = b

nuqtalarda o'tkazilgan urinmalar parallel bo'lsa,

f(1)

ni hisoblang.

b = 2

bo'lsa

f'(1)

ni hisoblang.

39.

y = x^2

parabolaga va ikkita aylana uringan (

A

B

nuqtalar) va ikkita aylana o'zaro uringan (

C

nuqtada).

O_1

(0; y_1)

nuqtalar orasidagi masofani toping.

O_1A = r = 2023

bo'lsa,

O_2B

ni toping.

40.

Funksiya grafigidan

f(2) = a

f(5) = b

berilgan.

\int_2^5 f(x) \cdot f'(x)\,dx = 13{,}5

a + b = 9

bo'lsa,

\frac{a \cdot b}{2}

ni hisoblang.

\int_2^5 f(x)\,dx + \int_a^b f^{-1}(x)\,dx

ni hisoblang (

f^{-1}(x)

—

f(x)

ning teskari funksiyasi).

41.

ABC

uchburchakka aylana ichki chizilgan.

M, N, K

nuqtalar aylananing

AB, AC, BC

tomonlariga mos ravishda urinish nuqtalari. Bunda

BK = 3

KC = 5

\angle ABC = 60^\circ

ga teng.

ABC

uchburchakning yuzini toping.

MN

kesmani toping.

42.

ABCD

parallelogram berilgan.

BH:BC = ED:AD = 1:5

CG:BC = AL:AD = 3:7

bo'lsa,

a) Bo'yalgan soha yuzini parallelogram yuziga nisbatini toping.

ABHOL

beshburchak yuzini parallelogram yuziga nisbatini toping.

43.

ABCDE

beshburchak berilgan.

AB = BC = CD = DE

\angle B = 96^\circ

\angle C = \angle D = 108^\circ

bo'lsa,

\angle A

ning ichki burchagini toping.

\angle E

ning ichki burchagini toping.

44.

Rasmda tasvirlangan shar va kub umumiy qismining (sharning kub ichidagi qismi) hajmi shar hajmining

\frac{1}{12}

, kub hajmining esa

\frac{1}{16}

qismini tashkil qiladi. Muntazam tetraedr va kub umumiy qismining (muntazam tetraedrning kub ichidagi qismi) hajmi muntazam tetraedr hajmining

\frac{1}{10}

, kub hajmining esa

\frac{1}{20}

qismini tashkil qiladi.

a) Shar hajmining muntazam tetraedr hajmiga nisbatini toping.

b) Agar muntazam tetraedrning qirrasi

6\sqrt{2}

ga teng bo'lsa, kub hajmini toping.

45.

Rasmda doiradan markaziy burchagi

\alpha

ga teng bo'lgan sektor qirqib olindi. Qolgan qismidan konus yasaldi (bo'yalgan qism). (

\pi \approx 3

\sqrt{6} \approx 2{,}45

deb oling).

\alpha = 60^\circ

R = 12

ga teng bo'lsa, konus hajmini toping.

b) Konus eng katta hajmga ega bo'lsa,

\alpha

burchakni toping va butun qismigacha yaxlitlang.