Year 12 WJEC Mathematics: Quick Reference Formula & Theorem Handbook | WJEC 数学公式定理速查手册

📚 Year 12 WJEC Mathematics: Quick Reference Formula & Theorem Handbook | WJEC 数学公式定理速查手册

This handbook compiles the essential formulas, identities, and theorems required for the Year 12 WJEC Mathematics specification. It serves as a quick revision aid covering Pure Mathematics, Statistics, and Mechanics. Use it alongside past papers and your class notes to reinforce your understanding.

本手册汇集了 Year 12 WJEC 数学考试所需的核心公式、恒等式和定理，涵盖纯数学、统计与力学，供快速复习使用。请结合历年真题与课堂笔记，查漏补缺，巩固理解。

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1. Algebraic Fundamentals | 代数基础

The quadratic formula: For ax² + bx + c = 0, the solutions are given by x = (–b ± √b² – 4ac) / (2a).

二次公式：对于 ax² + bx + c = 0，解为 x = (–b ± √b² – 4ac) / (2a)。

The discriminant Δ = b² – ct real roots; if Δ = 0 there is one repeated real root; if Δ < 0 there are no real roots.

判别式 Δ = b² – 4ac 决定了根的性质：Δ > 0 时有两个不等实根；Δ = 0 时有一个重根；Δ < 0 时无实根。

Completing the square transforms x² + bx into (x + b/2)² – (b/2)². This is useful for solving quadratics and finding the vertex of a parabola.

配方法：x² + bx = (x + b/2)² – (b/2)²，常用于解二次方程和求抛物线的顶点。

Sum and product of roots: if α and β are the roots of ax² + bx + c = 0, then α + β = –b/a and αβ = c/a.

根的和与积：若 α 和 β 是 ax² + bx + c = 0 的根，则 α + β = –b/a，αβ = c/a。

Factor theorem: (x – f(x) if and only if f(p) = 0. The remainder theorem states that when f(x) is divided by (x – p) the remainder is f(p).

因式定理：(x – p) 是多项式 f(x) 的因式当且仅当 f(p) = 0。余数定理指出，f(x) 除以 (x – p) 的余数为 f(p)。

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2. Quadratics and the Discriminant | 二次函数与判别式

Quadratic function in completed-square form: a(x + p)² + q. The vertex is at (–p, q) and the line of symmetry is x = –p.

二次函数的完全平方形式：a(x + p)² + q，顶点为 (–p, q)，对称轴为 x = –p。

Using the discriminant to identify the number of intersections between a line and a curve: substitute the line equation into the quadratic, form a new quadratic, and examine its discriminant.

利用判别式判断直线与二次曲线的交点个数：将直线方程代入二次式，得到关于 x 的二次方程，计算其判别式即可。

Quadratic inequalities such as ax² + bx + c > 0 are solved by finding critical values where the expression equals zero, then testing intervals or using a sketch.

二次不等式如 ax² + bx + c > 0 的解法：先求出等于零的临界值，再通过区间测试或草图确定解集。

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3. Indices and Logarithms | 指数与对数

Index laws for a > 0, b > 0:

指数律 (a > 0, b > 0)：

• aᵐ × aⁿ = aᵐ⁺ⁿ
• aᵐ ÷ aⁿ = aᵐ⁻ⁿ
• (aᵐ)ⁿ = aᵐⁿ
• a⁰ = 1
• a⁻ⁿ = 1/aⁿ
• a¹⁄ₙ = ⁿ√a
• (ab)ⁿ = aⁿbⁿ

The logarithm logₐ x is the power to which a must be raised to give x, i.e. a^{logₐ x} = x. The natural logarithm ln x has base e.

对数 logₐ x 表示以 a 为底 x 的指数，即 a^{logₐ x} = x。自然对数 ln x 以 e 为底。

Laws of logarithms for any valid base:

对数的基本运算法则（任意有效底数）：

• log a + log b = log(ab)
• log a – log b = log(a/b)
• n log a = log(aⁿ)
• logₐ a = 1, logₐ 1 = 0
• Change of base: logₐ x = log_b x / log_b a

The exponential function y = eˣ and the natural log are inverses: e^{ln x} = x for x > 0, and ln(eˣ) = x.

指数函数 y = eˣ 与自然对数互为反函数：对于 x > 0 有 e^{ln x} = x，且 ln(eˣ) = x。

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4. Sequences and the Binomial Expansion | 数列与二项展开

Arithmetic sequence: n-th term uₙ = a + (n – 1)d, sum of first n terms Sₙ = n/2 [2a + (n – 1)d] = n/2 (a + l), where l is the last term.

等差数列：第 n 项 uₙ = a + (n – 1)d，前 n 项和 Sₙ = n/2 [2a + (n – 1)d] = n/2 (a + l)，其中 l 为末项。

Geometric sequence: n-th term uₙ = arⁿ⁻¹, sum of first n terms Sₙ = a(1 – rⁿ)/(1 – r) for r ≠ 1. Sum to infinity exists when |r| < 1 and is S∞ = a/(1 – r).

等比数列：第 n 项 uₙ = arⁿ⁻¹，前 n 项和 Sₙ = a(1 – rⁿ)/(1 – r)，r ≠ 1。当 |r| < 1 时存在无穷和 S∞ = a/(1 – r)。

Binomial expansion for positive integer n: (a + b)ⁿ = Σⁿr=0 ⁿCᵣ aⁿ⁻ʳ bʳ, where ⁿCᵣ = n! / (r!(n – r)!). For (1 + x)ⁿ the expansion is 1 + nx + n(n–1)/2! x² + …

正整数 n 的二项展开：(a + b)ⁿ = Σⁿr=0 ⁿCᵣ aⁿ⁻ʳ bʳ，其中 ⁿCᵣ = n! / (r!(n – r)!)。(1 + x)ⁿ = 1 + nx + n(n–1)/2! x² + …

For rational n, the expansion (1 + x)ⁿ is valid for |x| < 1 and gives an infinite series: 1 + nx + n(n–1)/2! x² + n(n–1)(n–2)/3! x³ + …

对于有理数 n，(1 + x)ⁿ 在 |x| < 1 时展开为无穷级数：1 + nx + n(n–1)/2! x² + n(n–1)(n–2)/3! x³ + …

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5. Coordinate Geometry | 坐标几何

Distance between two points (x₁, y₁) and (x₂, y₂): √[(x₂ – x₁)² + (y₂ – y₁)²]. Midpoint: ((x₁ + x₂)/2, (y₁ + y₂)/2).

两点间距离：√[(x₂ – x₁)² + (y₂ – y₁)²]；中点坐标：((x₁ + x₂)/2, (y₁ + y₂)/2)。

Gradient of a straight line: m = (y₂ – y₁)/(x₂ – x₁). Equation of a line: y – y₁ = m(x – or y = mx + c. Parallel lines have equal gradients; perpendicular lines satisfy m₁ × m₂ = –1.

直线斜率：m = (y₂ – y₁)/(x₂ – x₁)。直线方程：y – y₁ = m(x – x₁) 或 y = mx + c。平行线斜率相等；垂直线斜率之积为 –1。

The equation of a circle with centre (a, b) and radius

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