# English mathematics: original derivations and adaptation record

Created 2026-10-04 (Japan time). Author: english_math_author. These 300 individually written English adaptations use 300 reviewed Japanese IDs and families: 56 from each of the original five 80-question topic blocks plus all twenty foundation questions ma-0401 through ma-0420. They preserve each concept, numerical condition, revision, correct option ID and canonical subject/topic/level/difficulty. They do not add 300 new conceptual families to the Japanese catalogue. All English entries are drafts awaiting independent review.

The mathematical conditions, fictional data, problem wording and diagrams are original project work. No external textbook, exam question, dataset or image was copied. This English record reconstructs the derivations from the stated conditions; it is author evidence, not independent approval. The base bank SHA-256 is d482f2589cb090a492c1b03b0ac781e12e5bfbeb1caa7d2f22be7628d575fd39. The original Japanese derivation documents were read for condition and concept consistency: docs/quiz/math-authoring.md and docs/quiz/math-basics-20261004-authoring.md. The daily data authoring record was also read to avoid treating pending daily drafts as reviewed catalogue material.

## Source scope and conventions

The original source supports the mathematical derivations and original fictional conditions below. It does not independently establish external scientific, historical, legal or societal facts. In ma-0279, the stated correlation is a hypothetical condition, and temperature is an example of a possible common factor rather than a claimed observed causal relationship. Logical weather statements are propositions used to practise transformations, not meteorological guarantees. Survey selection questions compare coverage of a defined target population, not actual survey results. Currency amounts and pricing rules are fictional conditions, not current prices or financial guidance.

Standard unit relations used in conversion exercises are 1 m = 100 cm and 1 L = 1000 mL; the exercises apply these conventional definitions, not a measurement of real objects. ma-0416 retains an external source for the established mass/length unit names. The BIPM SI base-units page was actually opened and read on 2026-10-04: its table lists mass as kilogram (kg) and length as metre (m). URL: https://www.bipm.org/en/measurement-units/si-base-units . The original derivation source does not claim to prove that an international standard selected these names. Capacity and area are distinguished as different mathematical kinds of quantity.

## Shared derivations behind geometric formulas

For a Euclidean triangle, a line through a vertex parallel to the opposite side expresses its three interior angles as a straight angle, hence 180 degrees. A midpoint segment follows by taking half of each endpoint vector. The Pythagorean relation follows by arranging four identical right triangles inside a square of side a+b: the central square has area c² and the remainder gives c² = (a+b)² − 2ab = a²+b².

The circumference relation C = 2πr follows from the definition of π as circumference divided by diameter. For area, integrating circular rings gives ∫₀ʳ 2πt dt = πr². A circle sector uses its angle as a fraction of a full turn. For a sphere, a surface of revolution calculation with y = √(r² − x²) gives surface strip area 2πy√(1+(dy/dx)²) dx = 2πr dx on −r < x < r. Integrating from −r to r gives 4πr². Thus ma-0200 uses a mathematical derivation, rather than an unexplained external surface-area fact.

For a point on a circle, join it to the centre to make isosceles triangles. Their equal base angles and the triangle angle sum show that an inscribed angle is half the central angle subtending the same arc. Opposite inscribed angles in a cyclic convex quadrilateral therefore sum to 180 degrees. The tangent-radius perpendicularity follows because the perpendicular is the shortest segment from the centre to that line. For the cosine rule, resolve a side into components b cos C and b sin C; the opposite side's squared length is (a − b cos C)² + (b sin C)² = a²+b²−2ab cos C. A chord subtending angle A has length 2R sin A by bisecting its associated isosceles triangle, yielding the extended sine rule.

## Selection and difficulty boundaries

Each original topic contributes exactly 56 independent families. Foundational additions contribute nine number/calculation, one expression/function, four geometry, two probability/statistics and four logic/application families. Selection favours basic and standard classifications: no advanced question was selected. Arithmetic repeats and several more specialized proofs were omitted to make room for place value, fractions, logarithms, complex numbers, functions, calculus, geometry, linear algebra, descriptive statistics, probability, elementary inference, practical rates, logic, sets and graphs. A translation is an alternative language presentation of the same exercise. It is not counted as a new Japanese question or as a numerical/synonym variant. The existing Japanese catalogue and its 2,000-question acceptance baseline remain untouched. Learning-level labels remain the canonical Japanese prerequisites; they are not asserted as alignment to any English-speaking country's curriculum. Difficulty has not been calibrated with user data.

All diagram geometry, element order, colours/tones and numerical or letter labels are preserved. Only descriptions and Japanese text labels are translated. No answer value or solution step is added to a diagram. The blind review descriptions communicate the same input conditions as the actual diagrams.

## Per-question derivations

### ma-0001 — ma-place-value (revision 1)

Condition: What is the hundreds digit in 30405?

Derivation: From the right, the places are ones, tens, then hundreds. The digit 4 is in the hundreds place and represents 400.

Options in canonical ID order: a: 4; b: 0; c: 3; d: 5. Derived answer: a.

### ma-0002 — ma-decimal-order (revision 1)

Condition: Which is largest: 0.7, 0.65, 0.09 or 0.705?

Derivation: Write the numbers with three decimal places: 0.700, 0.650, 0.090 and 0.705. Comparing place by place shows that 0.705 is largest.

Options in canonical ID order: a: 0.09; b: 0.705; c: 0.7; d: 0.65. Derived answer: b.

### ma-0005 — ma-multiply-meaning (revision 1)

Condition: There are four bags with six items in each bag. Which expression gives the total number of items?

Derivation: There are four equal groups of six, so multiply 6 by 4. Addition of 6 and 4 would count neither the groups nor their contents correctly.

Options in canonical ID order: a: 6×4; b: 6＋4; c: 6−4; d: 6÷4. Derived answer: a.

### ma-0006 — ma-division-remainder (revision 1)

Condition: Pack 29 items into boxes holding six items each. How many full boxes are there, and how many items remain?

Derivation: 29 = 6 × 4 + 5. Four boxes are full and five items remain. A remainder must be smaller than the six items needed for another full box.

Options in canonical ID order: a: 3 boxes, 11 items; b: 4 boxes, 5 items; c: 5 boxes, 1 item; d: 4 boxes, 3 items. Derived answer: b.

### ma-0008 — ma-order-operations (revision 1)

Condition: What is 8 + 3 × 4?

Derivation: Multiplication comes before addition: 3 × 4 = 12, then 8 + 12 = 20.

Options in canonical ID order: a: 44; b: 32; c: 15; d: 20. Derived answer: d.

### ma-0010 — ma-fraction-part (revision 1)

Condition: A loaf is divided into eight equal pieces. What fraction of the whole is three pieces?

Derivation: The denominator counts all eight equal pieces. The numerator counts the three selected pieces, giving 3/8.

Options in canonical ID order: a: 5/8; b: 3/8; c: 8/3; d: 3/5. Derived answer: b.

### ma-0011 — ma-fraction-equivalence (revision 1)

Condition: Which fraction equals 2/3?

Derivation: Multiplying both numerator and denominator by 4 gives 8/12. Scaling both by the same nonzero number preserves the fraction's value.

Options in canonical ID order: a: 6/8; b: 3/2; c: 8/12; d: 2/6. Derived answer: c.

### ma-0012 — ma-fraction-add (revision 1)

Condition: What is 1/4 + 1/6?

Derivation: Use denominator 12: 1/4 = 3/12 and 1/6 = 2/12. Their sum is 5/12.

Options in canonical ID order: a: 2/10; b: 1/10; c: 7/12; d: 5/12. Derived answer: d.

### ma-0013 — ma-fraction-multiply (revision 1)

Condition: What is three fifths of two thirds of a whole?

Derivation: Taking a fraction of an amount means multiplying: (2/3) × (3/5) = 6/15 = 2/5 of the original whole.

Options in canonical ID order: a: 2/5; b: 5/8; c: 6/5; d: 5/2. Derived answer: a.

### ma-0014 — ma-fraction-divide (revision 1)

Condition: How many portions of 1/8 L can be made from 3/4 L?

Derivation: Divide the total by the size of one portion: (3/4) ÷ (1/8) = (3/4) × 8 = 6 portions.

Options in canonical ID order: a: 12 portions; b: 6 portions; c: 3 portions; d: 8 portions. Derived answer: b.

### ma-0015 — ma-decimal-fraction (revision 1)

Condition: Write 0.125 as a fraction in lowest terms.

Derivation: 0.125 = 125/1000. Dividing numerator and denominator by 125 gives 1/8.

Options in canonical ID order: a: 1/5; b: 1/125; c: 1/8; d: 1/4. Derived answer: c.

### ma-0016 — ma-percent (revision 1)

Condition: What is 25% of a group of 40 people?

Derivation: 25% is one quarter. One quarter of 40 is 40 ÷ 4 = 10 people.

Options in canonical ID order: a: 15 people; b: 20 people; c: 25 people; d: 10 people. Derived answer: d.

### ma-0017 — ma-ratio-share (revision 1)

Condition: Divide 30 items in the ratio 2:3. How many items are in the smaller share?

Derivation: There are 2 + 3 = 5 ratio units. Each unit is 30 ÷ 5 = 6 items, so the smaller share is 2 × 6 = 12 items.

Options in canonical ID order: a: 12 items; b: 10 items; c: 15 items; d: 18 items. Derived answer: a.

### ma-0018 — ma-rounding (revision 1)

Condition: Round 3.746 to two decimal places using the rule that a next digit of 5 or more rounds up.

Derivation: The third decimal digit is 6, so increase the second decimal digit from 4 to 5. The result is 3.75.

Options in canonical ID order: a: 3.70; b: 3.75; c: 3.74; d: 3.8. Derived answer: b.

### ma-0020 — ma-inverse-check (revision 1)

Condition: Which equation checks that 72 ÷ 8 = 9?

Derivation: Multiplying the quotient by the divisor should recover the dividend: 9 × 8 = 72.

Options in canonical ID order: a: 72＋8＝80; b: 9−8＝1; c: 72×8＝576; d: 9×8＝72. Derived answer: d.

### ma-0021 — ma-signed-add (revision 1)

Condition: What is −7 + 12?

Derivation: Starting at −7 and moving 12 units to the right gives 5. For opposite signs, subtract the smaller absolute value from the larger.

Options in canonical ID order: a: 5; b: −5; c: 19; d: −19. Derived answer: a.

### ma-0022 — ma-signed-subtract (revision 1)

Condition: What is −4 − (−9)?

Derivation: Subtracting a negative is adding its opposite: −4 + 9 = 5.

Options in canonical ID order: a: −5; b: 5; c: −13; d: 13. Derived answer: b.

### ma-0023 — ma-signed-product (revision 1)

Condition: What is (−3) × (−5) × 2?

Derivation: Two negative factors give a positive product: (−3) × (−5) = 15. Then 15 × 2 = 30.

Options in canonical ID order: a: 10; b: −10; c: 30; d: −30. Derived answer: c.

### ma-0024 — ma-absolute-value (revision 1)

Condition: What is the distance between −6 and 2 on a number line?

Derivation: Distance is the absolute difference. |2 − (−6)| = |8| = 8, so it cannot be negative.

Options in canonical ID order: a: 4; b: −8; c: −4; d: 8. Derived answer: d.

### ma-0025 — ma-power-sign (revision 1)

Condition: What are the values of −3² and (−3)², in that order?

Derivation: In −3², square 3 first and then apply the minus sign, giving −9. In (−3)², the negative number itself is squared, giving 9.

Options in canonical ID order: a: −9 and 9; b: 9 and 9; c: −9 and −9; d: 9 and −9. Derived answer: a.

### ma-0026 — ma-reciprocal (revision 1)

Condition: What is the reciprocal of −2/5?

Derivation: The reciprocal must multiply the original number to give 1. (−2/5) × (−5/2) = 1, so it is −5/2.

Options in canonical ID order: a: 2/5; b: −5/2; c: 5/2; d: −2/5. Derived answer: b.

### ma-0027 — ma-prime (revision 1)

Condition: Which number is prime?

Derivation: A prime is an integer greater than 1 whose only positive divisors are 1 and itself. For 29, testing primes up to √29 requires only 2, 3 and 5; none divides it. The other choices are composite or 1.

Options in canonical ID order: a: 21; b: 49; c: 29; d: 1. Derived answer: c.

### ma-0028 — ma-factorization (revision 1)

Condition: What is the prime factorization of 84?

Derivation: 84 = 4 × 21 = 2² × 3 × 7. All factors in a prime factorization must be prime; a factor of 4 is not yet fully factored.

Options in canonical ID order: a: 2×3²×7; b: 2³×7; c: 3×4×7; d: 2²×3×7. Derived answer: d.

### ma-0029 — ma-gcd (revision 1)

Condition: What is the greatest common divisor of 18 and 30?

Derivation: 18 = 2 × 3² and 30 = 2 × 3 × 5. Their common prime factors give 2 × 3 = 6.

Options in canonical ID order: a: 6; b: 3; c: 12; d: 90. Derived answer: a.

### ma-0030 — ma-lcm (revision 1)

Condition: What is the least common multiple of 12 and 18?

Derivation: 12 = 2² × 3 and 18 = 2 × 3². Taking each prime at its larger exponent gives 2² × 3² = 36.

Options in canonical ID order: a: 216; b: 36; c: 6; d: 72. Derived answer: b.

### ma-0031 — ma-divisibility-nine (revision 1)

Condition: Which number is divisible by 9?

Derivation: A number is divisible by 9 exactly when its digit sum is. For 342, 3 + 4 + 2 = 9. The other digit sums are 10, 11 and 8.

Options in canonical ID order: a: 343; b: 344; c: 342; d: 341. Derived answer: c.

### ma-0032 — ma-terminating (revision 1)

Condition: Which fraction in lowest terms has a terminating decimal expansion?

Derivation: A reduced denominator containing only factors 2 and 5 divides some power of 10. Since 40 = 2³ × 5, 7/40 terminates. Denominators 3, 7 and 6 contain other prime factors.

Options in canonical ID order: a: 1/3; b: 2/7; c: 5/6; d: 7/40. Derived answer: d.

### ma-0033 — ma-repeating (revision 1)

Condition: Which number equals 0.333… with the digit 3 repeating forever?

Derivation: Let x = 0.333…. Then 10x = 3.333…, so subtracting gives 9x = 3 and x = 1/3. A finite decimal such as 0.333 is different.

Options in canonical ID order: a: 1/3; b: 3/10; c: 33/100; d: 333/1000. Derived answer: a.

### ma-0034 — ma-root (revision 1)

Condition: What is √81?

Derivation: The square-root symbol denotes the nonnegative square root. Since 9² = 81, √81 = 9. The equation x² = 81, by contrast, has two solutions.

Options in canonical ID order: a: 40.5; b: 9; c: ±9; d: −9. Derived answer: b.

### ma-0035 — ma-root-simplify (revision 2)

Condition: Simplify √72.

Derivation: 72 = 36 × 2, so √72 = √36 × √2 = 6√2.

Options in canonical ID order: a: 8√3; b: 36√2; c: 6√2; d: 3√2. Derived answer: c.

### ma-0036 — ma-root-rationalize (revision 1)

Condition: Which expression is 1/√3 with a rational denominator?

Derivation: Multiply numerator and denominator by √3. The denominator becomes 3, giving √3/3.

Options in canonical ID order: a: √3; b: 3/√3; c: 1/3; d: √3/3. Derived answer: d.

### ma-0037 — ma-irrational (revision 1)

Condition: Which number is irrational?

Derivation: √2 cannot equal a ratio of integers: a reduced ratio with square 2 would force both numerator and denominator to be even. The other choices are 1/4, −7 and 3, all rational.

Options in canonical ID order: a: √2; b: 0.25; c: −7; d: √9. Derived answer: a.

### ma-0039 — ma-negative-exponent (revision 2)

Condition: What is 2⁻³?

Derivation: A negative exponent means a reciprocal: 2⁻³ = 1/2³ = 1/8. It does not make the value negative.

Options in canonical ID order: a: −6; b: 8; c: 1/8; d: −8. Derived answer: c.

### ma-0040 — ma-scientific (revision 1)

Condition: Write 0.00042 in scientific notation with a coefficient at least 1 and less than 10.

Derivation: Move the decimal point four places to obtain 4.2, then multiply by 10⁻⁴ to preserve the value: 4.2 × 10⁻⁴.

Options in canonical ID order: a: 4.2×10⁴; b: 42×10⁻⁴; c: 0.42×10⁻³; d: 4.2×10⁻⁴. Derived answer: d.

### ma-0041 — ma-exponent-product (revision 1)

Condition: What is 3² × 3⁴?

Derivation: For powers with the same base, multiplication adds the exponents: 3² × 3⁴ = 3⁶.

Options in canonical ID order: a: 3⁶; b: 3⁸; c: 9⁶; d: 6⁶. Derived answer: a.

### ma-0042 — ma-fractional-power (revision 1)

Condition: What is 16^(3/4)?

Derivation: Take the positive fourth root of 16, which is 2, and cube it: 2³ = 8.

Options in canonical ID order: a: 64; b: 8; c: 12; d: 4. Derived answer: b.

### ma-0043 — ma-log-definition (revision 1)

Condition: What is log₂ 32?

Derivation: A logarithm asks for an exponent. Since 2⁵ = 32, log₂ 32 = 5.

Options in canonical ID order: a: 30; b: 1/5; c: 5; d: 16. Derived answer: c.

### ma-0044 — ma-log-product (revision 1)

Condition: For x > 0 and y > 0, which expression equals log₁₀(xy)?

Derivation: Writing x = 10ᵘ and y = 10ᵛ gives xy = 10ᵘ⁺ᵛ. Therefore the logarithm of the product is log₁₀x + log₁₀y. Positivity is required for these real logarithms.

Options in canonical ID order: a: log₁₀x×log₁₀y; b: log₁₀x−log₁₀y; c: log₁₀x÷log₁₀y; d: log₁₀x＋log₁₀y. Derived answer: d.

### ma-0045 — ma-log-change (revision 2)

Condition: Let a, b, c > 0, with a ≠ 1 and c ≠ 1. Which is the change-of-base formula for logarithms?

Derivation: If t = log_a b, then aᵗ = b. Taking base-c logarithms gives t log_c a = log_c b, so t = log_c b / log_c a. The denominator is nonzero because a ≠ 1.

Options in canonical ID order: a: log_a b＝log_c b/log_c a; b: log_a b＝log_c a/log_c b; c: log_a b＝log_c a＋log_c b; d: log_a b＝log_c a×log_c b. Derived answer: a.

### ma-0046 — ma-complex-unit (revision 1)

Condition: For the imaginary unit i, what is i²?

Derivation: The imaginary unit is defined by i² = −1.

Options in canonical ID order: a: −i; b: −1; c: 1; d: i. Derived answer: b.

### ma-0047 — ma-complex-product (revision 1)

Condition: What is (1 + 2i)(3 − i)?

Derivation: Expand to 3 − i + 6i − 2i². Using i² = −1 gives 5 + 5i.

Options in canonical ID order: a: 5−5i; b: 3＋6i; c: 5＋5i; d: 1＋5i. Derived answer: c.

### ma-0048 — ma-complex-conjugate (revision 1)

Condition: What is the complex conjugate of z = 3 − 4i?

Derivation: Complex conjugation keeps the real part and reverses the sign of the imaginary part, giving 3 + 4i.

Options in canonical ID order: a: −3＋4i; b: −3−4i; c: 4−3i; d: 3＋4i. Derived answer: d.

### ma-0049 — ma-complex-modulus (revision 1)

Condition: What is the modulus of 3 + 4i?

Derivation: The modulus is distance from the origin in the complex plane: √(3² + 4²) = √25 = 5.

Options in canonical ID order: a: 5; b: 7; c: 25; d: 1. Derived answer: a.

### ma-0050 — ma-complex-division (revision 1)

Condition: Which expression equals 1/(1 + i)?

Derivation: Multiply numerator and denominator by 1 − i. The denominator is (1 + i)(1 − i) = 2, giving (1 − i)/2.

Options in canonical ID order: a: −i; b: （1−i）/2; c: 1−i; d: （1＋i）/2. Derived answer: b.

### ma-0051 — ma-binomial-coefficient (revision 1)

Condition: What is the coefficient of x³y² in (x + y)⁵?

Derivation: Choose which two of the five factors supply y; the others supply x. There are 5 choose 2 = 10 choices.

Options in canonical ID order: a: 20; b: 15; c: 10; d: 5. Derived answer: c.

### ma-0052 — ma-factorial-zero (revision 1)

Condition: What value is assigned to 0!?

Derivation: The factorial recurrence 1! = 1 × 0! is preserved by setting 0! = 1. It also counts the one way to arrange an empty collection.

Options in canonical ID order: a: 0; b: Undefined; c: −1; d: 1. Derived answer: d.

### ma-0053 — ma-division-algorithm (revision 1)

Condition: Write −17 = 5q + r with integers q and r and 0 ≤ r < 5. What is r?

Derivation: Choosing q = −4 gives −17 = −20 + 3. Thus r = 3, which satisfies the specified nonnegative remainder convention.

Options in canonical ID order: a: 3; b: −2; c: 2; d: −3. Derived answer: a.

### ma-0054 — ma-mod-product (revision 1)

Condition: What is the remainder when 7 × 8 is divided by 5?

Derivation: Modulo 5, 7 is congruent to 2 and 8 is congruent to 3. Their product is congruent to 6, which is congruent to 1 modulo 5. Equivalently, 56 = 5 × 11 + 1, so the remainder is 1.

Options in canonical ID order: a: 4; b: 1; c: 2; d: 3. Derived answer: b.

### ma-0057 — ma-significant-figures (revision 1)

Condition: How many significant figures are written in 0.00450?

Derivation: The leading zeros locate the decimal point. The digits 4, 5 and the final zero indicate the stated precision, giving three significant figures.

Options in canonical ID order: a: 3; b: 2; c: 5; d: 6. Derived answer: a.

### ma-0058 — ma-relative-error (revision 1)

Condition: A true length of 200 cm is measured as 202 cm. What is the absolute relative error?

Derivation: The absolute error is |202 − 200| = 2 cm. Divide by the true length: 2/200 = 0.01 = 1%.

Options in canonical ID order: a: 101％; b: 1％; c: 2％; d: 0.5％. Derived answer: b.

### ma-0059 — ma-floor (revision 1)

Condition: What is ⌊−1.2⌋, the greatest integer less than or equal to −1.2?

Derivation: −1 is too large, since −1 > −1.2. The greatest allowed integer is −2. Rounding toward zero would give a different result.

Options in canonical ID order: a: 1; b: 2; c: −2; d: −1. Derived answer: c.

### ma-0061 — ma-countability (revision 1)

Condition: Compare the set of all natural numbers with the set of all even natural numbers as infinite sets. Which statement is correct?

Derivation: The map n ↦ 2n pairs every natural number with exactly one even natural number, and every even natural number is reached. This is a bijection, so the sets have the same cardinality.

Options in canonical ID order: a: There is a one-to-one correspondence; b: The even numbers form a finite set; c: There are necessarily twice as many natural numbers; d: No correspondence can be constructed. Derived answer: a.

### ma-0065 — ma-modular-inverse (revision 1)

Condition: What is the multiplicative inverse of 3 modulo 7?

Derivation: An inverse must give remainder 1 when multiplied by 3. Since 3 × 5 = 15 = 7 × 2 + 1, the inverse is 5 modulo 7.

Options in canonical ID order: a: 5; b: 3; c: 2; d: 6. Derived answer: a.

### ma-0073 — ma-binary (revision 2)

Condition: Convert the binary number 10110 to decimal.

Derivation: The place values are 16, 8, 4, 2 and 1. Thus 1 × 16 + 0 × 8 + 1 × 4 + 1 × 2 + 0 × 1 = 22.

Options in canonical ID order: a: 22; b: 18; c: 26; d: 10110. Derived answer: a.

### ma-0076 — ma-parity-proof (revision 2)

Condition: For every integer n, what must be true of n(n + 1)?

Derivation: Of two consecutive integers, one is even. Their product therefore contains a factor of 2 and is even.

Options in canonical ID order: a: It is odd; b: It is prime; c: It is a multiple of 4; d: It is even. Derived answer: d.

### ma-0080 — ma-field-inverse (revision 2)

Condition: Which is the only real number without a multiplicative inverse?

Derivation: For every nonzero real x, x × (1/x) = 1. Multiplying 0 by any real number gives 0, so 0 has no multiplicative inverse.

Options in canonical ID order: a: 1; b: −1; c: √2; d: 0. Derived answer: d.

### ma-0081 — ma-unknown-addend (revision 1)

Condition: If □ + 18 = 45, what is □?

Derivation: Subtract 18 from both sides: □ = 45 − 18 = 27.

Options in canonical ID order: a: 27; b: 63; c: 37; d: 23. Derived answer: a.

### ma-0082 — ma-unknown-divisor (revision 1)

Condition: If 24 ÷ □ = 6, what is □?

Derivation: The unknown divisor must satisfy 6 × □ = 24. Dividing by 6 gives □ = 4.

Options in canonical ID order: a: 30; b: 4; c: 18; d: 144. Derived answer: b.

### ma-0083 — ma-price-expression (revision 1)

Condition: Each pencil costs 80 yen. You buy x pencils and one eraser costing 100 yen. What is the total cost?

Derivation: The pencils cost 80x yen and the eraser adds a fixed 100 yen, so the total is 80x + 100 yen.

Options in canonical ID order: a: (80 + 100x) yen; b: (80x − 100) yen; c: (80x + 100) yen; d: 180x yen. Derived answer: c.

### ma-0086 — ma-table-proportion (revision 1)

Condition: When x is 1, 2 and 3, y is 4, 8 and 12 respectively. If y = kx, what is k?

Derivation: Divide y by x for each pair: 4/1 = 8/2 = 12/3 = 4. This common ratio is k.

Options in canonical ID order: a: 12; b: 4; c: 3; d: 2. Derived answer: b.

### ma-0087 — ma-not-proportion (revision 1)

Condition: When x is 1, 2 and 3, y is 3, 5 and 7 respectively. Why is y not directly proportional to x?

Derivation: Direct proportionality requires y/x to be constant. The ratios here are 3, 5/2 and 7/3, which differ. A constant increase in y alone is insufficient.

Options in canonical ID order: a: x is an integer; b: The increase in y is 2; c: y/x is not constant; d: y increases. Derived answer: c.

### ma-0088 — ma-inverse-table (revision 1)

Condition: When x is 1, 2 and 4, y is 12, 6 and 3 respectively. Which quantity is constant?

Derivation: The products are 1 × 12 = 2 × 6 = 4 × 3 = 12. A constant product expresses inverse proportionality.

Options in canonical ID order: a: x＋y; b: y−x; c: y/x; d: xy. Derived answer: d.

### ma-0089 — ma-matchsticks (revision 1)

Condition: Squares are placed in a row sharing adjacent sides. One square needs 4 sticks, two need 7 and three need 10. How many sticks are needed for n squares?

Derivation: Start with 4 sticks. Each additional square shares one side and adds 3 sticks, so the total is 4 + 3(n − 1) = 3n + 1.

Options in canonical ID order: a: 3n + 1 sticks; b: 4n sticks; c: n + 3 sticks; d: 3n sticks. Derived answer: a.

### ma-0093 — ma-linear-reading (revision 1)

Condition: The amount of water after t minutes is y = 5t + 20 L. What does 20 represent?

Derivation: At t = 0, the formula gives y = 20 L. Thus 20 is the starting amount; 5 is the increase per minute.

Options in canonical ID order: a: The amount of water at the start; b: The increase per minute; c: The finishing time; d: The container's capacity. Derived answer: a.

### ma-0094 — ma-speed-graph (revision 1)

Condition: On a graph with time on the horizontal axis and distance travelled on the vertical axis, a section is horizontal. What does it represent?

Derivation: Time increases while the distance travelled stays unchanged, so there is no movement during that interval.

Options in canonical ID order: a: Suddenly speeding up; b: Being stationary; c: Moving at a constant nonzero speed; d: Moving back. Derived answer: b.

### ma-0095 — ma-coordinate-read (revision 2)

Condition: How do you move from the origin to reach the point (3, −2)?

Derivation: The first coordinate is horizontal and the second is vertical. Positive 3 means three units right; negative 2 means two units down.

Options in canonical ID order: a: 2 right and 3 up; b: 3 right and 2 up; c: 3 right and 2 down; d: 3 left and 2 down. Derived answer: c.

### ma-0096 — ma-piecewise-price (revision 1)

Condition: Shipping costs 300 yen for orders below 2000 yen and 0 yen for orders of 2000 yen or more. What is the shipping cost for an order of exactly 2000 yen?

Derivation: The boundary value 2000 belongs to the '2000 or more' category, so the shipping cost is 0 yen.

Options in canonical ID order: a: 300 yen; b: 200 yen; c: 2000 yen; d: 0 yen. Derived answer: d.

### ma-0101 — ma-like-terms (revision 1)

Condition: Simplify 3x + 2 − x + 5.

Derivation: Combine like terms: 3x − x = 2x and 2 + 5 = 7. The result is 2x + 7.

Options in canonical ID order: a: 2x＋7; b: 4x＋7; c: 2x＋3; d: 9x. Derived answer: a.

### ma-0102 — ma-distributive (revision 1)

Condition: Expand −2(3x − 4).

Derivation: Multiply −2 by each term: (−2)(3x) = −6x and (−2)(−4) = +8. The result is −6x + 8.

Options in canonical ID order: a: −2x＋8; b: −6x＋8; c: −6x−8; d: 6x＋8. Derived answer: b.

### ma-0103 — ma-linear-equation (revision 1)

Condition: Solve 3x − 7 = 11.

Derivation: Add 7 to both sides to get 3x = 18, then divide by 3: x = 6.

Options in canonical ID order: a: −6; b: 18; c: 6; d: 4. Derived answer: c.

### ma-0104 — ma-inequality-sign (revision 1)

Condition: Solve −2x > 6.

Derivation: Divide both sides by −2. Dividing by a negative reverses the inequality, so x < −3.

Options in canonical ID order: a: x＞−3; b: x＜3; c: x＞3; d: x＜−3. Derived answer: d.

### ma-0105 — ma-simultaneous (revision 1)

Condition: Solve x + y = 7 and x − y = 1.

Derivation: Adding the equations gives 2x = 8, so x = 4. Substituting into x + y = 7 gives y = 3.

Options in canonical ID order: a: x＝4, y＝3; b: x＝3, y＝4; c: x＝6, y＝1; d: x＝1, y＝6. Derived answer: a.

### ma-0106 — ma-slope (revision 1)

Condition: A straight line passes through (1, 3) and (3, 7). What is its slope?

Derivation: Slope is change in y divided by change in x: (7 − 3)/(3 − 1) = 4/2 = 2.

Options in canonical ID order: a: 3; b: 2; c: 4; d: 1/2. Derived answer: b.

### ma-0107 — ma-intercept (revision 1)

Condition: What is the y-intercept of y = −2x + 5?

Derivation: At the y-axis, x = 0. Substitution gives y = 5, so the y-intercept is 5.

Options in canonical ID order: a: −5; b: 2.5; c: 5; d: −2. Derived answer: c.

### ma-0108 — ma-line-intersection (revision 1)

Condition: Where do y = 2x and y = −x + 6 intersect?

Derivation: At the intersection their y-values agree, so 2x = −x + 6. Thus x = 2 and y = 4.

Options in canonical ID order: a: (4,2); b: (3,6); c: (0,6); d: (2,4). Derived answer: d.

### ma-0110 — ma-difference-squares (revision 1)

Condition: Factor x² − 25.

Derivation: Use the difference of squares: x² − 5² = (x − 5)(x + 5).

Options in canonical ID order: a: （x＋5）²; b: （x−5）（x＋5）; c: （x−5）²; d: （x−25）（x＋1）. Derived answer: b.

### ma-0112 — ma-zero-product (revision 1)

Condition: What are all real solutions of (x − 2)(x + 4) = 0?

Derivation: A product is zero when at least one factor is zero. Solving x − 2 = 0 or x + 4 = 0 gives x = 2 or x = −4.

Options in canonical ID order: a: x = −2 or 4; b: Only x = 2; c: Only x = −4; d: x = 2 or −4. Derived answer: d.

### ma-0113 — ma-quadratic-negative (revision 1)

Condition: What are the real solutions of x² = −9?

Derivation: Every real square is nonnegative, so no real x can satisfy x² = −9. Allowing complex numbers would change the domain.

Options in canonical ID order: a: None; b: 3; c: −3; d: ±3. Derived answer: a.

### ma-0114 — ma-quadratic-axis (revision 2)

Condition: What is the axis of symmetry of y = (x − 2)² + 1?

Derivation: Inputs equally far to either side of x = 2 give the same squared value, so the vertical symmetry axis is x = 2.

Options in canonical ID order: a: y＝2; b: x＝2; c: y＝1; d: x＝−2. Derived answer: b.

### ma-0115 — ma-quadratic-min (revision 2)

Condition: What is the minimum of y = −x² + 4 for −1 ≤ x ≤ 2?

Derivation: On this interval, x² is largest at x = 2, where it is 4. Therefore −x² + 4 is smallest there and equals 0. At x = −1 it is 3.

Options in canonical ID order: a: 4; b: −4; c: 0; d: 3. Derived answer: c.

### ma-0116 — ma-function-domain (revision 1)

Condition: Which x-value must be excluded from the domain of y = 1/(x − 3)?

Derivation: Division by zero is undefined. The denominator vanishes at x = 3, so that value must be excluded.

Options in canonical ID order: a: 0; b: 1; c: −3; d: 3. Derived answer: d.

### ma-0117 — ma-absolute-equation (revision 1)

Condition: Which real x-values have absolute value 2?

Derivation: Absolute value measures distance from zero. Both −2 and 2 are two units from zero.

Options in canonical ID order: a: −2 and 2; b: Only 2; c: Only −2; d: 0 and 2. Derived answer: a.

### ma-0119 — ma-identity-vs-equation (revision 1)

Condition: For which real x does 2(x + 1) = 2x + 2 hold?

Derivation: Expanding the left side gives the right side for every real x. This is an identity, rather than an equation with only one solution.

Options in canonical ID order: a: Only x = 1; b: For no real x; c: For every real x; d: Only x = 0. Derived answer: c.

### ma-0121 — ma-complete-square (revision 1)

Condition: What is the minimum of x² − 4x + 7 over all real x?

Derivation: Complete the square: x² − 4x + 7 = (x − 2)² + 3. A square is at least zero, so the minimum is 3 at x = 2.

Options in canonical ID order: a: 3; b: 7; c: −4; d: 2. Derived answer: a.

### ma-0122 — ma-discriminant (revision 1)

Condition: How many distinct real solutions does x² + 2x + 5 = 0 have?

Derivation: Complete the square to get (x + 1)² + 4 = 0. The left side is always positive, so there are no real solutions. Equivalently, the discriminant is −16.

Options in canonical ID order: a: Infinitely many; b: 0; c: 1; d: 2. Derived answer: b.

### ma-0124 — ma-quadratic-inequality (revision 1)

Condition: Solve x² − x − 6 < 0 over the real numbers.

Derivation: Factor as (x + 2)(x − 3). The factors have opposite signs exactly between −2 and 3, so −2 < x < 3. The endpoints are excluded by the strict inequality.

Options in canonical ID order: a: x < −2 or x > 3; b: −3 < x < 2; c: All real numbers; d: −2 < x < 3. Derived answer: d.

### ma-0125 — ma-remainder-theorem (revision 1)

Condition: What is the remainder when P(x) = x³ − 2x + 4 is divided by x − 2?

Derivation: Writing P(x) = (x − 2)Q(x) + r and substituting x = 2 gives r = P(2) = 8 − 4 + 4 = 8.

Options in canonical ID order: a: 8; b: 4; c: 0; d: 12. Derived answer: a.

### ma-0126 — ma-exponential-equation (revision 1)

Condition: Solve 2^(x + 1) = 16 over the real numbers.

Derivation: 16 = 2⁴. Since the real exponential with base 2 is one-to-one, x + 1 = 4 and x = 3.

Options in canonical ID order: a: 2; b: 3; c: 4; d: 5. Derived answer: b.

### ma-0128 — ma-sin-radian (revision 1)

Condition: What is sin(π/6)?

Derivation: π/6 radians is 30°. Splitting an equilateral triangle in half gives a right triangle with the side opposite 30° half the hypotenuse, so the sine is 1/2.

Options in canonical ID order: a: √3/2; b: √2/2; c: 1; d: 1/2. Derived answer: d.

### ma-0129 — ma-trig-identity (revision 1)

Condition: For every real angle θ, what is sin²θ + cos²θ?

Derivation: On the unit circle, the coordinates are (cos θ, sin θ). Their squared sum is the squared radius, which is 1.

Options in canonical ID order: a: 1; b: 0; c: 2; d: sinθ＋cosθ. Derived answer: a.

### ma-0131 — ma-sine-addition (revision 1)

Condition: Which is the angle-addition formula for sin(α + β)?

Derivation: Rotating the unit-circle vector through β takes its vertical coordinate to sin α cos β + cos α sin β. This is sin(α + β).

Options in canonical ID order: a: sinα sinβ＋cosα cosβ; b: sinα cosβ−cosα sinβ; c: sinα cosβ＋cosα sinβ; d: sinα＋sinβ. Derived answer: c.

### ma-0132 — ma-arith-series (revision 1)

Condition: An arithmetic sequence has first term 3, common difference 2 and 10 terms. What is its sum?

Derivation: The last term is 3 + 9 × 2 = 21. Pairing first with last gives an average term of (3 + 21)/2 = 12. Ten terms sum to 120.

Options in canonical ID order: a: 100; b: 110; c: 210; d: 120. Derived answer: d.

### ma-0135 — ma-infinite-geometric (revision 1)

Condition: What is the sum of the infinite series 1 + 1/2 + 1/4 + …?

Derivation: The partial sum through 1/2ⁿ is 2 − 1/2ⁿ. As n increases without bound, the final term tends to zero, so the sum tends to 2.

Options in canonical ID order: a: Infinity; b: 3/2; c: 2; d: 1. Derived answer: c.

### ma-0136 — ma-function-composition (revision 1)

Condition: Let f(x) = x² and g(x) = x + 1. What is f(g(2))?

Derivation: Apply the inner function first: g(2) = 3. Then f(3) = 3² = 9.

Options in canonical ID order: a: 5; b: 6; c: 8; d: 9. Derived answer: d.

### ma-0137 — ma-inverse-function (revision 1)

Condition: What is the inverse function of f(x) = 3x − 2?

Derivation: Solve y = 3x − 2 for x: x = (y + 2)/3. Swapping input and output labels gives f⁻¹(x) = (x + 2)/3. An inverse function is different from a reciprocal.

Options in canonical ID order: a: f⁻¹(x)＝(x＋2)/3; b: f⁻¹(x)＝(x−2)/3; c: f⁻¹(x)＝1/(3x−2); d: f⁻¹(x)＝3x＋2. Derived answer: a.

### ma-0138 — ma-even-function (revision 1)

Condition: Which function on the real numbers is even?

Derivation: An even function satisfies f(−x) = f(x). For x⁴ + 1, the fourth power is unchanged by reversing the sign of x. The other choices fail this identity.

Options in canonical ID order: a: f(x)＝2x; b: f(x)＝x⁴＋1; c: f(x)＝x³; d: f(x)＝x＋1. Derived answer: b.

### ma-0139 — ma-period (revision 1)

Condition: What is the smallest positive period of sin(2x)?

Derivation: Sine repeats after an angle increase of 2π. Increasing x by π increases 2x by 2π, so the smallest positive period is π.

Options in canonical ID order: a: π/2; b: 4π; c: π; d: 2π. Derived answer: c.

### ma-0141 — ma-limit-polynomial (revision 2)

Condition: What is the limit of x² + 1 as x approaches 2?

Derivation: Polynomials are continuous, so substitute x = 2: 2² + 1 = 5.

Options in canonical ID order: a: 5; b: 3; c: 4; d: Does not exist. Derived answer: a.

### ma-0142 — ma-limit-removable (revision 2)

Condition: What is the limit of (x² − 1)/(x − 1) as x approaches 1?

Derivation: For x ≠ 1, the expression simplifies to x + 1. Its nearby values approach 2. The expression need not be defined at the point for the limit to exist.

Options in canonical ID order: a: Does not exist; b: 2; c: 0; d: 1. Derived answer: b.

### ma-0143 — ma-one-sided (revision 2)

Condition: Let f(x) = 1 for x < 0 and f(x) = 2 for x ≥ 0. What is its two-sided limit as x approaches 0?

Derivation: The left-hand limit is 1 and the right-hand limit is 2. Since they differ, no two-sided limit exists.

Options in canonical ID order: a: 2; b: 3/2; c: Does not exist; d: 1. Derived answer: c.

### ma-0144 — ma-continuity (revision 2)

Condition: Which condition means that f is continuous at x = a?

Derivation: Continuity requires f(a) to be defined, the limit as x approaches a to exist, and that limit to equal f(a). Differentiability is a stronger condition and is not required.

Options in canonical ID order: a: Only that f(a) exists; b: The left and right limits differ; c: The derivative must be zero; d: The limit exists and equals f(a). Derived answer: d.

### ma-0145 — ma-derivative-definition (revision 2)

Condition: Which limit defines the derivative of f at a?

Derivation: The difference quotient compares the output change f(a + h) − f(a) with the input change h. Its limit as h tends to zero, when it exists, is the instantaneous rate of change.

Options in canonical ID order: a: As h → 0, [f(a + h) − f(a)]/h; b: As h → 0, f(a + h)/h; c: As h → ∞, f(h)/h; d: As h → 0, f(a) + h. Derived answer: a.

### ma-0146 — ma-power-derivative (revision 2)

Condition: What is the derivative of f(x) = x⁴?

Derivation: Expanding (x + h)⁴ − x⁴ and dividing by h gives 4x³ + 6x²h + 4xh² + h³. As h tends to zero, this approaches 4x³.

Options in canonical ID order: a: x⁵/5; b: 4x³; c: x³; d: 4x⁴. Derived answer: b.

### ma-0147 — ma-product-derivative (revision 2)

Condition: What is the derivative of f(x) = x² sin x?

Derivation: Use the product rule: differentiate each factor once while keeping the other. The result is 2x sin x + x² cos x.

Options in canonical ID order: a: x²cos x; b: 2x sin x; c: 2x sin x＋x²cos x; d: 2x cos x. Derived answer: c.

### ma-0148 — ma-chain-rule (revision 2)

Condition: What is the derivative of f(x) = sin(x²)?

Derivation: The chain rule multiplies the derivative of the outer sine by the derivative of the inner x²: cos(x²) × 2x.

Options in canonical ID order: a: cos(x²); b: 2x sin(x²); c: cos(2x); d: 2x cos(x²). Derived answer: d.

### ma-0149 — ma-log-derivative (revision 2)

Condition: For x > 0, what is the derivative of ln x?

Derivation: The natural logarithm is the inverse of eˣ. Differentiating e^(ln x) = x gives x × (ln x)′ = 1, hence (ln x)′ = 1/x.

Options in canonical ID order: a: 1/x; b: ln x; c: x; d: 1/ln x. Derived answer: a.

### ma-0150 — ma-exponential-derivative (revision 2)

Condition: What is the derivative of eˣ?

Derivation: In the difference quotient, factor out eˣ. The remaining factor (eʰ − 1)/h tends to 1, giving derivative eˣ.

Options in canonical ID order: a: 1/x; b: eˣ; c: xeˣ; d: eˣ⁻¹. Derived answer: b.

### ma-0151 — ma-tangent-line (revision 2)

Condition: What is the tangent line to y = x² at x = 2?

Derivation: The point is (2, 4) and the derivative 2x gives slope 4 there. Thus y − 4 = 4(x − 2), or y = 4x − 4.

Options in canonical ID order: a: y＝4x＋4; b: y＝x＋2; c: y＝4x−4; d: y＝2x. Derived answer: c.

### ma-0153 — ma-stationary-not-extremum (revision 2)

Condition: What kind of point is x = 0 for f(x) = x³?

Derivation: The derivative 3x² is zero at 0, making it stationary. But x³ is negative just left of 0 and positive just right, so 0 is neither a local maximum nor a local minimum.

Options in canonical ID order: a: A stationary point but not a local extremum; b: A local maximum; c: A local minimum; d: A point of discontinuity. Derived answer: a.

### ma-0154 — ma-second-derivative (revision 2)

Condition: What is the second derivative of f(x) = x³ − 3x?

Derivation: The first derivative is 3x² − 3. Differentiating again gives 6x.

Options in canonical ID order: a: x⁴/4−3x²/2; b: 6x; c: 3x²−3; d: 6. Derived answer: b.

### ma-0155 — ma-antiderivative (revision 2)

Condition: What is an indefinite integral of x²?

Derivation: The derivative of x³/3 is x². Add an arbitrary constant C because its derivative is zero.

Options in canonical ID order: a: x³＋C; b: x²/2＋C; c: x³/3＋C; d: 2x＋C. Derived answer: c.

### ma-0156 — ma-definite-integral (revision 2)

Condition: What is ∫₀² x dx?

Derivation: An antiderivative is x²/2. Evaluate at the endpoints and subtract: 2²/2 − 0²/2 = 2.

Options in canonical ID order: a: 4; b: 1; c: 0; d: 2. Derived answer: d.

### ma-0161 — ma-triangle-edges (revision 1)

Condition: How many sides does a triangle have?

Derivation: A triangle is a closed figure formed by three line segments, so it has three sides and three vertices.

Options in canonical ID order: a: 3; b: 2; c: 4; d: 5. Derived answer: a.

### ma-0162 — ma-rectangle-definition (revision 1)

Condition: What kind of angles are all four angles of a rectangle?

Derivation: A rectangle is a quadrilateral with four right angles. Its sides need not all have the same length.

Options in canonical ID order: a: 60°; b: Right angles; c: Acute angles; d: Obtuse angles. Derived answer: b.

### ma-0163 — ma-square-inclusion (revision 1)

Condition: Which statement must be true of a square?

Derivation: A square has four right angles, so it meets the definition of a rectangle as well as having four equal sides. Its diagonals are longer than its sides.

Options in canonical ID order: a: Its interior angles are 60°; b: Its opposite sides intersect; c: It is also a rectangle; d: Its diagonals equal its sides in length. Derived answer: c.

### ma-0164 — ma-parallel (revision 1)

Condition: What is the relationship between two distinct straight lines in the same plane that never meet, however far they are extended?

Derivation: Two distinct coplanar lines that have no point of intersection are parallel. The condition that they lie in the same plane matters.

Options in canonical ID order: a: Perpendicular; b: Congruent; c: Symmetric; d: Parallel. Derived answer: d.

### ma-0166 — ma-angle-turn (revision 1)

Condition: How many degrees is three quarters of a full turn?

Derivation: A full turn is 360°. Three quarters is 360 × 3/4 = 270°.

Options in canonical ID order: a: 360°; b: 270°; c: 90°; d: 180°. Derived answer: b.

### ma-0167 — ma-triangle-angle (revision 1)

Condition: Two angles of a triangle are 50° and 60°. What is its remaining angle?

Derivation: The angles of a triangle sum to 180°, so the remaining angle is 180 − 50 − 60 = 70°.

Options in canonical ID order: a: 130°; b: 80°; c: 70°; d: 110°. Derived answer: c.

### ma-0169 — ma-triangle-area (revision 1)

Condition: A triangle has base 8 cm and perpendicular height 5 cm to that base. What is its area?

Derivation: Two copies form a parallelogram with the same base and height. The triangle's area is half of 8 × 5, or 20 cm².

Options in canonical ID order: a: 20 cm²; b: 40 cm²; c: 13 cm²; d: 26 cm². Derived answer: a.

### ma-0170 — ma-parallelogram-area (revision 1)

Condition: A parallelogram has base 6 cm, perpendicular height 4 cm and sloping side 5 cm. What is its area?

Derivation: Rearranging a triangular end gives a rectangle with base 6 and height 4. The area is 6 × 4 = 24 cm². The sloping side is not the height.

Options in canonical ID order: a: 15 cm²; b: 24 cm²; c: 30 cm²; d: 20 cm². Derived answer: b.

### ma-0173 — ma-circle-circumference (revision 1)

Condition: Express the circumference of a circle of radius 3 cm in terms of π.

Derivation: Circumference is π times the diameter. The diameter is twice the radius, giving 2π × 3 = 6π cm.

Options in canonical ID order: a: 6π cm; b: 3π cm; c: 9π cm; d: 12π cm. Derived answer: a.

### ma-0174 — ma-circle-area (revision 1)

Condition: Express the area of a circle of diameter 10 cm in terms of π.

Derivation: The radius is half the diameter, or 5 cm. The area is πr² = 25π cm².

Options in canonical ID order: a: 50π cm²; b: 25π cm²; c: 100π cm²; d: 10π cm². Derived answer: b.

### ma-0176 — ma-cube-surface (revision 1)

Condition: What is the surface area of a cube with edge length 4 cm?

Derivation: Each of its six square faces has area 4² = 16 cm². The total is 6 × 16 = 96 cm². The quantity 4³ would be volume.

Options in canonical ID order: a: 64 cm²; b: 48 cm²; c: 16 cm²; d: 96 cm². Derived answer: d.

### ma-0178 — ma-reflection-symmetry (revision 1)

Condition: How many lines of symmetry does an equilateral triangle have?

Derivation: Each vertex and the midpoint of the opposite side determine a reflection line. There are three vertices, giving three such lines.

Options in canonical ID order: a: 6; b: 3; c: 1; d: 2. Derived answer: b.

### ma-0179 — ma-point-symmetry (revision 1)

Condition: Which rotation about the intersection of the diagonals maps every parallelogram onto itself?

Derivation: The diagonals bisect each other, so a half-turn takes each vertex to its opposite vertex. A half-turn is 180°. Other listed turns require additional shape restrictions.

Options in canonical ID order: a: 60°; b: 45°; c: 180°; d: 90°. Derived answer: c.

### ma-0181 — ma-vertical-angles (revision 1)

Condition: Two lines intersect and one angle is 35°. What is its vertically opposite angle?

Derivation: Both angles are supplementary to the same adjacent angle, so they are equal. The vertically opposite angle is 35°; an adjacent one is 145°.

Options in canonical ID order: a: 35°; b: 55°; c: 145°; d: 180°. Derived answer: a.

### ma-0182 — ma-corresponding-angles (revision 1)

Condition: A transversal crosses two parallel lines. What is true of corresponding angles?

Derivation: Corresponding angles occupy the same relative position at the two crossings. Parallelism makes those angles equal.

Options in canonical ID order: a: Their difference must be 180°; b: They are equal; c: They must be 90°; d: Their sum must be 90°. Derived answer: b.

### ma-0183 — ma-polygon-angle-sum (revision 1)

Condition: What is the sum of the interior angles of a hexagon?

Derivation: A simple hexagon can be divided into four triangles, so its interior angle sum is 4 × 180° = 720°.

Options in canonical ID order: a: 900°; b: 1080°; c: 720°; d: 540°. Derived answer: c.

### ma-0184 — ma-exterior-angle-sum (revision 1)

Condition: Take one exterior angle at each vertex of a convex polygon, turning in the same direction. What is their sum?

Derivation: Walking once around the boundary changes direction by one complete turn. The exterior angles therefore sum to 360°.

Options in canonical ID order: a: 180°; b: Number of sides × 180°; c: 90°; d: 360°. Derived answer: d.

### ma-0185 — ma-congruence-sss (revision 1)

Condition: Which condition guarantees that two triangles are congruent?

Derivation: Three matching side lengths fix both the shape and size of a triangle. Matching angles alone only guarantees similarity; matching area or perimeter alone is insufficient.

Options in canonical ID order: a: All three pairs of corresponding sides are equal; b: Only all three pairs of corresponding angles are equal; c: Only their areas are equal; d: Only their perimeters are equal. Derived answer: a.

### ma-0188 — ma-triangle-inequality (revision 1)

Condition: Can a triangle have side lengths 2, 3 and 6?

Derivation: The two shorter sides sum to 5, which is less than 6. They cannot reach around to form a closed triangle, so the triangle inequality fails.

Options in canonical ID order: a: Yes, an acute triangle; b: Yes, a right triangle; c: Yes, an equilateral triangle; d: No. Derived answer: d.

### ma-0190 — ma-similar-area (revision 2)

Condition: Two similar plane figures have corresponding length ratio 3:4. What is their area ratio?

Derivation: Scaling lengths multiplies each of two dimensions by the scale factor. Areas scale by its square, giving 3²:4² = 9:16.

Options in canonical ID order: a: 27：64; b: 9：16; c: 3：4; d: 6：16. Derived answer: b.

### ma-0191 — ma-similar-volume (revision 1)

Condition: Two similar solids have corresponding length ratio 1:3. What is their volume ratio?

Derivation: All three dimensions scale by 3, so volume scales by 3³ = 27. The ratio is 1:27.

Options in canonical ID order: a: 1：9; b: 3：1; c: 1：27; d: 1：3. Derived answer: c.

### ma-0192 — ma-midsegment (revision 1)

Condition: In a triangle, how does the segment joining the midpoints of two sides relate to the third side?

Derivation: If the shared vertex is taken as the origin, the two midpoint vectors are half the endpoint vectors. Their difference is half the third-side vector, so the segment is parallel and half as long.

Options in canonical ID order: a: Perpendicular and half as long; b: Parallel and equally long; c: It always intersects the third side; d: Parallel and half as long. Derived answer: d.

### ma-0193 — ma-pythagoras (revision 1)

Condition: A right triangle has perpendicular sides 5 cm and 12 cm. What is the hypotenuse length?

Derivation: By the Pythagorean theorem, its squared length is 5² + 12² = 169. The positive length is √169 = 13 cm.

Options in canonical ID order: a: 13 cm; b: 17 cm; c: 7 cm; d: √17 cm. Derived answer: a.

### ma-0195 — ma-inscribed-angle (revision 1)

Condition: The central angle subtending an arc of a circle is 100°. What is an inscribed angle subtending the same arc?

Derivation: An inscribed angle subtending the same arc is half the central angle. Half of 100° is 50°.

Options in canonical ID order: a: 200°; b: 80°; c: 50°; d: 100°. Derived answer: c.

### ma-0196 — ma-cyclic-quadrilateral (revision 2)

Condition: The vertices of a convex quadrilateral ABCD lie in that order on a circle. If angle A is 110°, what is angle C?

Derivation: Angles A and C subtend the two arcs that together form the full circle. Their inscribed angles sum to 180°, so C = 180° − 110° = 70°.

Options in canonical ID order: a: 110°; b: 55°; c: 90°; d: 70°. Derived answer: d.

### ma-0197 — ma-tangent-radius (revision 1)

Condition: What is the angle between a circle's tangent and the radius to its point of contact?

Derivation: The radius reaches the closest point on the tangent to the centre. That shortest segment is perpendicular to the line, so the angle is 90°.

Options in canonical ID order: a: 90°; b: 0°; c: 45°; d: 60°. Derived answer: a.

### ma-0200 — ma-sphere-area (revision 1)

Condition: What is the surface area of a sphere of radius 2 cm?

Derivation: For a sphere, surface area is 4πr². With r = 2, this gives 4π × 4 = 16π cm². The expression 4πr³/3 measures volume instead.

Options in canonical ID order: a: 8π cm²; b: 4π cm²; c: 32π/3 cm²; d: 16π cm². Derived answer: d.

### ma-0202 — ma-midpoint-coordinate (revision 1)

Condition: A segment has endpoints A(−2, 4) and B(6, 0). What is its midpoint?

Derivation: Average each coordinate: ((−2 + 6)/2, (4 + 0)/2) = (2, 2).

Options in canonical ID order: a: (−4,2); b: (2,2); c: (4,4); d: (2,4). Derived answer: b.

### ma-0204 — ma-circle-equation (revision 1)

Condition: What is the equation of a circle with centre (2, −1) and radius 3?

Derivation: The squared distance to its centre is (x − 2)² + (y + 1)². Setting this equal to the squared radius gives (x − 2)² + (y + 1)² = 9.

Options in canonical ID order: a: (x＋2)²＋(y−1)²＝9; b: (x−2)²＋(y＋1)²＝3; c: x²＋y²＝9; d: (x−2)²＋(y＋1)²＝9. Derived answer: d.

### ma-0205 — ma-perpendicular-slope (revision 1)

Condition: What is the slope of a line perpendicular to a line of slope 2?

Derivation: Direction vectors (1, 2) and (1, m) are perpendicular when their dot product 1 + 2m is zero. Thus m = −1/2.

Options in canonical ID order: a: −1/2; b: 1/2; c: −2; d: 2. Derived answer: a.

### ma-0206 — ma-point-line-distance (revision 1)

Condition: What is the distance from (0, 0) to the line 3x + 4y − 10 = 0?

Derivation: The normal vector (3, 4) has length 5. The line is 10/5 = 2 units from the origin in that normal direction.

Options in canonical ID order: a: √10; b: 2; c: 10; d: 5. Derived answer: b.

### ma-0207 — ma-cosine-law (revision 1)

Condition: Two sides of a triangle are 3 and 4, and their included angle is 60°. What is the square of the remaining side length?

Derivation: The cosine rule gives 3² + 4² − 2 × 3 × 4 × cos 60° = 9 + 16 − 12 = 13.

Options in canonical ID order: a: 7; b: 49; c: 13; d: 25. Derived answer: c.

### ma-0208 — ma-sine-law (revision 1)

Condition: A triangle has side a = 5 opposite angle A = 30°. What is its circumradius R?

Derivation: The extended sine rule gives a = 2R sin A. Hence 5 = 2R × 1/2 = R, so R = 5.

Options in canonical ID order: a: 10; b: 2.5; c: 5√3; d: 5. Derived answer: d.

### ma-0211 — ma-vector-add (revision 1)

Condition: What is the vector sum (2, −1) + (−3, 4)?

Derivation: Add corresponding components: (2 − 3, −1 + 4) = (−1, 3).

Options in canonical ID order: a: (−6,−4); b: (−1,−5); c: (−1,3); d: (5,3). Derived answer: c.

### ma-0212 — ma-vector-norm (revision 1)

Condition: What is the length of the vector (1, 2, 2)?

Derivation: The squared length is 1² + 2² + 2² = 9. Taking the nonnegative square root gives 3.

Options in canonical ID order: a: 5; b: 9; c: √5; d: 3. Derived answer: d.

### ma-0213 — ma-dot-product (revision 1)

Condition: What is the dot product of (1, 2) and (3, −1)?

Derivation: Multiply corresponding components and add: 1 × 3 + 2 × (−1) = 1.

Options in canonical ID order: a: 1; b: 5; c: −2; d: 7. Derived answer: a.

### ma-0214 — ma-orthogonal-vector (revision 1)

Condition: Which vector is perpendicular to (2, 1)?

Derivation: Perpendicular vectors have zero dot product. For (1, −2), the dot product is 2 × 1 + 1 × (−2) = 0. The other listed products are nonzero.

Options in canonical ID order: a: (−2,−1); b: (1,−2); c: (2,1); d: (1,2). Derived answer: b.

### ma-0215 — ma-projection (revision 1)

Condition: What is the orthogonal projection of (3, 4) onto the x-axis?

Derivation: Projection keeps the horizontal component and removes the perpendicular vertical component, giving (3, 0).

Options in canonical ID order: a: (3,4); b: (5,0); c: (3,0); d: (0,4). Derived answer: c.

### ma-0216 — ma-triangle-centroid (revision 1)

Condition: A triangle has vertices (0, 0), (6, 0) and (0, 3). What is its centroid?

Derivation: The centroid is the average of the three vertex vectors: ((0 + 6 + 0)/3, (0 + 0 + 3)/3) = (2, 1).

Options in canonical ID order: a: (3,1.5); b: (6,3); c: (1,2); d: (2,1). Derived answer: d.

### ma-0218 — ma-ellipse (revision 1)

Condition: What is the full length of the major axis of x²/9 + y²/4 = 1?

Derivation: The larger denominator is 9, giving semimajor axis √9 = 3 along the x-axis. The full major axis is twice this, or 6.

Options in canonical ID order: a: 4; b: 6; c: 3; d: 9. Derived answer: b.

### ma-0221 — ma-matrix-size (revision 1)

Condition: What is the size of the product of a 2-by-3 matrix and a 3-by-4 matrix, in that order?

Derivation: The inner dimensions both equal 3, so multiplication is defined. Each of the two rows is paired with each of the four columns, producing a 2-by-4 matrix.

Options in canonical ID order: a: 2 rows, 4 columns; b: 3 rows, 3 columns; c: 2 rows, 3 columns; d: 4 rows, 2 columns. Derived answer: a.

### ma-0222 — ma-matrix-product (revision 1)

Condition: A has rows (1, 2) and (0, 1); B has rows (3, 0) and (4, 5). What is the first row of AB?

Derivation: Pair A's first row with B's columns. The entries are 1 × 3 + 2 × 4 = 11 and 1 × 0 + 2 × 5 = 10, giving (11, 10).

Options in canonical ID order: a: (11,5); b: (11,10); c: (3,10); d: (7,5). Derived answer: b.

### ma-0223 — ma-determinant (revision 1)

Condition: What is the determinant of the 2-by-2 matrix with rows (2, 1) and (3, 4)?

Derivation: The 2-by-2 determinant is ad − bc: 2 × 4 − 1 × 3 = 5.

Options in canonical ID order: a: 8; b: −5; c: 5; d: 11. Derived answer: c.

### ma-0224 — ma-inverse-matrix (revision 1)

Condition: What is the inverse of the matrix with rows (1, 1) and (0, 1)?

Derivation: Multiplying it by the matrix with rows (1, −1) and (0, 1) in either order gives the identity. The second matrix reverses the first matrix's shear.

Options in canonical ID order: a: Rows (1, 1), (0, 1); b: Rows (1, 0), (1, 1); c: Rows (−1, 1), (0, −1); d: Rows (1, −1), (0, 1). Derived answer: d.

### ma-0225 — ma-singular-matrix (revision 1)

Condition: Which statement is correct about the matrix with rows (1, 2) and (2, 4)?

Derivation: Its second row is twice its first, so its determinant is 1 × 4 − 2 × 2 = 0. It has rank 1 and cannot have an inverse.

Options in canonical ID order: a: It has no inverse; b: It is its own inverse; c: Its determinant is 2; d: Its rank is 2. Derived answer: a.

### ma-0226 — ma-rank (revision 1)

Condition: What is the rank of the matrix with rows (1, 0, 2) and (0, 1, 3)?

Derivation: The rows are linearly independent: the first two coordinates force both coefficients of a zero linear combination to be zero. There are two independent rows, so the rank is 2.

Options in canonical ID order: a: 3; b: 2; c: 1; d: 0. Derived answer: b.

### ma-0227 — ma-nullspace (revision 1)

Condition: For A with rows (1, 1) and (2, 2), which nonzero vector x satisfies Ax = 0?

Derivation: Both equations reduce to the sum of the two components being zero. The vector (1, −1) satisfies this; the other listed vectors do not.

Options in canonical ID order: a: (0,1); b: (2,1); c: (1,−1); d: (1,1). Derived answer: c.

### ma-0228 — ma-rank-nullity (revision 1)

Condition: A linear map from R⁵ to R³ has rank 2. What is the dimension of its kernel?

Derivation: The domain dimension equals rank plus kernel dimension. Thus the kernel dimension is 5 − 2 = 3; use the domain's dimension, not the codomain's.

Options in canonical ID order: a: 1; b: 2; c: 5; d: 3. Derived answer: d.

### ma-0229 — ma-linear-independence (revision 1)

Condition: Which pair of vectors is linearly independent in R²?

Derivation: A linear combination of (1, 0) and (0, 1) is zero only when both coefficients are zero. Each other pair contains either a zero vector or two scalar multiples.

Options in canonical ID order: a: (1, 0) and (0, 1); b: (1, 2) and (2, 4); c: (0, 0) and (1, 1); d: (1, 1) and (−1, −1). Derived answer: a.

### ma-0230 — ma-basis (revision 1)

Condition: How many vectors are in a basis of R³?

Derivation: A basis is both linearly independent and spans the space. R³ has three independent coordinate directions, so every basis has three vectors.

Options in canonical ID order: a: Any number; b: 3; c: 2; d: 4. Derived answer: b.

### ma-0231 — ma-subspace (revision 1)

Condition: Which set is a vector subspace of R²?

Derivation: The solutions of x + y = 0 include the origin and are closed under addition and real scalar multiplication. The other sets fail at least one of these requirements.

Options in canonical ID order: a: The points with x > 0; b: The circle of radius 1 centred at the origin; c: The points with x + y = 0; d: The points with x + y = 1. Derived answer: c.

### ma-0232 — ma-eigenvalue (revision 1)

Condition: What are the eigenvalues of the matrix with rows (2, 0) and (0, 5)?

Derivation: The coordinate directions are multiplied by 2 and 5 respectively. Equivalently, det(A − λI) = (2 − λ)(5 − λ), whose zeros are 2 and 5.

Options in canonical ID order: a: 0 and 7; b: Only 10; c: −2 and −5; d: 2 and 5. Derived answer: d.

### ma-0234 — ma-trace (revision 1)

Condition: What is the trace of a 3-by-3 matrix whose diagonal entries are 1, 4 and −2?

Derivation: Trace is the sum of diagonal entries, so it is 1 + 4 − 2 = 3. The off-diagonal entries do not affect it.

Options in canonical ID order: a: 2; b: 3; c: −8; d: 7. Derived answer: b.

### ma-0236 — ma-orthogonal-matrix (revision 1)

Condition: Which condition defines a real orthogonal matrix Q?

Derivation: QᵀQ = I means the columns have unit length and are mutually perpendicular. It also ensures that multiplication by Q preserves dot products.

Options in canonical ID order: a: Q²＝0; b: Qᵀ＝0; c: det Q＝0; d: QᵀQ＝I. Derived answer: d.

### ma-0237 — ma-rotation-matrix (revision 1)

Condition: Rotate (2, 1) counterclockwise through 90° about the origin. What is the resulting point?

Derivation: A counterclockwise quarter-turn maps (x, y) to (−y, x). Thus (2, 1) becomes (−1, 2).

Options in canonical ID order: a: (−1,2); b: (1,−2); c: (−2,−1); d: (2,−1). Derived answer: a.

### ma-0238 — ma-positive-definite (revision 1)

Condition: A is a real matrix with diagonal entries 2 and 3 and all other entries zero. Is A positive definite?

Derivation: For x = (x₁, x₂), xᵀAx = 2x₁² + 3x₂². When x is nonzero, at least one square is positive, so the whole expression is positive.

Options in canonical ID order: a: Yes, because xᵀAx = 0 for every x; b: Yes, because xᵀAx > 0 for every nonzero x; c: No, because its determinant is positive; d: No, because its diagonal entries differ. Derived answer: b.

### ma-0242 — ma-frequency-share (revision 1)

Condition: Six of 20 people walk. What percentage of the group walks?

Derivation: Divide the part by the whole: 6/20 = 0.3. Multiplying by 100 gives 30%.

Options in canonical ID order: a: 60％; b: 30％; c: 6％; d: 20％. Derived answer: b.

### ma-0243 — ma-bar-chart (revision 1)

Condition: Which graph compares the numbers of people in different categories using the heights of rectangles?

Derivation: A bar chart uses separate bars on a common scale to compare category counts. A scatter plot instead shows paired numerical observations.

Options in canonical ID order: a: Scatter plot; b: A diagram of a circle’s circumference; c: Bar chart; d: Line graph. Derived answer: c.

### ma-0246 — ma-mean (revision 1)

Condition: What is the arithmetic mean of 2, 4 and 9?

Derivation: Add the three values and divide by their number: (2 + 4 + 9)/3 = 15/3 = 5.

Options in canonical ID order: a: 15; b: 5; c: 4; d: 9. Derived answer: b.

### ma-0247 — ma-median-odd (revision 1)

Condition: What is the median of 1, 3, 8, 10 and 20?

Derivation: The five values are already ordered. The middle value is the third, 8. A median uses position rather than the sum of values.

Options in canonical ID order: a: 10; b: 8.4; c: 8; d: 3. Derived answer: c.

### ma-0249 — ma-mode (revision 1)

Condition: What is the mode of 1, 2, 2, 3 and 5?

Derivation: The mode is the most frequent value. Here 2 appears twice and every other value appears once, so the mode is 2.

Options in canonical ID order: a: 2; b: 1; c: 3; d: 2.6. Derived answer: a.

### ma-0250 — ma-range (revision 1)

Condition: What is the range, defined as maximum minus minimum, of 4, 7, 9 and 15?

Derivation: The largest value is 15 and the smallest is 4. Their difference is 15 − 4 = 11.

Options in canonical ID order: a: 35; b: 11; c: 15; d: 4. Derived answer: b.

### ma-0252 — ma-outlier-mean (revision 1)

Condition: The data are 1, 2 and 3. If 100 is added, how does the arithmetic mean change?

Derivation: The original mean is 6/3 = 2. After adding 100, the mean is 106/4 = 26.5. A very large observation can strongly affect the mean.

Options in canonical ID order: a: It stays at 2; b: It becomes 3; c: It must equal the median; d: It rises from 2 to 26.5. Derived answer: d.

### ma-0253 — ma-weighted-mean (revision 1)

Condition: Three people have an average score of 10 points and two others have an average of 20 points. What is the combined average?

Derivation: The group totals are 3 × 10 = 30 and 2 × 20 = 40 points. Divide their total 70 by all five people to get 14 points. Group averages must be weighted by group size.

Options in canonical ID order: a: 14 points; b: 15 points; c: 30 points; d: 12 points. Derived answer: a.

### ma-0256 — ma-survey-denominator (revision 1)

Condition: A survey of 40 people allows multiple choices. A receives 25 votes and B receives 20. What follows from their total of 45 votes?

Derivation: If no one chose both, A and B together could account for at most 40 votes. The extra five require at least five people choosing both. The overlap can be larger if some people chose neither.

Options in canonical ID order: a: There were 45 respondents; b: 45 people chose exactly one of A and B; c: Everyone chose both; d: At least five people chose both. Derived answer: d.

### ma-0258 — ma-representative-choice (revision 1)

Condition: You want to survey commuting methods in a region. Which sampling method helps reduce selection bias?

Derivation: A random sample from all residents covers the target population. Restricting sampling to drivers, station users or bicycle-shop customers excludes other groups and can bias the commuting distribution.

Options in canonical ID order: a: Ask only driving-licence holders; b: Randomly sample the region's residents; c: Ask only station users; d: Ask only bicycle-shop customers. Derived answer: b.

### ma-0259 — ma-small-sample (revision 1)

Condition: Red was drawn three times in four draws. What can be concluded about the probability of red?

Derivation: The observed proportion is 3/4, but this does not determine the underlying probability exactly. Different probabilities can produce the same four outcomes.

Options in canonical ID order: a: The next draw must be blue; b: There must be three times as many red items as blue; c: These results alone do not determine the exact probability; d: The probability must be 3/4. Derived answer: c.

### ma-0260 — ma-graph-axis (revision 1)

Condition: A bar chart's vertical axis starts at 90 instead of 0. What can this do to the displayed difference between values 100 and 110?

Derivation: The data still differ by 10. But their visible bar heights above 90 become 10 and 20, which can make the difference appear more dramatic.

Options in canonical ID order: a: Change the numerical difference to 20; b: Necessarily double the data ratio; c: Change the original data; d: Visually exaggerate the difference. Derived answer: d.

### ma-0261 — ma-equiprobability (revision 1)

Condition: What is the probability of an even result when rolling one fair six-sided die numbered 1 to 6?

Derivation: The six outcomes are equally likely. Three, namely 2, 4 and 6, are even, so the probability is 3/6 = 1/2.

Options in canonical ID order: a: 1/2; b: 1/6; c: 2/3; d: 1/3. Derived answer: a.

### ma-0262 — ma-sample-space (revision 1)

Condition: Two distinguishable coins are tossed independently, each with equally likely heads and tails. How many outcomes are possible?

Derivation: The ordered outcomes are HH, HT, TH and TT, giving four possibilities. HT and TH are distinct because the coins are distinguishable.

Options in canonical ID order: a: 8; b: 4; c: 2; d: 3. Derived answer: b.

### ma-0263 — ma-at-least-one (revision 1)

Condition: A fair coin is tossed independently twice. What is the probability of at least one head?

Derivation: The four equally likely outcomes are HH, HT, TH and TT. Only TT has no head, so the probability is 1 − 1/4 = 3/4.

Options in canonical ID order: a: 1/4; b: 1; c: 3/4; d: 1/2. Derived answer: c.

### ma-0265 — ma-without-replacement (revision 1)

Condition: Two items are sampled uniformly without replacement from a collection of two red items and one blue item. What is the probability that their colours differ?

Derivation: There are three equally likely unordered pairs of distinct items. Two contain the blue item and one red item, so the probability is 2/3.

Options in canonical ID order: a: 2/3; b: 1/3; c: 4/9; d: 1/2. Derived answer: a.

### ma-0266 — ma-dice-sum (revision 1)

Condition: Two fair six-sided dice are rolled independently. What is the probability that their sum is 7?

Derivation: There are 6 × 6 = 36 equally likely ordered pairs. Six have sum 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). The probability is 6/36 = 1/6.

Options in canonical ID order: a: 1/12; b: 1/6; c: 1/7; d: 7/36. Derived answer: b.

### ma-0269 — ma-relative-frequency (revision 1)

Condition: There are 37 successes in 100 trials. What is the relative frequency of success?

Derivation: Relative frequency is the observed count divided by all trials: 37/100 = 0.37. This observed proportion is different from an exactly known theoretical probability.

Options in canonical ID order: a: 0.37; b: 37; c: 0.63; d: 1/37. Derived answer: a.

### ma-0270 — ma-independent-history (revision 1)

Condition: A fair coin is tossed independently and has shown heads five times in a row. What is the probability that the next toss is tails?

Derivation: Independence means earlier outcomes do not change the next toss's probabilities. A fair coin still has probability 1/2 of tails.

Options in canonical ID order: a: 5/6; b: 1/2; c: 1; d: 1/6. Derived answer: b.

### ma-0271 — ma-addition-disjoint (revision 1)

Condition: Events A and B are mutually exclusive, with P(A) = 0.2 and P(B) = 0.3. What is P(A or B)?

Derivation: Because the events do not overlap, their probabilities add: 0.2 + 0.3 = 0.5.

Options in canonical ID order: a: 0.1; b: 0.8; c: 0.5; d: 0.06. Derived answer: c.

### ma-0272 — ma-overlap-probability (revision 1)

Condition: P(A) = 0.6, P(B) = 0.5 and P(A and B) = 0.2. What is P(A or B)?

Derivation: Adding P(A) and P(B) counts the overlap twice. Subtract it once: 0.6 + 0.5 − 0.2 = 0.9.

Options in canonical ID order: a: 1.1; b: 0.3; c: 0.7; d: 0.9. Derived answer: d.

### ma-0274 — ma-conditional-count (revision 1)

Condition: Of 20 people, eight wear glasses and two of those eight are left-handed. For a uniformly chosen person known to wear glasses, what is the probability of being left-handed?

Derivation: The condition restricts the possible people to the eight glasses wearers. Two are left-handed, so the conditional probability is 2/8 = 1/4.

Options in canonical ID order: a: 1/2; b: 1/4; c: 1/10; d: 2/5. Derived answer: b.

### ma-0276 — ma-histogram-vs-bar (revision 1)

Condition: What does a histogram primarily represent?

Derivation: A histogram groups numerical observations into intervals and displays their frequency distribution. Adjacent bars correspond to adjacent numerical intervals.

Options in canonical ID order: a: Only changes over time; b: The lengths of category names; c: Pairs of points for two variables; d: A frequency distribution of numerical values grouped into intervals. Derived answer: d.

### ma-0277 — ma-class-boundary (revision 1)

Condition: Which value belongs to the interval 'at least 10 and less than 20'?

Derivation: The lower boundary is included and the upper boundary is excluded. Thus 10 belongs, while 20, 9.9 and 21 do not.

Options in canonical ID order: a: 10; b: 20; c: 9.9; d: 21. Derived answer: a.

### ma-0278 — ma-histogram-density (revision 2)

Condition: In a histogram, bar areas are proportional to frequencies. Intervals [0, 10) and [10, 30) each have frequency 10. What is the first bar's height divided by the second bar's height?

Derivation: Height is proportional to frequency divided by interval width. The heights are proportional to 10/10 = 1 and 10/20 = 1/2, so their ratio is 2:1.

Options in canonical ID order: a: 4：1; b: 2：1; c: 1：1; d: 1：2. Derived answer: b.

### ma-0279 — ma-correlation-causation (revision 1)

Condition: Ice-cream sales and heatstroke case counts have a positive correlation. What can be concluded immediately from that correlation alone?

Derivation: Correlation does not identify a cause. A common factor such as temperature could affect both variables, so the correlation alone cannot establish that ice cream causes heatstroke.

Options in canonical ID order: a: Stopping sales would make the case count zero; b: The correlation cannot be observed; c: Ice cream cannot be established as the cause from this alone; d: Ice cream must be the cause. Derived answer: c.

### ma-0281 — ma-permutation (revision 1)

Condition: In how many ways can four distinct books be arranged in a row?

Derivation: There are 4 choices for the first position, then 3, then 2, then 1. Multiplying gives 4! = 24.

Options in canonical ID order: a: 24; b: 16; c: 12; d: 4. Derived answer: a.

### ma-0282 — ma-combination (revision 1)

Condition: How many unordered pairs can be chosen from five people?

Derivation: There are 5 × 4 ordered choices. Each unordered pair appears twice, so divide by 2: 20/2 = 10.

Options in canonical ID order: a: 5; b: 10; c: 20; d: 25. Derived answer: b.

### ma-0283 — ma-identical-items (revision 1)

Condition: How many distinct strings can be formed by arranging the letters A, A and B in a row?

Derivation: The strings are AAB, ABA and BAA. Equivalently, 3! arrangements are divided by 2! because swapping the identical A's changes nothing.

Options in canonical ID order: a: 2; b: 9; c: 3; d: 6. Derived answer: c.

### ma-0284 — ma-circular-permutation (revision 1)

Condition: Four distinct people sit around a circular table. Rotations count as the same arrangement and reflections count as different. How many arrangements are there?

Derivation: Fix one person's position to remove equivalent rotations. The remaining three people can be ordered in 3! = 6 ways. Reflections are not identified.

Options in canonical ID order: a: 24; b: 3; c: 12; d: 6. Derived answer: d.

### ma-0285 — ma-independence-equation (revision 1)

Condition: Which condition defines independence of events A and B?

Derivation: Independence means the probability of their intersection is the product of their separate probabilities: P(A and B) = P(A)P(B). It is different from being mutually exclusive.

Options in canonical ID order: a: P(A and B) = P(A)P(B); b: P(A and B) = 0; c: P(A) = P(B); d: P(A or B) = 1. Derived answer: a.

### ma-0287 — ma-expected-value (revision 1)

Condition: X equals 0 with probability 1/4 and 4 with probability 3/4. What is E[X]?

Derivation: Weight each value by its probability and add: 0 × 1/4 + 4 × 3/4 = 3.

Options in canonical ID order: a: 4; b: 1; c: 3; d: 2. Derived answer: c.

### ma-0288 — ma-variance-discrete (revision 1)

Condition: X equals 0 or 2, each with probability 1/2. What is its variance?

Derivation: The mean is 1. Both deviations, −1 and 1, have squared value 1, so their weighted average is 1.

Options in canonical ID order: a: 2; b: 4; c: 0; d: 1. Derived answer: d.

### ma-0289 — ma-stddev (revision 1)

Condition: If the variance is 9, what is the standard deviation?

Derivation: Standard deviation is the nonnegative square root of variance. √9 = 3, restoring the original unit of the observations.

Options in canonical ID order: a: 3; b: 9; c: 81; d: 4.5. Derived answer: a.

### ma-0290 — ma-variance-shift (revision 1)

Condition: Let Y = X + 10. How does Var(Y) relate to Var(X)?

Derivation: The mean also increases by 10. Therefore Y − E[Y] = X − E[X], so the squared deviations and their average stay the same.

Options in canonical ID order: a: 0; b: Var(X); c: Var(X)＋10; d: 100Var(X). Derived answer: b.

### ma-0291 — ma-variance-scale (revision 1)

Condition: Let Y = 3X. How does Var(Y) relate to Var(X)?

Derivation: The deviations from the mean are multiplied by 3. Their squares are multiplied by 9, so Var(Y) = 9Var(X).

Options in canonical ID order: a: Var(X)＋3; b: Var(X)/3; c: 9Var(X); d: 3Var(X). Derived answer: c.

### ma-0292 — ma-binomial-count (revision 1)

Condition: Four independent trials each have success probability 1/2. What is the probability of exactly two successes?

Derivation: There are 4 choose 2 = 6 ways to select the two successful positions. Each four-outcome sequence has probability (1/2)⁴ = 1/16. The total is 6/16 = 3/8.

Options in canonical ID order: a: 1/4; b: 1/2; c: 1/8; d: 3/8. Derived answer: d.

### ma-0293 — ma-binomial-mean (revision 1)

Condition: X has binomial distribution Bin(10, 0.3). What is its mean?

Derivation: Write X as the sum of ten success indicators, each with expectation 0.3. Expectations add, giving 10 × 0.3 = 3.

Options in canonical ID order: a: 3; b: 0.3; c: 7; d: 2.1. Derived answer: a.

### ma-0294 — ma-binomial-variance (revision 1)

Condition: X has binomial distribution Bin(8, 1/2). What is its variance?

Derivation: Each independent success indicator has variance p(1 − p). Add eight such variances: 8 × (1/2) × (1/2) = 2.

Options in canonical ID order: a: 1/2; b: 2; c: 4; d: 8. Derived answer: b.

### ma-0295 — ma-normal-symmetry (revision 1)

Condition: For X with normal distribution N(μ, σ²), where σ > 0, what is P(X ≤ μ)?

Derivation: The continuous normal distribution is symmetric about μ. Half the probability lies on each side, and the single point μ has probability zero, so P(X ≤ μ) = 1/2.

Options in canonical ID order: a: 1; b: σ; c: 1/2; d: 0. Derived answer: c.

### ma-0296 — ma-z-score (revision 1)

Condition: An observation is 70, the mean is 50 and the standard deviation is 10. What is its standardized score z?

Derivation: Subtract the mean and divide by the standard deviation: z = (70 − 50)/10 = 2.

Options in canonical ID order: a: 7; b: 20; c: 0.2; d: 2. Derived answer: d.

### ma-0297 — ma-quartile-iqr (revision 1)

Condition: The first quartile is 12 and the third quartile is 20. What is the interquartile range?

Derivation: The interquartile range is Q3 − Q1 = 20 − 12 = 8. It describes the span between these two quartiles.

Options in canonical ID order: a: 8; b: 16; c: 32; d: 12. Derived answer: a.

### ma-0298 — ma-boxplot-limits (revision 1)

Condition: Which quantity generally cannot be read exactly from a standard box plot?

Derivation: A standard box plot displays the median and quartiles, but does not give the arithmetic mean. The quartiles alone do not determine that mean.

Options in canonical ID order: a: Third quartile; b: Arithmetic mean; c: Median; d: First quartile. Derived answer: b.

### ma-0299 — ma-correlation-range (revision 1)

Condition: What is the possible range of a defined correlation coefficient r?

Derivation: The correlation is a normalized dot product of centred data. The Cauchy–Schwarz inequality bounds its absolute value by 1, so −1 ≤ r ≤ 1.

Options in canonical ID order: a: −100 ≤ r ≤ 100; b: All real numbers; c: −1 ≤ r ≤ 1; d: 0 ≤ r ≤ 1. Derived answer: c.

### ma-0301 — ma-pdf-not-probability (revision 1)

Condition: Which statement about a probability density f(x) of a continuous random variable is correct?

Derivation: A density is nonnegative and integrates to 1 over the whole line. Its height may exceed 1: a uniform density of height 2 over an interval of width 1/2 still has total area 1.

Options in canonical ID order: a: It may take values greater than 1; b: It must always be at most 1; c: f(x) itself is the probability at the point x; d: Negative values are allowed. Derived answer: a.

### ma-0302 — ma-uniform-probability (revision 1)

Condition: X is uniformly distributed on [0, 4]. What is P(1 ≤ X ≤ 3)?

Derivation: The density is 1/4. The requested interval has length 2, so its probability is 2 × 1/4 = 1/2.

Options in canonical ID order: a: 1; b: 1/2; c: 1/4; d: 3/4. Derived answer: b.

### ma-0303 — ma-cdf-definition (revision 1)

Condition: What is the definition of the cumulative distribution function F(x)?

Derivation: F(x) accumulates the probability of all values at most x: F(x) = P(X ≤ x). This definition applies to discrete and continuous random variables.

Options in canonical ID order: a: E[X]; b: Var(X); c: P(X≦x); d: P(X＝x). Derived answer: c.

### ma-0304 — ma-cdf-monotone (revision 1)

Condition: Which property must a cumulative distribution function have?

Derivation: As x increases, the event {X ≤ x} can only gain outcomes. Its probability cannot decrease, so F is nondecreasing. It can have jumps.

Options in canonical ID order: a: It must be decreasing; b: It is always differentiable; c: It is always a straight line; d: It is nondecreasing. Derived answer: d.

### ma-0305 — ma-poisson (revision 1)

Condition: X has a Poisson distribution with mean λ = 2. What is P(X = 0)?

Derivation: The Poisson probability is e⁻λ λᵏ/k!. Substituting λ = 2 and k = 0 gives e⁻² × 1/1 = e⁻².

Options in canonical ID order: a: e⁻²; b: 2e⁻²; c: 1−e⁻²; d: 1/2. Derived answer: a.

### ma-0306 — ma-exponential-memoryless (revision 2)

Condition: X has an exponential distribution with a positive rate, and s, t ≥ 0. Which statement expresses its memoryless property?

Derivation: Its survival probability has the form e⁻λu. The conditional probability is e⁻λ(s+t)/e⁻λs = e⁻λt, equal to P(X > t).

Options in canonical ID order: a: The waiting time must be constant; b: P(X > s + t given X > s) = P(X > t); c: P(X > s + t) = P(X > t); d: X is always zero. Derived answer: b.

### ma-0307 — ma-expectation-linearity (revision 2)

Condition: X and Y have finite expectations. Which equality always holds, even if they are dependent?

Derivation: Expectation is linear: adding the variables adds their expectations. Independence is unnecessary for E[X + Y] = E[X] + E[Y]. Independence is sufficient for E[XY] = E[X]E[Y], but without it that equality is not guaranteed.

Options in canonical ID order: a: Var(X + Y) = Var(X) + Var(Y); b: P(X = Y) = 0; c: E[X + Y] = E[X] + E[Y]; d: E[XY] = E[X]E[Y]. Derived answer: c.

### ma-0308 — ma-covariance (revision 2)

Condition: X and Y have finite variances. Which covariance term appears when Var(X + Y) is expanded?

Derivation: Centre both variables and square their sum. The two cross terms add to twice the expectation of the product of centred variables, giving 2Cov(X, Y).

Options in canonical ID order: a: Only Cov(X, Y); b: Cov(X, Y)²; c: No covariance term; d: 2Cov(X, Y). Derived answer: d.

### ma-0309 — ma-sample-mean-variance (revision 1)

Condition: There are n independent identically distributed observations, each with variance σ². What is the variance of their sample mean?

Derivation: Independence makes the variance of their sum nσ². Multiplying the sum by 1/n multiplies its variance by 1/n², giving σ²/n.

Options in canonical ID order: a: σ²/n; b: σ²; c: nσ²; d: σ²/n². Derived answer: a.

### ma-0310 — ma-unbiased-variance (revision 2)

Condition: For n independent identically distributed observations of finite variance, with n ≥ 2, what divisor makes the sum of squared deviations from the sample mean an unbiased variance estimate?

Derivation: The identity Σ(Xᵢ − X̄)² = Σ(Xᵢ − μ)² − n(X̄ − μ)² gives expected value nσ² − σ² = (n − 1)σ². Divide by n − 1 to obtain expectation σ².

Options in canonical ID order: a: √n; b: n−1; c: n; d: n＋1. Derived answer: b.

### ma-0312 — ma-standard-error-size (revision 1)

Condition: Independent identically distributed observations have constant variance. If the sample size is multiplied by four, what happens to the standard error of the sample mean?

Derivation: The standard error is σ/√n. Replacing n by 4n divides it by √4 = 2, so it is halved.

Options in canonical ID order: a: It becomes one quarter; b: It doubles; c: It becomes four times as large; d: It is halved. Derived answer: d.

### ma-0319 — ma-bernoulli-mle (revision 1)

Condition: Ten independent Bernoulli trials produce seven successes. What is the maximum-likelihood estimate of the success probability?

Derivation: The likelihood is proportional to p⁷(1 − p)³. The log-likelihood derivative 7/p − 3/(1 − p) is zero at p = 7/10, where the likelihood is maximized.

Options in canonical ID order: a: 7; b: 0.3; c: 0.7; d: 0.5. Derived answer: c.

### ma-0321 — ma-length-unit (revision 1)

Condition: How many centimetres are in 2.5 m?

Derivation: There are 100 cm in 1 m, so multiply by 100: 2.5 × 100 = 250 cm.

Options in canonical ID order: a: 250 cm; b: 25 cm; c: 2500 cm; d: 0.025 cm. Derived answer: a.

### ma-0322 — ma-capacity-unit (revision 1)

Condition: How many litres are in 750 mL?

Derivation: There are 1000 mL in 1 L, so divide by 1000: 750/1000 = 0.75 L.

Options in canonical ID order: a: 750000 L; b: 0.75 L; c: 7.5 L; d: 75 L. Derived answer: b.

### ma-0323 — ma-area-unit (revision 1)

Condition: How many square centimetres are in 1 m²?

Derivation: Each side of a one-metre square is 100 cm. Its area is 100 × 100 = 10000 cm². Area conversion squares the length-conversion factor.

Options in canonical ID order: a: 1000 cm²; b: 10 cm²; c: 10000 cm²; d: 100 cm². Derived answer: c.

### ma-0324 — ma-time-duration (revision 1)

Condition: How many minutes elapse from 14:45 to 16:10 on the same day?

Derivation: There are 15 minutes from 14:45 to 15:00 and 70 more to 16:10, making 85 minutes.

Options in canonical ID order: a: 65 minutes; b: 125 minutes; c: 75 minutes; d: 85 minutes. Derived answer: d.

### ma-0326 — ma-speed-distance (revision 1)

Condition: What distance is travelled in 30 minutes at a constant speed of 60 km/h?

Derivation: Convert 30 minutes to 0.5 hours. Distance equals speed times time: 60 × 0.5 = 30 km.

Options in canonical ID order: a: 2 km; b: 30 km; c: 120 km; d: 1800 km. Derived answer: b.

### ma-0328 — ma-discount (revision 1)

Condition: An item priced at 2400 yen is discounted by 25%. What is the amount paid?

Derivation: The discount is 2400 × 0.25 = 600 yen. Subtracting gives 1800 yen, or 75% of the original price.

Options in canonical ID order: a: 600 yen; b: 2375 yen; c: 3000 yen; d: 1800 yen. Derived answer: d.

### ma-0330 — ma-unit-price (revision 1)

Condition: Which offer is cheaper per 100 g: 300 g for 450 yen or 500 g for 700 yen?

Derivation: The first offer costs 450/3 = 150 yen per 100 g. The second costs 700/5 = 140 yen per 100 g, so the second is cheaper.

Options in canonical ID order: a: Not enough information; b: 500 g for 700 yen; c: 300 g for 450 yen; d: They cost the same per 100 g. Derived answer: b.

### ma-0331 — ma-packing-ceiling (revision 1)

Condition: Forty-one people need cars holding eight people each. What is the minimum number of cars?

Derivation: Five cars hold only 40 people. One extra person requires a sixth car, so round the needed number of cars up to 6.

Options in canonical ID order: a: 7; b: 8; c: 6; d: 5. Derived answer: c.

### ma-0332 — ma-spacing-ends (revision 1)

Condition: Trees are planted every 20 m along a straight 100 m path, including both endpoints. How many trees are there?

Derivation: There are 100/20 = 5 intervals but 6 endpoints: 0, 20, 40, 60, 80 and 100 m.

Options in canonical ID order: a: 5; b: 4; c: 7; d: 6. Derived answer: d.

### ma-0333 — ma-circular-spacing (revision 1)

Condition: Trees are planted every 20 m around a circular path with circumference 120 m. How many trees are there?

Derivation: There are 120/20 = 6 intervals. On a closed circle, the end returns to the starting tree, so the number of trees equals the number of intervals: 6.

Options in canonical ID order: a: 6; b: 7; c: 5; d: 8. Derived answer: a.

### ma-0335 — ma-concentration (revision 1)

Condition: Dissolve 5 g of salt in 95 g of water. What is the salt concentration as a percentage of the solution's mass?

Derivation: The solution has total mass 5 + 95 = 100 g. The salt fraction is 5/100, so the mass percentage is 5%. The denominator is the whole solution, not only the water.

Options in canonical ID order: a: 95％; b: 10％; c: 5％; d: 5/95％. Derived answer: c.

### ma-0336 — ma-average-speed (revision 2)

Condition: Travel 10 km out at 10 km/h and the same 10 km back at 20 km/h. What is the average speed, defined as total distance divided by total time?

Derivation: The outward trip takes 1 hour and the return takes 0.5 hours. Average speed is 20/1.5 = 40/3 km/h. Averaging the two speeds directly gives the wrong weighting.

Options in canonical ID order: a: 15 km/h; b: 30 km/h; c: 10 km/h; d: 40/3 km/h. Derived answer: d.

### ma-0337 — ma-meeting (revision 1)

Condition: Two people start simultaneously at opposite ends of a 900 m route and move towards each other at 60 and 90 m/min. After how many minutes do they meet?

Derivation: Their separation shrinks at 60 + 90 = 150 m/min. The time is 900/150 = 6 minutes.

Options in canonical ID order: a: 6 minutes; b: 10 minutes; c: 15 minutes; d: 30 minutes. Derived answer: a.

### ma-0338 — ma-overtake (revision 1)

Condition: A person walking at 60 m/min is 300 m ahead of someone following at 90 m/min in the same direction. How long until the follower catches up?

Derivation: The gap closes at 90 − 60 = 30 m/min. Closing 300 m takes 300/30 = 10 minutes.

Options in canonical ID order: a: 15 minutes; b: 10 minutes; c: 2 minutes; d: 5 minutes. Derived answer: b.

### ma-0339 — ma-map-scale (revision 1)

Condition: A map has scale 1:50000. What actual distance in kilometres corresponds to 2 cm on the map?

Derivation: The actual distance is 2 × 50000 = 100000 cm. Divide by 100 to obtain 1000 m, then by 1000 to obtain 1 km.

Options in canonical ID order: a: 0.1 km; b: 100 km; c: 1 km; d: 10 km. Derived answer: c.

### ma-0340 — ma-irrelevant-information (revision 2)

Condition: Buy three items costing 120 yen each and pay with 500 yen. The shop opens at 9:00. How much change is due?

Derivation: The items cost 3 × 120 = 360 yen. The change is 500 − 360 = 140 yen. The opening time is irrelevant to this calculation.

Options in canonical ID order: a: 360 yen; b: 131 yen; c: 149 yen; d: 140 yen. Derived answer: d.

### ma-0341 — ma-sequential-discounts (revision 1)

Condition: An item costs 1000 yen before a 20% discount, followed by a further 10% discount on the reduced price. What is the final price?

Derivation: The first discount leaves 1000 × 0.8 = 800 yen. The second leaves 800 × 0.9 = 720 yen. The percentages apply to different bases.

Options in canonical ID order: a: 720 yen; b: 700 yen; c: 800 yen; d: 900 yen. Derived answer: a.

### ma-0342 — ma-percentage-points (revision 1)

Condition: A percentage rises from 20% to 25%. What is the increase in percentage points?

Derivation: Subtract the percentages: 25 − 20 = 5 percentage points. The relative percentage increase would be (25 − 20)/20 = 25%, a different quantity.

Options in canonical ID order: a: 125 percentage points; b: 5 percentage points; c: 25 percentage points; d: A 5% relative increase. Derived answer: b.

### ma-0345 — ma-work-together (revision 1)

Condition: A alone finishes a job in 6 hours and B alone in 3 hours. At constant rates, how long do they take working together?

Derivation: Their hourly job fractions add: 1/6 + 1/3 = 1/2. Completing one whole job therefore takes 1 ÷ (1/2) = 2 hours.

Options in canonical ID order: a: 2 hours; b: 4.5 hours; c: 9 hours; d: 1 hour. Derived answer: a.

### ma-0346 — ma-water-tank (revision 1)

Condition: An initially empty tank gains water at 8 L/min and drains at 3 L/min simultaneously. Its capacity is at least 20 L. How long until it contains 20 L?

Derivation: The net rate is 8 − 3 = 5 L/min. Reaching 20 L takes 20/5 = 4 minutes.

Options in canonical ID order: a: 5 minutes; b: 4 minutes; c: 2.5 minutes; d: 20/11 minutes. Derived answer: b.

### ma-0347 — ma-linear-cost (revision 1)

Condition: Plan A charges 100 yen plus 20 yen per minute. Plan B charges 300 yen plus 10 yen per minute. At what duration are their costs equal?

Derivation: Set 100 + 20t = 300 + 10t. Subtracting 10t and 100 gives 10t = 200, so t = 20 minutes.

Options in canonical ID order: a: 30 minutes; b: 40 minutes; c: 20 minutes; d: 10 minutes. Derived answer: c.

### ma-0348 — ma-budget-integer (revision 1)

Condition: Your budget is 1000 yen, shipping is a fixed 200 yen and each item costs 150 yen. What is the maximum number of items you can buy?

Derivation: After shipping, 800 yen remains. Five items cost 750 yen, but six cost 900 yen, so the maximum is 5.

Options in canonical ID order: a: 6; b: 4; c: 7; d: 5. Derived answer: d.

### ma-0351 — ma-parity-invariant (revision 2)

Condition: Five coins all show heads. Each operation flips exactly two distinct coins. Can all five be made to show tails using only these operations?

Derivation: The number of tails changes by +2, −2 or 0 with each operation. Its parity stays even, starting from zero. Five tails is odd, so it cannot be reached.

Options in canonical ID order: a: Yes, in five operations; b: Yes, in ten operations; c: No; d: Yes, in three operations. Derived answer: c.

### ma-0352 — ma-distance-displacement (revision 1)

Condition: You walk 3 km east and then 1 km west. What are the total distance travelled and the displacement towards the east, in that order?

Derivation: Distance adds both travelled lengths: 3 + 1 = 4 km. Eastward displacement uses signed motion: 3 − 1 = 2 km.

Options in canonical ID order: a: 2 km and 4 km; b: 4 km and 4 km; c: 2 km and 2 km; d: 4 km and 2 km. Derived answer: d.

### ma-0353 — ma-unit-consistency (revision 1)

Condition: An object has density 2 g/cm³ and volume 5 cm³. What is its mass?

Derivation: Mass equals density times volume: 2 × 5 = 10 g. The cm³ units cancel, leaving grams.

Options in canonical ID order: a: 10 g; b: 2.5 g; c: 0.4 g; d: 7 g. Derived answer: a.

### ma-0354 — ma-scale-factor-change (revision 1)

Condition: A square's side length increases by 10%. By what percentage does its area increase?

Derivation: Each dimension is multiplied by 1.1, so area is multiplied by 1.1² = 1.21. The increase is 0.21 of the original area, or 21%.

Options in canonical ID order: a: 1％; b: 21％; c: 10％; d: 20％. Derived answer: b.

### ma-0356 — ma-calendar-modulo (revision 1)

Condition: Today is Monday. What day of the week is 100 days later?

Derivation: The week repeats every seven days. Since 100 = 7 × 14 + 2, advance two days from Monday to Wednesday.

Options in canonical ID order: a: Monday; b: Tuesday; c: Thursday; d: Wednesday. Derived answer: d.

### ma-0358 — ma-necessary-info (revision 1)

Condition: All you know about a rectangle is that its perimeter is 20 cm. What can be said about its area?

Derivation: Its side lengths sum to 10 cm, but their product is not fixed. Sides 1 and 9 give area 9 cm², while sides 5 and 5 give 25 cm². Both have perimeter 20 cm.

Options in canonical ID order: a: It must be 100 cm²; b: It is not uniquely determined; c: It must be 25 cm²; d: It must be 20 cm². Derived answer: b.

### ma-0359 — ma-logical-counterexample (revision 1)

Condition: Which number is a counterexample to 'every odd number is prime'?

Derivation: 9 is odd but equals 3 × 3, so it is composite. One such example disproves a statement about every odd number.

Options in canonical ID order: a: 3; b: 5; c: 9; d: 2. Derived answer: c.

### ma-0361 — ma-implication-false (revision 1)

Condition: When is the logical implication P ⇒ Q false?

Derivation: An implication fails exactly when its premise P is true but its conclusion Q is false. If the premise is false, it does not provide a counterexample to the implication.

Options in canonical ID order: a: P is true and Q is false; b: P is false and Q is true; c: Both are true; d: Both are false. Derived answer: a.

### ma-0362 — ma-contrapositive (revision 1)

Condition: What is the contrapositive of 'if it rains, the ground is wet'?

Derivation: For P ⇒ Q, the contrapositive is not Q ⇒ not P. Thus it is 'if the ground is not wet, it is not raining.' This is logically equivalent to the given implication.

Options in canonical ID order: a: If it rains, the ground is not wet; b: If the ground is not wet, it is not raining; c: If the ground is wet, it is raining; d: If it is not raining, the ground is not wet. Derived answer: b.

### ma-0363 — ma-converse (revision 1)

Condition: What is the converse of 'if x > 2, then x > 0'?

Derivation: The converse swaps premise and conclusion: if x > 0, then x > 2. It need not be true; x = 1 is a counterexample.

Options in canonical ID order: a: If x ≤ 2, then x ≤ 0; b: If x > 2, then x ≤ 0; c: If x > 0, then x > 2; d: If x ≤ 0, then x ≤ 2. Derived answer: c.

### ma-0364 — ma-necessary-sufficient (revision 1)

Condition: For real x, how does the condition x = 3 relate to the condition x² = 9?

Derivation: When x = 3, its square is 9, so x = 3 is sufficient. But x = −3 also has square 9, so x = 3 is not necessary.

Options in canonical ID order: a: Necessary but not sufficient; b: Necessary and sufficient; c: Neither necessary nor sufficient; d: Sufficient but not necessary. Derived answer: d.

### ma-0365 — ma-negation-all (revision 1)

Condition: What is the negation of 'P(n) holds for every integer n'?

Derivation: To deny a universal statement, it is enough that at least one integer fails it. The negation is that there exists an integer n for which P(n) is false.

Options in canonical ID order: a: There exists an integer n for which P(n) is false; b: P(n) is false for every integer; c: There exists an integer for which P(n) is true; d: No integers exist. Derived answer: a.

### ma-0366 — ma-negation-exists (revision 1)

Condition: What is the negation of 'at least one person passes'?

Derivation: The negation says that no person passes, equivalently that everyone fails. Having at least one failure would still allow someone else to pass.

Options in canonical ID order: a: Exactly one person passes; b: Everyone fails; c: At least one person fails; d: Everyone passes. Derived answer: b.

### ma-0367 — ma-demorgan (revision 2)

Condition: What is the negation of 'A and B'?

Derivation: For both A and B to fail to hold together, at least one must be false. De Morgan's law gives 'not A or not B.'

Options in canonical ID order: a: A or B; b: A and B; c: Not A or not B; d: Not A and not B. Derived answer: c.

### ma-0368 — ma-inclusive-or (revision 1)

Condition: In mathematical logic, which case does 'A or B' normally include?

Derivation: The usual logical 'or' is inclusive: it is true if either statement is true or both are true. 'Exactly one' needs an additional restriction.

Options in canonical ID order: a: Exactly one must be true; b: Only when both are false; c: Only when A is false; d: The case where both A and B are true. Derived answer: d.

### ma-0369 — ma-set-intersection (revision 1)

Condition: Let A = {1, 2, 3} and B = {2, 3, 4}. What is A ∩ B?

Derivation: An intersection contains the elements belonging to both sets. Here they are 2 and 3, giving {2, 3}.

Options in canonical ID order: a: {2,3}; b: {1,2,3,4}; c: {1,4}; d: {1}. Derived answer: a.

### ma-0370 — ma-set-difference (revision 1)

Condition: Let A = {1, 2, 3} and B = {2, 4}. What is the set obtained by removing B's elements from A?

Derivation: Keep the elements of A that do not belong to B. Removing 2 leaves {1, 3}; 4 was not in A to begin with.

Options in canonical ID order: a: {4}; b: {1,3}; c: {2}; d: {1,2,3,4}. Derived answer: b.

### ma-0371 — ma-empty-subset (revision 1)

Condition: Which statement about the empty set is correct?

Derivation: A set is a subset of A if all its elements belong to A. The empty set has no element that could violate this condition, so it is a subset of every set.

Options in canonical ID order: a: It contains the element 0; b: It has one element; c: It is a subset of every set; d: It is a subset of no set. Derived answer: c.

### ma-0372 — ma-power-set (revision 1)

Condition: How many subsets does a set with three elements have?

Derivation: Each element has two independent choices: included or excluded. There are 2³ = 8 subsets, including the empty set and the whole set.

Options in canonical ID order: a: 3; b: 6; c: 9; d: 8. Derived answer: d.

### ma-0373 — ma-induction-step (revision 1)

Condition: In mathematical induction, what must be shown in addition to the initial case?

Derivation: Show that whenever the statement holds at n, it also holds at n + 1. Together with the initial case, this carries the result through the subsequent natural numbers.

Options in canonical ID order: a: Assuming it holds at n, show it holds at n + 1; b: Check only one large n; c: Check only n = 1; d: Assume the conclusion and stop. Derived answer: a.

### ma-0374 — ma-proof-by-contradiction (revision 1)

Condition: What is the basic procedure for proof by contradiction?

Derivation: Assume the desired conclusion is false and derive a contradiction with the premises or an established fact. This rules out the negation and establishes the conclusion.

Options in canonical ID order: a: Discard all premises; b: Assume the conclusion's negation and derive a contradiction; c: Give one example of the conclusion; d: Ignore counterexamples. Derived answer: b.

### ma-0377 — ma-pigeonhole (revision 1)

Condition: Thirteen people are classified into the twelve birth months. What must be true?

Derivation: If each month contained at most one person, there would be at most twelve people. With thirteen, at least one month contains at least two people. This does not require a shared day of the month.

Options in canonical ID order: a: Two people share a birth month; b: Two people share a birthday; c: Every month has one person; d: Three people share a birth month. Derived answer: a.

### ma-0381 — ma-graph-handshake (revision 1)

Condition: In a finite undirected graph without self-loops, what is the sum of all vertex degrees?

Derivation: Each edge contributes one to the degree at each of its two endpoints. Summing degrees therefore counts every edge twice.

Options in canonical ID order: a: Twice the number of edges; b: The number of edges; c: The number of vertices; d: Always zero. Derived answer: a.

### ma-0383 — ma-tree-edges (revision 1)

Condition: How many edges does a finite tree with n vertices have?

Derivation: A one-vertex tree has zero edges. Removing a leaf from a larger tree removes one vertex and one edge while leaving a tree. Induction gives n − 1 edges.

Options in canonical ID order: a: n＋1; b: 2n; c: n−1; d: n. Derived answer: c.

### ma-0386 — ma-bipartite (revision 1)

Condition: Which statement must be true of a bipartite graph?

Derivation: Each edge crosses between the two parts. A cycle must alternate parts and return to its starting part, so every cycle has even length. An odd-length cycle is impossible.

Options in canonical ID order: a: It must contain a triangle; b: It contains no odd-length cycle; c: It has no edges; d: It must be a tree. Derived answer: b.

### ma-0387 — ma-planar-euler (revision 1)

Condition: A finite connected plane graph has V vertices, E edges and F faces, including the outside face. What relation holds?

Derivation: A tree has E = V − 1 and F = 1, giving V − E + F = 2. Removing a cycle edge merges two faces and reduces both E and F by one, preserving this value until a tree remains.

Options in canonical ID order: a: V−E＋F＝0; b: V＝E＝F; c: V−E＋F＝2; d: V＋E＋F＝2. Derived answer: c.

### ma-0388 — ma-graph-shortest (revision 1)

Condition: Which standard algorithm finds shortest paths in a graph whose edge weights are nonnegative?

Derivation: Dijkstra's algorithm repeatedly settles the smallest tentative distance and updates neighbouring distances. Nonnegative weights mean that a later extension cannot improve the already smallest unsettled distance through a longer route.

Options in canonical ID order: a: Bubble sort; b: The Euclidean algorithm; c: Binary search alone; d: Dijkstra's algorithm. Derived answer: d.

### ma-0389 — ma-dag-topological (revision 1)

Condition: When does a finite directed graph admit a topological ordering?

Derivation: A directed cycle would require each vertex to precede itself after following the cycle. Without directed cycles, there is a vertex with no incoming edge; remove it and repeat to construct an ordering.

Options in canonical ID order: a: When it has no directed cycle; b: When every vertex has degree 2; c: Only when it is connected; d: When all edge weights are 1. Derived answer: a.

### ma-0390 — ma-equivalence-relation (revision 1)

Condition: Which three properties define an equivalence relation?

Derivation: Reflexivity relates each element to itself; symmetry reverses related pairs; transitivity combines consecutive related pairs. Together they allow the elements to be grouped into equivalence classes.

Options in canonical ID order: a: Commutativity, distributivity and inverses; b: Reflexivity, symmetry and transitivity; c: Reflexivity, asymmetry and continuity; d: Monotonicity, continuity and differentiability. Derived answer: b.

### ma-0391 — ma-partial-order (revision 1)

Condition: Which properties define a partial order?

Derivation: A partial order is reflexive, antisymmetric and transitive. Antisymmetry says that a ≤ b and b ≤ a imply a = b. Not every pair has to be comparable.

Options in canonical ID order: a: Reflexivity, symmetry and transitivity; b: Only transitivity; c: Reflexivity, antisymmetry and transitivity; d: Symmetry, continuity and commutativity. Derived answer: c.

### ma-0393 — ma-function-injective (revision 1)

Condition: What does it mean for a function to be injective?

Derivation: Different inputs have different outputs. Equivalently, f(a) = f(b) implies a = b. This does not require every element of the codomain to be reached.

Options in canonical ID order: a: Different inputs map to different outputs; b: Every element of the codomain is reached; c: The domain and codomain are the same set; d: The output is always constant. Derived answer: a.

### ma-0394 — ma-function-surjective (revision 1)

Condition: What does it mean for a function to be surjective?

Derivation: Every element of the specified codomain is an output of at least one input. Two different inputs may still have the same output, so an inverse function is not guaranteed.

Options in canonical ID order: a: There is exactly one input; b: Its range equals its whole specified codomain; c: Different inputs always have different outputs; d: It must have an inverse function. Derived answer: b.

### ma-0397 — ma-algorithm-logarithm (revision 2)

Condition: Binary search starts with 1024 sorted candidates. How many halvings are needed to leave one candidate?

Derivation: 1024 = 2¹⁰. Halving ten times leaves one candidate. This counts candidate-set halvings, not an implementation's comparison operations.

Options in canonical ID order: a: 10; b: 1024; c: 512; d: 32. Derived answer: a.

### ma-0398 — ma-asymptotic (revision 1)

Condition: What is the asymptotic order of growth of n² + 3n + 1 as n tends to infinity?

Derivation: For n ≥ 1, n² ≤ n² + 3n + 1 ≤ 5n². These constant-factor bounds give Θ(n²); the lower-degree terms do not change the order.

Options in canonical ID order: a: Θ(2ⁿ); b: Θ(n²); c: Θ(n); d: Θ(1). Derived answer: b.

### ma-0401 — ma-queue-reverse-ordinal (revision 1)

Condition: Eight people stand in a line from front to back. Sakura is third from the front. What is her position counting from the back?

Derivation: Counting from both ends includes Sakura twice. Her position from the back is 8 − 3 + 1 = 6.

Options in canonical ID order: a: 5th; b: 7th; c: 6th; d: 3rd. Derived answer: c.

### ma-0402 — ma-analog-clock-reading (revision 1)

Condition: A clock's long hand points to 4, and its short hand is between 7 and 8. What time does it show?

Derivation: Each numbered step of the long hand represents five minutes, so 4 means 20 minutes. The short hand is still in the 7 o'clock hour, giving 7:20.

Options in canonical ID order: a: 7:20; b: 7:04; c: 4:07; d: 8:20. Derived answer: a.

### ma-0403 — ma-ruler-zero-origin (revision 2)

Condition: A ruler's zero mark is slightly inside its physical edge. How should you align an object so the number at its other end can be read directly as its length?

Derivation: Align one end of the object with the zero mark. Then the reading at the other end is its length. Starting at 1 cm would require subtracting 1 cm; the physical edge is not the zero mark.

Options in canonical ID order: a: Align one end with the ruler's physical edge; b: Align one end with the 1 cm mark; c: Align the object's centre with the zero mark; d: Align one end with the zero mark. Derived answer: d.

### ma-0404 — ma-diameter-segment-identification (revision 1)

Condition: A circle has centre O, and A, B and C lie on its circumference. Segment AB passes through O. Which segment is a diameter?

Derivation: A diameter joins two points on the circumference and passes through the centre. AB meets both conditions. OA, OB and OC each run only from the centre to the circumference and are radii.

Options in canonical ID order: a: OA; b: AB; c: OB; d: OC. Derived answer: b.

### ma-0405 — ma-unit-tile-irregular-area (revision 1)

Condition: Squares each have side length 1 cm. Stack two in the left column, one in the next column and one in the next, with their bottom edges aligned and no overlaps. What is the total area?

Derivation: Each square has area 1 cm². There are 2 + 1 + 1 = 4 squares, so the total area is 4 cm².

Options in canonical ID order: a: 3cm²; b: 4cm²; c: 5cm²; d: 8cm². Derived answer: b.

### ma-0406 — ma-shelf-two-direction-address (revision 1)

Condition: A shelf has columns A, B, C from left to right and rows 1, 2, 3 from bottom to top. A package is in the top row of column B. Write its position as the column letter followed by the row number.

Derivation: The column is B and the top row is 3, so combining the two independent position labels gives B3.

Options in canonical ID order: a: B1; b: C3; c: A3; d: B3. Derived answer: d.

### ma-0407 — ma-fold-reflected-vertex (revision 1)

Condition: A rectangle's vertices are A at top left, B at top right, C at bottom right and D at bottom left. Fold it along the vertical line halfway between its left and right sides. Which vertex meets A?

Derivation: The fold reflects the left half onto the right half without changing height. Top-left A therefore meets top-right B.

Options in canonical ID order: a: B; b: C; c: D; d: A. Derived answer: a.

### ma-0408 — ma-pictograph-legend-decoding (revision 1)

Condition: In a pictograph of borrowed books, each circle represents two books. Monday has three circles and Tuesday has two. How many books were borrowed on Monday?

Derivation: The number of symbols is different from the number of books. Three Monday symbols, each worth two books, represent 3 × 2 = 6 books.

Options in canonical ID order: a: 3 books; b: 2 books; c: 6 books; d: 5 books. Derived answer: c.

### ma-0409 — ma-line-graph-single-point-reading (revision 2)

Condition: Water in a container was recorded as 0 L after 0 minutes, 1 L after 1 minute, 4 L after 2 minutes and 5 L after 3 minutes. How many litres were there after 2 minutes?

Derivation: The entry paired with 2 minutes is 4 L. On the graph, find time 2 on the horizontal axis and read the height of its point on the litres axis.

Options in canonical ID order: a: 1L; b: 2L; c: 5L; d: 4L. Derived answer: d.

### ma-0410 — ma-fraction-different-wholes (revision 1)

Condition: Tape A is 20 cm long and tape B is 10 cm long. Each is cut into two equal-length pieces. How does half of A compare with half of B?

Derivation: Half of A is 20/2 = 10 cm, while half of B is 10/2 = 5 cm. The same fraction of different wholes need not be the same amount.

Options in canonical ID order: a: Half of B is longer; b: Half of A is longer; c: They have the same length; d: Their original lengths do not determine this. Derived answer: b.

### ma-0411 — ma-pairs-parity-classification (revision 1)

Condition: Which number of stones leaves exactly one stone over when the stones are grouped in pairs?

Derivation: 13 = 2 × 6 + 1, so one stone remains. This is an odd number. The other choices, 12, 14 and 16, divide into pairs with none left.

Options in canonical ID order: a: 13; b: 12; c: 14; d: 16. Derived answer: a.

### ma-0412 — ma-sharing-division-answer-unit (revision 1)

Condition: Twelve stickers are shared equally among four people. What does the answer to 12 ÷ 4 represent?

Derivation: The divisor 4 counts people. Dividing the total sticker count by that number gives the sticker count per person, which is 3.

Options in canonical ID order: a: The number of people; b: The total number of stickers; c: The number of leftover stickers; d: The number of stickers per person. Derived answer: d.

### ma-0413 — ma-decimal-tenfold-scale (revision 1)

Condition: Which number is ten times 0.6?

Derivation: 0.6 × 10 = 6. Multiplication by 10 moves the decimal place value one position; 0.06 would be one tenth as much, and 60 would be one hundred times as much.

Options in canonical ID order: a: 0.06; b: 0.16; c: 6; d: 60. Derived answer: c.

### ma-0414 — ma-mixed-number-meaning (revision 1)

Condition: Which expression represents the mixed number 'one and three quarters'?

Derivation: A mixed number combines an integer part and a fractional part by addition. Thus it is 1 + 3/4 = 7/4, rather than a product or quotient.

Options in canonical ID order: a: 1×3/4; b: 1＋3/4; c: 1÷3/4; d: 13÷4. Derived answer: b.

### ma-0415 — ma-addition-commutative-meaning (revision 1)

Condition: Which statement correctly compares the answers to 5 + 8 and 8 + 5?

Derivation: Both sums equal 13. Changing the order of two addends does not change their sum; this is the commutative property of addition.

Options in canonical ID order: a: They have the same answer; b: 5 + 8 is larger; c: 8 + 5 is larger; d: Their different order makes comparison impossible. Derived answer: a.

### ma-0416 — ma-quantity-unit-identification (revision 1)

Condition: Which unit is appropriate for a bag's weight in the everyday sense of its mass?

Derivation: Use kg for mass. The other units describe different kinds of quantities: cm for length, L for capacity and cm² for area. This question uses everyday weight to mean mass rather than a force.

Options in canonical ID order: a: cm; b: L; c: kg; d: cm². Derived answer: c.

### ma-0417 — ma-equal-bisection-meaning (revision 1)

Condition: What does it mean to divide a length of tape into two equal parts?

Derivation: There must be two pieces, and their lengths must be the same. Two unequal pieces are not equal parts, and three equal pieces would be thirds.

Options in canonical ID order: a: Cut it into two pieces of equal length; b: Cut it into one long and one short piece; c: Cut it into three pieces of equal length; d: Keep one part and discard the other. Derived answer: a.

### ma-0418 — ma-sphere-flat-faces (revision 1)

Condition: How many flat faces are there on the surface of a perfect sphere?

Derivation: A sphere's surface is curved everywhere. It has no planar face such as a box's side or a cylinder's flat end, so the number of flat faces is zero.

Options in canonical ID order: a: 1; b: 2; c: 6; d: 0. Derived answer: d.

### ma-0419 — ma-remainder-upper-bound (revision 1)

Condition: A whole number of items is divided into groups of seven. Which number cannot be the remainder?

Derivation: An integer remainder must be at least zero and less than seven. If seven items remained, they would form another full group. Thus 7 cannot be the remainder.

Options in canonical ID order: a: 0; b: 3; c: 7; d: 6. Derived answer: c.

### ma-0420 — ma-subunit-divisor-size-effect (revision 1)

Condition: How does the answer to 6 ÷ 0.5 compare with 6?

Derivation: There are twelve halves in six, so 6 ÷ 0.5 = 12, which is greater than 6. Dividing a positive number by a positive number smaller than 1 increases it.

Options in canonical ID order: a: It is smaller; b: It is larger; c: It is the same; d: The answer is undefined. Derived answer: b.

## Verification limits

Author checks assert 300 unique existing reviewed base IDs/families; 56 selections per original topic block; all twenty foundation IDs; exact preservation of stable metadata and answer-option IDs; draft status; English display fields; structurally valid illustrations; and exact answer-free blind exports. These are structural/provenance checks, not an independent correctness verdict. English explanations above are the author's derivations and require the separate blind solve and bilingual concept check coordinated by the parent. Browser rendering, user difficulty calibration, publishing and deployment are outside this authoring step.

## Repairs after independent blind review (2026-10-04)

The initial English bank and source bytes were preserved with exclusive creates under output/quiz-en-20261004/math-author/repair-before-20261004-1. These repairs clarify English wording and mathematical explanations without changing IDs, revisions, families, canonical classifications or correct option IDs. The initial blind packets remain unchanged. The author re-derived the following choices; independent final approval is still assigned to the reviewers.

- **ma-0013**: (3/5)(2/3) = 6/15 = 2/5. The three other values 5/8, 6/5 and 5/2 differ from 2/5. Correct option remains a; all four choices checked.
- **ma-0054**: 7 × 8 = 56 = 5 × 11 + 1. The unique remainder in [0,5) is 1; the alternatives 4, 2 and 3 do not satisfy this division identity. The intermediate residue representative 6 is not called a remainder. Correct option remains b; all four choices checked.
- **ma-0105**: Adding x+y=7 and x−y=1 yields x=4 and y=3. Other ordered choices (3,4), (6,1), (1,6) have differences −1, 5 and −5 instead of 1. Correct option remains a; all four choices checked.
- **ma-0183**: A simple six-vertex polygon triangulates into 6−2=4 triangles, whether convex or concave. Their summed interior angles are 720 degrees. The other choices correspond to different totals 900,1080,540. Correct option remains c; all four choices checked.
- **ma-0196**: Opposite inscribed angles subtend arcs partitioning the full 360-degree circle; halving gives A+C=180 degrees. With A=110 degrees, C=70 degrees. The other choices 110,55,90 do not satisfy the sum. Correct option remains d; all four choices checked.
- **ma-0243**: Only a bar chart compares categorical counts using rectangle heights. A scatter plot uses paired points, a diagram of a circumference shows a circular boundary, and a line graph connects points rather than comparing rectangle heights. Correct option remains c; all four choices checked.
- **ma-0265**: Label the individual objects R1,R2,B. Uniform sampling of two without replacement produces equally likely unordered pairs {R1,R2},{R1,B},{R2,B}. Two have different colours; probability 2/3. All other choices 1/3,4/9,1/2 differ. Correct option remains a; all four choices checked.
- **ma-0307**: E[X+Y]=E[X]+E[Y] follows by integrating or summing their pointwise sum and requires no independence. For X=Y taking 0 or 1 equally, Var(X+Y)=1 but Var(X)+Var(Y)=1/2; P(X=Y)=1; E[XY]=1/2 but E[X]E[Y]=1/4. These respectively refute options a,b,d. Independence is sufficient for factorization, not necessary: independent variables give the equality, while some dependent uncorrelated variables also give it. Correct option remains c; all four choices checked.