Secondary general education

Line UMK G.K. Muravina. Algebra and the beginnings of 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

The profile-level examination paper 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, teacher of mathematics of the highest category of the school, work experience of 20 years:

“In order to get a school certificate, a graduate must pass two mandatory exams in the form of the Unified State Examination, one of which is mathematics. In accordance with the Concept for the Development of Mathematical Education in the Russian Federation, the Unified State Exam in mathematics is divided into two levels: basic and specialized. Today we will consider options for the profile level.

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

Example 1 In the apartment where Petr lives, a cold water meter (meter) was installed. 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 the 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.

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 the advanced 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

Tasks of the Unified State Examination of profile mathematics divided into two blocks.

1. Part - short answers, includes 8 tasks that test basic mathematical training and the ability to apply knowledge of mathematics in everyday life.
2. 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 of difficulty- 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, when preparing for the exam, always reinforce the theory in mathematics by solving practical problems.

How will points be distributed?

The tasks of the first part of the KIMs in mathematics are close to the basic level USE tests, so it is impossible to score a high score on 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.

For the state exam in 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 to online math tests 2020. Choose and get the highest score!

• 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 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 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\(\begin(matrix)\begin(array)(lcl) (x-(2a+2))^2+(ya)^2=1 \\ y^2=x^2 \end( array)\end(matrix)\right.\)

     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.

19 : Numbers and their properties

THANK YOU

Projects

1. "Yagubov.RF" [Teachers]
2. "Yagubov.RF" [Mathematics]

Learn to spot grammatical errors. If you learn to confidently recognize them in the task, then you will not lose points in the essay. (Criterion 9 - "Compliance with language standards.") Also, an assignment for which you can get 5 points requires special treatment!

Task 7 USE in Russian

Task Formulation: Establish a correspondence between grammatical errors and sentences in which they are made: for each position of the first column, select the corresponding position from the second column.

Grammatical errors

suggestions

A) a violation in the construction of a sentence with participial turnover B) an error in the construction of a complex sentence C) violation in the construction of a sentence with an inconsistent application D) violation of the connection between the subject and the predicate E) violation of the aspect-temporal correlation of verb forms

1) I.S. Turgenev subjects Bazarov to the most difficult test - the "test of love" - ​​and this revealed the true essence of his hero. 2) Everyone who visited the Crimea took with him after parting with him vivid impressions of the sea, mountains, southern herbs and flowers. 3) The work "The Tale of a Real Man" is based on real events that happened to Alexei Maresyev. 4) S. Mikhalkov argued that the world of the merchant Zamoskvorechye can be seen on the stage of the Maly Theater thanks to the magnificent play of the actors. 5) In 1885 V.D. Polenov exhibited at a traveling exhibition ninety-seven sketches brought from a trip to the East. 6) The theory of eloquence for all kinds of poetic compositions was written by A.I. Galich, who taught Russian and Latin literature at the Tsarskoye Selo Lyceum. 7) In I. Mashkov's landscape "View of Moscow" there is a feeling of the sonorous colorfulness of a city street. 8) Happy are those who, after a long road with its cold and slush, see a familiar house and hear the voices of their loved ones. 9) Reading classical literature, you notice that how differently the “city of Petrov” is depicted in the works of A.S. Pushkin, N.V. Gogol, F.M. Dostoevsky.

Write in the table the selected numbers under the corresponding letters.

How to perform such a task? It is better to start from the left side. Find the named syntactic phenomenon (participial phrase, subject and predicate, etc.) in the sentences on the right and check if there is a grammatical error. Start with the ones that are easier to find and identify.

Let's analyze typical grammatical errors in the order in which they should be checked on the exam.

Inconsistent Application

An inconsistent appendix is ​​the title of a book, magazine, film, painting, etc., enclosed in quotation marks.

The sentence changes by case generic word, and the inconsistent application is in the initial form and does not change: v novel"War and Peace"; picture Levitan "Golden Autumn" at the station metro station "Tverskaya"

If there is no generic word in the sentence, the application itself changes in cases: heroes of "War and Peace"; I'm looking at Levitan's Golden Autumn, we'll meet at Tverskaya.

Grammar mistake : in the novel "War and Peace"; in the painting "Golden Autumn", at the Tverskaya metro station.

In the task, such an error occurred in sentence 3.

Direct and indirect speech.

A sentence with indirect speech is a complex sentence. Compare:

The conductor said: "I'll bring you tea" - The conductor said that he would bring us tea. Grammar mistake: The conductor said that I would bring you tea.(The personal pronoun should change.)

The passenger asked: "Can I open the window" - The passenger asked if he could open the window. Grammar mistake : The passenger asked if he could open the window.(The sentence has LI in the role of the union, the union WHAT is not allowed in the sentence.)

Participial

We find sentences with participial turnover, see if there are any errors in its construction.

1. The defined (main) word cannot get inside the participial turnover, it can come before or after it. Grammar mistake: who came spectators to meet with the director. Right: viewers who came to meet the director or viewers who came to meet with the director.

2. The participle must agree in gender, number and case with the main word, which is determined by meaning and by question: residents mountains (what?), frightened by a hurricane or residents mountains(what?), overgrown with fir trees. Grammar mistake: mountain dwellers frightened by the hurricane or inhabitants of the mountains, overgrown with firs.

Note: one of the things that happened last summer(we agree on the participle with the word ONE - we are talking about one event). I recall a number of events that happened last summer (we ask a question from EVENTS “what?”).

3. The sacrament has a present tense ( rule memorizing student), past tense ( student who memorized), but no future tense ( student who remembers the rule- grammar mistake).

In the task, such an error occurred in sentence 5.

Participial turnover

Remember: The participle calls the additional action, and the verb-predicate - the main one. The participle and the verb-predicate must refer to the same character!

We find the subject in the sentence and check whether it performs the action called the gerund. Going to the first ball, Natasha Rostova had a natural excitement. We argue: excitement arose - Natasha Rostova walked- Various characters. Correct option: Going to the first ball, Natasha Rostova experienced natural excitement.

In a definite personal sentence, it is easy to restore the subject: I, WE, YOU, YOU: When making an offer, consider(you) grammatical meaning of the word. We argue: you take into account and you make up- no error.

The verb-predicate can be expressed infinitive: When composing a sentence, it is necessary to take into account the grammatical meaning of the word.

We argue: After reading the sentence, it seems to me that there is no mistake. I cannot be the subject, because it is not in the initial form. This sentence has a grammatical error.

The grammatical connection between the subject and the predicate.

The error may be hidden in complex sentences built according to the model “THE WHO…”, “EVERYONE, WHO…”, “ALL, WHO…”, “NONE OF THOSE WHO…”, “MANY OF THOSE WHO…”, “ ONE OF THOSE WHO…” In each simple sentence, the complex subject will have its own subject, it is necessary to check whether they are consistent with their predicates. WHO, EVERYONE, NOBODY, ONE, combined with predicates in the singular; THOSE, ALL, MANY are combined with their predicates in the plural.

Analyzing the offer: None of those who visited there in the summer were not disappointed. NOBODY WAS - a grammatical error. WHO VISITED - there is no error. Those who did not come to the opening of the exhibition regretted it. THEY HAVE SORRY - there is no mistake. WHO DID NOT COME - a grammatical error.

In the task, such an error occurred in sentence 2.

Violation of the types of temporal correlation of verb forms.

Pay special attention to predicate verbs: incorrect use of the tense of the verb leads to confusion in the sequence of actions. I work inattentively, with stops, and as a result I made many ridiculous mistakes. Let's fix the error: I work inattentively, with stops, and as a result I make many ridiculous mistakes.(Both imperfective verbs are in the present tense.) I worked inattentively, with stops, and as a result I made many ridiculous mistakes.(Both verbs are in the past tense, the first verb - an imperfect form - indicates a process, the second - a perfect form - indicates a result.)

In the task, such an error occurred in sentence 1: Turgenev exposes and reveals...

Homogeneous members of a sentence

Grammar errors in conjunction sentences AND.

1. Union AND cannot link one of the members of a sentence to the whole sentence. I don't like to get sick and when i get two. Moscow is a city which was the birthplace of Pushkin and described in detail. When Onegin returned to Petersburg and having met Tatyana, he did not recognize her. Listened to a lecture on the importance of sports and why do they need to do. (Fix the bug: Listened to a lecture on the importance of sports and the benefits of sports. Or: Listened to a lecture on what is the importance of sport and why do they need to do .)
2. Union AND cannot connect homogeneous members expressed in the full and short form of adjectives and participles: He is tall and thin. She is smart and beautiful.
3. Union AND cannot link infinitive and noun: I love doing laundry, cooking and reading books. (Right: I love washing, cooking and reading books.)
4. It is difficult to recognize an error in such a syntactic construction: The Decembrists loved and admired the Russian people. In this sentence, the addition of the PEOPLE refers to both predicates, but is grammatically connected to only one of them: THE PEOPLE WERE ADMIRED (BY WHOM?). From the verb LOVE we ask the question WHO? Be sure to ask a question from each verb-predicate to the object. Here are typical mistakes: parents care and love children; I understand and sympathize with you; he learned and used the rule; I love and am proud of my son. Correcting such a mistake requires the introduction of various additions, each will be consistent with its verb-predicate: I love my son and I'm proud of him.

Using Compound Unions.

1. Learn to recognize the following conjunctions in a sentence: “NOT ONLY ..., BUT AND”; "HOW ..., SO AND". In these unions, you cannot skip individual words or replace them with others: Not only we, but our guests were surprised. The atmosphere of the era in comedy is created not only by actors, but also by off-stage characters. As during the day, so at night, work is in full swing.
2. Parts of the double union must be immediately before each of the homogeneous members . Incorrect word order leads to a grammatical error: We examined not only ancient cities, but also visited new areas.(Correct order: Not only did we see… but we also visited…)The essay should how about the main characters, so tell about artistic features. (Correct order: The essay should tell how about the main characters, as well as artistic features. )

Generalizing words with homogeneous terms

The generalizing word and the homogeneous members following it are in the same case: Do two sports:(how?) skiing and swimming.(Grammar mistake: Strong people have two qualities: kindness and modesty.)

Prepositions with homogeneous members

Prepositions in front of homogeneous members can only be omitted if these prepositions are the same: He visited v Greece, Spain, Italy, on the Cyprus. Grammar mistake: He visited v Greece, Spain, Italy, Cyprus.

Complex sentence

Mistakes related to the incorrect use of unions, allied words, demonstrative words are very common. There can be many options for errors, let's look at some of them.

Extra union: I was tormented by the question of whether I should tell my father everything. I didn't realize how far from the truth I was.

Mixing coordinating and subordinating conjunctions : When Murka got tired of messing with kittens, and she went somewhere to sleep.

Extra particle WOULD: He needs to come to me.

Index word missing: Your mistake is that you are in too much of a hurry.(Omitted IN VOL.)

The allied word WHICH is torn off from the word being defined: A warm rain moistened the earth, which the plants so needed.(Right: Warm rain in which needed plants, moistened the ground.)

In the task, such a mistake was made in sentence 9.

Incorrect use of the case form of a noun with a preposition

1. Prepositions THANK YOU, ACCORDING TO, DESPITE, AGAINST, AGAINST, LIKE + noun in DATIVE CASE: thanks to the skillYu , according to scheduleYu , contrary to the rulesam .

• The preposition PO can be used in the meaning "AFTER". In this case, the noun is in the prepositional case and has the ending AND: upon graduation (after graduation), upon arrival in the city (after arrival), upon expiration of the term (after the expiration of the term).

Remember: on arrival AND, at the end AND, upon completion AND, upon expiration AND, upon arrival E, upon arrival E.

• We remember the features of management in the following phrases:

To prove (what?) right

To marvel at (what?) patience

Give an example of (what?) error

Summarize (what?) work

Confess to (what?) a crime

Miss you, be sad (for whom?) for you

Pay attention to (what?) little things

Point out (what?) shortcomings

Blame (what?) for greed

Remember couples:

worry about son - worry about son

Believe in victory - confidence in victory

The question of construction - problems with construction

Generate rental income - Generate rental income

Ignorance of the problem - unfamiliarity with the problem

Offended by distrust - offended by distrust

pay attention to health pay attention to health

Business preoccupation - anxiety about business

pay the fare - pay the fare

Essay review - essay review

Service fee - service fee

Superiority over him - advantage over him

warn against danger - warn of danger

Distinguish between friends and foes - Distinguish between friends and foes

Surprised by patience - surprised by patience

Characteristic of him - characteristic of him

In task No. 7 of the profile level of the USE in mathematics, it is necessary to demonstrate knowledge of the function of the derivative and the antiderivative. In most cases, simply defining the concepts and understanding the meanings of the derivative is sufficient.

Analysis of typical options for tasks No. 7 USE in mathematics of a profile level

The first version of the task (demo version 2018)

The figure shows a graph of a differentiable function y = f(x). Nine points are marked on the x-axis: x 1 , x 2 , …, x 9 . Among these points, find all points where the derivative of the function y = f(x) is negative. In your answer, indicate the number of points found.

Solution algorithm:

1. Let's look at the graph of the function.
2. We are looking for points at which the function decreases.
3. We count their number.
4. We write down the answer.

Solution:

1. On the graph, the function periodically increases, periodically decreases.

2. In those intervals where the function decreases, the derivative has negative values.

3. These intervals contain points x 3 , x 4 , x 5 , x 9 . There are 4 such points.

The second version of the task (from Yaschenko, No. 4)

Solution algorithm:

1. Let's look at the graph of the function.
2. We consider the behavior of the function at each of the points and the sign of the derivative at them.
3. We find the points in the largest value of the derivative.
4. We write down the answer.

Solution:

1. The function has several intervals of decreasing and increasing.

2. Where the function decreases. The derivative has a minus sign. Such points are among those indicated. But there are points on the graph where the function increases. Their derivative is positive. These are the points with abscissas -2 and 2.

3. Consider a graph at points with x=-2 and x=2. At the point x = 2, the function goes up steeper, which means that the tangent at this point has a larger slope. Therefore, at the point with the abscissa 2. The derivative has the greatest value.

The third version of the task (from Yaschenko, No. 21)

Solution algorithm:

1. We equate the equations of the tangent and the function.
2. We simplify the obtained equality.
3. We find the discriminant.
4. Define the parameter a, for which the solution is unique.
5. We write down the answer.

Solution:

1. The coordinates of the tangent point satisfy both equations: the tangent and the function. So we can equate the equations. We will receive.