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100+ Free Osaka Public High School Entrance Examination — Mathematics Practice Questions

Prepare for the Osaka Prefectural Public High School Entrance Examination — Mathematics (大阪府公立高等学校入学者選抜学力検査 数学) exam with instant access — no signup required.

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Key Facts: Osaka Public High School Entrance Examination — Mathematics Exam

Practise for the Osaka Prefectural Public High School Entrance Exam in Mathematics with 100 four-option questions covering 数と式, 関数, 図形, and 確率・データ, weighted to the official 50/60-minute, 90-point paper sat on 11 March 2026. The official paper contains constructed-response items (計算/証明/作図/求め方記述) and comes in three tiers (基礎的/標準的/発展的), so this English-language MCQ bank is a study aid, not a format simulation.

Sample Osaka Public High School Entrance Examination — Mathematics Practice Questions

Try these sample questions to test your Osaka Public High School Entrance Examination — Mathematics exam readiness. Each question includes a detailed explanation. Start the interactive quiz above for the full 100+ question experience with AI tutoring.

1Calculate: (−3) × 4 − (−2).
A.−10
B.−14
C.10
D.−6
Explanation: (−3) × 4 = −12, and −12 − (−2) = −12 + 2 = −10. Subtracting a negative is adding the positive.
2Evaluate: −5 + 8 − 3.
A.0
B.16
C.−6
D.2
Explanation: Working left to right: −5 + 8 = 3, then 3 − 3 = 0.
3Compute 3/4 + 1/6 and give the answer in lowest terms.
A.11/12
B.4/10
C.2/5
D.4/24
Explanation: Common denominator 12: 3/4 = 9/12 and 1/6 = 2/12, so 9/12 + 2/12 = 11/12, already in lowest terms.
4Simplify: 2/3 ÷ 4/5.
A.5/6
B.8/15
C.3/4
D.8/9
Explanation: Division by a fraction equals multiplication by its reciprocal: 2/3 × 5/4 = 10/12 = 5/6.
5Simplify the square root: √72.
A.6√2
B.8√2
C.6√3
D.12
Explanation: √72 = √(36 × 2) = √36 × √2 = 6√2, since 36 is the largest perfect-square factor of 72.
6Simplify: √27 + √12.
A.5√3
B.3√39
C.√39
D.3√3
Explanation: √27 = 3√3 and √12 = 2√3, so √27 + √12 = 3√3 + 2√3 = 5√3. Like surds add.
7Solve for x: 3x − 7 = 2x + 5.
A.x = 12
B.x = 2
C.x = −12
D.x = 1/5
Explanation: Subtract 2x: x − 7 = 5. Add 7: x = 12. Checking: 3(12)−7 = 29 and 2(12)+5 = 29.
8Solve the simultaneous equations: x + y = 10, x − y = 4.
A.x = 7, y = 3
B.x = 6, y = 4
C.x = 7, y = 7
D.x = 3, y = 7
Explanation: Adding the two equations eliminates y: 2x = 14, so x = 7. Then 7 + y = 10 gives y = 3.
9Expand and simplify: (x + 3)(x − 5).
A.x² − 2x − 15
B.x² − 2x + 15
C.x² + 2x − 15
D.x² − 15
Explanation: Using FOIL: x·x = x², x·(−5) = −5x, 3·x = 3x, 3·(−5) = −15. Combine −5x + 3x = −2x, so x² − 2x − 15.
10Factorise: x² − 9.
A.(x − 3)(x + 3)
B.(x − 3)²
C.(x + 9)(x − 1)
D.x(x − 9)
Explanation: x² − 9 = x² − 3² is a difference of squares, factoring as (x − 3)(x + 3).

About the Osaka Public High School Entrance Examination — Mathematics Exam

The Osaka Prefectural Public High School Entrance Examination in Mathematics tests lower-secondary graduates on calculation, algebra, functions, geometry and data analysis. In the 令和8年度 (2026) cycle the general academic test (一般入学者選抜 学力検査) covers five subjects (国語, 社会, 数学, 理科, 英語), each worth 90 points (total 450), all sat on a single day on 11 March 2026 (1st 国語, 2nd 数学, 3rd 英語 with リスニング, 4th 理科, 5th 社会), with 合格発表 on 19 March 2026. The paper is aligned to the MEXT junior-high Course of Study (平成29年 中学校学習指導要領 数学編) and tests 知識・技能 plus 思考力・判断力・表現力, with constructed-response items including 計算, 証明, 作図, and 求め方記述. For 国語・数学・英語 the board sets three problem tiers — 基礎的問題, 標準的問題, 発展的問題 — and each school selects one; 理科 and 社会 use a single common paper. All prefectural schools use the common board-set paper; no school sets a self-made mathematics paper. Selection combines 学力検査 450 + 調査書 450 (中3 ×6, 中1・中2 ×2) into a 900-point 総合点 using a school-chosen 倍率タイプ (I–V); the top 110% form (Ⅰ)群, the top 90% of those are admitted, and the (Ⅱ)群 ボーダーゾーン is decided against each school's アドミッションポリシー. Since FY2014 (平成26年度) all Osaka prefectural public high schools form a single prefecture-wide school district (全府一学区 / 府内全域). There is no single prefecture-wide pass line.

Assessment

50-minute (基礎的/標準的問題) or 60-minute (発展的問題) paper worth 90 points, aligned to the MEXT 数学 Course of Study with constructed-response items including 計算, 証明, 作図, and 求め方記述. It is sat in the 2nd period on 11 March 2026. The five academic-test subjects (国語, 社会, 数学, 理科, 英語) each carry 90 points for a maximum of 450 points. The paper tests 知識・技能 plus 思考力・判断力・表現力. For 国語・数学・英語 the board provides three problem tiers (基礎的/標準的/発展的); each school chooses one tier; 理科 and 社会 use a single common paper.

Time Limit

50 minutes (基礎的/標準的問題) or 60 minutes (発展的問題). The 数学 paper is sat in the 2nd period on 11 March 2026; the full academic test is a single day — 1st 国語, 2nd 数学, 3rd 英語 (with リスニングテスト), 4th 理科, 5th 社会.

Passing Score

No fixed pass mark. Admission is decided by each school using a 総合点 out of 900: 学力検査 450 (5 subjects × 90) plus 調査書 450 (中3 評定 ×6, 中1・中2 評定 ×2), each multiplied by the school's chosen 倍率タイプ from types I–V (I: 1.4/0.6; II: 1.2/0.8; III: 1.0/1.0; IV: 0.8/1.2; V: 0.6/1.4). The top 110% of applicants form (Ⅰ)群; the top 90% of those are admitted; the remaining (Ⅱ)群 (ボーダーゾーン) is decided against the school's アドミッションポリシー using the 自己申告書 and 活動記録. There is no single prefecture-wide pass line.

Exam Fee

入学検定料 (entrance examination fee) for an Osaka prefectural public high school is ¥2,200 for the 全日制 (full-time) course, ¥950 for the 定時制 (part-time) course and ¥800 for the 通信制 (correspondence) course, paid via the online 出願システム. A separate 入学料 is ¥5,650 (全日制), ¥2,100 (定時制) or ¥500 (通信制). It is paid once for the whole five-subject 学力検査, not per subject paper. Source: 令和8年度 大阪府公立高等学校入学者選抜実施要項 第7 一般入学者選抜. (Osaka Prefectural Board of Education (大阪府教育委員会) — 教育庁教育振興室高等学校課学事グループ)

Osaka Public High School Entrance Examination — Mathematics Exam Content Outline

25%

Numbers & Algebraic Expressions (数と式)

Signed arithmetic, fractions, square roots, algebraic manipulation, linear and simultaneous equations, quadratic equations, and word-problem set-up.

25%

Functions (関数)

Direct and inverse proportionality, linear functions, the quadratic function y=ax², parabola vertex and intercepts, transformations, and graph/data reading.

33%

Geometry (図形)

Triangle congruence and similarity, the Pythagorean theorem, circle properties, inscribed angles, tangent lines, polygons, and solid-geometry volume and surface area.

17%

Data Analysis & Probability (資料の活用と確率)

Histograms and box plots, median, mean, mode and range, sample surveys, and probability with dice, coins, marbles and cards.

How to Pass the Osaka Public High School Entrance Examination — Mathematics Exam

What You Need to Know

  • Passing score: No fixed pass mark. Admission is decided by each school using a 総合点 out of 900: 学力検査 450 (5 subjects × 90) plus 調査書 450 (中3 評定 ×6, 中1・中2 評定 ×2), each multiplied by the school's chosen 倍率タイプ from types I–V (I: 1.4/0.6; II: 1.2/0.8; III: 1.0/1.0; IV: 0.8/1.2; V: 0.6/1.4). The top 110% of applicants form (Ⅰ)群; the top 90% of those are admitted; the remaining (Ⅱ)群 (ボーダーゾーン) is decided against the school's アドミッションポリシー using the 自己申告書 and 活動記録. There is no single prefecture-wide pass line.
  • Assessment: 50-minute (基礎的/標準的問題) or 60-minute (発展的問題) paper worth 90 points, aligned to the MEXT 数学 Course of Study with constructed-response items including 計算, 証明, 作図, and 求め方記述. It is sat in the 2nd period on 11 March 2026. The five academic-test subjects (国語, 社会, 数学, 理科, 英語) each carry 90 points for a maximum of 450 points. The paper tests 知識・技能 plus 思考力・判断力・表現力. For 国語・数学・英語 the board provides three problem tiers (基礎的/標準的/発展的); each school chooses one tier; 理科 and 社会 use a single common paper.
  • Time limit: 50 minutes (基礎的/標準的問題) or 60 minutes (発展的問題). The 数学 paper is sat in the 2nd period on 11 March 2026; the full academic test is a single day — 1st 国語, 2nd 数学, 3rd 英語 (with リスニングテスト), 4th 理科, 5th 社会.
  • Exam fee: 入学検定料 (entrance examination fee) for an Osaka prefectural public high school is ¥2,200 for the 全日制 (full-time) course, ¥950 for the 定時制 (part-time) course and ¥800 for the 通信制 (correspondence) course, paid via the online 出願システム. A separate 入学料 is ¥5,650 (全日制), ¥2,100 (定時制) or ¥500 (通信制). It is paid once for the whole five-subject 学力検査, not per subject paper. Source: 令和8年度 大阪府公立高等学校入学者選抜実施要項 第7 一般入学者選抜.

Keys to Passing

  • Complete 500+ practice questions
  • Score 80%+ consistently before scheduling
  • Focus on highest-weighted sections
  • Use our AI tutor for tough concepts

Osaka Public High School Entrance Examination — Mathematics Study Tips from Top Performers

1Drill signed arithmetic, fraction operations and square-root (根号) simplification to fluency, since the Osaka paper's short-answer calculation set rewards fast, accurate computation under a 50/60-minute limit.
2Practise simultaneous-equation (連立方程式) and quadratic-equation (二次方程式) word problems, including setting up equations from a written condition and checking both solutions.
3Learn to read 箱ひげ図 (box plots) and ヒストグラム (histograms) quickly for the 資料の活用 area: identify the median, quartiles and IQR from a box plot, and estimate frequencies from histogram bar areas.
4Master the parabola y=ax²: find the vertex, axis of symmetry and x-intercepts, and combine with 根号 calculations — the 放物線 area commonly mixes these two skills.
5Drill the Pythagorean theorem (三平方の定理) on right triangles, rectangles, squares and solid figures, and practise 相似 and 空間図形 problems, since 図形 is the largest content area at 33% of the paper.
6Use tree diagrams (樹形図) for two-trial probability (two dice, draws without replacement) and be able to list favourable outcomes systematically rather than guessing.
7Because your school may sit the 発展的問題 tier (60 minutes) or the 基礎的/標準的 tier (50 minutes), time your practice under both limits and attempt the calculation items first to bank marks before the longer constructed-response 大問.

Frequently Asked Questions

Is this question bank the same as the real Osaka mathematics exam?

No. The official paper is written in Japanese with constructed-response items including 計算, 証明, 作図, and items where the 求め方 (working) must be written out. This bank is an English-language multiple-choice study adaptation of the same syllabus content and cannot replace showing your own mathematical reasoning.

How long is the Osaka mathematics paper and what is it worth?

It lasts 50 minutes (for the 基礎的 or 標準的 problem tier) or 60 minutes (for the 発展的 tier), and is scored out of 90 points, one of five subjects that together total 450 points (each subject 90 points). Mathematics is sat in the 2nd period on 11 March 2026.

How is the selection structured in 2026?

Osaka combines the academic test (学力検査, 450 points) with the report card (調査書, 450 points, 中3 ×6 and 中1・中2 ×2) into a 900-point 総合点 using a school-chosen 倍率タイプ from types I–V (I: 1.4/0.6, II: 1.2/0.8, III: 1.0/1.0, IV: 0.8/1.2, V: 0.6/1.4). The top 110% form (Ⅰ)群, the top 90% of those are admitted, and the (Ⅱ)群 ボーダーゾーン is decided against each school's アドミッションポリシー. There is no prefecture-wide pass line.

How much is the examination fee?

The 入学検定料 (entrance examination fee) for an Osaka prefectural public high school is ¥2,200 for the 全日制 (full-time) course, ¥950 for the 定時制 (part-time) course and ¥800 for the 通信制 (correspondence) course, paid via the online 出願システム. A separate 入学料 (¥5,650 full-time) is due after acceptance. It is paid once for the whole five-subject 学力検査, not per subject paper.

When was the 2026 paper sat and when were results announced?

The 令和8年度 学力検査 was held on a single day, 11 March 2026 (1st 国語, 2nd 数学, 3rd 英語 with リスニング, 4th 理科, 5th 社会), with 合格発表 on 19 March 2026 via the online 出願システム.

What content areas does the mathematics paper cover?

The paper follows the MEXT junior-high Course of Study (平成29年 中学校学習指導要領 数学編) and covers 数と式 (signed arithmetic, fractions, square roots, equations), 関数 (proportion, linear and quadratic functions, parabolas), 図形 (similarity, Pythagorean theorem, circles, solid geometry), and 資料の活用 (histograms, box plots, sample surveys, probability).