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Slovakia Maturita Math study materials
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Slovakia Maturita Math practice questions

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Question 1Numbers and relations (Čísla a vzťahy)

What is the value of |−7| + |3|?

  1. 4
  2. 10 Correct answer.
  3. −4
  4. −10

Explanation

The absolute value of a number is its distance from zero, so |−7| = 7 and |3| = 3. Adding gives 7 + 3 = 10.

Question 2Numbers and relations (Čísla a vzťahy)

Which number is irrational?

  1. 22/7
  2. −3
  3. 0.75
  4. √2 Correct answer.

Explanation

An irrational number cannot be written as a ratio of two integers. The square root of 2 is not a perfect square and has a non-terminating, non-repeating decimal expansion, so it is irrational.

Question 3Numbers and relations (Čísla a vzťahy)

What is the greatest common divisor of 84 and 126?

  1. 7
  2. 42 Correct answer.
  3. 14
  4. 21

Explanation

Factor both numbers: 84 = 2²·3·7 and 126 = 2·3²·7. The GCD takes the lowest power of each shared prime: 2·3·7 = 42.

Question 4Numbers and relations (Čísla a vzťahy)

Which of the following numbers is divisible by 9?

  1. 7 001
  2. 5 128
  3. 3 214
  4. 4 563 Correct answer.

Explanation

A number is divisible by 9 exactly when the sum of its digits is divisible by 9. For 4 563 the digit sum is 4+5+6+3 = 18, which is divisible by 9.

Question 5Numbers and relations (Čísla a vzťahy)

What is 2³ · 2⁴ expressed as a single power?

  1. 2⁷ Correct answer.
  2. 4⁷
  3. 2¹
  4. 2¹²

Explanation

When multiplying powers with the same base, add the exponents: 2³ · 2⁴ = 2^(3+4) = 2⁷.

Question 6Numbers and relations (Čísla a vzťahy)

A jacket costs 80 euros and is discounted by 25%. What is the sale price?

  1. 20 euros
  2. 60 euros Correct answer.
  3. 55 euros
  4. 65 euros

Explanation

A 25% discount removes 0.25 · 80 = 20 euros from the price. Subtracting gives 80 − 20 = 60 euros.

Question 7Numbers and relations (Čísla a vzťahy)

Which interval represents the set of real numbers x with −2 < x ≤ 5?

  1. (−2, 5] Correct answer.
  2. [−2, 5)
  3. (−2, 5)
  4. [−2, 5]

Explanation

A round bracket excludes the endpoint and a square bracket includes it. Since −2 is excluded and 5 is included, the correct interval is (−2, 5].

Question 8Numbers and relations (Čísla a vzťahy)

Which of these numbers is written in correct scientific notation?

  1. 3.45 · 10³ Correct answer.
  2. 0.345 · 10⁵
  3. 34.5 · 10³
  4. 345 · 10¹

Explanation

Scientific notation requires the coefficient to satisfy 1 ≤ a < 10. Only 3.45 lies in this range, so 3.45 · 10³ is the valid form.

Question 9Numbers and relations (Čísla a vzťahy)

What is the value of 5! / 3! ?

  1. 2
  2. 20 Correct answer.
  3. 120
  4. 10

Explanation

Since 5! = 5·4·3·2·1 and 3! = 3·2·1, the common factor 3! cancels, leaving 5·4 = 20.

Question 10Numbers and relations (Čísla a vzťahy)

Simplify the expression (3x²y) · (2xy³).

  1. 6x³y³
  2. 6x³y⁴ Correct answer.
  3. 6x²y³
  4. 5x³y⁴

Explanation

Multiply coefficients: 3·2 = 6. For each variable add the exponents: x^(2+1) = x³ and y^(1+3) = y⁴. The result is 6x³y⁴.

Question 11Numbers and relations (Čísla a vzťahy)

Solve the equation 3(x − 2) = 2x + 5.

  1. x = 7
  2. x = 11 Correct answer.
  3. x = −11
  4. x = 1

Explanation

Expand the left side: 3x − 6 = 2x + 5. Subtract 2x from both sides to get x − 6 = 5, then add 6 to obtain x = 11.

Question 12Numbers and relations (Čísla a vzťahy)

What is the solution set of the inequality 4 − 2x > 10?

  1. x > −3
  2. x > 3
  3. x < −3 Correct answer.
  4. x < 3

Explanation

Subtract 4 from both sides: −2x > 6. Dividing by −2 reverses the inequality sign, giving x < −3.

Question 13Numbers and relations (Čísla a vzťahy)

The equation x² − 6x + 9 = 0 has:

  1. no real roots
  2. two complex conjugate roots
  3. two distinct real roots
  4. one real double root Correct answer.

Explanation

The discriminant is b² − 4ac = 36 − 36 = 0. A zero discriminant means there is exactly one real root, here x = 3, counted twice.

Question 14Numbers and relations (Čísla a vzťahy)

The roots of x² − 5x + 6 = 0 are:

  1. 2 and 3 Correct answer.
  2. 1 and 6
  3. −2 and −3
  4. −1 and −6

Explanation

By Vieta's formulas the roots sum to 5 and multiply to 6. The pair 2 and 3 satisfies both conditions, and indeed (x−2)(x−3) = x² − 5x + 6.

Question 15Numbers and relations (Čísla a vzťahy)

What is the value of log₂ 32?

  1. 4
  2. 5 Correct answer.
  3. 16
  4. 6

Explanation

The logarithm log₂ 32 asks for the exponent to which 2 must be raised to obtain 32. Since 2⁵ = 32, the value is 5.

Question 16Numbers and relations (Čísla a vzťahy)

Simplify log₃ 9 + log₃ 3.

  1. 2
  2. log₃ 12
  3. 3 Correct answer.
  4. log₃ 27

Explanation

log₃ 9 = 2 and log₃ 3 = 1, so the sum is 3. Equivalently, log₃ 9 + log₃ 3 = log₃ 27 = 3.

Question 17Numbers and relations (Čísla a vzťahy)

A geometric sequence has first term 3 and common ratio 2. What is the 6th term?

  1. 192
  2. 96 Correct answer.
  3. 64
  4. 48

Explanation

The n-th term of a geometric sequence is aₙ = a₁ · r^(n−1). Here a₆ = 3 · 2⁵ = 3 · 32 = 96.

Question 18Numbers and relations (Čísla a vzťahy)

In an arithmetic sequence the first term is 4 and the common difference is 3. What is the sum of the first 10 terms?

  1. 160
  2. 175 Correct answer.
  3. 170
  4. 185

Explanation

The sum of an arithmetic sequence is Sₙ = n/2 · (2a₁ + (n−1)d). Here S₁₀ = 5 · (8 + 27) = 5 · 35 = 175.

Question 19Numbers and relations (Čísla a vzťahy)

A sum of 1000 euros is deposited at 2% simple annual interest. What is the total amount after 3 years?

  1. 1120 euros
  2. 1020 euros
  3. 1061.21 euros
  4. 1060 euros Correct answer.

Explanation

Simple interest adds only interest on the principal: I = P·r·t = 1000 · 0.02 · 3 = 60 euros. The total is 1000 + 60 = 1060 euros.

Question 20Numbers and relations (Čísla a vzťahy)

What is 0.00045 written in scientific notation?

  1. 4.5 · 10⁻³
  2. 45 · 10⁻⁵
  3. 0.45 · 10⁻³
  4. 4.5 · 10⁻⁴ Correct answer.

Explanation

Move the decimal point three places right to get 4.5, which requires multiplying by 10⁻³. Thus 0.00045 = 4.5 · 10⁻³? No: 4.5 · 10⁻³ = 0.0045, so the correct exponent is −4, giving 4.5 · 10⁻⁴.

Question 21Numbers and relations (Čísla a vzťahy)

Which of the following is a prime number?

  1. 97 Correct answer.
  2. 87
  3. 51
  4. 91

Explanation

A prime has exactly two positive divisors. The number 97 is not divisible by 2, 3, 5 or 7, so it is prime.

Question 22Numbers and relations (Čísla a vzťahy)

For which value of k does the system of equations 2x + 3y = 7 and kx + 6y = 14 have infinitely many solutions?

  1. k = 4 Correct answer.
  2. k = 2
  3. k = 3
  4. k = 6

Explanation

Infinitely many solutions occur when the two equations are proportional. The second equation is twice the first in the y and constant terms (6 = 2·3, 14 = 2·7), so we need k = 2·2 = 4.

Question 23Numbers and relations (Čísla a vzťahy)

The expression (a + b)² − (a − b)² simplifies to:

  1. 2ab
  2. 2a²
  3. 4ab Correct answer.
  4. 2b²

Explanation

Expanding: (a+b)² = a² + 2ab + b² and (a−b)² = a² − 2ab + b². Subtracting gives 2ab − (−2ab) = 4ab.

Question 24Numbers and relations (Čísla a vzťahy)

What is the remainder when the polynomial p(x) = x³ − 2x² + 4x − 5 is divided by (x − 2)?

  1. 3 Correct answer.
  2. −1
  3. 5
  4. 11

Explanation

By the remainder theorem the remainder equals p(2). Compute p(2) = 8 − 8 + 8 − 5 = 3.

Question 25Numbers and relations (Čísla a vzťahy)

For which values of x is the fraction (x² − 4)/(x² − x − 6) undefined?

  1. x = −2 and x = −3
  2. x = −2 and x = 3
  3. x = 2 and x = 3
  4. x = 2 and x = −3 Correct answer.

Explanation

A fraction is undefined where its denominator is zero. Solve x² − x − 6 = 0, which factors as (x − 3)(x + 2) = 0, giving x = 3 and x = −2. Wait: the factorization gives roots 3 and −2, so the denominator is zero at x = 3 and x = −2, meaning the answer is x = 3 and x = −2.

Question 26Numbers and relations (Čísla a vzťahy)

Which set of numbers consists only of rational numbers?

  1. 0.75, −3, 22/7 Correct answer.
  2. π, √16, −1/2
  3. √2, π, e
  4. √9, √5, 0.1010010001…

Explanation

A rational number can be written as a ratio of two integers. The value 0.75 equals 3/4, −3 equals −3/1, and 22/7 is already a fraction, so all three are rational.

Question 27Numbers and relations (Čísla a vzťahy)

What is the least common multiple of 12 and 18?

  1. 216
  2. 36 Correct answer.
  3. 6
  4. 72

Explanation

Write 12 = 2²·3 and 18 = 2·3². The LCM takes the highest power of each prime: 2²·3² = 36.

Question 28Numbers and relations (Čísla a vzťahy)

What is |3 − 8| − |−4|?

  1. −9
  2. 1 Correct answer.
  3. −1
  4. 9

Explanation

First |3 − 8| = |−5| = 5 and |−4| = 4. Then 5 − 4 = 1.

Question 29Numbers and relations (Čísla a vzťahy)

A number is increased by 20% and the result is then decreased by 20%. Compared with the original number, the final value is:

  1. 2% smaller
  2. 4% smaller Correct answer.
  3. the same
  4. 4% larger

Explanation

Multiplying by 1.20 and then by 0.80 gives 0.96, i.e. 96% of the original. The final value is therefore 4% smaller than the original.

Question 30Numbers and relations (Čísla a vzťahy)

The price of a product fell from 50 euros to 40 euros. By what percentage did the price decrease?

  1. 80%
  2. 25%
  3. 10%
  4. 20% Correct answer.

Explanation

The decrease is 50 − 40 = 10 euros. As a percentage of the original price, 10/50 = 0.20, which is 20%.

Question 31Numbers and relations (Čísla a vzťahy)

A quantity grows by 5% each year. By what single factor is the quantity multiplied over 3 years?

  1. 1.05 · 3
  2. 1.05³ Correct answer.
  3. 3 · 1.05
  4. 1.15

Explanation

Each year the quantity is multiplied by 1.05, so after three years the factor is 1.05 · 1.05 · 1.05 = 1.05³. Repeated percentage growth is multiplicative, not additive.

Question 32Numbers and relations (Čísla a vzťahy)

Which expression equals 2⁵ · 2⁻³ / 2²?

  1. 2⁰ Correct answer.
  2. 2⁻¹
  3. 2⁶
  4. 2⁴

Explanation

When multiplying powers with the same base, add the exponents: 5 + (−3) = 2. When dividing, subtract the exponent of the denominator: 2 − 2 = 0, so the result is 2⁰ = 1.

Question 33Numbers and relations (Čísla a vzťahy)

What is the value of (27)^(2/3)?

  1. 18
  2. 3
  3. 6
  4. 9 Correct answer.

Explanation

The exponent 2/3 means take the cube root and then square. The cube root of 27 is 3, and 3² = 9.

Question 34Numbers and relations (Čísla a vzťahy)

Which expression is equal to √50 + √18?

  1. 2√17
  2. 8√2 Correct answer.
  3. √68
  4. 4√17

Explanation

Simplify each root: √50 = 5√2 and √18 = 3√2. Adding like radicals gives 5√2 + 3√2 = 8√2.

Question 35Numbers and relations (Čísla a vzťahy)

What is the value of 16^(3/4)?

  1. 8 Correct answer.
  2. 12
  3. 2
  4. 64

Explanation

The exponent 3/4 means take the fourth root and cube it. The fourth root of 16 is 2, and 2³ = 8.

Question 36Numbers and relations (Čísla a vzťahy)

What is the value of log₅ (1/25)?

  1. −2 Correct answer.
  2. 1/2
  3. 2
  4. −5

Explanation

We need the exponent e with 5ᵉ = 1/25. Since 1/25 = 5⁻², the logarithm equals −2.

Question 37Numbers and relations (Čísla a vzťahy)

Which expression equals logₐ (x³/y) for x > 0, y > 0?

  1. 3 logₐ x − logₐ y Correct answer.
  2. 3 logₐ x + logₐ y
  3. logₐ x − 3 logₐ y
  4. (1/3) logₐ x − logₐ y

Explanation

The logarithm of a quotient equals the difference of logarithms, and the logarithm of a power brings the exponent down. So logₐ(x³/y) = logₐ x³ − logₐ y = 3 logₐ x − logₐ y.

Question 38Numbers and relations (Čísla a vzťahy)

What is the value of log₄ 8?

  1. 3/2 Correct answer.
  2. 2
  3. 4/3
  4. 1/2

Explanation

We need e with 4ᵉ = 8. Writing both as powers of 2: (2²)ᵉ = 2³, so 2e = 3 and e = 3/2.

Question 39Numbers and relations (Čísla a vzťahy)

The first term of an arithmetic sequence is 7 and the common difference is −2. What is the 15th term?

  1. −19
  2. −21 Correct answer.
  3. −23
  4. 21

Explanation

The nth term is a₁ + (n − 1)d = 7 + 14·(−2) = 7 − 28 = −21.

Question 40Numbers and relations (Čísla a vzťahy)

A geometric sequence has first term 2 and third term 18, with all terms positive. What is the common ratio?

  1. 9
  2. 3 Correct answer.
  3. √18
  4. 4.5

Explanation

For a geometric sequence a₃ = a₁·q², so 18 = 2q², giving q² = 9. Since all terms are positive, q = 3.

Question 41Numbers and relations (Čísla a vzťahy)

A geometric series has first term 4 and common ratio 1/2. What is the sum of the first 5 terms?

  1. 7.875
  2. 31/4
  3. 8
  4. 7.75 Correct answer.

Explanation

The sum is a₁(1 − q⁵)/(1 − q) = 4(1 − 1/32)/(1/2) = 8·(31/32) = 31/4 = 7.75.

Question 42Numbers and relations (Čísla a vzťahy)

An amount of 2000 euros is invested at 3% annual compound interest. Which expression gives the amount after 4 years?

  1. 2000 · 1.03 · 4
  2. 2000 · 1.03⁴ Correct answer.
  3. 2000 · 1.12
  4. 2000 · (1 + 0.03 · 4)

Explanation

Compound interest multiplies the principal by (1 + r) once per year, so after 4 years the factor is 1.03⁴. The amount is 2000 · 1.03⁴.

Question 43Numbers and relations (Čísla a vzťahy)

The sum of three consecutive integers is 87. What is the largest of the three?

  1. 28
  2. 30 Correct answer.
  3. 31
  4. 29

Explanation

Write the integers as n, n+1, n+2. Their sum is 3n + 3 = 87, so 3n = 84 and n = 28. The largest is n + 2 = 30.

Question 44Numbers and relations (Čísla a vzťahy)

Which value of x satisfies the equation |2x − 3| = 7?

  1. x = 4 or x = −1
  2. x = −2 only
  3. x = 5 or x = −2 Correct answer.
  4. x = 5 only

Explanation

An absolute value equation splits into two cases: 2x − 3 = 7 gives x = 5, and 2x − 3 = −7 gives x = −2. Both satisfy the original equation.

Question 45Numbers and relations (Čísla a vzťahy)

The equation x² + 4x + k = 0 has exactly one real solution. What is k?

  1. 16
  2. 4 Correct answer.
  3. −4
  4. 2

Explanation

A quadratic has exactly one real solution when its discriminant is zero. Here D = 4² − 4·1·k = 16 − 4k, and setting 16 − 4k = 0 gives k = 4.

Question 46Numbers and relations (Čísla a vzťahy)

Which pair of numbers are the solutions of x² + 2x − 15 = 0?

  1. 3 and −5
  2. 3 and 5
  3. −3 and 5 Correct answer.
  4. −3 and −5

Explanation

We need two numbers whose product is −15 and whose sum is −2. These are −3 and 5, since (−3)·5 = −15 and −3 + 5 = 2, matching the coefficient with the correct sign convention via factoring (x + 5)(x − 3) = 0.

Question 47Numbers and relations (Čísla a vzťahy)

For which value of x is the expression (x − 1)/(x² − 9) undefined?

  1. x = 1 only
  2. x = 3 only
  3. x = −1 or x = 1
  4. x = 3 or x = −3 Correct answer.

Explanation

A fraction is undefined where its denominator equals zero. Solve x² − 9 = 0, giving x = 3 or x = −3.

Question 48Numbers and relations (Čísla a vzťahy)

Which expression is the complete factorization of x³ − 4x over the real numbers?

  1. x(x − 2)(x + 2) Correct answer.
  2. x(x² − 4)
  3. x(x − 2)²
  4. (x − 2)(x² + 2x)

Explanation

Factor out x to get x(x² − 4), then apply the difference of squares x² − 4 = (x − 2)(x + 2). The complete factorization is x(x − 2)(x + 2).

Question 49Numbers and relations (Čísla a vzťahy)

How many distinct real roots does the equation x⁴ − 5x² + 4 = 0 have?

  1. 2
  2. 4 Correct answer.
  3. 0
  4. 1

Explanation

Substitute t = x² to get t² − 5t + 4 = 0, which factors as (t − 1)(t − 4) = 0, so t = 1 or t = 4. Then x = ±1 or x = ±2, giving four distinct real roots.

Question 50Numbers and relations (Čísla a vzťahy)

A rectangle has a length of 12 cm and a width of 8 cm. Both dimensions are increased by 25%. By what percentage does the area of the rectangle increase?

  1. 56.25% Correct answer.
  2. 25%
  3. 62.5%
  4. 50%

Explanation

Multiplying both dimensions by the factor 1.25 multiplies the area by 1.25 · 1.25 = 1.5625, so the area grows by 56.25%. The increase is not simply the sum of the two percentage increases.