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Free Practice Questions for Practical Mathematics Skills Test (Suken)

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Key Facts: Practical Mathematics Skills Test (Suken) Exam

2 stages

Levels 1–5: same-day Stage 1 calculation and Stage 2 application

Mathematics Certification Institute of Japan, examination overview

70% / 60%

Approximate Stage 1 and Stage 2 passing bands for Levels 1–5

Suken examination overview and Level 2 grade page

6,500 JPY

2026 personal fee for Level 2

Suken personal sitting guide and Level 2 grade page

15 + 5

Level 2 item counts: 15 Stage 1 calculations; Stage 2 is 2 required plus 3 of 5

Suken Level 2 grade page

English dates

2026 English papers on 25 July and 25 October (personal Levels 1–8)

Suken examination overview, 英語版での受検について

Suken (数検) is Japan's mathematics certification. Level 2 is two stages: 15 calculator-free items in 50 minutes (~70% to pass) and a 90-minute application paper (~60%), for 6,500 JPY. Official answers are written, not multiple choice. This 100-question bank is independent English MCQ study.

Sample Practical Mathematics Skills Test (Suken) Practice Questions

Try these sample questions to review concepts for the Practical Mathematics Skills Test (Suken) exam. Each question includes a detailed explanation. Start the interactive quiz above for the full 100+ question experience with AI tutoring.

1Expand and simplify the algebraic expression: (2x - 3)(3x + 4) - (x - 2)^2. 次の式を展開して整理しなさい：(2x - 3)(3x + 4) - (x - 2)^2
A.5x^2 + 3x - 16
B.5x^2 - 5x - 8
C.5x^2 + 3x - 8
D.7x^2 - x - 16
Explanation: Expand the first product: (2x - 3)(3x + 4) = 6x^2 + 8x - 9x - 12 = 6x^2 - x - 12. Expand the second squared term: (x - 2)^2 = x^2 - 4x + 4. Subtract the two expressions: (6x^2 - x - 12) - (x^2 - 4x + 4) = 6x^2 - x^2 - x + 4x - 12 - 4 = 5x^2 + 3x - 16.
2Completely factor the polynomial expression: 2x^2 + 7xy + 3y^2 - 5x - 5y + 2. 次の式を因数分解しなさい：2x^2 + 7xy + 3y^2 - 5x - 5y + 2
A.(2x + y - 1)(x + 3y - 2)
B.(2x + 3y - 1)(x + y - 2)
C.(2x + y + 1)(x + 3y - 2)
D.(2x - y - 1)(x - 3y - 2)
Explanation: Group by powers of x: 2x^2 + (7y - 5)x + (3y^2 - 5y + 2). Factor the y quadratic: 3y^2 - 5y + 2 = (3y - 2)(y - 1). Using cross-multiplication for the quadratic in x: (2)(y - 1) + (1)(3y - 2) = 2y - 2 + 3y - 2 = 5y - 4, but we need 7y - 5. Pairing 2 with (3y - 2) and 1 with (y - 1): 2(3y - 2) + 1(y - 1) = 6y - 4 + y - 1 = 7y - 5. Thus, the factored form is (2x + y - 1)(x + 3y - 2).
3Simplify the radical fraction by rationalizing the denominator: (3 - √5) / (3 + √5). 次の式の分母を有理化して簡約しなさい：(3 - √5) / (3 + √5)
A.(7 - 3√5) / 2
B.(14 - 6√5) / 4
C.7 - 3√5
D.(7 + 3√5) / 2
Explanation: Multiply numerator and denominator by the conjugate (3 - √5): (3 - √5)^2 / [(3 + √5)(3 - √5)] = (9 - 6√5 + 5) / (9 - 5) = (14 - 6√5) / 4. Dividing numerator and denominator by 2 gives (7 - 3√5) / 2.
4Solve the quadratic equation for real roots: 3x^2 - 5x - 2 = 0. 2次方程式 3x^2 - 5x - 2 = 0 を解きなさい。
A.x = 2, -1/3
B.x = -2, 1/3
C.x = 1, -2/3
D.x = 3, -2/3
Explanation: Factor the quadratic: (3x + 1)(x - 2) = 0. Therefore, 3x + 1 = 0 => x = -1/3, or x - 2 = 0 => x = 2. The solutions are x = 2 and x = -1/3.
5For what values of m does the quadratic equation x^2 + 2(m - 1)x + (m^2 - 3) = 0 have two distinct real roots? 2次方程式 x^2 + 2(m - 1)x + (m^2 - 3) = 0 が異なる2つの実数解を持つような定数 m の値の範囲を求めなさい。
A.m < 2
B.m > 2
C.m < -2
D.m > -2
Explanation: For the equation to have two distinct real roots, the discriminant D must satisfy D/4 > 0. Here, D/4 = (m - 1)^2 - 1*(m^2 - 3) = (m^2 - 2m + 1) - (m^2 - 3) = -2m + 4. Setting -2m + 4 > 0 gives -2m > -4, which simplifies to m < 2.
6Solve the quadratic inequality: 2x^2 - 5x - 3 <= 0. 2次不等式 2x^2 - 5x - 3 <= 0 を解きなさい。
A.-1/2 <= x <= 3
B.x <= -1/2, x >= 3
C.-3 <= x <= 1/2
D.x <= -3, x >= 1/2
Explanation: Factor the quadratic: (2x + 1)(x - 3) <= 0. The parabola y = 2x^2 - 5x - 3 opens upwards and intersects the x-axis at x = -1/2 and x = 3. It takes values less than or equal to zero between the roots: -1/2 <= x <= 3.
7Express the complex number fraction in standard form a + bi: (3 + 4i) / (1 - 2i), where i^2 = -1. 複素数 (3 + 4i) / (1 - 2i) を a + bi の形で表しなさい。
A.-1 + 2i
B.1 + 2i
C.-1 - 2i
D.11/5 + (10/5)i
Explanation: Multiply numerator and denominator by the complex conjugate (1 + 2i): (3 + 4i)(1 + 2i) / [(1 - 2i)(1 + 2i)] = (3 + 6i + 4i + 8i^2) / (1 - 4i^2) = (3 + 10i - 8) / (1 + 4) = (-5 + 10i) / 5 = -1 + 2i.
8If alpha and beta are the roots of 2x^2 - 6x + 3 = 0, find the value of alpha^2 + beta^2. 2次方程式 2x^2 - 6x + 3 = 0 の2つの解を α, β とするとき、α^2 + β^2 の値を求めなさい。
A.6
B.9/2
C.15/2
D.3
Explanation: By Vieta's formulas, alpha + beta = -(-6)/2 = 3, and alpha * beta = 3/2. We know alpha^2 + beta^2 = (alpha + beta)^2 - 2*alpha*beta. Substituting the values: 3^2 - 2*(3/2) = 9 - 3 = 6.
9Find the remainder when the polynomial P(x) = 2x^3 - 5x^2 + 4x - 7 is divided by (x - 2). 多項式 P(x) = 2x^3 - 5x^2 + 4x - 7 を x - 2 で割ったときの余りを求めなさい。
A.-3
B.3
C.-7
D.5
Explanation: By the Remainder Theorem, the remainder when P(x) is divided by (x - 2) is equal to P(2). Calculate P(2) = 2(2)^3 - 5(2)^2 + 4(2) - 7 = 2(8) - 5(4) + 8 - 7 = 16 - 20 + 8 - 7 = -4 + 8 - 7 = 4 - 7 = -3.
10In an arithmetic progression, the 4th term is 17 and the 10th term is 41. Find the first term a_1 and common difference d. 等差数列において、第4項が17、第10項が41であるとき、初項 a と公差 d を求めなさい。
A.a_1 = 5, d = 4
B.a_1 = 4, d = 5
C.a_1 = 1, d = 4
D.a_1 = 5, d = 3
Explanation: The n-th term is a_n = a_1 + (n - 1)d. Thus: a_4 = a_1 + 3d = 17, and a_10 = a_1 + 9d = 41. Subtracting the first equation from the second: (a_1 + 9d) - (a_1 + 3d) = 41 - 17 => 6d = 24 => d = 4. Substitute d = 4 back: a_1 + 3(4) = 17 => a_1 + 12 = 17 => a_1 = 5.

About the Practical Mathematics Skills Test (Suken) Exam

The Practical Mathematics Skills Test (実用数学技能検定, 数検 / Suken) is Japan's mathematics and arithmetic certification, administered by the Mathematics Certification Institute of Japan with MEXT backing. Grades run from Level 11 (Grade 1 arithmetic) through Level 1 (undergraduate mathematics). Levels 1–5 are two written stages on the same day: Stage 1 tests calculator-free calculation skill, and Stage 2 tests applied reasoning with an allowed calculator (ordinary, scientific, or graphing; no phones). Official papers are constructed response with partial credit. Level 2 matches high-school year 2 (Mathematics II/B): 15 Stage 1 items in 50 minutes and a 90-minute Stage 2, with approximate 70%/60% passing bands, and a 6,500 JPY personal fee. The Institute offers English-language papers on named 2026 dates (personal Levels 1–8 on 25 July and 25 October 2026) at the same fee. OpenExamPrep provides independent English-language multiple-choice study with worked calculations. It is not an official translation, not a 記述式 simulation, and not a substitute for writing full solutions.

Exam sponsor: The Mathematics Certification Institute of Japan (公益財団法人 日本数学検定協会). The requirements and fees below concern the certification or admission exam, separate from our free practice resources.

Assessment

Levels 1–5 are two stages on the same day: Stage 1 計算技能検定 (no calculator) and Stage 2 数理技能検定 (calculator allowed). A single-stage pass can be reused. Arithmetic grades 6–11 are single papers. Official language is Japanese, with named English sittings in 2026.

Time Limit

Level 2: Stage 1 50 minutes, Stage 2 90 minutes. Pre-1/Level 1 Stage 1 is 60 minutes and Stage 2 is 120 minutes. Levels 3–5 Stage 2 is 60 minutes.

Passing Score

About 70% on Stage 1 and about 60% on Stage 2 for Levels 1–5.

Exam / Certification Fees

Personal Level 2: 6,500 JPY; Level 3: 4,900 JPY; Level 1: 8,500 JPY (tax included). A 1,000 JPY reduction applies when only one stage is retaken.

Exam sponsor website

Fees, eligibility, and exam policies can change. Confirm them with the exam sponsor before applying or paying.

Our practice resources: topics covered

We aim to reflect publicly available exam outlines and topic information in our study resources. Coverage, format, and difficulty may differ from the actual exam, and we cannot guarantee that every detail is accurate or current. Confirm exam requirements, fees, and policies with the official exam sponsor.

25% of this local practice set

Algebra, Numbers & Equations (数と式・方程式)

Expansion, factoring, quadratics, inequalities, sequences, and related Stage 1 calculation skill. Official grade pages show approximate structure charts, not a single mixed-bank percentage table.

25% of this local practice set

Functions & Analytic Geometry (関数とグラフ・図形と方程式)

Quadratic, exponential, and logarithmic functions plus lines and circles at Pre-2/Level 2 depth.

20% of this local practice set

Trigonometry & Vectors (三角比・三角関数・平面と空間のベクトル)

三角比 (Pre-2 content) and 三角関数 (Level 2 content) plus the Laws of Sines and Cosines. Vectors belong to Mathematics C, which the Institute maps to Pre-1, so those items reach above Level 2.

15% of this local practice set

Calculus & Rates of Change (微分と積分)

Polynomial calculus used from Level 2 (Mathematics II) upward.

15% of this local practice set

Probability, Statistics & Discrete Math (場合の数・確率・データの分析)

Counting, probability, and data analysis typical of Stage 2 application items.

Preparing for the Practical Mathematics Skills Test (Suken) Exam

What You Need to Know

  • Passing score: About 70% on Stage 1 and about 60% on Stage 2 for Levels 1–5.
  • Assessment: Levels 1–5 are two stages on the same day: Stage 1 計算技能検定 (no calculator) and Stage 2 数理技能検定 (calculator allowed). A single-stage pass can be reused. Arithmetic grades 6–11 are single papers. Official language is Japanese, with named English sittings in 2026.
  • Time limit: Level 2: Stage 1 50 minutes, Stage 2 90 minutes. Pre-1/Level 1 Stage 1 is 60 minutes and Stage 2 is 120 minutes. Levels 3–5 Stage 2 is 60 minutes.
  • Exam / certification fees: Personal Level 2: 6,500 JPY; Level 3: 4,900 JPY; Level 1: 8,500 JPY (tax included). A 1,000 JPY reduction applies when only one stage is retaken. Official sources

Using Our Practice Resources

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

Practical Mathematics Skills Test (Suken): Suggested Study Strategy

1Practice Stage 1 algebra by hand with no calculator until expansion, factoring, and sign errors are rare.
2Memorize the quadratic formula, sequence-sum formulas, and the Laws of Sines and Cosines so Stage 2 time is spent on setup rather than recall.
3Sketch graphs for function and locus items: vertex, intercepts, and circle center are usually the first quantities the mark scheme wants.
4On logarithms, check the domain (argument positive, base positive and not 1) before simplifying.

Frequently Asked Questions

What is the difference between Stage 1 and Stage 2 on Suken?

For Levels 1–5, Stage 1 (計算技能検定) is calculator-free calculation. Stage 2 (数理技能検定) is applied problem-solving. The Institute allows ordinary, scientific, or graphing calculators on Stage 2 for Levels 1–5, but not phones, tablets, or communicating devices.

Can a candidate pass Stage 1 and Stage 2 separately?

Yes. A pass in only one stage can be reused on a later sitting, with a 1,000 JPY fee reduction if the printed pass-certificate number is supplied at registration.

Which Suken level matches high-school mathematics?

The Institute maps Pre-2 to high-school year 1 (Mathematics I/A) and Level 2 to high-school year 2 (Mathematics II/B). Pre-1 is Mathematics III/C. Level 1 is undergraduate mathematics. MEXT's High School Equivalency Examination notice lists Suken Level 1, Pre-1, or Level 2 as a mathematics-subject exemption.

Is there an official English Suken paper?

Yes, on named dates. For 2026 the Institute states English papers on 25 July and 25 October; personal sittings cover Levels 1–8 at the same fee. This OpenExamPrep bank is still an independent English MCQ study adaptation, not those official constructed-response English papers.

How does OpenExamPrep adapt Suken into multiple-choice practice?

Official Suken is 記述式: candidates write methods and answers and may receive partial credit. This bank turns the same syllabus topics into four-option quantitative items with worked explanations. It is not a format simulation and does not replace writing full solutions.