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

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Key Facts: Yamagata Prefectural Public High School Entrance Examination — Mathematics Exam

Practise for the Yamagata Prefectural Public High School Entrance Exam in Mathematics (数学) with 100 four-option questions covering algebra, functions, geometry, data/probability, patterns and applied problems, each with a full worked solution, weighted to the official 50-minute paper (大問4題) sat second on 令和8年3月7日(土) (10:00-10:50). The official paper also has a 図形の証明 (geometric proof) and free-response items that this MCQ bank does not reproduce, so you must still practise proofs and 記述式 separately.

Sample Yamagata Prefectural Public High School Entrance Examination — Mathematics Practice Questions

Try these sample questions to test your Yamagata Prefectural 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.

1Solve for x: 3x − 5 = 16.

A.7

B.5

C.−7

D.21

Explanation: Add 5 to both sides: 3x = 21, then divide by 3: x = 7. Check: 3(7)−5 = 21−5 = 16.

2Solve: 2(x + 3) = 4x − 6. What is x?

A.6

B.0

C.−6

D.3

Explanation: Expand: 2x + 6 = 4x − 6 → 6 + 6 = 4x − 2x → 12 = 2x → x = 6. Check: 2(9) = 18 and 4(6)−6 = 18.

3Simplify: (2x + 3) + (x − 5) − (3x + 1).

A.−3

B.3x − 3

C.6x − 3

D.−3x − 3

Explanation: Combine like terms: 2x + x − 3x = 0x; 3 − 5 − 1 = −3. The x terms cancel, leaving the constant −3.

4Expand (x + 4)(x − 2).

A.x² + 2x − 8

B.x² − 2x + 8

C.x² + 6x − 8

D.x² − 6x − 8

Explanation: FOIL: x·x = x², x·(−2) = −2x, 4·x = 4x, 4·(−2) = −8. Sum: x² + (−2+4)x − 8 = x² + 2x − 8.

5Factor: x² − 9.

A.(x − 3)(x + 3)

B.(x − 9)(x + 1)

C.(x − 3)²

D.(x + 3)²

Explanation: This is a difference of squares: a² − b² = (a−b)(a+b) with a = x, b = 3, giving (x−3)(x+3).

6Factor: x² + 5x + 6.

A.(x + 2)(x + 3)

B.(x − 2)(x − 3)

C.(x + 1)(x + 6)

D.(x + 6)(x − 1)

Explanation: Find two numbers whose product is 6 and sum is 5: 2 and 3. So x² + 5x + 6 = (x + 2)(x + 3).

7Solve the simultaneous equations: x + y = 10, x − y = 4. What is x?

A.7

B.6

C.8

D.3

Explanation: Add the equations: 2x = 14 → x = 7. Then y = 10 − 7 = 3. Check: 7 + 3 = 10 and 7 − 3 = 4.

8Solve: 3x + 2y = 12, x = 2. What is y?

A.3

B.2

C.4

D.6

Explanation: Substitute x = 2: 3(2) + 2y = 12 → 6 + 2y = 12 → 2y = 6 → y = 3.

9If a:b = 3:4 and b:c = 4:5, what is a:c?

A.3:5

B.3:4

C.4:5

D.5:3

Explanation: Since b is common (4 in both ratios), a:b:c = 3:4:5 directly, so a:c = 3:5.

10A number divided by 4 gives quotient 7 and remainder 3. What is the number?

A.31

B.28

C.25

D.34

Explanation: Number = 4 × quotient + remainder = 4 × 7 + 3 = 28 + 3 = 31. Check: 31 ÷ 4 = 7 remainder 3.

About the Yamagata Prefectural Public High School Entrance Examination — Mathematics Exam

The Yamagata Prefectural Public High School Entrance Examination in Mathematics (数学) tests lower-secondary graduates on numbers & expressions, equations, functions, geometry (including a proof), and data & probability. From 2026 (令和8年度) a new two-stage system replaces the old 推薦選抜: a 前期（特色）選抜 (each school chooses 1-3 of 個人面接・集団面接・作文・発表・小論文・実技・口頭試問・学校独自学力検査, plus 調査書; A日程 1/20(火) or B日程 2/3(火)) and a 後期（一般）選抜 (five-subject 学力検査 plus 調査書). The general academic test (学力検査) is a prefecture-wide common paper covering five subjects (国語, 社会, 数学, 理科, 英語), each worth 100 points (合計 500点); for 令和8年度 the 後期学力検査 is sat on a single day, 3月7日(土). The 数学 common paper (10:00-10:50, sat second; 大問4題) spans 基礎～応用, includes a 図形の証明 and 関数の文章題, has relatively few sub-questions with high per-question weight, and mixes 記号選択 (multiple-choice) with 短答記述 and 記述式 (proof) answers, so it is not a pure multiple-choice paper. Selection combines the 学力検査 total (500) with the 調査書 (中3 45点 → 500点換算) on the school-chosen ratio (3:7/4:6/5:5/6:4/7:3), with each school setting its own比重 and some applying 傾斜配点 (1.5×). 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.

Assessment

100-point paper (大問4題) aligned to the MEXT 数学 Course of Study, covering algebra — equations & expressions (28%), functions (22%), geometry (30%), data & probability (10%), patterns & sequences (5%) and applied problems (5%). The prefectural common paper mixes 記号選択 items with 短答記述 and a 図形の証明 (geometric proof); it has relatively few sub-questions with high per-question weight. The five academic-test subjects each carry 100 points (合計 500点), and 数学 is sat second on the single 学力検査 day (令和8年3月7日(土), 10:00-10:50). No official per-topic percentage weighting is published for 数学; the percentages above are this bank's own curriculum-derived study weighting. The paper tests 知識・技能 plus 思考力・判断力・表現力, including reading conditions and constructing a proof. The Yamagata Prefectural Board of Education does not publish per-topic percentage weights for the 数学 paper, so the percentages listed here are this practice bank's own study weighting, derived from the MEXT Course of Study and the recurring 大問 structure (小問集合・関数と方程式・関数とグラフ・平面図形と証明).

Time Limit

Sat as part of the one-day five-subject 学力検査 (合計 500点, each subject 100 points) on 令和8年3月7日(土): 国語 (8:50-9:40), 数学 (10:00-10:50, sat second), 社会 (11:10-12:00), 理科 (12:50-13:40) and 英語 (14:00-14:50) in that order (each 50 minutes). The 前期（特色）選抜検査 is on A日程令和8年1月20日(火) or B日程令和8年2月3日(火) (each school chooses). Calculators are not permitted. Source: 令和8年度山形県公立高等学校入学者選抜実施要項, 山形県教育委員会.

Passing Score

No fixed prefecture-wide pass mark. The 数学 paper is scored out of 100 points. In the 後期（一般）選抜 the five subjects (国語・社会・数学・理科・英語) total 500 points (each subject 100 points); the 調査書 (内申点) uses 中3 9教科×5段階評定 = 45点満点 and is converted to a 500-point basis, then combined with the 500-point 学力検査 on the school-chosen ratio (3:7, 4:6, 5:5, 6:4 or 7:3). Each school may apply 傾斜配点 (a 1.5× weighting on one or two specified subjects; for 令和8年度 every school that uses it applies the 1.5× to 数学 and 外国語(英語)). Selection is relative to other applicants, not against a published pass line. Source: 令和8年度山形県公立高等学校入学者選抜実施要項, 山形県教育委員会.

Exam Fee

The 入学者選抜手数料 is ¥2,200 for a full-time (全日制) course (¥950 for 定時制, ¥300 for 通信制); it may be reduced or waived in cases of financial hardship. Source: 令和8年度山形県公立高等学校入学者選抜実施要項 / 山形県立学校の授業料等徴収条例, 山形県教育委員会. (Yamagata Prefectural Board of Education (山形県教育委員会), 教育局高校教育課, 〒990-8570 山形市松波二丁目8番1号 (TEL 023-630-3287, FAX 023-630-2774))

Yamagata Prefectural Public High School Entrance Examination — Mathematics Exam Content Outline

28%

Algebra — Equations & Expressions (数と式・方程式)

Linear and simultaneous equations, ratios, factoring, quadratics, identities, inequalities and radicals.

22%

Functions (関数)

Linear and quadratic functions, slope, vertex, direct and inverse proportion, exponential (y = 2^x), absolute value and transformations.

30%

Geometry (図形)

Angles, Pythagorean theorem, area/perimeter, circles, polygons, similarity and congruence, volume/surface, special triangles and locus — including proof-style reasoning.

10%

Data & Probability (資料の活用)

Mean, median, mode, probability, counting, outliers and reading data from tables.

Patterns & Sequences (規則性)

Arithmetic and geometric patterns, sums and figure-pattern reasoning.

Applied Problems (文章題)

Rates, percentages, scale, tax and real-world calculation problems.

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

What You Need to Know

Passing score: No fixed prefecture-wide pass mark. The 数学 paper is scored out of 100 points. In the 後期（一般）選抜 the five subjects (国語・社会・数学・理科・英語) total 500 points (each subject 100 points); the 調査書 (内申点) uses 中3 9教科×5段階評定 = 45点満点 and is converted to a 500-point basis, then combined with the 500-point 学力検査 on the school-chosen ratio (3:7, 4:6, 5:5, 6:4 or 7:3). Each school may apply 傾斜配点 (a 1.5× weighting on one or two specified subjects; for 令和8年度 every school that uses it applies the 1.5× to 数学 and 外国語(英語)). Selection is relative to other applicants, not against a published pass line. Source: 令和8年度山形県公立高等学校入学者選抜実施要項, 山形県教育委員会.
Assessment: 100-point paper (大問4題) aligned to the MEXT 数学 Course of Study, covering algebra — equations & expressions (28%), functions (22%), geometry (30%), data & probability (10%), patterns & sequences (5%) and applied problems (5%). The prefectural common paper mixes 記号選択 items with 短答記述 and a 図形の証明 (geometric proof); it has relatively few sub-questions with high per-question weight. The five academic-test subjects each carry 100 points (合計 500点), and 数学 is sat second on the single 学力検査 day (令和8年3月7日(土), 10:00-10:50). No official per-topic percentage weighting is published for 数学; the percentages above are this bank's own curriculum-derived study weighting. The paper tests 知識・技能 plus 思考力・判断力・表現力, including reading conditions and constructing a proof. The Yamagata Prefectural Board of Education does not publish per-topic percentage weights for the 数学 paper, so the percentages listed here are this practice bank's own study weighting, derived from the MEXT Course of Study and the recurring 大問 structure (小問集合・関数と方程式・関数とグラフ・平面図形と証明).
Time limit: Sat as part of the one-day five-subject 学力検査 (合計 500点, each subject 100 points) on 令和8年3月7日(土): 国語 (8:50-9:40), 数学 (10:00-10:50, sat second), 社会 (11:10-12:00), 理科 (12:50-13:40) and 英語 (14:00-14:50) in that order (each 50 minutes). The 前期（特色）選抜検査 is on A日程令和8年1月20日(火) or B日程令和8年2月3日(火) (each school chooses). Calculators are not permitted. Source: 令和8年度山形県公立高等学校入学者選抜実施要項, 山形県教育委員会.
Exam fee: The 入学者選抜手数料 is ¥2,200 for a full-time (全日制) course (¥950 for 定時制, ¥300 for 通信制); it may be reduced or waived in cases of financial hardship. Source: 令和8年度山形県公立高等学校入学者選抜実施要項 / 山形県立学校の授業料等徴収条例, 山形県教育委員会.

Keys to Passing

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

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

1Secure the basics first — the paper has few sub-questions with high per-question weight, so a single 基礎 miss is costly; drill linear/simultaneous equations, factoring and the Pythagorean theorem until automatic.

2For 関数, master slope, vertex and roots of quadratics, and practise translating a word problem into an equation before calculating.

3For 図形, prepare a proof template (仮定 → 既知の定理 → 結論) and rehearse the common routes: congruence (合同) for equal sides/angles, similarity (相似) for ratio problems, and the circle theorems.

4For 資料の活用, practise mean/median/mode and basic probability quickly; these are reliable points if you avoid arithmetic slips.

5Always check the conditions (x ≠ 2, etc.) and re-substitute your solution into the original equation to catch extraneous roots.

6Time the paper tightly — few questions mean little recovery time — and save the proof for when you have a clear route, not first.

Frequently Asked Questions

Is this question bank the same as the real Yamagata 数学 exam?

No. The official paper (大問4題) spans 基礎～応用 and includes a 図形の証明 (geometric proof) and 関数の文章題, with relatively few sub-questions and high per-question weight; its exact item count is not published. This bank is an English-language multiple-choice study adaptation covering algebra, functions, geometry, data/probability, patterns and applied problems, each with a worked solution; it does not reproduce the geometric-proof or free-response tasks and cannot replace them.

How long is the Yamagata 数学 paper and what is it worth?

It is scored out of 100 points, one of five subjects totalling 500 points (each subject 100 points). 数学 is sat second on the single 学力検査 day (令和8年3月7日(土), 10:00-10:50). The 前期（特色）選抜検査 is on A日程 1月20日(火) or B日程 2月3日(火). Calculators are not permitted.

Does the paper include a proof?

Yes. The 数学 paper includes a 図形の証明 (geometric proof) and 関数の文章題 alongside multiple-choice and 短答記述 items; it has relatively few sub-questions with high per-question weight. This MCQ bank covers only the calculations and reasoning behind those tasks; you must practise writing out proofs and 記述式 answers separately.

How is the selection structured?

From 2026 (令和8年度) Yamagata uses a two-stage system: 前期（特色）選抜 (each school chooses 1-3 of 面接・作文・発表・小論文・実技・口頭試問・学校独自学力検査, plus 調査書; A日程 1/20(火) or B日程 2/3(火)) and 後期（一般）選抜 (five-subject 学力検査 plus 調査書). The 後期 combines the 学力検査 total (5 subjects × 100 = 500 points) with the 調査書 (中3 9教科×5 = 45点, converted to a 500-point basis) on the school-chosen ratio (3:7/4:6/5:5/6:4/7:3). Each school may apply 傾斜配点 (a 1.5× weighting on one or two specified subjects; for 令和8年度 every school that uses it applies the 1.5× to 数学 and 外国語(英語)). Selection is relative to other applicants, not against a published pass line.

How much is the examination fee?

The 入学者選抜手数料 is ¥2,200 for a full-time (全日制) course (¥950 for 定時制, ¥300 for 通信制); it may be reduced or waived in cases of financial hardship.

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