Ukraine 10 клас Mathematics (profile level)
Chapters: 9
1. Algebra: functions, polynomials, equations and inequalities (36 h)
Sets · Numerical functions · Inverse functions · Equations and inequalities · Polynomials · Mathematical induction
- Sets: Representation, Types, Subsets, Venn Diagrams and Operations – A set is a well-defined collection of different objects. We write it in roster form {2, 4, 6} or set-builder form {x : x is even}. Sets can be empty, finite, infinite or equal. If every element of B is in A, B is a subset of A (B ⊂ A); a set with n elements has 2ⁿ subsets. Intervals like (a, b) and [a, b] are subsets of real numbers. The universal set U holds everything under study. With Venn diagrams we see union A ∪ B, intersection A ∩ B, difference A − B and complement A′ = U − A.
- 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.
- Inverse Functions – An inverse function undoes what a function does. If f sends a to b, then f⁻¹ sends b back to a. To find it, write y = f(x), swap x and y, and solve for y. The graph of f⁻¹ is the mirror image of the graph of f in the line y = x. Only one-to-one functions have an inverse; if a function is not one-to-one, limit its domain first (for example x² with x ≥ 0 has inverse √x).
- Inequalities: Rules, Intervals and Solving Them – An inequality says one amount is bigger or smaller than another, using <, >, ≤ or ≥. On a number line, the smaller number is on the left. You may add or subtract the same number on both sides, and multiply or divide by the same positive number, and the sign stays. If you multiply or divide by a negative number, the sign flips. The answer is usually a whole set of numbers, written as an interval such as (−∞, 4]. A quadratic inequality is solved from its roots and the shape of its graph. |x| < a means −a < x < a. Some inequalities are true for every number, like x² ≥ 0 and the AM–GM inequality.
- Remainder Theorem and Factor Theorem – When a polynomial p(x) is divided by (x − a), the remainder is p(a). So you can find the remainder without long division: just put x = a. If p(a) = 0, the remainder is 0 and (x − a) is a factor of p(x). This is the factor theorem.
- Proof by Mathematical Induction – Mathematical induction proves that a statement P(n) is true for every natural number n. Step 1 (base case): show P(1) is true. Step 2 (inductive step): assume P(k) is true for some k, and use it to show P(k + 1) is true. Then, like a line of dominoes, P(1) makes P(2) true, P(2) makes P(3) true, and so on for ever.
2. Algebra: power function (30 h)
nth roots · Rational exponents and power functions · Irrational equations and inequalities
- 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.
- Irrational Equations (Square Root Equations) – An irrational equation has the unknown inside a root. To solve it, put the root alone on one side, square both sides, solve the new equation, and then check every answer in the original. Squaring can add false roots, so the check is a must. Irrational inequalities need the domain and the sign of the other side.
3. Algebra: trigonometric functions (34 h)
Trigonometric functions of a number · Trigonometric formulas
- 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.
4. Algebra: trigonometric equations and inequalities (32 h)
Inverse trigonometric functions · Trigonometric equations and inequalities
- Inverse Trigonometric Functions (Class 12) – sin x, cos x and the other trig functions repeat, so they are not one-one and have no inverse on all of R. We cut each one to a piece (the principal value branch) where it is one-one and onto. On that piece it has an inverse: y = sin⁻¹x means sin y = x with y in [−π/2, π/2]. The graph of an inverse is the mirror image of the branch in the line y = x. Principal ranges: sin⁻¹ [−π/2, π/2], cos⁻¹ [0, π], tan⁻¹ (−π/2, π/2), cot⁻¹ (0, π), sec⁻¹ [0, π] − {π/2}, cosec⁻¹ [−π/2, π/2] − {0}.
5. Algebra: limits, continuity and derivative (54 h)
Limit and continuity · Derivative · Investigating functions
- Limits (Class 11): What a Function Gets Close To – A limit tells us which number f(x) gets close to when x gets close to a point. We walk towards the point from the left and from the right. If both sides reach the same number, that number is the limit. You will learn limits of polynomial, rational, trigonometric, exponential and log functions, with the standard results you must know.
- 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.
6. Geometry: introduction to solid geometry (15 h)
Axioms of solid geometry · First look at polyhedra
- 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.
- Polyhedra: Prisms, Pyramids and the Platonic Solids – A polyhedron is a closed solid whose surface is made only of flat polygons (faces). Faces meet along edges, and edges meet at vertices. Prisms have two equal parallel bases joined by parallelograms; pyramids have one base and triangles meeting at an apex; a frustum is a pyramid with its top cut off by a plane parallel to the base. For every convex polyhedron, Euler's formula holds: V − E + F = 2. There are exactly five regular (Platonic) polyhedra.
7. Geometry: parallelism in space (24 h)
Relative position of lines and planes · Parallel projection and sections
- 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).
- 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.
8. Geometry: perpendicularity in space (26 h)
Perpendicular lines and planes · Angles and distances · Orthogonal projection
- 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).
- 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.
9. Geometry: coordinates, vectors and transformations in space (22 h)
Coordinates in space · Vectors in space · Transformations 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.
- Transformations: Slide, Flip, Turn and Resize a Shape – A transformation moves a shape to a new place. Translation slides it, reflection flips it in a mirror line, rotation turns it about a centre, enlargement changes its size from a centre. The first three keep size and shape (congruent image). Enlargement keeps shape but changes size (similar image). Each has a simple coordinate rule.