📚 Edexcel International GCSE (9-1) Mathematics A Student Book 2: Core Topics and Exam Skills | Edexcel 国际 GCSE (9-1) 数学 A 学生用书 2：核心主题与应试技巧
This revision guide summarises the key areas covered in Edexcel International GCSE (9-1) Mathematics A Student Book 2. It focuses on higher-tier skills and problem-solving techniques needed for the written papers, especially the more challenging topics such as algebraic fractions, functions, circle theorems, vectors, and advanced trigonometry. Work through the summaries and examples here alongside your class notes and past papers to build confidence and accuracy.
本复习指南总结了 Edexcel 国际 GCSE (9-1) 数学 A 学生用书 2 涵盖的核心内容。它重点关注笔试中所需的高级技能与问题解决技巧，特别是代数分式、函数、圆定理、向量和高级三角学等较难主题。请将本文中的总结与示例和课堂笔记及历年真题结合使用，以增强信心并提高准确性。
1. Surds and Exact Values | 根式与精确值
Surds are irrational roots such as √2 or 3√5. You must be able to simplify expressions like √50 = 5√2 and to rationalise denominators. To rationalise 1/√3, multiply numerator and denominator by √3 to get √3/3. For expressions such as 2/(1 + √5), multiply by the conjugate (1 − √5) to remove the surd from the denominator.
根式是指如 √2 或 3√5 这样的无理根式。你需要能够化简诸如 √50 = 5√2 的式子，并会将分母有理化。要把 1/√3 有理化，请将分子与分母同时乘以 √3，得到 √3/3。对于 2/(1 + √5) 这类表达式，请乘以其共轭式 (1 − √5)，以去掉分母中的根号。
ct trig values are often tested without a calculator. Remember that sin 30° = ½, cos 30° = √3/2, tan 45° = 1, and sin 60° = √3/2. You should also know the values for 0°, 90°, and 45° in both degrees and radians where appropriate.
精确三角函数值常在无计算器的题目中考查。请记住 sin 30° = ½，cos 30° = √3/2，tan 45° = 1，以及 sin 60° = √3/2。你还应掌握 0°、90° 和 45° 的值，并在需要时掌握弧度制下的对应值。
- √a × √b = √ab
- √a ÷ √b = √a/b when b ≠ 0
- sin 45° = cos 45° = √2/2
2. Algebraic Fractions and Quadratic Expressions | 代数分式与二次表达式
+ 2), provided x ≠ 3. Cancelling common factors is only valid when the factor is not zero.
化简代数分式时，一定要先对分子和分母进行因式分解。例如， (x² − 9)/(x² − x − 6) = (x − 3)(x + 3)/((x − 3)(x + 2)) = (x + 3)/(x + 2)，前提是 x ≠ 3。只有当公因式不为零时，才能约去公因式。
ns can be solved by factorising, completing the square, or the quadratic formula x = (−b ± √b² − 4ac)/2a.
解分式方程时，先乘以公分母去分母，然后解所得方程，但一定要检查是否出现使原分母为零的增根。二次表达式可以通过因式分解、配方法或求根公式 x = (−b ± √b² − 4ac)/2a 来求解。
- Complete the square: x² + bx + c = (x + b/2)² − (b/2)² + c
- Difference of two squares: a² − b² = (a − b)(a + b)
- Discriminant b² − 4ac determines nature of roots
3. Functions and Inverse/Composite Functions | 函数及其反函数与复合函数
For example, if f(x) = 2x + 1 and g(x) = x², then fg(x) = 2x² + 1, but gf(x) = (2x + 1)².
函数将每个输入 x 映射到唯一的输出 f(x)。复合函数 fg(x) 表示 f(g(x))，即先应用 g，再应用 f。例如，若 f(x) = 2x + 1 且 g(x) = x²，则 fg(x) = 2x² + 1，但 gf(x) = (2x + 1)²。
ot all functions have an inverse over their whole domain; the function must be one-to-one.
求反函数 f⁻¹(x) 时，先写出 y = f(x)，交换 x 和 y，然后解出 y。y = f⁻¹(x) 的图像是 y = f(x) 关于直线 y = x 的反射。并非所有函数在整个定义域内都有反函数，函数必须是一一对应的。
- Domain: set of possible input values x
- Range: set of possible output values f(x)
- f⁻¹(f(x)) = x for all x in the domain of f
4. Inequalities and Graphical Regions | 不等式与图形区域
Solve linear inequalities such as 3x − 5 > 7 by rearranging, but remember to reverse the inequality sign when multiplying or dividing by a negative number. For example, −2x < 6 gives x > −3. Represent solutions on a number line with open or closed circles.
解诸如 3x − 5 > 7 的线性不等式时，只需移项，但请记住，当乘以或除以负数时，必须反转不等号方向。例如，−2x < 6 得到 x > −3。在数轴上表示解集时，使用空心圆或实心圆。
etch the parabola to identify where the expression is negative. The solution is 2 < x < 3.
对于诸如 x² − 5x + 6 < 0 的二次不等式，先因式分解为 (x − 2)(x − 3) < 0。检验临界值 x = 2 和 x = 3 附近的区间，或绘制抛物线图像以确定表达式为负的区域。其解为 2 < x < 3。
≤ x + 1, y > −2, and x + y ≤ 4. Use a solid line for ≤ or ≥ and a dashed line for < or >.
图形区域由多个不等式界定。请涂出满足所有约束条件的区域，例如 y ≤ x + 1、y > −2 和 x + y ≤ 4。对于 ≤ 或 ≥ 使用实线，对于 < 或 > 使用虚线。
5. Trigonometry: Sine Rule, Cosine Rule and Area | 三角学：正弦定理、余弦定理与面积公式
² = b² + c² − 2bc cos A when you know two sides and the included angle or all three sides.
对于非直角三角形，当已知两角一边或两边及一个非夹角时，使用正弦定理 a/sin A = b/sin B = c/sin C。当已知两边及其夹角或已知三边时，使用余弦定理 a² = b² + c² − 2bc cos A。
luded angle between sides a and b. This formula is particularly useful when the perpendicular height is not given.
任意三角形的面积可用公式 Area = ½ ab sin C 求得，其中 C 为边 a 与边 b 的夹角。当未给出垂直高度时，该公式尤其有用。
| Known information | Rule to use |
| Two angles and one side | Sine rule |
| Two sides and included angle | Cosine rule |
| Two sides and non-included angle | Sine rule, check for ambiguous case |
| Three sides | Cosine rule |
6. Circle Theorems | 圆定理
Circle theorems allow you to find missing angles in circle diagrams. The angle in a semicircle is always 90°. Angles in the same segment subtended by the same chord are equal. The angle at the centre is twice the angle at the circumference subtended by the same arc.
圆定理可用于求圆形图形中的缺失角。半圆中的圆周角始终为 90°。同一弦所对的同一弓形内的圆周角相等。圆心角是同一弧所对圆周角的两倍。
Opposite angles in a cyclic quadrilateral add up to 180°. A tangent to a circle meets the radius at 90°, and the tangent-chord theorem states that the angle between a tangent and a chord equals the angle in the alternate segment.
圆内接四边形的对角之和为 180°。圆的切线与半径相交成 90°。弦切角定理指出，切线与弦之间的夹角等于交替弓形中的圆周角。
- Angle in semicircle = 90°
- Angle at centre = 2 × angle at circumference
- Cyclic quadrilateral opposite angles sum to 180°
- Tangent meets radius at 90°
7. Vectors | 向量
A vector has both magnitude and direction. It can be written as a column vector, for example (3, −2), or using bold letters such as a and b. Vectors can be added, subtracted, and multiplied by a scalar. The zero vector is written 0.
向量既有大小又有方向。它可以用列向量表示，例如 (3, −2)，或用粗体字母如 a 和 b 表示。向量可以进行加法、减法和数乘运算。零向量记作 0。
d BC = (6, 2), then AB and BC are parallel and share point B, so A, B and C are collinear.
要证明三点共线，可证明连接它们的向量互为标量倍数。例如，若 AB = 2(3, 1) 且 BC = (6, 2)，则 AB 与 BC 平行且都经过点 B，因此 A、B、C 三点共线。
A + OB)/2. This is useful for geometric vector proofs involving ratios and parallelograms.
线段 AB 的中点向量为 (a + b)/2。若 M 是 AB 的中点，则 OM = (OA + OB)/2。这一结论在涉及比例和平行四边形的几何向量证明中非常有用。
8. Mensuration: Arcs, Sectors and 3D Solids | 测量：弧、扇形与三维体
or area is A = θ/360 × πr². Remember to use degrees unless the question specifies radians.
圆心角为 θ 的扇形弧长为 l = θ/360 × 2πr，其中 θ 以度为单位。扇形面积为 A = θ/360 × πr²。除非题目指定使用弧度制，否则请使用度。
| Solid | Volume | Surface area |
| Cylinder | πr²h | 2πr² + 2πrh |
| Cone | (1/3)πr²h | πrl + πr² |
| Sphere | (4/3)πr³ | 4πr² |
For composite solids, break the shape into simpler parts, calculate separately, then add or subtract volumes or surface areas as required. Always check whether the question asks for curved surface area or total surface area.
对于组合体，应将图形分解为简单部分分别计算，然后根据要求相加或相减体积或表面积。务必检查题目是求侧面积还是总表面积。
9. Sequences, Linear and Quadratic nth Term | 数列与线性、二次第 n 项
A linear sequence has a constant first difference. Its nth term is given by an + b, where a is the common difference. For example, the sequence 3, 7, 11, 15 has nth term 4n − 1 because the common difference is 4 and the zero term is −1.
线性数列的一阶差为常数。其第 n 项由 an + b 给出，其中 a 为公差。例如，数列 3, 7, 11, 15 的第 n 项为 4n − 1，因为公差为 4，第零项为 −1。
A quadratic sequence has a constant second difference. The nth term has the form an² + bn + c. Find a by halving the second difference, then use the first few terms to set up equations and solve for b and c. For example, 2, 6, 12, 20 has second difference 2, so nth term is n² + n.
二次数列的二阶差为常数。其第 n 项形式为 an² + bn + c。将二阶差除以 2 得到 a，再利用前几项建立方程解出 b 和 c。例如，2, 6, 12, 20 的二阶差为 2，因此第 n 项为 n² + n。
Other sequences include geometric progressions with a common ratio r, and Fibonacci-type sequences where each term is the sum of the previous two. Use the nth term to find any required term or to check whether a value appears in the sequence.
其他数列包括具有公比 r 的等比数列，以及每一项等于前两项之和的斐波那契型数列。利用第 n 项公式可求任意指定项，或检验某个值是否出现在数列中。
10. Probability: Tree Diagrams and Conditional Probability | 概率：树图与条件概率
l is 2/5, then the probability of red then blue is 3/5 × 2/5 = 6/25, assuming replacement.
对于独立事件，应沿树图的分支相乘概率。例如，若取出红球的概率为 3/5，取出蓝球的概率为 2/5，则在有放回条件下，先红后蓝的概率为 3/5 × 2/5 = 6/25。
ree diagram clearly with fractions or decimals. Conditional probability is found using P(A|B) = P(A ∩ B)/P(B).
在不放回抽样中，第二组分支上的概率会根据第一次结果而变化。务必用分数或小数清晰地标注树图。条件概率用公式 P(A|B) = P(A ∩ B)/P(B) 计算。
- Independent events: P(A and B) = P(A) × P(B)
- Mutually exclusive events: P(A or B) = P(A) + P(B)
- Conditional: P(A|B) = P(A ∩ B)/P(B)
11. Direct and Inverse Proportion | 正比例与反比例
Direct proportion means y ∝ ind k, then substitute to find unknown values. If y = 8 when x = 2, then k = 4, so y = 4x.
正比例表示 y ∝ x，因此 y = kx，其中 k 为常数。利用已知值求出 k，再代入求未知值。若 x = 2 时 y = 8，则 k = 4，所以 y = 4x。
Inverse proportion means y ∝ hen x = 3, then k = 18, so y = 18/x. This is different from a linear inverse relationship.
反比例表示 y ∝ 1/x，因此 y = k/x。其图像为双曲线。若 x = 3 时 y = 6，则 k = 18，所以 y = 18/x。这与线性反比关系不同。
Other power variations include y ∝ x² and y ∝ 1/x². Always state the constant of proportionality and show clearly how it is found. Exam questions may ask you to convert between proportion statements and algebraic equations.
其他幂次比例关系包括 y ∝ x² 和 y ∝ 1/x²。请始终写出比例常数并清晰地展示求解过程。考试题可能要求你在比例语句与代数方程之间进行转换。
12. Exam Technique and Common Pitfalls | 考试技巧与常见错误
In the IGCSE Mathematics A written papers, always show clear working because method marks are awarded even if the final answer is wrong. Use exact values such as √2 and π unless the question asks for a decimal approximation. Pay attention to units, money, and time conversions.
在 IGCSE 数学 A 笔试中，请始终展示清晰的解题过程，因为即使最终答案错误，也能获得方法分。除非题目要求给出小数近似值，否则请使用 √2 和 π 等精确值。注意单位、货币和时间换算。
Common pitfalls include losing negative signs when rearranging, mishandling algebraic fractions, confusing the sine and cosine rule conditions, and forgetting to square or cube in mensuration formulas. Double-check factorisation by expanding, and always reread the question to ensure the final answer answers what was asked.
常见错误包括移项时丢掉负号、处理代数分式出错、混淆正弦定理与余弦定理的使用条件，以及在测量公式中忘记平方或立方。通过展开来检查因式分解，并始终重读题目，确保最终答案回答了所问内容。
- Show all steps to gain method marks
- Use exact values unless told to round
- Check units and significant figures if required
- Manage time by attempting familiar questions first
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