Preparation for the Unified State Exam in mathematics (profile level): assignments, solutions and explanations. Grammatical means of communication

It is believed that the task on stereometry on the Profile Unified State Exam in mathematics is only for excellent students. That solving it requires special talents and mysterious “spatial thinking,” which only a few lucky people possess from birth.

Is it so?

Fortunately, everything is much simpler. What is so beautifully called “spatial thinking” most often means knowledge of the basics of stereometry and the ability to draw drawings.

First, you need knowledge of stereometry formulas. Our tables “Polyhedra” and “Bodies of rotation” contain all the formulas by which the volumes and surface areas of three-dimensional bodies are calculated.

Secondly, confidently solving geometry problems presented in Part 1 (the first 12 Unified State Exam problems). These are both planimetric and stereometric problems.

And most importantly, to solve problem 14 you will need the basic axioms and theorems of stereometry. It is best if you purchase a textbook on geometry for grades 10-11 (author - A.V. Pogorelov or L.S. Atanasyan), and answer the questions, the list of which is given below. Write down definitions and statements of theorems in your notebook. Make drawings. Try to prove theorems yourself.

While working on this task, formulate for yourself how they differ definition and sign. There is, for example, a definition of parallelism of a line and a plane - and a sign of parallelism of a line and a plane. What is the difference between them?

It is very good if you do the task yourself and then compare it with the answers. All answers can be found on our website, in this section.

Stereometry program.

1. A plane in space. Finish the sentence: A plane can be drawn through...

   (Give four possible answers.)

2. Location of planes in space. Finish the sentence: If two planes have a common point, then they...
3. Parallelism of a line and a plane. Definition and sign.
4. What is oblique and oblique projection. Drawing.
5. The angle between a straight line and a plane.
6. Perpendicularity of a line and a plane. Definition and sign.
7. Crossing straight lines. The angle between intersecting lines. Distance between crossing lines.
8. The distance from a straight line to a plane parallel to it.
9. Parallelism of planes. Definition and sign.
10. Perpendicularity of planes. Definition and sign.
11. Complete the sentence: a) Lines of intersection of two parallel planes with a third plane...

    b) Segments of parallel lines contained between parallel planes...

Here are a few simple rules for solving problems in stereometry:

There are two main ways to solve problems in stereometry on the Unified State Examination in mathematics. The first is classic: the practical application of definitions, theorems and characteristics, the list of which is given above. Second -

Task 2 of the Unified State Exam in Society: how to solve

The difficulty of this task 2 of the Unified State Exam in social studies is that it requires you to find a generalizing word for a specified number of terms. A generalizing word is a generic term or concept that includes in its meaning the meanings of other concepts and terms. As in other Unified State Examination tasks on society, the topics of the tasks can be very different: social sphere, political, spiritual, etc.

Here, for example, is a task from a real Unified State Exam test in society:

It immediately becomes clear to intelligent boys and girls that the proposed words relate to the topic “Spiritual Sphere of Society”, namely to the topic of religion. If you find it difficult to answer right away, I recommend reading my previous post "" . Having read the terms for the most knowledgeable, it immediately becomes clear that there are only two options left for the answer: cult and religion. What will be more generalizing? A cult is the worship of something.

You can experiment by placing a broom in the corner of your room. And pray to him every day, talk to him... In a month this will be the most valuable item for you :). Create a cult of the broom. What is religion? This is a specific form of worldview, awareness of the world. It is clear that the concept of “religion” includes the concept of “cult,” since a worldview may include the worship of various deities. For example, paganism among the Eastern Slavs: some had the cult of Perun (the god of thunder and lightning), others had the cult of the god of swamps, etc.

Or, for example, Orthodox Christianity: there is the cult of Jesus Christ, there is the cult of the Holy Spirit, there is the cult of the Most Holy Theotokos... Got it?

OK. So the correct answer is: religion

Recommendation 2. You need to have a good knowledge of terms and concepts from various topics in social studies. Understand which terms are related to which ones, and which ones follow from them. For this purpose in my paid video course "Social studies: Unified State Exam 100 points " I have given the structure of terms for all topics of Social Science. I also highly recommend your article about.

Let's look at another task 2 of the Unified State Exam in social studies:

We immediately understand that task 2 of the Unified State Exam examines the topic Social Sphere. If you have forgotten the topic, download my free video course. If you don't do this, you will most likely make a mistake. Some people's logic is so crooked that it's simply brutal! Meanwhile, the correct answer: “agent of socialization” is a group or association that participates in an individual’s mastery of the rules and norms of society, as well as social roles. If you are not familiar with these terms, I again highly recommend downloading my free video course.

Recommendation 3. Be extremely careful! Solve tasks 2 of the Unified State Exam in social studies again and again to do this qualitatively on the machine. Here is an example of a similar task that is more difficult:

The theme “Science” from the spiritual sphere of society. By the way, I had a detailed article on this topic. People who are not very attentive will immediately make a mistake by indicating in the answer: classification basis, or theoretical validity. Between the correct answer: scientific knowledge , which includes different classifications and theoretical validity!

In the following posts we will definitely look at other difficult tasks on society, so !

I have attached a couple of tasks for Unified State Examination 2 in society for you to decide:

In task No. 2 of the Unified State Examination in mathematics, it is necessary to demonstrate knowledge of working with power expressions.

Theory for task No. 2

The rules for handling degrees can be presented as follows:

In addition, you should remember about operations with fractions:

Now you can move on to analyzing typical options! 🙂

Analysis of typical options for tasks No. 2 of the Unified State Exam in basic level mathematics

First version of the task

Find the meaning of the expression

Execution algorithm:

1. Express a number with a negative exponent as a proper fraction.
2. Perform the first multiplication.
3. Represent powers of numbers as prime numbers, replacing powers by multiplication.
4. Perform multiplication.
5. Perform addition.

Solution:

That is: 10 -1 = 1/10 1 = 1/10

Let's perform the first multiplication, that is, multiplying a whole number by a proper fraction. To do this, multiply the numerator of the fraction by a whole number, and leave the denominator unchanged.

9 1/10 = (9 1)/10 = 9/10

The first power of a number is always the number itself.

The second power of a number is a number multiplied by itself.

10 2 = 10 10 = 100

Answer: 560.9

Second version of the task

Find the meaning of the expression

Execution algorithm:

1. Represent the first power of a number as an integer.
2. Represent negative powers of numbers as proper fractions.
3. Perform multiplication of integers.
4. Multiply whole numbers by proper fractions.
5. Perform addition.

Solution:

The first power of a number is always the number itself. (10 1 = 10)

To represent a negative power of a number as an ordinary fraction, you need to divide 1 by this number, but to a positive power.

10 -1 = 1/10 1 = 1/10

10 -2 = 1/10 2 = 1/(10 10) = 1/100

Let's multiply integers.

3 10 1 = 3 10 = 30

Let's multiply whole numbers by proper fractions.

4 10 -2 = 4 1/100 = (4 1)/100 = 4/100

2 10 -1 = 2 1/10 = (2 1)/10 = 2/10

Let us calculate the value of the expression, taking into account that

Answer: 30.24

Third version of the task

Find the meaning of the expression

Execution algorithm:

1. Represent powers of numbers in the form of multiplication and calculate the value of powers of numbers.
2. Perform multiplication.
3. Perform addition.

Solution:

Let's represent powers of numbers in the form of multiplication. In order to represent the power of a number in the form of multiplication, you need to multiply this number by itself as many times as it is contained in the exponent.

2 4 = 2 2 2 2 = 16

2 3 = 2 2 2 = 8

Let's do the multiplication:

4 2 4 = 4 16 = 64

3 2 3 = 3 8 = 24

Let's calculate the value of the expression:

Fourth version of the task

Find the meaning of the expression

Execution algorithm:

1. Perform the action in parentheses.
2. Perform multiplication.

Solution:

Let us represent the power of a number in such a way that we can take the common factor out of the bracket.

3 4 3 + 2 4 4 = 4 3 (3 + 2 4)

Let's perform the action in parentheses.

(3 + 2 4) = (3 + 8) = 11

4 3 = 4 4 4 = 64

Let us calculate the value of the expression, taking into account that

Fifth version of the task

Find the meaning of the expression

Execution algorithm:

1. Let us represent the power of a number in such a way that we can take the common factor out of the bracket.
2. Place the common factor out of brackets.
3. Perform the action in parentheses.
4. Represent the power of a number as a multiplication and calculate the value of the power of the number.
5. Perform multiplication.

Solution:

Let us represent the power of a number in such a way that we can take the common factor out of the bracket.

Let's take the common factor out of brackets

2 5 3 + 3 5 2 = 5 2 (2 5 + 3)

Let's perform the action in parentheses.

(2 5 + 3) = (10 + 3) = 13

Let's represent the power of a number in the form of multiplication. In order to represent the power of a number in the form of multiplication, you need to multiply this number by itself as many times as it is contained in the exponent.

5 2 = 5 5 = 25

Let us calculate the value of the expression, taking into account that

We perform multiplication in a column, we have:

Option for the second task from the Unified State Exam 2017 (1)

Find the meaning of the expression:

Solution:

In this task, it is more convenient to bring the values ​​to a more familiar form, namely, write the numbers in the numerator and denominator in standard form:

After this, you can divide 24 by 6, the result is 4.

Ten to the fourth power when divided by ten to the third power gives ten to the first, or simply ten, so we get:

Option for the second task from the Unified State Exam 2017 (2)

Find the meaning of the expression:

Solution:

In this case, we should note that the number 6 in the denominator is factored into factors 2 and 3 to the power of 5:

After this, you can perform reductions of degrees for two: 6-5 = 1, for three: 8-5 = 3.

Now we cube 3 and multiply by 2, getting 54.

Option for the second task of 2019 (1)

Execution algorithm

1. Apply to the numerator of holy powers (a x) y = a xy. We get 3 –6.
2. Apply to fractions of holy powers a x /a y =a x–y.
3. Raise 3 to the resulting power.

Solution:

(3 –3) 2 /3 –8 = 3 –6 /3 –8 = 3 –6–(–8)) = 3 –6+8 = 3 2 = 9

Option for the second task 2019 (2)

Execution algorithm

1. We use for the degree in the numerator (14 9) (ab) x =a x b x. Let us decompose 14 into the product of 2 and 7. We obtain the product of powers with bases 2 and 7.
2. Let's transform the expression into 2 fractions, each of which will contain powers with the same bases.
3. Apply to fractions of holy powers a x /a y =a x–y.
4. We find the resulting product.

Solution:

14 9 / 2 7 7 8 = (2 7) 9 / 2 7 7 8 = 2 9 7 9 / 2 7 7 8 = 2 9–7 7 9–8 = 2 2 7 1 = 4 ·7 = 28

Option for the second task 2019 (3)

Execution algorithm

1. We take the common factor 5 2 =25 out of brackets.
2. We multiply the numbers 2 and 5 in brackets. We get 10.
3. We add 10 and 3 in brackets. We get 13.
4. We multiply the common factor 25 and 13.

Solution:

2 5 3 +3 5 2 = 5 2 (2 5+3) = 25 (10+3) = 25 13 = 325

Option for the second task 2019 (4)

Execution algorithm

1. Square it (–1). We get 1, since it is raised to an even power.
2. Raise (–1) to the 5th power. We get –1, because raising to an odd power occurs.
3. We perform multiplication operations.
4. We get the difference of two numbers. We find her.

Solution:

6·(–1) 2 +4·(–1) 5 = 6·1+4·(–1) = 6+(–4) = 6–4 = 2

Option for the second task 2019 (5)

Execution algorithm

1. Let's convert the factors 10 3 and 10 2 into integers.
2. We find the products by moving the decimal point to the right by the appropriate number of decimal places.
3. Find the resulting sum.

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 of an increased level of complexity with a short answer and 7 tasks of a high level of complexity with a detailed answer.

The exam work in mathematics is allotted 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 instruments with reference materials printed on them. 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.

Lexical means of communication:

1. Lexical repetition- repetition of the same word. Around the city, forests spread across the low hills, mighty and untouched. In the forests there were large meadows and remote lakes with huge old pine trees along the banks.
2. Cognates. Of course, such a master knew his worth, felt the difference between himself and a less talented person, but he also knew perfectly well another difference - the difference between himself and a more talented person. Respect for the more capable and experienced is the first sign of talent.
3. Synonyms. We saw a moose in the forest. Sokhaty walked along the edge of the forest and was not afraid of anyone.
4. Antonyms. Nature has many friends. She has significantly fewer enemies.
5. Descriptive phrases. They built a highway. A noisy, fast-moving river of life connected the region with the capital.

Grammatical means of communication:

1. Personal pronouns. 1) And now I’m listening to the voice of an ancient stream. He coos like a wild dove. 2) The call for forest protection should be addressed primarily to young people. She should live and manage this land, she should decorate it. 3) He unexpectedly returned to his native village. His arrival delighted and frightened his mother.
2. Demonstrative pronouns(such, that, this) 1) A dark sky with bright, needle-like stars floated over the village. Such stars appear only in autumn. 2) The corncrakes screamed with distant, sweet twitching sounds. These corncrakes and sunsets are unforgettable; they were preserved forever by pure vision. – in the second text the means of communication are lexical repetition and the demonstrative pronoun “these”.
3. Pronominal adverbs(there, so, then, etc.) He [Nikolai Rostov] knew that this story contributed to the glorification of our weapons, and therefore it was necessary to pretend that you did not doubt it. That's what he did.
4. Unions(mostly composing) It was May 1945. Spring thundered. The people and the land rejoiced. Moscow saluted the heroes. And joy flew into the sky like lights. With the same chatter and laughter, the officers hastily began to get ready; again they put the samovar on dirty water. But Rostov, without waiting for tea, went to the squadron.”
5. Particles.
6. Introductory words and constructions(in one word, so, firstly, etc.) The young people spoke about everything Russian with contempt or indifference and, jokingly, predicted for Russia the fate of the Confederation of the Rhine. In short, the society was quite disgusting.
7. Unity of tense forms of verbs- the use of identical forms of grammatical tense, which indicate simultaneity or sequence of situations. Imitation of the French tone of the times of Louis XV was in vogue. Love for the fatherland seemed pedantry. The wise men of that time praised Napoleon with fanatical servility and joked about our failures. – all verbs are used in the past tense.
8. Incomplete sentences and ellipsis, referring to the previous elements of the text: Gorkin cuts the bread, distributes the slices. He puts it on me too: it’s huge, you’ll cover your whole face.
9. Syntactic parallelism– identical construction of several adjacent sentences. To be able to speak is an art. Listening is a culture.

Meaningful relationships expressed by coordinating conjunctions:

1. Connecting: and, yes (=and), and...and..., not only... but also, like... so and, also, too
2. Dividers: or, or, then...that, not that...not that, or...or, either...or
3. Nasty: a, but, yes (=but), however, but
4. Gradational: not only, but also, not so much... as, not really... but
5. Explanatory: that is, namely
6. Connecting: also, also, yes and, and moreover, and
7. too, yes and, that is, namely.

Meaningful relations expressed by subordinating conjunctions:

• Temporary: when, while, barely, only, while, just, barely, barely
• Causal: since, because, because, in view of the fact that, due to the fact that, due to the fact that, for (obsolete), due to the fact that
• Conditional: if (if only, if, if - obsolete), if, once, as soon
• Target: so that, in order to, in order to (obsolete), for the purpose of, in order to, then in order to
• Consequences: So
• Concessive: although, despite the fact that
• Comparative: as, as if, as if, exactly, than, as if, likewise, rather than (obsolete)
• Explanatory: what, how, to
• Conjunctions are not used at the beginning of a sentence: so, than, rather than, as well as explanatory conjunctions: what, how, so that.