AS Maths Unit 2 Mark Scheme Jan21: Essential Topic Review | AS数学单元2评分标准(2021年1月)知识点精讲

📚 AS Maths Unit 2 Mark Scheme Jan21: Essential Topic Review | AS数学单元2评分标准(2021年1月)知识点精讲

This article provides a detailed breakdown of the key topics covered in the AS Mathematics Unit 2 mark scheme for the January 2021 examination series. By understanding the marking points including method marks (M1), accuracy marks (A1) and follow-through marks (ft), students can develop effective exam strategies and avoid common errors. Each section below highlights a core topic, typical question styles, and how marks are allocated, ensuring you know exactly what examiners expect.

本文详细解析了2021年1月AS数学单元2评分标准中涵盖的核心知识点。通过理解方法分(M1)、答案分(A1)和后续标记(ft)等评分点，学生可以制定有效的应考策略并避免常见错误。以下每个小节重点介绍一个核心主题、典型题型以及分数分配方式，让你确切了解考官的期望。

1. Differentiation of Polynomials and Trigonometric Functions | 多项式与三角函数的求导

In many Unit 2 papers, you are asked to differentiate expressions like 3x⁴ − 5x² + 7x − 2 or y = 2 sin x + cos 2x. The mark scheme awards M1 for a correct attempt to use the power rule or the chain rule, and A1 for a fully correct derivative. Remember that d/dx (sin kx) = k cos kx and d/dx (cos kx) = −k sin kx. When simplifying, ensure each coefficient is accurate; a single sign error or missing factor of k will lose an A1 mark.

在许多单元2试卷中，你会被要求对类似3x⁴ − 5x² + 7x − 2 或 y = 2 sin x + cos 2x 的表达式求导。评分标准中，正确使用幂法则或链式法则可获得M1，完全正确的导数得A1。记住 d/dx (sin kx) = k cos kx, d/dx (cos kx) = −k sin kx。化简时务必保证每个系数准确；一个符号错误或遗漏的因子k都会导致A1扣分。

For example, differentiating y = 4x³ − 3 sin 2x yields dy/dx = 12x² − 6 cos 2x. The mark scheme often separates the derivative of the polynomial part and the trig part: M1 for reducing the power correctly, M1 for applying chain rule to sin 2x, A1 for 12x² − 6 cos 2x. Simplifying constants like 3 × 2 is crucial.

例如，对 y = 4x³ − 3 sin 2x 求导得到 dy/dx = 12x² − 6 cos 2x。评分标准通常将多项式部分和三角部分分开：正确降幂得M1，对 sin 2x 应用链式法则得M1，得到 12x² − 6 cos 2x 得A1。正确计算常数3×2至关重要。

2. Integration and Area Under a Curve | 积分与曲线下的面积

Integration questions ask you to find ∫(axⁿ + b sin x) dx or the area bounded by a curve and the x-axis. The mark scheme rewards M1 for integrating each term correctly (power rule +1 for n ≠ −1), and A1 for the correct integrated expression including the constant or the correct use of limits. When finding an area, you must evaluate the definite integral correctly; failure to substitute limits or a sign error in subtraction leads to loss of A1.

积分题会要求你求 ∫(axⁿ + b sin x) dx 或曲线与x轴围成的面积。评分标准对每个项正确积分（幂法则 n ≠ −1 时指数加1）给M1，正确积分表达式（包含常数或正确使用上下限）给A1。在求面积时，必须正确计算定积分；没有代入上下限或减法中的符号错误会导致A1扣分。

A typical mark scheme for ∫₀^{π/2} (2x + cos x) dx allocates M1 for writing x² + sin x, M1 for substituting the limits, and A1 for the final area value like (π²/4 + 1). Be careful: if you integrate cos x as −sin x, you will lose M1. Also, the mark scheme often gives a separate M1 for setting up the area as the integral of the function with the correct limits.

对于 ∫₀^{π/2} (2x + cos x) dx 的典型评分方案，写出 x² + sin x 得M1，代入上下限得M1，最终面积值如 (π²/4 + 1) 得A1。注意：如果将 cos x 积分成 −sin x，就会失去M1。此外，评分方案通常会给设置正确上下限的积分式单独的M1。

3. Solving Trigonometric Equations | 解三角方程

Trigonometric equations in Unit 2 often require you to solve equations like 3 sin θ = 2 cos 2θ for 0° ≤ θ ≤ 360°. The mark scheme awards M1 for using a correct identity (e.g., cos 2θ = 1 − 2 sin² θ), M1 for solving the resulting quadratic in sin θ, and A1 for each correct solution within the given interval. Extra solutions outside the range or missing solutions from the quadratic will lose marks.

单元2中的三角方程通常要求解如 3 sin θ = 2 cos 2θ ，其中 0° ≤ θ ≤ 360°。评分标准中，正确使用恒等式（如 cos 2θ = 1 − 2 sin² θ）得M1，解出关于 sin θ 的二次方程得M1，每个在给定区间内正确的解得A1。超出范围的额外解或遗漏二次方程的解都会扣分。

For example, solving 2 sin² θ + 3 sin θ − 2 = 0 yields (2 sin θ − 1)(sin θ + 2) = 0, giving sin θ = 1/2. The M1 is for factorising or using the quadratic formula. Then θ = 30°, 150° earns A1 each. The mark scheme also expects you to reject sin θ = −2 as no solution, and often awards an A1 for stating the reason or automatically not giving marks for that root. Always check the interval carefully; sin θ = 1/2 gives 30°, 150°.

例如，解 2 sin² θ + 3 sin θ − 2 = 0 得到 (2 sin θ − 1)(sin θ + 2) = 0，从而 sin θ = 1/2。因式分解或用求根公式得M1。然后 θ = 30°, 150° 各得A1。评分标准也要求你舍去 sin θ = −2 无解，通常会对说明理由给A1或自动不给该根分数。务必仔细检查区间；对于 sin θ = 1/2，解为30°, 150°。

4. Exponential and Logarithmic Equations | 指数与对数方程

Questions on exponentials and logs require you to solve equations like 2e²ˣ − 5eˣ + 2 = 0 or 3 ln x = 6 − ln(2x). The mark scheme emphasises correct application of log rules and algebraic manipulation. M1 is given for converting to a standard form (e.g., letting y = eˣ) and forming a quadratic, or for correctly combining logs. A1 marks follow for correct simplified solutions. Always check that the final answers satisfy the original domain, especially for log equations.

指数与对数题目要求解如 2e²ˣ − 5eˣ + 2 = 0 或 3 ln x = 6 − ln(2x) 的方程。评分标准强调正确应用对数和代数运算规则。转化为标准形式（例如设 y = eˣ）并构成二次方程，或正确合并对数可得M1。得出正确简单解答可得A1。务必检查最终答案是否满足原定义域，特别是对数方程。

For 2e²ˣ − 5eˣ + 2 = 0, substitute y = eˣ to get 2y² − 5y + 2 = 0 → (2y − 1)(y − 2) = 0 → y = 1/2 or y = 2. Then x = ln(1/2) = −ln 2, and x = ln 2. The mark scheme may award M1 for substitution and quadratic, M1 for solving quadratic, A1 for each correct x. Note: if a candidate gives x = ln 0.5, that is also acceptable.

对于 2e²ˣ − 5eˣ + 2 = 0，设 y = eˣ 得 2y² − 5y + 2 = 0 → (2y − 1)(y − 2) = 0 → y = 1/2 或 y = 2。那么 x = ln(1/2) = −ln 2,

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