ЦТ Mathematics study guide
ЦТ Mathematics · Work through your exam, one topic at a time.
ЦТ по математике (Централизованное тестирование по учебному предмету «Математика») is a standardized written test used in the Republic of Belarus as one of the results considered when applicants compete for places at universities and other higher-education institutions. It is administered by the Republican Institute for Knowledge Control (РИКЗ). The ЦТ Mathematics study guide has 10 sections and about 2,851 words.
01Numbers and computation: the ladder of number sets, powers, roots, proportions and percentages
Numbers form a ladder: naturals (counting), integers (adding zero and negatives), rationals (fractions p/q with q ≠ 0, including terminating and repeating decimals), irrationals (non-repeating infinite decimals such as √2, π), and the reals that close the ladder. Every computation question tests which rung you are standing on.
264 words, about 1 min02Algebraic expressions, equations, systems and identities
Algebraic competence means recognising a difference of squares, a perfect-square trinomial, a sum or difference of cubes, and grouping patterns before expanding blindly. The identities (a ± b)², a² − b², a³ ± b³ are shortcuts, not decoration.
305 words, about 2 min03Inequalities, systems of inequalities and solution sets on the number line
Two inequalities are equivalent if they have exactly the same solution set. Adding or subtracting a quantity preserves equivalence; multiplying or dividing by a negative number reverses the sign.
299 words, about 1 min04Functions and graphs: domain, range, monotonicity, extrema, transformations
For every function ask four questions in order: domain, range, monotonicity, extrema. Domain restrictions come from denominators (≠ 0), even roots (radicand ≥ 0), logarithms (argument 0) and tan (cos ≠ 0).
310 words, about 2 min05Trigonometry: identities, equations, inequalities, inverse functions
The backbone is sin²x + cos²x = 1, tan x = sin x/cos x, the double-angle formulas (sin 2x = 2 sin x cos x; cos 2x = cos²x − sin²x = 1 − 2sin²x = 2cos²x − 1) and the reduction formulas that convert any angle to a first-quadrant reference angle with the correct sign. Know the exact values at 0, π/6, π/4, π/3, π/2 cold.
310 words, about 2 min06Sequences: arithmetic and geometric progressions
An arithmetic progression has constant difference d: aₙ = a₁ + (n − 1)d and Sₙ = n(a₁ + aₙ)/2 = n(2a₁ + (n − 1)d)/2. A geometric progression has constant ratio q: bₙ = b₁q^(n−1), Sₙ = b₁(qⁿ − 1)/(q − 1) for q ≠ 1, and the infinite sum b₁/(1 − q) exists only when q < 1.
299 words, about 1 min07Planimetry: triangles, quadrilaterals, circles, similarity, areas, coordinates and vectors
Triangle toolkit: angle sum π; an exterior angle equals the sum of the two remote interior angles; the sine rule a/sin A = b/sin B = 2R; the cosine rule a² = b² + c² − 2bc cos A; area = ½ab sin C = ½ah = √(p(p − a)(p − b)(p − c)). Special lines: medians meet at the centroid, which divides each in ratio 2:1; the incenter is equidistant from the sides, the circumcenter from the vertices.
281 words, about 1 min08Stereometry: lines and planes in space, polyhedra, solids of revolution, areas and volumes
In space, two lines may be parallel, intersecting, or skew (neither parallel nor intersecting, non-coplanar). A line is perpendicular to a plane if it is perpendicular to two intersecting lines of that plane.
263 words, about 1 min09Combinatorics, probability theory and statistics
Multiplication principle: if a choice splits into independent stages with m and n options, the total is mn. Permutations count ordered arrangements: P(n, k) = n!/(n − k)!, and arranging all n objects in a row gives n!.
257 words, about 1 min10Applied and word problems: modelling, motion, work, mixtures, optimisation and estimation
Word problems are modelling problems: name the unknown, state the units, write the relation, solve, and check that the answer is meaningful in context (positive, integral where required, inside the stated range). Motion: distance = speed × time.
263 words, about 1 min