📚 Year 12 AQA Further Maths: Quick-Reference Formula & Theorem Handbook | AQA 进阶数学：公式定理速查手册
This handbook provides a concise, dual-language summary of the key formulas and theorems required for Year 12 AQA Further Mathematics. Each section pairs an English explanation with a Chinese translation to support revision and quick recall. Use it alongside your past paper practice to identify exactly which tool to apply in each problem.
本手册为 AQA 进阶数学（Year 12）提供中英双语的公式与定理速查。每个要点均采用英文解释和中文翻译对照呈现，方便复习和快速记忆。结合真题练习使用，能帮助你准确判断每道题所需的知识点。
1. Complex Numbers | 复数
A complex number is expressed as z = x + iy, where x, y ∈ ℝ and i² = –1. The real part is Re(z) = x, the imaginary part Im(z) = y.
复数表示为 z = x + iy，其中 x, y ∈ ℝ，且 i² = –1。实部 Re(z) = x，虚部 Im(z) = y。
The complex conjugate is z* = x – iy. It satisfies z z* = x² + y², which is purely real.
共轭复数为 z* = x – iy。它满足 z z* = x² + y²，为纯实数。
Modulus: |z| = √x² + y². Argument: arg(z) = θ , with the quadrant determined by the signs of x and y. Principal argument is usually in (–π, π].
模：|z| = √x² + y²。辐角：arg(z) = θ，满足 tanθ = y/x，象限由 x 和 y 的符号决定。主辐角通常取值在 (–π, π] 范围内。
Polar form: z = r(cosθ + i sinθ) where r = |z|, θ = arg(z). De Moivre's theorem: (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ) for integer n.
极坐标形式：z = r(cosθ + i sinθ)，其中 r = |z|，θ = arg(z)。德·莫伊夫定理：对于整数 n，(cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ)。
₂(cosθ₂ + i sinθ₂), then z₁z₂ = r₁r₂(cos(θ₁+θ₂) + i sin(θ₁+θ₂)) and z₁/z₂ = (r₁/r₂)(cos(θ₁–θ₂) + i sin(θ₁–θ₂)).
极坐标形式下的乘除法：若 z₁ = r₁(cosθ₁ + i sinθ₁)，z₂ = r₂(cosθ₂ + i sinθ₂)，则 z₁z₂ = r₁r₂(cos(θ₁+θ₂) + i sin(θ₁+θ₂))，且 z₁/z₂ = (r₁/r₂)(cos(θ₁–θ₂) + i sin(θ₁–θ₂))。
2. Matrices | 矩阵
atrix multiplication: if A is m × n and B is n × p, then the product C = AB is m × p, with Cᵢⱼ = ∑ₖ Aᵢₖ Bₖⱼ. Multiplication is not commutative (AB ≠ BA in general).
矩阵乘法：若 A 为 m × n，B 为 n × p，则乘积 C = AB 为 m × p，其中 Cᵢⱼ = ∑ₖ Aᵢₖ Bₖⱼ。乘法一般不满足交换律（通常 AB ≠ BA）。
s 1 on the main diagonal and 0 elsewhere, satisfying AI = IA = A for conformable matrices.
单位矩阵 I 主对角线元素为 1，其余为 0，满足 AI = IA = A（维度匹配时）。
Inverse of a 2 × 2 matrix: if A = [[a, b], [c, d]] and det(A) = ad – bc ≠ 0, then A⁻¹ = (1/det) [[d, –b], [–c, a]].
2 × 2 矩阵的逆：若 A = [[a, b], [c, d]] 且 det(A) = ad – bc ≠ 0，则 A⁻¹ = (1/det) [[d, –b], [–c, a]]。
zero determinant and no inverse. The determinant of a 3 × 3 matrix can be found by Sarrus' rule or expansion by minors.
行列式为零的矩阵是奇异矩阵，不存在逆。3 × 3 矩阵的行列式可以用萨鲁斯法则或按子式展开计算。
e plane: a 2 × 2 matrix represents a linear transformation. Common ones include rotation ( [[cosθ, –sinθ], [sinθ, cosθ]] ), reflection, scaling, and shear. The image of a point (x, y) is obtained by multiplying the matrix by the column vector [[x], [y]].
平面上的变换：2 × 2 矩阵表示线性变换。常见变换包括旋转（ [[cosθ, –sinθ], [sinθ, cosθ]] ）、反射、缩放和剪切。点 (x, y) 的像通过矩阵乘以列向量 [[x], [y]] 得到。
v satisfying Mv = λv is an eigenvector with eigenvalue λ. Eigenvalues are found from det(M – λI) = 0.
不变线和特征向量：对于矩阵 M，满足 Mv = λv 的非零向量 v 是特征向量，λ 为特征值。特征值由 det(M – λI) = 0 解出。
3. Roots of Polynomials | 多项式根
For a cubic equation ax³ + bx² + cx + d = 0 with roots α, β, γ: sum of roots α+β+γ = –b/a, sum of product pairs αβ+βγ+γα = c/a, product αβγ = –d/a.
对于三次方程 ax³ + bx² + cx + d = 0，根为 α, β, γ：根的和 α+β+γ = –b/a，两两乘积之和 αβ+βγ+γα = c/a，根的乘积 αβγ = –d/a。
For a quartic ax⁴ + bx³ + cx² + dx + e = 0 with roots α, β, γ, δ: sum = –b/a, pair-sum = c/a, triple-sum = –d/a, product = e/a.
对于四次方程 ax⁴ + bx³ + cx² + dx + e = 0，根为 α, β, γ, δ：和 = –b/a，两两乘积之和 = c/a，三三乘积之和 = –d/a，乘积 = e/a。
These relations can be used to form new equations whose roots are related to the original roots, for example α², 1/α, (α + k), etc. Use symmetric sums and substitution.
这些关系可用于构造新方程，使其根与原方程的根有特定关系，例如 α²、1/α、(α + k) 等。利用对称和与代换即可。
4. Summation of Series | 级数求和
Standard results you must know:
- ∑r=1n r = n(n+1)/2
- ∑r=1n r² = n(n+1)(2n+1)/6
- ∑r=1n r³ = n²(n+1)²/4
必须熟记的标准结果：
- ∑r=1n r = n(n+1)/2
- ∑r=1n r² = n(n+1)(2n+1)/6
- ∑r=1n r³ = n²(n+1)²/4
ions of r by splitting the sum into multiples of these standard series. For example, ∑(3r² – 2r) = 3∑r² – 2∑r. Factorise answers fully where possible.
利用这些公式对 r 的多项式函数求和时，将级数拆分为这些标准级数的倍数。例如 ∑(3r² – 2r) = 3∑r² – 2∑r。尽可能将答案完全因式分解。
of two expressions, then observe telescoping cancellation. Common forms: 1/(r(r+1)) = 1/r – 1/(r+1), and similar rational expressions.
含分式的级数可能需要使用差分法：将项表示为两个表达式的差，然后利用望远镜型抵消。常见形式：1/(r(r+1)) = 1/r – 1/(r+1) 以及类似的有理式。
5. Proof by Induction | 归纳法证明
tical induction proves a statement P(n) for all integers n ≥ k. The process: (1) Base case – verify P(k) is true. (2) Inductive step – assume P(n) is true, then prove P(n+1) is true. (3) Conclusion – by induction, P(n) holds for all n ≥ k.
数学归纳法用于证明对所有整数 n ≥ k 命题 P(n) 成立。步骤：(1) 基础情形——验证 P(k) 为真。(2) 归纳步骤——假设 P(n) 为真，然后证明 P(n+1) 为真。(3) 结论——由归纳法可知，P(n) 对所有 n ≥ k 成立。
Common applications: summation formulas, divisibility (e.g., n³ – n is divisible by 6), matrix powers (Aⁿ = ...), inequalities (2ⁿ > n²), and sequences (un+1 = f(un)). Clearly state the inductive hypothesis and manipulate the expression to match the target form.
常见应用：求和公式、整除性（如 n³ – n 能被 6 整除）、矩阵幂 (Aⁿ = ...)、不等式 (2ⁿ > n²) 以及递推数列 (un+1 = f(un))。需清楚写出归纳假设，并通过对表达式进行变形来匹配目标形式。
6. Maclaurin Series | 麦克劳林级数
The Maclaurin series expansion of a function f(x) about x = 0 is:
f(x) = f(0) + f '(0)x + f ''(0)x²/2! + f '''(0)x³/3! + … + f⁽ⁿ⁾(0)xⁿ/n! + …
函数 f(x) 在 x = 0 处的麦克劳林级数展开为：
f(x) = f(0) + f '(0)x + f ''(0)x²/2! + f '''(0)x³/3! + … + f⁽ⁿ⁾(0)xⁿ/n! + …
Key standard expansions:
- eˣ = 1 + x + x²/2! + x³/3! + …, valid for all x.
- sin x = x – x³/3! + x⁵/5! – …, all x.
- cos x = 1 – x²/2! + x⁴/4! – …, all x.
- ln(1+x) = x – x²/2 + x³/3 – x⁴/4 + …, valid for –1 < x ≤ 1.
- (1+x)ⁿ = 1 + nx + n(n–1)x²/2! + …, valid for |x| < 1 (unless n is a positive integer, then finite and valid for all x).
主要标准展开式：
- eˣ = 1 + x + x²/2! + x³/3! + …，对所有 x 有效。
- sin x = x – x³/3! + x⁵/5! – …，对所有 x 有效。
- cos x = 1 – x²/2! + x⁴/4! – …，对所有 x 有效。
- ln(1+x) = x – x²/2 + x³/3 – x⁴/4 + …，在 –1 < x ≤ 1 上有效。
- (1+x)ⁿ = 1 + nx + n(n–1)x²/2! + …，在 |x| < 1 时有效（除非 n 为正整数，此时为有限项且对所有 x 成立）。
To find the series for composite functions, substitute inside the standard expansion and simplify, ensuring the interval of validity is adjusted accordingly.
求复合函数的级数时，替换到标准展开式中并化简，同时注意相应调整有效区间。
7. Polar Coordinates | 极坐标
A point P is represented by (r, θ) where r is the distance from the pole O, and θ is the angle measured from the initial line (positive x-axis). Conversion: x = r cosθ, y = r sinθ, and r = √x² + y², tanθ = y/x.
点 P 用 (r, θ) 表示，r 是到极点 O 的距离，θ 是从极轴（正 x 轴）量起的角度。转换关系：x = r cosθ, y = r sinθ，r = √x² + y²，tanθ = y/x。
The area enclosed by a polar curve r = f(θ) between θ = α and θ = β is given by:
Area = ½ ∫_α^β r² dθ
极坐标曲线 r = f(θ) 在 θ = α 到 θ = β 之间围成的面积为：
面积 = ½ ∫_α^β r² dθ
orrect limits. For a full closed curve like the cardioid r = a(1+cosθ), use α = 0, β = 2π.
对于心形线、蜗线以及圆等曲线，先画出草图以确定正确的积分限。对于如心形线 r = a(1+cosθ) 这样的全封闭曲线，取 α = 0, β = 2π。
osθ. Tangent parallel to initial line occurs when dy/dθ = 0, perpendicular when dx/dθ = 0.
极坐标曲线的切线：斜率 dy/dx = (dy/dθ) / (dx/dθ)。利用 y = r sinθ，x = r cosθ 及乘积法则。平行于极轴的切线出现在 dy/dθ = 0 时，垂直于极轴则出现在 dx/dθ = 0 时。
8. Hyperbolic Functions | 双曲函数
Definitions:
- sinh x = (eˣ – e⁻ˣ)/2
- cosh x = (eˣ + e⁻ˣ)/2
- tanh x = sinh x / cosh x
定义：
- sinh x = (eˣ – e⁻ˣ)/2
- cosh x = (eˣ + e⁻ˣ)/2
- tanh x = sinh x / cosh x
Key identity: cosh²x – sinh²x = 1. Others: sinh(2x) = 2 sinh x cosh x, cosh(2x) = cosh²x + sinh²x = 2cosh²x – 1 = 1 + 2sinh²x.
核心恒等式：cosh²x – sinh²x = 1。其他：sinh(2x) = 2 sinh x cosh x，cosh(2x) = cosh²x + sinh²x = 2cosh²x – 1 = 1 + 2sinh²x。
h x = 1/cosh x. Inverse hyperbolic functions: arsinh, arcosh, artanh, used in integration.
导数：d/dx(sinh x) = cosh x, d/dx(cosh x) = sinh x, d/dx(tanh x) = sech²x，其中 sech x = 1/cosh x。反双曲函数：arsinh、arcosh、artanh，用于积分计算。
+ C. For inverse hyperbolic forms, the standard integrals are often given in the formula booklet, but recognising the pattern is essential.
双曲函数的积分：∫ sinh x dx = cosh x + C，∫ cosh x dx = sinh x + C。反双曲形式的标准积分通常在公式表中给出，但识别出相应的模式至关重要。
9. Further Calculus | 进阶微积分
Volumes of revolution about the x-axis: V = π ∫_ab y² dx. About the y-axis: V = π ∫_cd x² dy (convert the function to x in terms of y).
绕 x 轴旋转体的体积：V = π ∫_ab y² dx。绕 y 轴旋转：V = π ∫_cd x² dy（将函数改写为 x 关于 y 的表达式）。
For parametric equations (x(t), y(t)), volume about x-axis becomes V = π ∫_α^β y² (dx/dt) dt.
对于参数方程 (x(t), y(t))，绕 x 轴的体积变为 V = π ∫_α^β y² (dx/dt) dt。
nite limits or integrands that become infinite within the interval. Evaluate as limit: ∫_a∞ f(x) dx = limR→∞ ∫_aR f(x) dx. For infinite integrand at a, ∫_ab f(x) dx = limc→a⁺ ∫_cb f(x) dx.
反常积分：含无穷积分限或被积函数在积分区间内趋于无穷的积分。通过极限计算：∫_a∞ f(x) dx = limR→∞ ∫_aR f(x) dx。若在 a 处被积函数无穷，∫_ab f(x) dx = limc→a⁺ ∫_cb f(x) dx。
Mean value of a function: 1/(b–a) ∫_ab f(x) dx. Arc length for a function y = f(x): s = ∫_ab √(1 + (dy/dx)²) dx. For parametric: s = ∫√((dx/dt)² + (dy/dt)²) dt.
函数的平均值：1/(b–a) ∫_ab f(x) dx。函数 y = f(x) 的弧长：s = ∫_ab √(1 + (dy/dx)²) dx。参数形式：s = ∫√((dx/dt)² + (dy/dt)²) dt。
10. Vectors | 向量
Vector equation of a line: r = a + λd, where a is a point on the line, d is the direction vector, and λ is a scalar parameter. In 3D, this can be written in component form.
直线的向量方程：r = a + λd，其中 a 是直线上的一点，d 是方向向量，λ 是标量参数。在三维空间中，可写成分量形式。
Cartesian equation of a line in 3D: (x–a₁)/d₁ = (y–a₂)/d₂ = (z–a₃)/d₃ (= λ). If any component of d is zero, treat separately.
三维空间中直线的笛卡尔方程：(x–a₁)/d₁ = (y–a₂)/d₂ = (z–a₃)/d₃ (= λ)。若 d 的某个分量为零，需单独处理。
Equation of a plane: r·n = d lane, and d = a·n for any point a on the plane. The Cartesian form is n₁x + n₂y + n₃z = d.
平面的方程：r·n = d，其中 n 是平面的法向量，d = a·n 对平面上任一点 a 均成立。笛卡尔形式为 n₁x + n₂y + n₃z = d。
Scalar product: a·b = |a||b| cosθ = a₁b₁ + a₂b₂ + a₃b₃. To find the angle between two vectors: cosθ = (a·b) / (|a||b|).
数量积：a·b = |a||b| cosθ = a₁b₁ + a₂b₂ + a₃b₃。求两向量夹角：cosθ = (a·b) / (|a||b|)。
tion into the plane equation and solve for λ. The angle between a line and a plane: sinθ = |d·n| / (|d||n|), where d is direction, n is normal.
直线与平面的交点：将直线的参数方程代入平面方程，解出 λ。直线与平面的夹角：sinθ = |d·n| / (|d||n|)，其中 d 为方向向量，n 为法向量。
11. Differential Equations | 微分方程
dy = ∫ f(x) dx and integrate. Include a constant of integration and handle implicit forms.
一阶可分离微分方程：dy/dx = f(x)g(y)。重排为 ∫ (1/g(y)) dy = ∫ f(x) dx 并积分。包含积分常数，注意处理隐式形式。
factor method for linear first-order ODEs: dy/dx + P(x)y = Q(x). The integrating factor is I.F. = e∫ P(x) dx. Multiply through, then d/dx (y × I.F.) = Q(x) × I.F. Integrate and solve for y.
一阶线性常微分方程的积分因子法：dy/dx + P(x)y = Q(x)。积分因子为 I.F. = e∫ P(x) dx。全乘后，得到 d/dx (y × I.F.) = Q(x) × I.F. 积分并解出 y。
Contextual problems: exponential growth/decay, cooling, mixing, etc., often lead to differential equations. Translate the rate statement carefully and apply initial conditions to find particular solutions.
实际情境问题：指数增长与衰减、冷却、混合等，通常会导出微分方程。仔细将变化率语句转化为方程，并利用初始条件求特解。
12. Numerical Methods | 数值方法
, find an interval [a, b] where f(a) and f(b) have opposite signs. Repeatedly halve the interval and select the subinterval with a sign change until the required accuracy is reached.
区间二分法：为解 f(x) = 0，找到区间 [a, b] 使得 f(a) 与 f(b) 异号。反复将区间二分，并选取端点函数值异号的子区间，直到达到所需精度。
Newton-Raphson method: xn+1 = xn – f(xn) / f '(xn). It converges rapidly if the initial guess is close to the root and f '(x) ≠ 0 near the root. Watch for divergence or cycle.
牛顿-拉弗森法：xn+1 = xn – f(xn) / f '(xn)。若初始猜测足够接近根，且根附近 f '(x) ≠ 0，该方法收敛速度很快。注意可能发散或循环。
Numerical integration with the trapezium rule: for n strips of equal width h, ∫_ab y dx ≈ h/2 [y₀ + 2(y₁ + y₂ + … + yn–1) + yn], where h = (b–a)/n. Use ordinates given in a table.
梯形法则进行数值积分：对于 n 个等宽条带，宽度 h，∫_ab y dx ≈ h/2 [y₀ + 2(y₁ + y₂ + … + yn–1) + yn]，其中 h = (b–a)/n。使用表中给出的纵坐标。
Error in the trapezium rule reduces with increasing n; the approximate error can be estimated by comparing results for different n. The method overestimates for convex functions and underestimates for concave functions.
梯形法则的误差随着 n 增大而减小；可通过比较不同 n 的结果来估计近似误差。对于凸函数，该法会高估；对于凹函数，则会低估。
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