Tokyo Metropolitan Public High School Entrance Examination - Mathematics cheat sheet

Distributive law: a(b+c)=ab+ac and (a+b)(a−b)=a²−b²; these two expansions drive most simplification problems.

Perfect-square expansions: (a+b)²=a²+2ab+b², (a−b)²=a²−2ab+b².

Factorisation by grouping: for x²+(a+b)x+ab, the factors are (x+a)(x+b); find two numbers whose sum is the x-coefficient and product is the constant.

Rationalise a denominator by multiplying numerator and denominator by the conjugate: 1/(√a+√b)=(√a−√b)/(a−b).

√(a²b)=|a|√b when pulling factors out of a square root; the absolute value matters when a could be negative.

Laws of exponents: a^m·a^n=a^(m+n), (a^m)^n=a^(mn), a^(−n)=1/a^n, a^0=1 for a≠0.

Order of operations: parentheses, exponents and roots, multiplication and division left to right, then addition and subtraction.

Scientific notation expresses a number as a×10^n with 1≤|a|<10; adjust the exponent when moving the decimal point.

A linear function has the form y=ax+b; a is the slope (rate of change) and b is the y-intercept.

Slope through two points: a=(y₂−y₁)/(x₂−x₁); a positive slope rises left to right, a negative slope falls.

Parallel lines have equal slopes; perpendicular lines have slopes whose product is −1 (a₁a₂=−1).

Solve simultaneous equations by elimination (add or subtract multiples to cancel a variable) or by substitution.

A system has no solution when the lines are parallel with different intercepts, and infinitely many solutions when the equations describe the same line.

Point-slope form: y−y₁=a(x−x₁) gives the line through (x₁,y₁) with slope a.

The x-intercept is found by setting y=0 and solving for x; the y-intercept is b.

Distance–speed–time relations such as distance=speed×time often produce linear equations in word problems.

Quadratic formula: for ax²+bx+c=0 with a≠0, x=(−b±√(b²−4ac))/(2a).

The discriminant D=b²−4ac decides the number of real roots: D>0 two distinct roots, D=0 one repeated root, D<0 no real roots.

Completing the square: ax²+bx+c=a(x+b/2a)²+c−b²/4a, which reveals the vertex.

The graph y=ax²+bx+c is a parabola with vertex x-coordinate −b/(2a); it opens upward if a>0 and downward if a<0.

Axis of symmetry: x=−b/(2a); the vertex y-value is found by substituting this x back into the function.

Vieta's relations: for roots α, β of ax²+bx+c=0, α+β=−b/a and αβ=c/a.

A quadratic equation is solved by factoring when integer roots exist, otherwise by the quadratic formula or completing the square.

In y=a(x−p)²+q, the vertex is (p,q) and the axis of symmetry is x=p.

Angles on a straight line sum to 180°; angles around a point sum to 360°.

Vertical (opposite) angles are equal; alternate interior angles and corresponding angles are equal when two parallel lines are cut by a transversal.

Interior angle sum of an n-gon is (n−2)×180°; the exterior angle sum of any convex polygon is 360°.

Each interior angle of a regular n-gon measures (n−2)×180°/n.

Area formulas: triangle ½bh, parallelogram bh, trapezium ½(a+b)h, circle πr², sector (θ/360)πr².

Circumference of a circle is 2πr; arc length of a sector is (θ/360)×2πr.

Pythagorean theorem: in a right triangle with legs a, b and hypotenuse c, a²+b²=c²; conversely a²+b²=c² implies a right angle.

The inscribed angle subtended by a diameter is 90° (Thales' theorem), and inscribed angles on the same arc are equal.

Triangle congruence conditions: SSS, SAS, ASA (with the included side), and RHS for right triangles; AAA and SSA do not guarantee congruence.

Similar triangles satisfy AA, SAS-similarity, or SSS-similarity; corresponding angles are equal and corresponding sides are proportional.

If two triangles are similar with ratio k, their perimeters are in ratio k and their areas are in ratio k².

Similar solids have volumes in ratio k³ when corresponding lengths are in ratio k.

The midline (midsegment) of a triangle is parallel to the third side and half its length.

The angle bisector theorem: in triangle ABC, the bisector of angle A divides BC so that BD:DC=AB:AC.

Corresponding parts of congruent triangles are equal (CPCTC); use this to transfer lengths and angles between triangles.

Parallel lines cutting two transversals create proportional segments: if DE∥BC in triangle ABC, then AD:DB=AE:EC.

Mean = sum of values ÷ number of values; median is the middle value of ordered data; mode is the most frequent value.

For an even number of ordered values, the median is the average of the two middle values.

Range = maximum − minimum; it measures spread but is sensitive to outliers.

Relative frequency (experimental probability) = number of favourable outcomes ÷ total trials.

Theoretical probability of an event = favourable outcomes ÷ equally likely total outcomes; 0≤P≤1.

Probability of the complement: P(not A)=1−P(A).

For independent events, P(A and B)=P(A)×P(B); for mutually exclusive events, P(A or B)=P(A)+P(B).

In a histogram, frequency equals the bar height (frequency density) multiplied by the class width.