The video course “Get an A” includes all the topics necessary for successful passing the Unified State Exam in mathematics for 60-65 points. Completely all tasks 1-13 of the Profile Unified State Exam in mathematics. Also suitable for passing the Basic Unified State Examination in mathematics. If you want to pass the Unified State Exam with 90-100 points, you need to solve part 1 in 30 minutes and without mistakes!
Preparation course for the Unified State Exam for grades 10-11, as well as for teachers. Everything you need to solve Part 1 of the Unified State Exam in mathematics (the first 12 problems) and Problem 13 (trigonometry). And this is more than 70 points on the Unified State Exam, and neither a 100-point student nor a humanities student can do without them.
All the necessary theory. Quick ways solutions, pitfalls and secrets of the Unified State Exam. All current tasks of part 1 from the FIPI Task Bank have been analyzed. The course fully complies with the requirements of the Unified State Exam 2018.
The course contains 5 large topics, 2.5 hours each. Each topic is given from scratch, simply and clearly.
Hundreds of Unified State Exam tasks. Word problems and probability theory. Simple and easy to remember algorithms for solving problems. Geometry. Theory, reference material, analysis of all types of Unified State Examination tasks. Stereometry. Tricky solutions, useful cheat sheets, development of spatial imagination. Trigonometry from scratch to problem 13. Understanding instead of cramming. Clear explanations of complex concepts. Algebra. Roots, powers and logarithms, function and derivative. Basis for solution complex tasks 2 parts of the Unified State Exam.
Definition 1. Prismatic surface
Theorem 1. On parallel sections of a prismatic surface
Definition 2. Perpendicular section of a prismatic surface
Definition 3. Prism
Definition 4. Prism height
Definition 5. Right prism
Theorem 2. The area of the lateral surface of the prism
Parallelepiped:
Definition 6. Parallelepiped
Theorem 3. On the intersection of the diagonals of a parallelepiped
Definition 7. Right parallelepiped
Definition 8. Rectangular parallelepiped
Definition 9. Measurements of a parallelepiped
Definition 10. Cube
Definition 11. Rhombohedron
Theorem 4. On the diagonals of a rectangular parallelepiped
Theorem 5. Volume of a prism
Theorem 6. Volume of a straight prism
Theorem 7. Volume of a rectangular parallelepiped
Prism is a polyhedron whose two faces (bases) lie in parallel planes, and the edges that do not lie in these faces are parallel to each other.
Faces other than the bases are called lateral.
The sides of the side faces and bases are called prism ribs, the ends of the edges are called the vertices of the prism. Lateral ribs edges that do not belong to the bases are called. The union of lateral faces is called lateral surface of the prism, and the union of all faces is called the full surface of the prism. Prism height called the perpendicular dropped from the point of the upper base to the plane of the lower base or the length of this perpendicular. 
Direct prism called a prism whose side ribs are perpendicular to the planes of the bases. 
Correct called a straight prism (Fig. 3), at the base of which lies a regular polygon.
Designations:
l - side rib;
P - base perimeter;
S o - base area;
H - height;
P^ - perpendicular section perimeter;
S b - lateral surface area;
V - volume;
S p is the area of the total surface of the prism.
|
V=SH |
*It is assumed that every two successive planes intersect and that the last plane intersects the first
Theorem 1 . Sections of a prismatic surface by planes parallel to each other (but not parallel to its edges) are equal polygons.
Let ABCDE and A"B"C"D"E" be sections of a prismatic surface by two parallel planes. To make sure that these two polygons are equal, it is enough to show that triangles ABC and A"B"C" are equal and have the same direction of rotation and that the same holds for triangles ABD and A"B"D", ABE and A"B"E". But the corresponding sides of these triangles are parallel (for example, AC is parallel to AC) like the line of intersection of a certain plane with two parallel planes; it follows that these sides are equal (for example, AC is equal to A"C"), like opposite sides of a parallelogram, and that the angles formed by these sides are equal and have the same direction.
Definition 2 . A perpendicular section of a prismatic surface is a section of this surface by a plane perpendicular to its edges. Based on the previous theorem, all perpendicular sections of the same prismatic surface will be equal polygons.
Definition 3
. A prism is a polyhedron bounded by a prismatic surface and two planes parallel to each other (but not parallel to the edges of the prismatic surface)
The faces lying in these last planes are called prism bases; faces belonging to the prismatic surface - side faces; edges of the prismatic surface - side ribs of the prism. By virtue of the previous theorem, the base of the prism is equal polygons. All side faces prisms - parallelograms; all side ribs are equal to each other.
Obviously, if the base of the prism ABCDE and one of the edges AA" in size and direction are given, then it is possible to construct a prism by drawing edges BB", CC", ... equal and parallel to edge AA".
Definition 4 . The height of a prism is the distance between the planes of its bases (HH").
Definition 5
. A prism is called straight if its bases are perpendicular sections of the prismatic surface. In this case, the height of the prism is, of course, its side rib; the side edges will be rectangles.
Prisms can be classified according to the number of side faces, equal number sides of the polygon that serves as its base. Thus, prisms can be triangular, quadrangular, pentagonal, etc.
Theorem 2
. The area of the lateral surface of the prism is equal to the product of the lateral edge and the perimeter of the perpendicular section.
Let ABCDEA"B"C"D"E" be a given prism and abcde its perpendicular section, so that the segments ab, bc, .. are perpendicular to its lateral edges. The face ABA"B" is a parallelogram; its area is equal to the product of the base AA " to a height that coincides with ab; the area of the face ВСВ "С" is equal to the product of the base ВВ" by the height bc, etc. Consequently, the side surface (i.e. the sum of the areas of the side faces) is equal to the product of the side edge, in other words, the total length of the segments AA", ВВ", .., for the amount ab+bc+cd+de+ea.
Polyhedra
The main object of study of stereometry is spatial bodies. Body represents a part of space limited by a certain surface.
Polyhedron is a body whose surface consists of a finite number of flat polygons. A polyhedron is called convex if it is located on one side of the plane of every plane polygon on its surface. The common part of such a plane and the surface of a polyhedron is called edge. The faces of a convex polyhedron are flat convex polygons. The sides of the faces are called edges of the polyhedron, and the vertices are vertices of the polyhedron.
For example, a cube consists of six squares, which are its faces. It contains 12 edges (the sides of the squares) and 8 vertices (the tops of the squares).
The simplest polyhedra are prisms and pyramids, which we will study further.
Prism
Definition and properties of a prism
Prism is a polyhedron consisting of two flat polygons lying in parallel planes combined by parallel translation, and all segments connecting the corresponding points of these polygons. Polygons are called prism bases, and the segments connecting the corresponding vertices of the polygons are lateral edges of the prism.
Prism height is called the distance between the planes of its bases (). A segment connecting two vertices of a prism that do not belong to the same face is called prism diagonal(). The prism is called n-carbon, if its base is an n-gon.
Any prism has the following properties, resulting from the fact that the bases of the prism are combined by parallel translation:
1. The bases of the prism are equal.
2. The lateral edges of the prism are parallel and equal.
The surface of the prism consists of bases and lateral surface. The lateral surface of the prism consists of parallelograms (this follows from the properties of the prism). The area of the lateral surface of a prism is the sum of the areas of the lateral faces.
Straight prism
The prism is called direct, if its lateral edges are perpendicular to the bases. Otherwise the prism is called inclined.
The faces of a right prism are rectangles. The height of a straight prism is equal to its side faces.
Full prism surface is called the sum of the lateral surface area and the areas of the bases.
With the right prism called a right prism with a regular polygon at its base.
Theorem 13.1. The area of the lateral surface of a straight prism is equal to the product of the perimeter and the height of the prism (or, which is the same, by the lateral edge).
Proof. The lateral faces of a right prism are rectangles, the bases of which are the sides of the polygons at the bases of the prism, and the heights are the lateral edges of the prism. Then, by definition, the lateral surface area is:
,
where is the perimeter of the base of a straight prism.
Parallelepiped
If parallelograms lie at the bases of a prism, then it is called parallelepiped. All faces of a parallelepiped are parallelograms. In this case, the opposite faces of the parallelepiped are parallel and equal.
Theorem 13.2. The diagonals of a parallelepiped intersect at one point and are divided in half by the intersection point.
Proof. Consider two arbitrary diagonals, for example, and . Because the faces of a parallelepiped are parallelograms, then and , which means according to To there are two straight lines parallel to the third. In addition, this means that straight lines and lie in the same plane (plane). This plane intersects parallel planes and along parallel lines and . Thus, a quadrilateral is a parallelogram, and by the property of a parallelogram, its diagonals intersect and are divided in half by the intersection point, which was what needed to be proven.
A right parallelepiped whose base is a rectangle is called rectangular parallelepiped. All faces of a rectangular parallelepiped are rectangles. The lengths of the non-parallel edges of a rectangular parallelepiped are called its linear dimensions (dimensions). There are three such sizes (width, height, length).
Theorem 13.3. In a rectangular parallelepiped, the square of any diagonal equal to the sum squares of its three dimensions 
(proved by applying Pythagorean T twice).
A rectangular parallelepiped with all edges equal is called cube.
Tasks
13.1 How many diagonals does it have? n-carbon prism
13.2 In an inclined triangular prism, the distances between the side edges are 37, 13 and 40. Find the distance between the larger side edge and the opposite side edge.
13.3 A plane is drawn through the side of the lower base of a regular triangular prism, intersecting the side faces along segments with an angle between them. Find the angle of inclination of this plane to the base of the prism.
Definition. Prism is a polyhedron, all of whose vertices are located in two parallel planes, and in these same two planes lie two faces of the prism, which are equal polygons with, respectively parallel sides, and all edges not lying in these planes are parallel.
Two equal faces are called prism bases(ABCDE, A 1 B 1 C 1 D 1 E 1).
All other faces of the prism are called side faces(AA 1 B 1 B, BB 1 C 1 C, CC 1 D 1 D, DD 1 E 1 E, EE 1 A 1 A).
All side faces form lateral surface prisms .
All lateral faces of the prism are parallelograms .
The edges that do not lie at the bases are called the lateral edges of the prism ( AA 1, BB 1, CC 1, DD 1, EE 1).
Prism diagonal is a segment whose ends are two vertices of a prism that do not lie on the same face (AD 1).
The length of the segment connecting the bases of the prism and perpendicular to both bases at the same time is called prism height .
Designation:ABCDE A 1 B 1 C 1 D 1 E 1. (First, in the order of traversal, the vertices of one base are indicated, and then, in the same order, the vertices of another; the ends of each side edge are designated by the same letters, only the vertices lying in one base are designated by letters without an index, and in the other - with an index)
The name of the prism is associated with the number of angles in the figure lying at its base, for example, in Figure 1 there is a pentagon at the base, so the prism is called pentagonal prism. But because such a prism has 7 faces, then it heptahedron(2 faces - the bases of the prism, 5 faces - parallelograms, - its side faces)
Among straight prisms, a particular type stands out: regular prisms.
A straight prism is called correct, if its bases are regular polygons.
Parallelepiped
Parallelepiped is a quadrangular prism, at the base of which lies a parallelogram (an inclined parallelepiped). Right parallelepiped- a parallelepiped whose lateral edges are perpendicular to the planes of the base.Rectangular parallelepiped- a right parallelepiped whose base is a rectangle.
Properties and theorems:
Some properties of a parallelepiped are similar to the known properties of a parallelogram. A rectangular parallelepiped having equal dimensions is called cube
.A cube has all equal squares. The square of the diagonal is equal to the sum of the squares of its three dimensions 
,
where d is the diagonal of the square;
a is the side of the square.
An idea of a prism is given by:
- various architectural structures;
- children's toys;
- packaging boxes;
- designer items, etc.
The area of the total and lateral surface of the prism
Total surface area of the prism is the sum of the areas of all its faces Lateral surface area is called the sum of the areas of its lateral faces. The bases of the prism are equal polygons, then their areas are equal. That's whyS full = S side + 2S main,
Where S full- total surface area, S side-lateral surface area, S base- base area
The lateral surface area of a straight prism is equal to the product of the perimeter of the base and the height of the prism.
S side= P basic * h,
Where S side-area of the lateral surface of a straight prism,
P main - perimeter of the base of a straight prism,
h is the height of the straight prism, equal to lateral rib.
Prism volume
The volume of a prism is equal to the product of the area of the base and the height.