The figure shows a graph of the derivative of the function f(x) defined on the interval [–5; 6]. Find the number of points of the graph f (x), in each of which the tangent drawn to the graph of the function coincides or is parallel to the x-axis

The figure shows a graph of the derivative of a differentiable function y = f(x).

Find the number of points in the graph of the function that belong to the segment [–7; 7], in which the tangent to the graph of the function is parallel to the straight line given by the equation y = –3x.

Material point M starts from point A and moves in a straight line for 12 seconds. The graph shows how the distance from point A to point M changed over time. The abscissa shows the time t in seconds, the ordinate shows the distance s in meters. Determine how many times during the movement the speed of point M went to zero (ignore the beginning and end of the movement).

The figure shows sections of the graph of the function y \u003d f (x) and the tangent to it at the point with the abscissa x \u003d 0. It is known that this tangent is parallel to the straight line passing through the points of the graph with the abscissas x \u003d -2 and x \u003d 3. Using this, find the value of the derivative f "(o).

The figure shows a graph y = f'(x) - the derivative of the function f(x), defined on the segment (−11; 2). Find the abscissa of the point at which the tangent to the graph of the function y = f(x) is parallel to the x-axis or coincides with it.

The material point moves rectilinearly according to the law x(t)=(1/3)t^3-3t^2-5t+3, where x is the distance from the reference point in meters, t is the time in seconds measured from the beginning of the movement. At what point in time (in seconds) was her speed equal to 2 m/s?

The material point moves along a straight line from the initial to the final position. The figure shows a graph of its movement. Time in seconds is plotted on the abscissa axis, distance from the initial position of the point (in meters) is plotted on the ordinate axis. Find average speed dot movement. Give your answer in meters per second.

The function y \u003d f (x) is defined on the interval [-4; 4]. The figure shows a graph of its derivative. Find the number of points in the graph of the function y \u003d f (x), the tangent in which forms an angle of 45 ° with the positive direction of the Ox axis.

The function y \u003d f (x) is defined on the segment [-2; 4]. The figure shows a graph of its derivative. Find the abscissa of the point of the graph of the function y \u003d f (x), in which it takes the smallest value on the segment [-2; -0.001].

The figure shows the graph of the function y \u003d f (x) and the tangent to this graph, drawn at the point x0. The tangent is given by the equation y = -2x + 15. Find the value of the derivative of the function y = -(1/4)f(x) + 5 at the point x0.

Seven points are marked on the graph of the differentiable function y = f(x): x1,..,x7. Find all marked points where the derivative of the function f(x) is greater than zero. Enter the number of these points in your answer.

The figure shows the graph y \u003d f "(x) of the derivative of the function f (x), defined on the interval (-10; 2). Find the number of points at which the tangent to the graph of the function f (x) is parallel to the line y \u003d -2x-11 or matches it.

The figure shows a graph of y \u003d f "(x) - the derivative of the function f (x). Nine points are marked on the x-axis: x1, x2, x3, x4, x5, x6, x6, x7, x8, x9.
How many of these points belong to the intervals of decreasing function f(x) ?

The figure shows the graph of the function y \u003d f (x) and the tangent to this graph, drawn at the point x0. The tangent is given by the equation y = 1.5x + 3.5. Find the value of the derivative of the function y \u003d 2f (x) - 1 at the point x0.

The figure shows a graph y=F(x) of one of the antiderivatives of the function f (x). Six points with abscissas x1, x2, ..., x6 are marked on the graph. At how many of these points does the function y=f(x) take negative values?

The figure shows the schedule of the car along the route. Time is plotted on the abscissa axis (in hours), on the ordinate axis - the distance traveled (in kilometers). Find the average speed of the car on this route. Give your answer in km/h

The material point moves rectilinearly according to the law x(t)=(-1/6)t^3+7t^2+6t+1, where x is the distance from the reference point (in meters), t is the time of movement (in seconds). Find its speed (in meters per second) at time t=6 s

The figure shows a graph of the antiderivative y \u003d F (x) of some function y \u003d f (x), defined on the interval (-6; 7). Using the figure, determine the number of zeros of the function f(x) in a given interval.

The figure shows a graph y = F(x) of one of the antiderivatives of some function f(x) defined on the interval (-7; 5). Using the figure, determine the number of solutions to the equation f(x) = 0 on the segment [- 5; 2].

The figure shows a graph of a differentiable function y=f(x). Nine points are marked on the x-axis: x1, x2, ... x9. Find all marked points where the derivative of f(x) is negative. Enter the number of these points in your answer.

The material point moves rectilinearly according to the law x(t)=12t^3−3t^2+2t, where x is the distance from the reference point in meters, t is the time in seconds measured from the beginning of the movement. Find its speed (in meters per second) at time t=6 s.

The figure shows the graph of the function y=f(x) and the tangent to this graph drawn at the point x0. The tangent equation is shown in the figure. find the value of the derivative of the function y=4*f(x)-3 at the point x0.

The exam program, as in previous years, is made up of materials from the main mathematical disciplines. The tickets will include mathematical, geometric, and algebraic problems.

There are no changes in KIM USE 2020 in mathematics at the profile level.

Features of USE assignments in mathematics-2020

When preparing for the exam in mathematics (profile), pay attention to the basic requirements of the examination program. It is designed to test the knowledge of an in-depth program: vector and mathematical models, functions and logarithms, algebraic equations and inequalities.
Separately, practice solving tasks for.
It is important to show non-standard thinking.

Exam Structure

USE assignments profile mathematics divided into two blocks.

Part - short answers, includes 8 tasks that test basic mathematical training and the ability to apply knowledge of mathematics in everyday life.
Part - brief and detailed answers. It consists of 11 tasks, 4 of which require a short answer, and 7 - a detailed one with an argumentation of the actions performed.

Increased complexity- tasks 9-17 of the second part of KIM.
High level difficulties- tasks 18-19 –. This part of the exam tasks checks not only the level of mathematical knowledge, but also the presence or absence of a creative approach to solving dry "number" tasks, as well as the effectiveness of the ability to use knowledge and skills as a professional tool.

Important! Therefore, in preparation for USE theory in mathematics, always support the solution of practical problems.

How will points be distributed?

The tasks of the first part of KIMs in mathematics are close to USE tests basic level, That's why high score it's impossible to get them.

The points for each task in mathematics at the profile level were distributed as follows:

for correct answers to tasks No. 1-12 - 1 point each;
No. 13-15 - 2 each;
No. 16-17 - 3 each;
No. 18-19 - 4 each.

The duration of the exam and the rules of conduct for the exam

To complete the exam -2020 the student is assigned 3 hours 55 minutes(235 minutes).

During this time, the student should not:

be noisy;
use gadgets and other technical means;
write off;
try to help others, or ask for help for yourself.

For such actions, the examiner can be expelled from the audience.

On State exam mathematics allowed to bring only a ruler with you, the rest of the materials will be given to you immediately before the exam. issued on the spot.

Effective preparation is the solution online tests Math 2020. Choose and get the highest score!

I present the solution of task 7 of the OGE-2016 in informatics from the demo project. Compared to the 2015 demo, task 7 has not changed. This is a task for the ability to encode and decode information (Coding and decoding information). The answer to task 7 is a sequence of letters, which should be written in the answer field.

Screenshot of task 7.

Exercise:

The scout sent a radiogram to the headquarters
– – – – – – – –
This radiogram contains a sequence of letters in which only the letters A, D, G, L, T occur. Each letter is encoded using Morse code. There are no separators between letter codes. Write down the given sequence of letters in the answer.
The required fragment of Morse code is given below.

Answer: __

Such a task is best solved sequentially, closing each possible code.
1. (-) - - - - - - -, the first two positions can only be the letter A
2.
a) (-) (-) - - - - - -, the following three positions can be the letter D
b) (-) (-) - - - - - -, or one position letter L, but if we take the following combination (-) (-) (-) - - - - -, (letter T) then we will not choose more we can (there are simply no such combinations starting with two points), so we have reached a dead end and we conclude that this path is wrong
3. We return to option a)
(-) (- ) (- ) - - - - -, this is the letter Zh
4. (-) (-) (-) (-) - - - -, this is the letter L
5. (-) (-) (-) (-) (-) - - -, this is the letter D
6. ( -) (- ) ( - ) (-) (- ) (-) - -, and this is the letter L
7. (-) (- ) (- ) (-) (- ) (-) (-) -, letter A
8. (-) (- ) (- ) (-) (- ) (-) (-) (-), letter L
9. We collect all the letters that we got: AJLDLAL.

Answer: AJLDLAL

Average general education

UMK line G. K. Muravina. Algebra and beginnings mathematical analysis(10-11) (deep)

Line UMK Merzlyak. Algebra and the Beginnings of Analysis (10-11) (U)

Mathematics

Preparation for the exam in mathematics ( profile level): tasks, solutions and explanations

We analyze tasks and solve examples with the teacher

Examination paper profile level lasts 3 hours 55 minutes (235 minutes).

Minimum Threshold- 27 points.

The examination paper consists of two parts, which differ in content, complexity and number of tasks.

The defining feature of each part of the work is the form of tasks:

part 1 contains 8 tasks (tasks 1-8) with a short answer in the form of an integer or a final decimal fraction;
part 2 contains 4 tasks (tasks 9-12) with a short answer in the form of an integer or a final decimal fraction and 7 tasks (tasks 13-19) with a detailed answer (full record of the decision with the rationale for the actions performed).

Panova Svetlana Anatolievna, mathematic teacher the highest category schools, 20 years of work experience:

"In order to receive school certificate, the graduate must pass two mandatory exams in USE form, one of which is mathematics. In accordance with the Development Concept mathematics education V Russian Federation The USE in mathematics is divided into two levels: basic and specialized. Today we will consider options for the profile level.

Task number 1- checks with USE participants the ability to apply the skills acquired in the course of 5-9 classes in elementary mathematics in practical activities. The participant must have computer skills, be able to work with rational numbers, be able to round decimals be able to convert one unit of measurement to another.

Example 1 An expense meter was installed in the apartment where Petr lives cold water(counter). On the first of May, the meter showed an consumption of 172 cubic meters. m of water, and on the first of June - 177 cubic meters. m. What amount should Peter pay for cold water for May, if the price of 1 cu. m of cold water is 34 rubles 17 kopecks? Give your answer in rubles.

Solution:

1) Find the amount of water spent per month:

177 - 172 = 5 (cu m)

2) Find how much money will be paid for the spent water:

34.17 5 = 170.85 (rub)

Answer: 170,85.

Task number 2- is one of the simplest tasks of the exam. The majority of graduates successfully cope with it, which indicates the possession of the definition of the concept of function. Task type No. 2 according to the requirements codifier is a task for using acquired knowledge and skills in practical activities and Everyday life. Task No. 2 consists of describing, using functions, various real relationships between quantities and interpreting their graphs. Task number 2 tests the ability to extract information presented in tables, diagrams, graphs. Graduates need to be able to determine the value of a function by the value of the argument with various ways of specifying the function and describe the behavior and properties of the function according to its graph. It is also necessary to be able to find the largest or smallest value from the function graph and build graphs of the studied functions. The mistakes made are of a random nature in reading the conditions of the problem, reading the diagram.

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Example 2 The figure shows the change in the exchange value of one share of a mining company in the first half of April 2017. On April 7, the businessman purchased 1,000 shares of this company. On April 10, he sold three-quarters of the purchased shares, and on April 13 he sold all the remaining ones. How much did the businessman lose as a result of these operations?

Solution:

2) 1000 3/4 = 750 (shares) - make up 3/4 of all purchased shares.

6) 247500 + 77500 = 325000 (rubles) - the businessman received after the sale of 1000 shares.

7) 340,000 - 325,000 = 15,000 (rubles) - the businessman lost as a result of all operations.

1. A)$\frac(\pi )(2)+\pi k; \, \pm \frac(2\pi )(3)+2\pi k;\, k\in \mathbb(Z) $
  b)$\frac(9\pi )(2);\frac(14\pi )(3);\frac(16\pi )(3);\frac(11\pi )(2) $ A) Solve the equation $2\sin \left (2x+\frac(\pi )(6) \right)+ \cos x =\sqrt(3)\sin (2x)-1 $.
  b) Find its solutions that belong to the interval $\left $.
2. A)$\frac(\pi )(2)+\pi k; \, \pm \frac(\pi )(3)+2\pi k;\, k\in \mathbb(Z) $
  b)$\frac(5\pi )(2);\frac(7\pi )(2);\frac(11\pi )(3) $ A) Solve the equation $2\sin \left (2x+\frac(\pi )(6) \right)-\cos x =\sqrt(3)\sin (2x)-1 $.
  b) Find its solutions that belong to the interval $\left [\frac(5\pi )(2); 4\pi\right ] $.
3. A)
  b)$-\frac(5\pi )(2);-\frac(3\pi )(2);-\frac(5\pi )(4) $ A) Solve the equation $\sqrt(2)\sin\left (2x+\frac(\pi )(4) \right)+\sqrt(2)\cos x= \sin (2x)-1 $.
  b) Find its solutions that belong to the interval $\left [-\frac(5\pi )(2); -\pi \right ] $.
4. A)$\frac(\pi )(2)+\pi k; \, \pm \frac(5\pi )(6)+2\pi k;\, k\in \mathbb(Z) $
  b)$\frac(7\pi )(6);\frac(3\pi )(2);\frac(5\pi )(2) $ A) Solve the equation $\sqrt(2)\sin\left (2x+\frac(\pi )(4) \right)+\sqrt(3)\cos x= \sin (2x)-1 $.
  b) Find its solutions that belong to the interval $\left [ \pi; \frac(5\pi )(2) \right ] $.
5. A)$\pm \frac(\pi )(2)+2\pi k; \pm \frac(2\pi )(3)+2\pi k,k\in \mathbb(Z) $
  b)$-\frac(11\pi )(2); -\frac(16\pi )(3); -\frac(14\pi )(3); -\frac(9\pi )(2) \ ) A) Solve the equation \(\sqrt(2)\sin\left (2x+\frac(\pi )(4) \right)+\cos x= \sin (2x)-1 $.
  b) Find its solutions that belong to the interval $\left [-\frac(11\pi )(2); -4\pi \right ] $.
6. A)$\frac(\pi )(2)+\pi k; \, \pm \frac(\pi )(6)+2\pi k;\, k\in \mathbb(Z) $
  b)$-\frac(23\pi )(6);-\frac(7\pi )(2);-\frac(5\pi )(2) $ A) Solve the equation $2\sin\left (2x+\frac(\pi )(3) \right)-3\cos x= \sin (2x)-\sqrt(3) $.
  b) Find its solutions that belong to the interval $\left [-4\pi; -\frac(5\pi )(2) \right ] $.
7. A)$\frac(\pi )(2)+\pi k; \, \pm \frac(3\pi )(4)+2\pi k;\, k\in \mathbb(Z) $
  b)$\frac(13\pi )(4);\frac(7\pi )(2);\frac(9\pi )(2) $ A) Solve the equation $2\sin \left (2x+\frac(\pi )(3) \right)+\sqrt(6)\cos x=\sin (2x)-\sqrt(3) $.
  b) Find its solutions that belong to the interval $\left $.
1. A)$(-1)^k \cdot \frac(\pi)(4) +\pi k, k\in \mathbb(Z) $
  b)$-\frac(13\pi)(4) $ A) Solve the equation $\sqrt(2)\sin x+2\sin\left (2x-\frac(\pi)(6) \right)=\sqrt(3)\sin(2x)+1 $.
  b)
2. A)
  b)$2\pi; 3\pi; \frac(7\pi)(4) $ A) Solve the equation $\sqrt(2)\sin\left (2x+\frac(\pi)(4) \right)-\sqrt(2)\sin x=\sin(2x)+1 $.
  b) Find its solutions that belong to the interval $\left [ \frac(3\pi)(2); 3\pi \right ] $.
3. A)$\pi k, (-1)^k \cdot \frac(\pi)(3) +\pi k, k\in \mathbb(Z) $
  b)$-3\pi; -2\pi; -\frac(5\pi)(3) $ A) Solve the equation $\sqrt(3)\sin x+2\sin\left (2x+\frac(\pi)(6) \right)=\sqrt(3)\sin(2x)+1 $.
  b) Find its solutions that belong to the interval $\left [ -3\pi ; -\frac(3\pi)(2)\right ] $.
4. A)$\pi k; (-1)^(k) \cdot \frac(\pi)(6)+\pi k; k\in \mathbb(Z) $
  b)$-\frac(19\pi )(6); -3\pi ; -2\pi $ A) Solve the equation $\sin x+2\sin\left (2x+\frac(\pi)(6) \right)=\sqrt(3)\sin(2x)+1 $.
  b) Find its solutions that belong to the interval $\left [ -\frac(7\pi)(2); -2\pi \right ] $.
5. A)$\pi k; (-1)^(k+1) \cdot \frac(\pi)(6)+\pi k; k\in \mathbb(Z) $
  b)$\frac(19\pi )(6); 3\pi ; 2\pi $ A) Solve the equation $2\sin \left (2x+\frac(\pi )(3) \right)-\sqrt(3)\sin x = \sin (2x)+\sqrt(3) $.
  b) Find its solutions that belong to the interval $\left $.
6. A)$\pi k; (-1)^(k+1) \cdot \frac(\pi)(4) +\pi k, k\in \mathbb(Z) $
  b)$-3\pi; -\frac(11\pi)(4); -\frac(9\pi)(4); -2\pi $ A) Solve the equation $\sqrt(6)\sin x+2\sin \left (2x-\frac(\pi )(3) \right) = \sin (2x)-\sqrt(3) $.
  b) Find its solutions that belong to the interval $\left [ -\frac(7\pi)(2);-2\pi \right ] $.
1. A)$\pm \frac(\pi)(2)+2\pi k; \pm \frac(2\pi)(3)+2\pi k,k\in \mathbb(Z) $
  b)$\frac(7\pi)(2);\frac(9\pi)(2);\frac(14\pi)(3) $ A) Solve the equation $\sqrt(2)\sin(x+\frac(\pi)(4))+\cos(2x)=\sin x -1 $.
  b) Find its solutions that belong to the interval $\left [ \frac(7\pi)(2); 5\pi \right ]$.
2. A)$\pm \frac(\pi )(2)+2\pi k; \pm \frac(5\pi )(6) +2\pi k, k\in \mathbb(Z) $
  b)$-\frac(3\pi)(2);-\frac(5\pi)(2) ;-\frac(17\pi)(6) $A) Solve the equation $2\sin(x+\frac(\pi)(3))+\cos(2x)=\sin x -1 $.
  b)
3. A)$\frac(\pi)(2)+\pi k; \pm \frac(\pi)(3) +2\pi k,k\in \mathbb(Z) $
  b)$-\frac(5\pi)(2);-\frac(5\pi)(3);-\frac(7\pi)(3) $ A) Solve the equation $2\sin(x+\frac(\pi)(3))-\sqrt(3)\cos(2x)=\sin x +\sqrt(3) $.
  b) Find its solutions that belong to the interval $\left [ -3\pi;-\frac(3\pi)(2) \right ] $.
4. A)$\frac(\pi)(2)+\pi k; \pm \frac(\pi)(4) +2\pi k,k\in \mathbb(Z) $
  b)$\frac(5\pi)(2);\frac(7\pi)(2);\frac(15\pi)(4) $ A) Solve the equation $2\sqrt(2)\sin(x+\frac(\pi)(6))-\cos(2x)=\sqrt(6)\sin x +1 $.
  b) Find its solutions that belong to the interval $\left [\frac(5\pi)(2); 4\pi; \right ] $.
1. A)$(-1)^(k+1) \cdot \frac(\pi )(3)+\pi k ; \pi k, k\in \mathbb(Z) $
  b)$\frac(11\pi )(3); 4\pi ; 5\pi $ A) Solve the equation $\sqrt(6)\sin\left (x+\frac(\pi )(4) \right)-2\cos^(2) x=\sqrt(3)\cos x-2 $.
  b) Find its solutions that belong to the interval $\left [ \frac(7\pi )(2);5\pi \right ] $.
2. A)$\pi k; (-1)^k \cdot \frac(\pi )(4)+\pi k, k\in \mathbb(Z) $
  b)$-3\pi; -2\pi; -\frac(7\pi)(4) $ A) Solve the equation $2\sqrt(2)\sin\left (x+\frac(\pi )(3) \right)+2\cos^(2) x=\sqrt(6)\cos x+2 $ .
  b) Find its solutions that belong to the interval $\left [ -3\pi ; \frac(-3\pi )(2) \right ] $.
3. A)$\frac(3\pi)(2)+2\pi k, \frac(\pi)(6)+2\pi k, \frac(5\pi)(6)+2\pi k, k \in \mathbb(Z) $
  b)$-\frac(5\pi)(2);-\frac(11\pi)(6) ;-\frac(7\pi)(6) $ A) Solve the equation $2\sin\left (x+\frac(\pi)(6) \right)-2\sqrt(3)\cos^2 x=\cos x -\sqrt(3) $.
  b)
4. A)$2\pi k; \frac(\pi)(2)+\pi k,k\in \mathbb(Z) $
  b)$-\frac(7\pi)(2);;-\frac(5\pi)(2); -4\pi $ A) Solve the equation $\cos^2 x + \sin x=\sqrt(2)\sin\left (x+\frac(\pi)(4) \right) $.
  b) Find its solutions that belong to the interval $\left [ -4\pi; -\frac(5\pi)(2) \right ]$.
5. A)$\pi k; (-1)^(k+1) \cdot \frac(\pi)(6)+\pi k, k\in \mathbb(Z) $
  b)$-2\pi; -\pi ;-\frac(13\pi)(6) $ A) Solve the equation $2\sin\left (x+\frac(\pi)(6) \right)-2\sqrt(3)\cos^2 x=\cos x -2\sqrt(3) $.
  b) Find its solutions that belong to the interval $\left [ -\frac(5\pi)(2);-\pi \right ] $.
1. A)$\pi k; - \frac(\pi)(6)+2\pi k; -\frac(5\pi)(6) +2\pi k,k\in \mathbb(Z) $
  b)$-\frac(5\pi)(6);-2\pi; -\pi $ A) Solve the equation $2\sin^2 x+\sqrt(2)\sin\left (x+\frac(\pi)(4) \right)=\cos x $.
  b)
2. A)$\pi k; \frac(\pi)(4)+2\pi k; \frac(3\pi)(4) +2\pi k,k\in \mathbb(Z) $
  b)$\frac(17\pi)(4);3\pi; 4\pi $ A) Solve the equation $\sqrt(6)\sin^2 x+\cos x =2\sin\left (x+\frac(\pi)(6) \right) $.
  b) Find its solutions that belong to the interval $\left [ -2\pi;-\frac(\pi)(2) \right ]$.
1. A)$\pi k; \pm \frac(\pi)(3) +\pi k, k\in \mathbb(Z) $
  b)$3\pi; \frac(10\pi)(3);\frac(11\pi)(3);4\pi; \frac(13\pi)(3) $ A) Solve the equation $4\sin^3 x=3\cos\left (x-\frac(\pi)(2) \right) $.
  b) Find its solutions that belong to the interval $\left [ 3\pi; \frac(9\pi)(2) \right ] $.
2. A)
  b)$\frac(5\pi)(2); \frac(11\pi)(4);\frac(13\pi)(4);\frac(7\pi)(2);\frac(15 \pi)(4) $ A) Solve the equation $2\sin^3 \left (x+\frac(3\pi)(2) \right)+\cos x=0 $.
  b) Find its solutions that belong to the interval $\left [ \frac(5\pi)(2); 4\pi \right ] $.
1. A)$\frac(\pi)(2) +\pi k, \pm \frac(\pi)(4) +\pi k, k\in \mathbb(Z) $
  b)$-\frac(15\pi)(4);-\frac(7\pi)(2);-\frac(13\pi)(4);-\frac(11\pi)(4); -\frac(5\pi)(2);$ A) Solve the equation $2\cos^3 x=\sin \left (\frac(\pi)(2)-x \right) $.
  b) Find its solutions that belong to the interval $\left [ -4\pi; -\frac(5\pi)(2) \right ] $.
2. A)$\pi k, \pm \frac(\pi)(6) +\pi k, k\in \mathbb(Z) $
  b)$-\frac(19\pi)(6);-3\pi; -\frac(17\pi)(6);-\frac(13\pi)(6);-2\pi; $ A) Solve the equation $4\cos^3\left (x+\frac(\pi)(2) \right)+\sin x=0 $.
  b) Find its solutions that belong to the interval $\left [ -\frac(7\pi)(2); -2\pi \right ] $.
1. A)$\frac(\pi)(2)+\pi k; \frac(\pi)(4) +\pi k,k\in \mathbb(Z) $
  b)$-\frac(7\pi)(2);-\frac(11\pi)(4);-\frac(9\pi)(4) $ A) Solve the equation $\sin 2x+2\sin\left (2x-\frac(\pi)(6) \right)=\sqrt(3)\sin(2x)+1 $.
  b) Find its solutions that belong to the interval $\left [ -\frac(7\pi)(2); -2\pi \right ] $.
1. A)$\pi k; (-1)^k \cdot \frac(\pi)(6) +\pi k, k\in \mathbb(Z) $
  b)$-3\pi; -2\pi; -\frac(11\pi)(6) $ A) Solve the equation $2\sin\left (x+\frac(\pi)(3) \right)+\cos(2x)=1+\sqrt(3)\cos x $.
  b) Find its solutions that belong to the interval $\left [ -3\pi;-\frac(3\pi)(2) \right ] $.
2. A)$\pi k; (-1)^(k+1) \cdot \frac(\pi)(3) +\pi k, k\in \mathbb(Z) $
  b)$-3\pi;-\frac(8\pi)(3);-\frac(7\pi)(3);-2\pi $ A) Solve the equation $2\sqrt(3)\sin\left (x+\frac(\pi)(3) \right)-\cos(2x)=3\cos x -1 $.
  b) Find its solutions that belong to the interval $\left [ -3\pi;-\frac(3\pi)(2) \right ] $.

14 : Angles and distances in space

1. $\frac(420)(29)$
  A)
  b) Find the distance from the point $B$ to the line $AC_1 $, if $AB=21, B_1C_1=16, BB_1=12 $.
2. 12
  A) Prove that the angle $ABC_1 $ is a right angle.
  b) Find the distance from the point $B$ to the line $AC_1 $, if $AB=15, B_1C_1=12, BB_1=16 $.
3. $\frac(120)(17)$ In a cylinder, the generatrix is perpendicular to the plane of the base. Points $A$ and $B$ are chosen on the circle of one of the bases of the cylinder, and points $B_1 $ and $C_1 $ are chosen on the circle of the other base, and $BB_1 $ is the generatrix of the cylinder, and the segment $AC_1$ intersects the axis of the cylinder.
  A) Prove that the angle $ABC_1 $ is a right angle.
  b) Find the distance from the point $B$ to the line $AC_1 $, if $AB=8, B_1C_1=9, BB_1=12 $.
4. $\frac(60)(13)$ In a cylinder, the generatrix is perpendicular to the plane of the base. Points $A$ and $B$ are chosen on the circle of one of the bases of the cylinder, and points $B_1 $ and $C_1 $ are chosen on the circle of the other base, and $BB_1 $ is the generatrix of the cylinder, and the segment $AC_1$ intersects the axis of the cylinder.
  A) Prove that the angle $ABC_1 $ is a right angle.
  b) Find the distance from the point $B$ to the line $AC_1 $, if $AB=12, B_1C_1=3, BB_1=4 $.
1. $\arctan \frac(17)(6)$ In a cylinder, the generatrix is perpendicular to the plane of the base. Points $A$ and $B$ are chosen on the circle of one of the bases of the cylinder, and points $B_1 $ and $C_1 $ are chosen on the circle of the other base, and $BB_1 $ is the generatrix of the cylinder, and the segment $AC_1$ intersects the axis of the cylinder.
  A) Prove that the angle $ABC_1 $ is a right angle.
  b) Find the angle between the line $AC_1 $ and $BB_1 $, if $AB=8, B_1C_1=15, BB_1=6 $.
2. $\arctan \frac(2)(3)$ In a cylinder, the generatrix is perpendicular to the plane of the base. Points $A$ and $B$ are chosen on the circle of one of the bases of the cylinder, and points $B_1 $ and $C_1 $ are chosen on the circle of the other base, and $BB_1 $ is the generatrix of the cylinder, and the segment $AC_1$ intersects the axis of the cylinder.
  A) Prove that the angle $ABC_1 $ is a right angle.
  b) Find the angle between the line $AC_1 $ and $BB_1 $, if $AB=6, B_1C_1=8, BB_1=15 $.
1. 7.2 In a cylinder, the generatrix is perpendicular to the plane of the base. Points $A$ and $B$ are chosen on the circle of one of the bases of the cylinder, and points $B_1 $ and $C_1 $ are chosen on the circle of the other base, and $BB_1 $ is the generatrix of the cylinder, and the segment $AC_1$ intersects the axis of the cylinder.
  A)
  b) Find the distance between lines $AC_1$ and $BB_1$ if $AB = 12, B_1C_1 = 9, BB_1 = 8$.
2. In a cylinder, the generatrix is perpendicular to the plane of the base. Points $A$ and $B$ are chosen on the circle of one of the bases of the cylinder, and points $B_1 $ and $C_1 $ are chosen on the circle of the other base, and $BB_1 $ is the generatrix of the cylinder, and the segment $AC_1$ intersects the axis of the cylinder.
  A) Prove that the lines $AB$ and $B_1C_1$ are perpendicular.
  b) Find the distance between the lines $AC_1$ and $BB_1$ if $AB = 3, B_1C_1 = 4, BB_1 = 1$.
1. In a cylinder, the generatrix is perpendicular to the plane of the base. Points $A$ and $B$ are chosen on the circle of one of the bases of the cylinder, and points $B_1 $ and $C_1 $ are chosen on the circle of the other base, and $BB_1 $ is the generatrix of the cylinder, and the segment $AC_1$ intersects the axis of the cylinder.
  A) Prove that the lines $AB$ and $B_1C_1$ are perpendicular.
  b) Find the lateral surface area of the cylinder if $AB = 6, B_1C_1 = 8, BB_1 = 15$.
1. In a cylinder, the generatrix is perpendicular to the plane of the base. Points $A$ and $B$ are chosen on the circle of one of the bases of the cylinder, and points $B_1 $ and $C_1 $ are chosen on the circle of the other base, and $BB_1 $ is the generatrix of the cylinder, and the segment $AC_1$ intersects the axis of the cylinder.
  A) Prove that the lines $AB$ and $B_1C_1$ are perpendicular.
  b) Find the total surface area of the cylinder if $AB = 6, B_1C_1 = 8, BB_1 = 15$.
1. In a cylinder, the generatrix is perpendicular to the plane of the base. Points $A$ and $B$ are chosen on the circle of one of the bases of the cylinder, and points $B_1 $ and $C_1 $ are chosen on the circle of the other base, and $BB_1 $ is the generatrix of the cylinder, and the segment $AC_1$ intersects the axis of the cylinder.
  A) Prove that the lines $AB$ and $B_1C_1$ are perpendicular.
  b) Find the volume of the cylinder if $AB = 6, B_1C_1 = 8, BB_1 = 15$.
2. In a cylinder, the generatrix is perpendicular to the plane of the base. Points $A$ and $B$ are chosen on the circle of one of the bases of the cylinder, and points $B_1 $ and $C_1 $ are chosen on the circle of the other base, and $BB_1 $ is the generatrix of the cylinder, and the segment $AC_1$ intersects the axis of the cylinder.
  A) Prove that the lines $AB$ and $B_1C_1$ are perpendicular.
  b) Find the volume of the cylinder if $AB = 7, B_1C_1 = 24, BB_1 = 10$.
3. In a cylinder, the generatrix is perpendicular to the plane of the base. Points $A$ and $B$ are chosen on the circle of one of the bases of the cylinder, and points $B_1 $ and $C_1 $ are chosen on the circle of the other base, and $BB_1 $ is the generatrix of the cylinder, and the segment $AC_1$ intersects the axis of the cylinder.
  A) Prove that the lines $AB$ and $B_1C_1$ are perpendicular.
  b) Find the volume of the cylinder if $AB = 21, B_1C_1 = 15, BB_1 = 20$.
1. $\sqrt(5)$ In a cylinder, the generatrix is perpendicular to the plane of the base. Points $A$ , $B$ and $C$ are chosen on the circle of one of the bases of the cylinder, and the point $C_1$ is chosen on the circle of the other base, where $CC_1$ is the generatrix of the cylinder, and $AC$ - diameter of the base. It is known that the angle $ACB$ is equal to 30 degrees.
  A) Prove that the angle between lines $AC_1$ and $BC_1$ is 45 degrees.
  b) Find the distance from point B to the line $AC_1$ if $AB = \sqrt(6), CC_1 = 2\sqrt(3)$.
1. $4\pi$ In a cylinder, the generatrix is perpendicular to the plane of the base. Points $A$ , $B$ and $C$ are chosen on the circle of one of the bases of the cylinder, and the point $C_1$ is chosen on the circle of the other base, where $CC_1$ is the generatrix of the cylinder, and $AC$ - diameter of the base. It is known that the angle $ACB$ is equal to 30°, $AB = \sqrt(2), CC_1 = 2$.
  A) Prove that the angle between the lines $AC_1$ and $BC_1$ is 45 degrees.
  b) Find the volume of the cylinder.
2. $16\pi$ In a cylinder, the generatrix is perpendicular to the plane of the base. Points $A$ , $B$ and $C$ are chosen on the circle of one of the bases of the cylinder, and the point $C_1$ is chosen on the circle of the other base, where $CC_1$ is the generatrix of the cylinder, and $AC$ - diameter of the base. It is known that the angle $ACB$ is equal to 45°, $AB = 2\sqrt(2), CC_1 = 4$.
  A) Prove that the angle between lines $AC_1$ and $BC$ is 60 degrees.
  b) Find the volume of the cylinder.
1. $2\sqrt(3)$ In the cube $ABCDA_1B_1C_1D_1$ all edges are 6.
  A) Prove that the angle between the lines $AC$ and $BD_1$ is 60°.
  b) Find the distance between the lines $AC$ and $BD_1$.
1. $\frac(3\sqrt(22))(5) $
  A)
  b) Find $QP$, where $P$ is the intersection point of the plane $MNK$ and the edge $SC$, if $AB=SK=6 $ and $SA=8$.
1. $\frac(24\sqrt(39))(7) $ IN right pyramid$SABC$ the points $M$ and $N$ are the midpoints of the edges $AB$ and $BC$, respectively. A point $K$ is marked on the side edge $SA$. The section of the pyramid by the plane $MNK$ is a quadrilateral whose diagonals intersect at the point $Q$.
  A) Prove that the point $Q$ lies at the height of the pyramid.
  b) Find the volume of the pyramid $QMNB$ if $AB=12,SA=10 $ and $SK=2$.
1. $\arctan 2\sqrt(11) $ In a regular pyramid $SABC$, the points $M$ and $N$ are the midpoints of the edges $AB$ and $BC$, respectively. A point $K$ is marked on the side edge $SA$. The section of the pyramid by the plane $MNK$ is a quadrilateral whose diagonals intersect at the point $Q$.
  A) Prove that the point $Q$ lies at the height of the pyramid.
  b) Find the angle between the planes $MNK$ and $ABC$, if $AB=6, SA=12$ and $SK=3$.
1. $\frac(162\sqrt(51))(25) $ In a regular pyramid $SABC$, the points $M$ and $N$ are the midpoints of the edges $AB$ and $BC$, respectively. A point $K$ is marked on the side edge $SA$. The section of the pyramid by the plane $MNK$ is a quadrilateral whose diagonals intersect at the point $Q$.
  A) Prove that the point $Q$ lies at the height of the pyramid.
  b) Find the cross-sectional area of \u200b\u200bthe pyramid by the plane $MNK$, if $AB=12, SA=15 $ and $SK=6$.

15 : Inequalities

1. $(-\infty ;-12]\cup \left (-\frac(35)(8);0 \right ]$ Solve the inequality $\log _(11) (8x^2+7)-\log _(11) \left (x^2+x+1\right)\geq \log _(11) \left (\frac (x)(x+5)+7 \right) $.
2. $(-\infty ;-50]\cup \left (-\frac(49)(8);0 \right ]$ Solve the inequality $\log _(5) (8x^2+7)-\log _(5) \left (x^2+x+1\right)\geq \log _(5) \left (\frac (x)(x+7)+7 \right) $.
3. $(-\infty;-27]\cup \left (-\frac(80)(11);0 \right ]$ Solve the inequality $\log _7 (11x^2+10)-\log _7 \left (x^2+x+1\right)\geq \log _7 \left (\frac(x)(x+8)+ 10\right)$.
4. $(-\infty ;-23]\cup \left (-\frac(160)(17);0 \right ]$ Solve the inequality $\log _2 (17x^2+16)-\log _2 \left (x^2+x+1\right)\geq \log _2 \left (\frac(x)(x+10)+ 16\right)$.
1. $\left [\frac(\sqrt(3))(3); +\infty \right) $ Solve the inequality $2\log _2 (x\sqrt(3))-\log _2 \left (\frac(x)(x+1)\right)\geq \log _2 \left (3x^2+\frac (1)(x)\right)$.
2. $\left (0; \frac(1)(4) \right ]\cup \left [\frac(1)(\sqrt(3));1 \right) $ Solve the inequality $2\log_3(x\sqrt(3))-\log_3\left (\frac(x)(1-x) \right)\leq \log_3 \left (9x^(2)+\frac( 1)(x)-4 \right) $.
3. $\left (0; \frac(1)(5) \right ]\cup \left [ \frac(\sqrt(2))(2); 1 \right) $ Solve the inequality $2\log_7(x\sqrt(2))-\log_7\left (\frac(x)(1-x) \right)\leq \log_7 \left (8x^(2)+\frac( 1)(x)-5 \right) $.
4. $\left (0; \frac(1)(\sqrt(5)) \right ]\cup \left [\frac(1)(2);1 \right) $ Solve the inequality $2\log_2(x\sqrt(5))-\log_2\left (\frac(x)(1-x) \right)\leq \log_2 \left (5x^(2)+\frac( 1)(x)-2 \right) $.
5. $\left (0; \frac(1)(3) \right ]\cup \left [\frac(1)(2);1 \right) $ Solve the inequality $2\log_5(2x)-\log_5\left (\frac(x)(1-x) \right)\leq \log_5 \left (8x^(2)+\frac(1)(x) -3 \right) $.
1. $(0; 1] \cup \cup \left $ Solve the inequality $\log _5 (4-x)+\log _5 \left (\frac(1)(x)\right)\leq \log _5 \left (\frac(1)(x)-x+3 \right) $.
1. $(1; 1.5] \cup \cup \cup [ 3.5;+\infty) $ Solve the inequality $\log _5 (x^2+4)-\log _5 \left (x^2-x+14\right)\geq \log _5 \left (1-\frac(1)(x) \ right)$.
2. $(1; 1.5] \cup [ 4;+\infty) $ Solve the inequality $\log _3 (x^2+2)-\log _3 \left (x^2-x+12\right)\geq \log _3 \left (1-\frac(1)(x) \ right)$.
3. $\left (\frac(1)(2); \frac(2)(3) \right ] \cup \left [ 5; +\infty \right) $ Solve the inequality $\log _2 (2x^2+4)-\log _2 \left (x^2-x+10\right)\geq \log _2 \left (2-\frac(1)(x) \ right)$.
1. $(-3; -2]\cup $ Solve the inequality $\log_2 \left (\frac(3)(x)+2 \right)-\log_2(x+3)\leq \log_2\left (\frac(x+4)(x^2) \ right)$.
2. $[-2; -1)\cup (0; 9] $ Solve the inequality $\log_5 \left (\frac(2)(x)+2 \right)-\log_5(x+3)\leq \log_5\left (\frac(x+6)(x^2) \ right)$.
1. $\left (\frac(\sqrt(6))(3);1 \right)\cup \left (1; +\infty \right)$ Solve the inequality $\log _5 (3x^2-2)-\log _5 x
2. \(\left (\frac(2)(5); +\infty \right)$ Solve the inequality $\log_3 (25x^2-4) -\log_3 x \leq \log_3 \left (26x^2+\frac(17)(x)-10 \right) $.
3. $\left (\frac(5)(7); +\infty \right)$ Solve the inequality $\log_7 (49x^2-25) -\log_7 x \leq \log_7 \left (50x^2-\frac(9)(x)+10 \right) $.
1. $\left [ -\frac(1)(6); -\frac(1)(24) \right)\cup (0;+\infty) $ Solve the inequality $\log_5(3x+1)+\log_5 \left (\frac(1)(72x^(2))+1 \right)\geq \log_5 \left (\frac(1)(24x)+ 1\right)$.
2. $\left [ -\frac(1)(4); -\frac(1)(16) \right)\cup (0;+\infty) $ Solve the inequality $\log_3(2x+1)+\log_3 \left (\frac(1)(32x^(2))+1 \right)\geq \log_3 \left (\frac(1)(16x)+ 1\right)$.
1. $1$ Solve the inequality $\log _2 (3-2x)+2\log _2 \left (\frac(1)(x)\right)\leq \log _2 \left (\frac(1)(x^(2) )-2x+2 \right) $.
2. $(1; 3] $ Solve the inequality $\log _2 (x-1)+\log _2 \left (2x+\frac(4)(x-1)\right)\geq 2\log _2 \left (\frac(3x-1)( 2)\right)$.
3. $\left [ \frac(1+\sqrt(5))(2); +\infty \right) $ Solve the inequality $\log _2 (x-1)+\log _2 \left (x^2+\frac(1)(x-1)\right)\leq 2\log _2 \left (\frac(x^ 2+x-1)(2) \right) $.
4. $\left [ 2; +\infty \right) $ Solve the inequality $2\log _2 (x)+\log _2 \left (x+\frac(1)(x^2)\right)\leq 2\log _2 \left (\frac(x^2+x) (2) \right) $.
1. $\left [ \frac(-5+\sqrt(41))(8); \frac(1)(2) \right) $ Solve the inequality $\log _3 (1-2x)-\log _3 \left (\frac(1)(x)-2\right)\leq \log _3 (4x^2+6x-1) $.
1. $\left [ \frac(1)(6); \frac(1)(2) \right) $ Solve the inequality $2\log _2 (1-2x)-\log _2 \left (\frac(1)(x)-2\right)\leq \log _2 (4x^2+6x-1) $.
1. $(1; +\infty)$ Solve the inequality $\log _2 (x-1)+\log _2 \left (2x+\frac(4)(x-1)\right)\geq \log _2 \left (\frac(3x-1)(2 )\right)$.
1. $\left [ \frac(11+3\sqrt(17))(2); +\infty \right) $ Solve the inequality $\log_2 (4x^2-1) -\log_2 x \leq \log_2 \left (5x+\frac(9)(x)-11 \right) $.

18 : Equations, inequalities, systems with a parameter

1. $$ \left (-\frac(4)(3); -\frac(3)(4)\right) \cup \left (\frac(3)(4); 1\right)\cup \left ( 1;\frac(4)(3)\right)$$
  $\left\(\begin(matrix)\begin(array)(lcl) (x+ay-5)(x+ay-5a)=0 \\ x^2+y^2=16 \end(array )\end(matrix)\right.$
2. $$ \left (-\frac(3\sqrt(7))(7); -\frac(\sqrt(7))(3)\right) \cup \left (\frac(\sqrt(7)) (3); 1\right)\cup \left (1; \frac(3\sqrt(7))(7)\right)$$
  $\left\(\begin(matrix)\begin(array)(lcl) (x+ay-4)(x+ay-4a)=0 \\ x^2+y^2=9 \end(array )\end(matrix)\right.$
  The equation has exactly four different solutions.
3. $$ \left (-\frac(3\sqrt(5))(2); -\frac(2\sqrt(5))(15)\right) \cup \left (\frac(2\sqrt(5) ))(15); 1\right)\cup \left (1; \frac(3\sqrt(5))(2)\right)$$ Find all values of the parameter a, for each of which the system
  $\left\(\begin(matrix)\begin(array)(lcl) (x+ay-7)(x+ay-7a)=0 \\ x^2+y^2=45 \end(array )\end(matrix)\right.$
  The equation has exactly four different solutions.
4. $$ \left (-2\sqrt(2); -\frac(\sqrt(2))(4)\right) \cup \left (\frac(\sqrt(2))(4); 1\right )\cup \left (1; 2\sqrt(2) \right)$$ Find all values of the parameter a, for each of which the system
  $\left\(\begin(matrix)\begin(array)(lcl) (x+ay-3)(x+ay-3a)=0 \\ x^2+y^2=8 \end(array )\end(matrix)\right.$
  The equation has exactly four different solutions.
1. $$ (1-\sqrt(2); 0) \cup (0; 1.2) \cup (1.2; 3\sqrt(2)-3) $$ Find all values of the parameter a, for each of which the system
  $\left\(\begin(matrix)\begin(array)(lcl) x^2+y^2+2(a-3)x-4ay+5a^2-6a=0 \\ y^2= x^2 \end(array)\end(matrix)\right.$
  The equation has exactly four different solutions.
2. $$ (4-3\sqrt2; 1-\frac(2)(\sqrt5)) \cup (1-\frac(2)(\sqrt5); 1+\frac(2)(\sqrt5)) \cup (\frac(2)(3)+\sqrt2; 4+3\sqrt2) $$ Find all values of the parameter a, for each of which the system
  $\left\(\begin(matrix)\begin(array)(lcl) x^2+y^2-4ax+6x-(2a+2)y+5a^2-10a+1=0 \\ y ^2=x^2 \end(array)\end(matrix)\right.$
  The equation has exactly four different solutions.
3. $$ \left (-\frac(2+\sqrt(2))(3); -1 \right)\cup (-1; -0.6) \cup (-0.6; \sqrt(2)-2) $ $ Find all values of the parameter a, for each of which the system
  $\left\(\begin(matrix)\begin(array)(lcl) x^2+y^2-4(a+1)x-2ay+5a^2+8a+3=0 \\ y^ 2=x^2 \end(array)\end(matrix)\right.$
  The equation has exactly four different solutions.
4. $$ \left (\frac(2)(9); 2 \right) $$ Find all values of the parameter a, for each of which the system
  $\left\(\begin(matrix)\begin(array)(lcl) x^2+y^2-4(a+1)x-2ay+5a^2-8a+4=0 \\ y^ 2=x^2 \end(array)\end(matrix)\right.$
  The equation has exactly four different solutions.
5. $$ \left (3-\sqrt2; \frac(8)(5) \right) \cup \left (\frac(8)(5); 2 \right) \cup \left (2; \frac(3 +\sqrt2)( 2) \right) $$ Find all values of the parameter a, for each of which the system
  $\left\(\begin(matrix)\begin(array)(lcl) x^2+y^2-6(a-2)x-2ay+10a^2+32-36a=0 \\ y^ 2=x^2 \end(array)\end(matrix)\right.$
  The equation has exactly four different solutions.
6. $$ (1-\sqrt2; 0) \cup (0; 0.8) \cup (0.8; 2\sqrt2-2) $$ Find all values of the parameter a, for each of which the system
  $\left\(\begin(matrix)\begin(array)(lcl) x^2+y^2-2(a-4)x-6ay+10a^2-8a=0 \\ y^2= x^2 \end(array)\end(matrix)\right.$
  The equation has exactly four different solutions.
1. $$ (2; 4)\cup (6; +\infty)$$ Find all values of the parameter a, for each of which the system
  $\left\(\begin(matrix)\begin(array)(lcl) x^4-y^4=10a-24 \\ x^2+y^2=a \end(array)\end(matrix )\right.$
  The equation has exactly four different solutions.
2. $$ (2; 6-2\sqrt(2))\cup(6+2\sqrt(2);+\infty) $$ Find all values of the parameter a, for each of which the system
  $\left\(\begin(matrix)\begin(array)(lcl) x^4-y^4=12a-28 \\ x^2+y^2=a \end(array)\end(matrix )\right.$
  The equation has exactly four different solutions.
1. $$ \left (-\frac(3)(14)(\sqrt2-4); \frac(3)(5) \right ]\cup \left [ 1; \frac(3)(14)(\sqrt2 +4) \right) $$ Find all values of the parameter a, for each of which the system
  $\left\(\begin(matrix)\begin(array)(lcl) x^4+y^2=a^2 \\ x^2+y=|4a-3| \end(array)\end (matrix)\right.$
  The equation has exactly four different solutions.
2. $$ (4-2\sqrt(2);\frac(4)(3))\cup(4;4+2\sqrt(2)) $$ Find all values of the parameter a, for each of which the system
  $\left\(\begin(matrix)\begin(array)(lcl) x^4+y^2=a^2 \\ x^2+y=|2a-4| \end(array)\end (matrix)\right.$
  The equation has exactly four different solutions.
3. $$ (5-\sqrt(2);4)\cup (4;5+\sqrt(2))$$ Find all values of the parameter a, for each of which the system
  $\left\(\begin(matrix)\begin(array)(lcl) x^4+y^2=2a-7 \\ x^2+y=|a-3| \end(array)\end (matrix)\right.$
  The equation has exactly four different solutions.
4. $$ \left (\frac(1)(7)(4-\sqrt2); \frac(2)(5) \right) \cup \left (\frac(2)(5); \frac(1) (2) \right) \cup \left (\frac(1)(2) ; \frac(1)(7)(\sqrt2+4) \right) $$ Find all values of the parameter a, for each of which the system
  $\left\(\begin(matrix)\begin(array)(lcl) x^4+y^2=a^2 \\ x^2+y=|4a-2| \end(array)\end (matrix)\right.$
  The equation has exactly four different solutions.
1. $$ \left (\frac(-2-\sqrt(2))(3); -1 \right)\cup (-1; -0.6)\cup (-0.6; \sqrt(2)-2) $ $ Find all values of the parameter a, for each of which the system
  $\left\(\begin(matrix)\begin(array)(lcl) (x-(2a+2))^2+(y-a)^2=1 \\ y^2=x^2 \end( array)\end(matrix)\right.$
  The equation has exactly four different solutions.
2. $$(1-\sqrt(2); 0)\cup(0; 1.2) \cup (1.2; 3\sqrt(2)-3) $$ Find all values of the parameter a, for each of which the system
  $\left\(\begin(matrix)\begin(array)(lcl) (x-(3-a))^2+(y-2a)^2=9 \\ y^2=x^2 \ end(array)\end(matrix)\right.$
  The equation has exactly four different solutions.
1. $$(-9.25; -3)\cup (-3;3)\cup (3; 9.25)$$ Find all values of the parameter a, for each of which the system
  $\left\(\begin(matrix)\begin(array)(lcl) y=(a+3)x^2+2ax+a-3 \\ x^2=y^2 \end(array)\ end(matrix)\right.$
  The equation has exactly four different solutions.
2. $$(-4.25;-2)\cup(-2;2)\cup(2;4.25)$$ Find all values of the parameter a, for each of which the system
  $\left\(\begin(matrix)\begin(array)(lcl) y=(a+2)x^2-2ax+a-2 \\ y^2=x^2 \end(array)\ end(matrix)\right.$
  The equation has exactly four different solutions.
3. $$(-4.25; -2)\cup (-2;2)\cup (2; 4.25)$$ Find all values of the parameter a, for each of which the system
  $\left\(\begin(matrix)\begin(array)(lcl) y=(a-2)x^2-2ax-2+a \\ y^2=x^2 \end(array)\ end(matrix)\right.$
  The equation has exactly four different solutions.
1. $$ (-\infty ; -3)\cup (-3; 0)\cup (3;\frac(25)(8)) $$ Find all values of the parameter a, for each of which the system
  $\left\(\begin(matrix)\begin(array)(lcl) ax^2+ay^2-(2a-5)x+2ay+1=0 \\ x^2+y=xy+x \end(array)\end(matrix)\right.$
  The equation has exactly four different solutions.
1. $$\left [ 0; \frac(2)(3) \right ]$$ Find all values of the parameter a, for each of which the equation
  $\sqrt(x+2a-1)+\sqrt(x-a)=1 $
  Has at least one solution.