📚 Year 10 AQA Further Mathematics: Formula & Theorem Quick Reference Handbook | Year 10 AQA 进阶数学：公式定理速查手册
This quick reference handbook gathers every essential formula and theorem you need for the Year 10 AQA Further Mathematics course into one place. It is designed for rapid revision: algebra, surds, equations, sequences, coordinate geometry, trigonometry, vectors, matrices and introductory calculus are all covered, and each result is given in both English and Chinese so that you can check any formula at a glance.
本速查手册汇集了 Year 10 AQA 进阶数学课程必须掌握的所有核心公式与定理，旨在帮助您快速复习。内容涵盖代数、根式、方程、数列、坐标几何、三角学、向量、矩阵与微积分入门。每条公式均配有清晰的中英文解释，便于随时查阅。
1. Algebra Fundamentals: Index Laws | 代数基础：指数法则
Index laws are the backbone of the whole course. When you multiply two powers with the same base, the base stays the same and the exponents add. This is the first rule to memorise because it appears inside almost every later topic.
指数法则是整个课程的骨架。同底数幂相乘时，底数保持不变，指数相加。这是首先要牢记的法则，因为它几乎出现在后面每一个专题中。
aᵐ × aⁿ = aᵐ⁺ⁿ
When dividing powers with the same base, subtract the exponents. The restriction a ≠ 0 matters because division by zero is undefined, and it also links directly to the zero-exponent rule below.
同底数幂相除时，指数相减。限制条件 a ≠ 0 很重要，因为除以零没有意义，它也直接联系到下面的零指数法则。
aᵐ ÷ aⁿ = aᵐ⁻ⁿ (a ≠ 0)
When a power is raised to another power, multiply the exponents. Students most often confuse this with the multiplication law, so always ask yourself: are the brackets multiplying two factors, or raising one factor to a new power?
幂的乘方，指数相乘。学生最容易把它与乘法法则混淆，因此每次都要自问：括号内是两个因式相乘，还是把一个因式再取一次幂？
(aᵐ)ⁿ = aᵐⁿ
egative laws are not separate inventions - they are forced by the division law when m = n.
其余标准结论补全整套体系。请注意，零指数与负指数并非独立发明的规则——当 m = n 时，它们是由除法法则自然推出的结果。
- Zero exponent: a⁰ = 1 (a ≠ 0). 零指数：a⁰ = 1（a ≠ 0）。
- Negative exponent: a⁻ⁿ = 1 ÷ aⁿ. 负指数：a⁻ⁿ = 1 ÷ aⁿ。
- Power of a product: (ab)ⁿ = aⁿbⁿ. 积的乘方：(ab)ⁿ = aⁿbⁿ。
- Power of a quotient: (a ÷ b)ⁿ = aⁿ ÷ bⁿ (b ≠ 0). 商的乘方：(a ÷ b)ⁿ = aⁿ ÷ bⁿ（b ≠ 0）。
- Fractional exponent: a1/n = ⁿ√a, and am/n = ⁿ√aᵐ = (ⁿ√a)ᵐ. 分数指数：a1/n = ⁿ√a，am/n = ⁿ√aᵐ = (ⁿ√a)ᵐ。
A useful exam habit is to convert every root into a fractional index before simplifying. Expressions such as √a × ∛a become a1/2 × a1/3 = a5/6, which is far easier to handle than juggling two different radicals.
一个实用的考试习惯是：在化简之前，先把所有根式转换成分数指数。例如 √a × ∛a 可写成 a1/2 × a1/3 = a5/6，比同时处理两种根号容易得多。
2. Expanding, Factorising, Surds and Polynomials | 展开、因式分解、根式与多项式
Expanding brackets relies on the distributive law. For two linear brackets, the standard grid method gives four products, and the two middle terms usually combine.
展开括号依赖分配律。对于两个一次括号，标准网格法可给出四个乘积，中间两项通常可以合并。
(x + a)(x + b) = x² + (a + b)x + ab
The difference of two squares is the single most useful factorisation identity in Further Mathematics, because it also appears when rationalising denominators.
平方差公式是进阶数学中最有用的因式分解恒等式，因为它在分母有理化时同样会出现。
a² − b² = (a + b)(a − b)
Surds follow two multiplication rules and one simplification habit: always look for the largest square factor inside the root.
根式遵循两条乘法法则和一个化简习惯：始终先在根号内找出最大的平方因数。
- √a × √b = √ab. 根号相乘：√a × √b = √ab。
- √a ÷ √b = √a ÷ b (b > 0). 根号相除：√a ÷ √b = √a ÷ b（b > 0）。
- To rationalise 1 ÷ √a, multiply top and bottom by √a. 将 1 ÷ √a 有理化时，分子分母同乘 √a。
- To rationalise 1 ÷ (a + √b), multiply by the conjugate (a − √b). 将 1 ÷ (a + √b) 有理化时，同乘共轭式 (a − √b)。
ives zero, then (x − a) is a factor; otherwise the value you obtain is exactly the remainder.
因式定理与余数定理让你无需长除法也能处理三次多项式。若代入 x = a 得到零，则 (x − a) 是因式；否则所得数值恰好就是余数。
f(a) = 0 ⟺ (x − a) is a factor of f(x); remainder on dividing by (x − a) is f(a)
When simplifying algebraic fractions, factorise everything first, then cancel common brackets. Cancelling before factorising is the most common source of lost marks in this topic.
化简代数分式时，先全部分解因式，再约去公因式。未分解就约分是这一专题最常失分的原因。
3. Solving Equations and Inequalities | 解方程与不等式
Every quadratic can be solved by completing the square, factorising or the quadratic formula. The formula is the safest option when factorising is not obvious.
任何一元二次方程都可用配方法、因式分解或求根公式求解。当因式分解不明显时，公式法最稳妥。
x = (−b ± √b² − 4ac) ÷ (2a)
Completing the square rewrites the quadratic in vertex form, which immediately reveals the minimum or maximum point of the graph.
配方法把二次式改写为顶点形式，可立即看出抛物线的最高点或最低点。
x² + bx + c = (x + b ÷ 2)² + c − (b ÷ 2)²
The discriminant tells you the number of real roots before you solve anything, which saves time in multiple-choice and "show that" questions.
判别式让你在真正求解之前就知道实数根的个数，这在选择题和"证明"题中非常省时。
| Δ = b² − 4ac | Number of real roots | 实数根个数 |
| Δ > 0 | Two distinct real roots | 两个不同实根 |
| Δ = 0 | One repeated real root | 一个重根 |
| Δ < 0 | No real roots | 无实根 |
Linear simultaneous equations are solved by elimination or substitution. If the coefficient of x is a ÷ b in the first equation and b ÷ a in the second, multiplying the two equations together can eliminate x neatly.
一次联立方程用消元法或代入法求解。若第一个方程中 x 的系数为 a ÷ b，第二个方程中为 b ÷ a，则把两式相乘往往能干净地消去 x。
r. On a number line, use an open circle for < and > and a filled circle for ≤ and ≥.
解不等式时与解方程完全一致，但只要乘以或除以负数，就必须改变不等号方向。在数轴上，< 与 > 用空心圆表示，≤ 与 ≥ 用实心圆表示。
4. Sequences and Series | 数列与级数
An arithmetic sequence grows by adding a fixed common difference d. The nth term and the sum of the first n terms are both examinable formulae.
等差数列通过加上固定的公差 d 增长。第 n 项与前 n 项和都是考点公式。
aₙ = a + (n − 1)d
Sₙ = n ÷ 2 × [2a + (n − 1)d] = n ÷ 2 × (a + l)
A geometric sequence multiplies by a fixed common ratio r. Here you must be careful with the sign of r, because a negative ratio produces an alternating sequence.
等比数列通过乘以固定的公比 r 增长。此时必须留意 r 的符号，因为负公比会产生正负交替的数列。
aₙ = a rⁿ⁻¹
Sₙ = a(1 − rⁿ) ÷ (1 − r), r ≠ 1
For quadratic sequences the first difference changes linearly and the second difference is constant. If the second difference is 2k, the coefficient of n² is k, and the rest of the formula can be found by subtracting.
对于二次数列，一阶差分成线性变化，二阶差分恒定。若二阶差分为 2k，则 n² 的系数为 k，其余部分可通过相减求得。
| Sequence type | 数列类型 | Key idea | 核心思路 |
| Arithmetic | 等差 | Constant first difference | 一阶差分恒定 |
| Geometric | 等比 | Constant ratio between terms | 相邻项比值恒定 |
| Quadratic | 二次 | Constant second difference | 二阶差分恒定 |
5. Coordinate Geometry | 坐标几何
Coordinate geometry in Further Mathematics is built on three quantities: distance, midpoint and gradient. Almost every later part of a question reduces to one of these three.
进阶数学中的坐标几何建立在三个量之上：距离、中点与斜率。后面几乎所有小题都可归结为这三者之一。
Distance: d = √((x₂ − x₁)² + (y₂ − y₁)²)
Midpoint: M = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)
Gradient: m = (y₂ − y₁) ÷ (x₂ − x₁)
The equation of a straight line can be written in gradient-intercept form or in point-gradient form. Use the point-gradient form when the question gives you one point and a gradient.
直线方程可写成斜截式或点斜式。当题目给出一个点和斜率时，使用点斜式。
y = mx + c, and y − y₁ = m(x − x₁)
Parallel lines share the same gradient, while perpendicular lines have gradients whose product is negative one.
平行直线斜率相同，而垂直直线的斜率之积为负一。
Parallel: m₁ = m₂ Perpendicular: m₁ × m₂ = −1
The circle theorems are also part of this section. A circle with centre (a, b) and radius r satisfies a distance condition, which expands into the standard equation.
圆的方程也属于这一节。以 (a, b) 为圆心、r 为半径的圆满足一个距离条件，展开后即得标准方程。
(x − a)² + (y − b)² = r², and centre O gives x² + y² = r²
A common exam question asks you to show that a given line is a tangent to a circle, or to find the midpoint of a chord. Both reduce to substituting the line into the circle and inspecting the discriminant.
常见考题要求证明某条直线与圆相切，或求弦的中点。两者都可归结为把直线代入圆的方程并考察判别式。
6. Trigonometry: Rules, Identities and Exact Values | 三角学：定理、恒等式与特殊角
Right-angled trigonometry is summarised by the three ratios. In Further Mathematics they are usually the first step of a longer chain of reasoning rather than the final answer.
直角三角形三角学由三个比值概括。在进阶数学中，它们通常只是较长推理链的第一步，而不是最终答案。
sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse, tan θ = opposite ÷ adjacent
For non-right-angled triangles, the sine rule and cosine rule replace Pythagoras. Use the sine rule when you have a matching pair of side and opposite angle.
对于非直角三角形，正弦定理与余弦定理取代了毕达哥拉斯定理。当拥有"边与其对角"的一组对应关系时，使用正弦定理。
a ÷ sin A = b ÷ sin B = c ÷ sin C
The cosine rule is used when you have two sides and the included angle, or all three sides. It is the triangle version of Pythagoras with a correction term.
当已知两边及夹角，或已知三边时，使用余弦定理。它是带有修正项的三角形版毕达哥拉斯定理。
a² = b² + c² − 2bc cos A, cos A = (b² + c² − a²) ÷ (2bc)
The area of any triangle, given two sides and the included angle, does not require the perpendicular height.
已知两边及夹角时，求任意三角形的面积无须借助高的长度。
Area = ½ ab sin C
The two identities below link the three ratios and allow algebraic manipulation in proof questions.
下面两条恒等式把三个比值联系起来，并可在证明题中进行代数变形。
tan θ = sin θ ÷ cos θ sin²θ + cos²θ = 1
Exact values for 0°, 30°, 45°, 60° and 90° should be known without a calculator, because many non-calculator papers depend on them.
0°、30°、45°、60° 与 90° 的特殊角数值必须脱离计算器牢记，因为许多不可使用计算器的试卷直接依赖它们。
| θ | 0° | 30° | 45° | 60° | 90° |
| sin θ | 0 | 1 ÷ 2 | √2 ÷ 2 | √3 ÷ 2 | 1 |
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