📚 Year 11 OCR Further Maths: Vocabulary and Terminology Quick Memorisation Guide | 11年级OCR进阶数学：词汇术语速记指南
Mastering the terminology of OCR Level 2 Further Mathematics is the first step to confident problem-solving. Precise words carry marks: if a question says 'hence', the examiner wants you to use the previous part; if it says 'exact value', a decimal will lose the mark. This guide breaks down the key terms and concepts across every topic - algebra, polynomials, inequalities, binomial expansion, coordinate geometry, trigonometry, calculus, matrices, functions and vectors - and adds memory hooks so you can recall them fast in the exam hall.
掌握OCR二级进阶数学的术语是自信解题的第一步。用词精准就能拿分：题目写 'hence'（由此），考官要你用上一问的结果；写 'exact value'（精确值），写成小数就丢分。本指南逐一拆解所有主题的关键术语与概念——代数、多项式、不等式、二项式展开、坐标几何、三角学、微积分、矩阵、函数与向量——并附记忆钩子，助你在考场上快速回忆。
1. Command Words and Exam Language | 指令词与考试用语
Command words decide what the mark scheme actually rewards. 'Write down' needs no working. 'Show that' requires every step to be visible and to finish at the printed result. 'Hence' means you must use the previous part of the question; 'hence or otherwise' gives you a choice. 'Prove' demands a logical argument from known results, not a numerical check. 'Exact value' means a surd, a fraction or a logarithm - never a rounded decimal.
指令词决定评分标准真正给分的地方。'Write down'（写出）不需要过程。'Show that'（证明得到）要求每一步可见，且终点必须停在题目已给的结果上。'Hence'（由此）意味着必须使用本题上一问的结果；'hence or otherwise'（由此或其他方法）则给你选择权。'Prove'（证明）要求从已知结论出发的逻辑论证，而不是代入数字验证。'Exact value'（精确值）指根式、分数或对数，绝不是四舍五入后的小数。
A second family of words tells you how to present the answer: 'solve' (find all values), 'evaluate' (work out a numerical value), 'simplify' (write in the shortest equivalent form), 'sketch' (shape and key points, no need for graph paper accuracy), 'verify' (substitute to check a given statement), 'deduce' (reach a conclusion using the previous result).
第二类词告诉你如何呈现答案：'solve'（求解，找出所有值）、'evaluate'（求值，算出数值）、'simplify'（化简，写成最短等价形式）、'sketch'（画草图，只需形状与关键点，不必图格精确）、'verify'（验证，代入检验给定结论）、'deduce'（推出，利用上一问结果得出结论）。
- given that - a condition you may assume; 已知 - 可以直接使用的条件
- in terms of - the answer keeps that letter; 用某字母表示 - 答案中保留该字母
- correct to 3 significant figures - rounding instruction; 保留三位有效数字 - 四舍五入要求
- state - no working required; 陈述 - 无需过程
- explain - words, not just numbers; 解释 - 要用文字说明
2. Algebraic Manipulation and Polynomials | 代数运算与多项式
A polynomial is an expression built from non-negative integer powers of a variable with constant coefficients. The degree is the highest power present; the number multiplying that power is the leading coefficient. Names follow the degree: linear (1), quadratic (2), cubic (3), quartic (4).
多项式是由变量的非负整数次幂与常数系数构成的表达式。次数是其中最高的幂次；乘在该最高次幂前的数称为首项系数。按次数命名：一次（线性）、二次、三次、四次。
f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀
Other core vocabulary: term (each separate part), coefficient, constant term a₀, root or zero (a value where f(x) = 0), factorise, expand, collect like terms, and the index laws used to simplify powers (xᵃ × xᵇ = xᵃ⁺ᵇ, xᵃ ÷ xᵇ = xᵃ⁻ᵇ, (xᵃ)ᵇ = xᵃᵇ, x⁰ = 1, x⁻ⁿ = 1/xⁿ).
其他核心词汇：项（各个独立部分）、系数、常数项 a₀、根或零点（使 f(x) = 0 的值）、因式分解、展开、合并同类项，以及用于化简幂的指数律（xᵃ × xᵇ = xᵃ⁺ᵇ，xᵃ ÷ xᵇ = xᵃ⁻ᵇ，(xᵃ)ᵇ = xᵃᵇ，x⁰ = 1，x⁻ⁿ = 1/xⁿ）。
3. Factor Theorem, Remainder Theorem and Completing the Square | 因式定理、余数定理与配方法
Factor Theorem: This turns root-finding into factorisation and gives you a free route into long division.
因式定理：对于多项式 f(x)，若 f(k) = 0，则 (x − k) 是 f(x) 的一个因式；反之，若 (x − k) 是因式，则 f(k) = 0。这把"找根"转化为"分解因式"，也为长除法提供入口。
Remainder Theorem: when f(x) is divided by (x − a), the remainder equals f(a). When dividing by (ax − b), the remainder is f(b/a). It lets you find remainders instantly without long division.
余数定理：当 f(x) 除以 (x − a) 时，余数等于 f(a)；若除以 (ax − b)，余数为 f(b/a)。无需长除法即可立刻求出余数。
Completing the square rewrites a quadratic in vertex form, revealing the turning point:
补全平方（配方法）把二次式改写为顶点式，直接显示顶点坐标：
x² + bx + c = (x + b/2)² + c − b²/4
The minimum point is at (−b/2, c − b²/4). The discriminant real roots exist: Δ > 0 gives two, Δ = 0 gives one repeated root, Δ < 0 gives none.
其最低点位于 (−b/2, c − b²/4)。判别式 Δ = b² − 4ac 告诉你实根个数：Δ > 0 有两个，Δ = 0 有一个重根，Δ < 0 无实根。
4. Inequalities and Solution Sets | 不等式与解集
An inequality compares two expressions using <, >, ≤ or ≥. The first two are strict inequalities and are drawn with open circles on a number line; the second two are non-strict and use filled circles. The solution set is the collection of all values that satisfy every condition at once.
不等式用 <、>、≤ 或 ≥ 比较两个表达式。前两者是严格不等式，在数轴上画空心圆；后两者是非严格不等式，画实心圆。解集是同时满足所有条件的全部值的集合。
Key skills: remember that multiplying or dividing by a negative number reverses the inequality sign; find critical values by solving the matching equation, then test a value in each interval to decide where the inequality holds. A quadratic inequality such as x² − 5x + 6 < 0 has solution 2 < x < 3, written as an interval (2, 3).
关键技能：乘以或除以负数时，不等号反向；先解对应方程求出临界值，再在每个区间内取测试值判断不等式成立的范围。如 x² − 5x + 6 < 0 的解为 2 < x < 3，也可写作区间 (2, 3)。
- critical value - a boundary where the expression changes sign; 临界值 - 表达式变号的边界
- satisfies - makes the statement true; 满足 - 使该陈述成立
- region - area shaded on a graph showing all solutions; 区域 - 图上阴影部分，表示全部解
5. Binomial Expansion | 二项式展开
A binomial is an expression with exactly two terms, such as (a + b). A binomial expansion writes (a + b)ⁿ as a sum of terms, where each coefficient is a binomial coefficient written ₙCᵣ or read as 'n choose r'.
二项式是恰好含两项的表达式，如 (a + b)。二项式展开把 (a + b)ⁿ 写成各项之和，其中每个系数称为二项式系数，记作 ₙCᵣ，读作"n 选 r"。
(a + b)ⁿ = aⁿ + ₙC₁aⁿ⁻¹b + ₙC₂aⁿ⁻²b² + ... + bⁿ
where ₙCᵣ = n! / (r!(n − r)!). The coefficients of successive expansions form Pascal's triangle, in which each entry is the sum of the two above it: 1; 1 1; 1 2 1; 1 3 3 1; 1 4 6 4 1.
其中 ₙCᵣ = n! / (r!(n − r)!)。连续各次展开的系数构成帕斯卡三角，每个数等于其上方两数之和：1；1 1；1 2 1；1 3 3 1；1 4 6 4 1。
Exam vocabulary: a term is one complete product; the coefficient of x² means the number in front of x² after expansion; ascending powers start with the constant and climb, descending powers start with the highest power. Watch the sign when the bracket contains a minus, for example (2 − 3x)⁴.
考试词汇：项指一个完整的乘积；x² 的系数指展开后 x² 前面的数；升幂排列从常数项开始递增，降幂排列从最高次幂开始。括号内是减号时要格外注意符号，例如 (2 − 3x)⁴。
6. Coordinate Geometry of Lines and Circles | 直线与圆的坐标几何
Gradient x₂ − x₁). A horizontal line has gradient 0, a vertical line has an undefined gradient. The midpoint of two points is the average of the coordinates, and the distance between them is found from Pythagoras.
斜率（梯度）衡量陡峭程度：m = (y₂ − y₁)/(x₂ − x₁)。水平线斜率为 0，竖直线的斜率不存在（未定义）。两点的中点是各坐标的平均值，两点间距离由勾股定理求得。
d = √((x₂ − x₁)² + (y₂ − y₁)²)
Parallel lines share the same gradient; perpendicular dient-intercept), y − y₁ = m(x − x₁) (point-gradient), and ax + by + c = 0 (general form).
平行线斜率相同；垂直线满足 m₁ × m₂ = −1，即垂直斜率为负倒数。需要掌握的直线形式：y = mx + c（斜截式）、y − y₁ = m(x − x₁)（点斜式）与 ax + by + c = 0（一般式）。
For a circle, the equation (x − a)² + (y − b)² = r² has centre (a, b) and radius r. The tangent touches the circle once and is perpendicular to the radius at the point of contact; a chord joins two points on the circumference. The perpendicular bisector of a chord passes through the centre.
对于圆，方程 (x − a)² + (y − b)² = r² 的圆心为 (a, b)，半径为 r。切线与圆只有一个交点，且在切点处垂直于半径；弦连接圆周上两点。弦的垂直平分线必过圆心。
7. Trigonometry and Exact Values | 三角学与精确值
In a right-angled triangle, opposite and adjacent are defined relative to a chosen angle, and hypotenuse use, cos θ = adjacent/hypotenuse and tan θ = opposite/adjacent, remembered as SOH-CAH-TOA.
在直角三角形中，对边与邻边是相对于所选角而言的，斜边永远是最长边。基本比值为 sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边，可记为 SOH-CAH-TOA。
For non-right-angled triangles use the sine rule a/sin A = b/sin B = c/sin C, the cosine rule a² = b² + c² − 2bc cos A, and the area formula The cosine rule is the tool when you know three sides or two sides and the included angle.
非直角三角形使用正弦定理 a/sin A = b/sin B = c/sin C、余弦定理 a² = b² + c² − 2bc cos A，以及面积公式 面积 = ½ab sin C。已知三边，或已知两边及其夹角时，就用余弦定理。
Two identities appear constantly: sin²θ + cos²θ = 1 and tan θ = sin θ/cos θ. Exact values to memorise are sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3, sin 45° = cos 45° = 1/√2, and sin 60° = √3/2, cos 60° = ½, tan 60° = √3. Angles of elevation are measured upward from the horizontal, depression downward, and bearings are measured clockwise from north in three figures.
两个恒等式出现频率极高：sin²θ + cos²θ = 1 与 tan θ = sin θ/cos θ。需要背下的精确值有 sin 30° = ½、cos 30° = √3/2、tan 30° = 1/√3、sin 45° = cos 45° = 1/√2、sin 60° = √3/2、cos 60° = ½、tan 60° = √3。仰角自水平线向上量，俯角向下量，方位角则自正北顺时针量，写作三位数。
8. Calculus: Differentiation and Stationary Points | 微积分
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