📚 AS AQA Mathematics MA01 Pure Mathematics Unit 1 Complete Guide | AS AQA 数学 MA01 纯数学单元一 完整指南
The AQA International AS Mathematics MA01 Pure Mathematics Unit 1 forms the foundation of your mathematical studies, covering essential algebra, coordinate geometry, sequences, trigonometry, and the fundamentals of calculus. This guide consolidates every major topic into a single structured revision resource, aligned with the AQA 9660 specification.
AQA 国际 AS 数学 MA01 纯数学单元一是你数学学习的基石，涵盖基础代数、坐标几何、数列、三角学以及微积分基础。本指南将所有主要主题整合为一份结构化的复习资料，严格对齐 AQA 9660 考纲要求。
1. Algebra and Functions | 代数与函数
Quadratics lie at the heart of AS Algebra. A quadratic equation has the general form ax² + bx + c = 0, where a ≠ 0. You must be able to solve these by factorisation, completing the square, and the quadratic formula. The quadratic formula is the most reliable method and is stated below.
二次函数是 AS 代数部分的核心。二次方程的一般形式为 ax² + bx + c = 0，其中 a ≠ 0。你必须掌握因式分解法、配方法和二次公式三种解法。二次公式是最可靠的方法，如下所示。
x = (−b ± √(b² − 4ac)) / (2a)
Completing the square rewrites a quadratic in the form a(x + p)² + q. For example, x² + 6x + 5 = (x + 3)² − 4. This form is invaluable for finding the vertex of a parabola and solving inequalities.
配方法将二次式改写为 a(x + p)² + q 的形式。例如，x² + 6x + 5 = (x + 3)² − 4。这种形式在求抛物线顶点和解不等式时非常有用。
The discriminant, Δ = b² − 4ac, determines the nature of the roots of ax² + bx + c = 0:
判别式 Δ = b² − 4ac 决定 ax² + bx + c = 0 的根的性质：
- Δ > 0: two distinct real roots / 两个不相等的实数根
- Δ = 0: one repeated real root / 一个重根（两个相等实数根）
- Δ < 0: no real roots (two complex roots) / 无实数根（两个复数根）
For quadratic inequalities, factorise first, then sketch the parabola or test intervals. If (x − 2)(x + 3) > 0, the critical values are x = −3 and x = 2, and the solution is x < −3 or x > 2. If the inequality sign is reversed, the solution lies between the roots.
解二次不等式时，先因式分解，然后画出抛物线草图或测试区间。若 (x − 2)(x + 3) > 0，临界值为 x = −3 和 x = 2，解集为 x < −3 或 x > 2。若不等号方向相反，则解位于两根之间。
A function f maps every input value (domain) to exactly one output value (range). The inverse function f⁻¹ reverses this mapping. To find f⁻¹, rearrange y = f(x) to make x the subject and write x = f⁻¹(y). The composition fg(x) means apply g first, then apply f to the result. Always check that the output of g lies within the domain of f.
函数 f 将每个输入值（定义域）映射到唯一的一个输出值（值域）。反函数 f⁻¹ 逆转这一映射。求 f⁻¹ 时，将 y = f(x) 重新排列为以 x 为主项，并写成 x = f⁻¹(y)。复合函数 fg(x) 表示先应用 g，再将结果应用于 f。注意检查 g 的输出是否在 f 的定义域内。
2. Coordinate Geometry | 坐标几何
The equation of a straight line is expressed as y = mx + c, where m is the gradient and c is the y-intercept. Given two points (x₁, y₁) and (x₂, y₂), the gradient is:
直线方程表示为 y = mx + c，其中 m 是斜率，c 是 y 轴截距。已知两点 (x₁, y₁) 和 (x₂, y₂)，斜率为：
m = (y₂ − y₁) / (x₂ − x₁)
Parallel lines have equal gradients; perpendicular lines have gradients whose product is −1. The equation of a line through a known point with a known gradient can be written as y − y₁ = m(x − x₁). The distance between two points is given by √((x₂ − x₁)² + (y₂ − y₁)²), and the midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2).
平行线的斜率相等；垂线的斜率乘积为 −1。过已知点且斜率为 m 的直线方程可写为 y − y₁ = m(x − x₁)。两点间距离为 √((x₂ − x₁)² + (y₂ − y₁)²)，中点为 ((x₁ + x₂)/2, (y₁ + y₂)/2)。
The equation of a circle with centre (a, b) and radius r is (x − a)² + (y − b)² = r². When the equation is expanded, it can be written as x² + y² + 2gx + 2fy + c = 0. To find the centre and radius, complete the square for both x and y terms. For example, x² + y² − 6x + 4y − 12 = 0 becomes (x − 3)² + (y + 2)² = 25, giving centre (3, −2) and radius 5.
圆心为 (a, b)、半径为 r 的圆的方程为 (x − a)² + (y − b)² = r²。方程展开后可写为 x² + y² + 2gx + 2fy + c = 0。要求圆心和半径，需对 x 项和 y 项分别配方法。例如，x² + y² − 6x + 4y − 12 = 0 可化为 (x − 3)² + (y + 2)² = 25，圆心为 (3, −2)，半径为 5。
When solving intersections between a line and a curve, substitute the line equation into the curve equation to obtain a quadratic. Using the discriminant:
求直线与曲线的交点时，将直线方程代入曲线方程得到二次方程。利用判别式：
- Δ > 0: two intersection points / 两个交点
- Δ = 0: one point (tangency) / 一个交点（相切）
- Δ < 0: no intersection / 无交点
3. Sequences and Series | 数列与级数
An arithmetic sequence has a constant common difference d between consecutive terms. The n-th term is uₙ = a + (n − 1)d, where a is the first term. The sum of the first n terms is:
等差数列的相邻项之间存在常数公差 d。第 n 项为 uₙ = a + (n − 1)d，其中 a 是首项。前 n 项和为：
Sₙ = (n/2)[2a + (n − 1)d] = (n/2)(a + l)
where l is the last term. For example, the sum of the first 20 terms of 3, 7, 11, … is (20/2)[2(3) + 19(4)] = 10 × 82 = 820.
其中 l 是末项。例如，数列 3, 7, 11, … 前 20 项之和为 (20/2)[2(3) + 19(4)] = 10 × 82 = 820。
A geometric sequence has a constant common ratio r. The n-th term is uₙ = arⁿ⁻¹, and the sum of the first n terms is:
等比数列的相邻项之间存在常数公比 r。第 n 项为 uₙ = arⁿ⁻¹，前 n 项和为：
Sₙ = a(1 − rⁿ) / (1 − r) (r ≠ 1)
When |r| < 1, the infinite sum converges to S∞ = a / (1 − r). For instance, with a = 100 and r = 0.5, the sum to infinity is 200. A key AQA question will often ask you to determine whether a series converges and then evaluate the infinite sum.
当 |r| < 1 时，无穷级数收敛于 S∞ = a / (1 − r)。例如，a = 100，r = 0.5 时，无穷项和为 200。AQA 的常见考题会要求你判断级数是否收敛，然后求无穷项和。
The binomial theorem expands (a + b)ⁿ for a positive integer n:
二项式定理展开 (a + b)ⁿ（n 为正整数）：
(a + b)ⁿ = aⁿ + ⁿC₁ aⁿ⁻¹b + ⁿC₂ aⁿ⁻²b² + … + bⁿ
where ⁿCᵣ = n! / [r!(n − r)!]. For (1 + x)⁵ = 1 + 5x + 10x² + 10x³ + 5x⁴ + x⁵. In exam questions, always write out the first few terms and be careful with coefficients — a common mistake is forgetting the binomial coefficients. When asked in the form (a + bx)ⁿ, expand (a + bx)ⁿ = aⁿ(1 + (bx/a))ⁿ and then apply the binomial expansion.
其中 ⁿCᵣ = n! / [r!(n − r)!]。例如 (1 + x)⁵ = 1 + 5x + 10x² + 10x³ + 5x⁴ + x⁵。考试中务必写出前几项并注意系数——常见错误是漏写二项式系数。若题中给出 (a + bx)ⁿ 的形式，可先化为 aⁿ(1 + (bx/a))ⁿ 再展开。
4. Trigonometry | 三角学
In Pure Mathematics Unit 1, you must be proficient in both degrees and radians. Key conversions are π radians = 180° and 1 radian = 180°/π ≈ 57.3°. You should memorise common exact values such as sin 30° = ½, cos 60° = ½, tan 45° = 1, sin 90° = 1, and the corresponding radian measures (π/6, π/4, π/2, etc.).
在纯数学单元一中，你必须熟练使用角度制和弧度制。关键换算为 π 弧度 = 180°，1 弧度 = 180°/π ≈ 57.3°。你应熟记常见精确值，如 sin 30° = ½，cos 60° = ½，tan 45° = 1，sin 90° = 1，以及对应的弧度（π/6、π/4、π/2 等）。
The three fundamental trigonometric functions are defined for a right-angled triangle as sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, and tan θ = opposite/adjacent. The two identities you must know for this unit are:
三个基本三角函数在直角三角形中定义为 sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。本单元必须掌握的两个恒等式为：
sin²θ + cos²θ = 1 and tanθ = sinθ / cosθ
To solve trigonometric equations, for example 2sinθ = 1 in the interval 0° ≤ θ ≤ 360°, first find the principal value θ = 30°. Since sine is positive in the first and second quadrants, the full solution set is θ = 30° and θ = 150°. For cosecant, secant, and cotangent, use the identities csc θ = 1/sin θ, sec θ = 1/cos θ, and cot θ = 1/tan θ.
解三角方程时，例如在区间 0° ≤ θ ≤ 360° 内解 2sinθ = 1，首先求主值 θ = 30°。由于正弦在第一、第二象限为正，完整解集为 θ = 30° 和 θ = 150°。对于余割、正割和余切函数，使用恒等式 csc θ = 1/sin θ，sec θ = 1/cos θ，cot θ = 1/tan θ。
The graphs of y = sin x, y = cos x, and y = tan x have characteristic shapes you should be able to sketch. The sine and cosine graphs have amplitude 1 and period 360° (or 2π radians); the tangent graph has asymptotes at 90° and 270° and a period of 180° (π radians). Another very important skill is transforming these graphs — for instance, y = 3sin(2x) has amplitude 3 and period 180°.
y = sin x、y = cos x 和 y = tan x 的图像具有特征形状，你需要能准确画出草图。正弦和余弦图像的振幅为 1，周期为 360°（或 2π 弧度）；正切图像在 90° 和 270° 处有渐近线，周期为 180°（π 弧度）。另一个重要技能是图像变换——例如，y = 3sin(2x) 的振幅为 3，周期为 180°。
5. Exponentials and Logarithms | 指数与对数
An exponential function has the form y = aˣ where a > 0 and a ≠ 1. The most important exponential is y = eˣ, where e ≈ 2.718. Exponential models describe growth and decay in real-world contexts such as population, radioactive decay, and cooling processes.
指数函数的形式为 y = aˣ，其中 a > 0 且 a ≠ 1。最重要的指数函数是 y = eˣ，其中 e ≈ 2.718。指数模型用于描述人口增长、放射性衰变和冷却过程等现实场景中的增长与衰减。
A logarithm is the inverse operation of exponentiation. If y = aˣ, then log_a y = x. The natural logarithm ln x is defined as log_e x. The laws of logarithms are essential for solving exponential equations:
对数是指数运算的逆运算。若 y = aˣ，则 log_a y = x。自然对数 ln x 定义为 log_e x。对数运算法则是解指数方程的关键：
- log_a (xy) = log_a x + log_a y / 积的对数 = 对数之和
- log_a (x/y) = log_a x − log_a y / 商的对数 = 对数之差
- log_a (xⁿ) = n log_a x / 幂的对数 = 指数倍的对数
To solve an equation such as 3ˣ = 20, take logs of both sides: ln(3ˣ) = ln(20), so x ln 3 = ln 20, and x = ln 20 / ln 3 ≈ 2.727. Alternatively, you may use log₁₀, but be consistent — do not mix bases in a single calculation.
解方程 3ˣ = 20 时，两边取对数：ln(3ˣ) = ln(20)，因此 x ln 3 = ln 20，得 x = ln 20 / ln 3 ≈ 2.727。你也可以使用 log₁₀，但要保持一致性——在同一计算中不要混用底数。
When modelling exponential growth, the equation is often written as N = N₀eᵏᵗ. Taking natural logs of both sides converts the equation into a linear form: ln N = ln N₀ + kt, so plotting ln N against t gives a straight line with gradient k. This is a common AQA exam style question and is worth practising carefully.
在指数增长模型中，方程通常写成 N = N₀eᵏᵗ。两边取自然对数可将方程转化为线性形式：ln N = ln N₀ + kt，因此以 ln N 对 t 作图得到斜率为 k 的直线。这是 AQA 常见的考法，值得反复练习。
6. Differentiation | 微分
Differentiation measures the instantaneous rate of change of a function. The derivative of xⁿ is nxⁿ⁻¹. For example, if y = x³, then dy/dx = 3x². For a polynomial function, differentiate term by term: if y = 4x² − 3x + 7, then dy/dx = 8x − 3. This is the fundamental technique of Pure Mathematics Unit 1.
微分衡量函数的瞬时变化率。xⁿ 的导数为 nxⁿ⁻¹。例如，若 y = x³，则 dy/dx = 3x²。对于多项式函数，逐项求导：若 y = 4x² − 3x + 7，则 dy/dx = 8x − 3。这是纯数学单元一的基本技巧。
d/dx (xⁿ) = nxⁿ⁻¹
The derivatives of trigonometric functions are: d/dx (sin x) = cos x and d/dx (cos x) = −sin x (when x is measured in radians). You also need to know the derivatives of exponential and logarithmic functions: d/dx (eˣ) = eˣ and d/dx (ln x) = 1/x.
三角函数的导数为：d/dx (sin x) = cos x，d/dx (cos x) = −sin x（x 以弧度为单位）。你还需要掌握指数和对数函数的导数：d/dx (eˣ) = eˣ，d/dx (ln x) = 1/x。
The geometric interpretation of the derivative is the gradient of the tangent line at a point. The tangent to a curve y = f(x) at x = a has gradient f′(a), and its equation is y − f(a) = f′(a)(x − a). The normal is perpendicular to the tangent, so its gradient is −1/f′(a), provided f′(a) ≠ 0.
导数的几何意义是曲线上某点处切线的斜率。曲线 y = f(x) 在 x = a 处的切线斜率为 f′(a)，切线方程为 y − f(a) = f′(a)(x − a)。法线垂直于切线，其斜率为 −1/f′(a)（假设 f′(a) ≠ 0）。
Stationary points occur where dy/dx = 0. To classify them, use the second derivative test: if d²y/dx² > 0, it is a local minimum; if d²y/dx² < 0, it is a local maximum; if d²y/dx² = 0, the test is inconclusive and you should examine the sign of dy/dx on either side of the point.
驻点出现在 dy/dx = 0 处。用二阶导数判别法对其分类：若 d²y/dx² > 0，则为局部极小值；若 d²y/dx² < 0，则为局部极大值；若 d²y/dx² = 0，则该判别法失效，需考察 dy/dx 在该点两侧的符号。
Practical optimisation problems require setting up a function, differentiating, and finding stationary points. For example, to find the maximum area enclosing a fixed perimeter using a rectangular fence, write the area as a quadratic, differentiate, set to zero, and solve.
实际优化问题需要建立函数、求导并找驻点。例如，用固定长度的篱笆围出最大面积的矩形，需写出面积函数（二次式），求导后令其为零，再解方程。
7. Integration | 积分
Integration is the reverse process of differentiation. The indefinite integral of xⁿ is:
积分是微分的逆运算。xⁿ 的不定积分为：
∫xⁿ dx = xⁿ⁺¹ / (n + 1) + c (n ≠ −1)
where c is the constant of integration. For example, ∫x² dx = x³/3 + c. Always include the constant of integration for indefinite integrals. Similar rules apply to trig functions: ∫cos x dx = sin x + c and ∫sin x dx = −cos x + c. These can be extended to sums by integrating term by term.
其中 c 是积分常数。例如，∫x² dx = x³/3 + c。求不定积分时务必加上积分常数。三角函数的积分规则类似：∫cos x dx = sin x + c，∫sin x dx = −cos x + c。对和函数逐项积分即可推广这些规则。
A definite integral has limits and produces a numerical value:
定积分带有上下限，产生一个数值：
∫ₐᵇ f(x) dx = [F(x)]ₐᵇ = F(b) − F(a)
The definite integral of a function over an interval gives the signed area between the curve and the x-axis. Areas above the x-axis are positive; areas below are negative. To find the total area when the curve crosses the x-axis within the interval, split the integral at the x-intercepts and take absolute values.
函数在某个区间上的定积分表示
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