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Mathematics, matura

Topic revision, sample questions with explanations and tests. The questions are our own; official papers are on the CKE website.

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Date: 5 May 2027, 9:00
Working time: 180 min (basic level)
CKE trial paper: 18 December 2026

Standard time; it can be longer with special arrangements. CKE: komunikat o harmonogramie egzaminów w 2027 r. (20 sierpnia 2026) · CKE: informacja dyrektora o arkuszach przed egzaminami w 2027 r. (30 września 2026). Data checked: 2026-11-04.

Topics and revision

Real numbers, powers and roots

A power with a rational exponent: a^(m/n) = ⁿ√(aᵐ) for a > 0, e.g. 8^(2/3) = (∛8)² = 4. A negative exponent: a^(−n) = 1/aⁿ.
Rules of powers: aʳ · aˢ = aʳ⁺ˢ, aʳ : aˢ = aʳ⁻ˢ, (aʳ)ˢ = aʳ·ˢ.
A root of odd degree also exists for a negative number: ∛(−8) = −2. A square root is never negative: √((−3)²) = 3.
Monotonicity of powers: if a > 1 and x < y, then aˣ < aʸ; if 0 < a < 1 and x < y, then aˣ > aʸ.
The absolute value |x − a| is the distance of x from a on the number line. The equation |x − a| = c (c > 0) has solutions x = a − c and x = a + c.
A round bracket excludes the end of an interval and a square bracket includes it: 5 belongs to (−3, 5], but −3 does not.

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Logarithms

Definition: logₐ b = c if and only if aᶜ = b (a > 0, a ≠ 1, b > 0).
logₐ a = 1 and logₐ 1 = 0. The notation log b means the logarithm to base 10.
Logarithm of a product: logₐ (x · y) = logₐ x + logₐ y.
Logarithm of a quotient: logₐ (x / y) = logₐ x − logₐ y.
Logarithm of a power: logₐ xʳ = r · logₐ x.

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Percentages, deposits and loans

An increase of p%: multiply by (1 + p/100). A decrease of p%: multiply by (1 − p/100).
Compound interest: after n compounding periods Kₙ = K₀ · (1 + p/100)ⁿ.
Interest in a later period is also charged on earlier interest, so growth is faster than with simple addition.
A percentage point is a difference of percentages: a rise from 4% to 5% is 1 percentage point, but 25% of the starting value.
Returning to a value after a rise: after a 25% rise you must reduce by 20%, because 1.25 · 0.8 = 1.

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Divisibility and simple proofs

An even number: 2k, an odd number: 2k + 1, where k is an integer. A number leaving remainder r when divided by m: m · k + r.
Of two consecutive integers one is even, and of three consecutive integers one is divisible by 3.
Hence n(n + 1) is even, and (n − 1)n(n + 1) = n³ − n is divisible by 6.
The remainder of a product or sum depends only on the remainders of its factors or terms.
An example is not a proof: one counterexample disproves a statement “for every n”, but even many examples do not confirm it.

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Algebraic expressions

Short multiplication formulas: (a + b)² = a² + 2ab + b²; (a − b)² = a² − 2ab + b²; a² − b² = (a − b)(a + b).
(a + b)² is not a² + b²: the term 2ab is missing.
Taking out a common factor: 3x²y − 6xy² = 3xy(x − 2y).
A rational expression makes sense when the denominator is not 0. Only factors can be cancelled, not terms.
Dividing rational expressions means multiplying by the reciprocal; factorise the numerators and denominators first.

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Equations and inequalities

Multiplying or dividing an inequality by a negative number reverses its direction: −2x > 4 gives x < −2.
A linear equation may be an identity (every number is a solution) or a contradiction (no solutions).
Quadratic equation ax² + bx + c = 0: Δ = b² − 4ac. If Δ > 0: x = (−b ± √Δ)/(2a). If Δ = 0: x = −b/(2a). If Δ < 0: no real solutions.
Solve a quadratic inequality using the zeros and a sketch of the parabola: if a > 0, the negative values lie between the zeros.
An equation in factored form: a product equals 0 when one of its factors equals 0.
Do not divide both sides by an expression with x without checking its sign and zero: solutions can be lost. With fractions, state the assumption: the denominator is not 0.

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Systems of equations

Solve a system of two linear equations by substitution or by elimination.
A determined system has one solution: the lines intersect. An indeterminate system has infinitely many solutions: the lines coincide. An inconsistent system has no solutions: the lines are parallel and different.
Always check the solution in both equations.
In a word problem, name two unknowns and write two independent conditions.

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Functions and their properties

A function assigns exactly one value to each argument in its domain. The domain is the set of arguments; the range is the set of values of the function.
A zero of a function is an argument for which f(x) = 0. On the graph: the point where it crosses the x-axis.
The graph of y = f(x − a) is the graph of f shifted a to the right (for a > 0). The graph of y = f(x) + b is the graph of f shifted b up (for b > 0).
The function f(x) = a/x describes inversely proportional quantities: the product x · f(x) is constant and equals a.
The exponential function f(x) = aˣ (a > 0, a ≠ 1) is increasing for a > 1 and decreasing for 0 < a < 1; its values are positive.
The logarithmic function f(x) = logₐ x has domain (0, +∞) and the zero x = 1.

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The linear function

Formula: f(x) = ax + b. The coefficient a is the slope, and b is the y-coordinate of the point where the graph crosses the y-axis.
a > 0: increasing; a < 0: decreasing; a = 0: constant.
The zero (for a ≠ 0): x = −b/a.
A line through two points: a = (y₂ − y₁)/(x₂ − x₁), then find b by substituting one point.
In applications a is the change per unit (e.g. the cost per kilometre) and b is the starting value (e.g. the base fee).

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The quadratic function and optimisation

General form: f(x) = ax² + bx + c. Vertex form: f(x) = a(x − p)² + q, where W = (p, q) is the vertex. Factored form: f(x) = a(x − x₁)(x − x₂).
Vertex: p = −b/(2a), q = f(p) = −Δ/(4a).
a > 0: opens upwards, q is the least value. a < 0: opens downwards, q is the greatest value.
Look for the greatest and least values on a closed interval among the values at the ends of the interval and at p, if p is in the interval.
The number of zeros depends on Δ: Δ > 0 two, Δ = 0 one, Δ < 0 none.
Optimisation task: write the quantity as a quadratic function of one variable, set the domain, find the vertex and answer the question.

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Sequences

Arithmetic sequence: aₙ = a₁ + (n − 1)r; sum of the first n terms: Sₙ = (a₁ + aₙ)/2 · n.
Geometric sequence: aₙ = a₁ · qⁿ⁻¹; sum: Sₙ = a₁ · (1 − qⁿ)/(1 − q) for q ≠ 1.
Three consecutive terms: arithmetic if 2b = a + c; geometric if b² = a · c.
An increasing sequence: aₙ₊₁ > aₙ for every n. Check the sign of the difference aₙ₊₁ − aₙ.
A term from the sum: aₙ = Sₙ − Sₙ₋₁ for n ≥ 2.
A recursive sequence is computed term by term from the given start.

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Trigonometry

In a right-angled triangle: sin α = opposite leg / hypotenuse; cos α = adjacent leg / hypotenuse; tg α = opposite leg / adjacent leg (tg is the Polish notation for tan).
Values: sin 30° = cos 60° = 1/2; sin 45° = cos 45° = √2/2; sin 60° = cos 30° = √3/2; tg 30° = √3/3; tg 45° = 1; tg 60° = √3.
Identities: sin²α + cos²α = 1 and tg α = sin α / cos α.
For an obtuse angle: sin α > 0, cos α < 0, tg α < 0, and sin(180° − α) = sin α, cos(180° − α) = −cos α.
Cosine rule: c² = a² + b² − 2ab · cos γ.
Area of a triangle: A = (1/2) · a · b · sin γ, where γ is the angle between sides a and b.

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Plane geometry

An inscribed angle is half the central angle on the same arc. An inscribed angle on a diameter is a right angle.
A tangent is perpendicular to the radius at the point of tangency. Tangent segments drawn from one point are equal.
A sector with angle α: area = (α/360°) · πr², arc length = (α/360°) · 2πr.
Thales' theorem: parallel lines cut proportional segments on the arms of an angle.
Similar figures with scale k: perimeters are in the ratio k, and areas in the ratio k².
Type of triangle with longest side c: a² + b² > c² acute, a² + b² = c² right-angled, a² + b² < c² obtuse.
Special points: incentre (angle bisectors), circumcentre (perpendicular bisectors), centroid (medians, ratio 2 : 1), orthocentre (heights). In a right-angled triangle R = c/2 and r = (a + b − c)/2.
Area of a regular hexagon with side a: 6 · a²√3/4.

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Coordinate geometry

Distance between points: |AB| = √((x₂ − x₁)² + (y₂ − y₁)²). Midpoint: ((x₁ + x₂)/2, (y₁ + y₂)/2).
Lines y = a₁x + b₁ and y = a₂x + b₂: parallel if a₁ = a₂; perpendicular if a₁ · a₂ = −1.
The common point of two lines is the solution of the system of their equations.
A circle with centre (a, b) and radius r: (x − a)² + (y − b)² = r².
Symmetries of a point (x, y): in the x-axis: (x, −y); in the y-axis: (−x, y); in the origin: (−x, −y).
In a parallelogram ABCD the diagonals bisect each other: A + C = B + D (by coordinates).

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Solid geometry

Prism: V = B · H. Pyramid: V = (1/3) · B · H, where B is the base area.
Cylinder: V = πr²H, L = 2πrH. Cone: V = (1/3)πr²H, L = πrl, where l² = r² + H². Sphere: V = (4/3)πr³, S = 4πr².
Diagonal of a cube with edge a: a√3. Diagonal of a cuboid: √(a² + b² + c²).
The angle between a line and a plane is the angle between the line and its projection on the plane, e.g. the angle of a space diagonal with the base diagonal.
The angle between a lateral face and the base of a regular square pyramid: tg α = H : (a/2).
Similar solids with scale k: surface areas in the ratio k², volumes in the ratio k³.
Skew lines do not meet and are not parallel; they may be perpendicular, e.g. the edges AB and CG of a cube.

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Combinatorics

Multiplication rule: when a choice is made in successive stages, multiply the numbers of possibilities in the stages.
Addition rule: when cases exclude each other, add their numbers.
A three-digit number: the first digit cannot be zero, so it has 9 possibilities.
When a condition affects several places, start with the most restricted place, e.g. the last digit for evenness, and split the task into cases.

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Probability and statistics

Classical model: P(A) = number of favourable outcomes : number of all outcomes, when all are equally likely.
The complementary event: P(A′) = 1 − P(A). It is often easier to count the complement.
Rolling two dice has 6 · 6 = 36 outcomes, and the pairs are ordered: (1, 2) and (2, 1) are two outcomes.
Weighted mean: (w₁x₁ + ... + wₙxₙ) : (w₁ + ... + wₙ).
Median: the middle value of the ordered data; for an even amount of data, the mean of the two middle values.
The mode is the most frequent value.

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Sample questions

Question 1 · Real numbers, powers and roots

The number 27^(2/3) · 3^(−1) equals:

Answer: 3

27^(2/3) = (∛27)² = 3² = 9, and 3^(−1) = 1/3. The product: 9 · 1/3 = 3.

Question 2 · Logarithms

Calculate log₃ 81.

Answer: 4

3⁴ = 81, so log₃ 81 = 4.

Question 3 · Percentages, deposits and loans

5000 zł was placed on a one-year deposit at an interest rate of 4% per year. How many złoty will be on the deposit after a year? Ignore tax.

Answer: 5200

Interest: 4% · 5000 = 200 zł. Total: 5000 + 200 = 5200 zł.

Question 4 · Algebraic expressions

The expression (x − 3)² equals:

x² − 6x + 9
x² − 9
x² + 9
x² − 3x + 9

Answer: x² − 6x + 9

(a − b)² = a² − 2ab + b², so x² − 2 · x · 3 + 9 = x² − 6x + 9. The form x² − 9 leaves out the middle term.

Question 5 · Equations and inequalities

The solution set of the inequality 3x − 5 ≤ 7 is the interval:

(−∞, 4]
(−∞, 4)
[4, +∞)
(−∞, 2/3]

Answer: (−∞, 4]

3x ≤ 12, so x ≤ 4. The inequality is not strict, so 4 belongs to the set, hence the square bracket.

Official CKE papers

The page for each year has the papers and marking rules for every subject. We only link to them; CKE publishes the materials. These are Formula 2023 papers.

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