Average general education

UMK line G. K. Muravina. Algebra and beginnings 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

We analyze tasks and solve examples with the teacher

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 get school certificate, the graduate must pass two mandatory exams in the form of the Unified State Exam, one of which is mathematics. In accordance with the Development Concept mathematics education V 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 Examination participants’ 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. A flow meter was installed in the apartment where Peter lives cold water(counter). 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 by the value of its argument when in various ways specifying a function and describing 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.

In this section we are preparing for the Unified State Examination in mathematics at a basic, specialized level - we present analysis of problems, tests, description of the exam and useful recommendations. Using our resource, you will at least understand how to solve problems and be able to successfully pass the Unified State Exam in mathematics in 2019. Let's begin!

The Unified State Examination in mathematics is mandatory exam any student in 11th grade, so the information presented in this section is relevant for everyone. The mathematics exam is divided into two types - basic and specialized. In this section I provide an analysis of each type of task with detailed explanation for two options. Unified State Exam assignments strictly thematic, so for each issue you can give precise recommendations and provide the theory necessary specifically for solving this type of task. Below you will find links to assignments, by clicking on which you can study the theory and analyze examples. Examples are constantly replenished and updated.

Structure of the basic level of the Unified State Examination in mathematics

Examination paper in mathematics basic level consists of one piece , including 20 short-answer tasks. All tasks are aimed at testing the mastery of basic skills and practical application skills mathematical knowledge in everyday situations.

The answer to each of tasks 1–20 is integer, final decimal , or sequence of numbers .

A task with a short answer is considered completed if the correct answer is written down in answer form No. 1 in the form provided for in the instructions for completing the task.

Soon the fairy tale will tell, but not soon the deed will be done

Unfortunately, new teaching materials do not appear on the site as quickly as many visitors would like. It’s not surprising, because creating a high-quality lesson takes quite a lot of time. But what to do? Do you have a topic or a separate task that you did not find on my resource.... And we need to figure it out today!

Therefore, as the project developed, there was a need to create bank of ready-made solutions in various branches of higher mathematics. And you should definitely like this page, since the bank completely FREE! Right now you have the opportunity to download for free file archives with completed tasks various topics towers.

Source of assignments?

All tasks are completed by me personally, which guarantees high quality solutions.

A typical archive contains, as a rule, 200-300 tasks with ready-made solutions, and I tried to collect as diverse materials as possible, including rarely encountered tasks and examples of increased complexity. Solyanka consists of tests and individual assignments, completed to order by me at one time. In addition, I will please full-time students - the archives include standard calculations from Kuznetsov’s collection, IDZ from Ryabushko’s collection, as well as probability theory from Chudesenko’s collection. For reasons mentioned in the review Useful math sites, I'm giving them away completely free. Yes, of course, there are not very many calculations, but some students will be very lucky!

Just, comrades, let’s do it without complaints or offense. Yes, the quality is high, all examples and problems have been checked by teachers, but the solutions may contain typos, incorrect things and even errors. The tasks are presented “as is”. Doesn't make sense inform me about any bugs you find, because, firstly, they are technically difficult to fix, and secondly, there is simply no time (one repackaging and reupload of the archive is worth it). The exception is pdf files, which are located directly on the server, please write to me if you find a SERIOUS oversight in them.

There may be some misunderstandings and even misconceptions. But almost all site visitors have a non-zero level of mathematical training, so sample solutions will bring much more more benefits than harm; ideal if you already have a good understanding of a particular topic and want to find additional examples.

Each collection of problems is accompanied by short comments about the contents of the archive. Moreover, the bank is replenished with new materials from time to time.

In a word, come and visit us!

Ready-made tasks with solutions on the topic
Linear algebra and analytical geometry

The archive contains about three hundred tasks on linear algebra and analytical geometry, including several calculations by Kuznetsov and several Ryabushko IDZ. Almost all typical algebra problems have been solved: solving systems of equations, calculating determinants, studying systems for consistency, problems determining the rank of a matrix, matrix equations, eigenvalue/eigenvector problems. Analytical geometry is represented by both problems on the plane and in three-dimensional space.

Examples of solving limits

The archive contains a couple of hundred split limits, including sequence limits. Examples of solving limits using remarkable equivalences are considered. The collection contains 3 calculations by Kuztsev and 3 versions of Ryabushko’s IDZ. I won’t say which options exactly – enjoy playing the lottery and having fun! If you have read existing lessons on the topic, then understanding many of the limits of the selection will not be difficult

Differentiation of functions of one variable

The archive is represented mainly by standard calculations from Kuznetsov’s collection (10 options), there are 3 options from Ryabushko. The examples will allow you to significantly improve your differentiation technique, since the derivatives from Kuznetsov’s collection are quite heavy. Moreover, some derivatives have to be found in several stages! In addition to derivatives, there are typical problems: finding derivatives of an arbitrary, “nth” order, examples using the Lagrange formula; problems on constructing a tangent and a normal, and a few problems on finding a derivative by definition of a derivative.

Function studies

Perhaps the most time-consuming and painstaking selection. The solver contains 69 fully researched functions of one variable, including polynomial functions, fractional rational functions, functions with exponents, logarithms and other, rarer examples. There is a fairly high probability that you will find your own example or a very similar function. The piggy bank acts as an application of the topic “Functions and Graphs” and is intended for self-testing/training.

Ready-made solutions of indefinite integrals

Mostly, indefinite integrals, but there is a little definite integrals. The collection includes more than two hundred integrals, including very complex ones. Many examples were considered that did not fit into my lessons! The archive includes 6 standard calculations by Kuznetsov and 2 IDZ by Ryabushko. Kuznetsov's calculations contain problems of finding the volumes of bodies, arc lengths and surface areas, not only in the Cartesian, but also in the polar coordinate system, as well as for parametrically specified functions.

Problems on number and function series

Another heavy brick. Of course, research examples are available number series for convergence, and there are creative series that require experience and a non-standard approach. My pride is that there wasn’t a row that I couldn’t crack. The problems of finding the sum of a series and various tasks for approximate calculations using a series are considered. Function series are represented by the common problem of finding the region of convergence (again with creative cases). At your service is the popular problem of expanding a particular solution of a differential equation into a series. There are a dozen examples of solutions to Fourier series.
And even the disclosure of uncertainties within limits using series expansion. The archive contains 13 Kuznetsov types and a couple of Ryabushko variants. Bon appetit!

Ready-made solutions for differential equations

A good hundred diffusers. Used in problems various methods solutions differential equations first order, including those that I did not cover in class. There are diffusers of increased complexity - second-order and higher-order differentials, examples using the method of variation of arbitrary constants. Systems of differential equations are included. The archive contains 8 standard calculations by Kuznetsov and 2 variants by Ryabushko. In addition, I packed the collection exam papers, in addition to differential equations, they contain problems on series and multiple integrals.

Examples of solutions to multiple and curvilinear integrals

Double integral, changing the order of traversal of a region, calculating the area using double integral. Triple integral, calculating the volume of a body using triple integral. Examples of solutions not only in Cartesian, but also in polar, cylindrical coordinate systems. The main applications of multiple integrals are: finding the center of gravity of a flat plate, the center of mass of a body, and other problems. Curvilinear integrals along a curve, along a contour. Hit problem of finding the work of a force along a contour and using Green's formula.

Do you really believe that this is difficult?!

Examples of solving problems in field theory

Among other things, the collection includes the most popular problems with solutions of the following types: finding the directional derivative and gradient, calculating the flow vector field through a part of the plane, calculating the flow of a vector field through a closed surface using the Ostrogradsky-Gauss formula, calculating the circulation of a vector field along a contour. The archive contains several calculations from Kuznetsov’s collection.

Examples of solutions to problems in combinatorics and probability theory

In the following pdfs you can familiarize yourself with additional tasks for the lessons on combinatorics And probability theory. At the same time, I did not have the goal of collecting hundreds of examples; on the contrary, I tried to filter out obvious duplicates. In addition, in most files the tasks are ordered in order of increasing difficulty:

Problems with economic content

Many people know how difficult it is to “unwind” after long periods of winter holidays, and then one day after the holidays I found a great way to get back into the working rhythm - to create collections of ready-made tasks with a pronounced economic content, which are mothballed in my collection and which do not benefit anyone. These are some problems of economic and mathematical modeling, these are problems of financial mathematics (interest, rent, etc.), and finally, these are tasks related rather to economics. So let's improve the gloomy January weather with the following useful materials:

Problems of financial mathematics

In fact, it is said too loudly - in next file You will find urgent and practically useful tasks on interest and calculation of payments on deposits/loans.
I retouched the few solutions I found and made them as clear as possible! Moreover, along the way I figured out a few things myself =)

And, of course, a calculator for the tasks! Yes, at least in order to instantly find the percentage of interest from any number:

Input balance problem (Leontiev model)

One of the most famous economic and mathematical models that describes many sectors of the economy, part of the products of which are mutually consumed as a result of production, and the other part is a final product. Difficulties arose with direct/full cost matrices, painstakingly finding inverse matrix and matrix multiplication with fractional numbers? The following program will instantly carry out calculations that you would simply be tired of doing on a regular microcalculator:

This program was created in MS Excel and is available to active library users. Or for a reasonable fee.

Demo version you can see . Calculations were carried out for the most common educational case - three industries; Moreover, the solution can be printed and handed in!

And what’s especially nice is that there’s an example in front of you practical use matrix algebras.

Typical problems on economic indices

The following modest pdf, of course, does not cover the entire variety of economic indices, but I have thoroughly analyzed and systematized, perhaps, all varieties of the most popular problem in price/physical sales/turnover indices+ similar task to cost and total cost indices. Even a child can understand:

...and if he also uses it with this program, then he will never make mistakes!

Objectives of economic statistics

As in the previous file, all higher mathematics here is limited to the simplest arithmetic operations =) ...And it would be funny if it weren’t so sad (Economics is not mine specialized subject, but, nevertheless, your humble servant at one time had to endure several dozen tasks on socio-economic and production statistics, which were an “addition” to tests By mathematical statistics. In this regard, I do not guarantee super quality, although, on the other hand, there is no hack work either:

And finally, to the delight of full-time students, I am opening two more distributions:

Standard calculations on probability theory from Chudesenko's collection for free!

The first 22 problems have been solved, some more, some less. The first option for demonstration is posted directly on the site: The remaining options (2, 5, 7, 8, 9, 10, 11, 12, 15, 20, 21, 24, 25) due to their light weight, I moved from the file hosting service to the newly created library site. The archive can also be useful because Chudesenko’s collection contains quite difficult problems in probability theory, and perhaps you will find what you were looking for.

IDZ from the Ryabushko problem book - free!!

A solid addition to all the archives + tasks that have not yet been proposed on this page, in particular, examples with functions of several variables. Of course, not all 4 volumes are closed, but for some sections there is a good half of the options, or even more! Completed individual tasks can be used both “for their intended purpose” and as additional self-preparation/training. Long awaited Ryabushko's problem book was awarded a separate web page.

And a number of other works from various universities in the country:

4) MPEI tests(Moscow Economic Institute).

On at the moment That’s all for now, check back from time to time – I’ll come up with something else!