📚 Pure Mathematics Year 1: Core Topics | 纯数第一年核心知识点
Year 1 of A-Level Pure Mathematics builds the foundational skills that underpin every advanced area of the subject. This article covers the key topics: algebraic manipulation, quadratics, equations, inequalities, polynomials, graphs, binomial expansion, differentiation, integration, and vectors. Understanding these ideas thoroughly is essential for success in both mechanics and statistics components later.
A-Level 纯数第一年构建了支撑整个学科后续学习的基础技能。本文涵盖核心知识点：代数运算、二次函数、方程组、不等式、多项式、函数图像、二项式展开、微分、积分以及向量。透彻理解这些内容对今后力学和统计部分的学习至关重要。
1. Laws of Indices and Surds | 指数定律与根式运算
ic manipulation: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ, a¹/ⁿ = ⁿ√a. When simplifying surds, always look for square factors: √8 = 2√2. Rationalising denominators means removing surds from the bottom of fractions, e.g. 1/√2 becomes √2/2.
指数定律是代数运算的基础：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1，a⁻ⁿ = 1/aⁿ，a¹/ⁿ = ⁿ√a。化简根式时，总是先寻找平方因子：√8 = 2√2。有理化分母即从分数分母中消去根号，例如 1/√2 化为 √2/2。
Further practice includes combining surds: √a × √b = √ab, and expanding brackets like (√a + √b)² = a + 2√ab + b. Always express final answers in simplest surd form.
进一步练习包括根式的合并：√a × √b = √ab，以及展开括号如 (√a + √b)² = a + 2√ab + b。最终答案务必写成最简根式。
2. Quadratic Functions and Equations | 二次函数与方程
be found by factorising, completing the square, or using the quadratic formula: x = [−b ± √b² − 4ac 0 gives two distinct real roots, Δ = 0 gives one repeated root, Δ < 0 gives no real roots.
二次函数的一般形式为 ax² + bx + c = 0。可通过因式分解、配方法或求根公式求解：x = [−b ± √b² − 4ac] / 2a。判别式 Δ = b² − 4ac 揭示根的性质：Δ > 0 有两个相异实根，Δ = 0 有一个重根，Δ < 0 无实根。
. Its vertex lies at (−b/(2a), −Δ/(4a)). Completing the square rewrites the expression as a(x + p)² + q, which directly shows the coordinates of the turning point.
y = ax² + bx + c 的图像是一条抛物线。其顶点坐标为 (−b/(2a), −Δ/(4a))。配方法可将表达式改写为 a(x + p)² + q，直接显示转折点的坐标。
3. Simultaneous Equations | 联立方程组
Simultaneous equations are solved by elimination or substitution. For linear equations, eliminate one variable by adding/subtracting multiples. For one linear and one quadratic, substitute the linear expression into the quadratic. Always check for extraneous solutions, and interpret the results as intersection points of two graphs.
联立方程组可通过消元法或代入法求解。对于线性方程组，通过加减倍数消去一个未知数。若为一个线性、一个二次，则将线性表达式代入二次方程。始终注意验根，并将结果解释为两图像的交点。
2x + 1 → x² − x − 2 = 0 → (x−2)(x+1)=0, so x=2 or x=−1, yielding points (2,5) and (−1,−1).
例如：解 y = 2x + 1 与 y = x² + x − 1。代入得 x² + x − 1 = 2x + 1 → x² − x − 2 = 0 → (x−2)(x+1)=0，故 x=2 或 x=−1，对应交点 (2,5) 和 (−1,−1)。
4. Inequalities | 不等式
Linear inequalities are solved like equations, but remember to reverse the inequality sign when multiplying or dividing by a negative number. Quadratic inequalities require sketching the parabola to determine intervals. For example, x² − 4x + 3 < 0 factorises as (x−1)(x−3) < 0, giving 1 < x < 3.
线性不等式的解法与方程类似，但当乘以或除以负数时不等号方向需反转。二次不等式需借助抛物线草图确定区间。例如 x² − 4x + 3 < 0 因式分解为 (x−1)(x−3) < 0，得到 1 < x < 3。
s and sign tables. Always exclude values that make the denominator zero from the solution set.
对于分式不等式如 (x−2)/(x+5) ≥ 0，须考虑关键值并列出符号表。解集中不得包含使分母为零的值。
5. Polynomials and Algebraic Division | 多项式与代数除法
r. Algebraic division (long division or synthetic division) is used to factorise cubics and quartics. The remainder theorem gives the remainder when f(x) is divided by (x − a) as f(a).
多项式为诸如 2x³ − 5x² + x − 7 的表达式。因式定理指出，若 f(a)=0，则 (x − a) 为一因式。多项式除法（长除或综合除法）用于分解三次或四次多项式。余式定理给出 f(x) 除以 (x − a) 的余数为 f(a)。
To sketch a cubic, find its intercepts, turning points (using calculus), and end behaviour. Factorised form reveals roots and their multiplicity.
绘制三次函数草图时，需找到截距、利用微积分求转折点并观察函数在无穷远处的走向。因式乘积形式能显示根及其重数。
6. Graphs and Transformations | 函数图像与变换
Transformations map a base function f(x) to a new graph. Key types: f(x) + a vertical translation, f(x + a) horizontal shift, a·f(x) vertical stretch, f(ax) horizontal stretch (or compression), −f(x) reflection in the x‑axis, f(−x) reflection in the y‑axis. Combinations must be applied in the correct order.
函数变换将基本函数 f(x) 映射至新图像。主要类型：f(x) + a 垂直平移，f(x + a) 水平平移，a·f(x) 垂直伸缩，f(ax) 水平伸缩（或压缩），−f(x) 关于 x 轴的反射，f(−x) 关于 y 轴的反射。复合变换需按正确顺序执行。
Sketching reciprocal and proportional graphs like y = k/x, y = k/x² helps model real‑world situations. Asymptotes and intercepts are essential features.
绘制反比例函数图像如 y = k/x、y = k/x² 有助于模拟现实情景。渐近线及截距是关键特征。
7. Binomial Expansion | 二项式展开
For positive integer n, (a + b)ⁿ = Σₖ₌₀ⁿ (n choose k) aⁿ⁻ᵏ bᵏ, where (n choose k) = n!/(k!(n−k)! coefficients quickly for small n. The expansion is exact and finite. For rational index or |x| < 1, the expansion (1 + x)ⁿ ≈ 1 + nx + n(n−1)/2! x² + ... is infinite but valid within the radius of convergence.
对正整数 n，(a + b)ⁿ = Σₖ₌₀ⁿ (n choose k) aⁿ⁻ᵏ bᵏ，其中 (n choose k) = n!/(k!(n−k)!)。杨辉三角可快速给出较小 n 的系数。展开式为一精确且有限的多项式。若指数为有理数且 |x|<1，展开式 (1 + x)ⁿ ≈ 1 + nx + n(n−1)/2! x² + … 是一无穷级数，在收敛半径内有效。
Use expansions to approximate values, e.g. estimate √1 etting x = 0.02 in (1 + x)¹/². Always state the range of validity for infinite expansions.
利用展开式可近似求值，例如令 x = 0.02 代入 (1 + x)¹/² 估算 √1.02。无穷展开务必标明有效范围。
8. Differentiation | 微分
Differentiation gives the gradient function f' dx (xⁿ) = n xⁿ⁻¹. The derivative enables finding tangents and normals: tangent slope m = f'(a) at x = a, normal slope = −1/m. Second derivative f''(x) indicates concavity and helps classify turning points.
微分能给出曲线 y = f(x) 的梯度函数 f'(x)。对幂函数，d/dx (xⁿ) = n xⁿ⁻¹。导数用于求切线与法线：切线斜率 m = f'(a)（在 x = a 处），法线斜率 = −1/m。二阶导数 f''(x) 反映凹凸性并帮助判断驻点类型。
Stationary points occur where f'(x)=0. Use f''(x) test: if f''(x) > 0, it's a local minimum; if f''(x) < 0, a local maximum. Increasing/decreasing intervals are found from sign of f'(x).
驻点出现在 f'(x)=0 处。用二阶导数检验：若 f''(x) > 0 则为局部极小值；若 f''(x) < 0 则为局部极大值。单调递增或递减区间由 f'(x) 的正负决定。
9. Integration | 积分
e area under a curve between limits: ∫ₐᵇ f(x) dx = F(b) − F(a), where F is an antiderivative.
积分是微分的逆运算。不定积分：∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C，其中 n ≠ −1。定积分计算曲线下方与 x 轴之间的面积：∫ₐᵇ f(x) dx = F(b) − F(a)，F 为原函数。
When the area lies partly below the x‑axis, separate the intervals and take the absolute value of negative parts. Areas between curves require subtracting one function from the other.
若部分区域位于 x 轴下方，需分段并将负值部分取绝对值。两曲线间的面积需用一条函数减去另一条函数。
10. Vectors | 向量
ten as (x y) or xi + yj. Addition, subtraction, and scalar multiplication follow component‑wise rules. The magnitude (length) is |v| = √x² + y². Unit vector in direction v is v/|v|.
向量既有大小又有方向。在二维中，向量写作 (x y) 或 xi + yj。加减法与标量乘法按分量进行。向量大小（模）为 |v| = √x² + y²。沿 v 方向的单位向量为 v/|v|。
lems involving ratios, use the section formula to find coordinates of points dividing a line segment internally.
位置向量连接原点与点。向量 AB = OB − OA。涉及比例的问题可用定比分点公式求线段内分点的坐标。
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