IB AQA Maths: Last-Minute Revision Notes | IB AQA 数学：考前冲刺笔记

📚 IB AQA Maths: Last-Minute Revision Notes | IB AQA 数学：考前冲刺笔记

Preparing for your IB Mathematics exam with an AQA-style approach? This last-minute revision guide is designed to sharpen your recall on the most critical topics. We will walk through core concepts, essential formulas, and problem-solving strategies that often appear in exams. Keep this by your side in the final days before the test—each section pairs concise English explanations with Chinese translations to reinforce understanding.

正在用 AQA 风格备考 IB 数学考试吗？这份考前冲刺笔记旨在帮你快速唤醒最高频考点的记忆。我们将梳理核心概念、必备公式和常见题型的解题策略。在考前几天请把它放在手边——每个小节都配有精炼的英文解释和中文翻译，加深理解。

1. Algebraic Manipulation & Identities | 代数运算与恒等式

Mastering algebraic manipulation is the bedrock of IB Maths. Always check if an expression can be factorised by recognising common factors, difference of squares a² − b² = (a − b)(a + b), or perfect squares a² ± 2ab + b² = (a ± b)². For higher-degree polynomials, the factor theorem states that if f(p) = 0, then (x − p) is a factor. Use polynomial long division to fully factorise cubics and quartics.

代数运算是 IB 数学的基石。务必检查表达式是否可以因式分解：识别公因子、平方差 a² − b² = (a − b)(a + b) 或完全平方 a² ± 2ab + b² = (a ± b)²。对于高次多项式，因子定理指出，若 f(p) = 0，则 (x − p) 为因式。用多项式长除法来完全分解三次或四次多项式。

Remainder theorem: when f(x) is divided by (x − a), the remainder is f(a). Also, remember binomial expansion:

(1 + x)ⁿ = 1 + nx + n(n−1)/2! x² + … for |x| < 1, n ∈ ℚ

This expansion is valid for rational n, and you must know the factorial pattern for the coefficients.

余式定理：f(x) 除以 (x − a) 时，余数为 f(a)。此外，记住二项式展开：

(1 + x)ⁿ = 1 + nx + n(n−1)/2! x² + … （|x| < 1, n ∈ ℚ）

此展开对有理数 n 成立，系数需使用阶乘模式。

Simplify expressions step by step; never expand unless necessary.
逐步化简表达式，除非必要别急着展开。
Watch for hidden quadratics, e.g., 2²ˣ − 5·2ˣ + 4 = 0 can be set as y = 2ˣ.
留意隐藏的二次方程，如 2²ˣ − 5·2ˣ + 4 = 0，可设 y = 2ˣ。

2. Functions & Their Transformations | 函数及其变换

Functions are the language of IB Maths. Domain: all possible input values; Range: all possible output values. For composite function f(g(x)), ensure range of g is within domain of f. The inverse function f⁻¹(x) exists only if f is one-to-one; reflect in the line y = x.

函数是 IB 数学的语言。定义域：所有可能的输入值；值域：所有可能的输出值。对于复合函数 f(g(x))，确保 g 的值域在 f 的定义域内。反函数 f⁻¹(x) 仅在 f 为一一映射时存在；其图像关于直线 y = x 反射。

Transformations: y = a f(b(x + c)) + d

a: vertical stretch by factor a (if a < 0, reflection in x-axis).
a：垂直方向拉伸 a 倍（若 a < 0，则关于 x 轴反射）。
b: horizontal stretch by factor 1/b (if b < 0, reflection in y-axis).
b：水平方向拉伸 1/b 倍（若 b < 0，则关于 y 轴反射）。
c: horizontal translation by −c units.
c：水平平移 −c 个单位。
d: vertical translation by d units.
d：垂直平移 d 个单位。

Always apply transformations in order: horizontal shift, then stretch/reflection, then vertical shift. For trig functions, the period changes with b: period = 2π/|b| for sin, cos; π/|b| for tan.

始终按顺序进行变换：先水平平移，再伸缩/反射，最后垂直平移。对于三角函数，周期随 b 改变：正弦、余弦周期 = 2π/|b|；正切周期 = π/|b|。

3. Quadratic Equations & Inequalities | 二次方程与不等式

Quadratic formula: for ax² + bx + c = 0,

x = [ −b ± √(b² − 4ac) ] / (2a)

The discriminant Δ = b² − 4ac determines nature of roots: Δ > 0 → two distinct real roots; Δ = 0 → one repeated real root; Δ < 0 → no real roots.

二次方程求根公式：对于 ax² + bx + c = 0，

x = [ −b ± √(b² − 4ac) ] / (2a)

判别式 Δ = b² − 4ac 决定根的性质：Δ > 0 → 两个不等实根；Δ = 0 → 一个重根；Δ < 0 → 无实根。

Quadratic inequalities are solved by sketching the parabola. Factorise to find x-intercepts, then test intervals to see where ax² + bx + c > 0 or < 0. Remember to reverse the inequality sign when multiplying/dividing by a negative number.

解二次不等式需画出抛物线草图。因式分解求 x 轴交点，然后测试区间看 ax² + bx + c > 0 还是 < 0。记住乘以或除以负数时要翻转不等号方向。

Completing the square puts the quadratic in vertex form: a(x − h)² + k, where (h, k) is the vertex. This is key for finding maximum/minimum values.

配方法将二次式化为顶点式：a(x − h)² + k，其中 (h, k) 为顶点。这是求最大值/最小值的关键方法。

4. Exponential & Logarithmic Functions | 指数函数与对数函数

Exponentials and logarithms are inverses. Key relationship: y = aˣ ⇔ x = logₐ y. Natural logarithm ln x = logₑ x, with e ≈ 2.718. The natural exponential function eˣ is its own derivative.

指数函数与对数函数互为反函数。关键关系：y = aˣ ⇔ x = logₐ y。自然对数 ln x = logₑ x，其中 e ≈ 2.718。自然指数函数 eˣ 的导数等于其自身。

Log laws (base a):

logₐ (MN) = logₐ M + logₐ N
logₐ (M/N) = logₐ M − logₐ N
logₐ Mᵏ = k logₐ M
logₐ 1 = 0; logₐ a = 1
Change of base: logₐ b = logₓ b / logₓ a

对数运算律（底 a）：

logₐ (MN) = logₐ M + logₐ N
logₐ (M/N) = logₐ M − logₐ N
logₐ Mᵏ = k logₐ M
logₐ 1 = 0；logₐ a = 1
换底公式：logₐ b = logₓ b / logₓ a

Exponential growth/decay: y = A eᵏᵗ. If k > 0 growth, k < 0 decay. For interest, compound formula: A = P(1 + r/n)^(nt). Continuous compounding: A = P eʳᵗ. Be comfortable converting between exponential and log forms to solve equations.

指数增长/衰减：y = A eᵏᵗ。k > 0 增长，k < 0 衰减。利息复利公式：A = P(1 + r/n)^(nt)。连续复利：A = P eʳᵗ。熟练转换指数与对数形式来解方程。

5. Trigonometry: Ratios, Identities & Equations | 三角学：比值、恒等式与方程

Remember exact trig values for 0°, 30°, 45°, 60°, 90° (π/6, π/4, π/3, π/2 rad). For sin and cos, use the unit circle. Key identities:

tan θ = sin θ / cos θ

sin² θ + cos² θ = 1

熟记 0°, 30°, 45°, 60°, 90°（π/6, π/4, π/3, π/2 弧度）的精确三角比值。对 sin 和 cos，运用单位圆。核心恒等式：

tan θ = sin θ / cos θ

sin² θ + cos² θ = 1

Compound and double angle formulas are often provided in the IB formula booklet, but you must know when to use them:

sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)
sin 2A = 2 sin A cos A
cos 2A = cos² A − sin² A = 2cos² A − 1 = 1 − 2sin² A

和角与倍角公式通常在 IB 公式手册中提供，但你需要知道何时使用它们：

sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)
sin 2A = 2 sin A cos A
cos 2A = cos² A − sin² A = 2cos² A − 1 = 1 − 2sin² A

Solving trig equations: isolate the trig function, then find the principal value and use symmetry/coterminal angles to find all solutions within the given interval. Always check the domain and whether your calculator is in degree or radian mode.

解三角方程：先分离出三角函数，再求主值，利用对称性/同界角找出给定区间内的所有解。务必检查定义域以及计算器处于角度制还是弧度制。

6. Sequences & Series | 数列与级数

Arithmetic sequences have a common difference d. The nth term: uₙ = u₁ + (n − 1)d. Sum of first n terms: Sₙ = n/2 [2u₁ + (n − 1)d] or Sₙ = n/2 (u₁ + uₙ).

等差数列有公差 d。第 n 项：uₙ = u₁ + (n − 1)d。前 n 项和：Sₙ = n/2 [2u₁ + (n − 1)d] 或 Sₙ = n/2 (u₁ + uₙ)。

Geometric sequences have a common ratio r. uₙ = u₁ rⁿ⁻¹. Sum of n terms: Sₙ = u₁(1 − rⁿ)/(1 − r) for r ≠ 1. Infinite sum converges if |r| < 1: S∞ = u₁/(1 − r).

等比数列有公比 r。uₙ = u₁ rⁿ⁻¹。前 n 项和：Sₙ = u₁(1 − rⁿ)/(1 − r)（r ≠ 1）。无穷级数收敛当 |r| < 1：S∞ = u₁/(1 − r)。

Learn to distinguish between the two by checking either ratio of consecutive terms (constant → geometric) or difference (constant → arithmetic). Applications include compound interest, population growth, and repeating decimals.

学会通过检查相邻项的比值（常数→等比）或差值（常数→等差）来区分二者。应用包括复利、人口增长和循环小数化分数。

7. Differentiation: Rules & Applications | 微分：法则与应用

Differentiation gives the gradient of a curve at a point. Basic derivative: d/dx [xⁿ] = n xⁿ⁻¹. Key rules:

Product: d/dx [uv] = u’v + uv’
Quotient: d/dx [u/v] = (u’v − uv’) / v²
Chain: d/dx [f(g(x))] = f'(g(x))·g'(x)

微分可求出曲线上某点的梯度。基本导数：d/dx [xⁿ] = n xⁿ⁻¹。核心法则：

乘法法则：d/dx [uv] = u’v + uv’
除法法则：d/dx [u/v] = (u’v − uv’) / v²
链式法则：d/dx [f(g(x))] = f'(g(x))·g'(x)

Derivatives of special functions:

d/dx eˣ = eˣ ; d/dx eᵏˣ = k eᵏˣ
d/dx ln x = 1/x ; d/dx ln(kx) = 1/x
d/dx sin x = cos x ; d/dx cos x = −sin x ; d/dx tan x = sec² x

特殊函数导数：

d/dx eˣ = eˣ；d/dx eᵏˣ = k eᵏˣ
d/dx ln x = 1/x；d/dx ln(kx) = 1/x
d/dx sin x = cos x；d/dx cos x = −sin x；d/dx tan x = sec² x

Applications: find stationary points by setting f'(x) = 0. Use second derivative test: f”(x) > 0 → local minimum; f”(x) < 0 → local maximum; f''(x) = 0 → possibly inflection. Equation of tangent at x = a: y − f(a) = f'(a)(x − a). Normal: gradient = −1/f'(a).

应用：令 f'(x) = 0 求驻点。二阶导数检验：f”(x) > 0 → 局部极小；f”(x) < 0 → 局部极大；f''(x) = 0 → 可能为拐点。在 x = a 处的切线方程：y − f(a) = f'(a)(x − a)。法线：斜率 = −1/f'(a)。

8. Integration: Antiderivatives & Area | 积分：原函数与面积

Integration is the reverse of differentiation. Basic integral: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ −1. The constant of integration C is essential for indefinite integrals. For definite integrals, ∫ₐᵇ f(x) dx gives the net area between curve and x-axis.

积分是微分的逆运算。基本积分：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，n ≠ −1。积分常数 C 对于不定积分至关重要。定积分 ∫ₐᵇ f(x) dx 给出曲线与 x 轴之间的净面积。

Key integrals: ∫ 1/x dx = ln|x| + C; ∫ eˣ dx = eˣ + C; ∫ eᵏˣ dx = (1/k)eᵏˣ + C; ∫ cos x dx = sin x + C; ∫ sin x dx = −cos x + C; ∫ sec² x dx = tan x + C.

核心积分：∫ 1/x dx = ln|x| + C；∫ eˣ dx = eˣ + C；∫ eᵏˣ dx = (1/k)eᵏˣ + C；∫ cos x dx = sin x + C；∫ sin x dx = −cos x + C；∫ sec² x dx = tan x + C。

Area between curves: ∫ₐᵇ [f(x) − g(x)] dx, where f(x) ≥ g(x) on [a, b]. Volume of revolution around x-axis: V = π ∫ₐᵇ [f(x)]² dx. For parametric curves, or rotation around y-axis, adapt accordingly.

曲线间的面积：∫ₐᵇ [f(x) − g(x)] dx，其中在 [a, b] 上 f(x) ≥ g(x)。绕 x 轴旋转体积：V = π ∫ₐᵇ [f(x)]² dx。对于参数曲线或绕 y 轴旋转，需相应调整公式。

9. Probability & Statistics | 概率与统计

Probability basics: P(A) = n(A)/n(U). For mutually exclusive events: P(A ∪ B) = P(A) + P(B). Independent events: P(A ∩ B) = P(A)·P(B). Conditional probability: P(A|B) = P(A ∩ B)/P(B).

概率基础：P(A) = n(A)/n(U)。互斥事件：P(A ∪ B) = P(A) + P(B)。独立事件：P(A ∩ B) = P(A)·P(B)。条件概率：P(A|B) = P(A ∩ B)/P(B)。

Binomial distribution: X ~ B(n, p). P(X = r) = ⁿCᵣ pʳ (1−p)ⁿ⁻ʳ. Mean = np, variance = np(1−p). Normal distribution: X ~ N(μ, σ²). Use standardisation: z = (x − μ)/σ. The total area under the curve is 1; symmetry helps with probabilities.

二项分布：X ~ B(n, p)。P(X = r) = ⁿCᵣ pʳ (1−p)ⁿ⁻ʳ。均值 = np，方差 = np(1−p)。正态分布：X ~ N(μ, σ²)。标准化：z = (x − μ)/σ。曲线下总面积为 1；对称性有助于计算概率。

Know how to use your calculator for normal and binomial probabilities. For continuous data, median/quartiles from cumulative frequency graphs. Linear regression: line of best fit y = a + bx; Pearson’s correlation coefficient r measures linear association.

掌握如何使用计算器求解正态和二项概率。对于连续数据，从累积频率图中读取中位数/四分位数。线性回归：最佳拟合线 y = a + bx；皮尔逊相关系数 r 衡量线性相关程度。

10. Vectors in 2D & 3D | 二维与三维向量

Vectors have magnitude and direction. Notation: a = ⟨a₁, a₂⟩ in 2D, ⟨a₁, a₂, a₃⟩ in 3D. Magnitude |a| = √(a₁² + a₂²) or √(a₁² + a₂² + a₃²). Unit vector: â = a/|a|.

向量有大小和方向。记法：a = ⟨a₁, a₂⟩（二维），⟨a₁, a₂, a₃⟩（三维）。大小 |a| = √(a₁² + a₂²) 或 √(a₁² + a₂² + a₃²)。单位向量：â = a/|a|。

Scalar (dot) product: a·b = a₁b₁ + a₂b₂ (+ a₃b₃). Also a·b = |a||b| cos θ, where θ is the angle between vectors. Used to test perpendicularity: a·b = 0. Vector equation of a line: r = a + t b, where a is a point on line, b is direction.

标量积（点积）：a·b = a₁b₁ + a₂b₂ (+ a₃b₃)。同时 a·b = |a||b| cos θ，θ 为向量夹角。用于判断垂直：a·b = 0。直线的向量方程：r = a + t b，其中 a 为线上一点，b 为方向向量。

Find angle between vectors: cos θ = (a·b)/(|a||b|). For intersections, set two vector line equations equal and solve for parameters. In kinematics, displacement, velocity and acceleration are vectors; differentiate/integrate component-wise.

求向量夹角：cos θ = (a·b)/(|a||b|)。求交点时，令两条直线的向量方程相等并解参数。在运动学中，位移、速度和加速度皆为向量；按分量进行微分/积分。

11. Complex Numbers (AA HL) | 复数 (AA HL)

Complex number: z = a + bi, where i² = −1. Real part Re(z) = a, imaginary part Im(z) = b. Modulus |z| = √(a² + b²), argument arg(z) = θ (from positive real axis). Polar form: z = r(cos θ + i sin θ) = r cis θ.

复数：z = a + bi，其中 i² = −1。实部 Re(z) = a，虚部 Im(z) = b。模 |z| = √(a² + b²)，辐角 arg(z) = θ（从正实轴起）。极坐标形式：z = r(cos θ + i sin θ) = r cis θ。

Euler’s formula: eⁱᶿ = cos θ + i sin θ. De Moivre’s theorem: (r cis θ)ⁿ = rⁿ cis(nθ). This is powerful for finding powers and roots of complex numbers.

欧拉公式：eⁱᶿ = cos θ + i sin θ。棣莫弗定理：(r cis θ)ⁿ = rⁿ cis(nθ)。此定理在求复数的幂与方根时极为强大。

nth roots: zᵏ = r^(1/n) cis((θ + 2πk)/n), k = 0, 1, …, n−1. These roots are equally spaced on a circle. Conjugate: z̄ = a − bi. Useful identities: z·z̄ = |z|², and for polynomials with real coefficients, non-real roots occur in conjugate pairs.

n 次方根：zᵏ = r^(1/n) cis((θ + 2πk)/n)，k = 0, 1, …, n−1。这些根均匀分布在圆周上。共轭：z̄ = a − bi。常用恒等式：z·z̄ = |z|²；对于实系数多项式，非实根以共轭对形式出现。

12. Exam Technique & Common Pitfalls | 考试技巧与常见误区

Manage your time wisely. Read the entire question before starting; many marks are lost by misinterpreting what is asked. Show all working—IB awards method marks even if the final answer is wrong. For longer questions, parts often build on one another: if you cannot solve part (a), use the given result to attempt later parts.

合理分配时间。动笔前通读全题；许多失分源于误解题意。展示所有解题步骤——IB 即使最终答案有误，也会给方法分。对于长篇题目，各小问往往环环相扣：若不会解 (a)，可用题目给出的结果尝试后续部分。

Check your calculator mode (radians/degrees) before trig problems. Sketch diagrams for geometry, vectors, and functions; they unlock hidden information. When solving equations, always verify your solutions satisfy the original equation, especially after squaring or taking logs. For statistics, remember to state assumptions and interpret results in context.

做三角题前检查计算器模式（弧度/度）。在几何、向量和函数题中画草图，它们能揭示隐藏信息。解方程时，务必验证答案是否满足原方程，特别是平方或取对数之后。统计题中记得陈述假设并在上下文中解释结果。

Revise actively: work through past papers under timed conditions, mark them using official mark schemes, and note down recurring mistakes. Rest well before the exam; a clear mind is your best tool.

积极复习：在限时条件下刷历年真题，用官方评分方案批改，记录重复出现的错误。考前好好休息；清醒的头脑是你最好的工具。

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IB AQA Maths: Last-Minute Revision Notes | IB AQA 数学：考前冲刺笔记