In task No. 11 of the USE in mathematics at the profile level, it is required to solve a text problem. As a rule, the task comes down to compiling two equations with two unknowns, which must be expressed, substituted, calculated and answered! Let's get to the analysis, since there is no theory here.

Analysis of typical options for assignments No. 11 USE in mathematics at the profile level

The first version of the task (demo version 2018)

In spring, the boat goes against the current of the river 1 2/3 times slower than downstream. In summer, the current becomes 1 km / h slower. Therefore, in summer, the boat goes against the current 3/2 times slower than downstream. Find the speed of the current in spring (in km/h).

Solution algorithm:

1. We denote the unknown speeds by the variables x and y.
2. We compose a system of equations based on the condition.
3. We express x from one equation of the system in terms of the variable y.
4. We substitute the found expression into another equation of the system and
5. We write down the answer.

Solution:

1. Let the speed of the boat be x km/h and the speed of the current in the spring y km/h. Then y–1 (km/h) is the current speed in summer.

The speed of the boat downstream in spring is x + y km/h, and against the current x is y km/h. In summer, respectively, the speeds downstream and against are equal: x + (y - 1) and x - (y - 1) km / h.

2. According to the condition, the boat goes with the flow in the spring at a speed that is 5/3 of the speed against the current. We have: x + y \u003d (5/3) (x - y).

In summer, these speeds differ by 3/2 times. That is: x + (y - 1) \u003d (3/2) (x - (y - 1).

Let's make a system of equations:

3. Express x in terms of y from the first equation:

(5/3)(x-y)=(x+y),

(5/3)x - (5/3)y = x + y,

–(5/3)y – y = x – (5/3)x,

–(8/3)y = –(2/3)x,

4. Substitute the resulting value into another equality

(3/2)(4y–(y–1)) = 4y+(y–1).

(3/2)(4y-y+1) = 4y+y-1,

(3/2)(3y+1) = 5y–,

(9/2)y + 3/2 = 5y–1,

(9/2)y - 5y = -1 - 3/2,

(9/2)y - (10/2)y \u003d -1 - 3/2,

–(1/2)y = –5/2,

Therefore, the speed of the current in spring is 5 km/h.

The second version of the task

The distance between piers A and B is 77 km. A raft set off from A to B along the river, and after 1 hour a motor boat set off after it, which, having arrived at point B, immediately turned back and returned to A. By this time, the raft had sailed 40 km. Find the speed of the motorboat in still water if the speed of the river is 4 km/h. Give your answer in km/h.

Solution algorithm:

1. We denote the unknown speed by the variable x.
2. We compose an equation for solving the problem, given the condition.
3. We solve the resulting equation.
4. We draw a conclusion.
5. We write down the answer.

Solution:

1. Let the speed of the boat be x km/h. Then its speed downstream is x + 4 km/h, and against the current x is 4 km/h.

2. While the boat was going from point A to point B and back, the raft sailed 40 km along the river. The speed of the current is 4 km / h, you can set how long the raft moved: 40:4 = 10 h. The boat set off 1 hour later: 10 - 1 = 9 h. h, and against the current in hours. The time that the boat was in motion back and forth is 9 hours. We get the equation:

We simplify the resulting equation, and find x from it:

77(X– 4)+77(X+ 4)=9(X + 4)(X– 4)

77X– 77∙4 + 77X + 77∙4 = 9 (X 2 – 16)

154X – 9X 2 + 9∙16

– 9X 2 + 154X + 144 = 0

9X 2 – 154X – 144=0

We solve the quadratic equation through the discriminant, we get:

The speed cannot be negative, then the second root does not satisfy the condition. We get that the speed of the motor boat was 18 km/h.

The third version of the task (from Yaschenko, No. 31)

From point A to point B, the distance between which is 60 km, a motorist and a cyclist left at the same time. It is known that a motorist travels 30 km more per hour than a cyclist. Determine the speed of the cyclist if it is known that he arrived at point B 2 hours 40 minutes later than the motorist. Give your answer in kilometers per hour.

Solution algorithm:

1. Enter the variable x.
2. We make an equation based on the condition.
3. We solve the resulting equation.
4. We draw a conclusion.
5. We write down the answer.

Solution:

1. Denote by X the speed of the cyclist.

2. The speed of the motorist is 30 km / h higher, therefore, it is equal to X+30. A car travels 60 km in hours, and a cyclist in hours. By convention, the cyclist arrived 2 hours 40 minutes (8/3 hours) later at the destination than the motorist.

We get the equation

3. We solve the resulting equation. To do this, let's transform it:

We solve a quadratic equation, we get two roots

The speed cannot be negative, which means that the second root of the equation does not satisfy the condition. So the cyclist was moving at a speed of 15 km/h.

The eleventh task of the Unified State Examination in the Russian language can bring one primary point if it is correctly completed; to do this, you need to correctly insert the letter into the end of the verb or the participle suffix and write it out. Let's look at a theory that is useful in preparing for this exam task.

Theory for assignment No. 11 USE in the Russian language

Vowels in unstressed verb endings

In the personal endings of verbs in the present and future tenses, the vowels “e, y (u)” are written in the first conjugation, and “i, a (ya)” in the second. For example: you have, looks.

In the personal endings of the first and second conjugations imperative mood write the letter i: shout, shout; wipe - wipe; hold - hold.

Let's remember how to determine the conjugation of verbs:

Verbs whose personal forms have the endings of I and II conjugations are called differently conjugated. These include verbs want to run and their derivatives ( want to run away and etc.). verbs are specially conjugated eat, give and their derivatives ( eat, give and etc.).

Spelling participle suffixes

Valid participles- denote a sign of an object that itself performs (present tense) or performed (past tense) an action ( growing flowers growing flowers).

Passive participles- designate a sign of an object on which an action is performed or performed ( flowers grown by (someone) flowers grown by (someone)).

Vowels in present participle suffixes

Vowels in past participle suffixes

Valid participles	Before the suffixes "vsh / sh" the same vowel is written as in the indefinite form of the verb	dreaming - dreaming
		hearer - hear
Passive participles	Formed from verbs, in the indefinite form of which - the suffixes "at / yat"	Before the suffix "nn" is written a or i:
		lost - lose
	Formed from verbs, in the indefinite form of which - suffixes "it / et"	The suffix "enn" is written:
		filled - to fill

Task execution algorithm

1. Read the assignment carefully.
2. We find verbs and determine their conjugation by putting in an indefinite form. We determine which vowel is written in the personal ending of each verb.
3. We find the sacraments, determine their time and pledge. We determine which vowel is written in the suffix of each participle.
4. Write down the correct answer.

Analysis of typical options for task No. 11 USE in the Russian language

Eleventh task of the 2018 demo

1. worry..worrying
2. fell out .. sh
3. move..my
4. wrestling..shishing
5. open up..sh

Task execution algorithm:

1. Read the assignment carefully.
2. Let's look at verbs. Worried- formed from the verb of the second conjugation worry. Therefore, in the personal ending of the verb we write the vowel I. throw out- formed from the verb of the second conjugation to dump. We write the letter I at the end. Struggling- formed from the verb of the first conjugation fight. Therefore, you need to write the letter E at the end. Despite the fact that the answer is found, let's check the other options. Paste - formed from the verb of the second conjugation to paste, we write the letter I at the end.
3. Consider the sacraments. Movable- passive participle of the present tense, formed from the Old Russian verb of the second conjugation move, write the letter I.
4. Answer: fighting.

The first version of the task

Write down the word in which the letter E is written in place of the gap.

1. stuck ... stuck
2. met ... who
3. put it down
4. fell out ... ny
5. fit ... sh

Task execution algorithm:

1. Read the assignment carefully.
2. Let's first put the verbs in the indefinite form: rely on, lean on. Both of these verbs refer to second conjugation: the first has the suffix “it”, and the second is derived from the exception verb “drive”. So they have the letter "i" in them.
3. Consider the sacraments. IN actual past participle“glued” the letter “and” is written, since the verb from which it is formed is to glue, the letter is preserved. In the word "acquainted" is written "and" for the same reason; it is derived from the verb "get acquainted". But in the word "dumped", formed from the verb "dump", the suffix "enn" is written.
4. Answer: dumped.

The second version of the task

1. filled
2. jumped out ... who
3. alarmed ...
4. hesitant ... my
5. stab ... sh

Task execution algorithm:

1. Read the assignment carefully.
2. There is only one verb here; put it in indefinite form: stab. It belongs to the first conjugation; insert the letter "e" - stab.
3. Consider the sacraments. " Filled" is derived from the verb " fill»; as you can see, the suffix of the verb is “it”, which means that in the participle we write the suffix “enn”. Absolutely similarly comes out with the word " alarmed", formed from the word" alarm". But the word " jumped out" is derived from the verb " jump out»; according to the rule, the letter "i" is preserved before the participle suffix "vsh". The correct answer is found, but you can check the remaining word. " shaken" is derived from the verb " hesitate”, Referring to the first conjugation, therefore, in the passive participle of the present tense, the suffix “eat” is written.
4. Answer: jumped out.

The third version of the task

Write down the word in which the letter I is written in place of the gap.

1. hated ... who
2. breathe...sh
3. glued ...
4. rumble ... sh
5. collecting

Task execution algorithm:

1. Read the assignment carefully.
2. Verbs in the indefinite form: breathe, rumble. The first verb is an exception, and belongs to the second conjugation; the second is the usual verb of the first conjugation. Insert letters according to the rule: breathe, roar. Of course, the word breathe' is the correct answer, but other options can be checked.
3. In the present participle " hated" before the suffix "vsh" the letter from the indefinite form of the verb " hate». Passive Communion past tense " glued" is derived from the verb " glue", in which the suffix "it" is written; therefore, in the sacrament - the suffix "enn". Word " collecting"- the real participle of the present tense, formed from the verb" gather". The verb belongs to the first conjugation, which means that the participle suffix is ​​“yusch”.
4. Answer: breathe.

A motorboat, moving up the river, got from point A to point B in 4 hours. It is known that a raft from point B to point A will float along the river in 8 hours. How long does it take the boat to travel from B to A? Enter your answer in hours.

The bus traveled from point A to point B at a speed of 40 km/h, from point B to point C the bus traveled at a speed of 60 km/h, and from point C to point D at a speed of 24 km/h. The distances between points are the same. At what average speed would the bus have to travel to travel from point A to point D in the same time it took?

The pool can be filled through the first tap in 6 hours, through the second tap it can be filled in 8 hours, and through the third tap it can be emptied in 4 hours. How long will it take to fill the pool if all three taps are opened at the same time?

Two two-digit numbers are given. First, zero was assigned to the larger two-digit number on the right, followed by a smaller two-digit number, then zero was attributed to the smaller number on the right, and then a larger two-digit number. The larger five-digit number is divided by the smaller five-digit number. The quotient turned out to be 2, and the remainder is 590. Find the smaller two-digit number if the sum of twice the larger number and three times the smaller number is 72.

Four positive numbers form an increasing geometric progression. The sum of the extreme terms of the progression is 27, and the sum of the middle terms of the progression is 18. Find the first term of the specified progression.

Part of the journey from point A to point B consists of going uphill, part of the way coming down the mountain, and part of the way going on a level road. It is known that a bus on a flat road has a speed of 48 km/h, goes uphill at a speed of 40 km/h, and goes down a mountain at a speed of 60 km/h. Find the distance between points A and B if it took the bus 5 hours to get from point A to point B and back. Specify the distance in km.

The train left point A for point B. Having traveled 450 km, which was 75% of the total distance, it was delayed at the semaphore for 30 minutes. After that, to catch up, the speed of the train was increased by 15 km/h. The train arrived at point B on schedule. Find the speed of the train on the second section of the track after stopping at the semaphore. Give your answer in km/h.

Two workers completed some work in 11 days, and only the first worker worked for the last three days. It is known that in the first 7 days they completed 80% of the work together. In how many days can the first worker complete all the work by working independently?

A cyclist traveled at a constant speed from city A to city B, the distance between them being 180 km. The next day he went back at a speed of 8 km / h more than before. On the way he made a stop for 8 hours. As a result, he spent as much time on the way back as on the way from A to B. Find the speed of the cyclist on the way from A to B. Give the answer in km/h.

The distance between cities A and B is 150 km. A car left city A for city B, and 30 minutes later a motorcyclist left behind it at a speed of 90 km/h, caught up with the car in city C and turned back. When he returned to A, the car arrived at B. Find the distance from A to C. Give your answer in kilometers.

Lyceum graduates received 600 more applications for the Fundamental and Applied Linguistics program than gymnasium graduates. There are 5 times more girls among lyceum graduates than girls among gymnasium graduates. And there are n times more boys among lyceum graduates than boys among gymnasium graduates, and 6< n < 12 (n - целое число). Определить общее количество заявлений, если среди выпускников гимназий юношей на 20 больше, чем девушек.

In spring, the boat goes against the current of the river 1 2/3 times slower than downstream. In summer, the current becomes 1 km / h slower. Therefore, in summer, the boat goes against the current 1 1/2 times slower than downstream. Find the speed of the current in spring (in km/h).

The boat at 11:00 left point A for point B, located 30 km from A. After staying at point B for 2 hours and 40 minutes, the boat went back and returned to point A at 19:00 on the same day. Determine (in km/h) the own speed of the boat if it is known that the speed of the river is 3 km/h.

The motorboat at 11:00 left point A for point B, located 30 km from A. Having stayed at point B at 2 hours and 30 minutes, the boat went back and returned to point A at 21:00. Determine (in km/h) the own speed of the boat if it is known that the speed of the river is 3 km/h.

The kayak at 10:00 left point A for point B, located 15 km from A. After staying at point B for 1 hour and 20 minutes, the kayak went back and returned to point A at 18:00. Determine (in km/h) the own speed of the kayak if it is known that the speed of the river is 3 km/h.

Client A. made a deposit in the bank in the amount of 2500 rubles. Interest on the deposit is calculated once a year and is added to the current deposit amount. Exactly one year later, under the same conditions, B made the same deposit in the same bank. Exactly one year later, clients A. and B. closed their deposits and took all the accumulated money. At the same time, client A. received 275 rubles more than client B. What percentage per annum did the bank charge on these deposits?

The first pipe passes 1 liter of water per minute less than the second. How many liters of water per minute does the second pipe pass if it fills a 252-liter tank 9 minutes faster than the first pipe fills a 420-liter tank?

Petya and Vanya perform the same test. Petya answers 20 test questions in an hour, and Vanya answers 21 questions. They started answering the test questions at the same time, and Petya finished his test 5 minutes later than Vanya. How many questions does the test contain?

The cost of making strawberry jam consists of the cost of strawberries and the cost of sugar. In June, strawberries fell 60% and sugar rose 20% from April, resulting in a 50% reduction in the cost of making jam. What percentage of the cost of making jam in April was the cost of strawberries?

The price of a refrigerator in the store decreases annually by the same number of percent from the previous price. Determine by what percentage the price decreased each year if a refrigerator put up for sale for 20,000 rubles was sold for 15,842 rubles two years later.

The store put up a product for sale with some extra charge in relation to the purchase price. After selling 4/5 of the entire product, the store reduced the listed price by 40% and sold out the remaining product. As a result, the store's profit amounted to 38% of the purchase price of the goods. What percentage of the purchase price was the store's original markup?

Vera needs to sign 640 postcards. Every day she signs the same number of postcards more than the previous day. It is known that on the first day Vera signed 10 postcards. Determine how many postcards were signed on the fourth day if all the work was completed in 16 days.

Two conveyor lines for packing finished products pack 6,000 units of products per hour of joint work. The first of these lines needs an hour more to pack 6,000 units than it takes the second line to pack 8,000 units. How many units of product does the second line pack per hour?

The meat processing plant produces a pate consisting of pork, beef and offal, the masses of which are related as 3:5:2, respectively. It is planned to increase the output of this pate by 2.5 times, while the consumption of pork and beef is planned to be increased by 100% and 120%, respectively. Determine what percentage of the mass of the pate will be by-products if this plan is implemented.

Two motorcyclists start simultaneously in the same direction from two diametrically opposite points of a circular track, the length of which is 14 km. In how many minutes will the motorcyclists catch up for the first time if the speed of one of them is 21 km/h more than the speed of the other?

There are two vessels. The first contains 30 kg, and the second - 20 kg of an acid solution of various concentrations. If these solutions are mixed, you get a solution containing 68% acid. If you mix equal masses of these solutions, you get a solution containing 70% acid. How many kilograms of acid are contained in the first vessel?

In 2008, 40,000 people lived in the city block. In 2009, as a result of the construction of new houses, the number of inhabitants increased by 8%, and in 2010 - by 9% compared to 2009. How many people began to live in the quarter in 2010?

Mitya, Anton, Gosha and Boris established a company with an authorized capital of 200,000 rubles. Mitya contributed 14% of the authorized capital, Anton - 42,000 rubles, Gosha - 0.12 of the authorized capital, and Boris contributed the rest of the capital. The founders agreed to share the annual profit in proportion to the contribution made to the authorized capital. What amount of the profit of 1,000,000 rubles is due to Boris? (Give your answer in rubles.)

By sea, two dry cargo ships follow parallel courses in one direction: the first is 120 meters long, the second is 80 meters long. First, the second bulk carrier lags behind the first, and at some point in time, the distance from the stern of the first bulk carrier to the bow of the second is 400 meters. 12 minutes after that, the first bulk carrier lags behind the second so that the distance from the stern of the second bulk carrier to the bow of the first is 600 meters. By how many kilometers per hour is the speed of the first cargo ship less than the speed of the second?

The snail crawls from one tree to another. Every day she crawls the same distance more than the previous day. It is known that for the first and last days the snail crawled a total of 10 meters. Determine how many days the snail spent on the whole journey if the distance between the trees is 150 meters.

The first hour the car drove at a speed of 60 km/h, then 2 hours at a speed of 110 km/h, and the next 2 hours at a speed of 120 km/h. Find the average speed of the car for the whole journey. Express your answer in km/h.

Purchased goods of two varieties: the first for 4500 rubles. and the second for 2100 rubles. The second grade was bought 2 kg less than the first and it costs 200 rubles cheaper. How many kilograms of first-class goods were bought? (If there are several solutions, then write the largest in response)

Two masons working together can complete a task in 12 hours. The labor productivity of the first and second masons are related as 1:3. The masons agreed to work alternately. How long does the first bricklayer have to work for the task to be completed in 20 hours.

For the manufacture of 468 parts, the first worker spends 8 hours less than the second worker for the production of 520 parts. It is known that the first worker makes 6 more parts per hour. than the second. How many parts per hour does the first worker make?

By mixing 84% and 96% acid solutions and adding 10 kg of pure water, an 84% acid solution was obtained. If, instead of 10 kg of water, 10 kg of a 50% solution of the same acid were added, then an 89% acid solution would be obtained. How many kilograms of the 84% solution were used to make the mixture?

The student read a book of 480 pages, reading the same number of pages daily. If he read 16 more pages every day, he would have read the book five days earlier. How many days did the student read the book?

On December 31 at 8 o'clock in the morning, in anticipation of a new option, Victor left the house for a walk. At 8.20 on the site Alex.larin 178 option appeared, which did not go unnoticed by the poodle Roma. Roma immediately ran out of the house after the owner to tell him the good news. At 8.30 Victor heard the familiar voice of his friend behind him, realized that something had happened, and immediately turned back. After another 5 minutes, Roma met with Victor, instantly told him important news, turned around and, together with the owner, began to return home. Determine how much % Roma's speed dropped after meeting with the owner. (Viktor is known to always walk at a constant speed)

The elevator received 2 million 296 thousand tons of grain: wheat, rye and barley, and rye turned out to be 10% more than wheat, and barley - 30% less than rye. How many tons of barley entered the elevator?

An experimental machine for cutting fish, installed on a floating base, allows you to cut 15 pieces per minute. more fish than on the old equipment. How many pieces of fish are cut per minute by a new machine, if it is known that the catch is 26,000 pieces. processed 1 hour 15 minutes faster than before?

A diesel locomotive must cover a distance of 200 km in a certain time. When he traveled 45% of the way, he was delayed for 10 minutes at the semaphore. To arrive on time, the locomotive increased its speed by 5 km/h. Calculate the initial speed of the locomotive. Give your answer in km/h.

Two cyclists set out on a 130-kilometer run at the same time. The first one was driving at a speed 3 km/h higher than the second one, and arrived at the finish line 3 hours earlier than the second one. Find the speed of the cyclist who came to the finish line second.
Give your answer in km/h.

To preserve 10 kg of eggplant, 0.5 l of table vinegar (10% acetic acid solution) is needed. The hostess has vinegar essence (80% solution of acetic acid). How many milliliters of vinegar essence will the hostess need to preserve 20 kg of eggplant?

From points A and B, the distance between which is 210 km, two cars simultaneously leave towards each other. After the meeting, one of them has to be on the road for 2 hours, and the other 9/8 hours. Find the speed of cars.

From points A and B simultaneously towards each other
two pedestrians left and met after 3 hours. How much time did each pedestrian spend on the journey if it is known that one of them spent 2.5 hours more than the other on the entire journey?

At some point, the clock shows 2 minutes less than it should, although it goes forward. If they showed 3 minutes less than they should, but would go 0.5 minutes ahead a day ahead than they leave, then they will be the right time would show a day earlier than they show. By how many minutes does the clock advance in a day?

A boat left from A to B. When he traveled 4 km, a boat left A for B, which arrived at B 1.5 hours earlier than the steamer. What is the distance between A and B if the speed of the boat is 16 km/h and the speed of the boat is 36 km/h? Give your answer in km.

The boat passes against the current of the river to the destination for 120 km and after a short stay returns to the point of departure. Find the speed of the boat in still water, if the speed of the current is 3 km/h, the stay lasts 20 minutes, and the boat returns to the point of departure 17 hours after leaving it. Give your answer in km/h.

The pool is filled with four pipes in 4 hours. The first, second and fourth pipes fill the pool in 6 hours. The second, third and fourth - in 5 hours. How long will it take for the first and third pipes to fill the pool? Give your answer in hours.

There are two solutions with different percentages of salt. If you mix 1 kg of the first solution and 3 kg of the second, then the resulting solution will contain 32.5% salt. If you mix 3.5 kg of the first solution and 4 kg of the second, then the resulting solution will contain 26% salt. What will be the percentage of salt in the solution if you mix equal masses of the first and second solutions?

A motorcyclist left point A for point B and at the same time a motorist left B for A. The motorcyclist arrived at B 2 hours after the meeting, and the motorist arrived at A 30 minutes after the meeting. How many hours was the motorcyclist on the road?

The car traveled the first half of the way at a speed of 40 km/h, and the second half of the way at a speed of 60 km/h. Find the average speed of the car for the entire journey. Give your answer in kilometers per hour.

Andrey, in preparation for the Unified State Examination, set himself the task of solving 5 more problems every day than the previous one. For the first day he solved 7 problems, and for the last day - 37 problems. How many problems did he solve in total?

An order of 180 parts is completed by the first worker 3 hours faster than the second worker. How many parts per hour does the second worker make if it is known that he makes 3 less parts per hour than the first worker?

Igor and Pasha can paint the fence in 30 hours. Pasha and Volodya can paint the same fence in 36 hours, and Volodya and Igor in 45 hours. How many hours will it take the boys to paint the fence with three of them?

The distance between cities A and B is 440 km. The first car left city A for city B, and two hours later the second car left city B towards it at a speed of 90 km/h. Find the speed of the first car if the cars met at a distance of 260 km from city A. Give your answer in km/h.

Two people leave from the same place for a walk to the edge of the forest, located 4.3 km from the starting point. One is walking at 4 km/h and the other is walking at 4.6 km/h. Having reached the edge, the second one returns at the same speed. At what distance from the starting point will they meet? Give your answer in kilometers.

Three automatic machines of different power must produce 800 parts each. First, the first machine was launched, after 20 minutes - the second, and after another 35 minutes - the third. Each of them worked without failures and stops, and in the course of work there was a moment when each machine completed the same part of the task. How many minutes before the second machine finished the work of the third, if the first completed the task 1 hour 28 minutes after the third?

A boat and a raft left the pier at the same time. After 9 km, the boat turned around and, having traveled another 13 km, caught up with the raft. Find the speed of the river if the speed of the boat is 22 km/h. Give your answer in km/h.

Father and son have to dig a garden. The productivity of the father is three times less than that of the son. Working together, they can dig a vegetable garden in 3 hours. However, together they worked for only one hour, then one father worked for a while, and one son finished work. How long did the father work in total if all the work in the garden was completed in 7 hours?

From pier A to pier B, the first ship set off at a constant speed, and 3 hours later, the second one set off after it at a speed of 3 km / h more. The distance between the piers is 154 km. Find the speed of the second ship if it arrived at point B at the same time as the first. Give your answer in km/h.

There are two vessels. The first contains 100 kg, and the second - 60 kg of an acid solution of various concentrations. If these solutions are mixed, you get a solution containing 41% acid. If you mix equal masses of these solutions, you get a solution containing 50% acid. How many kilograms of acid are contained in the first vessel?

By mixing 45% and 97% acid solutions and adding 10 kg of pure water, a 62% acid solution was obtained. If, instead of 10 kg of water, 10 kg of a 50% solution of the same acid were added, then a 72% acid solution would be obtained. How many kilograms of a 45% solution were used to make the mixture?

The tiler has to lay 187 m^2 of tiles. If he lays 6 m^2 more per day than planned, he will finish the job 6 days earlier. How many square meters Does the tiler plan to lay tiles per day?

Two teams together must collect 400 tons of carrots. The first collected 15% more than the plan, and the second - 5% less than the plan. As a result, together they collected 428 tons of carrots. How many tons of carrots was the second brigade supposed to collect according to the plan?

The first pipe fills the tank 48 minutes longer than the second. Both pipes, working simultaneously, fill the same tank in 45 minutes. How many minutes does it take one second pipe to fill this tank?

Two excavators working together can dig a pit in 48 hours. If the first works for 40 hours and the second for 30 hours, then 75% of the work will be completed. How long can the second excavator dig a pit, working separately?

A team of loaders was assigned to transport 120 containers. After transporting 36 containers, the vehicle was replaced with a more powerful one, the carrying capacity of which is 10 more containers. As a result, the total number of flights was halved compared to originally planned. How many containers did the first car transport in one trip?

Three masons of different qualifications laid out brick wall, and the first one worked 6 hours, the second - 4 hours, and the third - 7 hours. If the first bricklayer worked 4 hours, the second - 2 hours and the third - 5 hours, then 2/3 of all the work would be done. How many hours would it take the masons to finish the brickwork if they all worked together at the same time?

From pier A to pier B, the distance between which is 99 km, the first motor ship set off at a constant speed, and 2 hours later, the second one set off after it, at a speed of 2 km / h more. Find the speed of the first ship if both ships arrive at point B at the same time. Give your answer in km/h.

The first worker spends 8 hours less to make 20 parts than the second worker to make 60 of the same parts. It is known that the first worker makes 4 more parts per hour than the second. How many parts per hour does the second worker make?

The ship passes along the river from point A to point B, the distance between which is 120 km, and after parking returns to point A. Find the speed of the ship in still water, if the speed of the current is 2 km / h, the parking lasts 5 hours, and at point departure, the ship returns 30 hours after departure from it. Give your answer in km/h.

The barge at 10:00 left point A for point B, located 30 km from A. After staying at point B for 4 hours, the barge set off and returned to point A at 22:00 on the same day. Determine (in km/h) the own speed of the barge if it is known that the speed of the river is 2 km/h.

The tiler has to lay 300 m^2 tiles. If he lays 5 m ^ 2 more per day than planned, he will finish the work 5 days earlier than scheduled. How many square meters of tiles per day does the tiler plan to lay?

Three kilograms of cherries cost the same as five kilograms of cherries, and three kilograms of cherries cost the same as two kilograms of strawberries. By what percent is a kilogram of strawberries cheaper than a kilogram of cherries?

Cities A, B and C are connected by a straight highway, and city B is located between cities A and C. A car left city A towards city C, and at the same time a truck left city B towards city C. In how many hours after the departure will the passenger car catch up with the truck if the speed passenger car 28 km / h more than the speed of the truck, and the distance between cities A and B is 112 km?

Half of the time spent on the road, the car was traveling at a speed of 60 km/h, and the second half of the time - at a speed of 46 km/h. Find the average speed of the car for the entire journey. Give your answer in km/h.

There are two alloys. The first contains 5% nickel, the second - 20% nickel. From these two alloys, a third alloy weighing 225 kg was obtained containing 15% nickel. By how many kilograms is the mass of the first alloy less than the mass of the second?

There are two vessels. The first contains 100 kg, and the second - 50 kg of an acid solution of various concentrations. If these solutions are mixed, you get a solution containing 28% acid. If you mix equal masses of these solutions, you get a solution containing 36% acid. How many kilograms of acid are contained in the first vessel?

Two racers are racing. They have to drive 70 laps along the 4.4 km long ring road. Both riders started at the same time, and the first one came to the finish line earlier than the second by 30 minutes. What was acb the average speed of the second rider if it is known that the first rider overtook the second one for the first time by a lap in 24 minutes? Give your answer in km/h.

For the manufacture of 780 parts, the first worker spends 4 hours less than the second worker for the production of 840 of the same parts. It is known that the first worker makes 2 more parts per hour than the second. How many parts per hour does the first worker make?

The first worker makes 5 more parts per hour than the second worker, and finishes work on an order of 570 parts 5 hours later than the second worker completes an order of 350 of the same parts. How many parts does the first worker make per hour?

The first cyclist left the village at a speed of 17 kilometers per hour. An hour later, at a speed of 13 kilometers per hour, a second cyclist left the same village in the same direction, and an hour after that, a third. Find the speed of the third cyclist if he first caught up with the second, and after 3 hours 10 minutes after that he caught up with the first. Give your answer in kilometers per hour.

For the first 4 days, 13 workers worked on the construction of the facility, after which three more workers joined them, and after 3 days, six workers were transferred to another facility. How long will this facility be built if six workers can complete this task in 20 days?

A cyclist left point A of the circular track, and after 30 minutes a motorcyclist followed him. 10 minutes after departure, he caught up with the cyclist for the first time, and 30 minutes after that he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 30 km. Give your answer in km/h.

Two riders will have to drive 85 laps along the 8 km long ring track. Both riders started at the same time, and the first one came to the finish line earlier than the second by 17 minutes. What was the average speed of the second rider if it is known that the first rider overtook the second for the first time by a lap in 48 minutes? Give your answer in km/h.

A tourist left point A for point B at a speed of 5 km/h. Towards him, at the same time, a cyclist left at a speed of 12 km / h. After 2 hours of travel, the distance between them was one third of the total distance between A and B. find the length of section AB

The motor boat passed 24 km against the current and returned back, having spent 1 hour less on the way back than when moving against the current. Find the speed (in km/h) of the boat in still water if the speed of the current is 2 km/h.

Two riders will have to drive 85 laps along the 8 km long ring track. Both riders started at the same time, and the first one came to the finish line earlier than the second by 17 minutes. What was the average speed of the second rider if it is known that the first rider overtook the second for the first time by a lap in 48 minutes? Give your answer in km/h

The first pipe passes 4 liters of water per minute less than the second. How many liters of water per minute does the first pipe pass if it fills a 336-liter tank for a minute longer than the second pipe fills a 375-liter tank?

From cities A and B, the distance between which is 280 km, two motorcyclists left at the same time towards each other and met after 4 hours at a distance of 80 km from city B. Find the speed of the motorcyclist who left city A. Give the answer in km / h.

From cities A and B, the distance between which is 270 km, two buses left at the same time towards each other, which met at a distance of 140 km from A. Find the speed of the bus (in km / h) that left point B, if the buses met after 2, 5 o'clock.

By mixing 70% and 60% acid solutions and adding 2 kg of pure water, a 50% acid solution was obtained. If, instead of 2 kg of water, 2 kg of a 90% solution of the same acid were added, then a 70% acid solution would be obtained. How many kilograms of a 70% solution were used to make the mixture?

The first alloy contains 5% copper, the second - 11% copper. The mass of the second alloy is greater than the mass of the first by 4 kg. From these two alloys, a third alloy containing 10% copper was obtained. Find the mass of the third alloy. Give your answer in kilograms.

Two motorists left A for B at the same time. The first traveled at a constant speed all the way. The second traveled the first half of the journey at a speed less than the speed of the first by 14 km/h, and the second half of the journey at a speed of 99 km/h, as a result of which he arrived at B at the same time as the first motorist. Find the speed of the first motorist, if it is known that it is more than 50 km/h. Give your answer in km/h.

A cyclist left A for B at a constant speed. The distance between A and B is 224 km. Having rested, he went back to A, increasing his speed by 2 km / h. On the way, he made a stop for 2 hours, as a result of which he spent the same time on the way back as on the way from A to B. Find the speed of the cyclist from A to B .

On Thursday, the company's shares rose in price by a certain number of percent, and on Friday they fell in price by the same number of percent. As a result, they began to cost 9% cheaper than at the opening of trading on Thursday. By what percent did the company's shares rise in price on Thursday?

The first pipe passes 5 liters of water per minute less than the second. How many liters of water per minute does the second pipe pass if it fills a 375 liter tank 10 minutes faster than the first pipe fills a 500 liter tank?

By mixing 24% and 67% acid solutions and adding 10 kg of pure water, a 41% acid solution was obtained. If, instead of 10 kg of water, 10 kg of a 50% solution of the same acid were added, then a 45% acid solution would be obtained. How many kilograms of a 24% solution were used to make the mixture?

From point A to point B, the distance between which is 250 km, a bus left. An hour later, a car left after him, which arrived at point B 40 minutes earlier than the bus. Calculate the average speed of the bus, if it is known that it is 1.5 times less average speed car

The first cyclist left the village at a speed of 12 km / h. An hour after him, at a speed of 10, a second cyclist left the same village in the same direction, and after another hour or a third. Find the speed of the third if he first caught up with the second, and after 2 hours then caught up with the first

Two motorists leave at the same time from point A to point B. The first traveled at a constant speed all the way. The second traveled the first half of the journey at a speed less than the speed of the first by 14 km/h, and the second half of the journey at a speed of 105 km/h, as a result of which he arrived at B at the same time as the first motorist. Find the speed of the first motorist, if it is known that it is more than 50 km/h. Give your answer in km/h

The first worker spends 12 hours less to make 540 parts than the second worker to make 600 parts. It is known that the first worker makes 10 more parts per hour than the second. How many parts per hour does the first worker make?

The ship passes along the river to the destination 483 km and after parking returns to the point of departure. Find the speed of the current if the speed of the ship in still water is 22 km/h, the stay lasts 2 hours, and the ship returns to the point of departure 46 hours after leaving it. Give your answer in kilometers per hour.

Two cars left point A for point B at the same time. The first traveled at a constant speed all the way. The second traveled the first half of the way at a speed less than the speed of the first by 13 km/h, and the second half of the way at a speed of 78 km/h, as a result of which he arrived at point B at the same time as the first car. Find the speed of the first car if it is known to be greater than 48 km/h. Give your answer in km/h.

The kayak at 10:00 left point A for point B, located 15 km from A. After staying at point B for 45 minutes, the kayak went back and returned to point A at 16:00 on the same day. Determine in (km/h) the own speed of the kayak if it is known that the speed of the river is 3 km/h.

From point A to point B, the distance between which is 60 km, a motorcyclist and a cyclist left at the same time. It is known that in an hour a motorcyclist travels 50 km more than a cyclist. Determine the speed of the cyclist if it is known that he arrived at point B 5 hours later than the motorcyclist. Give your answer in km/h.

The volumes of monthly gas production at the first, second and third fields are related as 7: 6: 14. It is planned to reduce the monthly gas production at the first field by 14% and at the second - also by 14%. By what percentage should monthly gas production be increased at the third field so that the total volume of gas produced per month does not change?

During a country trip, a car consumes 2 liters of gasoline less for every 100 km of travel than in the city. The driver left with a full tank, drove 120 km in the city and 210 km along the suburban highway to refuel. Filling up the car, he found that 42 liters of gasoline had entered the tank. How many liters of gasoline does a car use per 100 km in the city?

The first pipe fills the tank with a volume of 600 liters, and the second pipe fills the tank with a volume of 900 liters. It is known that one of the pipes passes 3 liters of water per minute more than the other. How many liters of water per minute does the second pipe pass if the tanks were filled at the same time?

From point A to point B downstream of the river, a motorboat and a kayak set off at the same time. The speed of the river is 3 km/h. For the last 1/7 part of the way, the motorboat went with the engine off, and its speed relative to the shore was equal to the speed of the current. On that part of the way, where the motorboat went with the engine on, its speed was 2 km/h more than the speed of the kayak. Find the speed of the kayak in still water if the kayak and the motorboat arrive at point B at the same time.

Average general education

Line UMK G.K. Muravina. Algebra and the beginnings of mathematical analysis (10-11) (deep)

Line UMK Merzlyak. Algebra and the Beginnings of Analysis (10-11) (U)

Mathematics

Preparation for the exam in mathematics (profile level): tasks, solutions and explanations

We analyze tasks and solve examples with the teacher

Examination paper profile level 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 tasks:

• part 1 contains 8 tasks (tasks 1-8) with a short answer in the form of an integer 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 (full record of the decision with the rationale for the actions performed).

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

“In order to get 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 Russian Federation The USE in mathematics is divided into two levels: basic and specialized. Today we will consider options for the profile level.

Task number 1- checks the ability of USE participants to apply the skills acquired in the course of 5-9 grades in elementary mathematics in practical activities. The participant must have computational skills, be able to work with rational numbers, be able to round decimal fractions, be able to convert one unit of measurement to another.

Example 1 An expense meter was installed in the apartment where Petr lives cold water(counter). On the first of May, the meter showed an 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 for May, if the price of 1 cu. 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 (cu m)

2) Find how much money will be paid for the spent water:

34.17 5 = 170.85 (rub)

Answer: 170,85.

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

Solution:

2) 1000 3/4 = 750 (shares) - make up 3/4 of all purchased shares.

6) 247500 + 77500 = 325000 (rubles) - the businessman received after the sale of 1000 shares.

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

Answer: 15000.

Task number 3- is a task of the basic level of the first part, checks the ability to perform actions with geometric shapes on the content of the course "Planimetry". Task 3 tests the ability to calculate the area of ​​a figure on checkered paper, the ability to calculate degree measures of angles, calculate perimeters, etc.

Example 3 Find the area of ​​a rectangle drawn on checkered paper with a cell size of 1 cm by 1 cm (see figure). Give your answer in square centimeters.

Solution: To calculate the area of ​​this figure, you can use the Peak formula:

To calculate the area of ​​this rectangle, we use the Peak formula:

S= B +	G
	2

where V = 10, G = 6, therefore

S = 18 +	6
	2

Answer: 20.

See also: Unified State Examination in Physics: solving vibration problems

Task number 4- the task of the course "Probability Theory and Statistics". The ability to calculate the probability of an event in the simplest situation is tested.

Example 4 There are 5 red and 1 blue dots on the circle. Determine which polygons are larger: those with all red vertices, or those with one of the blue vertices. In your answer, indicate how many more of one than the other.

Solution: 1) We use the formula for the number of combinations from n elements by k:

all of whose vertices are red.

3) One pentagon with all red vertices.

4) 10 + 5 + 1 = 16 polygons with all red vertices.

whose vertices are red or with one blue vertex.

whose vertices are red or with one blue vertex.

8) One hexagon whose vertices are red with one blue vertex.

9) 20 + 15 + 6 + 1 = 42 polygons that have all red vertices or one blue vertex.

10) 42 - 16 = 26 polygons that use the blue dot.

11) 26 - 16 = 10 polygons - how many polygons, in which one of the vertices is a blue dot, are more than polygons, in which all vertices are only red.

Answer: 10.

Task number 5- the basic level of the first part tests the ability to solve the simplest equations (irrational, exponential, trigonometric, logarithmic).

Example 5 Solve Equation 2 3 + x= 0.4 5 3 + x .

Solution. Divide both sides of this equation by 5 3 + X≠ 0, we get

2 3 + x	= 0.4 or		2		3 + X	=	2	,
5 3 + X			5				5

whence it follows that 3 + x = 1, x = –2.

Answer: –2.

Task number 6 in planimetry for finding geometric quantities (lengths, angles, areas), modeling real situations in the language of geometry. The study of the constructed models using geometric concepts and theorems. The source of difficulties is, as a rule, ignorance or incorrect application of the necessary theorems of planimetry.

Area of ​​a triangle ABC equals 129. DE- median line parallel to side AB. Find the area of ​​the trapezoid ABED.

Solution. Triangle CDE similar to a triangle CAB at two corners, since the corner at the vertex C general, angle CDE equal to the angle CAB as the corresponding angles at DE || AB secant AC. Because DE is the middle line of the triangle by the condition, then by the property of the middle line | DE = (1/2)AB. So the similarity coefficient is 0.5. The areas of similar figures are related as the square of the similarity coefficient, so

Hence, S ABED = S Δ ABC – S Δ CDE = 129 – 32,25 = 96,75.

Task number 7- checks the application of the derivative to the study of the function. For successful implementation, a meaningful, non-formal possession of the concept of a derivative is necessary.

Example 7 To the graph of the function y = f(x) at the point with the abscissa x 0 a tangent is drawn, which is perpendicular to the straight line passing through the points (4; 3) and (3; -1) of this graph. Find f′( x 0).

Solution. 1) Let's use the equation of a straight line passing through two given points and find the equation of a straight line passing through points (4; 3) and (3; -1).

(y – y 1)(x 2 – x 1) = (x – x 1)(y 2 – y 1)

(y – 3)(3 – 4) = (x – 4)(–1 – 3)

(y – 3)(–1) = (x – 4)(–4)

–y + 3 = –4x+ 16| · (-1)

y – 3 = 4x – 16

y = 4x– 13, where k 1 = 4.

2) Find the slope of the tangent k 2 which is perpendicular to the line y = 4x– 13, where k 1 = 4, according to the formula:

3) The slope of the tangent is the derivative of the function at the point of contact. Means, f′( x 0) = k 2 = –0,25.

Answer: –0,25.

Task number 8- checks the knowledge of elementary stereometry among the exam participants, the ability to apply formulas for finding surface areas and volumes of figures, dihedral angles, compare the volumes of similar figures, be able to perform actions with geometric figures, coordinates and vectors, etc.

The volume of a cube circumscribed around a sphere is 216. Find the radius of the sphere.

Solution. 1) V cube = a 3 (where A is the length of the edge of the cube), so

A 3 = 216

A = 3 √216

2) Since the sphere is inscribed in a cube, it means that the length of the diameter of the sphere is equal to the length of the edge of the cube, therefore d = a, d = 6, d = 2R, R = 6: 2 = 3.

Task number 9- requires the graduate to transform and simplify algebraic expressions. Task number 9 advanced level Difficulty with short answers. Tasks from the section "Calculations and transformations" in the USE are divided into several types:

transformations of numerical rational expressions;

transformations of algebraic expressions and fractions;

transformations of numerical/letter irrational expressions;

actions with degrees;

transformation of logarithmic expressions;

1. conversion of numeric/letter trigonometric expressions.

Example 9 Calculate tgα if it is known that cos2α = 0.6 and

3π	< α < π.
4

Solution. 1) Let's use the double argument formula: cos2α = 2 cos 2 α - 1 and find

tan 2 α =	1	– 1 =	1	– 1 =	10	– 1 =	5	– 1 = 1	1	– 1 =	1	= 0,25.
	cos 2 α		0,8		8		4		4		4

Hence, tan 2 α = ± 0.5.

3) By condition

3π	< α < π,
4

hence α is the angle of the second quarter and tgα< 0, поэтому tgα = –0,5.

Answer: –0,5.

#ADVERTISING_INSERT# Task number 10- checks the ability of students to use the acquired early knowledge and skills in practical activities and everyday life. We can say that these are problems in physics, and not in mathematics, but all the necessary formulas and quantities are given in the condition. The tasks are reduced to solving a linear or quadratic equation, or a linear or quadratic inequality. Therefore, it is necessary to be able to solve such equations and inequalities, and determine the answer. The answer must be in the form of a whole number or a final decimal fraction.

Two bodies of mass m= 2 kg each, moving at the same speed v= 10 m/s at an angle of 2α to each other. The energy (in joules) released during their absolutely inelastic collision is determined by the expression Q = mv 2 sin 2 α. At what smallest angle 2α (in degrees) must the bodies move so that at least 50 joules are released as a result of the collision?
Solution. To solve the problem, we need to solve the inequality Q ≥ 50, on the interval 2α ∈ (0°; 180°).

mv 2 sin 2 α ≥ 50

2 10 2 sin 2 α ≥ 50

200 sin2α ≥ 50

Since α ∈ (0°; 90°), we will only solve

We represent the solution of the inequality graphically:

Since by assumption α ∈ (0°; 90°), it means that 30° ≤ α< 90°. Получили, что наименьший угол α равен 30°, тогда наименьший угол 2α = 60°.

Task number 11- is typical, but it turns out to be difficult for students. The main source of difficulties is the construction of a mathematical model (drawing up an equation). Task number 11 tests the ability to solve word problems.

Example 11. During spring break, 11-grader Vasya had to solve 560 training problems to prepare for the exam. On March 18, on the last day of school, Vasya solved 5 problems. Then every day he solved the same number of problems more than the previous day. Determine how many problems Vasya solved on April 2 on the last day of vacation.

Solution: Denote a 1 = 5 - the number of tasks that Vasya solved on March 18, d– daily number of tasks solved by Vasya, n= 16 - the number of days from March 18 to April 2 inclusive, S 16 = 560 - the total number of tasks, a 16 - the number of tasks that Vasya solved on April 2. Knowing that every day Vasya solved the same number of tasks more than the previous day, then you can use the formulas for finding the sum of an arithmetic progression:

560 = (5 + a 16) 8,

5 + a 16 = 560: 8,

5 + a 16 = 70,

a 16 = 70 – 5

a 16 = 65.

Answer: 65.

Task number 12- check students' ability to perform actions with functions, be able to apply the derivative to the study of the function.

Find the maximum point of a function y= 10ln( x + 9) – 10x + 1.

Solution: 1) Find the domain of the function: x + 9 > 0, x> –9, that is, x ∈ (–9; ∞).

2) Find the derivative of the function:

4) The found point belongs to the interval (–9; ∞). We define the signs of the derivative of the function and depict the behavior of the function in the figure:

The desired maximum point x = –8.

Download for free the work program in mathematics to the line of UMK G.K. Muravina, K.S. Muravina, O.V. Muravina 10-11 Download free algebra manuals

Task number 13- an increased level of complexity with a detailed answer, which tests the ability to solve equations, the most successfully solved among tasks with a detailed answer of an increased level of complexity.

a) Solve the equation 2log 3 2 (2cos x) – 5log 3 (2cos x) + 2 = 0

b) Find all the roots of this equation that belong to the segment.

Solution: a) Let log 3 (2cos x) = t, then 2 t 2 – 5t + 2 = 0,

	log3(2cos x) =	2	⇔		2cos x = 9	⇔		cos x =	4,5	⇔ because |cos x| ≤ 1,
	log3(2cos x) =	1			2cos x = √3			cos x =	√3
		2							2

then cos x =	√3
	2

	x =	π	+ 2π k
		6
	x = –	π	+ 2π k, k ∈ Z
		6

b) Find the roots lying on the segment .

It can be seen from the figure that the given segment has roots

11π	And	13π	.
6		6

Answer: A)	π	+ 2π k; –	π	+ 2π k, k ∈ Z; b)	11π	;	13π	.
	6		6		6		6

Task number 14- advanced level refers to the tasks of the second part with a detailed answer. The task tests the ability to perform actions with geometric shapes. The task contains two items. In the first paragraph, the task must be proved, and in the second paragraph, it must be calculated.

The circumference diameter of the base of the cylinder is 20, the generatrix of the cylinder is 28. The plane intersects its bases along chords of length 12 and 16. The distance between the chords is 2√197.

a) Prove that the centers of the bases of the cylinder lie on the same side of this plane.

b) Find the angle between this plane and the plane of the base of the cylinder.

Solution: a) A chord of length 12 is at a distance = 8 from the center of the base circle, and a chord of length 16, similarly, is at a distance of 6. Therefore, the distance between their projections on a plane parallel to the bases of the cylinders is either 8 + 6 = 14, or 8 − 6 = 2.

Then the distance between chords is either

= = √980 = = 2√245

= = √788 = = 2√197.

According to the condition, the second case was realized, in which the projections of the chords lie on one side of the axis of the cylinder. This means that the axis does not intersect this plane within the cylinder, that is, the bases lie on one side of it. What needed to be proven.

b) Let's denote the centers of the bases as O 1 and O 2. Let us draw from the center of the base with a chord of length 12 the perpendicular bisector to this chord (it has a length of 8, as already noted) and from the center of the other base to another chord. They lie in the same plane β perpendicular to these chords. Let's call the midpoint of the smaller chord B, greater than A, and the projection of A onto the second base H (H ∈ β). Then AB,AH ∈ β and, therefore, AB,AH are perpendicular to the chord, that is, the line of intersection of the base with the given plane.

So the required angle is

∠ABH = arctan	AH	= arctg	28	= arctg14.
	BH		8 – 6

Task number 15- an increased level of complexity with a detailed answer, checks the ability to solve inequalities, the most successfully solved among tasks with a detailed answer of an increased level of complexity.

Example 15 Solve the inequality | x 2 – 3x| log 2 ( x + 1) ≤ 3x – x 2 .

Solution: The domain of definition of this inequality is the interval (–1; +∞). Consider three cases separately:

1) Let x 2 – 3x= 0, i.e. X= 0 or X= 3. In this case, this inequality becomes true, therefore, these values ​​are included in the solution.

2) Let now x 2 – 3x> 0, i.e. x∈ (–1; 0) ∪ (3; +∞). In this case, this inequality can be rewritten in the form ( x 2 – 3x) log 2 ( x + 1) ≤ 3x – x 2 and divide by a positive expression x 2 – 3x. We get log 2 ( x + 1) ≤ –1, x + 1 ≤ 2 –1 , x≤ 0.5 -1 or x≤ -0.5. Taking into account the domain of definition, we have x ∈ (–1; –0,5].

3) Finally, consider x 2 – 3x < 0, при этом x∈ (0; 3). In this case, the original inequality will be rewritten in the form (3 x – x 2) log 2 ( x + 1) ≤ 3x – x 2. After dividing by positive expression 3 x – x 2 , we get log 2 ( x + 1) ≤ 1, x + 1 ≤ 2, x≤ 1. Taking into account the area, we have x ∈ (0; 1].

Combining the obtained solutions, we obtain x ∈ (–1; –0.5] ∪ ∪ {3}.

Answer: (–1; –0.5] ∪ ∪ {3}.

Task number 16- advanced level refers to the tasks of the second part with a detailed answer. The task tests the ability to perform actions with geometric shapes, coordinates and vectors. The task contains two items. In the first paragraph, the task must be proved, and in the second paragraph, it must be calculated.

In an isosceles triangle ABC with an angle of 120° at the vertex A, a bisector BD is drawn. Rectangle DEFH is inscribed in triangle ABC so that side FH lies on segment BC and vertex E lies on segment AB. a) Prove that FH = 2DH. b) Find the area of ​​the rectangle DEFH if AB = 4.

Solution: A)

1) ΔBEF - rectangular, EF⊥BC, ∠B = (180° - 120°) : 2 = 30°, then EF = BE due to the property of the leg lying opposite the angle of 30°.

2) Let EF = DH = x, then BE = 2 x, BF = x√3 by the Pythagorean theorem.

3) Since ΔABC is isosceles, then ∠B = ∠C = 30˚.

BD is the bisector of ∠B, so ∠ABD = ∠DBC = 15˚.

4) Consider ΔDBH - rectangular, because DH⊥BC.

2x	=	4 – 2x
2x(√3 + 1)		4

1	=	2 – x
√3 + 1		2

√3 – 1 = 2 – x

x = 3 – √3

EF = 3 - √3

2) S DEFH = ED EF = (3 - √3 ) 2(3 - √3 )

S DEFH = 24 - 12√3.

Answer: 24 – 12√3.

Task number 17- a task with a detailed answer, this task tests the application of knowledge and skills in practical activities and everyday life, the ability to build and explore mathematical models. This task is a text task with economic content.

Example 17. The deposit in the amount of 20 million rubles is planned to be opened for four years. At the end of each year, the bank increases the deposit by 10% compared to its size at the beginning of the year. In addition, at the beginning of the third and fourth years, the depositor annually replenishes the deposit by X million rubles, where X - whole number. Find highest value X, at which the bank will add less than 17 million rubles to the deposit in four years.

Solution: At the end of the first year, the contribution will be 20 + 20 · 0.1 = 22 million rubles, and at the end of the second - 22 + 22 · 0.1 = 24.2 million rubles. At the beginning of the third year, the contribution (in million rubles) will be (24.2 + X), and at the end - (24.2 + X) + (24,2 + X) 0.1 = (26.62 + 1.1 X). At the beginning of the fourth year, the contribution will be (26.62 + 2.1 X), and at the end - (26.62 + 2.1 X) + (26,62 + 2,1X) 0.1 = (29.282 + 2.31 X). By condition, you need to find the largest integer x for which the inequality

(29,282 + 2,31x) – 20 – 2x < 17

29,282 + 2,31x – 20 – 2x < 17

0,31x < 17 + 20 – 29,282

0,31x < 7,718

x <	7718
	310

x <	3859
	155

x < 24	139
	155

The largest integer solution to this inequality is the number 24.

Answer: 24.

Task number 18- a task of an increased level of complexity with a detailed answer. This task is intended for competitive selection to universities with increased requirements for the mathematical preparation of applicants. A task of a high level of complexity is not a task for applying one solution method, but for a combination of different methods. For the successful completion of task 18, in addition to solid mathematical knowledge, a high level of mathematical culture is also required.

At what a system of inequalities

	x 2 + y 2 ≤ 2ay – a 2 + 1
	y + a ≤ |x| – a

has exactly two solutions?

Solution: This system can be rewritten as

	x 2 + (y– a) 2 ≤ 1
	y ≤ |x| – a

If we draw on the plane the set of solutions to the first inequality, we get the interior of a circle (with a boundary) of radius 1 centered at the point (0, A). The set of solutions of the second inequality is the part of the plane that lies under the graph of the function y = | x| – a, and the latter is the graph of the function
y = | x| , shifted down by A. The solution of this system is the intersection of the solution sets of each of the inequalities.

Consequently, this system will have two solutions only in the case shown in Fig. 1.

The points of contact between the circle and the lines will be the two solutions of the system. Each of the straight lines is inclined to the axes at an angle of 45°. So the triangle PQR- rectangular isosceles. Dot Q has coordinates (0, A), and the point R– coordinates (0, – A). In addition, cuts PR And PQ are equal to the circle radius equal to 1. Hence,

QR= 2a = √2, a =	√2	.
	2

Answer: a =	√2	.
	2

Task number 19- a task of an increased level of complexity with a detailed answer. This task is intended for competitive selection to universities with increased requirements for the mathematical preparation of applicants. A task of a high level of complexity is not a task for applying one solution method, but for a combination of different methods. To successfully complete task 19, you must be able to search for a solution by choosing different approaches from among the known, modifying the studied methods.

Let sn sum P members of an arithmetic progression ( a p). It is known that S n + 1 = 2n 2 – 21n – 23.

a) Give the formula P th member of this progression.

b) Find the smallest modulo sum S n.

c) Find the smallest P, at which S n will be the square of an integer.

Solution: a) Obviously, a n = S n – S n- 1 . Using this formula, we get:

S n = S (n – 1) + 1 = 2(n – 1) 2 – 21(n – 1) – 23 = 2n 2 – 25n,

S n – 1 = S (n – 2) + 1 = 2(n – 1) 2 – 21(n – 2) – 23 = 2n 2 – 25n+ 27

Means, a n = 2n 2 – 25n – (2n 2 – 29n + 27) = 4n – 27.

B) because S n = 2n 2 – 25n, then consider the function S(x) = | 2x 2 – 25x|. Her graph can be seen in the figure.

It is obvious that the smallest value is reached at the integer points located closest to the zeros of the function. Obviously these are points. X= 1, X= 12 and X= 13. Since, S(1) = |S 1 | = |2 – 25| = 23, S(12) = |S 12 | = |2 144 – 25 12| = 12, S(13) = |S 13 | = |2 169 – 25 13| = 13, then the smallest value is 12.

c) It follows from the previous paragraph that sn positive since n= 13. Since S n = 2n 2 – 25n = n(2n– 25), then the obvious case when this expression is a perfect square is realized when n = 2n- 25, that is, with P= 25.

It remains to check the values ​​​​from 13 to 25:

S 13 = 13 1, S 14 = 14 3, S 15 = 15 5, S 16 = 16 7, S 17 = 17 9, S 18 = 18 11, S 19 = 19 13 S 20 = 20 13, S 21 = 21 17, S 22 = 22 19, S 23 = 23 21, S 24 = 24 23.

It turns out that for smaller values P full square is not achieved.

Answer: A) a n = 4n- 27; b) 12; c) 25.

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*Since May 2017, the DROFA-VENTANA joint publishing group has been part of the Russian Textbook Corporation. The corporation also included the Astrel publishing house and the LECTA digital educational platform. Alexander Brychkin, a graduate of the Financial Academy under the Government of the Russian Federation, candidate of economic sciences, head of innovative projects of the DROFA publishing house in the field of digital education, has been appointed General Director ( electronic forms textbooks, "Russian Electronic School", digital educational platform LECTA). Prior to joining the DROFA publishing house, he held the position of Vice President for Strategic Development and Investments of the EKSMO-AST publishing holding. Today, the Russian Textbook Publishing Corporation has the largest portfolio of textbooks included in the Federal List - 485 titles (approximately 40%, excluding textbooks for remedial school). The corporation's publishing houses own the sets of textbooks in physics, drawing, biology, chemistry, technology, geography, astronomy, most in demand by Russian schools - the areas of knowledge that are needed to develop the country's production potential. The corporation's portfolio includes textbooks and study guides For elementary school awarded the Presidential Prize in Education. These are textbooks and manuals on subject areas that are necessary for the development of the scientific, technical and industrial potential of Russia.