📚 Year 11 Eduqas Further Maths: Key Terms Quick Memorisation Guide | Year 11 Eduqas 进阶数学：词汇术语速记指南
In Year 11 Eduqas Further Mathematics, a confident command of technical vocabulary is the scaffolding upon which problem‑solving is built. To help you internalise these terms, this guide pairs every definition with a memorable trick – from visual mnemonics to word associations – so you can recall them quickly in the exam room.
在 Year 11 Eduqas 进阶数学中，熟练的专业术语是解题的脚手架。本指南为每个定义都搭配了难忘的巧记方法——从视觉助记到词语联想——帮助你在考场中迅速调取这些术语。
1. Algebraic Terminology | 代数术语
A variable is a symbol, often a letter, that stands for a number we do not yet know. In 5x + 2, 'x' is the variable. Memory trick: think of 'variable' as 'can vary'.
变量是一个符号（通常是字母），代表尚未确定的数值。在 5x + 2 中，'x' 就是变量。记忆窍门：“变”即“可变”。
A coefficient is the number multiplying a variable. In 7a, 7 is the coefficient and 1 is the invisible coefficient of a in the term a. Trick: 'co‑' means 'together with' – the number works together with the variable.
系数是乘以变量的那个数字。在 7a 中，7 是系数；而单项 a 中隐藏的系数是 1。巧记：“系”如同“系着”变量，共同出现。
A constant is a number on its own; its value never changes. In 3n + 4, 4 is the constant term. Think: a constant friend is always the same – the number stays put.
常数指单独出现的数字，其值固定不变。在 3n + 4 中，4 就是常数项。联想：常数如同老友，总是老样子。
An expression is a collection of terms using operations, but it has no equals sign. 4ab + 2c − 7 is an expression. Remember: an expression is like a phrase – it expresses something but does not make a full statement.
表达式由字母、数字和运算符号组合而成，但没有等号。4ab + 2c − 7 是表达式。记忆：表达式像短语，它表达含义但并不做出完整断言。
An equation has an equals sign and states that two expressions are equal. 2x − 3 = 11 is an equation. Trick: 'equa‑' echoes 'equal'.
方程含有等号，表示左右两边相等。2x − 3 = 11 是方程。巧记：方程中有“等”，即相等。
An identity is an equation that is true for all values of the variable, like x + x ≡ 2x. The triple bar ≡ shows an identity. Think: the extra bar reinforces 'always true'.
恒等式是对于变量所有取值都成立的等式，如 x + x ≡ 2x。三横线 ≡ 标明恒等。联想：多一道横线，强调“恒久”为真。
2. Polynomials & Factor Theorem | 多项式与因式定理
A polynomial is a finite sum of terms where the variable has whole‑number exponents, such as 4x³ − 2x + 7.
多项式是所有变量指数均为非负整数的若干项之和，例如 4x³ − 2x + 7。
The degree of a polynomial is the highest power of the variable. 6x⁴ + x² − 3 has degree 4. Memory: 'degree' sounds like 'highest step' – like climbing to the top floor.
多项式的次数是变量指数的最大值。6x⁴ + x² − 3 的次数是 4。巧记：“次”可想象成爬楼梯的最高一级。
The factor theorem states: (x − a) is a factor of polynomial p(x) if and only if p(a) = 0. Trick: plug 'a' into the polynomial – if you get zero, you've found a factor.
因式定理：（x − a）是多項式 p(x) 的因式当且仅当 p(a) = 0。记忆：把 a 代进多项式，结果为零，就找到了因式。
The remainder theorem says: when p(x) is divided by (x − a), the remainder equals p(a). Handy tip: the remainder theorem springs from the factor theorem – if p(a) ≠ 0, that number is the remainder.
余数定理：p(x) 除以 (x − a) 时，余数就是 p(a)。窍门：从因式定理延伸而来——若 p(a) ≠ 0，该值即为余数。
3. Functions | 函数
A function is a rule that assigns each input exactly one output. We write f(x) = 2x + 1. Trick: imagine a machine that takes an input x and produces a single output y.
函数是一种规则，每个输入只对应一个输出。写作 f(x) = 2x + 1。巧记：想象一台机器，投入 x 即产出一个 y。
The domain is the set of all allowed input (x) values. The range is the set of all possible output (y) values. Mnemonic: 'd' comes before 'r' in the alphabet – input before output. Also, domain is like 'dominion' of x, range is how y 'ranges'.
定义域是允许输入的 x 值集合，值域是所有可能输出的 y 值集合。助记：字母“定”在前，“值”在后——先有输入，后有输出。
An inverse function reverses the original function, usually written as f⁻¹(x). It swaps input and output. Memory: the superscript −1 is like 'undo' – if f takes a → b, then f⁻¹ takes b → a.
反函数逆转原函数，通常记作 f⁻¹(x)，它交换输入与输出。巧记：上标 −1 就像“撤销”键——若 f 把 a 变成 b，那么 f⁻¹ 把 b 变回 a。
A composite function applies one function to the result of another. For fg(x) you first apply g, then f. Tip: read fg(x) right to left – g first, then f – as if reading the recipe backwards.
复合函数把一个函数作用在另一个函数的结果上。fg(x) 意味着先执行 g，再执行 f。窍门：从右向左读 fg(x)——先 g 后 f。
A function that gives a unique output for each different input is one‑to‑one. It passes the horizontal line test. Recall: one‑to‑one means no two x share the same y.
若不同输入对应不同输出，则该函数是一一映射，它通过水平线测试。记忆：一一映射即没有两个 x 对应相同的 y。
4. Coordinate Geometry | 坐标几何
The gradient e measures its steepness. Between (x₁, y₁) and (x₂, y₂), gradient m = (y₂ − y₁)/(x₂ − x₁). Think: 'rise over run' – how much it climbs when you go across 1 unit.
斜率（梯度）度量直线的倾斜程度。两点间斜率 m = (y₂ − y₁)/(x₂ − x₁)。联想：“高差除以横移”——横走 1 单位时爬升多少。
The y‑intercept is where the line crosses the y‑axis (x = 0). In y = mx + c, c is the y‑intercept. Recall: 'intercept' means 'cut' – where the line cuts the axis.
y 截距是直线与 y 轴相交点的 y 坐标（此时 x = 0）。在 y = mx + c 中，c 就是 y 截距。巧记：“截”即截断——直线截断轴的位置。
The midpoint of two points is found by averaging their x‑ and y‑coordinates: M = ((x₁ + x₂)/2, (y₁ + y₂)/2). Trick: the midpoint is simply the 'middle' point – take the mean of the ends.
中点由两点的 x 坐标和 y 坐标分别取平均值得到：M = ((x₁ + x₂)/2, (y₁ + y₂)/2)。窍门：中点即“正中间”的点，两端取均值。
The distance between two points is √[(x₂ − x₁)² + (y₂ − y₁)²]. It's Pythagoras! Think of a right‑angled triangle whose legs are the horizontal and vertical differences.
两点间距离为 √[(x₂ − x₁)² + (y₂ − y₁)²]。这就是勾股定理！把横纵差值看作直角三角形的两直角边。
The equation of a circle centred at (a, b) with radius r is (x − a)² + (y − b)² = r². Memory: each coordinate is shifted by the centre, and the sum of squares equals radius squared – exactly like a stretched Pythagoras.
以 (a, b) 为圆心、半径为 r 的圆方程为 (x − a)² + (y − b)² = r²。记忆：坐标均按圆心平移，平方和等于半径平方——恰是扩大的勾股定理。
5. Trigonometry | 三角学
The three basic ratios – sine, cosine, tangent – are defined by the sides of a right‑ ngle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Use the classic mnemonic: SOH CAH TOA.
三个基本比——正弦（sin）、余弦（cos）、正切（tan）——由直角三角形的边定义。经典口诀：SOH CAH TOA（对/斜、邻/斜、对/邻）。
The sine rule a/sin A = b/sin B = c/sin C links sides to opposite angles in any triangle. Vision: the side and the sine of its opposite angle are 'married'.
正弦定理 a/sin A = b/sin B = c/sin C 联系任意三角形的边与其对角正弦。想象：边和它对角的正弦是一对“搭档”。
The cosine rule 2bc cos A helps when you have two sides and the included angle (SAS) or three sides (SSS). It's Pythagoras with a correction term −2bc cos A.
余弦定理 a² = b² + c² − 2bc cos A 适用于已知 SAS 或 SSS 的情况。它像是勾股定理加了一个修正项 −2bc cos A。
Exact trigonometric values for 0°, 30°, 45°, 60°, 90° should be memorised. Tip: use your fingers or a table of surds: ½, √2/2, √3/2 etc.
精确三角值（0°, 30°, 45°, 60°, 90°）需要熟记。窍门：可用手指法或记住根号分数列：½, √2/2, √3/2 等。
The key identity sin²θ + cos²θ ≡ 1 is derived from Pythagoras on the unit circle. Remember it as the 'trig Pythagoras'.
重点恒等式 sin²θ + cos²θ ≡ 1 源于单位圆上的勾股定理。记忆为“三角勾股定理”。
The CAST diagram tells you which trig ratios are positive in each quadrant. Starting from the top‑right and moving anticlockwise: A (All), S (Sine), T (Tangent), C (Cosine). Students say 'All Students Take Calculus'.
CAST 图表明各象限中哪些三角比为正。从右上开始逆时针：A（全正）、S（正弦正）、T（正切正）、C（余弦正）。口诀：A 爱 S 吃 T 葱 C。
6. Matrices & Transformations | 矩阵与变换
A matrix is a rectangular array of numbers. The order of a matrix is given as rows × columns. A 2×3 matrix has 2 rows and 3 columns. Trick: think of a cinema: rows first, then columns – horizontal then vertical.
矩阵是数字的矩形排列。阶表示为行数 × 列数。2×3 矩阵有 2 行 3 列。巧记：先横后竖，如同电影院的排和座。
The determinant of a 2×2 matrix (a b; c d) is ad − bc. If the determinant is zero, the matrix is singular and has no inverse. Mnemonic: 'ad‑some‑bc' – like a subtraction in your head.
2×2 矩阵 (a b; c d) 的行列式为 ad − bc。若行列式为零，矩阵是奇异矩阵，没有逆矩阵。口诀：斜乘相减，ad 减 bc。
The inverse of a non‑ found by swapping a and d, changing the signs of b and c, and dividing by the determinant. Recall: swap main diagonal, negate the other diagonal, then multiply by 1/det.
非奇异 2×2 矩阵的逆矩阵计算：交换主对角线元素 a 和 d，将另一对角线上 b 和 c 变号，再除以行列式。记忆：主换号，负变号，再除 det。
Matrices represent geometric transformations. Multiplying a position vector by a matrix changes its position. Think: the matrix is the 'instruction manual' for the movement.
矩阵表示几何变换。将位置向量乘以一个矩阵，就改变了它的位置。想象：矩阵是运动的“操作说明书”。
Common transformations: rotation (turn about origin), reflection (flip across a line), enlargement (scale factor k), and shear (sliding layers parallel to an axis). Visual memory: each transformation has a recognisable 'signature' matrix.
常见变换：旋转（绕原点转）、反射（沿直线翻转）、放大（比例因子 k）和剪切（平行于轴的错位）。视觉记忆：每种变换有其标志性的矩阵形式。
7. Vectors | 向量
A vector has both magnitude (size) and direction. It can be written as a column matrix or using i, j unit vectors. Contrast with scalar: a scalar has only magnitude.
向量既有大小又有方向。可写为列矩阵或使用单位向量 i, j。与标量对比：标量仅有大小。
Magnitude of a vector (x, y) is √x² + y². It’s Pythagoras again – the length of the arrow.
向量 (x, y) 的模长为 √x² + y²。还是勾股定理——箭头的长度。
A unit vector has magnitude 1. Divide a vector by its magnitude to obtain a unit vector in the same direction. Tip: a unit vector retains direction but scales length to 1.
单位向量模长为 1。将向量除以其模长，即得同方向的单位向量。窍门：保留方向，把长度缩为 1。
Vectors are parallel if one is a scalar multiple of the other. They are equal if they have the same magnitude and direction. Parallel vectors have the same 'slope' in component form.
若一向量是另一向量的标量倍数，则二者平行。若大小和方向均相同，则二者相等。平行向量在分量形式上具有相同的“斜率”。
The resultant vector is the single vector that has the same effect as two or more vectors added together. Tip: add nose‑to‑tail, and the resultant goes from start to finish.
合向量是能产生同样效果的那个单一向量。将多个向量首尾相连，合向量从起点指向终点。
8. Calculus | 微积分
Differentiation finds the gradient of a curve. The derivative dy/dx of y = xⁿ is n xⁿ⁻¹. Mnemonic: 'bring the power down and reduce it by one'.
微分求曲线的斜率。y = xⁿ 的导数 dy/dx = n xⁿ⁻¹。口诀：指数向下搬，自身降一阶。
A stationary point occurs where dy/dx = 0. Its nature is determined by the second derivative d²y/dx². Recall: second derivative positive → Minimum; negative → Maximum.
驻点出现在 dy/dx = 0 处。其性质由二阶导数 d²y/dx² 判定：二阶导数为正→极小值；为负→极大值。
For increasing and decreasing functions: where dy/dx > 0, the function is increasing; where dy/dx < 0, it is decreasing. Think: slope up means increasing, slope down means decreasing.
函数递增与递减：dy/dx > 0 时函数递增，dy/dx < 0 时函数递减。联想：上坡递增，下坡递减。
Integration is the reverse process of differentiation. The indefinite integral of xⁿ is xⁿ⁺¹/(n+1) + c. Mnemonic: put the power up by one and divide by the new power, then add c.
积分是微分的逆运算。xⁿ 的不定积分为 xⁿ⁺¹/(n+1) + c。口诀：指数加一，除以新指数，再加 c。
The constant of integration c appears because indefinite integration gives a family of curves. Remember: without c, you're missing the whole family!
积分常数 c 出现的原因是，不定积分得到一族曲线。牢记：不加 c，
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