Mathematics. Preparation for the Unified State Exam 2013. Ed. Lysenko F.F., Kulabukhova S.Yu.

2012 - 416 p.

The manual contains required material for self-preparation for the Unified State Exam in mathematics:
30 new proprietary educational and training tests, compiled according to the Unified State Examination specifications, taking into account the experience of the 2012 exam;
a problem book (more than 1,700 problems), designed for more detailed practice of different types test tasks;
detailed solution one option;
brief theoretical reference book.

The book will allow graduates and applicants, without resorting to additional literature, to obtain the desired result on the Unified State Examination - from the minimum number of points required to pass the exam to the maximum possible, almost 100 points. The publication is addressed to graduates of educational institutions and teachers. The manual is part of the educational and methodological complex “Mathematics. Preparation for the Unified State Exam", including books such as "Mathematics. Reshebnik. Preparation for the Unified State Exam 2013", "Mathematics. Basic level of the Unified State Exam-2013. A manual for “dummies” (B1-B6 and B7-B14), etc.

Format: pdf

Size: 14.8 MB

Download: 14 .12.2018, links removed at the request of the Legion publishing house (see notes e)

To this textbook There are 2 solvers:

Part I. Solutions to variants of educational and training tests.(Chapter I)

Part II. Solutions to a collection of problems for preparing for the Unified State Exam.(Chapter II)

Table of contents
From the authors 8
Brief theoretical reference 11
§ 1. Legend 11
§ 2. Powers and roots 12
§ 3. Module and its properties 13
§ 4. Progressions 14
§ 5. Logarithms 14
§ 6. Probability theory 15
§ 7. Trigonometry 16
§ 8. Polynomials and their roots 20
§ 9. Equations 24
§ 10. Inequalities 26
§ 11. Functions 28
§ 12. Planimetry 41
§ 13. Stereometry 54
Chapter I. Educational and training tests 68
Option No. 1 68
Option No. 2 71
Option No. 3 74
Option No. 4 77
Option No. 5 80
Option No. 6 84
Option No. 7 87
Option No. 8 92
Option No. 9 96
Option No. 10 100
Option No. 11 105
Option No. 12 108
Option No. 13 112
Option No. 14 116
Option No. 15 119
Option No. 16 122
Option No. 17 125
Option No. 18 128
Option No. 19 131
Option No. 20 134
Option No. 21 137
Option No. 22 140
Option No. 23 143
Option No. 24 147
Option No. 25 151
Option No. 26 154
Option No. 27 157
Option No. 28 160
Option No. 29 164
Option No. 30 168
Solution of option No. 17 172
Chapter II. Collection of problems for preparing for Unified State Exam 180
Basic level (part B) 180
§ 1. Algebra and principles of analysis 180
1.1. Expressions and Transformations 180
1.1.1. Power with rational exponent 180
1.1.2. Powers and roots 180
1.1.3. Logarithmic and Exponential Expressions 182
1.1.4. Trigonometric expressions 183
1.1.5. Combination Expressions 185
1.2. Equations. Systems of equations 186
1.2.1. Logarithmic and exponential equations 186
1.2.2. Trigonometric Equations 188
1.2.3. Rational Equations 188
1.2.4. Irrational equations 189
1.3. Functions 191
1.3.1. Increasing, decreasing, extremum of a function (without finding the derivative) 191
1.3.2. Graph of function 192
1.3.3. Derivative of function 211
1.3.4. Antiderivative of function 241
§ 2. Arithmetic and algebra 242
2.1. Word problems 242
2.1.1. Percentages, alloys, mixtures 242
2.1.2. Movement 254
2.1.3. Work, productivity 262
2.1.4. Miscellaneous tasks 267
§ 3. Geometry 290
3.1. Planimetry 290
3.1.1. Inscribed and circumscribed circle, triangle 290
3.1.2. Right triangle 291
3.1.3. Triangle 291
3.1.4. Parallelogram. Square. Diamond 297
3.1.5. Trapezoid 298
3.1.6. n-gons 301
3.1.7. Circle, tangent, secant 301
3.1.8. Miscellaneous tasks 302
3.2. Stereometry 315
3.2.1. Pyramid 315
3.2.2. Prism. Parallelepiped 318
3.2.3. Cube 323
3.2.4. Cone 325
3.2.5. Cylinder 326
3.2.6. Ball 327
3.2.7. Body combinations 327
Increased level 1 (Cl, C2) 331
§ 4. Algebra and principles of analysis (C1) 331
4.1. Equations. Systems of equations 331
4.1.1. Logarithmic and Exponential Equations 331
4.1.2. Trigonometric Equations 332
4.1.3. Irrational Equations 333
4.1.4. Combined Equations 333
§ 5. Geometry (C2) 340
5.1. Stereometry, 340
5.1.1. Pyramid 340
5.1.2. Prism. Parallelepiped 342
5.1.3. Cube 345
5.1.4. Cone 345
5.1.5. Cylinder 346
5.1.6. Ball 346
5.1.7. Body combinations 347
Advanced level 2 (C3) 347
§ 6. Algebra and principles of analysis 347
6.1. Equations. Systems of equations 347
6.1.1. Irrational equations 347
6.1.2. Combined Equations 347
6.2. Inequalities 347
6.2.1. Logarithmic and exponential inequalities 347
6.2.2. Rational inequalities 351
6.2.3. Irrational inequalities 351
6.2.4. Inequalities containing modulus 352
6.2.5. Combined inequalities 352
Advanced level 3 (C4, C5) 354
§ 7. Algebra and principles of analysis (C5) 354
7.1. Equations. Systems of equations 354
7.1.1. Logarithmic and Exponential Equations 354
7.1.2. Trigonometric Equations 354
7.1.3. Rational Equations 354
7.1.4. Irrational Equations 355
7.1.5. Equations containing modulus 355
7.1.6. Combined Equations 356
7.2. Inequalities 364
7.2.1. Logarithmic and exponential inequalities 364
7.2.2. Rational inequalities 364
7.2.3. Irrational inequalities 364
7.2.4. Inequalities containing modulus 364
7.2.5. Combined inequalities 365
7.3. Functions 366
7.3.1. Function Domain 366
7.3.2. Combined problems 366
§ 8. Arithmetic and algebra (C5) 367
8.1. Progression problems 367
8.1.1. Arithmetic progression 367
8.1.2. Geometric progression 368
§ 9. Geometry (C4) 368
9.1. Planimetry 368
9.1.1. Inscribed and circumscribed circle, triangle 368
9.1.2. Triangle 369
9.1.3. Parallelogram. Square. Rhombus 371
9.1.4. Trapezoid 372
9.1.5. n-gons 376
9.1.6. Circle, tangent, secant 376
9.1.7. Inscribed and circumcircle, quadrilateral 378
Olympiad problems (C6) 379
§ 10. Algebra and principles of analysis 379
10.1. Equations. Systems of equations 379
10.1.1. Trigonometric Equations 379
10.1.2. Combined Equations 379
§ 11. Arithmetic and algebra 379
11.1. Word problems 379
11.1.1. Miscellaneous tasks 379
11.1.2. Decimal notation of the number 380
11.1.3. Divisibility 382
Answers to tests 388
Answers to the collection of problems 395
Literature 409

Table of contents
From the authors 8
Brief theoretical reference
§ 1. Conventions 11
§ 2. Powers and roots 12
§ 3. Module and its properties 13
§ 4. Progressions 14
§ 5. Logarithms 14
§ 6. Probability theory 15
§ 7. Trigonometry 16
§ 8. Polynomials and their roots 20
§ 9. Equations 24
§ 10. Inequalities 26
§eleven. Functions: 28
§ 12. Planimetry 41
§ 13. Stereometry 54
Chapter I. Educational and training tests 67
Option No. 1 67
Option No. 2 71
Option No. 3 75
Option No. 4 78
Option No. 5 82
Option No. 6 86.
Option No. 7 90
Option No. 8 94
Option No. 9 98
Option No. 10 101
Option No. 11 105
Option No. 12 108
Option No. 13 112
Option No. 14 116
Option No. 15 120
Option No. 16 123
Option No. 17 127
Option No. 18 131
Option No. 19 134
Option No. 20 138
Option No. 21 " 142
Option No. 22 146
Option No. 23 150
Option No. 24 153
Option No. 25 157
Option No. 26 160
Option No. 27 165
Option No. 28 168
Option No. 29 171
Option No. 30 174
Solution of option No. 5 178
Chapter II. Collection of problems for preparing for the Unified State Exam 186
Basic level (part B) 186
§ 1. Algebra and principles of analysis 186
1.1. Expressions and Transformations 186
1.1.1. Power with rational exponent 186
1.1.2. Powers and roots 186
1.1.3. Logarithmic and Exponential Expressions 188
1.1.4. Trigonometric expressions 189
1.1.5. Combination Expressions 191
1.2. Equations. Systems of equations 191
1.2.1. Logarithmic and Exponential Equations 191
1.2.2. Trigonometric Equations 193
1.2.3. Rational Equations 193
1.2.4. Irrational equations 194
1.3. Functions 195
1.3.1. Increasing, decreasing, extremum of a function (without finding the derivative) 195
1.3.2. Graph of function 197
1.3.3. Derivative of function 211
1.3.4. Antiderivative of function 238
§ 2. Arithmetic and algebra 239
2.1. Word problems 239
2.1.1. Percentages, alloys, mixtures 239
2.1.2. Movement 249
2.1.3. Work, productivity 256
2.1.4. Miscellaneous tasks 261
§ 3. Geometry 279
3.1. Planimetry 279
3.1.1. Inscribed and circumscribed circle, triangle 279
3.1.2. Right Triangle 281
3.1.3. Triangle 281
3.1.4. Parallelogram. Square. Diamond 286
3.1.5. Trapezoid 288
3.1.6. p-gons 290
3.1.7. Circle, tangent, secant 290
3.1.8. Miscellaneous tasks 291
3.2. Stereometry 301
3.2.1. Pyramid 301
3.2.2. Prism. Parallelepiped 303
3.2.3. Cube 307
3.2.4. Cone 307
3.2.5. Cylinder 308
3.2.6. Body combinations 309
§ 4. Probability theory and statistics 312
4.1. Probability theory 312
4.1.1. Classic definition probabilities 312
Increased level 1 (C1,C2) 318
§ 5. Algebra and principles of analysis (C1) 318
5.1. Equations. Systems of equations 318
5.1.1. Logarithmic and Exponential Equations 318.
5.1.2. Trigonometric Equations 318
5.1.3. Irrational Equations 320
5.1.4. Combined Equations 320
§ 6. Geometry (C2) 327
6.1. Stereometry 327
6.1.1. Pyramid 327
6.1.2. Prism. Parallelepiped 329
6.1.3. Cube 332
6.1.4. Cone 332
6.1.5. Cylinder 333
6.1.6. Ball 333
6.1.7. Body combinations 334
Advanced level 2 (NW) 334
§ 7. Algebra and principles of analysis 334
7.1. Equations. Systems of equations 334
7.1.1. Irrational equations 334
7.1.2. Combined Equations 334
7.2. Inequalities 334
7.2.1. Logarithmic and exponential inequalities 334
7.2.2. Rational inequalities 338
7.2.3. Irrational inequalities 338
7.2.4. Inequalities containing modulus 339
7.2.5. Combined inequalities 339
Advanced level 3 (C4, C5) 341
§ 8. Algebra and principles of analysis (C5) 341
8.1. Equations. Systems of equations 341
8.1.1. Logarithmic and Exponential Equations 341
8.1.2. Trigonometric Equations 341
8.1.3. Rational Equations 341
8.1.4. Irrational equations 342
8.1.5. Equations containing modulus 342
8.1.6. Combined Equations 343
8.2. Inequalities 351
8.2.1. Logarithmic and exponential inequalities 351
8.2.2. Rational inequalities 351
8.2.3. Irrational inequalities 351
8.2.4. Inequalities containing modulus 351
8.2.5. Combined inequalities 352
8.3. Functions 353
8.3.1. Function Domain 353
8.3.2. Combined problems 353
§ 9. Arithmetic and algebra (C5) 354
9.1. Progression problems 354
9.1.1. Arithmetic progression 354
9.1.2. Geometric progression 355
§ 10. Geometry (C4) 355
10.1. Planimetry 355
10.1.1. Inscribed and circumscribed circle, triangle 355
10.1.2. Triangle 356
10.1.3. Parallelogram. Square. Rhombus 358
10.1.4. Trapezoid 359
10.1.5. p-gons 363
10.1.6. Circle, tangent, secant 363
10.1.7. Inscribed and circumscribed circle, quadrilateral 365
Olympiad problems (C6) 366
§eleven. Algebra and beginnings of analysis 366
11.1. Equations. Systems of equations 366
11.1.1. Trigonometric Equations 366
11.1.2. Combined Equations 366
§ 12. Arithmetic and algebra 366
12.1. Word problems 366
12.1.1. Miscellaneous tasks 366
12.1.2. Decimal notation of the number 367
12.1.3. Divisibility 369
Answers to tests 375
Answers to the collection of problems 383
Literature 396m

Preface for the teacher

To carry out diagnostic work in mathematics, 8 versions of educational and training tests are offered, designed to prepare for the Unified State Exam. profile level in 2015. The tests are compiled in accordance with regulatory documents (codifiers of content elements and requirements for the level of training of graduates, specifications, demo version control measuring materials and system for assessing examination work), regulating the development of control measuring instruments Unified State Exam materials in mathematics at the profile level in 2015.

165 minutes (2 hours 45 minutes) are allotted to perform diagnostic work. It is recommended to carry out the work in the first half of the 11th grade in accordance with the teaching materials on algebra and the beginnings of analysis by A. G. Mordkovich and the teaching materials on geometry by L. S. Atanasyan.

The result of the work should be correctional work, including individual recommendations to students.

Diagnostic work plan (training test)

№ P/ P

Requirements (skills) tested by diagnostic work tasks

Analyze real numerical data and statistical information; carry out practical calculations using formulas, use estimates and estimates in practical calculations

Describe using functions various real relationships between quantities and interpret their graphs; extract information presented in tables, charts and graphs

Model real situations in the language of probability theory and statistics, calculate the probabilities of events in the simplest cases

Solve rational, irrational, exponential, logarithmic and trigonometric equations and their systems

Solve planimetric problems to find geometric quantities (lengths, angles, areas).

Model real situations in the language of geometry, explore constructed models using geometric concepts and theorems, algebra; decide practical problems related to finding geometric quantities

Determine the value of a function by the value of its argument when in various ways tasks; describe the behavior and properties of a function using a graph; Find from the graph of a function its maximum and minimum values.

Perform computational operations, combining oral and written techniques, find the value of the root of a natural degree, the value of a degree, a logarithm.

Calculate the values ​​of numeric and alphabetic expressions, carry out the necessary substitutions and transformations

Carry out, according to known formulas and rules, the transformation of literal expressions, including powers, radicals, logarithms, trigonometric functions

Describe using functions various real relationships between quantities and interpret their graphs; extract information presented in tables, charts and graphs.

Solve applied problems, including those of a socio-economic and physical nature, to find speed, find the largest and smallest values ​​of quantities

Solve the simplest stereometric problems to findgeometric quantities (lengths, angles, areas), use when solvingstereometric problems planimetric facts and methods

Model real situations in the language of algebra, create equations and inequalities to solve the problem; explore the constructed models in the language of algebraic apparatus

Calculate derivatives and antiderivatives of elementary functions.

Examine functions for monotonicity in the simplest cases, find the largest and smallest values

Solve rational, irrational, exponential, logarithmic and trigonometric equations and their systems.

Solve equations, the simplest systems of equations, using the properties of functions and their graphs; use a graphical method to approximately solve equations and inequalities.

Solve rational, exponential and logarithmic inequalities and their systems

Analyze real numerical data and statistical information; carry out practical calculations using formulas, use estimates and estimates in practical calculations

Option 1

Part 1

Vasya downloads a 30 MB file from the Internet to his computer in 29 seconds. Petya downloads a 28 MB file in 27 seconds, and Misha downloads a 32 MB file in 29 seconds. How many seconds will it take to download a 528 MB file on the computer with the fastest download speed?

The midline and height of the trapezoid are 7 and 2, respectively. Find the area of ​​the trapezoid.

Before the start of the first round of the championship table tennis Participants are randomly divided into gaming pairs using lots. In total, 36 athletes are participating in the championship, including 8 participants from Russia, including Ivan Papayev. Find the probability that in the first round Ivan Papayev will play with any athlete from Russia?

A car whose mass is equal to kg, begins to move with acceleration, which within t seconds remains unchanged, and the distance travels during this time meters. The value of the force (in newtons) applied to the car at this time is . Define longest time after the car starts moving, during which it will cover the specified distance, if it is known that the force F applied to the car is not less than 2720 N. Express your answer in seconds.

the lengths of the edges are known: , , , And .

The first worker spends 12 hours less to produce 493 parts than the second worker to produce 580 parts. It is known that the first worker makes 9 more parts in an hour than the second. How many parts does the first worker make per hour?

Part 2

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

Option 2

Part 1

Vasya downloads a 30 MB file from the Internet to his computer in 29 seconds. Petya downloads a 28 MB file in 26 seconds, and Misha downloads a 32 MB file in 30 seconds. How many seconds will it take to download a 504 MB file on the computer with the fastest download speed?

Find the area of ​​a square if its diagonal is 14.

Before the start of the first round of the chess championship, participants are randomly divided into playing pairs using lots. In total, 26 chess players are participating in the championship, including 11 participants from Russia, including Pyotr Trofimov. Find the probability that in the first round Pyotr Trofimov will play with any chess player from Russia?

In a vessel having a regular shape triangular prism, poured water. The water level reaches 343 cm. At what height will the water level be if it is poured into another similar vessel, whose side of the base is 7 times larger than the first? Express your answer in centimeters.

P , Where D m/s, A , determine the smallest possible diameter of the column if the pressure exerted on the support should not exceed 50,000 Pa. Express your answer in meters.

In a rectangular parallelepiped edge , edge , edge . Dot - middle of the rib , And .

The first worker spends 8 hours less to produce 65 parts than the second worker to produce 117 of the same parts. It is known that the first worker makes 4 more parts in an hour than the second. How many parts does the second worker make per hour?

Part 2

b) Find all the roots of this equation belonging to the segment [-3π;-2π].

17. (19) On December 31, 2014, Pavel took 8,599,000 rubles from the bank on credit at 14per 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 (i.e. increases the debt by 14), then Pavel transfers x rubles to the bank. What should the amount x be for Paul to repay the debt in three equal payments, i.e. for three years?

Option 3

Part 1

Vasya downloads a 30 MB file from the Internet to his computer in 28 seconds. Petya downloads a 28 MB file in 25 seconds, and Misha downloads a 32 MB file in 28 seconds. How many seconds will it take to download a 400 MB file on the computer with the fastest download speed?

The bases of the trapezoid are 4 and 37, the height is 2. Find the area of ​​the trapezoid.

Before the start of the first round of the badminton championship, participants are randomly divided into playing pairs using lots. In total, 26 badminton players are participating in the championship, including 6 participants from Russia, including Nikita Litvinov. Find the probability that Nikita Litvinov will play with any badminton player from Russia in the first round?

Before departure, the locomotive sounded a horn with a frequencyHz A little later, a diesel locomotive approaching the platform sounded its whistle. Due to the Doppler effect, the frequency of the second beepf greater than the first: it depends on the speed of the diesel locomotive according to the law(Hz), wherec - speed of sound in sound (in m/s). A person standing on a platform can distinguish signals by tone if they differ by at least 3 Hz. Determine the minimum speed at which the diesel locomotive approached the platform if the person was able to distinguish the signals andm/s. Express your answer in m/s.

In a rectangular parallelepipedthe lengths of the edges are known:, , . Find the area of ​​the section passing through the vertices, And.

The first worker spends 9 hours less to produce 442 parts than the second worker to produce 546 parts. It is known that the first worker makes 5 more parts in an hour than the second. How many parts does the first worker make per hour?

Find the greatest value

Part 2

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

Option 4

Part 1

Determine from the figure how many volts the voltage will drop during 15 hours of flashlight operation.

Vasya downloads a 30 MB file from the Internet to his computer in 29 seconds. Petya downloads a 28 MB file in 25 seconds, and Misha downloads a 32 MB file in 30 seconds. How many seconds will it take to download a 630 MB file on the computer with the fastest download speed?

The midline and height of the trapezoid are 10 and 4 respectively. Find the area of ​​the trapezoid.

There are 20 teams participating in the World Championship. Using lots, they need to be divided into five groups of four teams each. There are cards with group numbers mixed in the box:

1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5.

Team captains draw one card each. What is the probability that the Chinese team will be in the second group?

It is planned to use a cylindrical column to support the canopy. Pressure P (in pascals) exerted by the canopy and column on the support is determined by the formula , Where kg - total weight canopy and columns,D - column diameter (in meters). Calculating the acceleration of free fall m/s, A , determine the smallest possible diameter of the column if the pressure exerted on the support should not exceed 800,000 Pa. Express your answer in meters.

In a rectangular parallelepiped edge , edge , edge . Dot - middle of the rib . Find the area of ​​the section passing through the points , And .

Part 2

a) Solve the equation:cos2 x + sin 2 x = 0,25

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

16. (17) Solve inequality + .

17. (19) On January 1, 2014, Mikhail took out 1 million rubles on credit from the bank. The payment scheme is as follows: on the 1st of each month the bank credits 2by the remaining amount of the debt (i.e. increases the debt by 2), then Mikhail transfers the payment to the bank. For what minimum number of months can Mikhail take out a loan so that monthly payments are no more than 250 thousand rubles?

Option 5

Part 1

1. When paying for services through a payment terminal, a 3% commission is charged. The terminal accepts amounts that are multiples of 10 rubles. The monthly Internet fee is 600 rubles. What is the minimum amount to put into the receiving device of the terminal so that the account of the company providing Internet services ends up with an amount of at least 600 rubles?

Determine from the graph how many degrees the engine will heat up from the fifth to the eighth minute of warming up.

Vasya downloads a 30 MB file from the Internet to his computer in 26 seconds. Petya downloads a 28 MB file in 26 seconds, and Misha downloads a 32 MB file in 27 seconds. How many seconds will it take to download a 704 MB file on the computer with the fastest download speed?

10 teams are participating in the World Championship. Using lots, they need to be divided into two groups of five teams each. There are cards with group numbers mixed in the box:

1, 1, 1, 1, 1, 2, 2, 2, 2, 2.

Team captains draw one card each. What is the probability that the Brazilian team will be in the second group?

The first worker spends 4 hours less to produce 725 parts than the second worker to produce 783 parts. It is known that the first worker makes 2 more parts per hour than the second. How many parts does the first worker make per hour?

Part 2

a) Solve the equation: 6sin 2 x + 5 sin(π/2 - x) - 2 = 0

b) Find all the roots of this equation belonging to the segment [-5π ; -7 π /2]

Option 6

Part 1

Determine from the graph how many degrees the engine will heat up from the third to the fifth minute of warming up.

1. Vasya downloads a 30 MB file from the Internet to his computer in 27 seconds. Petya downloads a 28 MB file in 25 seconds, and Misha downloads a 32 MB file in 29 seconds. How many seconds will it take to download a 518 MB file on the computer with the fastest download speed?

There are 10 numbers on the phone keypad, from 0 to 9. What is the probability that a randomly pressed number will be even and greater than 3?

Water was poured into a vessel shaped like a regular triangular prism. The water level reaches 20 cm. At what height will the water level be if it is poured into another similar vessel, whose base side is 2 times larger than the first? Express your answer in centimeters.

In a rectangular

parallelepipededge, edge, edge. Dot- middle of the rib. Find the area of ​​the section passing through the points, And.

Part 2

1. Solve inequality (X – 3) ⎹ X – 3 ⎸ - ⎸ X – 1 ⎹ ≥ 0.

2,660,000 rubles. What amount did Sergei take from the bank if he paid off the debt in three equal payments (i.e., paid off the debt in three years)?

Option 7

Part 1

Determine from the figure how many volts the voltage will drop from the 6th to the 30th hour of operation of the flashlight.

Vasya downloads a 30 MB file from the Internet to his computer in 27 seconds. Petya downloads a 28 MB file in 27 seconds, and Misha downloads a 32 MB file in 29 seconds. How many seconds will it take to download a 435 MB file on the computer with the fastest download speed?

The first worker spends 8 hours less to produce 513 parts than the second worker to produce 540 parts. It is known that the first worker makes 7 more parts in an hour than the second. How many parts does the first worker make per hour?

Part 2

Option 8

Part 1

Determine from the figure how many volts the voltage will drop from the 6th to the 62nd hour of operation of the flashlight.

3. Vasya downloads a 30 MB file from the Internet to his computer in 29 seconds. Petya downloads a 28 MB file in 25 seconds, and Misha downloads a 32 MB file in 27 seconds. How many seconds will it take to download a 496 MB file on the computer with the fastest download speed?

Find the diagonal of a square if its area is 968.

What is the probability that a randomly selected natural number Is 67 to 88 divisible by 2?

Water was poured into a vessel shaped like a regular triangular prism. The water level reaches 294 cm. At what height will the water level be if it is poured into another similar vessel, whose base side is 7 times larger than the first? Express your answer in centimeters.