📚 Year 1 Edexcel Pure Mathematics: Core Topics and Exam Tips | Edexcel 数学 Year 1 核心主题与备考要点
Year 1 Edexcel Pure Mathematics builds the essential algebraic, trigonometric and calculus skills you will need for the full A Level. This revision guide covers the core topics that appear most often in AS and Year 1 exams, with clear reminders of common errors and exam tips. Each section below pairs an English summary with its Chinese equivalent, so you can review both the concepts and the key terminology.
Edexcel 数学 Year 1 纯数部分为你打下代数、三角和微积分的基础，这些技能对完整的 A Level 课程至关重要。本复习指南覆盖 AS 和 Year 1 考试中最常见的核心主题，并清晰提示常见错误与应试技巧。以下每一节都提供英文总结和对应的中文说明，方便你同时复习概念和关键术语。
1. Algebraic Expressions and Indices | 代数表达式与指数
Simplify expressions by collecting like terms, using index laws, and rationalising denominators. Remember that (a + b)² expands to a² + 2ab + b², not a² + b². When simplifying surds, factor out the largest square number from under the root.
通过合并同类项、运用指数律和分母有理化来简化表达式。记住 (a + b)² 展开为 a² + 2ab + b²，而不是 a² + b²。简化根式时，把根号下最大的平方数因数提出来。
Key index laws: x² × x³ = x⁵, (x²)³ = x⁶, x⁻¹ = 1/x, x¹ᐟ² = √x. These appear in almost every algebra question, so write them clearly before starting.
关键指数律：x² × x³ = x⁵、(x²)³ = x⁶、x⁻¹ = 1/x、x¹ᐟ² = √x。这些规则几乎出现在每一道代数题中，因此开始解题前先清晰地写出来。
xᵃ × xᵇ = xᵃ⁺ᵇ, (xᵃ)ᵇ = xᵃᵇ, x⁻ⁿ = 1/xⁿ
2. Quadratics | 二次函数
two distinct roots if Δ > 0, one repeated root if Δ = 0, and no real roots if Δ < 0.
通过配方法求二次函数的顶点并解方程。判别式 Δ = b² - 4ac 可以判断实根的个数：Δ > 0 时有两个不同实根，Δ = 0 时有一个重根，Δ < 0 时没有实根。
y whether the parabola opens upwards (a > 0) or downwards (a < 0), then mark the vertex, the y-intercept and any x-intercepts.
画 y = ax² + bx + c 的图像时，先判断抛物线开口向上 (a > 0) 还是向下 (a < 0)，然后标出顶点、y 轴截距和所有 x 轴截距。
x = [-b ± √b² - 4ac] / 2a
3. Equations and Inequalities | 方程与不等式
Solve simultaneous equations by substitution or elimination. For one linear and one quadratic equation, substitute the linear expression into the quadratic, then solve the resulting quadratic. Always check both solutions satisfy the original equations.
用代入法或消元法解联立方程。对于一条一次方程和一条二次方程，先把一次表达式代入二次方程，再解所得的一元二次方程。务必检验两个解是否都满足原方程。
For inequalities, treat them like equations when rearranging, but reverse the inequality sign if you multiply or divide by a negative number. Express solutions using set notation such as {x : x < -2} ∪ {x : x > 3}.
解不等式时，移项过程与方程类似，但如果乘以或除以负数，必须反转不等号。最终答案用集合符号表示，例如 {x : x < -2} ∪ {x : x > 3}。
If a < b and c < 0, then ac > bc
4. Graphs and Transformations | 函数图像与变换
s a horizontal stretch. Pay close attention to the sign and the factor inside the bracket.
y = f(x) 的图像变换遵循固定规则：y = f(x) + a 是竖直平移，y = f(x + a) 是水平平移，y = a f(x) 是竖直伸缩，y = f(ax) 是水平伸缩。要特别注意括号内的符号和倍数。
reflects it in the y-axis. Always sketch the original graph first, then apply transformations one at a time in the correct order.
对于对称变换，y = -f(x) 将图像关于 x 轴对称，y = f(-x) 关于 y 轴对称。务必先画出原图像，然后按正确顺序逐步进行变换。
y = f(x) → y = a f(b(x + c)) + d
5. Straight Line Graphs and Circles | 直线方程与圆
rallel lines have equal gradients; perpendicular lines have gradients whose product is -1.
经过点 (x₁, y₁) 和 (x₂, y₂) 的直线斜率为 m = (y₂ - y₁)/(x₂ - x₁)。直线方程可以写成 y - y₁ = m(x - x₁) 或 y = mx + c。平行直线斜率相等；垂直直线斜率乘积为 -1。
². To find the tangent at a point on the circle, first find the gradient of the radius, then use the negative reciprocal as the tangent gradient.
圆心为 (a, b)、半径为 r 的圆方程为 (x - a)² + (y - b)² = r²。要求圆上一点的切线，先求该点处半径的斜率，再取其负倒数作为切线斜率。
(x - a)² + (y - b)² = r²
6. Algebraic Methods and Binomial Expansion | 代数方法与二项式展开
en (x - p) is a factor of f(x). Polynomial division or equating coefficients can then be used to factorise cubic or quartic expressions fully.
使用因式定理：若 f(p) = 0，则 (x - p) 是 f(x) 的因式。之后可用多项式除法或比较系数法对三次或四次表达式进行完全因式分解。
The binomial expansion for positive integer n is (a + b)ⁿ = Σ nCr aⁿ⁻ʳ bʳ u will usually expand expressions such as (1 + x)ⁿ up to a given power, often the x³ term.
正整数 n 的二项式展开为 (a + b)ⁿ = Σ nCr aⁿ⁻ʳ bʳ，其中 nCr 是二项式系数。在 Year 1 中，你通常需要将 (1 + x)ⁿ 展开到指定次幂，常见到 x³ 项。
(1 + x)ⁿ = 1 + nx + n(n-1)x²/2! + n(n-1)(n-2)x³/3! + ...
7. Trigonometric Ratios | 三角比
ving non-right-angled triangles: a/sin A = b/sin B = c/sin C and a² = b² + c² - 2bc cos A.
熟记 0°、30°、45°、60° 和 90° 的 sin、cos、tan 精确值。正弦定理和余弦定理是解非直角三角形的重要工具：a/sin A = b/sin B = c/sin C 以及 a² = b² + c² - 2bc cos A。
The area of a triangle can be found using ½ab sin C, which is particularly useful when you know two sides and the included angle. Draw a clear diagram and label all known sides and angles before applying any rule.
三角形面积可用 ½ab sin C 求得，当你已知两边及其夹角时尤其有用。先画出清晰的图形并标出所有已知边和角，再应用相应公式。
Area = ½ab sin C
8. Trigonometric Identities and Equations | 三角恒等式与方程
These allow you to simplify expressions and solve equations by reducing different trigonometric functions to a single one.
必须掌握的两个恒等式是 tan θ = sin θ / cos θ 和 sin²θ + cos²θ = 1。利用它们可以将表达式化简，并通过把不同的三角函数化为同一种来解方程。
When solving trigonometric equations, find the principal value using your calculator, then use the graph or CAST diagram to find all solutions in the required interval. Always check whether the range is in degrees or radians.
解三角方程时，先用计算器求出主值，再利用图像或 CAST 图找出指定区间内的所有解。务必确认题目使用的是角度制还是弧度制。
sin²θ + cos²θ = 1
9. Vectors | 向量
A vector has both magnitude and direction. In column form, the vector ai + bj has magnitude √a² + b². Unit vectors have magnitude 1 and are found by dividing a vector by its magnitude.
向量既有大小又有方向。用列向量表示时，向量 ai + bj 的大小为 √a² + b²。单位向量的大小为 1，可由原向量除以其大小得到。
Vector problems often require you to find AB = b - a, where a and b are position vectors.
位置向量描述某一点相对于原点的位置，位移向量描述从一点到另一点的移动。向量题经常要求你计算 AB = b - a，其中 a 和 b 是位置向量。
|ai + bj| = √a² + b²
10. Differentiation | 微分
is applied term by term to polynomials. The derivative gives the gradient of the tangent to the curve at any point, so it can be used to find equations of tangents and normals.
若 y = xⁿ，则 dy/dx = n xⁿ⁻¹。该法则逐项应用于多项式。导数给出曲线在任意点处切线的斜率，因此可用于求切线和法线的方程。
y/dx = 0 to find stationary points, then test the sign of d²y/dx² or use a gradient table.
二阶导数 d²y/dx² 可以判断函数在某点递增还是递减，并帮助识别局部极大值和极小值。令 dy/dx = 0 求驻点，然后检验 d²y/dx² 的符号或使用斜率表判断。
d/dx (xⁿ) = n xⁿ⁻¹
11. Integration | 积分
Integration is the reverse of differentiation. The indefinite integral of xⁿ is xⁿ⁺¹/(n+1) + C for n ≠ -1. The constant of integration C must be included unless you are evaluating a definite integral.
积分是微分的逆运算。xⁿ 的不定积分为 xⁿ⁺¹/(n+1) + C，其中 n ≠ -1。除非计算定积分，否则必须包含积分常数 C。
f(x) from a to b. If the curve lies below the x-axis, the integral is negative, so use absolute values when finding total area.
从 x = a 到 x = b 的定积分给出曲线 y = f(x) 下方从 a 到 b 的面积。若曲线位于 x 轴下方，积分值为负，因此求总面积时要取绝对值。
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ -1
12. Exponentials and Logarithms | 指数与对数
The natural logarithm ln x is the inverse of eˣ, so ln(eˣ) = x and eln x ogarithms are logₐ(xy) = logₐx + logₐy, logₐ(x/y) = logₐx - logₐy, and logₐ(xⁿ) = n logₐx.
自然对数 ln x 是 eˣ 的反函数，因此 ln(eˣ) = x 且 eln x = x。对数运算律为 logₐ(xy) = logₐx + logₐy、logₐ(x/y) = logₐx - logₐy、logₐ(xⁿ) = n logₐx。
To solve equations such as e²ˣ ecay models, identify the initial amount, the growth factor and the time period carefully.
解 e²ˣ⁺¹ = 5 这类方程时，两边同时取自然对数，再运用 ln(eᵃ) = a 的性质。对于指数增长或衰减模型，要仔细辨别初始量、增长因子和时间周期。
ln(eˣ) = x, eln x = x
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