The video course "Get an A" includes all the topics necessary for the successful passing of the exam in mathematics by 60-65 points. Completely all tasks 1-13 of the Profile USE in mathematics. Also suitable for passing the Basic USE in mathematics. If you want to pass the exam with 90-100 points, you need to solve part 1 in 30 minutes and without mistakes!

Preparation course for the exam for grades 10-11, as well as for teachers. Everything you need to solve part 1 of the exam in mathematics (the first 12 problems) and problem 13 (trigonometry). And this is more than 70 points on the Unified State Examination, and neither a hundred-point student nor a humanist can do without them.

All the necessary theory. Quick solutions, traps and secrets of the exam. All relevant tasks of part 1 from the Bank of FIPI tasks have been analyzed. The course fully complies with the requirements of the USE-2018.

The course contains 5 large topics, 2.5 hours each. Each topic is given from scratch, simply and clearly.

Hundreds of exam tasks. Text problems and probability theory. Simple and easy to remember problem solving algorithms. Geometry. Theory, reference material, analysis of all types of USE tasks. Stereometry. Cunning tricks for solving, useful cheat sheets, development of spatial imagination. Trigonometry from scratch - to task 13. Understanding instead of cramming. Visual explanation of complex concepts. Algebra. Roots, powers and logarithms, function and derivative. Base for solving complex problems of the 2nd part of the exam.

4. The formula for the radius of a circle, which is described about a rectangle through the diagonal of a square:

5. The formula for the radius of a circle, which is described near a rectangle through the diameter of a circle (circumscribed):

6. The formula for the radius of a circle, which is described near a rectangle through the sine of the angle that is adjacent to the diagonal, and the length of the side opposite this angle:

7. The formula for the radius of a circle, which is described about a rectangle in terms of the cosine of the angle that is adjacent to the diagonal, and the length of the side at this angle:

8. The formula for the radius of a circle, which is described near a rectangle through the sine of an acute angle between the diagonals and the area of ​​the rectangle:

Angle between a side and a diagonal of a rectangle.

Formulas for determining the angle between the side and the diagonal of a rectangle:

1. The formula for determining the angle between the side and the diagonal of a rectangle through the diagonal and the side:

2. The formula for determining the angle between the side and the diagonal of a rectangle through the angle between the diagonals:

The angle between the diagonals of the rectangle.

Formulas for determining the angle between the diagonals of a rectangle:

1. The formula for determining the angle between the diagonals of a rectangle through the angle between the side and the diagonal:

β = 2α

2. The formula for determining the angle between the diagonals of a rectangle through the area and the diagonal.

Rectangle. Since the rectangle has two axes of symmetry, its center of gravity is located at the intersection of the axes of symmetry, i.e. at the point of intersection of the diagonals of the rectangle.

Triangle. The center of gravity lies at the point of intersection of its medians. It is known from geometry that the medians of a triangle intersect at one point and divide in a ratio of 1:2 from the base.

A circle. Since the circle has two axes of symmetry, its center of gravity is at the intersection of the axes of symmetry.

Semicircle. The semicircle has one axis of symmetry, then the center of gravity lies on this axis. Another coordinate of the center of gravity is calculated by the formula: .

Many structural elements are made from standard rolled products - angles, I-beams, channels and others. All dimensions, as well as the geometric characteristics of rolled profiles, are tabular data that can be found in the reference literature in standard assortment tables (GOST 8239-89, GOST 8240-89).

Example 1 Determine the position of the center of gravity of the figure shown in the figure.

Decision:

We select the coordinate axes so that the Ox axis passes along the extreme lower overall dimension, and the Oy axis - along the extreme left overall dimension.

We break a complex figure into the minimum number of simple figures:

rectangle 20x10;

triangle 15x10;

circle R=3 cm.

We calculate the area of ​​each simple figure, its coordinates of the center of gravity. The results of the calculations are entered in the table

Answer: C(14.5; 4.5)

Example 2 . Determine the coordinates of the center of gravity of a composite section consisting of a sheet and rolled profiles.

Decision.

We select the coordinate axes, as shown in the figure.

We denote the figures by numbers and write out the necessary data from the table:

We calculate the coordinates of the center of gravity of the figure using the formulas:

Answer: C(0; 10)

Laboratory work No. 1 "Determining the center of gravity of composite flat figures"

Target: Determine the center of gravity of a given flat complex figure by experimental and analytical methods and compare their results.

Work order

Draw in notebooks your flat figure in size, indicating the coordinate axes.

Determine the center of gravity analytically.

1. Break the figure into the minimum number of figures, the centers of gravity of which, we know how to determine.

   Indicate the numbers of areas and the coordinates of the center of gravity of each figure.

   Calculate the coordinates of the center of gravity of each figure.

   Calculate the area of ​​each figure.

   Calculate the coordinates of the center of gravity of the entire figure using the formulas (put the position of the center of gravity on the drawing of the figure):

Installation for experimental determination of the coordinates of the center of gravity by suspension consists of a vertical rack 1 (see fig.) to which the needle is attached 2 . flat figure 3 Made of cardboard, which is easy to pierce a hole. holes BUT and AT pierced at randomly located points (preferably at the most distant distance from each other). A flat figure is hung on a needle, first at a point BUT , and then at the point AT . With the help of a plumb 4 , fixed on the same needle, a vertical line is drawn on the figure with a pencil corresponding to the plumb line. Center of gravity With figure will be located at the intersection of the vertical lines drawn when hanging the figure at points BUT and AT .