Tokyo Metropolitan Public High School Entrance Examination - Mathematics flashcards

33 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 underlies expanding products such as (x+2)(x−3)=x²−x−6.

Difference of squares

The identity a²−b²=(a+b)(a−b). It factors expressions like x²−9 into (x+3)(x−3) and simplifies products of conjugate binomials.

Rationalising a denominator

Rewriting a fraction so no root remains in the denominator, usually by multiplying numerator and denominator by the conjugate or by the surd itself.

Laws of exponents

Rules such as a^m·a^n=a^(m+n), (a^m)^n=a^(mn), a^(−n)=1/a^n and a^0=1 for nonzero a. They let you simplify powers quickly.

Factoring x²+(a+b)x+ab

Find two numbers with sum equal to the x-coefficient and product equal to the constant term; the expression factors as (x+a)(x+b).

Slope of a line

For a line y=ax+b, the slope a=(y₂−y₁)/(x₂−x₁). It measures the change in y per unit change in x.

Parallel and perpendicular slopes

Parallel lines have equal slopes. Perpendicular lines have slopes whose product is −1, for example 2 and −1/2.

Solving simultaneous equations

Use elimination to cancel a variable by adding or subtracting multiples of the equations, or substitution to express one variable in terms of the other.

Point-slope form

The equation y−y₁=a(x−x₁) describes the line with slope a passing through the point (x₁,y₁). Expand it to get y=ax+b.

No-solution system

A pair of linear equations has no solution when the lines are parallel with different intercepts: equal slopes but unequal constants after rearrangement.

Quadratic formula

For ax²+bx+c=0 with a≠0, the roots are x=(−b±√(b²−4ac))/(2a). It solves any quadratic, factorable or not.

Discriminant

The value D=b²−4ac for ax²+bx+c=0. If D>0 there are two real roots, if D=0 one repeated root, if D<0 none.

Vertex of a parabola

For y=ax²+bx+c, the vertex has x-coordinate −b/(2a); substitute this value back to obtain the y-coordinate. The axis of symmetry is x=−b/(2a).

Completing the square

Rewriting ax²+bx+c as a(x+b/2a)²+c−b²/4a. This form shows the vertex and solves quadratics without the formula.

Vieta's relations

For roots α and β of ax²+bx+c=0, the sum α+β=−b/a and the product αβ=c/a. Useful for finding expressions in the roots.

Vertex form

The form y=a(x−p)²+q, whose graph is a parabola with vertex (p,q) and axis of symmetry x=p.

Angles on a straight line

Adjacent angles forming a straight line add to 180°. Angles meeting around a single point add to 360°.

Parallel line angles

When a transversal cuts two parallel lines, alternate interior angles are equal and corresponding angles are equal. Co-interior angles sum to 180°.

Interior angle sum of a polygon

A convex n-gon has interior angles summing to (n−2)×180°. The exterior angles of any convex polygon sum to 360°.

Sector area and arc length

For a sector with central angle θ in a circle of radius r: area=(θ/360)πr² and arc length=(θ/360)×2πr.

Pythagorean theorem

In a right triangle with legs a and b and hypotenuse c, a²+b²=c². The converse confirms a right angle when the relation holds.

Triangle congruence conditions

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

Similar triangle conditions

Triangles are similar under AA (two equal angles), SAS-similarity (two proportional sides with equal included angle) or SSS-similarity.

Similarity area and volume ratios

If corresponding lengths are in ratio k, perimeters are in ratio k, areas in ratio k², and volumes in ratio k³.

Triangle midline theorem

The segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.

Angle bisector theorem

In triangle ABC, the bisector of angle A meets BC at D so that BD:DC=AB:AC. It gives a ratio of segments.

Mean, median and mode

The mean is the sum divided by the count, the median is the middle of the ordered data, and the mode is the most frequent value.

Range of data

The range is the maximum value minus the minimum value. It measures spread but is strongly affected by a single outlier.

Theoretical probability

When all outcomes are equally likely, P(A) equals the number of favourable outcomes divided by the total number of outcomes.

Complement rule

For any event A, the probability that A does not occur is P(not A)=1−P(A). Often easier than counting directly.

Independent events

Two events are independent when one does not affect the other; then P(A and B)=P(A)×P(B).

Histogram frequency

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

Relative frequency

The experimental probability of an event is its observed count divided by the total number of trials; it approaches theoretical probability as trials increase.