"Various ways solving tasks No. 13 of the exam "
Meeting of the regional methodological association
math teachers " Professional competence a teacher as a condition for high-quality training of students for GIA "
Vorobieva Olga Alexandrovna,
teacher of mathematics, secondary school №3
Analyzing uSE results in mathematics, it should be noted that many students do not start completing assignments from group C, and if they do, they often make mistakes. There are many reasons for this. One of them is an insufficient number of independently solved tasks, mistakes made are not analyzed, and, as a rule, the knowledge gained is superficial, since basically only tasks of the same type are considered, and only standard methods of solution are considered.
- Analyzing the results of the exam in mathematics, it should be noted that many students do not start completing assignments from group C, and if they do, then they often make mistakes. There are many reasons for this. One of them is an insufficient number of independently solved tasks, mistakes made are not analyzed, and, as a rule, the knowledge gained is superficial, since mainly only tasks of the same type are considered, and only standard methods of solution.
- In task 13 of the USE in mathematics of the profile level, it is required to solve the equation and select its roots that satisfy a certain condition.
- The selection of roots is an additional item of the problem conditions or logically follows from the structure of the equation itself. And experience shows that these restrictions are precisely the main difficulty for students.
- Arithmetic way
- Algebraic way
- Geometric way
- Functional-graphic way
- Direct substitution of roots in the equation and existing restrictions
- Iterating over integer parameter values \u200b\u200band calculating roots
- Solving an Inequality for an Unknown Integer Parameter and Calculating the Roots
- Study of an equation with two integer parameters (used when solving a system of equations)
- Selecting the roots of the trigonometric equation on number circle
- Selecting the roots of a trigonometric equation on the number line
In the 13th task of the profile level of the exam in mathematics, it is necessary to solve the equation, but already of an increased level of complexity, since tasks of the former level C begin from 13 tasks, and this task can be called C1. Let's move on to examining examples typical assignments.
Analysis of typical options for tasks No. 13 of the USE in mathematics of the profile level
The first version of the task (demo version 2018)
a) Solve the equation cos2x \u003d 1-cos (n / 2-x)
b) Find all the roots of this equation, belonging to the gap [-5p / 2; -p].
Solution algorithm:
- t
- We do the reverse change and solve the simplest trigonometric equations.
- We build a number axis.
- We put roots on it.
- We mark the ends of the segment.
- We select those values \u200b\u200bthat lie within the interval.
- We write down the answer.
Decision:
1. We transform the right side of the equality using the reduction formula cos ( π/ 2−x) \u003d sin x... We have:
сos2x \u003d 1 - sin x.
Transform the left side of the equation using the double argument cosine formula using sine:
cos (2x) \u003d 1−2sin 2x
We get the following equation: 1 − sin 2 x\u003d 1− sin x
Now the equation contains only one trigonometric function sin x.
2. Introduce a replacement: t \u003d sin x... We solve the resulting quadratic equation:
1−2t 2 =1−t,
−2t 2 +t=0,
t(−2t+1)=0,
t \u003d 0or -2t + 1 \u003d 0,
t 1 \u003d 0 t 2 \u003d 1/2.
3. We do the reverse replacement:
sin x\u003d 0 or sin x = ½
We solve these equations:
sin x =0↔x=πn, nЄZ
sin ( x)=1/2↔x\u003d (-1) n ∙ ( π / 6)+ πn, nЄZ.
Therefore, we obtain two families of solutions.
1. In the previous section, two families were obtained, each of which has infinitely many solutions. It is necessary to find out which of them are in the given interval. To do this, build a number line.
2. We put on it the roots of both families, marking them with green (first) and blue (second).
3. Mark the ends of the gap in red.
4. In the indicated interval there are three roots that are three roots: −2 π ;−11π/ 6 and −7 π/ 6.
and) πn, nЄZ;(-1) n ∙ ( π / 6)+ πn, nЄZ
b) −2 π ;−11π 6;−7π 6
Second variant of the task (from Yashchenko, no. 1)
a) Solve the equation.
Solution algorithm:
- Replace this function with a variable tand solve the resulting quadratic equation.
- We make the reverse substitution and solve the simplest exponential, then trigonometric equations.
- We build a coordinate plane and a circle of unit radius on it.
- Mark the points that are the ends of the segment.
- We select those values \u200b\u200bthat lie inside the segment.
- We write down the answer.
Decision:
1. Introduce the replacement t \u003d 4 cos x. then the equation will take the form:
We solve the quadratic equation using the discriminant and root formulas:
D \u003d b 2 - c \u003d 81 - 4 ∙ 4 ∙ 2 \u003d 49,
t 1 \u003d (9 - 7) / 8 \u003d ¼, t 2 \u003d (9 + 7) / 8 \u003d 2.
1. Build a coordinate plane and a circle of unit radius on it.
2. Mark the points that are the ends of the segment.
3. Select the values \u200b\u200bthat lie inside the segment.
These are the roots. There are two of them.
and) 
The third variant of the task (from Yashchenko, no. 6)
a) Solve the equation 
.
b) Find all the roots of this equation that belong to the segment.
Solution algorithm:
- Using trigonometric formulas, we bring the equation to a form containing only one trigonometric function.
- Replace this function with a variable tand solve the resulting quadratic equation.
- We make the reverse substitution and solve the simplest exponential, and then trigonometric equations.
- We solve inequalities for each case.
- We write down the answer.
Decision:
1. According to the reduction formulas 
.
2. Then this equation will take the form:
3. Introducing a replacement 
... We get:
We solve the usual quadratic equation using the discriminant and root formulas:
Both roots are positive.
3. Back to variable x:
For the correct completion of the thirteenth task of the Unified State Exam in the Russian language, graduates can receive one primary score... To do this, you need to write out two words from the sentence that satisfy the condition. The assignment contains only such parts of speech as conjunctions, particles, prepositions, pronouns, adverbs and allied combinations. It is important to be able to distinguish between the meanings of homonymous words, which will help the theoretical material below.
Theory for the task number 13 of the exam in the Russian language
Spelling of different parts of speech
| Together | Apart | Hyphenated | |
|---|---|---|---|
| Alliances | Because, because, because, because, but, why, also, too, moreover, moreover, as if, as if, so that, then, if | As if, that is, since, if, as it were, or in connection with the fact that, then ... then, | |
| Allied combinations | Due to the fact that, due to the fact that, | not that ... not that | |
| though | |||
| Pronouns | Nobody´, nothing´, | Someone from something | Someone (-something -something -something) |
| Somebody, nothing, nothing | with no one, no one, | ||
| the same as; same, same, same | |||
| Adverbs | From everywhere, from now on, partly, in general, | Tirelessly, side by side, eye to eye, in general, still, abroad, abroad, everything is the same, exactly the same | Somehow (-something, -some, -something, something), |
| soon, in an undertone, alone, in advance, at first, at first, at first, little by little, at all, why, then, immediately, immediately | in Russian, in my opinion, | ||
| in a hare, first, | |||
| barely, barely, barely, slightly, today or tomorrow, | |||
| de jure, de facto | |||
| Prepositions | As a consequence, in view of | During, in continuation, in conclusion, in contrast, for purposes, in force, in measure, in the form, in the area, | From behind, from under, over-over |
| towards, about, | throughout, in respect, except, | ||
| like, following, | at the expense, | ||
| over, instead of, | not to mention | ||
| inside, because | |||
| like, | |||
| despite, | |||
| regardless of | |||
| Particles | Even, really, really | Whether, would, the same, | Still, let me |
| just what |
Derivative prepositions / nouns with a preposition
| Derived preposition | Noun with a preposition |
|---|---|
| During | During |
| - during winter | - along the river \u003d in the fast flow of the river |
| -in two hours | |
| In continuation | In continuation |
| Answers the questions “How long? When?" | You can put an adjective or participle between a preposition and a noun |
| - for a whole week | -in the sequel \u003d in the long-awaited sequel |
| -for two months | |
| Due to | In consequence, in consequence |
| Can be replaced with the preposition "because of"; answers the question "Why?" | You can put an adjective or participle between a preposition and a noun |
| - due to illness | -in the investigation of the case \u003d in the long investigation of the case |
| - in the investigation of the case \u003d in the sensational investigation of the case | |
| Towards | To a meeting |
| Can be replaced with the preposition "k"; answers the question "Where?" | |
| -to meet him | - to a meeting with a friend \u003d for a long-awaited meeting with a friend |
| In view of | Keep in mind |
| Can be replaced with the prepositions “because of” and “for a reason”; answers the question “Why? From what?" | Stable expression |
| - due to bad weather | |
| About | To the account |
| Can be replaced with the preposition "about (about)" | You can put an adjective between a preposition and a noun |
| - agreed on a trip | -to the fund account \u003d to the bank account of the fund |
| Like | Like |
| Can be replaced with the preposition "like" | This refers to the geometric term |
| -like lunch | |
| Finally | In custody |
| Can be replaced by the words "ending, in the end, at the end" | You can put an adjective between a preposition and a noun |
| -in conclusion \u003d in strict conclusion | |
| Following | Trace in trail |
| It matters "where" | Means "for" |
| -we looked after the departing train | -children followed the trail of each other |
It is important to remember the spelling of the following words: in connection with, as opposed to, following (\u003d behind), in the middle, near, instead of, throughout
Derivative prepositions / participles
Conjunctions / pronouns with particle
| Union | Pronoun with particle |
|---|---|
| To | What would |
| Can be replaced by unions “in order to, in order to”; “Would” cannot be removed from the phrase | Can be replaced with a noun; "Would" can be rearranged |
| -I came to tell you \u003d I came to tell you | - I asked what else I would like to read \u003d I asked what book I should read \u003d I asked what else I would like to read |
| Also, also | Same, same |
| You can replace the union "and" by placing it at the beginning of the sentence; “Same” cannot be removed from the phrase | "Same" can be rearranged |
| -I saw this movie too \u003d and I saw this movie | -I also read this book \u003d I read the same book |
| But | For that |
| Can be replaced with the union "but" | "That" can be replaced by a noun, adjective or adverb |
| - we rarely see each other, but we often call each other \u003d we rarely see each other, but often call each other | -no need to take on what you don’t like \u003d you don’t need to take up what you don’t like |
| so | So |
| Can be replaced by the words "summing up, therefore" | Means "very" |
| -so, we can say that \u003d summing up, we can say that | -we were tired and so hungry that we decided to stay at the hotel |
| That is why | From that |
| Means "since" | Can be replaced with a noun with a preposition |
| -it happened because \u003d it happened because | -from what he will do \u003d from the deed, what he will do |
| Moreover, moreover | Moreover, at what |
| Can be replaced with the words "moreover", "along with that" | "Volume" can be replaced with an adjective |
| -it works quickly, and also efficiently \u003d it works quickly, moreover, it also works efficiently | -the building has a beautiful garden \u003d the tall building has a beautiful garden |
Algorithm for the task
- We carefully read the task.
- We analyze each sentence, opening the brackets in accordance with the spelling rules of the Russian language.
- We write down the correct answer.
Analysis of typical options for assignment number 13 of the exam in the Russian language
The thirteenth task of the demo 2018
- (ON) THAT was the way L.N. Tolstoy, his relatives could guess, (ON) HOW hard is his brain working now.
- From the first pages I experienced a strange sensation: WOULD BE) from the gloomy world, I (THAT) HOUR was transferred to another world - sunny and bright.
- (C) CONSEQUENCES researchers have repeatedly said that the apotheosis of Russian glory is the painting "Heroes", in which V.M. Vasnetsov expressed his romantic and deeply civil understanding of Russia at the SAME time.
- The physical properties of interstellar gas significantly depend (FROM) ONE, whether it is in comparative proximity to hot stars or, (ON) REVOLUTION, is sufficiently distant from them.
Algorithm for the task:
- We carefully read the task.
- (ON) THAT was the way L.N. Tolstoy, his relatives could guess (ON) HOW hard is his brain working now.BY THIS we write separately, since this is a preposition with a demonstrative pronoun; HOW MUCH we write together, since it can be replaced by the pronoun HOW.
- (B) EFFECTS Scientists have established that magnesium plays an important role in the regulation of potassium levels in the body, and ALSO (SAME) regulates the adrenal glands.- AS A CONSEQUENCE, we write together, since this is an adverb that can be replaced with an adverb THEN; We ALSO write together, because you cannot omit the SAME particle without losing the meaning.
- From the first pages I experienced a strange sensation: WOULD BE) from the gloomy world, I (THAT) HOUR was transferred to another world - sunny and bright.LET'S write separately, since the particle can be omitted without loss of meaning; DOTCH we write together, as you can replace IN THE SAME MIG.
- (C) CONSEQUENCES researchers have repeatedly said that the apotheosis of Russian glory is the painting "Heroes", in which V.M. Vasnetsov expressed his romantic and deeply civil understanding of Russia at the SAME time.AS A CONSEQUENCE, we write together, as it is possible to replace THEN; We write the SAME separately, since you can omit the particle without losing the meaning.
- The physical properties of interstellar gas significantly depend (FROM) ONE, whether it is in comparative proximity to hot stars or, (ON) REVOLUTION, is sufficiently distant from them.FROM THAT we write separately, since this is a preposition and a demonstrative pronoun, which can be replaced by a noun; CONVERSE we write together, since it can be replaced with the adverb OPPOSITE.
- We write down the correct answer: subsequently, also.
The first variant of the task
Algorithm for the task:
- We carefully read the task.
- We analyze each sentence, expanding the brackets in accordance with the spelling rules of the Russian language:
- We decided to go (ON) THIS alley, BECAUSE (BECAUSE) it is quiet: there is no transport at all. - both words are written separately; “Because” is a union, and “on this” is written separately, since this is not an expression of the consequence and is the word “lane”.
- (C) DEPENDING on the communication situation, people behave (IN) DIFFERENTLY. - “depending on” is always spelled separately, and “differently” is hyphenated.
- (NOT) ALWAYS we understand the meaning of toponyms, (FOR) OFTEN sounding strange to the ear of a modern person. - "not always" is never written together; “often” is written together, but the task says that both words in a sentence must be written together.
- (THIS) HOUR he seeks to achieve his at whatever cost. - “now” is written together, replaced by the combination “at the moment”; but "by all means" is always written separately.
- Set the alarm clock, (THAT) WOULD NOT oversleep and get up (ON) EARLIER. - “to” is written together, as it is replaced by the combination “to”. "Early" is an adverb that is always written together.
- We write down the correct answer: so early (do not forget that on the exam you must write the answers without spaces and punctuation marks).
Second variant of the task
Algorithm for the task:
- We carefully read the task.
- We analyze each sentence, expanding the brackets in accordance with the spelling rules of the Russian language:
- Nobody drives (THIS) the way, BECAUSE (THAT) the road is in an emergency condition."Along this path" - separately; can be replaced, for example, by the combination "the old way". “Because” - again separately.
- (TO) THE MEETING we were late, although (OUT) FOR the snow we left the house an hour earlier.“For a meeting” - separately, since it can be replaced with “for a business meeting”, or “for an important meeting”, and “because of” is written with a hyphen, as it indicates the reason.
- AND (SO), (B) FINISHING Let me thank you for your cooperation. - "So" is written together, but "in conclusion" - separately.
- (C) IN SEE of the unstable political situation, the trip to Egypt planned by us (ON) SPEKH had to be postponed. - "In view" is written as one - is replaced by the word "because of"; "Hastily" - an adverb, written together.
- (B) GAVE from civilization, you AS (WILL) realize all the imperfection of our modern world. - “far” is written together (replaced by “far”), but “as if” is always separate.
- We write down the correct answer: in view of haste.
The third variant of the task
Define a sentence in which both highlighted words are spelled LITTLE. Expand the brackets and write out these two words.
Algorithm for the task:
- We carefully read the task.
- We analyze each sentence, expanding the brackets in accordance with the spelling rules of the Russian language:
- WHAT (WOULD) say about Russia abroad, but the country has long been (NOT) the same as it was in the 90s. - “Whatever” and “not that” are written separately: in the first case, “would” can be rearranged - “whatever they say about Russia,” and in the second it is simply impossible to write together.
- I tell you the SAME thing that Andrei, THAT (WOULD) you have the same information. - "The same thing" is written separately, but "to" - together, we replace it with "in order to".
- (BY) VISIBLY, Alexey did not hear what I answered him, (BY) the fact that he repeated his question. - "Apparently" - an adverb that we write through a hyphen; “Because” - as a part of the union “because”.
- Pushkin (TOT) HOUR became addicted to billiards, although he did not become a serious player (SO). - "Immediately" is an adverb spelled together; We write "so and" separately.
- My sister sang (B) a half-voice, I (SAME) began to sing along softly. - The last sentence remains; "In an undertone" is always written together; We also write "also" together - we replace it with the union "and".
- We write down the correct answer: in an undertone, too.
Unified State Exam. Russian language.
Task 13. How easy it is to complete?
Task number 13- one of the most difficult. This is due to the fact that it is necessary to know a lot of the rules of continuous, separate, hyphenated spelling of words. In addition, there are many words that you just need to remember. So there are difficulties.
I suggest the easiest way to accomplish this task.
Algorithm for the task number 13
Fused, separate, hyphenated spelling words
Read the assignment carefully. It is necessary to find a sentence of five proposed, in which the selected words are written together or apart... Even if the books you study mostly suggest finding solid spelling words, an exam is an exam, you need to be prepared for anything. So it is with a careful reading of the assignment that its implementation begins.
In each sentence, exclude words that are written with hyphen... Most often these are:
Words with suffixes EITHER, ANYTHING and the prefix CFU
The words all the same, exactly the same.
Prefixed adverbs BYand suffixes OMU, HIM, SKI, LI:
in our opinion, foxes.
Adjectives for shades of colors, taste (bright red, sweet and sour)
Cardinal points: southwest.
Rooted words floor: start with L(half a lemon), with a vowel (half an apple), with a capital letter(half of Europe).
Adjectives derived from homogeneous members, between them you can put a union AND(magazine and newspaper - that is, magazine and newspaper)
The first step has been taken. Certainly, some sentence will contain a word that is spelled with a hyphen. Therefore, the number of offers is decreasing.
As if
In view of
Keep in mind
During
In continuation
Due to
Subsequently
Because
Whereas
I.e
In order to
Despite
Regardless of
At once
As if
The third step is the most important. You need to clearly distinguish the words spelled together or apart.
To- what would
Same - same
Also - the same
But - for that
Why - from what
Because - from that
Because - because
And - what does it have to do with it?
About (\u003d about) - to the account (in the bank)
Remember:if a logical stress falls on a word, you highlight it with intonation, it is pronounced firmly, with a slight slowdown in intonation, and most importantly, you can concretely imagine something, then this word is written APART.
If none of the above is present, then this is a normal union, it is written LITTLE.
Compare.
WHAT WOULD should I give you for your birthday? (The emphasis falls on the word, we represent the gift that we want to buy).
We met, TO discuss current affairs. (The word is pronounced quickly, as if in passing, we can not imagine anything, saying the word FOR, we cannot)
FOR THAT I received the task five.
He prepared for a long time, BUT passed the exam well.
Remember: if after SO SAME there is AS AND, then it is always written separately. (The work was done as high quality as always.)
Word SO spelled together if it is usual introductory word, sums up something. ( SO, work was completed before vacation)
If, however, we have an adverb and a union, then it is written separately, you can ask a question as?(So he spent all his free time (HOW did he spend? - SO).
Remember that negative adverbs are always written together: nowhere, in any way, not at all, nowhere, nowhereetc.
These are the main cases to remember first.
All the rules are on this site. Pay special attention to the tables with the spelling of adverbs, memorize the words.
EXAMPLE
Determine the sentence in which both highlighted words are spelled LITTLE.Expand the brackets and write out these two words.
Everything was (ON) AS IS, (THAT) IS absolutely not changed.
(WHAT) WOULD arrive on time (AT) MEETING, we left early in the morning.
(CFU) WHERE (IN) DALY were seen the lights of the huts.
He disappeared (AS) suddenly as he appeared.
(AND) SO let's start with the fact that I (AT) THE END met with you.
EXPLANATION
We find sentences in which words are written with a hyphen. This is the first and third - CFM-WHERE, STILL... We exclude them. There are 3 offers left.
We find such words in separate writing which you have no doubt about. it I.E (the first sentence, however, has already been deleted)
There are 3 sentences left in which words can be written correctly, pondering their meaning.
2nd sentence: where did we go? - TO THE MEETING(for example, for a long-awaited meeting). That is, we clearly imagine the meeting to which our heroes go. We write apart. Word TO here it is written together, since the lexical meaning in the word "what" not).
4 sentence - easy, it contains AS WELL AS, so I write the word separately.
Remains number 5 - this is the correct answer: SO- introductory word, FINALLY- adverb, when?
Complete more tasks, and you will definitely succeed
Good luck!
Prepared by: Melnikova Vera Aleksandrovna
Unified State Exam. Russian language.
Task 13. How easy it is to complete?
Task number 13- one of the most difficult. This is due to the fact that it is necessary to know a lot of the rules of continuous, separate, hyphenated spelling of words. In addition, there are many words that you just need to remember. So there are difficulties.
I suggest the easiest way to accomplish this task.
Algorithm for the task number 13
Merged, separate, hyphenated spelling of words
Read the assignment carefully. It is necessary to find a sentence of five proposed, in which the selected words are written together or apart... Even if the books you study mostly suggest finding solid spelling words, an exam is an exam, you need to be prepared for anything. So it is with a careful reading of the assignment that its implementation begins.
In each sentence, exclude words that are written with hyphen... Most often these are:
Words with suffixes EITHER, ANYTHING and the prefix CFU
The words all the same, exactly the same.
Prefixed adverbs BYand suffixes OMU, HIM, SKI, LI:
in our opinion, foxes.
Adjectives for shades of colors, taste (bright red, sweet and sour)
Cardinal points: southwest.
Rooted words floor: start with L(half a lemon), with a vowel (half an apple), with a capital letter(half of Europe).
Adjectives formed from homogeneous members, you can put a union between them AND(magazine and newspaper - that is, magazine and newspaper)
The first step has been taken. Certainly, some sentence will contain a word that is spelled with a hyphen. Therefore, the number of offers is decreasing.
As if
In view of
Keep in mind
During
In continuation
Due to
Subsequently
Because
Whereas
I.e
In order to
Despite
Regardless of
At once
As if
The third step is the most important. You need to clearly distinguish the words spelled together or apart.
To- what would
Same - same
Also - the same
But - for that
Why - from what
Because - from that
Because - because
And - what does it have to do with it?
About (\u003d about) - to the account (in the bank)
Remember:if a logical stress falls on a word, you highlight it with intonation, it is pronounced firmly, with a slight slowdown in intonation, and most importantly, you can concretely imagine something, then this word is written APART.
If none of the above is present, then this is a normal union, it is written LITTLE.
Compare.
WHAT WOULD should I give you for your birthday? (The emphasis falls on the word, we represent the gift that we want to buy).
We met, TO discuss current affairs. (The word is pronounced quickly, as if in passing, we can not imagine anything, saying the word FOR, we cannot)
FOR THAT I received the task five.
He prepared for a long time, BUT passed the exam well.
Remember: if after SO SAME there is AS AND, then it is always written separately. (The work was done as high quality as always.)
Word SO is written together, if this is an ordinary introductory word, something is summed up. ( SO, work was completed before vacation)
If, however, we have an adverb and a union, then it is written separately, you can ask a question as?(So he spent all his free time (HOW did he spend? - SO).
Remember that negative adverbs are always written together: nowhere, in any way, not at all, nowhere, nowhereetc.
These are the main cases to remember first.
All the rules are on this site. Pay special attention to the tables with the spelling of adverbs, memorize the words.
EXAMPLE
Determine the sentence in which both highlighted words are spelled LITTLE.Expand the brackets and write out these two words.
Everything was (ON) AS IS, (THAT) IS absolutely not changed.
(WHAT) WOULD arrive on time (AT) MEETING, we left early in the morning.
(CFU) WHERE (IN) DALY were seen the lights of the huts.
He disappeared (AS) suddenly as he appeared.
(AND) SO let's start with the fact that I (AT) THE END met with you.
EXPLANATION
We find sentences in which words are written with a hyphen. This is the first and third - CFM-WHERE, STILL... We exclude them. There are 3 offers left.
We find such words, which you have no doubt about being written separately. it I.E (the first sentence, however, has already been deleted)
There are 3 sentences left in which words can be written correctly, pondering their meaning.
2nd sentence: where did we go? - TO THE MEETING(for example, for a long-awaited meeting). That is, we clearly imagine the meeting to which our heroes go. We write apart. Word TO here it is written together, since the lexical meaning in the word "what" not).
4 sentence - easy, it contains AS WELL AS, so I write the word separately.
Remains number 5 - this is the correct answer: SO- introductory word, FINALLY- adverb, when?
Complete more tasks, and you will definitely succeed
Good luck!
Prepared by: Melnikova Vera Aleksandrovna