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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 an integer or finite number decimal;
• 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, mathematic teacher highest category schools, 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 of development of mathematics education in 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- checks with Unified State Exam participants the ability to apply the skills acquired in the course of grades 5 - 9 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 greatest or smallest value 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.

Series “Unified State Exam. FIPI - school" was prepared by the developers of control measuring materials (CMM) of the unified state exam. The collection contains:
36 standard exam options, compiled in accordance with the draft demo version of the KIM Unified State Exam in mathematics at the profile level in 2017;
instructions for implementation exam paper;
answers to all tasks;
solutions and evaluation criteria for assignments 13-19.
Completing the tasks of standard examination options provides students with the opportunity to independently prepare for the state final certification, as well as to objectively assess the level of their preparation.
Teachers can use standard exam options to organize monitoring of student learning results educational programs average general education and intensive preparation of students for the Unified State Exam.

Examples.
30 athletes compete at the diving championship, including 3 divers from Holland and 9 divers from Colombia. The order of performances is determined by drawing lots. Find the probability that a jumper from Holland will compete eighth.

Mixing 25% and 95% acid solutions and adding 20 kg clean water, received a 40% acid solution. If instead of 20 kg of water we added 20 kg of a 30% solution of the same acid, we would get a 50% acid solution. How many kilograms of the 25% solution were used to prepare the mixture?

20 athletes are competing at the diving championship, including 7 divers from Holland and 10 divers from Colombia. The order of performances is determined by drawing lots. Find the probability that a jumper from Holland will compete eighth.

Content
Introduction
Map individual achievements student
Instructions for performing the work
Standard Unified State Exam answer forms
Option 1
Option 2
Option 3
Option 4
Option 5
Option 6
Option 7
Option 8
Option 9
Option 10
Option 11
Option 12
Option 13
Option 14
Option 15
Option 16
Option 17
Option 18
Option 19
Option 20
Option 21
Option 22
Option 23
Option 24
Option 25
Option 26
Option 27
Option 28
Option 29
Option 30
Option 31
Option 32
Option 33
Option 34
Option 35
Option 36
Answers
Solutions and evaluation criteria for assignments 13-19.

Free download e-book in a convenient format, watch and read:
Download the Unified State Exam book, Mathematics, Profile level, Model exam options, 36 options, Yashchenko I.V., 2017 - fileskachat.com, fast and free download.

• I will pass the Unified State Exam, Mathematics, Self-study course, Technology for solving problems, Profile level, Part 3, Geometry, Yashchenko I.V., Shestakov S.A., 2018
• I will pass the Unified State Exam, Mathematics, Self-study course, Technology for solving problems, Profile level, Part 2, Algebra and the beginnings of mathematical analysis, Yashchenko I.V., Shestakov S.A., 2018
• I will pass the Unified State Exam, Mathematics, Self-study course, Technology for solving problems, Basic level, Part 3, Geometry, Yashchenko I.V., Shestakov S.A., 2018
• I will pass the Unified State Exam, Mathematics, Profile level, Part 3, Geometry, Yashchenko I.V., Shestakov S.A., 2018

The following textbooks and books.

Unified State Exam 2017 Trial version

Profile level
Problem conditions with

The examination paper consists of two parts, including 19 tasks. You are given 3 hours and 55 minutes to complete the examination paper in mathematics. Answers to tasks 1–12 are written as a whole number or a final decimal fraction. When completing tasks 13–19, you need to write down the complete solution.

Part 1

Answer to tasks 1-12 is an integer or a terminating decimal fraction. The answer should be written down in answer form No. 1 to the right of the number of the corresponding task,starting from the first cell. Write each number, minus sign and decimal point inin a separate box in accordance with the samples given in the form. There is no need to write units of measurement.

1 . At a gas station, one liter of gasoline costs 33 rubles. 20 kopecks The driver poured 10 liters of gasoline into the tank and bought a bottle of water for 41 rubles. How many rubles will he receive in change from 1000 rubles?

2 . The figure shows a graph of precipitation in Kaliningrad from February 4 to February 10, 1974. Days are plotted on the x-axis, and precipitation in mm is plotted on the y-axis. Determine from the figure how many days from this period there was from 2 to 8 mm of precipitation.

3 . Two circles are depicted on checkered paper. The area of ​​the inner circle is 2. Find the area of ​​the shaded figure.

4 . The probability that student Petya will correctly solve more than 8 problems during a history test is 0.76. The probability that Petya will correctly solve more than 7 problems is 0.88. Find the probability that Petya solves exactly 8 problems correctly.

5 . Solve the equation. If an equation has more than one root, answer with the smaller one.

6 . Circle inscribed in isosceles triangle, divides one of the lateral sides at the point of contact into two segments, the lengths of which are 10 and 1, counting from the vertex opposite the base. Find the perimeter of the triangle.

7 . The figure shows a graph of the derivative of the function , defined on the interval (–8; 9). Find the number of minimum points of the function , belonging to the segment [–4; 8].

8 . Find the lateral surface area of ​​the correct triangular prism, inscribed in a cylinder whose base radius is equal to and whose height is equal to .

9 . Find the meaning of the expression

10 . Distance from an observer at altitude h m above the ground, expressed in kilometers, to the horizon line visible to them is calculated by the formula, where R= 6400 km is the radius of the Earth. A person standing on the beach sees the horizon 4.8 kilometers away. There is a staircase leading to the beach, each step of which is 10 cm high. What is the least number of steps a person needs to climb in order for him to see the horizon at a distance of at least 6.4 kilometers?

11 . Two people go from the same house for a walk to the edge of the forest, located 1.1 km from the house. One goes at a speed of 2.5 km/h, and the other at a speed of 3 km/h. Having reached the edge, the second one returns back at the same speed. At what distance from the departure point will they meet? Give your answer in kilometers.

12 . Find the minimum point of the function belonging to the interval.

To record solutions and answers to tasks 13-19 use answer sheet No. 2.Write down first the number of the task being performed, and then the full reasoned decision andanswer.

13 . a) Solve the equation. b) Determine which of its roots belong to the segment.

14 . In a parallelepiped ABCDA 1 B 1 C 1 D 1 dot M mid-rib C 1 D 1 and dot K divides an edge A.A. 1 regarding AK:KA = 1:3. Through dots K And M a plane α is drawn parallel to the line BD and intersecting diagonal A 1 C at the point O.
a) Prove that the plane α divides the diagonal A 1 C in a relationship A 1 O: OC = 3:5.
b) Find the angle between the plane α and the plane ( ABC), if it is known that ABCDA 1 B 1 C 1 D 1- cube

15 . Solve the inequality .

16 . Parallelogram ABCD and the circle are arranged so that the side AB touches the circle CD is a chord, and the sides D A and B.C. intersect the circle at points P And Q respectively.
a) Prove that near a quadrilateral ABQP can describe a circle.
b) Find the length of the segment DQ, if it is known that AP= a, B.C.= b, BQ= c.

17 . Vasya took out a bank loan in the amount of 270,200 rubles. The loan repayment scheme is as follows: at the end of each year, the bank increases the remaining amount of the debt by 10%, and then Vasya transfers his next payment to the bank. It is known that Vasya repaid the loan in three years, and each of his subsequent payments was exactly three times larger than the previous one. What amount did Vasya pay for the first time? Give your answer in rubles.

18 . Find all such values ​​of the parameter for each of which the equation has solutions on the interval ..

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 Exam assignments 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- problems 18-19 –. This part of the exam tasks checks not only the level mathematical knowledge, but also the presence or absence of a creative approach to solving dry “digital” tasks, as well as the effectiveness of the ability to use knowledge and skills as a professional tool.

Important! Therefore, in preparation for Unified State Exam theory In mathematics, always support them 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.

On State exam 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- this is the solution online tests in mathematics 2020. Choose and get the maximum score!

Assessment

two parts, including 19 tasks. Part 1 Part 2

3 hours 55 minutes(235 minutes).

Answers

But you can make a compass Calculators on the exam not used.

passport), pass and capillary or! Allowed to take with myself water(in a transparent bottle) and I'm going

The examination paper consists of two parts, including 19 tasks. Part 1 contains 8 tasks of a basic difficulty level with a short answer. Part 2 contains 4 tasks increased level difficulties with short answer and 7 tasks high level Difficulty with a detailed answer.

The time allocated for completing the examination work in mathematics is 3 hours 55 minutes(235 minutes).

Answers for tasks 1–12 are written down as a whole number or finite decimal fraction. Write the numbers in the answer fields in the text of the work, and then transfer them to answer form No. 1, issued during the exam!

When performing work, you can use the ones issued along with the work. Only a ruler is allowed, but it's possible make a compass with your own hands. Do not use tools with markings on them. reference materials. Calculators on the exam not used.

You must have an identification document with you during the exam ( passport), pass and capillary or gel pen with black ink! Allowed to take with myself water(in a transparent bottle) and I'm going(fruit, chocolate, buns, sandwiches), but they may ask you to leave them in the corridor.