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USE TEST - 2015 IN MATHEMATICS
PROFILE LEVEL
OPTION 4
PART 1
1. A bottle of shampoo costs 190 rubles. Which largest number bottles can be bought for 1000 rubles during the sale, when the discount is 35%?
2. The diagram shows the average air temperature in Simferopol for each month in 1988. Months are indicated horizontally, mean temperature is indicated vertically in degrees Celsius. Determine from the diagram how many months there were with a negative average temperature in Simferopol in 1988.
3. In three salons cellular communication the same phone is sold on credit for different conditions. Conditions are given in the table.
| Salon |
Price phone, |
An initial fee, as a percentage of the price |
Term credit, |
Sum monthly payment, rub. |
| Epsilon | 10500 | 10 | 6 | 1960 |
| Delta | 11600 | 5 | 6 | 2040 |
| Omicron | 12700 | 20 | 12 | 860 |
Determine in which of the salons the purchase will cost the most (taking into account the overpayment), and in response write this the largest amount in rubles.
4. Find the area of the trapezoid shown in checkered paper with a cell size of 1cm x 1cm (see fig.). Give your answer in square centimeters.
5. In a random experiment, a symmetrical coin is tossed twice. Find the probability that it comes up tails exactly once.
6. Find the root of the equation
7. In right triangle angle between altitude and median drawn from a vertex right angle, is equal to 26 ° . Find the bigger one sharp corners this triangle. Give your answer in degrees.
8. The figure shows the graph of the function y \u003d f (x) and the tangent to it at the point with the abscissa x 0. Find the value of the derivative of the function f (x) at the point x 0.
9. Find the volume of the polyhedron shown in the figure (all dihedral angles straight lines).
PART 2
10. Find the value of the expression
11. To determine effective temperature stars use the Stefan-Boltzmann law, according to which the radiation power of a heated body P, measured in watts, is directly proportional to its surface area and the fourth power of temperature:
where σ = 5.7 10 -8 is a constant, the area S is measured in square meters, and the temperature T is in degrees Kelvin. It is known that some star has an area
and the power P radiated by it is equal to 4.104 10 27 W. Determine the temperature of this star. Express your answer in degrees Kelvin.
12. In the right triangular pyramid SABC point M is the middle of the edge BC, S is the vertex. It is known that AB = 6, and the lateral surface area is 45. Find the length of the segment SM.
13. Two cars left point A for point B at the same time. The first one went from constant speed all the way. The second traveled the first half of the journey at a speed of 44 km/h, and the second half of the journey at a speed 21 km/h greater than the speed of the first, as a result of which it arrived at B at the same time as the first car. Find the speed of the first car. Give your answer in km/h.
14. Find highest value functions
15. a) Solve the equation 4sin 4 2x + 3cos4x −1 = 0.
b) Find all the roots of this equation, belonging to the segment[P; 3p/2].
16. The base area of a regular quadrangular pyramid SABCD is 64.
a) Construct the line of intersection of the plane SAC and the plane passing through the vertex S of this pyramid, the midpoint of the side AB and the center of the base.
b) Find the area of the lateral surface of this pyramid if the area of the section of the pyramid by the SAC plane is 64.
17. Solve the inequality
18. Medians AA 1 , BB 1 and SS 1 triangle ABC intersect at point M. Points A 2 , B 2 and C 2 are the midpoints of the segments MA, MB and MS, respectively.
a) Prove that the area of the hexagon A 1 B 2 C 1 A 2 B 1 C 2 is twice less area triangle ABC.
b) Find the sum of the squares of all sides of this hexagon if it is known that AB = 4, BC = 7 and AC = 8.
19. On December 31, 2014, Dmitry borrowed 4,290,000 rubles from a bank at 14.5% per annum. The loan repayment scheme is as follows - December 31 of each next year the bank charges interest on the remaining amount of the debt (that is, increases the debt by 14.5%), then Dmitry transfers X rubles to the bank. What should be the amount X for Dmitry to pay off the debt in two equal payments (that is, for two years)?
20. Find all parameter values a , for each of which the equation
has at least one root on the segment .
21. Increasing finite arithmetic progression consists of various non-negative numbers. The mathematician calculated the difference between the square of the sum of all members of the progression and the sum of their squares. Then the mathematician added the next term to this progression and again calculated the same difference.
a) Give an example of such a progression if the difference was 40 more the second time than the first time.
b) The second time the difference turned out to be 1768 more than the first time. Could the progression have originally consisted of 13 members?
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1 MATHEMATICS, Grade 11 Option 1, April 015 Regional diagnostic work in MATHEMATICS OPTION 1 Instructions for performing work 90 minutes are given to complete the regional diagnostic work in mathematics. The work consists of two parts, including 14 tasks. Part 1 contains 9 tasks (tasks 1-9) basic level complexity, checking the availability of practical mathematical knowledge and skills. Part contains 5 tasks (tasks 10-14) elevated levels based on the material of the mathematics course high school checking the level of profile mathematical training. The answer to each of tasks 1-14 is an integer or a final decimal. All USE forms are filled in with bright black ink. You can use a gel, capillary or fountain pen. When completing assignments, you can use a draft. Please note that entries in the draft will not be taken into account when evaluating the work. We advise you to complete the tasks in the order in which they are given. To save time, skip the task that you can't complete right away and move on to the next one. If after completing all the work you have time left, you can return to the missed tasks. The points you get for completed tasks are summed up. Try to complete as many tasks as possible and score the largest number points. We wish you success! Part I The answers to tasks 1-9 must be an integer or a final decimal fraction. The answer should be written in the answer sheet 1 to the right of the number of the task being performed, starting from the first cell. Write each number, minus sign and comma in a separate box in accordance with the samples given in the form. Units of measurement are not required. 1. The club has five tourist tents. Which smallest number do you need to take tents on a hike with 6 people? MATHEMATICS, Grade 11 Option 1, April 015. When the aircraft is in level flight, the lift acting on the wings depends only on the speed. The figure shows this dependence for some aircraft. On the abscissa axis, the speed is plotted (in kilometers per hour), on the ordinate axis, the force (in tons of force). Determine from the figure what is the lift force (in tons of force) at a speed of 00 km / h? 3. In three mobile phone stores, the same phone is sold on credit under different conditions. Conditions are given in the table. Salon Phone price (rub.) Down payment (in % of the price) Loan term (months) Monthly payment amount (rub.) Gamma Delta Omega Determine which of the salons the purchase will cost the least (taking into account the overpayment). In response, write down this amount in rubles.
2 MATHEMATICS, grade 11 Option 1, April Find the area of the trapezoid shown in the figure. 5. Dice thrown twice. How many elementary outcomes of experience favor the event "A = sum of points equals 5"? MATHEMATICS, Grade 11 Option 1, April 015 Part II The answer to the tasks must be an integer or a final decimal fraction. The answer should be written in the answer sheet 1 to the right of the number of the task being performed, starting from the first cell. Write each number, minus sign and comma in a separate box in accordance with the samples given in the form. Units of measurement are not required. 10. Find tg() π α + if tgα = 5 log 6. Solve the equation 4(x 8) 4 = log A crane is fixed in the side wall of a tall cylindrical tank near the bottom. After it is opened, water begins to flow out of the tank, while the height of the water column in it, expressed in meters, changes according to the law H (t) \u003d at + bt + H 0, where 1 H 0 \u003d m First level water, a = m/min 1, and b = m/min 51 8 are constants, t is the time in minutes elapsed since the tap was opened. How long will water flow out of the tank? Give your answer in minutes. 7. The intersection point of the bisectors of two angles of the parallelogram ABC adjacent to one side belongs to opposite side. smaller side parallelogram is 5. Find its longest side. E 1. The height of the cone is 8, and the length of the generatrix is 10. Find the area axial section this cone Material point moves rectilinearly according to the law x(t) = t + 9t + 16, where x is the distance from the reference point in meters, t is the time in seconds measured from the beginning of the movement. Find its speed (in meters per second) at time t=4 s. 9. How many times will the volume increase regular tetrahedron if all its edges are tripled? 13. The motorboat passed 4 km against the current of the river and returned to the point of departure, having spent on Return trip an hour less. Find the speed of the boat in still water if the speed of the current is 1 km/h. Give your answer in km/h Find the maximum point of the function y = x 48x + 17.
3 MATHEMATICS, grade 11 Option, April 015 Regional diagnostic work in MATHEMATICS OPTION Instructions for performing work 90 minutes are given to complete the regional diagnostic work in mathematics. The work consists of two parts, including 14 tasks. Part 1 contains 9 tasks (tasks 1-9) of the basic level of complexity, checking the availability of practical mathematical knowledge and skills. The part contains 5 tasks (tasks 10-14) of advanced levels based on the material of the secondary school mathematics course, checking the level of specialized mathematical training. The answer to each of tasks 1-14 is an integer or a final decimal fraction. All USE forms are filled in with bright black ink. You can use a gel, capillary or fountain pen. When completing assignments, you can use a draft. Please note that entries in the draft will not be taken into account when evaluating the work. We advise you to complete the tasks in the order in which they are given. To save time, skip the task that you can't complete right away and move on to the next one. If after completing all the work you have time left, you can return to the missed tasks. The points you get for completed tasks are summed up. Try to complete as many tasks as possible and score the most points. We wish you success! MATHEMATICS, Grade 11 Option, April 015. The graph shows the dependence of the torque of an automobile engine on the number of revolutions per minute. The number of revolutions per minute is plotted on the x-axis. On the y-axis, the torque is in N m. For the car to start moving, the torque must be at least 60 N m. What is the smallest number of engine revolutions per minute sufficient for the car to start moving? Part I The answers to tasks 1-9 must be an integer or a final decimal fraction. The answer should be written in the answer sheet 1 to the right of the number of the task being performed, starting from the first cell. Write each number, minus sign and comma in a separate box in accordance with the samples given in the form. Units of measurement are not required. 1. To prepare a marinade for cucumbers, 18 g of citric acid is required per 1 liter of water. Lemon acid sold in bags of 10 g. What is the smallest number of bags that the hostess needs to buy to prepare 7 liters of marinade? 3. Ceramic tiles of the same trademark three different sizes. Tiles are packed in packs. Need to buy tiles to cover the floor square room with a side of 3 m. The dimensions of the tile, the number of tiles in a pack and the cost of a pack are shown in the table. Tile size (smcm) Number of tiles in a pack 0 to r. How much will the cheapest purchase option cost (tiles are sold in whole packs)?
4 MATHEMATICS, grade 11 Option, April Find the area of a trapezoid whose vertices have coordinates (,), (8, 4), (8, 8), (, 10). 5. In a random experiment, a symmetrical coin is tossed three times. Find the probability of getting at least two tails. 6. Solve the equation () log 5x + 11 = 7. The acute angle of a right triangle is 50. Find the angle between the height H and the median M drawn from the vertex of the right angle. Give your answer in degrees. 8. The line y 5x 4 6 = + is parallel to the tangent to the graph of the function y = x + 3x + 6. Find the abscissa of the tangent point. 9. The circumference of the base of the cone is 6, the generatrix is equal. Find the area of the lateral surface of the cone. M N MATHEMATICS, grade 11 Option, April 015 Part II The answer to the tasks must be an integer or a final decimal fraction. The answer should be written in the answer sheet 1 to the right of the number of the task being performed, starting from the first cell. Write each number, minus sign and comma in a separate box in accordance with the samples given in the form. Units of measurement are not required. 10. Find the value of the expression 4sin 8 cos8. sin Rating R of an online store is calculated by the formula r = suc r R r ex suc m (K + 1), where 0.0K m =, r suc + 0.1 suc r average rating store by customers (from 0 to 1), r ex-assessment of the store by experts (from 0 to 0.7) and K is the number of customers who rated the store. Find the rating of the Alpha online store, if the number of buyers who left a review about the store is 6, their average rating is 0.68, and the expert rating is 0.3. 1. Ribs cuboid emanating from one vertex are 5 and 7. Find its surface area. B 1 C 1 A The road between points A and B consists of an ascent and a descent, and its length is 8 km. A tourist traveled from A to B in 5 hours. The time of its movement on the descent was 1 hour. With what speed did the tourist walk on the descent, if the speed of his movement on the ascent is less than the speed of the movement on the descent by 3 km/h? A B C 14. Find the minimum point of the function y = (x 10x + 10) e x 10
5 MATHEMATICS, grade 11 Option 3, April 015 Regional diagnostic work in MATHEMATICS OPTION 3 Instructions for performing work 90 minutes are given to complete the regional diagnostic work in mathematics. The work consists of two parts, including 14 tasks. Part 1 contains 9 tasks (tasks 1-9) of the basic level of complexity, checking the availability of practical mathematical knowledge and skills. The part contains 5 tasks (tasks 10-14) of advanced levels based on the material of the secondary school mathematics course, checking the level of specialized mathematical training. The answer to each of tasks 1-14 is an integer or a final decimal fraction. All USE forms are filled in with bright black ink. You can use a gel, capillary or fountain pen. When completing assignments, you can use a draft. Please note that entries in the draft will not be taken into account when evaluating the work. We advise you to complete the tasks in the order in which they are given. To save time, skip the task that you can't complete right away and move on to the next one. If after completing all the work you have time left, you can return to the missed tasks. The points you get for completed tasks are summed up. Try to complete as many tasks as possible and score the most points. We wish you success! Part I The answers to tasks 1-9 must be an integer or a final decimal fraction. The answer should be written in the answer sheet 1 to the right of the number of the task being performed, starting from the first cell. Write each number, minus sign and comma in a separate box in accordance with the samples given in the form. Units of measurement are not required. 1. Installation of two water meters (cold and hot) costs 300 rubles. Before the installation of water meters, they paid 800 rubles a month for water. After installing the meters, the monthly payment for water began to be 600 rubles. In what least number of months will the savings in water bills exceed the cost of installing meters if the water tariffs do not change? MATHEMATICS, grade 11 Option 3, April 015. The figure shows the change in air temperature over three days. The date and time are indicated horizontally, the temperature value in degrees Celsius is indicated vertically. Determine from the figure the highest air temperature on April 7. Give your answer in degrees Celsius. 3. The rating agency determines the value for money rating of electric hair dryers. The rating is calculated based on the average price P and scores on functionality F, quality Q and design. Each individual indicator is evaluated by experts on a five-point scale with integers from 0 to 4. The final rating is calculated by the formula R=3(F+Q)+-0.01P. The table gives estimates of each indicator for several models of hair dryers. Determine which model has the lowest rating. In response, write down the value of this rating. Hair dryer model average price Functionality Quality Design A B C D
6 MATHEMATICS, grade 11 Option 3, April Find the area of a square whose vertices have coordinates (9; 0), (10; 9), (1; 10), (0; 1). MATHEMATICS, Grade 11 Option 3, April 015 Part II The answer to the tasks must be an integer or a final decimal fraction. The answer should be written in the answer sheet 1 to the right of the number of the task being performed, starting from the first cell. Write each number, minus sign and comma in a separate box in accordance with the samples given in the form. Units of measurement are not required. 5. The shooter shoots at the target once. In case of a miss, the shooter fires a second shot at the same target. The probability of hitting the target with one shot is 0.7. Find the probability that the target will be hit (either by the first or the second shot). 6. Solve the equation 6 \u003d x 1 7. Two angles of a quadrilateral inscribed in a circle are 8 and 58. Find the largest of the remaining angles. Give your answer in degrees. 8. The material point moves rectilinearly according to the law x(t) = t 3t + 15, where x is the distance from the reference point in meters, t is the time in seconds, measured from the beginning of the movement. At what point in time (in seconds) was her speed equal to 11 m/s? 9. How many times will the surface area of the pyramid increase if all its edges are increased by 40 times? O 10. Find the value of the expression 50sin30 cos30. sin Locator bathyscaphe, evenly plunging vertically down, emits ultrasonic pulses with a frequency of 749 MHz. The bathyscaphe's descent speed, f f0 expressed in m/s, is determined by the formula ν = c, where c = 1500 /s is the speed f + f 0 of sound in water, f 0 is the frequency of emitted pulses (in MHz), f is the frequency of the signal reflected from the bottom registered by the receiver (in MHz). Determine the highest possible frequency of the reflected signal f if the submersible sinking speed should not exceed m/s. 1. Two edges of a cuboid coming out of the same vertex are equal, 3. The volume of the cuboid is 36. Find its diagonal. B 1 C 1 A The first and second pumps fill the pool in 10 minutes, the second and third in 15 minutes, and the first and third in 4 minutes. How many minutes will it take for these three pumps to fill the pool working together? 14. Find the largest value of the function y = ln(5 x) 5x + 11 on the segment 1 1 [ ; ] 10. A B C
7 MATHEMATICS, grade 11 Option 4, April 015 Regional diagnostic work in MATHEMATICS OPTION 4 Instructions for performing work 90 minutes are given to complete the regional diagnostic work in mathematics. The work consists of two parts, including 14 tasks. Part 1 contains 9 tasks (tasks 1-9) of the basic level of complexity, checking the availability of practical mathematical knowledge and skills. The part contains 5 tasks (tasks 10-14) of advanced levels based on the material of the secondary school mathematics course, checking the level of specialized mathematical training. The answer to each of tasks 1-14 is an integer or a final decimal fraction. All USE forms are filled in with bright black ink. You can use a gel, capillary or fountain pen. When completing assignments, you can use a draft. Please note that entries in the draft will not be taken into account when evaluating the work. We advise you to complete the tasks in the order in which they are given. To save time, skip the task that you can't complete right away and move on to the next one. If after completing all the work you have time left, you can return to the missed tasks. The points you get for completed tasks are summed up. Try to complete as many tasks as possible and score the most points. We wish you success! Part I The answers to tasks 1-9 must be an integer or a final decimal fraction. The answer should be written in the answer sheet 1 to the right of the number of the task being performed, starting from the first cell. Write each number, minus sign and comma in a separate box in accordance with the samples given in the form. Units of measurement are not required. 1. Installation of two water meters (cold and hot) costs 500 rubles. Before the installation of water meters, they paid 800 rubles a month for water. After installing the meters, the monthly payment for water began to be 600 rubles. In what least number of months will the savings in water bills exceed the cost of installing meters if the water tariffs do not change? MATHEMATICS, grade 11 Option 4, April 015. The graph shows the process of warming up the engine of a car. The abscissa shows the time in minutes that has elapsed since the engine was started, and the ordinate shows the engine temperature in degrees Celsius. Determine from the schedule how many minutes the engine warmed up from a temperature of 60 to a temperature From home to the dacha you can get by bus, train or taxi. The table shows the time that needs to be spent on each section of the path. What is the shortest travel time? Give your answer in hours. By bus By train By taxi 1 3 From home to the bus station 10 min. From home to station railway 0 min. From home to stop fixed-route taxi 5 minutes. Bus on the way: h Train on the way: 1 h 45 min. From the bus stop to the cottage on foot 10 minutes. From the station to the cottage on foot 10 minutes. Fixed-route taxi From a stop on the road: fixed-route taxi 1 h 5 min. to the cottage on foot 35 min.
8 MATHEMATICS, grade 11 Option 4, April Find the area of a trapezoid whose vertices have coordinates (,), (8, 4), (8, 8), (, 10). 5. The biathlete shoots at the targets five times. The probability of hitting the target with one shot is 0.8. Find the probability that the biathlete hit the targets the first three times and missed the last two. Round the result to the nearest hundredth. 6. Solve the equation () 1x 18 0.5 = The sum of the two angles of the parallelogram is 100. Find one of the remaining angles. Give your answer in degrees The material point moves in a straight line according to the law x(t) = t + 4t + 19, where 4 x is the distance from the reference point in meters, t is the time in seconds measured from the beginning of the movement. At what point in time (in seconds) was her speed equal to 6 m/s? 9. Find the volume of the polyhedron whose vertices are points A, B, C, E, F, 1 of the correct hexagonal prism ABCEFA1BC 1 11E 1F 1, whose base area is 4, and side rib equals 3. MATHEMATICS, Grade 11 Option 4, April 015 Part II The answer to the tasks must be an integer or a final decimal fraction. The answer should be written in the answer sheet 1 to the right of the number of the task being performed, starting from the first cell. Write each number, minus sign and comma in a separate box in accordance with the samples given in the form. Units of measurement are not required. 10. Find the value of the expression 6cos59 sin A stone-throwing machine shoots stones at some sharp angle to the horizon. The flight path of the stone is described by the formula y = ax + bx, where 1 1 a = m, b=1 are constant parameters, x(m) is the horizontal displacement of the stone, 100 y(m) is the height of the stone above the ground. At what greatest distance (in meters) from a fortress wall 8 m high should a car be positioned so that the stones fly over the wall at a height of at least 1 meter? 1. Find the height of a regular triangular pyramid, the sides of the base of which are equal, and the volume is equal to The ship, whose speed in still water is 15 km / h, passes along the river and after parking returns to its starting point. The current speed is 3 km/h, the stay lasts 3 hours, and the ship returns to the starting point 58 hours after departure from it. How many kilometers did the ship travel during the entire voyage? A S C B 14. Find the maximum point of the function y = (15 x) e x+ 15
9 MATHEMATICS, grade 11 Option 5, April 015 Regional diagnostic work in MATHEMATICS OPTION 5 Instructions for performing work 90 minutes are given to complete the regional diagnostic work in mathematics. The work consists of two parts, including 14 tasks. Part 1 contains 9 tasks (tasks 1-9) of the basic level of complexity, checking the availability of practical mathematical knowledge and skills. The part contains 5 tasks (tasks 10-14) of advanced levels based on the material of the secondary school mathematics course, checking the level of specialized mathematical training. The answer to each of tasks 1-14 is an integer or a final decimal fraction. All USE forms are filled in with bright black ink. You can use a gel, capillary or fountain pen. When completing assignments, you can use a draft. Please note that entries in the draft will not be taken into account when evaluating the work. We advise you to complete the tasks in the order in which they are given. To save time, skip the task that you can't complete right away and move on to the next one. If after completing all the work you have time left, you can return to the missed tasks. The points you get for completed tasks are summed up. Try to complete as many tasks as possible and score the most points. MATHEMATICS, Grade 11 Option 5, April 015. In the figure, bold dots show the daily amount of precipitation that fell in Kazan from February 3 to February 15, 1909. The dates of the month are indicated horizontally, and the amount of precipitation on the corresponding day, in millimeters, is indicated vertically. For clarity, bold dots in the figure are connected by a line. Determine from the figure how many days from this period there was no precipitation. We wish you success! Part I The answers to tasks 1-9 must be an integer or a final decimal fraction. The answer should be written in the answer sheet 1 to the right of the number of the task being performed, starting from the first cell. Write each number, minus sign and comma in a separate box in accordance with the samples given in the form. Units of measurement are not required. 1. Renovation of an apartment requires 51 rolls of wallpaper. How many packs of wallpaper glue do you need to buy if one pack of glue is designed for 4 rolls? 3. When building a rural house, one of two types of foundation can be used: stone or concrete. For a stone foundation, 9 tons of natural stone and 9 bags of cement are needed. For a concrete foundation, 7 tons of crushed stone and 50 bags of cement are needed. A ton of stone costs roubles, crushed stone costs 780 rubles per ton, and a bag of cement costs 30 rubles. How many rubles will the material for the foundation cost if you choose the cheapest option?
10 MATHEMATICS, Grade 11 Option 5, April What radius should be the circle centered at the point P (5; 1) so that it touches the y-axis? 5. There are two payment machines in the store. Each of them can be faulty with a probability of 0.05, regardless of the other automaton. Find the probability that in random moment time, both automata will be operational at the same time Solve the equation 10 x + \u003d 0, In a triangle, angle C is a straight line. H is the height, bisector, O is the point of intersection of lines H and, the angle is 6. Find the angle O. Give the answer in degrees. The line y = x + 14 is tangent to the graph of the function y = x 4x + 3x Find the abscissa of the point of contact. 9. The area of the lateral surface of the cylinder is 40π and the diameter of the base is 5. Find the height of the cylinder. N O MATHEMATICS, Grade 11 Option 5, April 015 Part II The answer to the tasks must be an integer or a final decimal fraction. The answer should be written in the answer sheet 1 to the right of the number of the task being performed, starting from the first cell. Write each number, minus sign and comma in a separate box in accordance with the samples given in the form. Units of measurement are not required. 10. Find the value of the expression Rating R of an online store is calculated by the formula r = so r R r ex so, where r 0.0K so far is the average rating of the store by customers (K + 1) r som + 0.1 (from 0 to 1), r ex-assessment of the store by experts (from 0 to 0.7) and K is the number of buyers who rated the store. Find the rating of the online store "Beta", if the number of customers who left a review of the store is 0, their average rating is 0.65, and the expert rating is 0, The side of the base of a regular hexagonal pyramid is 4, and the angle between the side face and the base is 45. Find the volume of the pyramid. 13. From one point circular track, whose length is 1 km, two cars start simultaneously in the same direction. The speed of the first car is 106 km/h, and 48 minutes after the start it was one lap ahead of the second car. Find the speed of the second car. Give your answer in km/h Find the minimum point of the function y = + x + 1. x
11 MATHEMATICS, grade 11 Option 6, April 015 Regional diagnostic work in MATHEMATICS OPTION 6 Instructions for performing the work MATHEMATICS, grade 11 Option 6, April 015. The diagram shows the average monthly air temperature in Nizhny Novgorod(Bitter) for each month of 1994. Months are indicated horizontally, temperatures are indicated vertically in degrees Celsius. Determine from the diagram how many months there were with a positive average monthly temperature. 90 minutes are given to complete the regional diagnostic work in mathematics. The work consists of two parts, including 14 tasks. Part 1 contains 9 tasks (tasks 1-9) of the basic level of complexity, checking the availability of practical mathematical knowledge and skills. The part contains 5 tasks (tasks 10-14) of advanced levels based on the material of the secondary school mathematics course, checking the level of specialized mathematical training. The answer to each of tasks 1-14 is an integer or a final decimal fraction. All USE forms are filled in with bright black ink. You can use a gel, capillary or fountain pen. When completing assignments, you can use a draft. Please note that entries in the draft will not be taken into account when evaluating the work. We advise you to complete the tasks in the order in which they are given. To save time, skip the task that you can't complete right away and move on to the next one. If after completing all the work you have time left, you can return to the missed tasks. The points you get for completed tasks are summed up. Try to complete as many tasks as possible and score the most points. We wish you success! Part I The answers to tasks 1-9 must be an integer or a final decimal fraction. The answer should be written in the answer sheet 1 to the right of the number of the task being performed, starting from the first cell. Write each number, minus sign and comma in a separate box in accordance with the samples given in the form. Units of measurement are not required. 1. One roll of wallpaper is enough to cover a strip from floor to ceiling with a width of 5 m. How many rolls of wallpaper do you need to buy to cover a rectangular room measuring 1.3 m by 4 m? 3. For glazing museum showcases, you need to order 4 identical glasses from one of the three companies. The area of each glass is 0.35 m. The table shows the prices for glass and glass cutting. How much will the cheapest order cost? Company Price of glass (rubles per 1 m) Glass cutting (rubles per glass) Additional terms When ordering more than 3000 rubles. cutting free.
12 MATH, Grade 11 Option 6, April 015 MATH, Grade 11 Option 6, April Find the ordinate of the center of a circle circumscribed about a triangle whose vertices have coordinates (8, 0), (0, 6), (8, 6). y 6 O 8 x Part II The answers to the tasks must be an integer or a final decimal fraction. The answer should be written in the answer sheet 1 to the right of the number of the task being performed, starting from the first cell. Write each number, minus sign and comma in a separate box in accordance with the samples given in the form. Units of measurement are not required. 10. Find the value of the expression (log9 81) (log 64) 5. In a random experiment, a symmetrical coin is tossed twice. Find the probability that heads comes up the first time and tails the second. 6. Solve the equation x = The side of an isosceles triangle is 1, the angle at the vertex opposite the base is 10. Find the diameter of the circumscribed circle of this triangle. The material point moves in a straight line according to the law x(t) = t t + 18, where 6 x is the distance from the point reference in meters, t time in seconds, measured from the beginning of the movement. At what point in time (in seconds) was her speed equal to 1 m/s? 9. How many times will the volume of the pyramid increase if its height is quadrupled? O 11. A weight of 0.8 kg oscillates on a spring with a speed that varies according to the law ν (t) = 0.9sinπt, where time is in seconds. The kinetic energy of the load, mv measured in joules, is calculated by the formula E =, where m is the mass of the load (in kg), ν is the speed of the load (in m/s). Determine what fraction of the time from the first second after the start of the movement kinetic energy load will be at least 1, J. Express your answer as a decimal fraction, if necessary, round to hundredths. 1. Find the volume of a regular hexagonal prism, the base sides of which are equal to 1, and the side edges are equal. To make 588 parts, the first worker spends 7 hours less than the second worker to make 67 parts. It is known that the first worker makes 4 more parts per hour than the second. How many parts per hour does the first worker make? Find the largest value of the function y = x 3x + 4 on the interval [ ;0]
13 MATHEMATICS, grade 11 Option 7, April 015 Regional diagnostic work in MATHEMATICS OPTION 7 Instructions for performing work 90 minutes are given to complete the regional diagnostic work in mathematics. The work consists of two parts, including 14 tasks. Part 1 contains 9 tasks (tasks 1-9) of the basic level of complexity, checking the availability of practical mathematical knowledge and skills. The part contains 5 tasks (tasks 10-14) of advanced levels based on the material of the secondary school mathematics course, checking the level of specialized mathematical training. The answer to each of tasks 1-14 is an integer or a final decimal fraction. All USE forms are filled in with bright black ink. You can use a gel, capillary or fountain pen. When completing assignments, you can use a draft. Please note that entries in the draft will not be taken into account when evaluating the work. We advise you to complete the tasks in the order in which they are given. To save time, skip the task that you can't complete right away and move on to the next one. If after completing all the work you have time left, you can return to the missed tasks. The points you get for completed tasks are summed up. Try to complete as many tasks as possible and score the most points. We wish you success! MATHEMATICS, grade 11 Option 7, April 015. The diagram shows the number of visitors to the RIA Novosti website on all days from November 10 to November 9, 009. Days of the month are indicated horizontally, the number of visitors to the site for a given day is indicated vertically. Determine from the diagram how many times the largest number of visitors exceeds the smallest number of visitors per day. Part I The answers to tasks 1-9 must be an integer or a final decimal fraction. The answer should be written in the answer sheet 1 to the right of the number of the task being performed, starting from the first cell. Write each number, minus sign and comma in a separate box in accordance with the samples given in the form. Units of measurement are not required. 1. In the dormitory of the institute, four people can be accommodated in each room. What is the smallest number of rooms needed to accommodate 59 out-of-town students? 3. You can get from the house to the dacha by bus, by train or by fixed-route taxi. The table shows the time that needs to be spent on each section of the path. What is the shortest travel time? Give your answer in hours. By bus By train By taxi 1 3 From home to the bus station 5 min. From the house to the railway station 30 min. From the house to the bus stop 0 min. Bus on the way: h 5 min. Train on the way: 1 h 40 min. Shuttle taxi on the road: 1 h 30 min. From the bus stop to the cottage on foot 10 minutes. From the station to the cottage on foot 5 minutes. From the bus stop to the cottage on foot 35 minutes.
14 MATHEMATICS, Grade 11 Option 7, April Find the abscissa of the center of a circle circumscribed about a rectangle whose vertices have coordinates (-, -), (6, -), (6, 4), (-, 4), respectively. 5. Before starting football match The referee tosses a coin to determine which team will start the ball game. Team "Physicist" plays three matches with different teams. Find the probability that in these games "Physicist" wins the lot exactly twice. log 6. Solve the equation 11(7x 5) 11 = log In a triangle, the angle is 60, the angle is 8., E and F are the heights intersecting at point O. Find the angle OF. Give your answer in degrees. The line y = 4x + 11 is parallel to the tangent to the graph of the function y = x + 5x 6. Find the abscissa of the point of contact. 9. How many times will the area of the lateral surface of the cone decrease if the radius of its base decreases by 1.5 times, and the generatrix remains the same? y 4 O E F 6 x MATHEMATICS, Grade 11 Option 7, April 015 Part II The answers to the tasks must be an integer or a final decimal fraction. The answer should be written in the answer sheet 1 to the right of the number of the task being performed, starting from the first cell. Write each number, minus sign and comma in a separate box in accordance with the samples given in the form. Units of measurement are not required. 10. Find the value of the expression (log515) (log 416) 11. The dependence of the volume of demand q (units per month) for the products of a monopoly enterprise on the price p (thousand rubles) is given by the formula q = 85 5 p. The company's revenue for the month r (in thousand rubles) is calculated by the formula r(p) = q p. Determine highest price p, at which the monthly revenue r(p) will be at least 10 thousand rubles. Give the answer in thousand rubles. 1. The volume of a regular hexagonal pyramid 6. The side of the base is 1. Find the side edge. 13. The car drove the first third of the track at a speed of 60 km/h, the second third at a speed of 10 km/h, and the last one at a speed of 110 km/h. Find the average speed of the car for the entire journey. Give your answer in km/h. x Find the maximum point of the function y =. x
15 MATHEMATICS, grade 11 Option 8, April 015 Regional diagnostic work in MATHEMATICS OPTION 8 Instructions for performing work 90 minutes are given to complete the regional diagnostic work in mathematics. The work consists of two parts, including 14 tasks. Part 1 contains 9 tasks (tasks 1-9) of the basic level of complexity, checking the availability of practical mathematical knowledge and skills. The part contains 5 tasks (tasks 10-14) of advanced levels based on the material of the secondary school mathematics course, checking the level of specialized mathematical training. The answer to each of tasks 1-14 is an integer or a final decimal fraction. All USE forms are filled in with bright black ink. You can use a gel, capillary or fountain pen. When completing assignments, you can use a draft. Please note that entries in the draft will not be taken into account when evaluating the work. We advise you to complete the tasks in the order in which they are given. To save time, skip the task that you can't complete right away and move on to the next one. If after completing all the work you have time left, you can return to the missed tasks. The points you get for completed tasks are summed up. Try to complete as many tasks as possible and score the most points. We wish you success! Part I The answers to tasks 1-9 must be an integer or a final decimal fraction. The answer should be written in the answer sheet 1 to the right of the number of the task being performed, starting from the first cell. Write each number, minus sign and comma in a separate box in accordance with the samples given in the form. Units of measurement are not required. 1. The cost of a semi-annual subscription to the journal is 550 rubles, and the cost of one issue of the journal is 9 rubles. For six months, Anya bought 5 issues of the magazine. How many rubles less would she spend if she subscribed to the magazine? MATHEMATICS, Grade 11 Option 8, April 015. In the figure, bold dots show the average daily air temperature in Brest every day from July 6 to July 19, 1981. Dates of the month are indicated horizontally, temperatures in degrees Celsius are indicated vertically. For clarity, bold dots are connected by a line. Determine from the figure what the temperature was on July 15. Give your answer in degrees Celsius. 3. The client wants to rent a car for a day for a trip of 400 km. The table shows the characteristics of three cars and the rental price. Car Fuel Fuel consumption (l per 100 km) Rent (rub. for 1 day) A Diesel B Gasoline C Gas In addition to the rental, the client is obliged to pay fuel for the car for the entire trip. The price of diesel fuel is 19 rubles per liter, gasoline is 3 rubles per liter, gas is 16 rubles per liter. What amount in rubles will the client pay for rent and fuel if he chooses the cheapest option?
16 MATHEMATICS, grade 11 Option 8, April Points O(0, 0), (6,), A(6, 8) and B are the vertices of the parallelogram. Find the ordinate of the point. y One group from each of the declared countries performs at the rock festival. The order of performance is determined by lot. What is the probability that a band from Denmark will perform after a band from Sweden and after a band from Norway? Round the result to the nearest hundredth. 6. Solve the equation () x 5 0.01 = Lines and intersecting given circle, intersect at the point Q (see figure). Find the angle Q if the inscribed angles and are based on arcs of a circle, the degree values of which are 60 and 0, respectively. Give your answer in degrees. 8. The line y 6x 3 = is tangent to the graph of the function y = x 5x + x 5. Find the abscissa of the point of contact. 9. Two edges of a cuboid coming out of the same vertex are equal to 7 and. The volume of the box is 11. Find the third edge of the box that comes out of the same vertex. O A B 6 B 1 C 1 A 1 1 x C Q MATHEMATICS, grade 11 Option 8, April 015 Part II The answer to the tasks must be an integer or a final decimal fraction. The answer should be written in the answer sheet 1 to the right of the number of the task being performed, starting from the first cell. Write each number, minus sign and comma in a separate box in accordance with the samples given in the form. Units of measurement are not required. 10. Find the value of the expression log6 90 log6. To obtain an enlarged image of a light bulb on the screen, a converging lens with a main focal length f \u003d 50 cm. The distance d 1 from the lens to the bulb can vary from 60 to 80 cm, and the distance d from the lens to the screen ranges from 150 to 175 cm. The image on the screen will be clear if the + = ratio is met. Specify on which d1 d f the shortest distance a light bulb can be placed from the lens so that its image on the screen is clear. Express your answer in centimeters. 1. Find the volume of a regular triangular pyramid whose sides of the base are 11, and the height is Mixed a certain amount of a 16% solution of a certain substance with the same amount of an 18% solution of this substance. What percentage is the concentration of the resulting solution? Find the minimum point of the function y = x + 5x + 7x 5. A S C B
17 MATHEMATICS, grade 11 Option 9, April 015 Regional diagnostic work in MATHEMATICS OPTION 9 Instructions for performing work 90 minutes are given to complete the regional diagnostic work in mathematics. The work consists of two parts, including 14 tasks. Part 1 contains 9 tasks (tasks 1-9) of the basic level of complexity, checking the availability of practical mathematical knowledge and skills. The part contains 5 tasks (tasks 10-14) of advanced levels based on the material of the secondary school mathematics course, checking the level of specialized mathematical training. The answer to each of tasks 1-14 is an integer or a final decimal fraction. All USE forms are filled in with bright black ink. You can use a gel, capillary or fountain pen. When completing assignments, you can use a draft. Please note that entries in the draft will not be taken into account when evaluating the work. We advise you to complete the tasks in the order in which they are given. To save time, skip the task that you can't complete right away and move on to the next one. If after completing all the work you have time left, you can return to the missed tasks. The points you get for completed tasks are summed up. Try to complete as many tasks as possible and score the most points. We wish you success! Part I The answers to tasks 1-9 must be an integer or a final decimal fraction. The answer should be written in the answer sheet 1 to the right of the number of the task being performed, starting from the first cell. Write each number, minus sign and comma in a separate box in accordance with the samples given in the form. Units of measurement are not required. 1. For painting 1 square. m ceiling requires 160 g of paint. The paint is sold in cans per kg. What is the smallest number of cans of paint you need to buy to paint a ceiling of 54 square meters. m? MATHEMATICS, Grade 11 Option 9, April 015. In the figure, bold dots show the daily amount of precipitation that fell in Tomsk from January 8 to January 4, 005. The dates of the month are indicated horizontally, and the amount of precipitation on the corresponding day, in millimeters, is indicated vertically. For clarity, bold dots in the figure are connected by a line. Determine from the figure what the greatest amount of precipitation fell in the period from 13 to 0 January. Give your answer in millimeters. 3. The table shows the average prices (in rubles) for some basic foodstuffs in three Russian cities (as of the beginning of 010). Name of product Petrozavodsk Pavlovsk Tver Wheat bread (loaf) Milk (1 liter) Potatoes (1 kg) Cheese (1 kg) Meat (beef) Sunflower oil (1 liter) wheat bread, kg of beef, 1 liter of sunflower oil. In response, write down the cost of this set of products in this city (in rubles).
18 MATHEMATICS, Grade 11 Option 9, April The points O(0, 0), (6,), (0, 6) and are the vertices of the parallelogram. Find the ordinate of the point. y 6 5. In a random experiment, three dice. Find the probability of getting 6 in total. Round the result to the nearest hundredth. 6. Solve the equation 4 = 16 4 x Find the acute angle between the bisectors of the acute angles of a right triangle. Give your answer in degrees. 8. The line y 6x 4 3 = + is tangent to the graph of the function y = x 3x + 9x + 3. Find the abscissa of the point of contact. 9. In a regular triangular pyramid S, the point L is the middle of the edge, S is the vertex. It is known that = 5, and the lateral surface area is 180. Find the length of the segment SL. O A L 6 1 S B x 1 C MATHEMATICS, Grade 11 Option 9, April 015 Part II The answer to the tasks must be an integer or a final decimal fraction. The answer should be written in the answer sheet 1 to the right of the number of the task being performed, starting from the first cell. Write each number, minus sign and comma in a separate box in accordance with the samples given in the form. Units of measurement are not required. 10. Find tg() π α if tgα = 11. According to Ohm's law for complete chain current strength, measured in amperes, is equal to ε I =, where ε source emf(in volts), r=1 ohm its internal R + r resistance, R circuit resistance (in ohms). At what least resistance circuit current strength will be no more than 0% of the short circuit current strength ε I kz =? (Express your answer in Ohms.) r 1. The base of the pyramid is a rectangle with sides 3 and 4. Its volume is 16. Find the height of this pyramid. 13. Two cars left point to point 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 of 4 km/h, and the second half of the way at a speed 8 km/h greater than the speed of the first, as a result of which he arrived at B at the same time as the first car. Find the speed of the first car. Give your answer in km/h Find the maximum point of the function y = (x 15x + 15) e x
19 MATHEMATICS, grade 11 Option 10, April 015 Regional diagnostic work in MATHEMATICS OPTION 10 Instructions for performing work 90 minutes are given to complete the regional diagnostic work in mathematics. The work consists of two parts, including 14 tasks. Part 1 contains 9 tasks (tasks 1-9) of the basic level of complexity, checking the availability of practical mathematical knowledge and skills. The part contains 5 tasks (tasks 10-14) of advanced levels based on the material of the secondary school mathematics course, checking the level of specialized mathematical training. The answer to each of tasks 1-14 is an integer or a final decimal fraction. All USE forms are filled in with bright black ink. You can use a gel, capillary or fountain pen. When completing assignments, you can use a draft. Please note that entries in the draft will not be taken into account when evaluating the work. We advise you to complete the tasks in the order in which they are given. To save time, skip the task that you can't complete right away and move on to the next one. If after completing all the work you have time left, you can return to the missed tasks. The points you get for completed tasks are summed up. Try to complete as many tasks as possible and score the most points. We wish you success! Part I The answers to tasks 1-9 must be an integer or a final decimal fraction. The answer should be written in the answer sheet 1 to the right of the number of the task being performed, starting from the first cell. Write each number, minus sign and comma in a separate box in accordance with the samples given in the form. Units of measurement are not required. 1. There are 50 sheets of A4 paper in a pack. 1100 sheets are consumed per week in the office. What is the smallest amount of reams of paper that will last 4 weeks? MATHEMATICS, grade 11 Option 10, April 015. During chemical reaction amount starting material(reagent), which has not yet reacted, gradually decreases over time. In the figure, this dependence is represented by a graph. The abscissa shows the time in minutes that has elapsed since the start of the reaction, the ordinate shows the mass of the remaining reagent that has not yet reacted (in grams). Determine from the graph how many grams of the reagent reacted in three minutes? 3. For a group of foreign guests, it is required to buy 10 guidebooks. The necessary guides are found in three internet-shops. Terms of purchase and delivery are given in the table. Online store Price of one guide (rub.) Delivery cost (rub.) Additional conditions A No B C Delivery is free if the order amount exceeds 3000 rubles. Delivery is free if the order amount exceeds 500 rubles. Determine which store total amount purchases including delivery will be the smallest. In response, write down the smallest amount in rubles.
20 MATHEMATICS, Grade 11 Option 10, April What radius should be the circle centered at the point P (9, 8) so that it touches the y-axis? 5. Before the start of a volleyball match, the team captains draw fair lots to determine which team will start the ball game. The Stator team takes turns playing with the Rotor, Motor and Starter teams. Find the probability that Stator will start only the first and last games. 6. Solve the equation () log 5x 1 + = log 8 7. In a right triangle, the angle between the height H and the bisector N drawn from the vertex of the right angle is 1. Find the smaller angle given triangle. Give your answer in degrees. 8. The line y = 7x + 11 is parallel to the tangent to the graph of the function y = x + 8x + 6. Find the abscissa of the point of contact. 9. The area of the lateral surface of the cylinder is 36π, and the height is 4. Find the diameter of the base. N N MATHEMATICS, grade 11 Option 10, April 015 Part II The answer to the tasks must be an integer or a final decimal fraction. The answer should be written in the answer sheet 1 to the right of the number of the task being performed, starting from the first cell. Write each number, minus sign and comma in a separate box in accordance with the samples given in the form. Units of measurement do not need to be written.,1 0.6 10. Find the value of the expression 0. To support the canopy, it is planned to use a cylindrical column. The pressure P (in pascals) exerted by the canopy and the column on the support is determined by 4mg according to the formula P =, where m=1350 kg total weight canopy and columns, diameter π of the column (in meters). Counting acceleration free fall g=10 m/s and π=3, determine the smallest possible diameter of the column if the pressure exerted on the support should not exceed Pa. Express your answer in meters. 1. In a regular quadrangular pyramid, the height is 8, the side edge is 10. Find its volume. 13. In a vessel containing 6 liters of 11% aqueous solution some substance, add 5 liters of water. What percentage is the concentration of the resulting solution? Find the minimum point of the function y = (x + 1) e x. A B S C
21 MATHEMATICS, grade 11 Option 11, April 015 Regional diagnostic work in MATHEMATICS OPTION 11 Instructions for performing work 90 minutes are given to complete the regional diagnostic work in mathematics. The work consists of two parts, including 14 tasks. Part 1 contains 9 tasks (tasks 1-9) of the basic level of complexity, checking the availability of practical mathematical knowledge and skills. The part contains 5 tasks (tasks 10-14) of advanced levels based on the material of the secondary school mathematics course, checking the level of specialized mathematical training. The answer to each of tasks 1-14 is an integer or a final decimal fraction. All USE forms are filled in with bright black ink. You can use a gel, capillary or fountain pen. When completing assignments, you can use a draft. Please note that entries in the draft will not be taken into account when evaluating the work. We advise you to complete the tasks in the order in which they are given. To save time, skip the task that you can't complete right away and move on to the next one. If after completing all the work you have time left, you can return to the missed tasks. The points you get for completed tasks are summed up. Try to complete as many tasks as possible and score the most points. We wish you success! Part I The answers to tasks 1-9 must be an integer or a final decimal fraction. The answer should be written in the answer sheet 1 to the right of the number of the task being performed, starting from the first cell. Write each number, minus sign and comma in a separate box in accordance with the samples given in the form. Units of measurement are not required. MATHEMATICS, Grade 11 Option 11, April 015. In the figure, bold dots show the price of oil at the close of exchange trading on all business days from 17 to 31 August 004. The dates of the month are indicated horizontally, the price of a barrel of oil in US dollars is indicated vertically. For clarity, bold dots in the figure are connected by a line. Based on the figure, determine the lowest oil price at the close of trading in the specified period (in US dollars per barrel). 3. For a group of foreign guests, it is required to buy 30 guidebooks. The necessary guides were found in three online stores. The price of the guide and the terms of delivery of the entire purchase are shown in the table. Internet shop Price of one guidebook (rub.) Delivery cost (rub.) Additional conditions Alpha no 1. 59 rolls of wallpaper are required to renovate an apartment. How many packs of wallpaper glue do you need to buy if one pack of glue is designed for 6 rolls? Beta Vector Delivery is free if the order amount exceeds 8000 rubles. Delivery is free if the order amount exceeds 7500 rubles. How much will the cheapest purchase option with delivery cost?
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USE SOLUTIONS IN MATHEMATICS - 2013
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You can put a link to this page.Our system of testing and preparation for the exam I DECIDE the Unified State Examination of the Russian Federation.
From 2001 to 2009, an experiment began in Russia to combine final exams from schools with entrance exams to higher educational establishments. In 2009, this experiment was completed, and since then the unified state exam has become the main form of control of school preparation.
In 2010, the old exam writing team was replaced by a new one. Together with the developers, the structure of the exam has also changed: the number of tasks has decreased, the number of geometric problems, a task of the Olympiad type appeared.
An important innovation was the preparation of an open bank of examination tasks, in which the developers placed about 75,000 tasks. No one can solve this abyss of problems, but this is not necessary. In fact, the main types of tasks are represented by the so-called prototypes, there are about 2400 of them. All other tasks are derived from them using computer cloning; they differ from prototypes only in specific numerical data.
Continuing, we present to your attention the solutions to all the prototype exam tasks that exist in open jar. After each prototype, a list of clone tasks compiled on its basis for independent exercises is given.
