Toyama Prefectural Public High School Entrance Examination - Mathematics cheat sheet

Sign rules for products: (+)(+)=+, (−)(−)=+, (+)(−)=−; the same rules apply to quotients.

For fractions a/b + c/d, rewrite over the common denominator bd: (ad + bc)/bd.

Subtracting a negative is adding: a − (−b) = a + b; distributing a minus sign flips every sign inside the parentheses.

Laws of exponents: a^m × a^n = a^(m+n), a^m ÷ a^n = a^(m−n), (a^m)^n = a^(mn), (ab)^n = a^n b^n.

Distributive law: a(b + c) = ab + ac, and (a + b)(a − b) = a² − b².

Expansion identities: (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b².

Factorisation: common factor first, then x² + (a + b)x + ab = (x + a)(x + b).

Rationalising a denominator: multiply by the conjugate, e.g. 1/(√a + √b) = (√a − √b)/(a − b).

Square roots: √(a²b) = a√b for a > 0; √a × √b = √(ab) and √a ÷ √b = √(a/b).

Adding or subtracting the same quantity on both sides of an equation or inequality preserves the solution set.

Multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign.

Quadratic formula: solutions of ax² + bx + c = 0 are x = (−b ± √(b² − 4ac))/(2a).

The discriminant D = b² − 4ac gives two real roots if D > 0, one repeated root if D = 0, no real roots if D < 0.

A quadratic with roots p and q factors as a(x − p)(x − q), so p + q = −b/a and pq = c/a.

Simultaneous equations: eliminate one variable by adding or subtracting scaled equations, then back-substitute.

A system has no solution when the two lines are parallel (same slope, different intercept).

Compound inequalities such as −2 < x ≤ 3 are read as the overlap of x > −2 and x ≤ 3.

Always verify a solution by substituting it back into the original equation; squaring can introduce extraneous roots.

A function is proportional (y = ax) exactly when its graph is a straight line through the origin.

For a linear function y = ax + b, a is the slope (change in y per unit change in x) and b is the y-intercept.

Slope through two points: a = (y₂ − y₁)/(x₂ − x₁); parallel lines share the same slope.

For the parabola y = ax², the axis of symmetry is x = 0 and the vertex is the origin; a > 0 opens upward, a < 0 downward.

The graph of y = a(x − p)² + q has vertex (p, q) and axis of symmetry x = p.

Rate of change of y = ax² over x from p to q is a(p + q), not a constant.

Inverse proportion y = a/x gives a hyperbola with branches in quadrants I and III when a > 0.

To find intersections of y = ax² and y = mx + n, solve ax² = mx + n, i.e. ax² − mx − n = 0.

Triangle angle sum is 180°; the exterior angle equals the sum of the two non-adjacent interior angles.

An exterior angle of a polygon with n sides is 360°/n, so the interior angle is 180° − 360°/n.

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

Area formulas: triangle (1/2)bh, parallelogram bh, trapezium (1/2)(a + b)h, circle πr².

Similar figures with ratio k: lengths scale by k, areas by k², volumes by k³.

Inscribed angle theorem: an inscribed angle is half the central angle subtending the same arc; an angle in a semicircle is 90°.

Volume formulas: prism and cylinder (base area × height), pyramid and cone (1/3 × base area × height), sphere (4/3)πr³.

Surface area of a sphere is 4πr²; lateral surface of a cone is πrl where l is the slant height.

Mean = sum of values ÷ number of values; median is the middle value of ordered data (average of the two middle values when n is even).

Mode is the most frequent value; the range is maximum minus minimum.

Relative frequency = frequency ÷ total, and all relative frequencies sum to 1.

Probability of an event = (number of favourable outcomes)/(total equally likely outcomes).

Complement rule: P(not A) = 1 − P(A).

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

A histogram's bar area, not its height alone, represents frequency when class widths differ.

Correlation describes the direction of association between two variables, not that one causes the other.

A statement and its contrapositive are logically equivalent; the converse need not be true.

Proof by contradiction: assume the negation of the claim and derive an impossibility.

Counterexample rule: one example disproving a universal claim is enough; one example never proves it.

In a rate problem, distance = speed × time; for two objects meeting, the sum of distances equals the initial separation.

Work-rate problems: combined rate = sum of individual rates, so 1/T = 1/a + 1/b.

Consecutive integers can be written n, n + 1, n + 2; even and odd integers as 2n and 2n + 1.

Percentage change = (new − old)/old × 100; successive changes of p% and q% multiply as (1 + p/100)(1 + q/100).

When setting up an equation from a word problem, define the variable in words first and state the units of the answer.