The Unified State Examination practical work in mathematics is intended both for classroom work and for self-testing of knowledge.
The proposed manual contains training versions of test items of the Unified State Exam (USE) in mathematics (profile level), compiled taking into account all the features and requirements of the Unified State Exam, aimed at those students for whom mathematics is a compulsory subject upon admission to the chosen university.

The workshop is intended for teachers and methodologists who use tests to prepare students for the Unified State Exam; it can also be used by students for self-preparation and self-control.
By Order No. 729 of the Ministry of Education and Science of the Russian Federation, textbooks from the Ekzamen publishing house are approved for use in general education organizations.

2. A promotion has been announced in a household appliances store: if a buyer purchases a product worth more than 20,000 rubles, he receives a certificate for 4,000 rubles, which can be exchanged in the same store for any product worth less than 4,000 rubles. If the buyer participates in the promotion, he loses the right to return the product to the store. Buyer A. wants to buy a vacuum cleaner worth 19,400 rubles, a mixer worth 2,300 rubles. and a fan costing 3200 rubles.
In which case will A. pay the least for the purchase:
1)A. will buy all three things;
2)A. will buy a vacuum cleaner and mixer, and will receive a fan for a certificate;
3)A. buy a vacuum cleaner and a fan, and get a mixer for a certificate?
Find the amount that A. will pay for the purchase in the required case.

3)A. In a random experiment, a symmetrical coin is tossed 4 times.
Find the probability that heads will appear at least once.

CONTENT
INSTRUCTIONS FOR PERFORMING THE WORK.
OPTION 1
Part 1.
Part 2.
OPTION 2
Part 1.
Part 2.
OPTION 3
Part 1.
Part 2.
OPTION 4
Part 1.
Part 2.
OPTION 5
Part 1.
Part 2.
OPTION in
Part 1.
Part 2.
OPTION 7
Part 1.
Part 2.
OPTION 8
Part 1.
Part 2.
OPTION 9
Part 1.
Part 2.
OPTION 10
Part 1.
Part 2.
ANSWERS.
SOLUTION OPTION 5
Part 1.
Part 2.

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• Unified State Exam, Mathematics, Preparation for the Unified State Exam, Expert in the Unified State Exam, Lappo L.D., Popov M.A., 2015
• Unified State Exam 2015, Mathematics, Exam tests, Basic level, Workshop on completing standard test tasks, Lappo L.D., Popov M.A.
• Unified State Exam, mathematics, profile level, independent preparation for the Unified State Exam, universal materials with methodological recommendations, solutions and answers, Lappo L.D., Popov M.A., 2015
• Unified State Exam 2015, Mathematics, exam tests, basic level, workshop on completing standard Unified State Exam test tasks, Lappo L.D., Popov M.A.

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Unified State Exam 2015. Mathematics. Exam tests. Profile level. Workshop. Lappo L.D., Popov M.A.

M.: 2015 - 48 pp.

The Unified State Examination practical work in mathematics is intended both for classroom work and for self-testing of knowledge. The proposed manual contains training versions of test items of the Unified State Exam (USE) in mathematics (profile level), compiled taking into account all the features and requirements of the Unified State Exam, aimed at those students for whom mathematics is a compulsory subject upon admission to the chosen university. The workshop is intended for teachers and methodologists who use tests to prepare students for the Unified State Exam; it can also be used by students for self-preparation and self-control.

Format: pdf

Size: 1.3 MB

Watch, download: yandex.disk

CONTENT
INSTRUCTIONS FOR PERFORMING WORK 4
OPTION 1
Part 1 5
Part 2 6
OPTION 2
Part 1 8
Part 2 9
OPTION 3
Part 1 11
Part 2 12
OPTION 4
Part 1 14
Part 2 15
OPTION 5
Part 1 17
Part 2 18
OPTION 6
Part 1 20
Part 2 21
OPTION 7
Part 1 23
Part 2 24
OPTION 8
Part 1 . 26
Part 2 27
OPTION 9
Part 1 29
Part 2 30
OPTION 10
Part 1 32
Part 2 33
ANSWERS 35
SOLUTION OPTION 5
Part 1 41
Part 2 43

The work consists of 15 tasks of increased and high levels of complexity and is intended to test the mastery of mathematics at a specialized level. The work is intended for students aimed at using mathematics in their future professional activities.
The work consists of two parts. The first part includes 8 short answer tasks 1-8. The answer to each of them is a whole number or a finite decimal fraction.
Part 2 contains 7 tasks (9-15) based on the material from the high school mathematics course. Each of these tasks requires writing down a complete solution and answer.
When completing assignments, you can use a draft. 3 hours 55 minutes (235 minutes) are given to complete the work. We advise you to complete the tasks in the order in which they are given. To save time, skip a task that you cannot complete immediately and move on to the next one. If you have time left after completing all the work, you can return to the missed tasks.
The points you receive for completed tasks are summed up. Try to complete as many tasks as possible and score the most points.

Assessment

two parts, including 19 tasks. Part 1 Part 2

3 hours 55 minutes(235 minutes).

Answers

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

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

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

The exam work in mathematics is allotted 3 hours 55 minutes(235 minutes).

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

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

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

And decisions

Part 1

1 . The train departed from St. Petersburg at 23:50 and arrived in Moscow at 7:50 the next day. How many hours did the train travel?

2 . The dots in the figure show the average air temperature in Sochi for each month of 1920. Months are indicated horizontally, temperatures in degrees Celsius are indicated vertically. For clarity, the points are connected by a line. Determine from the figure how many months of this period the average temperature was more than 18 degrees Celsius.

3 . A construction contractor plans to purchase 15 tons of facing bricks from one of three suppliers. One brick weighs 5 kg. The price of bricks and delivery conditions for the entire purchase are shown in the table. How many rubles will the cheapest purchase option cost, including delivery?

4 . Find the area of ​​a rhombus depicted on checkered paper with a cell size of 1 cm x 1 cm. Give your answer in cm 2.

5 . There are only 25 tickets in the collection of biology tickets, two of them contain a question about mushrooms. At the exam, the student receives one randomly selected ticket from this collection. Find the probability that this ticket will not contain a question about mushrooms.

6 . Find the root of the equation

7 . Triangle ABC is inscribed in a circle with center O. Find angle BOC if angle BAC is 32 o. Give your answer in degrees.

8 . The figure shows a graph of a differentiable function. Nine points are marked on the abscissa axis: . Among these points, find all the points at which the derivative of the function is negative. In your answer, indicate the number of points found.

9 . In a cylindrical vessel, the liquid level reaches 16 cm. At what height will the liquid level be if it is poured into a second cylindrical vessel, the diameter of the base of which is 2 times larger than the diameter of the base of the first? Express your answer in cm.

Part 2

10 . Find if and

11 . The locator of a bathyscaphe, uniformly plunging vertically downwards, emits an ultrasonic signal with a frequency of 749 MHz. The receiver records the frequency of the signal reflected from the ocean floor. The submersion speed of the bathyscaphe (in m/s) and frequencies are related by the relation , where c = 1500 m/s is the speed of sound in water; f0 - frequency of the emitted signal (in MHz); f is the frequency of the reflected signal (in MHz). Find the frequency (in MHz) of the reflected signal if the submersible sinks at a speed of 2 m/s.

12 . A sphere is described around the cone (the sphere contains the circumference of the base of the cone and its vertex). The center of the sphere coincides with the center of the base of the cone. The radius of the sphere is . Find the generatrix of the cone.

13 . In the spring, the boat moves against the river flow several times slower than downstream. In summer, the current becomes 1 km/h slower. Therefore, in summer the boat goes against the current several times slower than with the current. Find the speed of the current in spring (in km/h).

14 . Find the maximum point of the function

15 . a) Solve the equation ; b) Find all the roots of this equation belonging to the interval

16 . At the base of a straight prism ABCDA 1 B 1 C 1 D 1 lies square ABCD with side 2, and the height of the prism is 1. Point E lies on the diagonal BD 1, and BE is equal to 1. a) Construct a section of the prism by the plane A 1 C 1 E. b) Find the angle between the section plane and the plane ABC.

17 . Solve the inequality

18 . Two circles touch externally at point K. Line AB touches the first circle at point A, and the second at point B. Line BK intersects the first circle at point D, line AK intersects the second circle at point C. a) Prove that lines AD and BC are parallel. b) Find the area of ​​triangle AKB if it is known that the radii of the circles are 4 and 1.

19 . On December 31, 2013, Sergey took out 9,930,000 rubles on credit from the bank at 10% per annum. The loan repayment scheme is as follows: on December 31 of each next year, the bank charges interest on the remaining amount of the debt (that is, increases the debt by 10%), then Sergey transfers a certain amount of the annual payment to the bank. What should be the amount of the annual payment for Sergei to pay off the debt in three equal annual payments?