📚 International AS Mathematics 9660-MA01 2017 Mark Scheme: Core Concepts Explained | 国际AS数学9660-MA01 2017评分方案：核心知识点精讲
The 9660-MA01 paper is the Pure Mathematics unit of the Edexcel International AS Mathematics qualification. The 2017 mark scheme (version 2) reveals exactly what examiners are looking for: clear logical steps, correct notation, and precise application of standard techniques. This article unpacks each core topic tested in that session, offering bilingual explanation of the essential methods and marking points. Mastering these will not only help you score full marks on similar questions but also build a solid foundation for A2 study.
9660-MA01 是爱德思国际 AS 数学考试中的纯数学单元。2017 年第二版评分方案明确指出了考官的给分点：清晰的逻辑步骤、正确的符号运用以及标准方法的精准执行。本文深入拆解该次考试涉及的全部核心知识点，用中英双语详解关键方法与评分要点。掌握这些内容，不仅能帮助你在类似题目中拿到满分，更能为 A2 阶段的学习打下扎实基础。
1. Algebraic Manipulation and Surds | 代数运算与根式化简
Surds and rationalising denominators are frequently examined. In the 2017 paper, candidates were required to simplify expressions like √48 + √27 and then rationalise a fraction. The mark scheme rewards intermediate steps, such as expressing each surd in its simplest form first: √48 = √(16×3) = 4√3, √27 = √(9×3) = 3√3, giving 7√3. For rationalising, multiplying numerator and denominator by the conjugate is essential.
根式化简与分母有理化是高频考点。2017 年试卷要求考生化简类似 √48 + √27 的表达式，并对分式进行有理化。评分方案非常看重中间步骤，比如先将每个根式写成最简形式：√48 = √(16×3) = 4√3，√27 = √(9×3) = 3√3，得到 7√3。在进行有理化时，分子分母同乘共轭根式是关键得分步骤。
A common error is stopping at an unsimplified surd. Always check whether the integer inside the root has a square factor. The mark scheme also insists on writing the final answer in the form a + b√c where a, b, c are integers with no common factor.
常见错误是未将根式化简到底。一定要检查根号内的整数是否含有平方因子。评分方案还要求最终答案写成 a + b√c 的形式，其中 a、b、c 为整数且无公因数。
2. Quadratics and the Discriminant | 二次方程与判别式
Quadratic functions appear in many forms. The 2017 mark scheme shows that completing the square, using the discriminant, and solving quadratic inequalities were tested. For example, to prove that a curve is always above the x-axis, you need to show the discriminant b² – 4ac < 0 and that the coefficient of x² is positive.
二次函数以多种形式出现。2017 评分方案显示，配方、判别式以及解二次不等式均为考点。例如，要证明某条曲线恒在 x 轴上方，需要展示判别式 b² – 4ac < 0，且 x² 系数为正。
Discriminant Δ = b² – 4ac
Learners often forget to reference the sign of the leading coefficient. The mark scheme explicitly awards a mark for stating ‘a > 0’ after finding Δ < 0 when proving a quadratic is always positive.
考生经常忘记提及二次项系数的符号。评分方案明确规定，在证明二次函数恒为正时，求出 Δ < 0 后还需明确写出 'a > 0′ 才能获得相应分数。
3. Simultaneous Equations | 联立方程组
Solving linear and quadratic simultaneous equations by substitution is a core skill. The 2017 mark scheme gave credit for rearranging the linear equation for y (or x), substituting into the quadratic, and simplifying to a standard quadratic in one variable. Factorising or using the formula then yields the solutions.
通过代入法求解一次与二次联立方程组是一项核心技能。2017 评分方案对将线性方程改写为 y（或 x）并代入二次式、化简为标准一元二次方程的过程给予步骤分。随后因式分解或用公式法得出解。
Many candidates lose marks by failing to pair the x and y values correctly. The mark scheme usually requires presenting the final answer as coordinate pairs (x₁, y₁) and (x₂, y₂).
许多考生因未正确配对 x 和 y 值而失分。评分方案通常要求最终答案以坐标对 (x₁, y₁) 和 (x₂, y₂) 的形式呈现。
4. Graphs and Transformations | 函数图像与变换
Transformations of graphs, such as translation, stretch, and reflection, are standard. In 2017, candidates were asked to sketch y = f(x – 2) given f(x), or to identify the sequence of transformations that map one curve onto another. The mark scheme accepts descriptions like ‘translation by vector (3, 0)’ or ‘stretch in the y-direction with scale factor 2’.
图像的平移、伸缩、反射等变换是常规题目。2017 年试题要求根据 f(x) 画出 y = f(x – 2) 的草图，或识别将一个曲线映射到另一个的一系列变换。评分方案接受类似“按向量 (3, 0) 平移”或“沿 y 轴方向伸缩，倍数为 2”这样的描述。
Always state the direction and the magnitude. For a horizontal stretch, it is vital to write ‘stretch in the x-direction with scale factor 1/k’, not just ‘stretch’. The mark scheme penalises vague language.
描述时必须说明方向和大小。对于水平伸缩，务必写出“沿 x 轴方向伸缩，倍数为 1/k”，而不是仅仅写“伸缩”。评分方案会扣减模糊表述的分数。
5. Differentiation: Tangents, Normals, Stationary Points | 微分：切线、法线与驻点
Differentiation of polynomials is tested every session. The 2017 paper featured finding the equation of a tangent or normal at a given point, and determining stationary points. The gradient of the curve is dy/dx, evaluated at the x-coordinate. The normal gradient is -1/(dy/dx).
多项式微分每次考试都会出现。2017 年试卷涉及求曲线在某给定点的切线或法线方程，以及确定驻点。曲线斜率由 dy/dx 在给定 x 坐标处的值给出，法线斜率为 -1/(dy/dx)。
For stationary points, set dy/dx = 0 and solve. The mark scheme requires the second derivative test or a sign diagram to classify maxima and minima. Simply stating d²y/dx² > 0 for a minimum earns the final mark.
对于驻点，令 dy/dx = 0 并求解。评分方案要求用二阶导数检验或符号表来区分极大值和极小值。明确指出 d²y/dx² > 0 时为极小值即可拿到最后一分。
6. Integration and Area Under a Curve | 积分与曲线下方面积
Indefinite and definite integration are standard. The 2017 mark scheme looked for accurate integration of terms like 3x² → x³ (+c). For definite integrals, substitution of limits must be shown clearly. When finding the area bounded by a curve and a line, the procedure is: integrate the difference of the functions between intersection points.
不定积分与定积分都是标准考点。2017 评分方案要求准确积分，如 3x² → x³ (+c)。对于定积分，必须清晰展示代入上下限的过程。计算曲线与直线所围成的面积时，步骤应为：先求交点，再对两函数之差在交点区间上积分。
A frequent omission is forgetting the +c in indefinite integration. Though sometimes not penalised in the first part, it may be needed for finding the equation of the curve given a point.
一个常见遗漏是不定积分忘记加 +c。虽然有时第一部分不扣分，但在给定点求曲线方程时，+c 必不可少。
7. Arithmetic Sequences and Series | 等差数列与级数
The 2017 series questions used the formulas for the nth term uₙ = a + (n-1)d and the sum Sₙ = n/2 [2a + (n-1)d]. Candidates were expected to form two equations from given information and solve for a and d simultaneously. Logical working is critical: state the formulas, substitute, simplify, solve.
2017 年关于级数的题目用到了通项公式 uₙ = a + (n-1)d 和求和公式 Sₙ = n/2 [2a + (n-1)d]。考生需要根据已知条件建立两个方程，联立求解 a 和 d。清晰的逻辑推导至关重要：列出公式、代入数值、简化、求解。
The mark scheme penalises misuse of indices, for instance confusing n with n-1. Always double-check which term corresponds to which n-value.
评分方案对索引的误用会扣分，例如混淆 n 与 n-1。务必反复确认所给条件对应的是第几项，n 值是否正确。
8. Binomial Expansion | 二项式展开
Expanding (a + bx)ⁿ for an integer n uses the binomial theorem. In 2017, candidates were asked to expand (2 – 3x)⁵ up to the term in x³, or to find a coefficient when a term is unknown. The mark scheme expects clear use of nCr notation or Pascal’s triangle, with each term simplified correctly.
对整数 n 展开 (a + bx)ⁿ 使用二项式定理。2017 年试题要求展开 (2 – 3x)⁵ 至 x³ 项，或在一项系数未知时求出该项。评分方案期望清晰使用 nCr 符号或帕斯卡三角形，并正确化简每一项。
(r+1)th term = ⁿCᵣ aⁿ⁻ʳ (bx)ʳ
Watch for negative signs: in (2 – 3x)⁵, b = -3, so the powers of -3 must be handled carefully. Missing a sign accounts for many lost marks.
注意负号：在 (2 – 3x)⁵ 中，b = -3，因此 -3 的幂次必须小心处理。符号错误是常见失分点。
9. Trigonometry: Equations and Identities | 三角学：方程与恒等式
Trigonometric equations within a given range are a staple. The 2017 mark scheme tested solving 2sin²θ – cosθ = 1 for 0° ≤ θ ≤ 360°. The method required using sin²θ + cos²θ = 1 to rewrite in terms of a single trig function, factorising, and solving. All solutions within the interval must be given.
在给定区间内求解三角方程是必考题。2017 评分方案考查了在 0° ≤ θ ≤ 360° 范围内解方程 2sin²θ – cosθ = 1。解题方法需要利用 sin²θ + cos²θ = 1 将方程化为仅含一种三角函数的式子，然后因式分解并求解。必须给出区间内的所有解。
The mark scheme deducts marks for missing secondary values. Using a CAST diagram or graph sketches is recommended to ensure no solution is omitted. Also, check the domain mode (degrees or radians) as stated in the question.
评分方案会因遗漏次要值而扣分。建议使用 CAST 图或函数草图以保证解不缺失。同时，注意审清题目要求的单位是角度制还是弧度制。
10. Exponentials and Logarithms | 指数与对数
Laws of logarithms and solving exponential equations were prominent. Typical tasks: solve 3²ˣ = 5, giving the answer in terms of natural logs, or simplify 2logₐx – logₐy. The mark scheme rewards stating the log law explicitly (e.g., logₐ(xⁿ) = nlogₐx).
对数运算律和解指数方程是重点。典型题目：解 3²ˣ = 5，答案用自然对数表示，或化简 2logₐx – logₐy。评分方案对明确写出对数律的步骤给予肯定（例如 logₐ(xⁿ) = nlogₐx）。
When taking logs of both sides, remember that log(ab) = log a + log b, and ln(e) = 1. Many candidates incorrectly apply the laws when terms are added. The mark scheme looks for a correct linear equation in x after applying logs.
在对等式两边取对数时，记住 log(ab) = log a + log b，且 ln(e) = 1。很多考生在加法项上错误使用对数律。评分方案要求取对数后得到关于 x 的正确线性方程。
11. Coordinate Geometry: Circles | 解析几何：圆
Circle geometry questions typically involve finding the centre and radius from an equation, or the equation of a tangent. The 2017 paper asked for the centre of the circle x² + y² – 6x + 4y = 12 by completing the square. The standard form is (x – a)² + (y – b)² = r², with centre (a, b). The tangent gradient is the negative reciprocal of the radius gradient.
圆的几何题目通常要求从方程中找出圆心和半径，或求切线方程。2017 年试卷要求通过配方求出圆 x² + y² – 6x + 4y = 12 的圆心。标准形式为 (x – a)² + (y – b)² = r²，圆心为 (a, b)。切线斜率是半径斜率的负倒数。
Explaining that the normal to the tangent is the radius helps justify the gradient relationship. The mark scheme often awards a method mark for stating ‘gradient of radius = …’, then ‘gradient of tangent = …’.
说明切线的法线即为半径，有助于论证斜率关系。评分方案常对“半径斜率 = … ”和“切线斜率 = … ”的步骤给予方法分。
12. Practical Tips from the 2017 Mark Scheme | 2017 评分方案的实战提分要点
Examiners’ reports for 2017 emphasise that clear, systematic working is more important than elegance. ‘SOLVE’ means show all steps: state the formula, substitute numbers, simplify. Never skip the final answer line. In ‘show that’ questions, you must demonstrate each algebraic manipulation; omitted steps lose the proof mark even if the result is correct. Finally, always answer in the format requested: exact values (in terms of π or surds), to 3 significant figures, or in simplest fraction form.
2017 年的考官报告强调，清晰系统的推导过程比解题技巧的优雅更重要。“求解”意味着展示全部步骤：列出公式、代入数值、化简。切勿省略最终答案行。在“证明……”类题目中，必须展示每一步代数变形；即使结果正确，省略步骤也会失去证明分。最后，务必按照题目要求的格式作答：精确值（含 π 或根式）、保留三位有效数字、或最简分数形式。
Applying these insights while practising past papers will dramatically improve your scores. Treat the mark scheme as a checklist: each bullet point represents a potential mark. If your solution matches the logical flow of the mark scheme, you are on the right track.
在练习历年真题时应用这些心得，将显著提高你的分数。把评分方案当成核对清单：每一个要点都对应一个潜在得分点。如果你的解题步骤与评分方案的逻辑流程一致，就说明走在正确的轨道上。
Published by TutorHao | Pure Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply