AS Further Mathematics Unit 2 June 2019 Mark Scheme Breakdown | AS进阶数学单元2 2019年6月评分标准题型解析

📚 AS Further Mathematics Unit 2 June 2019 Mark Scheme Breakdown | AS进阶数学单元2 2019年6月评分标准题型解析

Understanding the mark scheme is crucial for maximising your score in AS Further Mathematics Unit 2. The June 2019 paper covered key topics such as complex numbers, matrices, series, hyperbolic functions, vectors, and differential equations. This article breaks down the question types and how marks were awarded, enabling you to focus on the essential steps that examiners look for.

理解评分标准对在 AS 进阶数学单元 2 中拿高分至关重要。2019 年 6 月的试卷涵盖了复数、矩阵、级数、双曲函数、向量和微分方程等重点内容。本文解析题型与得分要点，帮助你把握考官看重的关键步骤。

1. Complex Numbers: Arithmetic and Conjugates | 复数：运算与共轭

Questions on complex numbers often begin with finding the sum, product, or quotient of two given complex numbers. Marks are awarded for correct real and imaginary parts and for multiplying by the conjugate when dividing.

复数题目通常先要求计算两个给定复数的和、积或商。得分点包括正确的实部和虚部，以及在做除法时乘以共轭。

For example, to divide (3 + 4i) / (1 – 2i), multiply numerator and denominator by (1 + 2i). Simplifying to the form a + bi earns full method marks. Any mistake in sign or imaginary part can lose an accuracy mark. Mark schemes often require the final answer to be expressed with a single real and single imaginary term, so always separate them clearly.

例如，计算 (3 + 4i)/(1 – 2i) 时，分子分母同乘 (1 + 2i)。化简成 a + bi 形式可得全部分数；如果符号或虚部出错，会失去准确性分。评分标准常要求最终答案以单个实部和单个虚部的形式给出，因此要清晰分离。

Beware of replacing i² with -1 correctly. In longer problems, like solving quadratic equations with real coefficients, complex roots must be given as conjugate pairs. The mark scheme often gives a method mark for using the quadratic formula and an accuracy mark for writing the roots correctly.

注意把 i² 正确替换为 -1。在更复杂的问题中，如解实系数二次方程，复根必须以共轭对形式给出。评分标准通常对使用求根公式给方法分，对写出正确的根给准确性分。

2. Argand Diagrams and Loci | 阿尔冈图与轨迹

Loci questions require sketching a circle or perpendicular bisector on an Argand diagram. The mark scheme gives marks for identifying the centre and radius for |z – a| = r, or drawing the correct line for |z – a| = |z – b|. Always label the diagram with the relevant complex numbers or points.

轨迹题要求在阿尔冈图中画出圆或垂直平分线。评分标准为正确指出 |z – a| = r 的圆心和半径，或画出 |z – a| = |z – b| 的直线给分。务必在图上标出相关复数的点。

When shading a region, marks are allocated for clearly indicating the boundary (solid or dashed) and shading the correct side. For example, the region |z – 3| ≤ 2 requires a solid circle and shading inside the circle. The mark scheme penalises missing the boundary distinction or shading the wrong area.

阴影区域题，边界（实线或虚线）分明且阴影正确才得分。例如区域 |z – 3| ≤ 2 需要用实线圈并阴影在圆内。评分方案会扣掉边界区分缺失或阴影区域错误的分。

Intersection of loci or half-lines must also be handled precisely. The examiners look for clear construction lines and the correct point of intersection marked with a small cross or dot.

轨迹或半射线的交点也需精确处理。考官希望看到清晰的作图线，并在正确交点处标记小十字或圆点。

3. Matrix Operations and Determinant | 矩阵运算与行列式

Matrix multiplication questions reward correct multiplication and addition of elements. The determinant of a 2×2 matrix is often part of finding an inverse. In June 2019, credit was given for showing the formula det(M) = ad – bc and substituting correctly. If the matrix is singular (det = 0), the mark scheme expects you to state that no inverse exists and may ask for the geometric implication, such as a transformation that maps the plane onto a line.

矩阵乘法题目要求正确相乘并相加各元素。2×2 矩阵的行列式常是求逆的一部分。2019 年 6 月的评分中，写出 det(M) = ad – bc 并正确代入即可得分。若矩阵奇异（行列式为零），评分标准要求你指出逆不存在，可能还要求说明几何意义，例如变换将平面映射到一条直线。

For the inverse, you must write 1/det(M) multiplied by the adjugate matrix. Marks are split between determinant, forming the adjugate, and final simplification. Leaving the inverse as a scalar multiple of a matrix with fractions is fine, but simplifying to a single matrix without common fractions often yields the accuracy mark.

求逆矩阵时，先写出行列式倒数乘伴随矩阵。得分点分布在行列式计算、构造伴随和最终化简上。允许答案保留为标量乘矩阵的形式，但化简为不含公分母的单个矩阵常能拿到准确性分。

When using matrices to solve simultaneous equations, the mark scheme grants method marks for writing the system in matrix form AX = B and then attempting X = A⁻¹B. Careful substitution of values avoids sign errors.

用矩阵解联立方程组

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