📚 A-Level Further Maths Command Words | A-Level 进阶数学指令词精讲
Command words are the instructional terms used in A-Level Further Mathematics examination questions. They tell you exactly what the examiner expects you to do. Many marks are lost each year because students misinterpret these words and give incomplete or irrelevant answers. This article explains the most important command words you will meet in topics such as complex numbers, matrices, hyperbolic functions, polar coordinates, differential equations and proof. Understanding each term clearly will help you present your solutions in the way that gains full marks.
指令词（command words）是 A-Level 进阶数学试题中使用的指导性词汇，它们准确地告诉你考官希望你完成什么任务。每年都有很多考生因为误解这些单词，给出了不完整或无关的答案而失分。本文将解释你在复数、矩阵、双曲函数、极坐标、微分方程和证明等题目中会遇到的最重要的指令词。清晰地理解每一个术语，将帮助你以能够获得满分的方式呈现解答。
1. State / Write Down | 写出 / 直接写出
These command words require little or no working. You are expected to provide the answer directly, often from memory or by simple inspection. For example, you might be asked to ‘State the definition of cosh x’ or ‘Write down the polar form of the complex number 1 + i√3’. No intermediate steps need to be shown, but you must be precise.
这些指令词几乎不需要或完全不需要书写步骤。你需要直接给出答案，通常凭借记忆或简单的观察就能得到。例如，可能要求你“写出 cosh x 的定义”或者“直接写出复数 1 + i√3 的极坐标形式”。不需要展示中间步骤，但答案必须精确。
In Further Maths, ‘State’ often appears in the first part of a question to test basic knowledge, such as stating Maclaurin series or standard matrix transformations. Even though no working is required, a careless slip can cost you a mark that should have been guaranteed.
在进阶数学中，“写出”常出现在题目的第一部分，用于测试基础知识，比如写出麦克劳林级数或标准矩阵变换。虽然不要求写过程，但一次粗心的小错就可能让你丢掉本应稳拿的分数。
2. Find / Determine | 求出 / 确定
‘Find’ and ‘Determine’ are two of the most common command words. They require you to work out a value, expression or result, usually showing enough steps to justify your answer. In many cases, the method carries marks, so you should present a logical sequence of reasoning. For instance, ‘Find the general solution of the differential equation d²y/dx² − 4y = 0’ expects you to write down the auxiliary equation, find its roots and construct the complementary function.
“求出”和“确定”是最常见的指令词之一。它们要求你计算出某个值、表达式或结果，通常需要展示足够的步骤来支持你的答案。在很多情况下，解题方法占有分值，所以你应当展示一个逻辑严密的推理过程。例如，“求出微分方程 d²y/dx² − 4y = 0 的通解”就期望你写出辅助方程，求出根并构造余函数。
When an examiner asks you to ‘determine’ something in Further Maths, they often want you to make a decision based on given conditions – such as determining whether a matrix is singular, or determining the range of values for which an improper integral converges. Always show how you arrive at your conclusion.
当考官要求你“确定”某个结论时，他们通常希望你基于给定条件做出判断——例如确定一个矩阵是否是奇异矩阵，或确定一个反常积分收敛时参数的范围。一定要展示你是如何得出结论的。
3. Calculate / Evaluate | 计算 / 求值
These words indicate that a numerical or exact symbolic result is required after performing mathematical operations. ‘Calculate’ is often used when a sequence of arithmetic or algebraic steps leads to a single value, e.g. ‘Calculate the product of the matrices A and B’. ‘Evaluate’ suggests that you must substitute numbers or expressions into a formula and simplify, such as ‘Evaluate ∫₀¹ x eˣ dx’. Both expect you to show the key steps of the calculation.
这类词表示，经过数学运算后，你需要得出一个数值的或精确的符号化结果。“计算”常用于一系列算术或代数步骤最终得出单个值的情况，例如“计算矩阵 A 与 B 的乘积”。“求值”则暗示你需要将数字或表达式代入公式并化简，比如“求定积分 ∫₀¹ x eˣ dx 的值”。两者都期望你展示关键的计算步骤。
In topics like further calculus and numerical methods, ‘evaluate’ might refer to evaluating a limit or the sum of a series. Where an exact answer is required, leaving an answer in a simplified surd or logarithmic form is often expected, rather than a decimal approximation.
在进阶微积分和数值方法等主题中，“求值”可能指的是求极限或求级数的和。当要求精确答案时，通常希望答案以化简的根式或对数形式呈现，而不是小数近似值。
4. Prove / Show That | 证明 / 求证
‘Prove’ and ‘Show that’ are central to Further Mathematics and test your ability to construct rigorous arguments. When you ‘prove’ something, you derive the result from axioms, definitions or previously established theorems. ‘Show that’ typically gives you the answer in the question – your job is to provide a chain of reasoning that leads convincingly to that given statement. For example, ‘Show that cosh² x − sinh² x = 1’ requires you to start from the exponential definitions and simplify algebraically.
“证明”和“求证”是进阶数学的核心，考查你构建严格论证的能力。当你“证明”某个结论时，你需要从公理、定义或之前确立的定理推导出结果。“求证”则通常在题目中给出了答案，你的任务是提供一条令人信服的推理链，最终到达那个给定的陈述。例如，“求证 cosh² x − sinh² x = 1”要求你从指数形式的定义出发进行代数化简。
In proof questions, every line must be logically connected to the previous one. Using the words ‘hence’, ‘since’ and ‘therefore’ helps make your reasoning clear. Marks are often awarded for a clear structure, not just the final line. Induction, contradiction and direct proof are the main methods encountered.
在证明题中，每一行都必须与前一行有逻辑联系。使用“因此”、“因为”和“所以”可以使你的推理更清晰。分值往往分配给清晰的结构，而不仅仅给最后一行。数学归纳法、反证法和直接证明是常见的证明方法。
5. Hence / Hence or Otherwise | 因此 / 因而或用其他方法
‘Hence’ tells you that you must use the result you have just obtained in the previous part of the question. This is extremely common in multi-step Further Maths problems. If the question says ‘Hence find the exact value of …’, you are expected to use the preceding expression to make the calculation easier, rather than starting from scratch. Ignoring ‘hence’ and using an alternative method may cost you marks, even if the final answer is correct, because you have not followed the intended pathway.
“因此”要求你必须使用刚刚在上一小问中获得的结果。这在多步的进阶数学题中极为常见。如果题目说“因此求……的精确值”，你就应该用之前的表达式使计算更简单，而不是从头开始。忽视“因此”而使用其它方法可能会导致失分，即便最终答案正确，因为你没有遵循预设的解题路径。
‘Hence or otherwise’ gives you more freedom: you can use the previous result if it helps, but you are also allowed to use any valid alternative method. However, the ‘hence’ route is often designed to be much quicker, so always try to see the connection first.
“因而或用其他方法”给了你更多自由：如果前面的结果有用，你可以使用它，但也允许你使用任何有效的其他方法。然而，“因而”的路径往往设计得更加快捷，所以总是先尝试找出其中的联系。
Example:
Given that z = 1 − i is a root of z² + pz + q = 0, find p and q.
Hence find the other root.
示例：
已知 z = 1 − i 是 z² + pz + q = 0 的一个根，求 p 和 q。
因此求出另一个根。
6. Solve | 求解
‘Solve’ means you must find the value(s) of the variable that satisfy a given equation or inequality. This command appears in virtually every topic: solving polynomial equations, systems of linear equations using matrices, differential equations, trigonometric and hyperbolic equations. Always give your answers in the required form, e.g. general solution, principal values, interval-based solutions, or exact values involving π.
“求解”意味着你必须找出满足给定方程或不等式的变量值。这个指令几乎出现在每一个主题中：求解多项式方程、用矩阵解线性方程组、解微分方程、三角方程和双曲函数方程。始终按照要求的形式给出答案，例如通解、主值解、区间解或含有 π 的精确值。
In Further Maths, solving a second-order differential equation will involve finding both the complementary function and the particular integral. The command ‘solve’ implies you should combine these into the full general solution. In matrix algebra, ‘solve’ might mean finding the inverse matrix and multiplying, or using row operations. Be systematic.
在进阶数学中，求解一个二阶微分方程既包含求出余函数，也包含求出特解。指令“求解”意味着你应当将二者组合成完整的通解。在矩阵代数中，“求解”可能意味着求逆矩阵再相乘，或者使用行变换。解题要系统化。
7. Sketch / Draw | 绘制草图 / 绘制
‘Sketch’ requires a diagram that shows the main features of a graph or geometric figure without necessarily plotting points accurately to scale. You should label intercepts, asymptotes, turning points and any other key coordinates. In polar coordinates, for instance, ‘Sketch the curve r = 2 + cos θ’ expects you to identify symmetry, maximum and minimum r values, and the general shape. A sketch does not need to be a perfect artwork, but it must be clear and correctly proportioned.
“绘制草图”要求你画出一个能够展示图形或几何体主要特征的示意图，而不必精确按比例标出所有点。你应当标出截距、渐近线、转折点及其他关键坐标。例如，在极坐标中，“绘制曲线 r = 2 + cos θ 的草图”期望你识别对称性、r 的最大最小值以及整体形状。草图不需要是一件完美的艺术品，但必须清晰且比例大致正确。
‘Draw’ may sometimes imply a more accurate construction, potentially using a ruler and compass, but in the context of an A-Level exam, ‘draw’ and ‘sketch’ are often used interchangeably for graphs. However, for Argand diagrams and loci, precision in labelling axes and regions matters greatly.
“绘制” 有时可能指更精确的作图，或许要使用尺规，但在 A-Level 考试语境中，对于图形而言，“绘制”和“绘制草图”常常可以互换。不过对于阿尔冈图和轨迹，精确标注坐标轴和区域非常重要。
8. Explain / Describe | 解释 / 描述
These words require you to write a short piece of prose (usually one or two sentences) to demonstrate your understanding of a concept or your interpretation of a result. In Further Maths, you might be asked to ‘Explain why the matrix is non-invertible’ or ‘Describe the transformation represented by M’. Your answer should use precise mathematical language, referring to the determinant, rank, nullity, or geometric transformations such as rotations, reflections and stretches.
这类词要求你写一小段文字（通常一两句话），以展示你对某个概念的理解或对某个结果的解释。在进阶数学中，你可能会被要求“解释为什么该矩阵不可逆”或“描述矩阵 M 所代表的变换”。你的答案应当使用精确的数学语言，涉及行列式、秩、零度或几何变换，如旋转、反射和拉伸。
Good explanations often link symbolic results to physical or geometric interpretations. For example, when explaining why a particular solution is extraneous, you could refer to the domain of the original function or the context of the problem. Always aim for clarity and conciseness.
好的解释常常将符号化结果与几何或现实意义联系起来。例如，在解释为什么某个解是增根时，你可以提到原函数的定义域或问题背景。力求清晰简洁。
9. Verify | 验证
‘Verify’ means you need to check that a given value, expression or statement is true by substituting it into an equation or by performing a logical check. Unlike ‘prove’, verification usually starts with a candidate answer and confirms it satisfies the original conditions. For instance, ‘Verify that y = e²ˣ sin x is a solution of d²y/dx² − 4 dy/dx + 5y = 0’ requires you to differentiate the given function twice, substitute and simplify to show the left-hand side equals zero.
“验证”意味着你需要通过代入方程或执行逻辑检查，来确认某个给定值、表达式或陈述是正确的。与“证明”不同，验证通常从一个候选答案开始，确认它满足原始条件。例如，“验证 y = e²ˣ sin x 是方程 d²y/dx² − 4 dy/dx + 5y = 0 的一个解”，要求你对给定函数进行两次微分，代入并化简，以展示等号左边等于零。
This command is particularly common in differential equations and in series approximations, where you verify that a truncated Taylor polynomial approximates a function up to a certain error. Always write down the substitution clearly. If the two sides simplify to the same expression, you have verified it.
这个指令在微分方程和级数逼近中特别常见，例如验证一个截断的泰勒多项式在某个误差范围内逼近一个函数。每次都要清楚地写出代入过程。如果两边化简为相同的表达式，你就完成了验证。
10. Express … in the form … | 将……表示为……形式
‘Express … in the form …’ is an instruction to rewrite a given mathematical expression in a specific structure. This frequently occurs with complex numbers (e.g. ‘Express −1 + i√3 in modulus-argument form’), hyperbolic functions (e.g. ‘Express 2 sinh x + 3 cosh x in exponential form’), and trigonometric combinations (e.g. ‘Express 3 sin θ + 4 cos θ in the form R sin(θ + α)’). The key is to follow the target format exactly and give all constants in exact form if required.
“将……表示为……形式”是指把给定的数学表达式重写成某种特定结构。这常见于复数（例如“将 −1 + i√3 表示为模-辐角形式”），双曲函数（例如“将 2 sinh x + 3 cosh x 表示为指数形式”），以及三角组合（例如“将 3 sin θ + 4 cos θ 表示为 R sin(θ + α) 的形式”）。关键是要严格遵循目标格式，并且在需要时以精确值给出所有常数。
When a question asks you to express something in partial fractions, you must write it as a sum of simpler rational functions, clearly stating the values of the constants you have found. Marks are awarded for correct algebraic decomposition and for stating the result clearly.
当题目要求你将一个有理式表示为部分分式时，你必须将其写成简单有理函数之和，并清晰地给出你所求出的常数。正确的代数分解以及清晰地陈述结果都会得分。
11. Deduce / Hence Deduce | 推断 / 从而推断
‘Deduce’ signals that you must reach a conclusion by reasoning from known facts or results already established. It is very similar to ‘hence’, but often places more emphasis on logical implication rather than direct algebraic substitution. For example, after proving that the derivative of tanh x is sech² x, a question may ask ‘Deduce the integral of sech² x dx’. You are expected to recognise the inverse relationship between differentiation and integration.
“推断”表示你必须从已知事实或已确立的结果出发，通过推理得出结论。它与“因此”非常相似，但往往更强调逻辑蕴含，而不是直接的代数代入。例如，在证明了 tanh x 的导数为 sech² x 之后，题目可能会问“推断 sech² x dx 的积分”。此时你需要识别微分与积分之间的互逆关系。
In more advanced contexts, you might deduce properties of a conic section from the eccentricity, or deduce the range of stability for a numerical method from the error analysis. Always cite the premise you are using as your starting point.
在更高级的语境中，你可能需要从离心率推断圆锥曲线的性质，或从误差分析推断数值方法的稳定范围。一定要引用你用作起点的前提条件。
12. Use … to … | 用……来……
This command explicitly instructs you which method or tool must be applied. For instance, ‘Use the substitution u = √x to evaluate the integral’ or ‘Use proof by induction to show that …’. When you see ‘Use … to …’, you must follow the specified technique; even if another method is more familiar to you, the marks are tied to the examiner’s chosen approach. The working must clearly reference the given substitution, formula or theorem.
这个指令明确指示你必须应用哪一种方法或工具。例如，“用换元 u = √x 计算该积分”或者“用数学归纳法证明……”。当你看到“用……来……”的时候，必须遵循指定的技巧；哪怕你更熟悉另一种方法，分数也只与考官的指定方法挂钩。解答过程必须清晰地体现所给的换元、公式或定理。
This command is especially frequent in numerical methods (e.g. ‘Use the Newton-Raphson method with initial value x₀ = 1.5 to find the root correct to 2 decimal places’) and in complex number problems where De Moivre’s theorem or a specific form is required. Make sure every step reflects the prescribed method.
这个指令在数值方法中尤其常见（例如“用牛顿-拉夫森方法，以初始值 x₀ = 1.5 求根，精确到两位小数”），也见于要求使用棣莫弗定理或指定形式的复数问题中。确保每一步都反映所规定的方法。
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