📚 Year 11 AQA Further Maths: Formula & Theorem Quick Reference Handbook | Year 11 AQA 进阶数学：公式定理速查手册
This quick-reference handbook compiles essential formulas, theorems, and key concepts for the Year 11 AQA Level 2 Certificate in Further Mathematics. Mastering these will strengthen your problem-solving skills and prepare you for both calculator and non-calculator papers.
本速查手册汇编了 Year 11 AQA 进阶数学二级证书所需的关键公式、定理和核心概念。熟练掌握这些内容将增强您的解题能力，为计算器和非计算器试卷做好准备。
1. Algebra & Factor Theorem | 代数与因式定理
The Remainder Theorem states that when a polynomial f(x) is divided by (x – a), the remainder is f(a).
余数定理指出，当多项式 f(x) 除以 (x – a) 时，余数为 f(a)。
The Factor Theorem says (x – a) is a factor of f(x) if and only if f(a) = 0.
因式定理表明，(x – a) 是 f(x) 的因式当且仅当 f(a) = 0。
For quadratic ax² + bx + c = 0, roots are given by x = [–b ± √b² – 4ac] / 2a.
对于二次方程 ax² + bx + c = 0，根为 x = [–b ± √b² – 4ac] / 2a。
Completing the square: x² + bx = (x + b/2)² – (b/2)².
配方法：x² + bx = (x + b/2)² – (b/2)²。
Simultaneous equations can be solved by elimination or substitution; one linear and one quadratic often lead to a quadratic equation.
联立方程可通过消元法或代入法求解；一个线性与一个二次方程通常得到一个二次方程。
Inequalities: multiply or divide by a negative number reverses the inequality sign.
不等式：乘以或除以负数时，不等号方向改变。
2. Binomial Expansion | 二项式展开
Binomial coefficient: ⁿCᵣ = n! / [r!(n – r)!], also written as (n choose r).
二项式系数：ⁿCᵣ = n! / [r!(n – r)!]，也写作 (n 选 r)。
For positive integer n: (a + b)ⁿ = Σ (ⁿCᵣ) aⁿ⁻ͬ bͬ, from r = 0 to n.
对于正整数 n：(a + b)ⁿ = Σ (ⁿCᵣ) aⁿ⁻ͬ bͬ，r 从 0 到 n。
Pascal’s triangle helps find coefficients for small n: row n starts with 1 and ends with 1, each inner number is sum of two above.
帕斯卡三角形用于求解小 n 的系数：第 n 行以 1 开始和结束，内部每个数是上方两数之和。
When expanding (1 + kx)ⁿ for rational n, the series is infinite if n is not a positive integer; valid for |kx| < 1.
当展开 (1 + kx)ⁿ 且 n 为有理数时，若 n 非正整数则级数无穷；有效范围 |kx| < 1。
3. Matrices | 矩阵
Matrix multiplication: AB exists if number of columns of A equals number of rows of B. Element (i, j) of product is dot product of row i of A and column j of B.
矩阵乘法：若 A 的列数等于 B 的行数，则 AB 存在。乘积中第 (i, j) 元素为 A 第 i 行与 B 第 j 列的点积。
Identity matrix I = [[1,0],[0,1]]; AI = IA = A.
单位矩阵 I = [[1,0],[0,1]]；AI = IA = A。
Determinant of 2×2 matrix M = [[a,b],[c,d]] is det(M) = ad – bc.
2×2 矩阵 M = [[a,b],[c,d]] 的行列式为 det(M) = ad – bc。
Inverse of M is M⁻¹ = 1/(ad – bc) [[d,–b],[–c,a]], exists only if ad – bc ≠ 0.
M 的逆矩阵为 M⁻¹ = 1/(ad – bc) [[d,–b],[–c,a]]，仅当 ad – bc ≠ 0 时存在。
Matrix transformations: rotation by θ (anticlockwise) is [[cosθ,–cosθ]]; reflection in y = x is [[0,1],[1,0]]; enlargement scale factor k is [[k,0],[0,k]].
矩阵变换：逆时针旋转 θ 矩阵为 [[cosθ,–sinθ],[sinθ,cosθ]]；关于直线 y = x 的反射矩阵为 [[0,1],[1,0]]；缩放因子 k 为 [[k,0],[0,k]]。
4. Coordinate Geometry & Circles | 坐标几何与圆
Equation of a circle with centre (a, b) and radius r: (x – a)² + (y – b)² = r².
圆心 (a, b) 半径为 r 的圆方程：(x – a)² + (y – b)² = r²。
General form: x² + y² + 2gx + 2fy + c = 0; centre (–g, –f), radius √g² + f² – c.
一般形式：x² + y² + 2gx + 2fy + c = 0；圆心 (–g, –f)，半径 √g² + f² – c。
The perpendicular from the centre to a chord bisects the chord.
圆心到弦的垂线平分该弦。
The tangent to a circle at point (x₁, y₁) is perpendicular to the radius at that point; equation can be found using gradient of radius and point-slope form, or using discriminant approach for intersection.
圆在点 (x₁, y₁) 处的切线与该点半径垂直；方程可通过半径斜率及点斜式求得，或通过判别式法求交线得出。
Distance between two points (x₁, y₁) and (x₂, y₂) is √[(x₂ – x₁)² + (y₂ – y₁)²]. Midpoint: ((x₁ + x₂)/2, (y₁ + y₂)/2).
两点 (x₁, y₁) 和 (x₂, y₂) 的距离为 √[(x₂ – x₁)² + (y₂ – y₁)²]。中点：((x₁ + x₂)/2, (y₁ + y₂)/2)。
5. Differentiation | 微分
Derivative of xⁿ: d/dx (xⁿ) = nxⁿ⁻¹, for any real n.
xⁿ 的导数：d/dx (xⁿ) = nxⁿ⁻¹，n 为任意实数。
Constant multiple rule: d/dx [k·f(x)] = k·f'(x). Sum rule: derivative of sum is sum of derivatives.
常数倍法则：d/dx [k·f(x)] = k·f'(x)。和的导数等于导数之和。
Increasing function where f'(x) > 0; decreasing where f'(x) < 0.
若 f'(x) > 0，函数递增；若 f'(x) < 0，函数递减。
Stationary points occur when f'(x) = 0. Nature tested by second derivative: if f''(x) > 0, minimum; f''(x) < 0, maximum; f''(x) = 0 could be a point of inflection (check gradient change).
驻点发生在 f'(x) = 0 处。用二阶导数判定性质：若 f''(x) > 0，极小值；f''(x) < 0，极大值；f''(x) = 0 可能为拐点（需检验斜率变化）。
Equation of tangent at x = a: y – f(a) = f'(a)(x – a). Normal gradient = –1/f'(a).
x = a 处的切线方程：y – f(a) = f'(a)(x – a)。法线斜率为 –1/f'(a)。
6. Integration | 积分
Integration as reverse of differentiation: ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C, n ≠ –1.
积分作为微分的逆运算：∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C，n ≠ –1。
∫ k·f(x) dx = k ∫ f(x) dx; integral of sum = sum of integrals.
∫ k·f(x) dx = k ∫ f(x) dx；和的积分等于积分之和。
ite integral ∫ₐᵇ f(x) dx gives signed area under curve y = f(x) from a to b. Evaluate F(b) – F(a) where F is antiderivative.
定积分 ∫ₐᵇ f(x) dx 给出从 a 到 b 曲线 y = f(x) 下方的带符号面积。计算 F(b) – F(a)，其中 F 是原函数。
0 is ∫ₐᵇ f(x) dx; if curve goes below axis, split and take absolute values for total area.
当 f(x) ≥ 0 时，曲线与 x 轴之间的面积为 ∫ₐᵇ f(x) dx；若曲线位于轴下方，分段并取绝对值求和得到总面积。
Area between two curves y = f(x) and y = g(x) from a to b is ∫ₐᵇ [upper – lower] dx.
两条曲线 y = f(x) 和 y = g(x) 在 a 到 b 之间的面积为 ∫ₐᵇ [上方函数 – 下方函数] dx。
7. Trigonometry | 三角函数
Exact trigonometric values (know for 0°, 30°, 45°, 60°, 90°):
特殊角三角函数精确值（识记 0°, 30°, 45°, 60°, 90°）：
| θ (degrees) | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | ½ | √3/2 | 1/√3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | ½ | √3 |
| 90° | 1 | 0 | undefined |
= 1, and from it tan²θ + 1 = sec²θ (though sec not heavily used, the relationship helps).
关键恒等式：sin²θ + cos²θ = 1，以及由此可得 tan²θ + 1 = sec²θ（尽管考试不强调 sec，但关系式有帮助）。
Sine rule: a/sin A = b/sin B = c/sin C. Cosine rule: a² = b² + c² – 2bc cos A.
正弦定理：a/sin A = b/sin B = c/sin C。余弦定理：a² = b² + c² – 2bc cos A。
Area of triangle: ½ ab sin C. Solving trig equations: use graph or CAST diagram; find principal value then general solutions within given interval.
三角形面积：½ ab sin C。解三角方程：使用图像或 CAST 图；先求主值，再求给定区间内的所有解。
8. Functions | 函数
maps each input (domain) to exactly one output (range). Notation f: x → ... or f(x) = ...
函数将每个输入（定义域）映射到唯一输出（值域）。记号 f: x → ... 或 f(x) = ...
Domain: set of allowed x values. Range: set of possible y values produced.
定义域：允许的 x 值集合。值域：产生的可能 y 值的集合。
Composite function gf(x) means apply f first, then g; gf(x) = g(f(x)). Order matters.
复合函数 gf(x) 表示先应用 f，再应用 g；gf(x) = g(f(x))。顺序很重要。
equals range of f. Find by writing y = f(x) and rearranging for x, then swapping x and y.
反函数 f⁻¹(x) 逆转 f 的作用；f⁻¹ 的定义域等于 f 的值域。求法为设 y = f(x)，解出 x，再互换 x 和 y。
Graph of inverse is reflection of original graph in line y = x.
反函数图像是原函数图像关于直线 y = x 的反射。
Examples: linear f(x) = 2x+3 has inverse f⁻¹(x) = (x – 3)/2; quadratic needs restricted domain for inverse to be a function.
例子：线性 f(x) = 2x+3 的反函数为 f⁻¹(x) = (x – 3)/2；二次函数需限制定义域，其反函数才能成为函数。
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