📚 Year 11 WJEC Maths: Formula Theorems Quick Reference Handbook | Year 11 WJEC 数学：公式定理速查手册
Welcome to your ultimate quick-reference handbook for Year 11 WJEC Mathematics. This guide distils every essential formula and theorem you will need across number, algebra, geometry, trigonometry, statistics and probability. Use it for rapid revision, targeted practice, and as a reliable memory aid before your exams.
欢迎使用这本 Year 11 WJEC 数学终极速查手册。我们为你浓缩了从数与代数到几何、三角、统计与概率等各个板块必需的全部公式和定理。你可以用它快速复习、有针对地练习，并在考前作为可靠的记忆辅助工具。
1. Number Essentials | 数的基本运算
a < 10 and n is an integer. This compact notation simplifies calculations with extremely big or tiny quantities and appears frequently in measurement and science contexts.
a × 10ⁿ, 1 ≤ a < 10, n ∈ ℤ
标准形将极大或极小的数表示为 a × 10ⁿ, 其中 a 满足 1 ≤ a < 10, n 为整数。这种紧凑记法能大幅简化涉及极端量值的计算, 常用于测量和科学背景中。
The laws of indices underpin almost all algebraic manipulation. The table below captures the most important rules for multiplying, dividing and raising powers.
| xᵐ × xⁿ = xᵐ⁺ⁿ |
| xᵐ ÷ xⁿ = xᵐ⁻ⁿ |
| (xᵐ)ⁿ = xᵐⁿ |
| x⁻ⁿ = 1/xⁿ |
| x⁰ = 1 (x ≠ 0) |
指数法则是整个代数运算的基石。上表概括了幂的乘、除与乘方最关键的规则, 务必熟练应用。
Surds involve roots that cannot be simplified to integers. Simplify surds by extracting square factors: √a²b = a√b. You can multiply and divide surds using √a × √b = √ab, and rationalise denominators by multiplying numerator and denominator by the conjugate or the same surd.
√a × √b = √ab, √a / √b = √a/b
无理根式是不能再化为整数的根式。化简方法为提取平方因子: √a²b = a√b。根式可相乘除: √a × √b = √ab; 有理化分母时, 分子分母同乘共轭根式或相同根式。
2. Algebra: Manipulating Expressions | 代数：表达式变换
(x + b) = x² + (a + b)x + ab. Recognising these patterns speeds up factorising quadratics.
(x + a)(x + b) = x² + (a + b)x + ab
去括号基于分配律: a(b + c) = ab + ac。两个二项式相乘时可用 FOIL 法则: (x + a)(x + b) = x² + (a + b)x + ab。熟练识别这类模式能让你快速因式分解二次式。
Factorising reverses expansion. For a quadratic x² + bx + c, find two numbers that multiply to c and add to b. For quadratics that cannot be factored easily, completing the square transforms the expression into a(x + p)² + q form, which reveals the vertex and helps solve equations.
x² + bx + c = (x + p)² + q, p = b/2, q = c – p²
因式分解是展开的逆运算。对二次式 x² + bx + c, 寻找乘积为 c、和为 b 的两个数。对于难以直接因式分解的二次式, 配方法将其化为 a(x + p)² + q 的形式, 从而找到顶点并方便解方程。
he quadratic formula solves any equation ax² + bx + c = 0 (a ≠ 0). The discriminant Δ = b² – Δ > 0 gives two real roots, Δ = 0 gives one repeated root, and Δ < 0 gives no real roots.
x = (-b ± √b² – 4ac) / (2a)
二次公式可解任何方程 ax² + bx + c = 0 (a ≠ 0)。判别式 Δ = b² – 4ac 揭示根的性质: Δ > 0 有两个不等实根, Δ = 0 有一个重根, Δ < 0 则无实根。
3. Graphs and Linear Equations | 图形与线性方程
on of a straight line can be written as y = mx + c, where m is the gradient and c is the y‑cept. Given two points (x₁, y₁) and (x₂, y₂), the gradient is calculated as rise over run.
m = (y₂ – y₁) / (x₂ – x₁)
直线方程可写为 y = mx + c, 其中 m 是斜率, c 是 y 轴截距。已知两点 (x₁, y₁) 与 (x₂, y₂) 时, 斜率通过纵差除以横差计算得出。
Parallel lines have equal gradients. Perpendicular lines have gradients that multiply to –1: m₁ × m₂ = – is relationship is vital for coordinate geometry questions involving altitudes or normals.
m₁ × m₂ = –1 (perpendicular lines)
平行直线斜率相等。垂直直线的斜率乘积为 –1: m₁ × m₂ = –1。这一关系在涉及高度或法线的坐标几何题中至关重要。
Simultaneous equations can be solved by elimination or substitution. For a pair of linear equations, you are finding the intersection point of two lines. Graphical solutions can also be used, but accurate algebra is preferred for exact answers.
联立方程可通过消元或代入法求解。一对线性方程实际上是求两条直线的交点。图解亦可, 但要得到精确答案, 应优先使用准确的代数方法。
4. Pythagoras and Trigonometry | 勾股定理与三角学
In any right‑angled triangle, Pythagoras' theorem links the squares of the shorter sides to the hypotenuse. It is used for finding missing side lengths and verifying if a triangle is right‑angled.
a² + b² = c² (c is the hypotenuse)
在任何直角三角形中, 勾股定理联系了两条直角边的平方与斜边平方。它用于求未知边长或验证三角形是否为直角三角形。
The three trigonometric ratios define relationships between angles and side lengths in right‑angled triangles. Remember SOH CAH TOA to recall which ratio to use.
opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent
三个三角比定义了直角三角形中角与边的关系。记住 SOH CAH TOA 就能迅速确定该用哪个比。
For non‑right‑angled triangles (Higher tier), the sine rule and cosine rule are essential. Use the sine rule when you know either two angles and a side, or two sides and a non‑included angle. The cosine rule is best when you have two sides and the included angle or all three sides.
a/sin A = b/sin B = c/sin C
a² = b² + c² – 2bc cos A
对于非直角三角形（高阶内容）, 正弦定理与余弦定理不可或缺。已知两角一边或两边一对角时用正弦定理; 已知两边及其夹角或三边时则用余弦定理。
The area of a triangle can also be found using trigonometry when two sides and the included angle are known.
Area = ½ ab sin C
当已知两边及其夹角时, 还可用三角方法计算三角形面积: 面积 = ½ ab sin C。
5. Area and Perimeter | 面积与周长
Here are the fundamental area and perimeter formulas you must memorise. Always use consistent units and remember that area is measured in square units while perimeter is a length.
| Rectangle: A = lw, P = 2(l + w) |
