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Key Facts: Scio NSZ Math Exam

The Scio NSZ Matematika is a 35-task, 90-minute standardized university admissions math exam with 4 choices and -1/3 penalty; no calculators allowed; scored in percentiles.

Sample Scio NSZ Math Practice Questions

Try these sample questions to test your Scio NSZ Math exam readiness. Each question includes a detailed explanation. Start the interactive quiz above for the full 100+ question experience with AI tutoring.

1Solve the linear equation in the set of real numbers R: (2x - 1)/3 - (3x - 2)/4 = (x + 1)/6.

A.x = 0

B.x = 2

C.x = -2

D.x = 1/2

Explanation: Multiply the entire equation by the common denominator 12: 4(2x - 1) - 3(3x - 2) = 2(x + 1). Expanding both sides yields 8x - 4 - 9x + 6 = 2x + 2, which simplifies to -x + 2 = 2x + 2. Subtracting 2 from both sides gives -x = 2x, so 3x = 0 and x = 0. Verifying in the original equation: (0 - 1)/3 - (0 - 2)/4 = -1/3 + 1/2 = 1/6, and (0 + 1)/6 = 1/6.

2For all real numbers x != 2, simplify the algebraic fraction: (2x^2 - 8) / (x^2 - 4x + 4).

A.2

B.2(x + 2) / (x - 2)

C.2(x - 2) / (x + 2)

D.(x + 2) / (x - 2)

Explanation: Factor the numerator: 2x^2 - 8 = 2(x^2 - 4) = 2(x - 2)(x + 2). Factor the denominator: x^2 - 4x + 4 = (x - 2)^2. For x != 2, cancel the common factor (x - 2): [2(x - 2)(x + 2)] / (x - 2)^2 = 2(x + 2) / (x - 2).

3Simplify the numerical radical expression: sqrt(50) + 2*sqrt(18) - sqrt(72).

A.7*sqrt(2)

B.3*sqrt(2)

C.5*sqrt(2)

D.4*sqrt(2)

Explanation: Simplify each radical: sqrt(50) = sqrt(25 * 2) = 5*sqrt(2); 2*sqrt(18) = 2*sqrt(9 * 2) = 2 * 3*sqrt(2) = 6*sqrt(2); sqrt(72) = sqrt(36 * 2) = 6*sqrt(2). Therefore, sqrt(50) + 2*sqrt(18) - sqrt(72) = 5*sqrt(2) + 6*sqrt(2) - 6*sqrt(2) = 5*sqrt(2).

4Find the product of all real roots of the quadratic equation 3x^2 - 11x - 4 = 0.

A.11/3

B.-11/3

C.4/3

D.-4/3

Explanation: By Vieta's formulas, for a quadratic equation ax^2 + bx + c = 0 with real roots, the product of the roots is x1 * x2 = c/a. Here a = 3, b = -11, and c = -4. The discriminant is D = (-11)^2 - 4(3)(-4) = 121 + 48 = 169 > 0, confirming real roots exist. Thus, x1 * x2 = -4/3.

5How many non-negative integers (x in {0, 1, 2, ...}) satisfy the linear inequality 3 - 2(x + 1) >= 5x - 18?

A.3

B.2

C.4

D.1

Explanation: Expand the left side: 3 - 2x - 2 >= 5x - 18, which simplifies to 1 - 2x >= 5x - 18. Adding 2x to both sides gives 1 >= 7x - 18. Adding 18 gives 19 >= 7x, so x <= 19/7 ≈ 2.714. The non-negative integers satisfying this condition are {0, 1, 2}, which is exactly 3 integers.

6Solve the equation |2x - 5| = 3x - 1 in the set of real numbers R.

A.x = -4

B.x = 6/5

C.x = -4 and x = 6/5

D.No real solution

Explanation: Because an absolute value is non-negative, we must have 3x - 1 >= 0 => x >= 1/3. Case 1: 2x - 5 >= 0 (x >= 5/2). Then 2x - 5 = 3x - 1 => x = -4. But -4 does not satisfy x >= 5/2, so it is an extraneous solution. Case 2: 2x - 5 < 0 (x < 5/2). Then -(2x - 5) = 3x - 1 => -2x + 5 = 3x - 1 => 5x = 6 => x = 6/5 = 1.2. Since 1/3 <= 1.2 < 2.5, x = 6/5 is the unique valid solution. Verification: |2(1.2) - 5| = |-2.6| = 2.6, and 3(1.2) - 1 = 3.6 - 1 = 2.6.

7Find the solution set in R for the absolute value inequality |3x + 2| <= 8.

A.[-8/3, 2]

B.(-oo, -10/3] U [2, oo)

C.[-10/3, 2]

D.[-2, 2]

Explanation: The inequality |A| <= B (for B >= 0) is equivalent to the double inequality -B <= A <= B. Thus: -8 <= 3x + 2 <= 8. Subtract 2 from all parts: -10 <= 3x <= 6. Divide by 3: -10/3 <= x <= 2. In interval notation, this is [-10/3, 2].

8For all real numbers x not in {-2, 2}, simplify the rational expression: (x^3 - 8) / (x^2 - 4).

A.(x^2 - 2x + 4) / (x + 2)

B.x - 2

C.(x^2 + 4) / (x + 2)

D.(x^2 + 2x + 4) / (x + 2)

Explanation: Use the difference of cubes identity on the numerator: a^3 - b^3 = (a - b)(a^2 + ab + b^2). Here x^3 - 8 = (x - 2)(x^2 + 2x + 4). Use the difference of squares on the denominator: x^2 - 4 = (x - 2)(x + 2). Canceling the factor (x - 2) gives (x^2 + 2x + 4) / (x + 2).

9Find the value of x + y for the solution (x, y) to the system of linear equations: 3x - 4y = 11 2x + 5y = -8

A.-1

B.1

C.3

D.-3

Explanation: Multiply the first equation by 2 and the second by 3: 6x - 8y = 22 and 6x + 15y = -24. Subtract the first from the second: 23y = -46 => y = -2. Substitute y = -2 into the first equation: 3x - 4(-2) = 11 => 3x + 8 = 11 => 3x = 3 => x = 1. Thus (x, y) = (1, -2), and x + y = 1 + (-2) = -1.

10What is the sum of all positive real roots of the biquadratic equation x^4 - 13x^2 + 36 = 0?

A.13

B.5

C.0

D.6

Explanation: Substitute u = x^2: u^2 - 13u + 36 = 0. Factoring gives (u - 4)(u - 9) = 0, so u = 4 or u = 9. Returning to x: x^2 = 4 gives x = +-2; x^2 = 9 gives x = +-3. The positive real roots are 2 and 3. Their sum is 2 + 3 = 5.

About the Scio NSZ Math Exam

The Scio NSZ Matematika is the national comparative mathematics examination used for university admissions by STEM, economics, business, and informatics faculties across the Czech Republic and Slovakia (including faculties of ČVUT, VUT, VŠE, and Charles University). The examination contains 35 multiple-choice tasks to be completed in 90 minutes. Calculators and standard mathematical tables are prohibited — Scio notes that every calculation can be completed without one — but blank note pages and a summary of key formulas are included in the test booklet. The syllabus follows the gymnázium mathematics curriculum: basic knowledge, algebra, functions and equations, plane and solid geometry, analytic geometry, combinatorics, probability and statistics. Scio publishes these areas without percentage weightings. Each question offers 4 options, and incorrect answers incur a penalty of -1/3 point. This practice bank is an English-language study adaptation of that scope; the official test is administered in Czech (in Slovak at Slovak venues).

Questions

35 scored questions

Time Limit

90 minutes.

Passing Score

Harmonized percentile score (0–100); admissions cutoffs set by participating faculties (e.g., FIT ČVUT, FIS VŠE, MFF UK).

Exam Fee

920 CZK (Mini), 1,290 CZK (Standard) or 1,490 CZK (Komplet) per exam date; free with Scio's social discount or a faculty voucher. (Scio s.r.o. (Národní srovnávací zkoušky — NSZ))

Scio NSZ Math Exam Content Outline

25%

Algebra a rovnice (Algebra & Equations)

Algebraic transformations, fractional expressions, quadratic equations, systems of equations, modulus inequalities, and logarithmic equations.

25%

Funkce a posloupnosti (Functions & Sequences)

Function properties (monotonicity, parity, boundedness), trigonometric equations, arithmetic/geometric progressions, and infinite series sums.

25%

Geometrie a analytika (Geometry & Analytic Geometry)

Triangle and circle theorems, trigonometry (sine/cosine laws), solid geometry formulas, and analytic geometry of lines and conic sections.

25%

Kombinatorika a pravdepodobnost (Combinatorics & Probability)

Combinatorial counting principles, binomial expansions, classical probability, independent events, and statistical measures.

How to Pass the Scio NSZ Math Exam

What You Need to Know

Passing score: Harmonized percentile score (0–100); admissions cutoffs set by participating faculties (e.g., FIT ČVUT, FIS VŠE, MFF UK).
Exam length: 35 questions
Time limit: 90 minutes.
Exam fee: 920 CZK (Mini), 1,290 CZK (Standard) or 1,490 CZK (Komplet) per exam date; free with Scio's social discount or a faculty voucher.

Keys to Passing

Work through all 100 available questions
Review every answer and explanation
Track weak areas and revisit them
Use our AI tutor for tough concepts

Scio NSZ Math Study Tips from Top Performers

1Practice doing all arithmetic by hand: practice fast mental multiplication, division, and factoring without relying on a calculator.

2Familiarize yourself with the exact formulas provided on the official Scio formula sheet so you don't waste time looking up formulas that aren't included.

3In analytic geometry problems, sketch the coordinate system and geometric figures roughly on scratch paper to verify whether your calculated slope and intercepts make visual sense.

4Be disciplined with time: since questions are worth the same 1 point, skip lengthy geometric proof-like problems and secure points on rapid algebra and sequence questions first.

5Check definitions and conditions carefully: always verify domain restrictions (e.g., denominators not equal to zero, arguments of logarithms greater than zero, radices non-negative).

Frequently Asked Questions

Are calculators allowed in the Scio NSZ Matematika test?

No. Calculators of any kind are strictly prohibited. All calculations must be performed by hand using mental math and scratch paper.

What aids are provided during the exam?

Scio provides an official one-page formula sheet (přehled vzorců) attached to the test booklet containing basic algebraic, trigonometric, and geometric formulas.

What is the penalty for incorrect answers?

There is a -1/3 point penalty (-0.33) for each wrong answer. Correct answers add 1 point, and unanswered questions receive 0 points.

How long is the exam and how many questions are there?

The examination consists of 35 multiple-choice questions to be completed in 90 minutes (an average of about 2 minutes and 34 seconds per question).