📚 Support Pack 2: Contents | 支持包2：内容
This support pack brings together the core topics and common question types tested in AQA A-level Mathematics (7357). It is structured so that you can revise pure mathematics, statistics and mechanics in one place, with concise explanations paired in English and Chinese.
本支持包汇总了 AQA A-level 数学（7357）中考查的核心主题和常见题型。它让你可以在一个地方复习纯数学、统计和力学，并配有中英文对照的简明解释。
1. How to Use This Support Pack | 如何使用本支持包
Start by reading each section summary, then attempt the checkpoint questions before checking the solutions. Keep the AQA formula booklet beside you so that you learn where every formula appears in the real exam.
先阅读每节总结，然后在核对答案前尝试完成检查题。将 AQA 公式手册放在手边，以便熟悉每个公式在真实考试中的位置。
For each topic there are three levels: recall, fluency and exam-style. Tick off each level only when you can complete it without notes and within a realistic time limit.
每个主题都分为三个层次：回忆、熟练和考试题型。只有在不看笔记且能在合理时间内完成时，才勾选该层次。
Work through the sections in order because later mechanics and statistics questions often require pure techniques such as differentiation, integration and logarithms.
请按顺序学习各节，因为后面的力学和统计题经常需要纯数技巧，例如微分、积分和对数。
2. Pure Core: Algebra and Functions | 纯数核心：代数与函数
This section revises indices, surds, quadratic functions, simultaneous equations and inequalities. You must be able to complete the square and use the discriminant to determine the number of real roots.
本节复习指数、根式、二次函数、联立方程和不等式。你必须掌握配方法，并能用判别式判断实根的数量。
real roots, when Δ = 0 there is one repeated root, and when Δ < 0 there are no real roots.
方程 ax² + bx + c = 0 的判别式为 Δ = b² - 4ac。当 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。
x = (-b ± √b² - 4ac) / 2a
Sketching a quadratic graph requires you to identify the y-intercept, the line of symmetry and the vertex. Completing the square gives the vertex form a(x - h)² + k.
绘制二次函数图像需要确定 y 轴截距、对称轴和顶点。配方法可得到顶点式 a(x - h)² + k。
Watch out: when solving inequalities, multiplying or dividing by a negative number reverses the inequality sign. Always check your answer by substituting a test value.
注意：解不等式时，乘以或除以负数会改变不等号方向。始终代入一个检验值来检查答案。
3. Coordinate Geometry and Graphs | 坐标几何与图像
You need to work confidently with straight-line equations, parallel and perpendicular gradients, circles and parametric curves. A common exam task is to find the equation of a tangent or normal to a circle.
你需要熟练处理直线方程、平行与垂直斜率、圆和参数曲线。常见考题是求圆的切线或法线方程。
x₁). The equation of a circle with centre (a, b) and radius r is (x - a)² + (y - b)² = r².
对于两点 (x₁, y₁) 和 (x₂, y₂)，斜率 m = (y₂ - y₁) / (x₂ - x₁)。以 (a, b) 为圆心、r 为半径的圆方程为 (x - a)² + (y - b)² = r²。
m₂ = -1. The tangent to a circle at a point is perpendicular to the radius at that point.
两条直线垂直时，其斜率 m₁ 和 m₂ 满足 m₁ m₂ = -1。圆上一点的切线垂直于该点处的半径。
btain a Cartesian equation. For example, x = t + 1 and y = t² - 2 become y = (x - 1)² - 2.
处理参数方程时，消去参数 t 得到直角坐标方程。例如，x = t + 1 和 y = t² - 2 可化为 y = (x - 1)² - 2。
4. Trigonometry and Identities | 三角学与恒等式
This topic includes radians, arc length, sector area, sine and cosine rules, and solving trigonometric equations. You should also know the Pythagorean identities and the double-angle formulae.
本主题包括弧度制、弧长、扇形面积、正弦定理与余弦定理，以及解三角方程。你还应掌握毕达哥拉斯恒等式和二倍角公式。
a² = b² + c² - 2bc cos A. Arc length is rθ and sector area is ½ r²θ when θ is in radians.
正弦定理为 a / sin A = b / sin B = c / sin C。余弦定理为 a² = b² + c² - 2bc cos A。当 θ 以弧度为单位时，弧长为 rθ，扇形面积为 ½ r²θ。
ot²θ = cosec²θ. Double-angle formulae are sin 2θ = 2 sinθ cosθ and cos 2θ = cos²θ - sin²θ.
毕达哥拉斯恒等式包括 sin²θ + cos²θ = 1，1 + tan²θ = sec²θ 和 1 + cot²θ = cosec²θ。二倍角公式为 sin 2θ = 2 sinθ cosθ 和 cos 2θ = cos²θ - sin²θ。
Remember to set your calculator to radians for calculus questions involving trigonometric functions. AQA often gives angles in radians in pure mathematics papers.
记住在涉及三角函数的微积分题中将计算器设置为弧度模式。AQA 在纯数学试卷中通常使用弧度制给出角度。
5. Exponentials and Logarithms | 指数与对数
Exponential growth and decay models appear regularly in both pure and applied questions. You must be able to switch between index and logarithmic forms and use log laws to solve equations.
指数增长和衰减模型在纯数和应用题中经常出现。你必须能够在指数形式与对数形式之间转换，并运用对数法则解方程。
/y) = logₐx - logₐy and logₐ(xⁿ) = n logₐx. The natural logarithm ln x is the inverse of eˣ.
关键法则为 logₐ(xy) = logₐx + logₐy，logₐ(x/y) = logₐx - logₐy，logₐ(xⁿ) = n logₐx。自然对数 ln x 是 eˣ 的反函数。
ᵏᵗ, where N₀ is the initial amount and k is the growth constant. For decay, k is negative.
指数增长模型可写为 N = N₀eᵏᵗ，其中 N₀ 是初始量，k 是增长常数。对于衰减，k 为负值。
When solving e²ˣ tural logs on both sides: 2x = ln 5, so x = ½ ln 5. Avoid rounding until the final answer.
解 e²ˣ = 5 时，两边取自然对数：2x = ln 5，因此 x = ½ ln 5。在最终答案之前不要四舍五入。
6. Differentiation | 微分
This section focuses on the derivative as a rate of change, differentiation from first principles, and the rules for polynomials, exponentials, logarithms and trigonometric functions.
本节重点是将导数理解为变化率、从第一性原理求导，以及多项式、指数、对数和三角函数的求导法则。
The product rule is d/dx (uv) = u'v + uv'. The quotient rule is d/dx (u/v) = (u'v - uv') / v². The chain rule is dy/dx = dy/du × du/dx.
乘法法则为 d/dx (uv) = u'v + uv'。除法法则为 d/dx (u/v) = (u'v - uv') / v²。链式法则为 dy/dx = dy/du × du/dx。
² to classify them: if it is positive the point is a minimum, if negative it is a maximum.
驻点出现在 dy/dx = 0 处。使用二阶导数 d²y/dx² 判断其类型：若为正则是极小值点，若为负则是极大值点。
Common derivatives include d/dx (sin x) = cos x, d/dx (cos x) = -sin x, d/dx (eˣ) = eˣ and d/dx (ln x) = 1/x.
常见导数包括 d/dx (sin x) = cos x，d/dx (cos x) = -sin x，d/dx (eˣ) = eˣ，d/dx (ln x) = 1/x。
7. Integration | 积分
Integration is the reverse of differentiation. You should be confident with indefinite and definite integrals, areas under curves, and integration by substitution and by parts.
积分是微分的逆运算。你应熟练计算不定积分和定积分、曲线下面积，以及换元积分法和分部积分法。
The fundamental theorem gives ∫ f(x) dx = F(x) + C where F'(x) = f(x). The area between a curve and the x-axis from a to b is ∫ₐᵇ f(x) dx.
微积分基本定理给出 ∫ f(x) dx = F(x) + C，其中 F'(x) = f(x)。曲线与 x 轴在 a 到 b 之间的面积为 ∫ₐᵇ f(x) dx。
ding to the LIATE order: logarithmic, inverse trig, algebraic, trigonometric, exponential.
分部积分法使用 ∫ u dv = uv - ∫ v du。根据 LIATE 顺序选择 u：对数、反三角、代数、三角、指数。
When integrating by substitution, replace dx using du/dx and change the limits if it is a definite integral. Do not forget to return to the original variable for indefinite integrals.
使用换元积分法时，用 du/dx 替换 dx，如果是定积分还要更换上下限。对于不定积分，不要忘记换回原变量。
8. Sequences and Series | 序列与级数
Arithmetic and geometric sequences are the main focus, together with sigma notation and the binomial expansion. You need to know the condition for convergence of an infinite geometric series.
等差和等比数列是本节重点，还包括求和符号和二项展开式。你需要知道无穷等比级数收敛的条件。
For an arithmetic sequence, the nth term is a + (n - 1)d and the sum of n terms is n/2 [2a + (n - 1)d]. For a geometric sequence, the nth term is arⁿ⁻¹ and the sum is a(1 - rⁿ) / (1 - r) for r ≠ 1.
等差数列第 n 项为 a + (n - 1)d，前 n 项和为 n/2 [2a + (n - 1)d]。等比数列第 n 项为 arⁿ⁻¹，前 n 项和为 a(1 - rⁿ) / (1 - r)，其中 r ≠ 1。
S∞ = a / (1 - r), valid for |r| < 1
The binomial expansion of (1 + x)ⁿ is 1 + nx + n(n - 1)x² / 2! + n(n - 1)(n - 2)x³ / 3! + ... and is valid for |x| < 1 when n is not a positive integer.
(1 + x)ⁿ 的二项展开式为 1 + nx + n(n - 1)x² / 2! + n(n - 1)(n - 2)x³ / 3! + ...，当 n 不是正整数时，其有效范围为 |x| < 1。
Use sigma notation carefully: Σ from r = 1 to n of
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