106+ Free Toyama Prefectural Public High School Entrance Examination — Mathematics Practice Questions

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

Practise for the Toyama Prefectural Public High School Entrance Exam in Mathematics (数学) with 100 four-option questions covering 数と式, 関数, 図形 and 資料の活用, including genuine worked calculations, weighted to the official 40-point, 50-minute common paper sat on 令和8年3月6日(金). The official paper mixes 記号選択 items with 短答, 記述 and 作図 answers requiring 途中の計算過程, so this English-language MCQ bank drills the underlying methods — it is not a format simulation, and you still need to practise writing solutions out in Japanese.

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

Try these sample questions to test your Toyama Prefectural Public High School Entrance Examination — Mathematics exam readiness. Each question includes a detailed explanation. Start the interactive quiz above for the full 106+ question experience with AI tutoring.

1What is the prime factorisation of 360?

A.2³ × 3² × 5

B.2² × 3² × 5

C.2³ × 3 × 5

D.2² × 3³ × 5

Explanation: 360 = 8 × 45 = 2³ × 9 × 5 = 2³ × 3² × 5. Dividing successively by primes confirms 360 = 2³ × 3² × 5.

2Find the greatest common divisor of 84 and 126.

A.14

B.21

C.42

D.6

Explanation: 84 = 2² × 3 × 7 and 126 = 2 × 3² × 7, so the GCD is 2 × 3 × 7 = 42. 42 divides both 84 (42×2) and 126 (42×3) exactly.

3Find the least common multiple of 12 and 18.

A.6

B.24

C.36

D.72

Explanation: 12 = 2² × 3 and 18 = 2 × 3², so the LCM is 2² × 3² = 36. 36 is the smallest positive number divisible by both 12 and 18.

4Solve for x: 3x − 7 = 2x + 5.

A.x = 12

B.x = 2

C.x = −2

D.x = 1

Explanation: Subtract 2x from both sides to get x − 7 = 5, then add 7 to obtain x = 12. Checking: 3(12)−7 = 29 = 2(12)+5.

5Solve the simultaneous equations: x + y = 10, 2x − y = 5.

A.x = 5, y = 5

B.x = 3, y = 7

C.x = 5, y = 3

D.x = 7, y = 3

Explanation: Adding the equations eliminates y: 3x = 15, so x = 5. Then y = 10 − 5 = 5. Checking in the second equation: 2(5) − 5 = 5.

6A shop sells pens at 3 for ¥220 and erasers at 4 for ¥260. How much do 6 pens and 8 erasers cost altogether?

A.¥480

B.¥960

C.¥1,000

D.¥740

Explanation: 6 pens are two 3-pen bundles, costing 2 × ¥220 = ¥440. 8 erasers are two 4-eraser bundles, costing 2 × ¥260 = ¥520. The total is ¥440 + ¥520 = ¥960. Working in whole bundles avoids the non-integer unit prices of ¥220 ÷ 3 and ¥260 ÷ 4.

7Solve for x: x² − 5x + 6 = 0.

A.x = 2 or x = 3

B.x = −2 or x = −3

C.x = 1 or x = 6

D.x = −1 or x = −6

Explanation: Look for two numbers whose product is the constant term 6 and whose sum is 5, the size of the middle coefficient: those numbers are 2 and 3. Factoring gives (x − 2)(x − 3) = 0, so x = 2 or x = 3. Substituting either value back into x² − 5x + 6 gives 0.

8Solve for x: x² − 7x + 12 = 0.

A.x = 3 or x = 4

B.x = 2 or x = 6

C.x = 1 or x = 12

D.x = −3 or x = −4

Explanation: Factoring gives (x − 3)(x − 4) = 0, so x = 3 or x = 4. The pair 3 and 4 multiplies to 12 and sums to 7.

9Solve the inequality 2(x − 3) < x + 1 and express the solution in the simplest form.

A.x < 7

B.x > 7

C.x < 5

D.x > 5

Explanation: Expanding gives 2x − 6 < x + 1. Subtracting x and adding 6 yields x < 7. The inequality sign does not flip because no negative multiplication or division is applied.

10Which integer values of x satisfy both 2x − 1 ≥ 3 and x + 2 < 8?

A.x = 2, 3, 4, 5

B.x = 1, 2, 3, 4, 5

C.x = 2, 3, 4

D.x = 3, 4, 5

Explanation: From 2x − 1 ≥ 3 we get x ≥ 2. From x + 2 < 8 we get x < 6. Combining, 2 ≤ x < 6, so the integers are 2, 3, 4, 5.

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

The Toyama Prefectural Public High School Entrance Examination in Mathematics (数学) tests lower-secondary graduates on numbers and expressions, functions, geometry, and data and probability. The general academic test (学力検査) is a prefecture-wide common paper covering five subjects (国語, 数学, 社会, 理科, 英語[リスニング含む]), each worth 40 points (合計 200点); for 令和8年度 (2026 admission) the 一般入学者選抜学力検査 is 3月5日(木) and 3月6日(金) with 合格発表 on 3月13日(金). The 数学 common paper (2日目, 50 minutes) combines 記号選択 (multiple-choice) items with 短答, 記述 and 作図 (constructed-response and construction) questions requiring 途中の計算過程 and proofs, so it is not a pure multiple-choice paper. Selection combines the 学力検査 total (200) with the 調査書評定点 (150) for a 350-point total, with some schools applying 傾斜配点. 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), calculators are not permitted, and it tests 知識・技能 plus 思考力・判断力・表現力.

Assessment

40-point paper aligned to the MEXT 数学 Course of Study, covering 数と式 (25%), 関数 (25%), 図形 (30%) and 資料の活用 (20%). The prefectural common paper mixes 記号選択 items with 短答, 記述 and 作図 answers that earn partial credit for 途中の計算過程; the five academic-test subjects each carry 40 points (合計 200点), sat over two days as prefectural common papers; 数学 runs 50 minutes on 2日目. Problems are at junior-high graduation level (中学校卒業程度); calculators are not permitted. The paper tests 知識・技能 plus 思考力・判断力・表現力.

Time Limit

Sat as part of the two-day five-subject 学力検査 (合計 200点, each subject 40 points): 1日目 (3月5日(木)) covers 社会, 国語 and 理科 (each 50 minutes); 2日目 (3月6日(金)) covers 英語 (聞き取り about 10 minutes plus 筆記 40 minutes) and 数学 (50 minutes). For 令和8年度 the 追検査 is 3月10日(火) and 一般入学者選抜合格発表 is 3月13日(金). Calculators are not permitted. Source: 令和8年度富山県立高等学校入学者選抜実施要領 / 入学者選抜日程, 富山県教育委員会.

Passing Score

No fixed prefecture-wide pass mark. The 数学 paper is scored out of 40 points. In the 一般入学者選抜 the five subjects (国語・数学・社会・理科・英語) total 200 points (each subject 40 points); the 調査書評定点 (内申点) adds 150 points (中二 9教科×5=45, 中三 9教科×5×2=90, 特別活動等 15), for a combined 350-point total. Some schools apply 傾斜配点 to emphasise certain subjects. If either the 学力検査 or the 調査書 result places an applicant within the top 10% of 定員, they may be admitted regardless of the other score. Selection is relative, not against a published pass line. Source: 令和8年度富山県立高等学校入学者選抜実施要領, 富山県教育委員会.

Exam Fee

The 入学考査手数料 is ¥2,200 for a 全日制 course and ¥950 for a 定時制 course; it may be reduced or waived in cases of financial hardship. Source: 富山県立高等学校入学者選抜に係る入学考査手数料収納代行業務仕様書 (富山県教育委員会教育みらい室県立高校課) and 令和8年度富山県立高等学校入学者選抜実施要領. (Toyama Prefectural Board of Education (富山県教育委員会), 教育みらい室県立高校課, 〒930-8501 富山市新総曲輪1-7 県庁南別館4階 (TEL 076-444-3448, FAX 076-444-4437))

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

25%

Numbers & Expressions (数と式)

Integers, prime factorisation, linear equations, simultaneous equations (連立方程式), quadratic equations, inequalities, algebraic expressions and identities, polynomials, exponents, and square roots.

25%

Functions (関数)

Linear functions, proportional and inverse proportional relationships, quadratic functions y=ax² (二次関数), parabolas, vertex and axis of symmetry, intersections of parabolas with lines, and function–geometry fusion problems.

30%

Geometry (図形)

Congruence and similarity, triangles, quadrilaterals, circles, arcs and chords, inscribed and central angles (円周角の定理), the Pythagorean theorem (ピタゴラスの定理 / 三平方の定理), parallel-line proportionality, surface area and volume of solids, and constructions (作図).

20%

Data & Probability (資料の活用)

Mean, median, mode and range, histograms, frequency distributions, relative and cumulative frequency, quartiles, interquartile range and box-and-whisker plots (箱ひげ図), sampling surveys (標本調査), and probability. Variance and standard deviation are 高校数学I content and are not tested here.

How to Pass the Toyama 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 40 points. In the 一般入学者選抜 the five subjects (国語・数学・社会・理科・英語) total 200 points (each subject 40 points); the 調査書評定点 (内申点) adds 150 points (中二 9教科×5=45, 中三 9教科×5×2=90, 特別活動等 15), for a combined 350-point total. Some schools apply 傾斜配点 to emphasise certain subjects. If either the 学力検査 or the 調査書 result places an applicant within the top 10% of 定員, they may be admitted regardless of the other score. Selection is relative, not against a published pass line. Source: 令和8年度富山県立高等学校入学者選抜実施要領, 富山県教育委員会.
Assessment: 40-point paper aligned to the MEXT 数学 Course of Study, covering 数と式 (25%), 関数 (25%), 図形 (30%) and 資料の活用 (20%). The prefectural common paper mixes 記号選択 items with 短答, 記述 and 作図 answers that earn partial credit for 途中の計算過程; the five academic-test subjects each carry 40 points (合計 200点), sat over two days as prefectural common papers; 数学 runs 50 minutes on 2日目. Problems are at junior-high graduation level (中学校卒業程度); calculators are not permitted. The paper tests 知識・技能 plus 思考力・判断力・表現力.
Time limit: Sat as part of the two-day five-subject 学力検査 (合計 200点, each subject 40 points): 1日目 (3月5日(木)) covers 社会, 国語 and 理科 (each 50 minutes); 2日目 (3月6日(金)) covers 英語 (聞き取り about 10 minutes plus 筆記 40 minutes) and 数学 (50 minutes). For 令和8年度 the 追検査 is 3月10日(火) and 一般入学者選抜合格発表 is 3月13日(金). Calculators are not permitted. Source: 令和8年度富山県立高等学校入学者選抜実施要領 / 入学者選抜日程, 富山県教育委員会.
Exam fee: The 入学考査手数料 is ¥2,200 for a 全日制 course and ¥950 for a 定時制 course; it may be reduced or waived in cases of financial hardship. Source: 富山県立高等学校入学者選抜に係る入学考査手数料収納代行業務仕様書 (富山県教育委員会教育みらい室県立高校課) and 令和8年度富山県立高等学校入学者選抜実施要領.

Keys to Passing

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

1Master exact arithmetic without a calculator — common fractions, square roots and simplification — because the paper is sat calculator-free.

2For 連立方程式 (simultaneous equations), drill both the mechanical solution and the applied word-problem set-up, since applied items test translation from text to equations.

3For geometry, write proofs in the standard assertion-reason format, citing the congruence/similarity condition explicitly, because 記述 proof problems and 作図 (construction) items reward structure.

4For 二次関数 y=ax², memorise how a changes the parabola's width and direction, and practise finding the vertex, axis of symmetry and intersections with a line — function–geometry fusion items are among the hardest.

5For 資料の活用, compute mean, median, relative frequency, interquartile range and probability fluently, practise reading histograms and 箱ひげ図 (box-and-whisker plots) quickly, and be ready to estimate a population from a 標本調査.

6Practise both 記号選択 (multiple-choice) and 短答 / 記述 working — Toyama's common paper uses both — and time yourself on full 50-minute papers, leaving the longest proof, 作図 or applied item for last.

Frequently Asked Questions

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

No. The official paper is worked in Japanese and combines 記号選択 (multiple-choice) items with 短答 (short-answer), 記述 (constructed-response) and 作図 (construction) questions that require 途中の計算過程 and proofs; its exact item count is not published. This bank is an English-language multiple-choice study adaptation of the same junior-high syllabus (数と式, 関数 including y=ax², 図形 with 相似・円周角・三平方の定理・作図, and 資料の活用) and cannot replace writing out your own calculations in Japanese under timed conditions.

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

It is scored out of 40 points, one of five subjects totalling 200 points (each subject 40 points). 数学 runs 50 minutes on 2日目 (令和8年3月6日(金)), with 一般入学者選抜合格発表 on 3月13日(金); the 推薦入学者選抜検査 (面接等) is 2月9日(月) with 内定通知 on 2月12日(木), and a 追検査 is 3月10日(火).

Are calculators allowed?

No. The mathematics paper is sat without a calculator, in line with standard prefectural academic-test rules; numerical reasoning and exact computation are expected.

How is the selection structured?

Toyama's 一般入学者選抜 combines the 学力検査 total (5 subjects × 40 = 200 points, with some schools applying 傾斜配点 to emphasise certain subjects) with the 調査書評定点 (内申点, 150 points: 中二 45 + 中三 90 + 特別活動等 15) for a 350-point total. If either the 学力検査 or 調査書 result places an applicant in the top 10% of 定員, they may be admitted regardless of the other score. Some schools add 面接 (interview) or 実技検査. Selection is relative to other applicants, not against a published pass line.

How much is the examination fee?

The 入学考査手数料 is ¥2,200 for a 全日制 course (¥950 for 定時制); it may be reduced or waived in cases of financial hardship.

106+ Free Toyama Prefectural Public High School Entrance Examination — Mathematics Practice Questions

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