Parametric Equations — Mark Scheme Patterns

Mark Allocation Overview

Question Type	Total Marks	M marks	A marks	B marks
First derivative $\frac{dy}{dx}$	2–3	1	1–2	0
Second derivative $\frac{d^2y}{dx^2}$	3	1–2	1–2	0
Parametric differentiation combined	5	2	3	0
Arc length	5–6	2–3	3	0

Pattern: First Derivative (2–3 marks)

• M1: Find $\frac{dx}{dt}$ and $\frac{dy}{dt}$ correctly
• A1: Correct $\frac{dy}{dx}$ expression (may be unsimplified)
• A1: Correct simplified form (if required)

常见扣分

若 $\frac{dy}{dx}$ 未简化到最简形式，可能丢 A1。如 $\frac{t^2}{2t}$ 应简化为 $\frac{t}{2}$。

Pattern: Second Derivative (3 marks, often part of a larger question)

• M1: Differentiate $\frac{dy}{dx}$ with respect to $t$
• A1: Correct $\frac{d}{dt}\left(\frac{dy}{dx}\right)$
• A1: Correct $\frac{d^2y}{dx^2}$ (divide by $\frac{dx}{dt}$)

评分标志

M1 仅给到"对 $t$ 求导"这一步。如果直接用 $\frac{\ddot{y}}{\ddot{x}}$ 则不得分——必须展示除以 $\frac{dx}{dt}$ 的步骤。

Pattern: Arc Length (5–6 marks)

• M1: Find $\frac{dx}{dt}$ and $\frac{dy}{dt}$
• M1: Form $\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2$
• A1: Simplify integrand (e.g. $8(1-\cos t)$ or $16\sin^2(t/2)$)
• M1: Set up arc length integral $L = \int \sqrt{(\dot{x}^2 + \dot{y}^2)}\,dt$
• A1: Correct square root simplification
• A1: Correct final answer

评分关键点

步骤	得分点	常见错误
$\frac{dx}{dt}$, $\frac{dy}{dt}$	M1	导数计算错误
平方和	M1	漏项或展开错误
化简	A1	未用半角公式
积分设置	M1	积分变量写错
开方	A1	忽略绝对值
最终答案	A1	数值或代数错误

Follow-Through Rules

• 如果 $\frac{dx}{dt}$ 或 $\frac{dy}{dt}$ 错一个，后续的 $\frac{dy}{dx}$ 可 ft
• 弧长问题中，平方和化简错则后续不得分
• $\frac{d^2y}{dx^2}$ 中，$\frac{dy}{dx}$ 的错可 ft，但方法必须正确