Toyama Prefectural Public High School Entrance Examination - Mathematics flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Distributive law

Multiplying a sum by a number: a(b + c) = ab + ac. It also expands products of binomials, e.g. (x + 2)(x + 3) = x² + 5x + 6.

Difference of two squares

The identity a² − b² = (a + b)(a − b). It factorises expressions such as x² − 9 into (x + 3)(x − 3) and simplifies products quickly.

Perfect square expansion

(a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b². The middle term is always twice the product of the two terms.

Laws of exponents

For positive base a: a^m × a^n = a^(m+n), a^m ÷ a^n = a^(m−n), (a^m)^n = a^(mn). These rules let you simplify powers without expanding them.

Rationalising a denominator

Rewriting a fraction so no surd remains below the line. For 1/√a multiply by √a/√a; for 1/(√a + √b) multiply by the conjugate (√a − √b).

Quadratic formula

The solutions of ax² + bx + c = 0 with a ≠ 0 are x = (−b ± √(b² − 4ac))/(2a). It solves any quadratic, including ones that do not factorise.

Discriminant

D = b² − 4ac for ax² + bx + c = 0. If D > 0 there are two distinct real roots, if D = 0 one repeated root, if D < 0 no real roots.

Inequality sign reversal

Multiplying or dividing both sides of an inequality by a negative number flips the direction of the sign. For example, −2x > 6 becomes x < −3.

Sum and product of roots

For ax² + bx + c = 0 with roots p and q: p + q = −b/a and pq = c/a. These follow from a(x − p)(x − q) = 0.

Solving simultaneous equations

Scale one or both equations so a variable's coefficients match, then add or subtract to eliminate it. Substitute the found value back to get the other variable.

Slope of a line

For y = ax + b the slope is a, equal to (y₂ − y₁)/(x₂ − x₁) between two points. Parallel lines have equal slopes; perpendicular lines multiply to −1.

Proportional relationship

y is proportional to x when y = ax for a constant a ≠ 0. Its graph is a straight line passing through the origin, and y/x is constant.

Inverse proportion

y = a/x with a ≠ 0, so xy is constant. The graph is a hyperbola with two branches lying in opposite quadrants.

Vertex of a parabola

The graph of y = a(x − p)² + q has vertex (p, q) and axis of symmetry x = p. If a > 0 the vertex is the minimum point.

Rate of change of y = ax²

Between x = p and x = q the average rate of change of y = ax² is a(p + q). It depends on the interval, so the graph is not linear.

Intersection of line and parabola

To find where y = ax² meets y = mx + n, set ax² = mx + n and solve the resulting quadratic. The number of real solutions equals the number of intersection points.

Triangle angle sum

The three interior angles of any triangle add to 180°. An exterior angle equals the sum of the two interior angles not adjacent to it.

Pythagorean theorem

In a right triangle the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². Its converse tests for a right angle.

Similar figure ratios

If two similar figures have length ratio k, their areas are in ratio k² and their volumes in ratio k³. This links scale models to real objects.

Inscribed angle theorem

An angle inscribed in a circle is half the central angle standing on the same arc. An angle inscribed in a semicircle is therefore 90°.

Volume of a cone

A cone with base radius r and height h has volume (1/3)πr²h, one third of the cylinder with the same base and height.

Sphere volume and surface

A sphere of radius r has volume (4/3)πr³ and surface area 4πr². Both depend only on the radius.

Exterior angle of a polygon

For a regular polygon with n sides each exterior angle is 360°/n, so each interior angle is 180° − 360°/n. Exterior angles always sum to 360°.

Mean, median and mode

The mean is the sum divided by the count; the median is the middle of the ordered data; the mode is the most frequent value. Each summarises the data differently.

Probability of an event

When outcomes are equally likely, P(A) = (favourable outcomes)/(total outcomes). Every probability lies between 0 and 1 inclusive.

Complement rule

For any event A, P(not A) = 1 − P(A). It is useful when counting the complement is easier than counting the event itself.

Independent events

Two events are independent when one's occurrence does not change the other's probability. Then P(A and B) = P(A) × P(B), as with repeated coin tosses.

Histogram area and frequency

In a histogram, frequency equals the bar's area, i.e. class width × height. When class widths differ, compare areas, not heights.

Proof by contradiction

To prove a claim, assume it is false and deduce a logical impossibility. Concluding the assumption fails proves the original statement must be true.

Counterexample

A single case that satisfies the conditions but breaks a claim. One counterexample disproves a universal statement, while examples alone never prove one.

Converse and contrapositive

The converse of 'if P then Q' is 'if Q then P' and may be false. The contrapositive 'if not Q then not P' is always logically equivalent to the original.

Combined work rate

If two workers alone finish a job in a and b hours, together they take T hours where 1/T = 1/a + 1/b. Rates add when working simultaneously.