Hi, my name is Evgeniy, and I have been preparing for the Unified State Exam and Basic State Exam in mathematics for 13 years. In this video, we will analyze the Unified State Exam 2025 version for 100 points. The version is made up of problems that have already been included in the Unified State Exam and from FIPI, so the versions are of the same difficulty level as the real Unified State Exam 👍 LINKS: Download the version: https://vk.com/wall-40691695_101034 VK group: https://vk.com/shkolapifagora Video courses: https://vk.com/market-40691695 How I passed the Unified State Exam: https://vk.com/wall-40691695_66680 Reviews: https://vk.com/wall-40691695_98328 Insta: / shkola_pifagora 🔥 TIMECODES: Start – 00:00 Problem 1 – 00:58 The acute angle B of a right triangle is 66°. Find the angle between the bisector CD and the median CM, drawn from the vertex of the right angle. Give your answer in degrees. Problem 2 – 03:36 The lengths of vectors a ⃗ and b ⃗ are 3 and 5, and the angle between them is 60°. Find the scalar product a ⃗∙b ⃗. Problem 3 – 06:23 Find the volume of the polyhedron whose vertices are points A, D, A_1, B, C, B_1 of the rectangular parallelepiped ABCDA_1 B_1 C_1 D_1, for which AB=3, AD=4, AA_1=5. Problem 4 – 08:30 The probability that student A. will correctly solve more than 9 problems on a math test is 0.63. The probability that A. will correctly solve more than 8 problems is 0.75. Find the probability that A. will correctly solve exactly 9 problems. Problem 5 – 11:14 The room is lit by three lamps. The probability of each lamp burning out within a year is 0.9. The lamps burn out independently of each other. Find the probability that at least one lamp will not burn out within a year. Problem 6 – 15:27 Find the root of the equation 3^log_9⁡(4x+1) =9. Problem 7 – 18:48 Find the value of the expression √(754^2-304^2 ). Problem 8 – 23:21 A material point moves in a straight line according to the law x(t)=1/6 t^3-2t^2+6t+250, where x is the distance from the reference point in meters, t is the time in seconds measured from the start of the motion. At what point in time (in seconds) was its speed equal to 96 m/s? Problem 9 – 25:04 In an adiabatic process for an ideal gas, the law pV^k=6.4∙10^6 Pa∙m^5 is satisfied, where p is the pressure in the gas (in Pa), V is the volume of the gas (in m^3), k=5/3. Find what volume V (in m^3) will the gas occupy at a pressure p equal to 2∙10^5 Pa. Problem 10 – 29:27 The first garden pump pumps 8 liters of water in 2 minutes, the second pump pumps the same volume of water in 7 minutes. How many minutes must these two pumps work together to pump 36 liters of water? Problem 11 – 33:26 The figure shows the graphs of functions of the types f(x)=k/x and g(x)=ax+b, intersecting at points A and B. Find the abscissa of point B. Problem 12 – 39:51 Find the greatest value of the function y=ln⁡(8x)-8x+7 on the interval [1/16;5/16]. Problem 13 – 44:19 a) Solve the equation 16^sin⁡x =(1/4)^(2 sin⁡2x ). b) Indicate the roots of this equation that belong to the interval [2π;7π/2]. Problem 15 – 53:44 Solve the inequality log_0.5⁡(x^3-3x^2-9x+27)≤log_0.25⁡〖(x-3)^4 〗. Analysis of errors 15 – 01:03:10 Problem 16 – 01:15:15 In July 2020, it is planned to take out a loan from a bank for a certain amount. The conditions for its repayment are as follows: - every January, the debt increases by r% compared to the end of the previous year; - from February to June of each year, it is necessary to repay part of the debt in one payment. If you pay 1,464,100 rubles annually, the loan will be fully repaid in 4 years, and if you pay 2,674,100 rubles annually, the loan will be fully repaid in 2 years. Find r. Problem 18 – 01:30:55 For what values ​​of the parameter a does the equation (4x-x-3-a)/(x^2-xa)=0 have exactly 2 different solutions. Problem 19 – 01:48:11 The following operation is performed with a three-digit number: the sum of its digits is subtracted from it, and then the resulting difference is divided by 3. a) Could the number 300 be obtained as a result of such an operation? b) Could the number 151 be obtained as a result of such an operation? c) How many different numbers can be obtained as a result of such an operation from numbers from 100 to 600 inclusive? Problem 17 – 01:57:00 Heights AK and CM are drawn in an acute-angled triangle ABC. Perpendiculars ME and KH are dropped to them from points M and K, respectively. a) Prove that lines EH and AC are parallel. b) Find the ratio of EH to AC if ∠ABC=45°. Problem 14 – 02:11:54 A right prism ABCA_1 B_1 C_1 is given, with an isosceles triangle ABC with base AB as its base. On AB a point P is marked such that AP:PB=3:1. Point Q bisects edge B_1 C_1. Point M bisects edge BC. Through point M a plane α is drawn, perpendicular to PQ. a) Prove that line AB is parallel to plane α. b) Find the ratio in which plane α divides segment PQ, if AA_1=5, AB=12, cos⁡〖∠ABC〗=3/5. #OptionsUSEprofilePythagorasSchool