Ukraine 10 клас Mathematics (algebra and beginnings of analysis; geometry)
Chapters: 6
1. Algebra: functions, their properties and graphs (15 h)
Numerical functions · nth roots and rational exponents · Power functions
- Relations and Functions: Cartesian Product, Domain, Range, Graphs – An ordered pair (a, b) has a first and a second place, so (1, 2) ≠ (2, 1). The Cartesian product A × B is the set of all ordered pairs (a, b) with a ∈ A and b ∈ B, and n(A × B) = n(A) × n(B). A relation from A to B is any subset of A × B; its domain is the set of first elements and its range the set of second elements, while B is the co-domain. A function is a special relation where every element of A has exactly one image. We study constant, identity, polynomial, rational, modulus, signum, exponential, log and greatest integer functions with their graphs, and add, subtract, multiply and divide functions point by point.
- Radicals and nth Roots – The nth root of a number a is the number that, multiplied by itself n times, gives a. We write it ⁿ√a, and it is the same as a^(1/n). Radicals can be multiplied, divided and simplified by pulling out perfect powers, and a root in a denominator can be removed by rationalising.
- Power Functions: y = a·xⁿ – A power function has the form y = a·xⁿ, where a is a number and n is a fixed exponent. Even whole powers (x², x⁴) make U shapes that are symmetric about the y-axis. Odd whole powers (x³, x⁵) make S shapes that are symmetric about the origin. All y = xⁿ with n > 0 pass through (0, 0) and (1, 1). Negative powers (x⁻¹ = 1/x) have asymptotes and are not defined at x = 0. Fractional powers are roots: x^(1/2) = √x, x^(1/3) = ∛x. The root function is the inverse of the matching power.
2. Algebra: trigonometric functions (18 h)
Angles and radian measure · Trigonometric identities · Graphs of trigonometric functions · Simplest trigonometric equations
- Trigonometric Functions: Radians, Unit Circle, Graphs and Identities – An angle of one radian cuts an arc equal to the radius, so π radians = 180°. On a unit circle, the point at angle x is P = (cos x, sin x), which gives sin²x + cos²x = 1 and extends sine and cosine to every real number. The signs follow 'All, Sin, Tan, Cos' in quadrants I to IV; sin x and cos x repeat every 2π and stay between −1 and 1. Compound-angle formulas such as cos(A + B) = cos A cos B − sin A sin B lead to tan(A + B), cot(A + B), sum-to-product, double-angle and triple-angle identities.
3. Algebra: the derivative and its applications (14 h)
Derivative · Using the derivative
- Derivatives (Class 11): Slope, Rate of Change and Rules – A derivative tells how fast something changes at one moment. On a graph it is the slope of the tangent line. We find it with a limit: take a tiny step h, find the average change, and let h go to 0. Then we learn the quick rules for xⁿ, sin x, cos x, sums, differences, products and quotients.
- Application of Derivatives – The derivative measures how fast one quantity changes compared with another. Its sign tells whether a function goes up (f′ > 0, increasing) or down (f′ < 0, decreasing). Where f′ = 0 the tangent is flat: these critical points may be a local maximum or minimum, checked by the first derivative test (sign change) or the second derivative test (sign of f″). This lets us solve real problems like the biggest box or the cheapest tank.
4. Geometry: parallel lines and planes in space (17 h)
Axioms of solid geometry · Lines and planes in space · Parallel projection
- Solid Geometry: Points, Lines and Planes in Space – Solid geometry studies figures in three dimensions. Three points not on one line fix a plane. Two lines in space can be parallel, intersecting or skew (not in one plane). A line can lie in a plane, cut it, or be parallel to it; it is perpendicular to a plane if it is perpendicular to two intersecting lines of that plane. Angles in space are found by projecting onto a plane and using right triangles.
- Lines and Planes in Space – Two points fix a line; three points not on one line fix a plane. Two lines in space meet, are parallel or are skew. A line lies in a plane, is parallel to it or cuts it at one point; two planes are parallel or meet in a line. A line is perpendicular to a plane if it is perpendicular to two crossing lines of the plane. With normal vector n = (a, b, c) the plane is ax + by + cz = d, a line is r = a + t·u, and the distance from (x₀, y₀, z₀) to the plane is |ax₀ + by₀ + cz₀ − d| ÷ √(a² + b² + c²).
5. Geometry: perpendicular lines and planes (17 h)
Perpendicularity in space · Distances and angles in space
- Lines and Planes in Space – Two points fix a line; three points not on one line fix a plane. Two lines in space meet, are parallel or are skew. A line lies in a plane, is parallel to it or cuts it at one point; two planes are parallel or meet in a line. A line is perpendicular to a plane if it is perpendicular to two crossing lines of the plane. With normal vector n = (a, b, c) the plane is ax + by + cz = d, a line is r = a + t·u, and the distance from (x₀, y₀, z₀) to the plane is |ax₀ + by₀ + cz₀ − d| ÷ √(a² + b² + c²).
6. Geometry: coordinates and vectors in space (10 h)
Coordinates in space · Vectors in space
- Introduction to Three-dimensional Geometry (Class 11) – In space we use three mutually perpendicular axes x, y, z through the origin O. Each pair makes a coordinate plane: XY (z = 0), YZ (x = 0) and ZX (y = 0). The three planes divide space into eight octants, named by the signs of x, y, z. A point P(x, y, z) is reached by moving x along the x-axis, y parallel to the y-axis and z parallel to the z-axis; x, y, z are its distances from the YZ, ZX and XY planes. Points on the x-axis are (x, 0, 0); points on the XY-plane are (x, y, 0). The distance between P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) is PQ = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²).
- Vector Algebra – A vector has a size (magnitude) and a direction. In 3D we write it as a = xî + yĵ + zk̂. Its length is |a| = √(x² + y² + z²). Its direction cosines are l = x/|a|, m = y/|a|, n = z/|a|, and l² + m² + n² = 1. Vectors are added head-to-tail (triangle law) or component by component. ka stretches a by k and flips it if k is negative. The point dividing AB in m : n has position vector (mb + na)/(m + n) inside and (mb − na)/(m − n) outside. Dot product a·b = |a||b|cosθ gives a number and tells the angle and the projection. Cross product a×b = |a||b|sinθ n̂ gives a vector at right angles to both; its length is the area of the parallelogram on a and b.