📚 AQA AS Maths Unit 1 Mark Scheme (Jan 21): Key Lessons and Revision Guide | AQA AS 数学 Unit 1 评分方案（2021年1月）要点与复习指南
A mark scheme is much more than a set of final answers; it is a detailed record of how AQA examiners award method, accuracy and communication marks in AS Unit 1. Working through the January 2021 Unit 1 mark scheme helps you see exactly where marks are won and lost, especially in pure algebra, coordinate geometry and calculus.
评分方案远不止是一组最终答案；它详细记录了 AQA 考官如何在 AS Unit 1 中给方法分、准确分和表达分。仔细研究 2021 年 1 月 Unit 1 的评分方案，可以帮助你准确看到分数在哪里获得、在哪里丢失，尤其是在纯代数、坐标几何和微积分部分。
1. How the Unit 1 Mark Scheme Works | Unit 1 评分方案如何运作
In AQA AS Mathematics, Unit 1 is a pure-heavy paper that mixes short procedural questions with multi-step problem solving. The mark scheme assigns each mark a code: M for a correct method, A for final accuracy, B for an independent statement, and ft for follow-through from a previous error. This means many marks are awarded for what you show, not just for the final line.
在 AQA AS 数学中，Unit 1 是一份以纯数学为主的试卷，既有简短计算题，也有多步骤应用题。评分方案给每一分标注代码：M 表示正确方法，A 表示最终准确，B 表示独立结论，ft 表示继续沿用前一错误的后续分数。这意味着许多分数是根据你展示的步骤给出的，而不仅仅是最后一行答案。
The January 2021 mark scheme also confirms that AQA uses 'isw' (ignore subsequent working). If you obtain a correct answer and then write extra incorrect steps afterwards, you often keep the mark. However, contradictory working can still lose accuracy marks, so keep your final line clear and consistent.
2021 年 1 月的评分方案还确认 AQA 使用 'isw'（忽略后续计算）。如果你得到正确答案后又写了额外错误步骤，通常仍能保留分数。但是，相互矛盾的计算过程仍可能丢失准确分，因此请保持最后一行清晰且前后一致。
2. Command Words and Mark Allocation | 指令词与分值分配
Different command words tell you what skill is being assessed and how many marks are likely to be attached. The table below summarises typical Unit 1 command words and what the mark scheme expects you to show.
不同的指令词告诉你考查的技能是什么，以及可能附带多少分数。下表总结了 Unit 1 中常见指令词以及评分方案要求你展示的内容。
| Command word | 指令词 | Meaning | 含义 | Mark scheme focus | 评分关注点 |
|---|---|---|
| Simplify | 化简 | Reduce to standard exact form | A marks for correct exact answer; M marks for valid intermediate step |
| Find | 求 | Compute a value or expression | M for method, A for value, often no justification required |
| Show that | 证明 | Verify a given result | Marks for each required manipulation, final A for exact given form |
| Hence / otherwise | 由此 / 其他方法 | Use previous result or an alternative | 'Hence' often has method marks tied to earlier part; 'otherwise' allows alternative route |
When a question says 'give your answer in exact form', the mark scheme will usually reject a rounded decimal. For example, √8 must be written as 2√2, not 2.83, to receive the accuracy mark.
当题目说“以精确形式给出答案”时，评分方案通常会拒绝四舍五入的小数。例如，√8 必须写成 2√2，而不是 2.83，才能获得准确分。
3. Algebra and Indices: Working with Exact Values | 代数与指数：精确值运算
Unit 1 mark schemes consistently reward correct simplification of surds and indices. A typical method mark is given for expanding a bracket such as (√50 + √18)², and an accuracy mark is given for reaching 128 in exact form.
Unit 1 评分方案一贯奖励正确化简根式和指数。例如展开 (√50 + √18)² 这类括号通常给方法分，而准确分则授予得到精确值 128 的解答。
(√50 + √18)² = 50 + 2√900 + 18 = 50 + 60 + 18 = 128
The mark scheme often contains the note 'allow 128 for 2√900 = 60 seen' do . This means you can earn the accuracy mark even if you do not simplify √900 to 30 immediately, as long as you eventually reach the correct total.
评分方案中常出现“若看到 2√900 = 60 则给 128 分”的注释。这意味着即使你没有立即把 √900 化简为 30，只要最终得到正确结果，就可以获得准确分。
For index questions such as simplifying (16a⁴)⁻¹/², the January 2021 style mark scheme awards one mark for applying the negative power and one for the fractional power, leading to 1/(4a²).
对于指数题，如化简 (16a⁴)⁻¹/²，2021 年 1 月风格的评分方案通常给负指数一步 1 分，给分数指数一步 1 分，最终结果为 1/(4a²)。
4. Quadratics and the Discriminant | 二次函数与判别式
Quadratic questions in Unit 1 frequently ask you to complete the square, solve by factorising, or use the discriminant. Mark schemes award method marks for the correct setup and accuracy marks for the final roots or completed-square form.
Unit 1 中的二次函数题经常要求你完成平方、用因式分解求解或使用判别式。评分方案对正确设定给方法分，对最终的根或完成平方形式给准确分。
y gives M1 for taking out the factor 2 correctly, A1 for writing 2[(x − 2)² − 4] + 5, and A1 for the final form 2(x − 2)² − 3.
例如，对 y = 2x² − 8x + 5 完成平方，评分方案通常给正确提取因子 2 的方法分 M1，给写出 2[(x − 2)² − 4] + 5 的准确分 A1，再给最终形式 2(x − 2)² − 3 的准确分 A1。
The discriminant D = b² − 4ac is a high-frequency topic. A question such as 'Find the values of k for which x² + kx + 9 = 0 has equal roots' ed as is marked as follows: M1 for setting D = 0, A1 for k² − 36 = 0, and A1 for both k = 6 and k = −6.
判别式 D = b² − 4ac 是高频考点。例如“求使 x² + kx + 9 = 0 有相等实根的 k 值”这类题按以下方式给分：设 D = 0 得 M1，得到 k² − 36 = 0 得 A1，得到 k = 6 和 k = −6 两个值得 A1。
D = k² − 4(1)(9) = k² − 36 = 0 → k = ±6
es both values for the final A mark, so always check for ± roots in discriminant problems.
常见的准确分丢失发生在学生只写 k = 6 而忘记 k = −6 时。评分方案明确规定两个值都要写出才能拿到最后的 A 分，因此判别式问题中一定要检查 ± 根。
5. Coordinate Geometry and Circles | 坐标几何与圆
Circle questions in AS Unit 1 often require you to find the centre and radius from an expanded equation using completing the square. The mark scheme splits marks between rearranging terms, completing each square, and identifying the centre and radius correctly.
AS Unit 1 的圆方程题通常要求你通过完成平方从展开式中求出圆心和半径。评分方案将分数分配在移项、完成各部分平方以及正确识别圆心和半径上。
olution shows the two completed squares, the standard form, and the final centre (−3, 2) with radius 5.
对于 x² + y² + 6x − 4y − 12 = 0，满分解答应展示两个完成平方的步骤、标准形式，以及最终圆心 (−3, 2) 和半径 5。
(x + 3)² − 9 + (y − 2)² − 4 − 12 = 0 → (x + 3)² + (y − 2)² = 25
The mark scheme also rewards correct use of the tangent condition: a tangent to a circle is perpendicular to the radius at the point of contact. If the radius gradient is m, the tangent gradient is −1/m, and using this relationship usually earns a method mark.
评分方案还奖励正确使用切线与半径垂直这一条件：切线与圆在切点处的半径互相垂直。如果半径的斜率为 m，那么切线斜率为 −1/m，使用这一关系通常能获得方法分。
Coordinate geometry questions involving two points often have a built-in follow-through mark. If you misread one coordinate but apply the correct distance or midpoint formula, the mark scheme may allow ft marks for subsequent parts.
涉及两点的坐标几何题通常设有跟随分。如果你读错了一个坐标，但后续正确应用了距离或中点公式，评分方案可能会给后续部分 ft 分。
6. Differentiation: Tangents, Normals and Stationary Points | 微分：切线、法线与驻点
Differentiation is one of the most mark-rich areas in Unit 1. The mark scheme often follows a three-step pattern: differentiate, substitute the given x-value, and use the gradient to form a line equation or identify a stationary point.
微分是 Unit 1 中分值最高的部分之一。评分方案通常遵循三步：求导、代入给定的 x 值、利用梯度建立直线方程或判断驻点。
1. The mark scheme gives M1 for differentiating correctly, M1 for setting the derivative to zero, and A1 for both x-values; the corresponding y-values then gain further accuracy marks.
对于 y = x³ − 3x + 2，其导数为 dy/dx = 3x² − 3。令 dy/dx = 0 得 x = ±1。评分方案给正确求导 M1，令导数为零 M1，得到两个 x 值得 A1；对应的 y 值再获得进一步准确分。
scheme expects the derivative 2x, gradient 4, point (2, 5), and final equation y = 4x − 3.
切线和法线题也很常见。要求 y = x² + 1 在 x = 2 处的切线，评分方案期望写出导数 2x、梯度 4、点 (2, 5) 以及最终方程 y = 4x − 3。
dy/dx = 2x → gradient at x = 2 is 4 → y − 5 = 4(x − 2) → y = 4x − 3
A very common mark scheme trap is forgetting to find the y-coordinate before writing the line equation. Candidates who find the correct gradient but use an incorrect y-value often lose the final A mark, so always substitute the given x into the original equation first.
一个非常常见的评分陷阱是在写直线方程前忘记求 y 坐标。如果考生找到正确梯度但使用了错误的 y 值，通常会丢失最后的 A 分，因此务必先把给定 x 代入原方程求出 y。
7. Integration and Area Under a Curve | 积分与曲线下面积
Integration questions in Unit 1 usually begin with an indefinite integral. The mark scheme awards M1 for applying the power rule correctly, A1 for each correctly integrated term, and an additional A1 for including the constant '+c'.
Unit 1 中的积分题通常从不定积分开始。评分方案对正确应用幂法则给 M1，对每个正确积分的项给 A1，对写出常数 '+c' 再给一个 A1。
For ∫(4x³ − 2x) dx, a full-mark solution is x⁴ − x² + c. Omitting '+c' l accuracy mark in an indefinite integral, even if the rest of the integration is perfect.
对于 ∫(4x³ − 2x) dx，满分解为 x⁴ − x² + c。在不定积分中漏写 '+c' 会丢失最后的准确分，即使其余积分完全正确。
∫(4x³ − 2x) dx = x⁴ − x² + c
For area under a curve, the January 2021 style mark scheme expects you to integrate between the given limits, substitute both limits carefully, and subtract the lower value from the upper value. A sign error in substitution is a common cause of lost accuracy marks.
对于曲线下面积，2021 年 1 月风格的评分方案期望你在给定上下限之间积分，仔细代入两个极限，并用上限值减去下限值。代入时符号错误是丢失准确分的常见原因。
If the curve crosses the x-axis, the mark scheme typically requires separate integrals for the positive and negative regions, because a single integral may give a zero or incorrect net area. State clearly which regions you are combining.
如果曲线穿过 x 轴，评分方案通常要求对正区域和负区域分别积分，因为一个积分可能得到零或错误的净面积。请清楚说明你在合并哪些区域。
8. Trigonometry: Exact Values and Identities | 三角学：精确值与恒等式
Exact trigonometric values are tested regularly in AS Unit 1. The mark scheme requires the exact surd or fractional form for 30°, 45° and 60°; rounded decimals do not gain accuracy marks.
精确三角值在 AS Unit 1 中经常考查。评分方案要求 30°、45° 和 60° 的精确根式或分数形式；四舍五入的小数不能获得准确分。
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