HKDSE Maths (Core) flashcards

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

Quadratic formula

For ax² + bx + c = 0 with a ≠ 0, the roots are x = [−b ± √(b² − 4ac)]/(2a). Use it when the quadratic does not factorise easily.

Discriminant

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

Sum and product of roots

If α and β are roots of ax² + bx + c = 0, then α + β = −b/a and αβ = c/a. This lets you form a quadratic from its roots.

Factor theorem

(x − a) is a factor of polynomial P(x) exactly when P(a) = 0. Dividing by (x − a) leaves remainder P(a), which is the remainder theorem.

Change of base

log_a M = log_b M ÷ log_b a, valid for positive bases not equal to 1 and positive M. It converts a logarithm to any convenient base, such as 10.

Compound interest formula

Amount A = P(1 + r%)^n, where P is principal, r% the interest rate per period and n the number of periods. Interest earned equals A − P.

Geometric series sum

For first term a and ratio r ≠ 1, S(n) = a(1 − r^n)/(1 − r). The infinite sum a/(1 − r) exists only when |r| < 1.

Arithmetic sequence

nth term T(n) = a + (n − 1)d, where d is the common difference. Sum of n terms S(n) = n/2 × (first + last).

Direct and inverse variation

y ∝ x means y = kx, a straight line through the origin. y ∝ 1/x means y = k/x. Find k from one data pair, then substitute.

Inequality sign reversal

Multiplying or dividing both sides of an inequality by a negative number reverses the sign. For quadratic inequalities, find roots then test intervals.

Exterior angle of triangle

An exterior angle of a triangle equals the sum of the two interior opposite angles. This follows from the 180° interior angle sum.

Polygon angle sums

Interior angle sum of an n-sided polygon is (n − 2) × 180°. Exterior angles always sum to 360°, so each exterior angle of a regular n-gon is 360°/n.

Similar figure ratios

If two similar figures have side ratio k, their areas are in ratio k² and their volumes in ratio k³. Corresponding angles are equal in similar figures.

Angle in semicircle

The angle subtended by a diameter at any point on the circle is 90°. This is a special case of the angle at the centre being twice the angle at the circumference.

Cyclic quadrilateral property

Opposite angles of a cyclic quadrilateral sum to 180°. Conversely, if a pair of opposite angles of a quadrilateral sum to 180°, its vertices lie on a circle.

Tangent-radius theorem

A tangent to a circle is perpendicular to the radius drawn to the point of contact. Also, two tangents drawn from an external point are equal in length.

Pythagoras theorem

In a right-angled triangle, a² + b² = c², where c is the hypotenuse, the side opposite the right angle. It is used to find unknown sides.

Trigonometric ratios

In a right-angled triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Remember SOH-CAH-TOA for these three definitions.

Sine rule

In any triangle, a/sin A = b/sin B = c/sin C, where each side is opposite the matching angle. Use it with two angles and one side, or two sides and a non-included angle.

Cosine rule

c² = a² + b² − 2ab cos C, where C is the angle between sides a and b. Use it when given three sides or two sides and the included angle.

Triangle area formula

Area of a triangle = (1/2)ab sin C, using two sides and the included angle. It reduces to (1/2) × base × height when C = 90°.

Three-figure bearings

A bearing is measured clockwise from north and written with three digits, for example 045° or 210°. North is 000°, east 090°, south 180°, west 270°.

Slope of a line

m = (y₂ − y₁)/(x₂ − x₁). Parallel lines have equal slopes; perpendicular lines have slopes whose product is −1, so one slope is the negative reciprocal of the other.

Distance and midpoint

Distance between two points = √[(x₂ − x₁)² + (y₂ − y₁)²]. Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2). Both come from coordinate geometry.

Equation of a circle

Centre (h, k), radius r: (x − h)² + (y − k)² = r². In general form x² + y² + Dx + Ey + F = 0, centre is (−D/2, −E/2).

Mean, median, mode

Mean = total ÷ number of data. Median is the middle value of ordered data. Mode is the most frequent value. Each measures central tendency differently.

Standard deviation shift rule

Adding a constant to every datum shifts the mean and median but leaves the standard deviation unchanged. Multiplying every datum by k multiplies the standard deviation by |k|.

Probability addition rule

P(A or B) = P(A) + P(B) − P(A and B). Subtract the overlap so outcomes counted in both events are not double-counted.

Independent events

Events A and B are independent when P(A and B) = P(A) × P(B). Then the occurrence of one event gives no information about the other.

Complementary probability

P(not E) = 1 − P(E), where not E is the complement of event E. Often easier to compute the complement probability and subtract from one.

Percentage change

Percentage change = (new value − old value)/old value × 100%. A positive result is an increase; a negative result is a decrease.

Exact trigonometric values

sin 30° = 1/2, cos 60° = 1/2, tan 45° = 1, sin 45° = cos 45° = √2/2, sin 60° = cos 30° = √3/2. These arise from 30-60-90 and 45-45-90 triangles.