一、二项式展开：帕斯卡三角形与二项式定理 | Binomial Expansion: Pascal’s Triangle and the Binomial Theorem
AS 阶段纯数学的第一个核心工具是二项式展开。对于形如 (a + b)n 的表达式，其中 n 是正整数，我们可以直接展开，也可以用二项式定理写出任意一项。理解这项内容的关键，是记住系数来自帕斯卡三角形，而各项中 a 与 b 的指数之和始终等于 n。
The first core tool in AS pure mathematics is the binomial expansion. For an expression of the form (a + b)n, where n is a positive integer, you can either expand it directly or write down any individual term using the binomial theorem. The key to understanding this topic is to remember that the coefficients come from Pascal’s triangle, and that the powers of a and b in any term always add up to n.
二项式定理的一般形式是 (1 + x)n = 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + …。当 n 是正整数时，展开式在 xn 项处停止，一共 n + 1 项。例如 (1 + x)⁴ = 1 + 4x + 6x² + 4x³ + x⁴，系数 1、4、6、4、1 正是帕斯卡三角形第四行的数字。
The general binomial theorem takes the form (1 + x)n = 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + … . When n is a positive integer, the expansion stops at the xn term, giving n + 1 terms in total. For example, (1 + x)⁴ = 1 + 4x + 6x² + 4x³ + x⁴, and the coefficients 1, 4, 6, 4, 1 are exactly the numbers in the fourth row of Pascal’s triangle.
考试中更常见的题型是求某一项的系数。例如求 (2 + 3x)⁵ 中 x³ 项的系数：先用通项公式 Tr+1 = C(5,r) × 25−r × (3x)r，令 r = 3，得到 C(5,3) × 2² × 3³ = 10 × 4 × 27 = 1080。这种”找系数”的方法比展开全部六项要快得多。
A more common exam question asks you to find the coefficient of a single term. For example, to find the coefficient of x³ in (2 + 3x)⁵, use the general term Tr+1 = C(5,r) × 25−r × (3x)r, set r = 3, and get C(5,3) × 2² × 3³ = 10 × 4 × 27 = 1080. This “find the coefficient” method is much faster than writing out all six terms.
二、三角恒等式：sin²θ + cos²θ = 1 与倍角公式 | Trigonometric Identities: sin²θ + cos²θ = 1 and the Double-Angle Formulae
AS 纯数学的三角部分围绕一条基本恒等式展开：sin²θ + cos²θ = 1。它是勾股定理在单位圆上的直接体现，几乎所有三角化简题最终都会回到这条式子。由它出发，两边同时除以 cos²θ 可以得到 tan²θ + 1 = sec²θ，这是解含 tan 的方程的常用工具。
The trigonometry section of AS pure mathematics is built around one fundamental identity: sin²θ + cos²θ = 1. This is a direct reflection of Pythagoras’ theorem on the unit circle, and almost every trigonometric simplification question eventually comes back to this equation. Dividing both sides by cos²θ gives tan²θ + 1 = sec²θ, which is a common tool for solving equations involving tan.
倍角公式同样高频出现：sin 2θ = 2 sinθ cosθ，cos 2θ 有三种等价写法（cos²θ − sin²θ、2cos²θ − 1、1 − 2sin²θ）。在解题时，cos 2θ 的三种形式需要根据题目结构灵活选择。例如要积分 cos²θ 时，把 cos 2θ = 2cos²θ − 1 改写成 cos²θ = (1 + cos 2θ)/2，就可以直接积分了。
The double-angle formulae are equally high-frequency: sin 2θ = 2 sinθ cosθ, and cos 2θ has three equivalent forms (cos²θ − sin²θ, 2cos²θ − 1, and 1 − 2sin²θ). In problem solving, you choose among the three forms of cos 2θ according to the structure of the question. For example, when integrating cos²θ, rewrite cos 2θ = 2cos²θ − 1 as cos²θ = (1 + cos 2θ)/2, and the integral can then be evaluated directly.
解三角方程时务必注意：方程 sinθ = 1/2 在 0° 到 360° 之间有两个解（30° 和 150°），而不是一个。画出 sin 曲线或单位圆，用”对称性”找到所有解，再根据题目给定的区间筛选。漏解是三角题最常见的失分原因。
When solving trigonometric equations, always remember that sinθ = 1/2 has two solutions between 0° and 360° (30° and 150°), not one. Draw the sine curve or the unit circle, use symmetry to find all solutions, and then filter them according to the interval given in the question. Missing solutions is the most common reason for losing marks in trigonometry.
三、二次函数：判别式与配方法 | Quadratic Functions: The Discriminant and Completing the Square
二次函数 y = ax² + bx + c 是 AS 数学里反复出现的基础对象。判别式 Δ = b² − 4ac 直接告诉我们方程 ax² + bx + c = 0 有多少个实根：Δ > 0 有两个不相等实根，Δ = 0 有一个重根，Δ < 0 没有实根。这个工具在"曲线与直线交点数"问题上特别有用。
The quadratic function y = ax² + bx + c is a foundational object that appears again and again in AS mathematics. The discriminant Δ = b² − 4ac tells us directly how many real roots the equation ax² + bx + c = 0 has: Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated root, and Δ < 0 gives no real roots. This tool is especially useful in "how many points of intersection" questions.
配方法是把一般式转化为顶点式 a(x − h)² + k 的过程，它直接给出抛物线的顶点坐标 (h, k) 和对称轴 x = h。例如 y = 2x² − 8x + 3，配方得到 y = 2(x − 2)² − 5，顶点为 (2, −5)。配方法同时也是解二次方程、推导判别式和完成积分的重要基础。
Completing the square converts the general form into the vertex form a(x − h)² + k, which directly gives the vertex (h, k) and the axis of symmetry x = h. For example, y = 2x² − 8x + 3 becomes y = 2(x − 2)² − 5 after completing the square, so the vertex is (2, −5). Completing the square is also the foundation for solving quadratics, deriving the discriminant, and carrying out certain integrations.
求根公式 x = [−b ± √(b² − 4ac)] / 2a 必须烂熟于心，但考试中更常要求你”先配方，再求根”，因为配方过程能同时展示顶点、对称轴和最值。熟悉这两种路径，并知道何时用哪一种，是二次函数拿满分的关键。
The quadratic formula x = [−b ± √(b² − 4ac)] / 2a must be known by heart, but exam questions more often ask you to “complete the square first, then find the roots”, because the completing process simultaneously reveals the vertex, the axis of symmetry, and the maximum or minimum value. Being fluent in both routes, and knowing when to use each, is the key to full marks on quadratic functions.
四、直线方程与圆的方程：坐标几何基础 | Straight Lines and Circles: Coordinate Geometry Essentials
坐标几何中，两点间的距离公式 d = √[(x₂ − x₁)² + (y₂ − y₁)²] 和斜率公式 m = (y₂ − y₁)/(x₂ − x₁) 是基础中的基础。两条直线平行当且仅当斜率相等，垂直当且仅当斜率互为负倒数（m₁ × m₂ = −1）。中点的坐标是两端点坐标的平均值。
In coordinate geometry, the distance formula d = √[(x₂ − x₁)² + (y₂ − y₁)²] and the gradient formula m = (y₂ − y₁)/(x₂ − x₁) are the absolute foundations. Two lines are parallel if and only if their gradients are equal, and perpendicular if and only if their gradients are negative reciprocals (m₁ × m₂ = −1). The midpoint is simply the average of the coordinates of the two endpoints.
圆的方程有两种写法：标准式 (x − a)² + (y − b)² = r² 直接显示圆心 (a, b) 和半径 r，一般式 x² + y² + 2gx + 2fy + c = 0 则需要通过配方还原。例如 x² + y² − 6x + 4y − 12 = 0 配方后得到 (x − 3)² + (y + 2)² = 25，圆心 (3, −2)、半径 5。
The equation of a circle has two forms: the standard form (x − a)² + (y − b)² = r² directly shows the centre (a, b) and radius r, while the general form x² + y² + 2gx + 2fy + c = 0 must be converted back by completing the square. For example, x² + y² − 6x + 4y − 12 = 0 becomes (x − 3)² + (y + 2)² = 25 after completing the square, giving centre (3, −2) and radius 5.
“直线与圆相交”是高频综合题：判断直线是否与圆相交、相切还是相离，可以用判别式（把直线方程代入圆方程得到一个关于 x 或 y 的二次方程）或比较圆心到直线的距离与半径的大小。距离公式 |ax₁ + by₁ + c| / √(a² + b²) 在这里是核心工具。
“Line meets circle” is a high-frequency combined question: to decide whether a line intersects, touches, or misses a circle, you can either use the discriminant (substitute the line equation into the circle equation to get a quadratic in x or y) or compare the distance from the centre to the line with the radius. The perpendicular distance formula |ax₁ + by₁ + c| / √(a² + b²) is the central tool here.
五、等差数列与等比数列：通项公式与求和 | Arithmetic and Geometric Sequences: nth Term and Sum Formulae
数列在 AS 阶段分两类。等差数列（arithmetic sequence）的公差 d 固定，通项公式是 un = a + (n − 1)d，前 n 项和公式是 Sn = n/2 × (2a + (n − 1)d)，也可以写成 Sn = n/2 × (首项 + 末项)。这两条公式是整个数列章节的基石。
Sequences at AS level come in two types. An arithmetic sequence has a fixed common difference d, with nth term un = a + (n − 1)d and sum of the first n terms Sn = n/2 × (2a + (n − 1)d), which can also be written as Sn = n/2 × (first term + last term). These two formulae are the bedrock of the entire sequences chapter.
等比数列（geometric sequence）的公比 r 固定，通项公式是 un = arn−1，前 n 项和公式是 Sn = a(1 − rn)/(1 − r)。当公比满足 |r| < 1 时，无穷等比数列收敛，其和为 S∞ = a/(1 − r)。这个”无限求和”公式在 AS 阶段就要求学生理解并应用。
A geometric sequence has a fixed common ratio r, with nth term un = arn−1 and sum Sn = a(1 − rn)/(1 − r). When the common ratio satisfies |r| < 1, the infinite geometric series converges, with sum S∞ = a/(1 − r). This “sum to infinity” formula is something AS students are expected to understand and apply.
解题时的第一要务是判断题目属于哪一类：出现”每项加同一个数”用等差，出现”每项乘同一个数”用等比。很多失分源于把两类公式混用。判断清楚类型后，把已知条件代入相应的通项和求和公式，联立方程求解即可。
The first priority when solving problems is to decide which type the question belongs to: “add the same number each time” means arithmetic, “multiply by the same number each time” means geometric. Many marks are lost from mixing up the two sets of formulae. Once the type is clear, substitute the given information into the relevant nth-term and sum formulae and solve the resulting simultaneous equations.
六、指数与对数：e、ln 与换底公式 | Exponentials and Logarithms: e, ln, and the Change-of-Base Formula
指数函数 y = ax 与对数函数 y = logax 互为反函数，这是理解整个对数章节的出发点。对数把乘法变成加法：loga(xy) = logax + logay，把除法变成减法，把幂变成乘法 loga(xn) = n logax。这三条运算律是解对数方程的基础。
The exponential function y = ax and the logarithmic function y = logax are inverse functions, and this is the starting point for understanding the entire logarithms chapter. Logarithms turn multiplication into addition: loga(xy) = logax + logay, turn division into subtraction, and turn powers into multiplication: loga(xn) = n logax. These three laws are the foundation for solving logarithmic equations.
自然对数 ln 以 e ≈ 2.718 为底，ex 的导数和积分都等于它本身，这让 e 在微积分中地位特殊。换底公式 logab = logcb / logca 用于在不同底数之间转换，而最常用的一对结论是 alogax = x 和 ln(ex) = x，它们体现了指数与对数的互逆关系。
The natural logarithm ln has base e ≈ 2.718, and the derivative and integral of ex are both ex itself, which gives e a special place in calculus. The change-of-base formula logab = logcb / logca is used to convert between bases, while the most useful pair of results are alogax = x and ln(ex) = x, which capture the inverse relationship between exponentials and logarithms.
解指数方程的关键步骤是”两边同时取对数”：例如解 3x = 20，两边取 ln 得到 x ln 3 = ln 20，从而 x = ln 20 / ln 3。解对数方程则要小心定义域，任何 logax 中的 x 必须为正，求出候选解后必须代回检验，排除使真数为负或零的增根。
The key step in solving an exponential equation is “take logarithms on both sides”: for example, to solve 3x = 20, take ln on both sides to get x ln 3 = ln 20, hence x = ln 20 / ln 3. When solving logarithmic equations, be careful with the domain: any argument inside a logarithm must be positive, so after finding candidate solutions you must substitute them back to reject extraneous roots that would make the argument negative or zero.
七、微分：幂法则、乘积法则与链式法则 | Differentiation: The Power, Product, and Chain Rules
微分的核心是求变化率，即曲线在某一点的切线斜率。最基本的规则是幂法则：若 y = xn，则 dy/dx = nxn−1。这是所有其他微分规则的基石，例如 y = 5x³ 的导数是 15x²，y = 2/x（即 2x−1）的导数是 −2/x²。
The core of differentiation is finding a rate of change, that is, the gradient of a curve at a point. The most basic rule is the power rule: if y = xn, then dy/dx = nxn−1. This is the foundation of all other differentiation rules. For example, the derivative of y = 5x³ is 15x², and the derivative of y = 2/x (that is, 2x−1) is −2/x².
当函数是乘积或复合形式时，需要更高级的规则。乘积法则：若 y = uv，则 dy/dx = u(dv/dx) + v(du/dx)。链式法则：若 y = f(g(x))，则 dy/dx = f′(g(x)) × g′(x)，常记作”外函数求导乘以内函数求导”。例如 y = (2x + 1)⁵ 的导数是 5(2x + 1)⁴ × 2 = 10(2x + 1)⁴。
When a function is a product or a composition, more advanced rules are needed. The product rule: if y = uv, then dy/dx = u(dv/dx) + v(du/dx). The chain rule: if y = f(g(x)), then dy/dx = f′(g(x)) × g′(x), often remembered as “derivative of the outer function times derivative of the inner function”. For example, the derivative of y = (2x + 1)⁵ is 5(2x + 1)⁴ × 2 = 10(2x + 1)⁴.
微分在应用中的典型问题是求切线方程和驻点。曲线 y = f(x) 在 x = a 处的切线斜率是 f′(a)，切线方程为 y − f(a) = f′(a)(x − a)。驻点是 f′(x) = 0 的点，通过二阶导数（或导数的符号变化）判断是极大值还是极小值。这些是”优化问题”（求最大面积、最小成本）的基础。
Typical applied differentiation problems involve finding tangents and stationary points. The tangent to y = f(x) at x = a has gradient f′(a) and equation y − f(a) = f′(a)(x − a). A stationary point is where f′(x) = 0, and the second derivative (or the sign change of the derivative) tells you whether it is a maximum or a minimum. These are the foundations of “optimisation problems” (maximum area, minimum cost).
八、积分：微分的逆运算与定积分 | Integration: The Reverse of Differentiation and Definite Integrals
积分是微分的逆运算。基本公式是：若 y = xn，则 ∫ xn dx = xn+1/(n+1) + C（n ≠ −1）。常数 C 称为积分常数，是”不定积分”区别于”确定函数”的标志。例如 ∫ 3x² dx = x³ + C，∫ (2x + 1) dx = x² + x + C。
Integration is the reverse of differentiation. The basic formula is: if y = xn, then ∫ xn dx = xn+1/(n+1) + C (n ≠ −1). The constant C is the constant of integration, and it is what distinguishes an “indefinite integral” from a single definite function. For example, ∫ 3x² dx = x³ + C, and ∫ (2x + 1) dx = x² + x + C.
定积分 ∫[a,b] f(x) dx 表示曲线 y = f(x) 与 x 轴在 x = a 到 x = b 之间所围成的”有符号面积”（x 轴下方为负）。计算方法是先求原函数 F(x)，再代入上下限做差：∫[a,b] f(x) dx = F(b) − F(a)。这个 F(b) − F(a) 的结构是微积分基本定理的核心。
The definite integral ∫[a,b] f(x) dx represents the “signed area” between the curve y = f(x) and the x-axis from x = a to x = b (negative where the curve is below the axis). It is evaluated by first finding an antiderivative F(x) and then subtracting the values at the limits: ∫[a,b] f(x) dx = F(b) − F(a). This F(b) − F(a) structure is the heart of the fundamental theorem of calculus.
求曲线与坐标轴或两条曲线之间的面积，是 AS 阶段积分的标准应用。注意当曲线在 x 轴下方时，定积分会得到负值，真实面积要取绝对值或分段计算。涉及两条曲线时，面积 = ∫ (上方曲线 − 下方曲线) dx，积分限是两曲线交点的横坐标。
Finding areas between a curve and the axes, or between two curves, is the standard application of integration at AS level. Note that when a curve lies below the x-axis the definite integral is negative, so the true area requires taking the absolute value or splitting into pieces. For two curves, area = ∫ (upper curve − lower curve) dx, with the limits being the x-coordinates of the intersection points.
九、向量：二维向量的加减与点积 | Vectors: Addition, Subtraction, and the Dot Product in 2D
向量是既有大小又有方向的量，在二维平面中通常写成列向量或 i、j 分量形式。向量的加法是分量对应相加，数乘是把每个分量乘以同一个标量。例如 (3i + 2j) + (i − 4j) = 4i − 2j，而 2(3i + 2j) = 6i + 4j。向量的模（长度）由勾股定理给出：|ai + bj| = √(a² + b²)。
A vector is a quantity with both magnitude and direction, written in two dimensions as a column vector or in i, j component form. Vector addition adds corresponding components, and scalar multiplication multiplies every component by the same scalar. For example, (3i + 2j) + (i − 4j) = 4i − 2j, while 2(3i + 2j) = 6i + 4j. The magnitude (length) of a vector is given by Pythagoras: |ai + bj| = √(a² + b²).
两个向量 a = a₁i + a₂j 与 b = b₁i + b₂j 的点积（数量积）定义为 a · b = a₁b₁ + a₂b₂，它也可以用角度表示：a · b = |a||b| cosθ，其中 θ 是两向量的夹角。把两条式子联系起来即可求夹角：cosθ = (a · b) / (|a||b|)。当 a · b = 0 时，两向量垂直。
The dot product (scalar product) of two vectors a = a₁i + a₂j and b = b₁i + b₂j is defined as a · b = a₁b₁ + a₂b₂, and it can also be expressed in terms of the angle: a · b = |a||b| cosθ, where θ is the angle between them. Linking the two forms lets you find the angle: cosθ = (a · b) / (|a||b|). When a · b = 0, the two vectors are perpendicular.
向量的几何应用包括：证明两条线平行（一个向量是另一个的数乘）、求位置向量、用向量描述几何图形（如平行四边形中两条对角线互相平分）。这些题目通常要求你把几何语言翻译成向量语言，再用代数方法简洁地证明结论。
Geometric applications of vectors include proving that two lines are parallel (one vector is a scalar multiple of the other), finding position vectors, and describing geometric shapes with vectors (such as the diagonals of a parallelogram bisecting each other). These questions usually ask you to translate geometric language into vector language, then prove the conclusion cleanly using algebra.
十、考试技巧：如何正确使用公式表 | Exam Technique: Using the Formula Booklet Correctly
AQA 的 AS 数学考试会随卷提供一份公式表（formula booklet），里面列出了三角恒等式、二项式展开、微积分公式和统计表等标准结果。用好这份公式表的前提是”知道每一条公式在哪一页、什么时候用”，而不是在考场上才第一次翻看。考前把公式表通读一遍，能帮你快速定位。
The AQA AS mathematics exam provides a formula booklet alongside the paper, listing standard results such as trigonometric identities, the binomial expansion, calculus formulae, and statistical tables. Using this booklet well means “knowing which page each formula is on and when to use it”, rather than opening it for the first time in the exam room. Reading the booklet through once before the exam helps you locate things quickly.
需要特别注意的是，公式表只覆盖”标准结果”，许多重要工具并不在里面，例如配方法、判别式的含义、链式法则的熟练运用、以及积分的”逆运算”思路。这些必须靠平时的练习内化。公式表是提示而非替代，扎实的基础才是拿分的关键。
It is important to note that the booklet only covers “standard results”; many essential tools are not in it, such as completing the square, the meaning of the discriminant, fluent use of the chain rule, and the “reverse of differentiation” way of thinking about integration. These must be internalised through practice. The booklet is a reminder, not a substitute; solid foundations are what win the marks.
答题时养成两个习惯：第一，写公式时先写下你正在使用的标准结果，再代入数字，这样即使算错也能拿到方法分；第二，对每一问的答案做合理性检查，例如求出的长度应为正、概率应在 0 到 1 之间、角度应在给定区间内。这种”回头检查”能帮你抓住不少被粗心偷走的分数。
Develop two habits while answering. First, write down the standard result you are using before substituting numbers, so that you earn method marks even if the arithmetic goes wrong. Second, sanity-check every answer: a length should be positive, a probability should lie between 0 and 1, and an angle should fall within the given interval. This kind of “look back” check recovers plenty of marks that carelessness would otherwise steal.
十一、多项式除法与因式定理 | Polynomial Division and the Factor Theorem
多项式与代数分式是 AS 纯数学中承上启下的内容。因式定理（factor theorem）指出：若 f(a) = 0，则 (x − a) 是多项式 f(x) 的因式。它的逆命题同样成立：若 (x − a) 是 f(x) 的因式，则 f(a) = 0。这条定理把”求多项式的根”与”分解因式”直接联系起来。
Polynomials and algebraic fractions are a bridging topic in AS pure mathematics. The factor theorem states that if f(a) = 0, then (x − a) is a factor of the polynomial f(x). The converse also holds: if (x − a) is a factor of f(x), then f(a) = 0. This theorem directly connects “finding roots of a polynomial” with “factorising it”.
例如 f(x) = x³ − 4x² + x + 6，代入 x = −1 得 f(−1) = −1 − 4 − 1 + 6 = 0，因此 (x + 1) 是 f(x) 的因式。用多项式除法（或综合除法）除以 (x + 1)，得到商 x² − 5x + 6，再分解为 (x − 2)(x − 3)，最终 f(x) = (x + 1)(x − 2)(x − 3)。多项式长除法与代数中的长除法思路完全一致。
For example, for f(x) = x³ − 4x² + x + 6, substituting x = −1 gives f(−1) = −1 − 4 − 1 + 6 = 0, so (x + 1) is a factor. Dividing by (x + 1) using polynomial long division (or synthetic division) gives the quotient x² − 5x + 6, which factorises further to (x − 2)(x − 3), so f(x) = (x + 1)(x − 2)(x − 3). Polynomial long division follows exactly the same idea as long division in ordinary arithmetic.
余数定理（remainder theorem）是因式定理的推广：f(x) 除以 (x − a) 的余数等于 f(a)。当 f(a) = 0 时余数为零，就退化成了因式定理。这两条定理配合长除法，构成了求解三次及以上多项式方程的完整工具链。
The remainder theorem is a generalisation of the factor theorem: the remainder when f(x) is divided by (x − a) is equal to f(a). When f(a) = 0 the remainder is zero, which reduces back to the factor theorem. Together with long division, these two theorems form the complete toolkit for solving polynomial equations of degree three and above.
十二、指数增长与衰减模型 | Exponential Growth and Decay Models
指数函数不只是抽象的代数对象，更是描述现实世界增长与衰减的模型。当某个量的变化率与它当前的大小成正比时，这个量就遵循指数增长或指数衰减。AS 阶段最常见的模型是 y = aekt：k > 0 表示增长，k < 0 表示衰减，a 是初始值（t = 0 时的值）。
Exponential functions are not just abstract algebraic objects; they are the models that describe real-world growth and decay. When the rate of change of a quantity is proportional to its current size, the quantity follows exponential growth or exponential decay. The most common model at AS level is y = aekt: k > 0 represents growth, k < 0 represents decay, and a is the initial value (the value when t = 0).
典型应用包括复利计算、细菌繁殖（增长）和放射性衰变、药物在体内的消除（衰减）。解题时通常给出两组数据，第一组确定初始值 a，第二组代入模型解出 k。例如某放射性物质初始 100 g，10 天后剩 80 g，代入 80 = 100e10k，两边除以 100 并取 ln，得 10k = ln 0.8，从而 k = (ln 0.8)/10 ≈ −0.0223。
Typical applications include compound interest, bacterial reproduction (growth), and radioactive decay or the elimination of a drug from the body (decay). Problem solving usually gives two pieces of data: the first determines the initial value a, and the second is substituted into the model to solve for k. For example, a radioactive substance starts at 100 g and falls to 80 g after 10 days, so substitute 80 = 100e10k, divide both sides by 100 and take ln to get 10k = ln 0.8, hence k = (ln 0.8)/10 ≈ −0.0223.
这类题目常要求学生用对数求解并解释参数含义：半衰期（衰减到一半所需时间）由 ekt = 1/2 解得 t = (ln 0.5)/k。理解参数 k 与”翻倍时间””半衰期”之间的关系，是建模题拿分的关键。
These questions often ask students to solve with logarithms and interpret the meaning of the parameters: the half-life (the time taken to decay to half) is found from ekt = 1/2 as t = (ln 0.5)/k. Understanding the relationship between the parameter k and concepts like “doubling time” and “half-life” is the key to scoring on modelling questions.
Summary | 总结
本文系统梳理了 AQA AS 数学纯数学部分的核心公式与关键技巧：二项式展开的通项与系数、三角恒等式与倍角公式、二次函数的判别式与配方法、坐标几何中的直线与圆、等差与等比数列的求和、指数与对数的运算律、微分的三大法则、积分的逆运算与面积计算，以及二维向量的点积与几何应用。掌握这些公式并理解其应用场景，是应对 AS 纯数学考试的基础。
This article has systematically organised the core formulae and key techniques of the pure mathematics component of AQA AS mathematics: the general term and coefficients of the binomial expansion, trigonometric identities and double-angle formulae, the discriminant and completing the square for quadratics, straight lines and circles in coordinate geometry, sums of arithmetic and geometric sequences, the laws of exponentials and logarithms, the three main rules of differentiation, integration as the reverse of differentiation with area calculations, and the dot product and geometric applications of two-dimensional vectors. Mastering these formulae and understanding when each applies is the foundation for tackling the AS pure mathematics paper.
建议的学习路径是：先用本文核对每条公式的适用条件，再针对性地做历年真题，尤其是”求系数””求切线方程””求面积”这三类高频题型。考试时善用公式表定位标准结果，同时牢记公式表之外的基本功要靠平时积累。扎实掌握这些核心内容后，AS 纯数学的高分将水到渠成。
The recommended study path is: first use this article to check the conditions under which each formula applies, then practise with targeted past-paper questions, especially the three high-frequency types of “find the coefficient”, “find the tangent equation”, and “find the area”. In the exam, use the formula booklet to locate standard results, while remembering that the fundamentals beyond the booklet must be built up through regular practice. Once these core ideas are firmly in hand, strong marks in AS pure mathematics will follow naturally.
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