EXAM QUICK REFERENCE

EJU Mathematics Course 1 exam quick reference

Course 1 serves humanities and some science applicants, and its scope is strictly the first seven syllabus domains. Chapters 1–12 open those domains, chapter 13 handles the mechanics of a guided mark-sheet paper, and chapter 14 addresses the difficulty international students actually face: the mathematics is familiar, the Japanese is not.

Strictly follows domains 1–7 of JASSO's 2026 Course 1 syllabus; domains 8 and later belong to Course 2 and are out of scope here.

Exam time: 80 minutes
Domains: 7 domains
Answer language: Japanese or English
Questions: 70

Question practice is open. These resources are available without signing in.

QUICK REFERENCE

Look up a term, grammar form, or method

Showing 72 items

Numbers and expressions I: reals and polynomials

001実数と数直線real numbers and the number lineEssential

Core note: Rational and irrational numbers together form the reals, in one-to-one correspondence with points on the number line; every repeating decimal is rational.
Exam area: Numbers and expressions I: reals and polynomials
Decision focus: Rationalizing and factoring must become automatic.

002平方根と有理化square roots and rationalizing denominatorsEssential

Core note: √a denotes the non-negative number whose square is a; multiply by the conjugate to clear a radical from a denominator.
Exam area: Numbers and expressions I: reals and polynomials
Decision focus: Rationalizing and factoring must become automatic.

003指数法則laws of exponentsFrequent

Core note: Products add exponents, quotients subtract them, powers multiply them, and a negative exponent denotes a reciprocal.
Exam area: Numbers and expressions I: reals and polynomials
Decision focus: Rationalizing and factoring must become automatic.

004展開と因数分解expansion and factorizationEssential

Core note: Multiplication formulas turn products into sums and, in reverse, rewrite a polynomial as a product of factors — the prerequisite for solving and simplifying.
Exam area: Numbers and expressions I: reals and polynomials
Decision focus: Rationalizing and factoring must become automatic.

005対称式symmetric expressionsFrequent

Core note: An expression unchanged by swapping its variables can always be written in terms of the sum a+b and the product ab.
Exam area: Numbers and expressions I: reals and polynomials
Decision focus: Rationalizing and factoring must become automatic.

006二重根号nested radicalsExtension

Core note: A form √(a±2√b) unnests to √m±√n once you find two numbers with sum a and product b.
Exam area: Numbers and expressions I: reals and polynomials
Decision focus: Rationalizing and factoring must become automatic.

Numbers and expressions II: inequalities and absolute values

007絶対値absolute valueEssential

Core note: |a| is the distance from a to the origin; removing the bars requires splitting into sign cases.
Exam area: Numbers and expressions II: inequalities and absolute values
Decision focus: Reversing the inequality when multiplying by a negative is the most common slip.

0081次不等式linear inequalitiesEssential

Core note: Solve by transposing and collecting terms; multiplying or dividing by a negative number reverses the inequality.
Exam area: Numbers and expressions II: inequalities and absolute values
Decision focus: Reversing the inequality when multiplying by a negative is the most common slip.

009連立不等式systems of inequalitiesFrequent

Core note: Solve each inequality, then intersect the solution sets, using a number line to check the overlap and whether endpoints are included.
Exam area: Numbers and expressions II: inequalities and absolute values
Decision focus: Reversing the inequality when multiplying by a negative is the most common slip.

010絶対値を含む方程式・不等式equations and inequalities with absolute valuesEssential

Core note: |x|<a means −a<x<a and |x|>a means x<−a or x>a; more complex forms are split at their break points.
Exam area: Numbers and expressions II: inequalities and absolute values
Decision focus: Reversing the inequality when multiplying by a negative is the most common slip.

011整数部分と小数部分integer and fractional partsFrequent

Core note: Bracket the irrational number between integers first; the fractional part is the number minus its integer part.
Exam area: Numbers and expressions II: inequalities and absolute values
Decision focus: Reversing the inequality when multiplying by a negative is the most common slip.

012近似値と有効数字approximate values and significant figuresExtension

Core note: Measured values carry error; significant figures express how far they can be trusted, and scientific notation makes the digit count explicit.
Exam area: Numbers and expressions II: inequalities and absolute values
Decision focus: Reversing the inequality when multiplying by a negative is the most common slip.

Sets and propositions

013集合の演算set operationsEssential

Core note: Union, intersection, and complement are best organized with a Venn diagram; count elements only after drawing it.
Exam area: Sets and propositions
Decision focus: Reduce every condition question to set inclusion.

014ド・モルガンの法則De Morgan's lawsFrequent

Core note: The complement of a union is the intersection of complements and vice versa — essential when negating compound conditions.
Exam area: Sets and propositions
Decision focus: Reduce every condition question to set inclusion.

015命題と真偽propositions and truth valuesEssential

Core note: A proposition is a statement with a definite truth value; one counterexample disproves it, while proving it requires covering every case.
Exam area: Sets and propositions
Decision focus: Reduce every condition question to set inclusion.

016必要条件・十分条件necessary and sufficient conditionsEssential

Core note: If p⇒q then p is sufficient for q and q is necessary for p; deciding by set inclusion is the reliable method.
Exam area: Sets and propositions
Decision focus: Reduce every condition question to set inclusion.

017逆・裏・対偶converse, inverse, and contrapositiveFrequent

Core note: A statement and its contrapositive share a truth value, as do the converse and inverse; prove the contrapositive when a direct proof is hard.
Exam area: Sets and propositions
Decision focus: Reduce every condition question to set inclusion.

018背理法proof by contradictionFrequent

Core note: Assume the conclusion fails, derive a contradiction, and thereby confirm it — the standard route for irrationality proofs.
Exam area: Sets and propositions
Decision focus: Reduce every condition question to set inclusion.

Quadratic functions I: graphs and extrema

019平方完成completing the squareEssential

Core note: Rewriting y=ax²+bx+c as y=a(x−p)²+q makes the vertex and the extreme value readable at a glance.
Exam area: Quadratic functions I: graphs and extrema
Decision focus: Fix the opening direction, then check whether the vertex lies inside the interval.

020頂点と軸vertex and axis of symmetryEssential

Core note: The vertex (p, q) is the highest or lowest point of the parabola, and the line x=p is its axis of symmetry.
Exam area: Quadratic functions I: graphs and extrema
Decision focus: Fix the opening direction, then check whether the vertex lies inside the interval.

021平行移動と対称移動translation and reflection of graphsFrequent

Core note: Shifting right by p replaces x with x−p and shifting up by q adds q; reflecting in the x-axis negates the function and reflecting in the y-axis replaces x with −x.
Exam area: Quadratic functions I: graphs and extrema
Decision focus: Fix the opening direction, then check whether the vertex lies inside the interval.

022最大値・最小値maximum and minimum valuesEssential

Core note: An upward parabola takes its minimum at the vertex and a downward one its maximum, so decide the opening direction first.
Exam area: Quadratic functions I: graphs and extrema
Decision focus: Fix the opening direction, then check whether the vertex lies inside the interval.

023定義域の制限extrema on a restricted domainEssential

Core note: When the vertex lies outside the interval the extremes occur at the endpoints, so compare the vertex against both endpoints.
Exam area: Quadratic functions I: graphs and extrema
Decision focus: Fix the opening direction, then check whether the vertex lies inside the interval.

0242次関数の決定determining a quadratic functionFrequent

Core note: Use a(x−p)²+q given the vertex, a(x−α)(x−β) given the x-intercepts, and the general form with simultaneous equations given three points.
Exam area: Quadratic functions I: graphs and extrema
Decision focus: Fix the opening direction, then check whether the vertex lies inside the interval.

Quadratic functions II: equations and inequalities

025解の公式the quadratic formulaEssential

Core note: The roots of ax²+bx+c=0 are x=(−b±√(b²−4ac))/(2a); the even-b reduced form saves time.
Exam area: Quadratic functions II: equations and inequalities
Decision focus: Root-placement problems require the discriminant, the axis, and the endpoints together.

026判別式the discriminantEssential

Core note: The sign of D=b²−4ac fixes the number of real roots: two if positive, one repeated if zero, none if negative.
Exam area: Quadratic functions II: equations and inequalities
Decision focus: Root-placement problems require the discriminant, the axis, and the endpoints together.

027解と係数の関係relations between roots and coefficientsEssential

Core note: The roots sum to −b/a and multiply to c/a, letting you evaluate symmetric expressions without solving.
Exam area: Quadratic functions II: equations and inequalities
Decision focus: Root-placement problems require the discriminant, the axis, and the endpoints together.

0282次不等式quadratic inequalitiesEssential

Core note: Find the x-intercepts first, then read the solution interval off the graph according to the opening direction.
Exam area: Quadratic functions II: equations and inequalities
Decision focus: Root-placement problems require the discriminant, the axis, and the endpoints together.

029解の配置placement of rootsFrequent

Core note: Constraining roots to an interval requires three simultaneous conditions: the discriminant, the position of the axis, and the sign at the endpoints.
Exam area: Quadratic functions II: equations and inequalities
Decision focus: Root-placement problems require the discriminant, the axis, and the endpoints together.

030放物線と直線の共有点intersections of a parabola and a lineFrequent

Core note: Eliminate to get a quadratic; its discriminant gives the number of intersections, with tangency at D=0.
Exam area: Quadratic functions II: equations and inequalities
Decision focus: Root-placement problems require the discriminant, the axis, and the endpoints together.

Figures and measurement I: trigonometric ratios

031三角比trigonometric ratiosEssential

Core note: In a right triangle, sine is opposite over hypotenuse, cosine adjacent over hypotenuse, and tangent opposite over adjacent.
Exam area: Figures and measurement I: trigonometric ratios
Decision focus: Judge signs on the unit circle across 0°–180° rather than relying on the right triangle alone.

032特別角の値values at special anglesEssential

Core note: The ratios at 30°, 45°, and 60° must be instantly available; they open most measurement problems.
Exam area: Figures and measurement I: trigonometric ratios
Decision focus: Judge signs on the unit circle across 0°–180° rather than relying on the right triangle alone.

033鈍角の三角比trigonometric ratios of obtuse anglesEssential

Core note: The unit circle extends the definition to 0°–180°, where sine stays positive while cosine and tangent turn negative.
Exam area: Figures and measurement I: trigonometric ratios
Decision focus: Judge signs on the unit circle across 0°–180° rather than relying on the right triangle alone.

034三角比の相互関係trigonometric identitiesEssential

Core note: The three identities sin²θ+cos²θ=1, tanθ=sinθ/cosθ, and 1+tan²θ=1/cos²θ convert between ratios.
Exam area: Figures and measurement I: trigonometric ratios
Decision focus: Judge signs on the unit circle across 0°–180° rather than relying on the right triangle alone.

03590°−θ・180°−θ の公式complementary and supplementary angle formulasFrequent

Core note: Conversions such as sin(90°−θ)=cosθ, sin(180°−θ)=sinθ, and cos(180°−θ)=−cosθ.
Exam area: Figures and measurement I: trigonometric ratios
Decision focus: Judge signs on the unit circle across 0°–180° rather than relying on the right triangle alone.

036三角比を含む方程式・不等式equations and inequalities with trigonometric ratiosFrequent

Core note: Read every solution off the unit circle within 0°≤θ≤180°, checking whether the endpoints qualify.
Exam area: Figures and measurement I: trigonometric ratios
Decision focus: Judge signs on the unit circle across 0°–180° rather than relying on the right triangle alone.

Figures and measurement II: theorems and solids

037正弦定理the law of sinesEssential

Core note: a/sinA=b/sinB=c/sinC=2R — used with two angles and a side, or to find the circumradius.
Exam area: Figures and measurement II: theorems and solids
Decision focus: In solids, first cut the plane that contains the segment you need.

038余弦定理the law of cosinesEssential

Core note: a²=b²+c²−2bc·cosA — finds the third side from two sides and the included angle, or an angle from three sides.
Exam area: Figures and measurement II: theorems and solids
Decision focus: In solids, first cut the plane that contains the segment you need.

039三角形の面積area of a triangleEssential

Core note: S=(1/2)bc·sinA is fastest given two sides and the included angle, and typically pairs with the law of cosines.
Exam area: Figures and measurement II: theorems and solids
Decision focus: In solids, first cut the plane that contains the segment you need.

040ヘロンの公式Heron's formulaExtension

Core note: Given three sides, S=√(s(s−a)(s−b)(s−c)) where s is half the perimeter.
Exam area: Figures and measurement II: theorems and solids
Decision focus: In solids, first cut the plane that contains the segment you need.

041内接円と外接円の半径inradius and circumradiusFrequent

Core note: The circumradius comes from the law of sines and the inradius from S=rs, with s the semiperimeter.
Exam area: Figures and measurement II: theorems and solids
Decision focus: In solids, first cut the plane that contains the segment you need.

042空間図形の計量measurement of solidsFrequent

Core note: Cut a solid along a plane containing the segment in question, then work inside that plane with trigonometric ratios.
Exam area: Figures and measurement II: theorems and solids
Decision focus: In solids, first cut the plane that contains the segment you need.

Counting

043和の法則・積の法則addition and multiplication principlesEssential

Core note: Add over mutually exclusive cases and multiply over successive steps; first decide whether you are classifying or sequencing.
Exam area: Counting
Decision focus: Decide whether order matters before you start counting.

044順列permutationsEssential

Core note: Arranging r of n items gives nPr=n!/(n−r)!, counting different orders as different outcomes.
Exam area: Counting
Decision focus: Decide whether order matters before you start counting.

045組合せcombinationsEssential

Core note: Choosing without ordering gives nCr=nPr/r!; distinguishing this from permutations is the first decision.
Exam area: Counting
Decision focus: Decide whether order matters before you start counting.

046円順列circular permutationsFrequent

Core note: Fix one position around the circle to get (n−1)!, halving again if reflections count as identical.
Exam area: Counting
Decision focus: Decide whether order matters before you start counting.

047重複順列・同じものを含む順列repeated permutations and multiset arrangementsFrequent

Core note: With repetition allowed the count is nʳ; with identical items divide n! by the factorials of each repeat count.
Exam area: Counting
Decision focus: Decide whether order matters before you start counting.

048包除原理inclusion-exclusionFrequent

Core note: n(A∪B)=n(A)+n(B)−n(A∩B) — the first thing to try whenever a question says "at least one".
Exam area: Counting
Decision focus: Decide whether order matters before you start counting.

Probability

049確率の定義definition of probabilityEssential

Core note: When all outcomes are equally likely, probability is the number of favorable outcomes divided by the total.
Exam area: Probability
Decision focus: Start from the complement whenever the question says "at least".

050排反事象と加法定理mutually exclusive events and the addition ruleEssential

Core note: For mutually exclusive events P(A∪B)=P(A)+P(B); otherwise subtract P(A∩B).
Exam area: Probability
Decision focus: Start from the complement whenever the question says "at least".

051余事象の確率probability of the complementEssential

Core note: P(A)=1−P(Ā); phrases like "at least one" or "not all" almost always call for the complement.
Exam area: Probability
Decision focus: Start from the complement whenever the question says "at least".

052独立試行と反復試行independent and repeated trialsEssential

Core note: Independent trials multiply, and exactly r successes in n trials has probability nCr·pʳ(1−p)ⁿ⁻ʳ.
Exam area: Probability
Decision focus: Start from the complement whenever the question says "at least".

053条件付き確率conditional probabilityFrequent

Core note: P(B|A)=P(A∩B)/P(A) — recount after shrinking the sample space to A.
Exam area: Probability
Decision focus: Start from the complement whenever the question says "at least".

054期待値expected valueFrequent

Core note: Sum each value times its probability to get the long-run average; the probabilities must total 1.
Exam area: Probability
Decision focus: Start from the complement whenever the question says "at least".

Properties of integers

055約数と倍数divisors and multiplesEssential

Core note: If b=ak then a divides b; memorize the divisibility tests for 2, 3, 4, 5, and 9.
Exam area: Properties of integers
Decision focus: Handle claims about every integer by classifying on remainders.

056素因数分解prime factorizationEssential

Core note: Every positive integer factors uniquely into primes, and the divisor count is the product of each exponent plus one.
Exam area: Properties of integers
Decision focus: Handle claims about every integer by classifying on remainders.

057最大公約数と最小公倍数greatest common divisor and least common multipleEssential

Core note: The product of two numbers equals gcd times lcm; take the minimum and maximum exponent of each prime.
Exam area: Properties of integers
Decision focus: Handle claims about every integer by classifying on remainders.

058ユークリッドの互除法the Euclidean algorithmFrequent

Core note: Repeatedly replace the larger number by the remainder; the last non-zero remainder is the gcd.
Exam area: Properties of integers
Decision focus: Handle claims about every integer by classifying on remainders.

0591次不定方程式linear Diophantine equationsFrequent

Core note: ax+by=c has integer solutions exactly when gcd(a,b) divides c; find one particular solution, then write the general form.
Exam area: Properties of integers
Decision focus: Handle claims about every integer by classifying on remainders.

060余りによる分類classification by remainderFrequent

Core note: Sorting integers by their remainder mod n handles claims that must hold for every integer.
Exam area: Properties of integers
Decision focus: Handle claims about every integer by classifying on remainders.

Properties of figures I: triangles

061角の二等分線と比the angle bisector and ratiosEssential

Core note: An internal bisector divides the opposite side in the ratio of the adjacent sides; the external bisector gives the external division point.
Exam area: Properties of figures I: triangles
Decision focus: For ratio problems, try Menelaus and Ceva first.

062三角形の五心the centers of a triangleEssential

Core note: The circumcenter meets perpendicular bisectors, the incenter angle bisectors, and the centroid divides each median 2:1.
Exam area: Properties of figures I: triangles
Decision focus: For ratio problems, try Menelaus and Ceva first.

063メネラウスの定理Menelaus's theoremFrequent

Core note: A transversal cutting the three sides (or extensions) makes the product of three ratios equal 1; multiply along a single circuit.
Exam area: Properties of figures I: triangles
Decision focus: For ratio problems, try Menelaus and Ceva first.

064チェバの定理Ceva's theoremFrequent

Core note: When three cevians are concurrent the product of three ratios equals 1; it pairs naturally with Menelaus.
Exam area: Properties of figures I: triangles
Decision focus: For ratio problems, try Menelaus and Ceva first.

065中点連結定理と相似the midline theorem and similarityFrequent

Core note: The segment joining two midpoints is parallel to the third side and half its length; areas scale as the square of the ratio.
Exam area: Properties of figures I: triangles
Decision focus: For ratio problems, try Menelaus and Ceva first.

066三角形の成立条件the triangle inequalityFrequent

Core note: Any two sides sum to more than the third — always return to this when a question asks for the range of a side.
Exam area: Properties of figures I: triangles
Decision focus: For ratio problems, try Menelaus and Ceva first.

Properties of figures II: circles and solids

067円周角の定理the inscribed angle theoremEssential

Core note: Inscribed angles on the same arc are equal and half the central angle; an angle in a semicircle is right.
Exam area: Properties of figures II: circles and solids
Decision focus: The moment concyclicity appears, check whether opposite angles sum to 180°.

068円に内接する四角形cyclic quadrilateralsEssential

Core note: Opposite angles sum to 180° and each interior angle equals the exterior angle at the opposite vertex — also a test for concyclicity.
Exam area: Properties of figures II: circles and solids
Decision focus: The moment concyclicity appears, check whether opposite angles sum to 180°.

069接弦定理the tangent-chord angleFrequent

Core note: The angle between a tangent and a chord equals the inscribed angle subtending that chord.
Exam area: Properties of figures II: circles and solids
Decision focus: The moment concyclicity appears, check whether opposite angles sum to 180°.

070方べきの定理the power of a pointEssential

Core note: Two secants through a point satisfy PA·PB=PC·PD, and for a tangent PT²=PA·PB.
Exam area: Properties of figures II: circles and solids
Decision focus: The moment concyclicity appears, check whether opposite angles sum to 180°.

0712円の位置関係relative position of two circlesFrequent

Core note: Comparing the center distance with the sum and difference of the radii yields five configurations.
Exam area: Properties of figures II: circles and solids
Decision focus: The moment concyclicity appears, check whether opposite angles sum to 180°.

072多面体とオイラーの多面体定理polyhedra and Euler's formulaExtension

Core note: Convex polyhedra satisfy v−e+f=2; there are exactly five Platonic solids, and converting between face and edge counts is a common ask.
Exam area: Properties of figures II: circles and solids
Decision focus: The moment concyclicity appears, check whether opposite angles sum to 180°.