📚 Year 12 CAIE Further Mathematics: Vocabulary & Terminology Quick Memory Guide | Year 12 CAIE 进阶数学：词汇术语速记指南
Mastering Year 12 CAIE Further Mathematics begins with a solid command of terminology. This guide breaks down the essential vocabulary from the FP1 syllabus into memorable chunks, pairing each key term with its Chinese equivalent and a clear explanation. Use these bite-sized definitions to speed up your revision and build confidence for exam questions.
掌握 Year 12 CAIE 进阶数学，首先要攻克术语关。本指南将 FP1 考纲中的核心词汇拆解成记忆友好的小块，每个关键术语配以中文对照和清晰解释。利用这些精炼定义，你可以加速复习，在考试中更加自信。
1. Polynomial Roots and Vieta’s Formulas | 多项式根与韦达公式
Sum of roots (Σα): For a quadratic ax² + bx + c = 0, the sum of the roots α and β is −b/a. For cubic ax³ + bx² + cx + d = 0, Σα = −b/a. This generalises to any polynomial degree.
根之和 (Σα)：对于二次方程 ax² + bx + c = 0，根 α 与 β 之和为 −b/a。对于三次方程 ax³ + bx² + cx + d = 0，Σα = −b/a。这一规律适用于任意次多项式。
Product of roots (Σαβ): The sum of pairwise products for the quadratic is c/a. For the cubic, the sum of products taken two at a time is c/a, and the product of all three roots is −d/a.
根之积 (Σαβ)：二次方程中根的两两乘积之和为 c/a。三次方程中，两根乘积之和为 c/a，三个根的总乘积为 −d/a。
Vieta’s formulas: These link coefficients to symmetric functions of roots. A common memory aid is “minus over a for sum, plus over a for product” in even degree sums, watching signs for odd-degree products.
韦达定理：将系数与根的对称函数联系起来。记忆口诀：对于和，符号取负除以 a；对于积，符号交替，奇数次积带负号。
Symmetric functions of roots: Expressions like Σα² and Σα²β that remain unchanged when roots are swapped. They can be expressed via Vieta without finding individual roots.
根的对称函数：如 Σα²、Σα²β 等，在交换根时保持不变的表达式。可借助韦达定理直接求得，无需解出单个根。
2. Complex Numbers – Basics and Argand Diagram | 复数基础与阿尔冈图
Complex number (z = a + bi): a is the real part (Re(z)) and b is the imaginary part (Im(z)). i is the imaginary unit with i² = −1. Complex numbers extend the real number system.
复数 (z = a + bi)：a 为实部 (Re(z))，b 为虚部 (Im(z))。i 为虚数单位，满足 i² = −1。复数扩展了实数系统。
Complex conjugate (z*): For z = a + bi, the conjugate is a − bi. Geometrically, it is a reflection in the real axis. Conjugates are used to divide complex numbers and find purely real denominators.
共轭复数 (z*)：若 z = a + bi，其共轭为 a − bi。几何意义是沿实轴反射。共轭用于复数除法，使分母实数化。
Argand diagram: A plane where the x-axis represents the real part and the y-axis represents the imaginary part. Points, vectors, and loci of complex numbers can be visualised here.
阿尔冈图：以 x 轴为实轴、y 轴为虚轴的平面。在此平面上可直观表示复数的点、向量和轨迹。
Locus in complex plane: A set of points satisfying a condition such as |z − (a+bi)| = r (circle) or |z − z₁| = |z − z₂| (perpendicular bisector). Describing loci is a common exam task.
复平面轨迹：满足条件的点集，如 |z − (a+bi)| = r 表示圆，|z − z₁| = |z − z₂| 表示垂直平分线。描述轨迹是常见题型。
3. Modulus, Argument and de Moivre’s Theorem | 模、辐角和棣莫弗定理
Modulus |z|: The distance of z from the origin on the Argand diagram. For z = a + bi, |z| = √(a² + b²). It is always a non-negative real number.
模 |z|：复数 z 在阿尔冈图上到原点的距离。若 z = a + bi，则 |z| = √(a² + b²)。模总是非负实数。
Argument arg(z): The angle θ that z makes with the positive real axis, usually given in radians between −π and π. tan θ = b/a, but quadrant must be considered.
辐角 arg(z)：复数 z 与正实轴的夹角 θ，通常以弧度表示，范围在 −π 到 π 之间。tan θ = b/a，但需根据象限确定。
Modulus-argument form: z = r(cos θ + i sin θ) where r = |z| and θ = arg(z). This is also written as r cis θ. It is essential for multiplication and powers.
模–辐角形式：z = r(cos θ + i sin θ)，其中 r = |z|，θ = arg(z)。也常写作 r cis θ。该形式便于乘法和乘方运算。
de Moivre’s theorem: (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ) for integer n. It allows rapid calculation of powers and roots of complex numbers.
棣莫弗定理：对整数 n，(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)。可用于快速计算复数的乘方与开方。
nth roots of unity: Solutions to zⁿ = 1, given by cos(2kπ/n) + i sin(2kπ/n) for k = 0,1,…,n−1. They are equally spaced on the unit circle.
单位根：方程 zⁿ = 1 的解，表示为 cos(2kπ/n) + i sin(2kπ/n)，k = 0 到 n−1。它们在单位圆上等距分布。
4. Matrices – Determinants and Inverses | 矩阵、行列式和逆矩阵
Square matrix of order 2 or 3: A matrix with equal rows and columns, e.g. 2×2 or 3×3. Operations include addition, multiplication by a scalar, and multiplication of matrices.
二阶或三阶方阵：行数与列数相等的矩阵，如 2×2 或 3×3。运算包括加法、数乘和矩阵乘法。
Determinant (det A): For a 2×2 matrix [[a,b],[c,d]], det = ad − bc. For 3×3, use the rule of Sarrus or expansion by minors. A zero determinant means the matrix is singular.
行列式 (det A)：对于 2×2 矩阵 [[a,b],[c,d]]，det = ad − bc。对于 3×3，可用萨吕法则或按子式展开。行列式为零则表示矩阵是奇异的。
Singular and non-singular matrices: A matrix is singular if det = 0; it has no inverse. A non-singular matrix has det ≠ 0 and possesses an inverse.
奇异与非奇异矩阵：若 det = 0，则矩阵为奇异矩阵，不存在逆矩阵。非奇异矩阵满足 det ≠ 0，且存在逆矩阵。
Inverse of a 2×2 matrix: For M = [[a,b],[c,d]], M⁻¹ = 1/(ad−bc) [[d,−b],[−c,a]]. Only works when ad−bc ≠ 0.
2×2 矩阵的逆：若 M = [[a,b],[c,d]]，则 M⁻¹ = 1/(ad−bc) [[d,−b],[−c,a]]。仅在 ad−bc ≠ 0 时成立。
5. Matrix Transformations and Invariant Lines | 矩阵变换与不变线
Linear transformation matrix: A 2×2 matrix represents a transformation of points in the plane: rotation, reflection, stretch, shear, or a composition. The image of a point (x,y) is obtained by multiplying the column vector [x;y] by the matrix.
线性变换矩阵：2×2 矩阵表示平面内点的变换：旋转、反射、拉伸、剪切或它们的复合。点 (x,y) 的像可通过矩阵乘以列向量 [x;y] 得到。
Rotation matrix: [[cos θ, -sin θ],[sin θ, cos θ]] rotates points about the origin by angle θ anticlockwise. Its determinant is 1, preserving area.
旋转矩阵：[[cos θ, -sin θ],[sin θ, cos θ]] 将点绕原点逆时针旋转角度 θ。行列式为 1，保持面积不变。
Reflection matrix: e.g. reflection in the line y = x is [[0,1],[1,0]]. Reflection in the x-axis is [[1,0],[0,−1]]. Determinant is −1, preserving area magnitude but reversing orientation.
反射矩阵：例如关于直线 y = x 反射的矩阵为 [[0,1],[1,0]]；关于 x 轴反射为 [[1,0],[0,−1]]。行列式为 −1，保持面积大小但反向。
Invariant lines: A line is invariant under a transformation if every point on the line maps to a point on the same line (not necessarily the same point). This leads to eigenvalue problems or setting y = mx + c and solving.
不变线：若一条直线上的每一点经变换后仍落在同一直线上（不必是同一点），则该直线为不变线。求解常归结为特征值问题，或设 y = mx + c 并求解。
6. Mathematical Induction | 数学归纳法
Principle of mathematical induction: A method to prove a statement P(n) for all positive integers n. It consists of a base case (n = 1 or another starting integer) and an inductive step: assume true for n = k, prove true for n = k+1.
数学归纳法原理：一种证明对所有正整数 n 命题 P(n) 成立的方法。包括基础情形（n = 1 或其他起始值）和归纳步骤：假设 n = k 时成立，证明 n = k+1 时也成立。
Base case: Verify the statement for the smallest value (often n = 1). This is the foundation of the inductive argument.
基础情形：验证命题对最小的值成立（通常 n = 1）。这是归纳论证的基石。
Inductive hypothesis: Assume P(k) is true for some arbitrary integer k ≥ base value. This assumption drives the induct step.
归纳假设：假设对某个任意整数 k（≥ 基础值）命题 P(k) 成立。该假设驱动归纳步骤的证明。
Inductive step: Using P(k) to prove P(k+1). Often requires algebraic manipulation, summation properties, or divisibility arguments. The conclusion is that P(n) holds for all n.
归纳步骤：利用 P(k) 证明 P(k+1)。常需进行代数变形、求和性质或整除性论证。最终结论为 P(n) 对所有 n 成立。
7. Summation of Series | 级数求和
Standard summation formulas: Σr = n(n+1)/2, Σr² = n(n+1)(2n+1)/6, Σr³ = n²(n+1)²/4. These are the building blocks for summing polynomials in r.
标准求和公式：Σr = n(n+1)/2，Σr² = n(n+1)(2n+1)/6，Σr³ = n²(n+1)²/4。这些是求含 r 的多项式级数和的基石。
Sum of a polynomial series: To sum Σ (ar³ + br² + cr + d), split into individually evaluated standard sums. Always factorise the final expression if possible.
多项式级数求和：求 Σ (ar³ + br² + cr + d) 时，拆分成各个标准求和再相加。最后尽可能因式分解结果。
Method of differences: Used when terms can be written as f(r) − f(r+1) or f(r+1) − f(r). Most terms cancel, leaving just a few end terms. The method is common with rational functions and trigonometric series.
差分法：当项可写为 f(r) − f(r+1) 或 f(r+1) − f(r) 时使用。大部分项相消，仅剩首尾几项。常用于有理函数和三角级数。
Sum to infinity: For a convergent series, the limit as n → ∞ is evaluated. This is especially relevant to series obtained via differences where the tail vanishes.
无穷级数和：对收敛级数，计算 n → ∞ 时的极限。由差分法得到的级数常常尾部消失，适用此法。
8. Polar Coordinates | 极坐标
Polar coordinates (r, θ): r is the radial distance from the pole (origin), θ is the angle measured from the initial line (positive x-axis). Conversion: x = r cos θ, y = r sin θ.
极坐标 (r, θ)：r 是点到极点（原点）的距离，θ 是从极轴（正 x 轴）量起的角度。转换公式：x = r cos θ，y = r sin θ。
Polar curves: Equations of the form r = f(θ) produce shapes like cardioids, roses, and spirals. Sketching requires evaluating r for key angles and noting symmetry.
极坐标曲线：形如 r = f(θ) 的方程产生心形线、玫瑰线、螺线等图形。画图需计算关键角度处的 r 并利用对称性。
Area enclosed by a polar curve: Area = ½ ∫ r² dθ integrated between appropriate limits. This formula comes from summing slices of circular sectors.
极坐标曲线围成面积：面积 = ½ ∫ r² dθ，在适当区间积分。该公式源于对扇形小片的求和。
Tangents to polar curves: Use dy/dx = (dy/dθ)/(dx/dθ) with x = r cos θ, y = r sin θ. Parallel and perpendicular to the initial line tangents can be found by setting dy/dθ = 0 or dx/dθ = 0.
极坐标曲线的切线：利用 dy/dx = (dy/dθ)/(dx/dθ)，其中 x = r cos θ，y = r sin θ。平行或垂直于极轴的切线可通过令 dy/dθ = 0 或 dx/dθ = 0 求得。
9. Vectors – Scalar and Vector Products | 向量点积与叉积
Position vector and direction vector: A position vector is tied to the origin; a direction vector gives the direction of a line. For a line through point A with direction d, vector equation is r = a + t d.
位置向量与方向向量：位置向量以原点为起点；方向向量给出直线的方向。过点 A 方向为 d 的直线向量方程为 r = a + t d。
Scalar (dot) product: a·b = |a||b| cos θ, and for components a·b = a₁b₁ + a₂b₂ + a₃b₃. It is used to find angles, perpendicularity (a·b = 0), and projections.
数量积（点积）：a·b = |a||b| cos θ，分量形式为 a₁b₁ + a₂b₂ + a₃b₃。用于求角、垂直判定 (a·b = 0) 和投影。
Vector (cross) product: a × b produces a vector perpendicular to both a and b, with magnitude |a||b| sin θ. In component form, a×b = determinant [[i,j,k],[a₁,a₂,a₃],[b₁,b₂,b₃]].
向量积（叉积）：a × b 得到垂直于 a 和 b 的向量，大小为 |a||b| sin θ。分量形式为行列式 [[i,j,k],[a₁,a₂,a₃],[b₁,b₂,b₃]]。
Area of triangle and parallelogram: Area of parallelogram = |a × b|; area of triangle = ½ |a × b|. This is very useful for geometric problems.
三角形与平行四边形面积：平行四边形面积 = |a × b|；三角形面积 = ½ |a × b|。这在几何问题中非常实用。
10. Rational Functions and Curve Sketching | 有理函数与曲线草图
Rational function: A function of the form f(x) = P(x)/Q(x) where P and Q are polynomials. Asymptotes and intercepts guide the graph.
有理函数：形如 f(x) = P(x)/Q(x) 的函数，其中 P 和 Q 为多项式。通过渐近线和截距可勾勒大致图像。
Vertical asymptote: Occurs where Q(x) = 0 but P(x) ≠ 0. The function tends to ±∞ near these x-values.
垂直渐近线：出现在 Q(x) = 0 而 P(x) ≠ 0 处。在这些 x 值附近，函数趋于 ±∞。
Horizontal/oblique asymptote: If degree of P < degree of Q, y = 0 is horizontal asymptote. If degrees are equal, y = leading coefficient ratio. If degree of P = degree of Q + 1, there is an oblique asymptote found by division.
水平/斜渐近线：若分子次数小于分母，有水平渐近线 y = 0。次数相等时，y = 首项系数比。若分子次数比分母多 1，则存在斜渐近线，由多项式除法求得。
Stationary points and curve behaviour: Differentiate to find turning points. Use second derivative or nature table. Combine with intercepts and asymptotes to sketch the full curve.
驻点与曲线走向：求导找出转折点，用二阶导数或单调性表判断性质。结合截距和渐近线，可完整勾勒曲线。
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