📚 AS Further Mathematics: End-of-Term Revision Checklist | AS 进阶数学：期末复习提纲
A successful revision for AS Further Mathematics requires not only recalling formulas but understanding deep connections between topics. This checklist guides you through the essential concepts, skills, and problem types that frequently appear in end-of-term assessments. Use it to identify strengths and weaknesses as you prepare for your exams.
AS 进阶数学的成功复习不仅需要记忆公式，更需要理解各主题之间的深层联系。本提纲将带您梳理常见的核心概念、解题技巧和期末试卷中经常出现的题型，帮助您在备考过程中明确强项与短板。
1. Complex Numbers | 复数
Know the definition z = a + bi, where a, b are real and i² = −1. You must be confident with adding, subtracting, multiplying and dividing complex numbers, especially using the complex conjugate to find the quotient.
掌握定义 z = a + bi，其中 a、b 为实数，i² = −1。你必须熟练掌握复数的加、减、乘、除运算，特别是利用共轭复数求商。
Represent complex numbers on an Argand diagram, compute modulus |z| = √(a² + b²) and argument arg(z), and express them in polar and exponential forms: z = |z| (cos θ + i sin θ) = |z| eiθ.
在 Argand 图上表示复数，计算模 |z| = √(a² + b²) 和辐角 arg(z)，并用极坐标形式和指数形式表示：z = |z| (cos θ + i sin θ) = |z| eiθ。
Apply de Moivre’s theorem: (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ) to find powers and roots of complex numbers. Use it to derive trigonometric identities and solve equations such as zⁿ = k.
应用 de Moivre 定理：(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ) 求复数的幂和根。用它推导三角恒等式并解形如 zⁿ = k 的方程。
2. Roots of Polynomials | 多项式根的性质
Recall the relationships between roots and coefficients for quadratics, cubics and quartics. For a cubic ax³ + bx² + cx + d = 0 with roots α, β, γ, know that α + β + γ = −b/a, αβ + βγ + γα = c/a and αβγ = −d/a.
回忆二次、三次和四次方程根与系数的关系。对于三次方程 ax³ + bx² + cx + d = 0，其根为 α、β、γ，则有 α + β + γ = −b/a，αβ + βγ + γα = c/a，αβγ = −d/a。
Use the fact that complex roots of real polynomials occur in conjugate pairs to deduce unknown coefficients or form polynomials with given roots. Transform roots via substitutions like w = α + k or w = kα to find new equations.
利用实系数多项式复根成对出现这一事实，推出未知系数或构建具有给定根的多项式。通过代换如 w = α + k 或 w = kα 对根进行变换，从而求出新的方程。
3. Summation of Series | 级数求和
Memorise standard results: Σr = ½n(n+1), Σr² = ⅙n(n+1)(2n+1), Σr³ = ¼n²(n+1)². Use them to evaluate sums of polynomials, e.g., Σ(3r² − 2r + 4), by separating into simpler sums.
熟记标准结果：Σr = ½n(n+1)，Σr² = ⅙n(n+1)(2n+1)，Σr³ = ¼n²(n+1)²。利用它们计算多项式的和，例如 Σ(3r² − 2r + 4)，只需拆分成更简单的求和。
Apply the method of differences to telescoping sums. Recognise expressions of the form 1/(r(r+1)) and rewrite using partial fractions to cancel terms. Be prepared to prove summation formulas by induction.
应用裂项法处理可以递推相消的和。能识别形如 1/(r(r+1)) 的式子，并利用部分分式改写以消去中间项。还要准备好用数学归纳法证明求和公式。
4. Matrices | 矩阵
Perform matrix addition, scalar multiplication and matrix multiplication. Know that matrix multiplication is not commutative. Determine the order of a product and find the transpose AT.
进行矩阵的加法、数乘和矩阵乘法。需要注意矩阵乘法不满足交换律。能确定乘积的阶数并求转置矩阵 AT。
Compute determinants of 2×2 and 3×3 matrices. For 2×2, det A = ad − bc; for 3×3, use the formula or the Sarrus method. Understand that a non-zero determinant implies the matrix is invertible.
计算 2×2 和 3×3 矩阵的行列式。对 2×2 矩阵，det A = ad − bc；对 3×3 矩阵，使用公式或 Sarrus 法则。理解行列式非零意味着矩阵可逆。
Find the inverse of a non-singular 2×2 or 3×3 matrix (using the adjoint method for 3×3). Solve systems of linear equations Ax = b by both inverse matrix and row reduction.
求非奇异 2×2 或 3×3 矩阵的逆（3×3 用伴随矩阵法）。通过逆矩阵或行变换解线性方程组 Ax = b。
5. Linear Transformations | 线性变换
Understand how a matrix M maps a vector (x, y) to a new vector. Recognise standard transformation matrices: reflections in the coordinate axes or lines y = ±x, rotations by θ (anticlockwise), stretches parallel to axes, and shears.
理解矩阵 M 如何将向量 (x, y) 映射为新向量。识别标准变换矩阵：关于坐标轴或直线 y = ±x 的反射、逆时针旋转 θ 角、沿坐标轴的伸缩以及剪切。
Combine transformations by multiplying matrices in the correct order (right to left). Find invariant points by solving Mx = x and invariant lines by solving M(x, y)T = λ(x, y)T for λ.
通过正确顺序（从右向左）的矩阵乘法组合变换。求不变点即解 Mx = x，求不变线即解 M(x, y)T = λ(x, y)T 以求 λ。
6. Hyperbolic Functions | 双曲函数
Define sinh x = (ex − e−x)/2, cosh x = (ex + e−x)/2 and tanh x = sinh x/cosh x. Know their graphs, domains, ranges, and the cosh evenness, sinh oddness.
会定义双曲函数：sinh x = (ex − e−x)/2，cosh x = (ex + e−x)/2，tanh x = sinh x/cosh x。知道它们的图像、定义域、值域，以及 cosh 为偶函数、sinh 为奇函数。
Prove and use hyperbolic identities, especially cosh² x − sinh² x = 1 and double-angle formulas. Solve equations involving hyperbolic functions, often by converting to exponentials or using identities.
证明并使用双曲恒等式，尤其要掌握 cosh² x − sinh² x = 1 和倍角公式。通过转化为指数形式或利用恒等式来解涉及双曲函数的方程。
Define inverse hyperbolic functions using logarithms: arsinh x = ln(x + √(x² + 1)), arcosh x = ln(x + √(x² − 1)) (x ≥ 1), artanh x = ½ ln((1+x)/(1−x)) (|x|<1).
用对数定义反双曲函数：arsinh x = ln(x + √(x² + 1))，arcosh x = ln(x + √(x² − 1))（x ≥ 1），artanh x = ½ ln((1+x)/(1−x))（|x|<1）。
7. Polar Coordinates | 极坐标
Switch between Cartesian (x, y) and polar (r, θ) using x = r cos θ, y = r sin θ and r² = x² + y². For curves given by r = f(θ), sketch typical shapes such as cardioids, roses, and circles with centre on the polar axis.
利用 x = r cos θ、y = r sin θ 以及 r² = x² + y² 在直角坐标和极坐标之间转换。对于给出的曲线 r = f(θ)，能描绘典型形状，如心形线、玫瑰线和圆心在极轴上的圆。
Calculate the area enclosed by a polar curve: A = ½ ∫αβ r² dθ. Set up the integral correctly, noting how limits and the curve’s symmetry can simplify the work.
计算极坐标曲线围成的面积：A = ½ ∫αβ r² dθ。正确设立积分，注意如何利用积分限和曲线的对称性简化计算。
8. First Order Differential Equations | 一阶微分方程
Solve separable equations by rearranging into g(y) dy = f(x) dx and integrating both sides. Include cases where substitution may first be needed, such as y’ = (y+x)/x leading to a homogeneous type.
通过将方程整理为 g(y) dy = f(x) dx 并两边积分来解可分离变量的一阶微分方程。包括可能需要先行代换的情形，例如 y’ = (y+x)/x 这类齐次方程。
For a linear first-order ODE of the form dy/dx + P(x)y = Q(x), find the integrating factor I = exp(∫ P(x) dx) and multiply through to obtain d/dx (Iy) = I Q(x). Then integrate and solve for y.
对形如 dy/dx + P(x)y = Q(x) 的一阶线性常微分方程，求出积分因子 I = exp(∫ P(x) dx)，方程两边同乘后得到 d/dx (Iy) = I Q(x)。再积分并解出 y。
9. Second Order Differential Equations | 二阶微分方程
Solve homogeneous second-order linear ODEs a d²y/dx² + b dy/dx + c y = 0 by forming the auxiliary equation am² + bm + c = 0. Write the general solution based on distinct real roots, repeated roots, or complex roots.
通过建立辅助方程 am² + bm + c = 0 来解齐次二阶线性常微分方程 a d²y/dx² + b dy/dx + c y = 0。根据两相异实根、重根或共轭复根写出通解。
For non-homogeneous equations, find the particular integral using the method of undetermined coefficients. Try a form similar to the right-hand side f(x): polynomial of degree n, exponential kepx, or combination sin/cos. The general solution is the sum of the complementary function and the particular integral.
对于非齐次方程，使用待定系数法求特解。根据右端 f(x) 的形式设定特解形式：n 次多项式、指数函数 kepx 或正余弦组合。通解为余函数与特解之和。
10. Proof by Induction | 数学归纳法
Structure a proof clearly: state the proposition P(n), verify the base case (usually n = 1), assume P(k) is true, and prove that P(k+1) holds, then conclude. Common applications include summation formulas, divisibility, inequalities, and matrix powers.
清晰组织证明：陈述命题 P(n)，验证基本情形（通常 n = 1），假设 P(k) 成立，证明 P(k+1) 成立，最后总结陈述。常见应用包括求和公式、可除性问题、不等式以及矩阵的幂。
For divisibility, express P(k+1) in terms of P(k) so that the inductive hypothesis can be used. For inequalities, carefully manipulate the k+1 expression to fit the assumed inequality. Practice writing coherent, logical steps without skipping algebra.
关于整除性，应将 P(k+1) 表示成能用上 P(k) 的形式。对于不等式，要小心处理 k+1 时的表达式以适配假设的不等式。多加练习，写出连贯、逻辑清晰的步骤，不跳过代数变形。
Be ready to prove recurrences, such as un+1 = f(un), and to show that a closed form matches the recurrence. This often appears in sequences defined recursively and matrix power problems.
准备好证明递推关系，
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