Mathematics Part I-1

Mathematics Part I-2

Mathematics Part I-3

Maxim threw a dice twice, the sides of which are numbered from 1 to 6, and built a rectangle with sides equal to the numbers drawn. What is the probability that the area of ​​this rectangle will be greater than 15? Round your answer to the nearest hundredth.

Mathematics Part I-4

Mathematics Part I-5

Mathematics Part I-6

Mathematics Part I-7

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

Mathematics Part I-8

Mathematics Part II-9

Mathematics Part II-10

Devices with a total resistance of R1 = 90 Ohms are connected to the power outlet. In parallel with them, an electric heater is supposed to be connected to the outlet. Determine the lowest possible resistance of this electric heater if it is known that when two conductors with resistances R1 Ohm and R2 Ohm are connected in parallel, their total resistance is given by the formula R_(total) = (R1*R2)/(R1+R2) (Ohm), and for For normal functioning of the electrical network, the total resistance in it must be at least 9 ohms. Express your answer in ohms.

Mathematics Part II-11

Mathematics Part II-12

Mathematics Part II-13

Mathematics Part II-14

The base of the pyramid SABCD is the parallelogram ABCD. Points K, L, M are located on the edges SA, SB, SC respectively, and at the same time

SK/SA = 1/2; SL/SB = 2/5; SM/SC = 2/3

A) Prove that the lines KM and LD intersect.

B) Find the ratio of the volume of the pyramid SKLMD to the volume of the pyramid SABCD.

Mathematics Part II-15

Mathematics Part II-16

In an isosceles trapezoid ABCD AD BC, AD = 21, AB = 10, BC = 9. The diagonals AC and BD divide the trapezoid into four overlapping triangles DAB, ABC, BCD, CDA. Circles w1, w2, w3, w4 are inscribed in each triangle, respectively, the centers of which are located at points O1, O2, O3, O4.

A) Prove that the quadrilateral O1O2O3O4 is a rectangle.

Mathematics Part II-17

On April 15, it is planned to take out a loan in the amount of 900 thousand rubles from the bank for 11 months.
The conditions for its return are as follows:
- on the 1st day of each month, the debt increases by p% compared to the end of the previous month;
- from the 2nd to the 14th of each month it is necessary to pay part of the debt in one payment;
- on the 15th day of each month from the 1st to the 10th month, the debt should be the same amount less than the debt on the 15th day of the previous month;
- On the 15th day of the 10th month, the debt amounted to 200 thousand rubles;
- by the 15th day of the 11th month, the debt must be repaid in full.
Find p if the bank was paid a total of 1021 thousand rubles.

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

There are no changes in the KIM Unified State Exam 2020 in mathematics at the profile level.

Features of Unified State Examination tasks in mathematics 2020

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

Exam structure

Unified State Examination tasks in specialized mathematics divided into two blocks.

1. Part - short answers, includes 8 problems that test basic mathematical preparation and the ability to apply mathematics knowledge in everyday life.
2. Part - short and detailed answers. It consists of 11 tasks, 4 of which require a short answer, and 7 - a detailed one with arguments for the actions performed.

• Advanced difficulty- tasks 9-17 of the second part of KIM.
• High level of difficulty- tasks 18-19 –. This part of the exam tasks tests not only the level of mathematical knowledge, but also the presence or absence of a creative approach to solving dry “numerical” tasks, as well as the effectiveness of the ability to use knowledge and skills as a professional tool.

Important! Therefore, when preparing for the Unified State Exam, always support your theory in mathematics by solving practical problems.

How will points be distributed?

The tasks in the first part of the KIM in mathematics are close to the basic level Unified State Exam 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 problems No. 1-12 - 1 point;
• No. 13-15 – 2 each;
• No. 16-17 – 3 each;
• No. 18-19 – 4 each.

Duration of the exam and rules of conduct for the Unified State Exam

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

During this time, the student should not:

• behave noisily;
• use gadgets and other technical means;
• write off;
• try to help others, or ask for help for yourself.

For such actions, the examinee may be expelled from the classroom.

For the state exam in mathematics allowed to bring Bring only a ruler with you; the rest of the materials will be given to you immediately before the Unified State Exam. are issued on the spot.

Effective preparation is the solution to online tests in mathematics 2020. Choose and get the maximum score!

Secondary general education

Line UMK G. K. Muravin. Algebra and principles of mathematical analysis (10-11) (in-depth)

UMK Merzlyak line. Algebra and beginnings of analysis (10-11) (U)

Mathematics

Preparation for the Unified State Exam in mathematics (profile level): assignments, solutions and explanations

The profile level examination 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 the tasks:

• part 1 contains 8 tasks (tasks 1-8) with a short answer in the form of a whole number 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 (a complete record of the solution with justification for the actions taken).

Panova Svetlana Anatolevna, mathematics teacher of the highest category of school, work experience 20 years:

“In order to receive 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 Examination in mathematics is divided into two levels: basic and specialized. Today we will look at profile-level options.”

Task No. 1- tests the Unified State Exam participants’ ability to apply the skills acquired in the 5th to 9th grade course in elementary mathematics in practical activities. The participant must have computational skills, be able to work with rational numbers, be able to round decimals, and be able to convert one unit of measurement to another.

Example 1. In the apartment where Peter lives, a cold water flow meter (meter) was installed. On May 1, the meter showed a 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 in May, if the price is 1 cubic meter? 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 (cubic m)

2) Let’s find how much money they will pay for wasted water:

34.17 5 = 170.85 (rub)

Answer: 170,85.

Task No. 2- is one of the simplest exam tasks. The majority of graduates successfully cope with it, which indicates knowledge of the definition of the concept of function. Type of task No. 2 according to the requirements codifier is a task on the use of 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 No. 2 tests the ability to extract information presented in tables, diagrams, and graphs. Graduates need to be able to determine the value of a function from the value of the argument in various ways of specifying the function and describe the behavior and properties of the function based on its graph. You also need to be able to find the largest or smallest value from a function graph and build graphs of the studied functions. Errors made are random 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 shares he purchased, and on April 13, he sold all the remaining shares. How much did the businessman lose as a result of these operations?

Solution:

2) 1000 · 3/4 = 750 (shares) - constitute 3/4 of all shares purchased.

6) 247500 + 77500 = 325000 (rub) - the businessman received 1000 shares after selling.

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

The Unified State Examination in mathematics is one of the main tests for school graduates before receiving a certificate and entering a higher education institution. This type of knowledge control is used to assess knowledge in disciplines acquired during schooling. The Unified State Exam takes the form of testing; tasks for the final test are prepared by Rosobrnadzor and other authorized bodies in the field of education. The passing grade in mathematics depends on the individual requirements of the university to which one is applying.graduate. Successfully passing the exam with a high grade is an important factor for success in admission.

Profile-level mathematics is required for admission to technical and economic universities. The basis of the exam tasks is the basic level, with more complex problems and examples added to it. Short and detailed answers are expected:

• The first tasks do not require in-depth knowledge - this is a test of basic level knowledge;
• The next 5 are more difficult, requiring an average to high level of mastery of the subject. These tasks are checked using a computer because the answer is short.

If a student chooses this level, this implies his desire to continue studying the exact sciences at a higher educational institution. The choice in favor of a specialized exam also indicates that the student’s level of knowledge is quite high, in other words, fundamental preparation is not needed.
The preparation process includes repetition of the main sections, solving problems of increased complexity that require a non-standard, creative approach.

Methods of preparation

• Basic training is carried out at school, where the student masters the basics, sometimes the teacher conducts additional electives for graduates. The main recommendation is to carefully and thoroughly master all topics, especially in graduate school.
• Independent work: this requires special self-discipline, will and self-control. You need to carefully read . The problem is in the direction - only a specialist can competently guide the future applicant to those topics that need attention.
• Tutoring: a professional specialist will help you solve complex tasks effectively and quickly.
• Courses and online learning: a modern and proven method that saves time and money. An important advantage: you can take tests online, quickly get answers, and practice on different tasks.

About the exam

The total number of participants in the main period of the Unified State Examination in mathematics at the specialized level in 2018 is more than 391 thousand people. The average test score in 2018 increased by more than 2 test points compared to 2017. The number of exam participants who scored 61 points or more increased over the year from 120.6 thousand to 125.6 thousand. At the same time, the greatest increase was shown by completing tasks with a short answer; tasks with a complete solution were generally completed somewhat worse than in 2017 .

As in previous years, the minimum primary score required to pass the exam was 6 (27 test points). In 2018, 7.48% of exam participants did not achieve the minimum score, in 2017 - 14.35%, this figure almost doubled. The upward trend in results applies to the results of the Unified State Exam both in the country as a whole and in most regions.

More detailed analytical and methodological materials for the 2018 Unified State Exam are available at.

EXAMINATION PLAN FOR THE USE IN MATHEMATICS 2019

Designation of the level of difficulty of the task: B - basic, P - advanced, V - high.

Correspondence between minimum raw scores and 2019 minimum test scores. Order on amendments to Appendix No. 1 to the order of the Federal Service for Supervision in Education and Science. .

OFFICIAL SCALE 2019

THRESHOLD SCORE
The order of Rosobrnadzor established a minimum number of points confirming that exam participants have mastered the basic general education programs of secondary (complete) general education in accordance with the requirements of the federal state educational standard of secondary (complete) general education.
MATHEMATICS THRESHOLD: 6 primary points (27 test points).

EXAMINATION FORMS
You can download the forms in high quality at