📚 AS AQA Mathematics 9660 International AS MA01 Pure Mathematics Unit 1 Revision | AS AQA 数学 9660 国际 AS MA01 纯数学单元1复习
This revision guide covers the core content of AQA International AS Mathematics (9660) Unit MA01 Pure Mathematics 1. It is designed to reinforce key techniques, common exam traps and efficient problem-solving strategies for the AS Pure paper.
本复习指南涵盖 AQA 国际 AS 数学（9660）单元 MA01 纯数学 1 的核心内容。旨在强化关键技巧、常见考试陷阱以及 AS 纯数学试卷的高效解题策略。
1. Surds and Indices | 根式与指数
Surds are irrational roots such as √2 or ³√5. They are simplified using the rules √ab = √a × √b and √a/b = √a / √b, and rationalised so the denominator contains no surd.
根式是无理根，例如 √2 或 ³√5。化简根式使用法则 √ab = √a × √b 以及 √a/b = √a / √b，并需分母有理化，使分母不含根号。
aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1 and a⁻ⁿ = 1/aⁿ. Fractional indices represent roots: a1/n = ⁿ√a and am/n = ⁿ√aᵐ.
指数法则包括 aᵐ × aⁿ = aᵐ⁺ⁿ、aᵐ ÷ aⁿ = aᵐ⁻ⁿ、(aᵐ)ⁿ = aᵐⁿ、a⁰ = 1 和 a⁻ⁿ = 1/aⁿ。分数指数表示根式：a1/n = ⁿ√a，am/n = ⁿ√aᵐ。
When simplifying expressions, always write answers with positive indices unless the question states otherwise. Rationalise denominators such as 1/√2 by multiplying by √2/√2.
化简表达式时，除非题目另有说明，答案应使用正指数。遇到如 1/√2 的分母有理化，可乘以 √2/√2。
2. Quadratic Functions and the Discriminant | 二次函数与判别式
be solved by factorising, completing the square, or using the quadratic formula x = (-b ± √b² - 4ac) / 2a. The discriminant Δ = b² - 4ac tells you the nature of the roots.
二次方程 ax² + bx + c = 0 可通过因式分解、配方法或求根公式 x = (-b ± √b² - 4ac) / 2a 求解。判别式 Δ = b² - 4ac 揭示根的性质。
Δ = b² - 4ac
al root; if Δ < 0 there are no real roots. This is often tested with unknown coefficients.
若 Δ > 0，有两个相异实根；若 Δ = 0，有一个重根；若 Δ < 0，无实根。此类问题常与未知系数结合考查。
Completing the square rewrites ax² + bx + c as a(x + p)² + q. This form gives the vertex (-p, q) and helps with sketching graphs and finding maximum or minimum values.
配方法将 ax² + bx + c 写成 a(x + p)² + q。该形式直接给出顶点 (-p, q)，便于画图和求最大或最小值。
3. Simultaneous Equations and Inequalities | 联立方程与不等式
Simultaneous equations can be solved by substitution or elimination. When one equation is linear and the other is quadratic, substitute the linear equation into the quadratic and solve the resulting quadratic equation.
联立方程可用代入法或消元法求解。当一个方程为线性、另一个为二次时，将线性方程代入二次方程，再解所得二次方程。
Always interpret the solutions graphically: a line and a curve may intersect at 0, 1 or 2 points. If the question asks for intersection points, give both coordinates.
一定要从图形上理解解的含义：直线与曲线可能相交于 0、1 或 2 个点。若题目要求交点，需写出两个坐标。
Linear and quadratic inequalities should be solved by rearranging to 0, factorising, and using a sign diagram. Remember to reverse the inequality sign when multiplying or dividing by a negative number.
线性与二次不等式可通过移项使右边为 0，因式分解后用符号表求解。注意乘或除以负数时，不等号方向要改变。
4. Coordinate Geometry: Straight Lines | 坐标几何：直线
- x₁). The equation of a straight line can be written as y - y₁ = m(x - x₁) or y = mx + c.
过两点 (x₁, y₁) 和 (x₂, y₂) 的直线斜率为 m = (y₂ - y₁)/(x₂ - x₁)。直线方程可写为 y - y₁ = m(x - x₁) 或 y = mx + c。
m = (y₂ - y₁) / (x₂ - x₁)
₁ × m₂ = -1. This is a frequent exam skill when finding equations of tangents and normals.
平行直线斜率相等。垂直直线斜率之积为 -1，即 m₁ × m₂ = -1。这是求切线和法线方程时经常考查的技能。
found by √((x₂ - x₁)² + (y₂ - y₁)²). Midpoint coordinates are ((x₁ + x₂)/2, (y₁ + y₂)/2).
两点间距离公式为 √((x₂ - x₁)² + (y₂ - y₁)²)。中点坐标为 ((x₁ + x₂)/2, (y₁ + y₂)/2)。
5. Coordinate Geometry: Circles | 坐标几何：圆
(y - b)² = r². The general form x² + y² + 2gx + 2fy + c = 0 has centre (-g, -f) and radius √g² + f² - c.
圆心为 (a, b)、半径为 r 的圆方程为 (x - a)² + (y - b)² = r²。一般式 x² + y² + 2gx + 2fy + c = 0 的圆心为 (-g, -f)，半径为 √g² + f² - c。
(x - a)² + (y - b)² = r²
To find the equation of a tangent or normal to a circle, first find the gradient of the radius, then use perpendicularity. The tangent is perpendicular to the radius at the point of contact.
求圆的切线或法线方程时，先求半径的斜率，再利用垂直关系。切线与过切点的半径垂直。
A line can intersect a circle at two points, touch it at one point, or miss it entirely. Use the discriminant after substituting the line equation to decide which case occurs.
直线与圆可能相交于两点、相切于一点或完全不相交。将直线方程代入圆方程后利用判别式可判断属于哪种情形。
6. Trigonometry: Ratios and Identities | 三角比与恒等式
n AS work, angles may be in degrees or radians, and exact values for 0, 30, 45, 60 and 90 degrees are essential.
对于锐角 θ，sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。在 AS 内容中，角度可用度数或弧度表示，0、30、45、60 和 90 度的精确值必须掌握。
sin² θ + cos² θ = 1
o sin² θ = 1 - cos² θ and cos² θ = 1 - sin² θ. You should also know tan θ = sin θ / cos θ.
恒等式 sin² θ + cos² θ = 1 可推出 sin² θ = 1 - cos² θ 和 cos² θ = 1 - sin² θ。还需掌握 tan θ = sin θ / cos θ。
e used for non-right-angled triangles. Use sine rule for two angles and a side, cosine rule for three sides or two sides with an included angle.
正弦定理 a/sin A = b/sin B = c/sin C 和余弦定理 a² = b² + c² - 2bc cos A 用于非直角三角形。已知两角一边用正弦定理，已知三边或两边一夹角用余弦定理。
7. Trigonometric Equations | 三角方程
graphs to locate all solutions in the given interval, usually 0° ≤ x ≤ 360° or 0 ≤ x ≤ 2π.
解如 sin x = 1/2 的方程时，先求主解，再利用正弦、余弦或正切图像的对称性，在给定区间（通常 0° ≤ x ≤ 360° 或 0 ≤ x ≤ 2π）内找出所有解。
Always check that solutions lie within the valid range -1 ≤ sin x ≤ 1 and -1 ≤ cos x ≤ 1.
对于含 sin x 或 cos x 的二次方程，先因式分解再分别求解。必须检查解是否满足 -1 ≤ sin x ≤ 1 和 -1 ≤ cos x ≤ 1 的有效范围。
Use identities to reduce equations to a single trigonometric ratio. For example, replace sin² x with 1 - cos² x to solve equations involving both sin x and cos x.
利用恒等式将方程化为单一三角比。例如，将 sin² x 替换为 1 - cos² x，可解同时含 sin x 和 cos x 的方程。
8. Differentiation: Rules and Applications | 微分：法则与应用
The derivative of xⁿ is n xⁿ⁻¹. This works for any real power n, including negative and fractional powers, so rewrite expressions like 1/x² as x⁻² and √x as x1/2 before differentiating.
xⁿ 的导数是 n xⁿ⁻¹。该法则适用于任意实数指数 n，包括负指数和分数指数，因此求导前应先将 1/x² 写成 x⁻²，将 √x 写成 x1/2。
d/dx (xⁿ) = n xⁿ⁻¹
y₁) is y - y₁ = m(x - x₁), where m = dy/dx evaluated at x₁. The normal has gradient -1/m.
曲线在某点的斜率是该点 dy/dx 的值。点 (x₁, y₁) 处的切线方程为 y - y₁ = m(x - x₁)，其中 m 为 dy/dx 在该点的值。法线的斜率为 -1/m。
the second derivative or sign change to classify maxima, minima and points of inflection.
当 dy/dx > 0 时函数递增，dy/dx < 0 时函数递减。驻点处 dy/dx = 0；可用二阶导数或符号变化判断极大值、极小值和拐点。
9. Integration: Definite and Indefinite | 积分：不定积分与定积分
xⁿ⁺¹ / (n + 1) + C, where C is the constant of integration. Rewrite terms such as 1/x² or √x using negative or fractional powers before integrating.
积分是微分的逆运算。对于 n ≠ -1，∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + C，其中 C 为积分常数。积分前应先将 1/x² 或 √x 改写为负指数或分数指数形式。
∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + C, n ≠ -1
reas below the x-axis are negative, so split the interval if the curve crosses the x-axis.
定积分 ∫ₐᵇ f(x) dx 表示曲线 y = f(x)、x 轴以及直线 x = a 和 x = b 之间所围的有向面积。x 轴下方的面积为负，因此若曲线穿过 x 轴，需要拆分区间。
To find the area between a curve and a straight line, integrate the difference between the two functions over the relevant interval. Always subtract the lower curve from the upper curve.
求曲线与直线之间的面积时，在相应区间上对两个函数的差进行积分。务必用上方曲线的函数减去下方曲线的函数。
10. Exponential and Logarithmic Functions | 指数与对数函数
The natural exponential function y = eˣ has derivative dy/dx = eˣ and integral ∫ eˣ dx = eˣ + C. The natural logarithm y = ln x is the inverse of eˣ, so ln eˣ = x and eln x = x.
自然指数函数 y = eˣ 的导数为 dy/dx = eˣ，积分为 ∫ eˣ dx = eˣ + C。自然对数 y = ln x 是 eˣ 的反函数，因此 ln eˣ = x 且 eln x = x。
logₐ(xy) = logₐx + logₐy
logₐy, logₐ(x/y) = logₐx - logₐy and logₐ(xⁿ) = n logₐx. These help solve equations like 2ˣ = 5 by taking logarithms of both sides.
对数法则为 logₐ(xy) = logₐx + logₐy、logₐ(x/y) = logₐx - logₐy 和 logₐ(xⁿ) = n logₐx。这些法则可用于解如 2ˣ = 5 的方程，方法是两边取对数。
Exponential growth and decay problems often use models such as y = a ekt. Substitute given values to find constants, then answer questions about initial amount, half-life or doubling time.
指数增长与衰减问题常用模型 y = a ekt。代入给定数据求出常数，然后回答关于初始量、半衰期或倍增时间的问题。
11. Binomial Expansion | 二项展开
The binomial expansion for positive integer n is (a + b)ⁿ = Σ [nCr aⁿ⁻ʳ bʳ], where nCr = n! / (r!(n - r)!). The coefficients can be found from Pascal's triangle or the nCr button on a calculator.
正整数指数 n 的二项展开式为 (a + b)ⁿ = Σ [nCr aⁿ⁻ʳ bʳ]，其中 nCr = n! / (r!(n - r)!)。系数可由杨辉三角或计算器上的 nCr 键得到。
(1 + x)ⁿ = 1 + nx + [n(n - 1) / 2!] x² + ... for |x| < 1
For rational or negative powers, the expansion (1 + x)ⁿ = 1 + nx + [n(n - 1) / 2!] x² + [n(n - 1)(n - 2) / 3!] x³ + ... is valid only when |x| < 1. Give the range of validity in exam answers.
对于有理数或负数指数，展开式 (1 + x)ⁿ = 1 + nx + [n(n - 1) / 2!] x² + [n(n - 1)(n - 2) / 3!] x³ + ... 仅在 |x| < 1 时有效。考试答案中需写出有效范围。
When expanding expressions like (2 + 3x)⁵, take out a factor to write it as 2⁵(1 + (3/2)x)⁵, so the standard form (1 + x)ⁿ can be used.
展开如 (2 + 3x)⁵ 的表达式时，可提取公因子写成 2⁵(1 + (3/2)x)⁵，以便使用标准形式 (1 + x)ⁿ。
12. Functions, Graphs and Transformations | 函数、图像与变换
function f⁻¹(x) reverses the effect of f, and its graph is a reflection in the line y = x.
函数 f 将每个输入 x 映射到唯一输出 f(x)。复合函数 fg(x) = f(g(x)) 先应用 g。反函数 f⁻¹(x) 逆转 f 的作用，其图像关于直线 y = x 对称。
Transformations of the graph y = f(x) include y = f(x) + a
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