Mastering Edexcel International GCSE (9-1) Further Pure Mathematics | 掌握 Edexcel 国际 GCSE (9-1) 进阶纯数学

📚 Mastering Edexcel International GCSE (9-1) Further Pure Mathematics | 掌握 Edexcel 国际 GCSE (9-1) 进阶纯数学

The Edexcel International GCSE (9-1) Further Pure Mathematics specification extends core IGCSE ideas into more formal algebra, trigonometry, vectors and calculus. This revision guide follows the structure of the Student Book by Mr Ali Datoo and focuses on the skills most often tested in Papers 1 and 2. Each section pairs an English explanation with a Chinese explanation so you can check understanding in both languages.

Edexcel 国际 GCSE (9-1) 进阶纯数学课程将核心 IGCSE 思想拓展到更正式的代数、三角、向量和微积分。本复习指南遵循 Mr Ali Datoo 学生用书的结构，聚焦试卷一和试卷二中最常考查的技能。每节均以英文说明和中文说明配对，帮助你用两种语言检验理解。

1. Indices and Surds | 指数与根式

The laws of indices are fundamental in Further Pure Mathematics. When multiplying powers with the same base, add the indices: a^m × aⁿ = a^m+n. When dividing, subtract the indices: a^m ÷ aⁿ = a^m−n. A power raised to another power multiplies the indices: (a^m)ⁿ = a^mn. A negative index gives a reciprocal, so a⁻ⁿ = 1/aⁿ. Fractional indices represent roots: a^1/2 = √a and a^m/n = ⁿ√a^m = (ⁿ√a)^m.

指数定律是进阶纯数学的基础。同底数幂相乘时，指数相加：a 的 m 次方乘以 a 的 n 次方等于 a 的 m+n 次方。相除时，指数相减：a 的 m 次方除以 a 的 n 次方等于 a 的 m−n 次方。幂的乘方，指数相乘：(a 的 m 次方)的 n 次方等于 a 的 mn 次方。负指数表示倒数，因此 a 的 −n 次方等于 1/a 的 n 次方。分数指数表示根式：a 的 1/2 次方等于 √a，a 的 m/n 次方等于 n 次根号下 a 的 m 次方，也等于 n 次根号下 a 的 m 次方。

To simplify surds, use the rules √ab = √a × √b and √a/b = √a / √b. Always look for the largest square factor inside the root. To rationalise a denominator such as 1/(√a + √b), multiply both numerator and denominator by the conjugate √a − √b. This removes the surd from the denominator and is often required for full marks in simplification questions.

化简根式时，使用公式 √ab = √a × √b 以及 √a/b = √a / √b。始终寻找根号内的最大平方因子。要将形如 1/(√a + √b) 的分母有理化，可将分子分母同时乘以共轭式 √a − √b。这样可消去分母中的根号，也是化简题中取得满分的关键步骤。

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2. Logarithmic Functions | 对数函数

If y = a^x then x = log_a y, where a > 0 and a ≠ 1. The three key laws of logarithms are: log_a (xy) = log_a x + log_a y; log_a (x/y) = log_a x − log_a y; and log_a xⁿ = n log_a x. These laws allow you to combine or split logarithmic expressions, which is essential when solving equations.

若 y = a 的 x 次方，则 x = log 以 a 为底 y 的对数，其中 a > 0 且 a ≠ 1。对数的三条关键法则是：log_a(xy) = log_a x + log_a y；log_a(x/y) = log_a x − log_a y；log_a xⁿ = n log_a x。这些法则使你能够合并或拆分对数表达式，是解方程时必不可少的工具。

The change of base formula is also part of the specification: log_a b = log_c b / log_c a for any positive base c. This is especially useful when using a calculator, because most calculators only have log₁₀ and ln keys. When solving exponential equations such as 3^x = 20, take logarithms of both sides: x log 3 = log 20, so x = log 20 / log 3.

换底公式同样属于考试范围：log_a b = log_c b / log_c a，其中 c 为任意正数。这在用计算器计算时尤其有用，因为大多数计算器只有 log₁₀ 和 ln 键。解指数方程如 3^x = 20 时，对两边同时取对数：x log 3 = log 20，因此 x = log 20 / log 3。

3. Quadratic Functions and the Discriminant | 二次函数与判别式

Any quadratic expression can be written in completed square form. For ax² + bx + c, completing the square gives a(x + b/(2a))² + (c − b²/(4a)). This form immediately reveals the vertex of the parabola: (−b/(2a), c − b²/(4a)). It is also the basis for deriving the quadratic formula x = (−b ± √b² − 4ac) / (2a).

任何二次式都可以写成配方法的形式。对于 ax² + bx + c，配方

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