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

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

This article provides a last-minute revision summary of key concepts, formulas, and problem-solving strategies for students taking IB (Analysis & Approaches / Applications & Interpretation) and OCR (A Level / Further Maths) mathematics exams. Use these notes to quickly review the most important topics before the exam and build confidence in your approach.

本文为正在备考 IB（分析与方法 / 应用与解释）和 OCR（A Level / 进阶数学）考试的学生提供考前冲刺总结，涵盖关键概念、公式和解题策略。使用这些笔记在考前快速回顾最重要的知识点，增强应试信心。

1. Algebraic Essentials | 代数要点

A solid command of indices, surds, and polynomial manipulation is vital. Remember the laws of indices: a^m × aⁿ = a^m+n and (a^m)ⁿ = a^mn. When rationalising denominators like 1/(√a + √b), multiply by the conjugate (√a – √b)/(√a – √b).

对指数、根式和多项式运算的扎实掌握至关重要。记住指数律：a^m × aⁿ = a^m+n，(a^m)ⁿ = a^mn。在分母有理化时，如 1/(√a + √b)，乘以其共轭式 (√a – √b)/(√a – √b)。

The quadratic formula solves ax² + bx + c = 0. Always compute the discriminant Δ = b² – 4ac to determine the nature of roots: two real, one repeated, or two complex.

二次公式解 ax² + bx + c = 0。始终计算判别式 Δ = b² – 4ac 以确定根的性质：两个实根、一个重根或两个复根。

x = (-b ± √(b² – 4ac)) / (2a)

The factor theorem states that (x – a) is a factor of f(x) if and only if f(a) = 0. Pair this with polynomial long division or synthetic division to factorise cubics and higher-degree polynomials.

因式定理指出，(x – a) 是 f(x) 的因式当且仅当 f(a) = 0。结合多项式长除法或综合除法，用于因式分解三次及更高次多项式。

Completing the square helps rewrite quadratics in vertex form a(x – h)² + k and is essential for finding maximum/minimum values and sketching curves.

配方法可将二次函数写成顶点式 a(x – h)² + k，对求最值和绘制曲线至关重要。

2. Functions & Transformations | 函数与变换

A function maps each input x to exactly one output f(x). The domain is the set of allowed x-values, and the range is all possible output values. For composite functions, f(g(x)) is read as ‘f of g of x’ and requires the range of g to be compatible with the domain of f.

函数将每一个输入 x 唯一映射到一个输出 f(x)。定义域是所有允许的 x 值的集合，值域是所有可能的输出值。复合函数 f(g(x)) 读作 ‘f of g of x’，要求 g 的值域与 f 的定义域兼容。

An inverse function f⁻¹(x) exists only if f is one-to-one. Graphically, the inverse is a reflection in the line y = x. To find an inverse, swap x and y, then solve for y.

反函数 f⁻¹(x) 仅在 f 是双射（一一对应）时才存在。反函数的图像关于直线 y = x 对称。求反函数时，交换 x 与 y，然后解出 y。

Transformations of graphs: y = f(x) + a is a vertical translation; y = f(x + a) is a horizontal shift (opposite direction). y = a f(x) stretches vertically; y = f(ax) compresses horizontally if a > 1.

图像变换：y = f(x) + a 是垂直平移；y = f(x + a) 是水平平移（方向相反）。y = a f(x) 垂直伸缩；若 a > 1，y = f(ax) 水平压缩。

The modulus function |x| gives the absolute value. Equations like |2x – 1| = 3 split into two cases: 2x – 1 = 3 and 2x – 1 = -3.

绝对值函数 |x| 给出非负值。如 |2x – 1| = 3 的方程拆分为两种情况：2x – 1 = 3 和 2x – 1 = -3。

3. Trigonometry & Circular Functions | 三角与圆函数

Exact trigonometric values for 30°, 45°, and 60° must be memorised. Know the identities: sin²θ + cos²θ = 1, dividing by cos²θ gives tan²θ + 1 = sec²θ.

必须熟记 30°、45° 和 60° 的精确三角函数值。掌握恒等式：sin²θ + cos²θ = 1，两边同除以 cos²θ 得 tan²θ + 1 = sec²θ。

The sine rule (a/sin A = b/sin B = c/sin C) and cosine rule (a² = b² + c² – 2bc cos A) are used for non-right-angled triangles. For area, use Area = ½ ab sin C.

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。面积公式用 Area = ½ ab sin C。

Radians: π rad = 180°. Arc length s = rθ and sector area A = ½ r²θ, where θ is in radians.

弧度制：π rad = 180°。弧长 s = rθ，扇形面积 A = ½ r²θ，其中 θ 以弧度为单位。

Solving trig equations: find base angle in the relevant quadrant using the ASTC diagram, then add periodicity. For sin x, add 2πn; for tan x, add πn.

解三角方程：利用 ASTC 图找到相关象限的基准角，然后加上周期性。sin x 加 2πn；tan x 加 πn。

Compound and double-angle formulas are exam favourites: sin(A ± B) = sin A cos B ± cos A sin B; cos(2θ) = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ.

和角与倍角公式是考试热点：sin(A ± B) = sin A cos B ± cos A sin B；cos(2θ) = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ。

4. Sequences, Series & Binomial Expansion | 数列、级数与二项式展开

Arithmetic sequences have a common difference d. The nth term u_n = a + (n-1)d and the sum of n terms S_n = n/2 [2a + (n-1)d] = n/2 (a + l), where l is the last term.

等差数列有公差 d。第 n 项 u_n = a + (n-1)d，前 n 项和 S_n = n/2 [2a + (n-1)d] = n/2 (a + l)，其中 l 为末项。

Geometric sequences have common ratio r. The nth term is ar^n-1, and the sum of n terms is S_n = a(1 – rⁿ)/(1 – r) for r ≠ 1. The sum to infinity exists when |r| < 1: S_∞ = a/(1 – r).

等比数列有公比 r。第 n 项为 ar^n-1，前 n 项和 S_n = a(1 – rⁿ)/(1 – r) (r ≠ 1)。当 |r| < 1 时存在无穷项和：S_∞ = a/(1 – r)。

Sigma notation Σ is used to write sums compactly. Always check the starting index carefully to avoid off-by-one errors.

求和记号 Σ 用于简洁地表示级数。务必仔细检查起始下标，避免差一错误。

The binomial expansion for (1 + x)ⁿ is 1 + nx + n(n-1)/2! x² + … for any rational n, where |x| < 1 if n is not a positive integer. For (a + b)ⁿ, use the general term C(n, r) a^n-r b^r.

(1 + x)ⁿ 的二项式展开为 1 + nx + n(n-1)/2! x² + …，对于任意有理数 n，若 n 非正整数需满足 |x| < 1。对于 (a + b)ⁿ，使用通项 C(n, r) a^n-r b^r。

5. Differentiation Techniques | 微分技巧

The derivative gives the gradient of a tangent to a curve. From first principles, f'(x) = limit_h→0 [f(x+h) – f(x)]/h. In practice, use rules: d/dx(xⁿ) = nx^n-1; derivative of constant is zero.

导数给出曲线切线的斜率。由第一原理：f'(x) = limit_h→0 [f(x+h) – f(x)]/h。实际运用规则：d/dx(xⁿ) = nx^n-1；常数的导数为零。

Product rule: d/dx(uv) = u’v + uv’. Quotient rule: d/dx(u/v) = (u’v – uv’)/v². Chain rule: if y = f(g(x)), then dy/dx = f'(g(x)) g'(x).

积的法则：d/dx(uv) = u’v + uv’。商的法则：d/dx(u/v) = (u’v – uv’)/v²。链式法则：若 y = f(g(x))，则 dy/dx = f'(g(x)) g'(x)。

Know derivatives of standard functions: d/dx(sin x) = cos x, d/dx(cos x) = -sin x, d/dx(e^x) = e^x, d/dx(ln x) = 1/x.

熟记基本函数的导数：d/dx(sin x) = cos x, d/dx(cos x) = -sin x, d/dx(e^x) = e^x, d/dx(ln x) = 1/x。

For parametric equations x = f(t), y = g(t), use dy/dx = (dy/dt) / (dx/dt). For implicit differentiation, treat y as a function of x and apply d/dx to both sides.

参数方程 x = f(t), y = g(t) 使用 dy/dx = (dy/dt) / (dx/dt)。隐函数微分时，把 y 视为 x 的函数，对等式两边进行微分。

6. Integration & Applications | 积分及其应用

Integration reverses differentiation. The basic rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ -1. For n = -1, ∫ (1/x) dx = ln|x| + C.

积分是微分的逆运算。基本规则：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1)。当 n = -1 时，∫ (1/x) dx = ln|x| + C。

∫ e^x dx = e^x + C, ∫ sin x dx = -cos x + C, ∫ cos x dx = sin x + C

Definite integrals compute the signed area between a curve and the x-axis. Area = ∫_a^b f(x) dx. Remember that area below the x-axis contributes negatively.

定积分计算曲线与 x 轴之间的有向面积。Area = ∫_a^b f(x) dx。记住 x 轴下方区域的面积贡献为负。

Integration by substitution is the reverse of the chain rule. Use u = g(x) and du = g'(x) dx. Integration by parts: ∫ u dv = uv – ∫ v du, useful for products like x e^x or x sin x.

换元积分法是链式法则的逆过程。设 u = g(x)，du = g'(x) dx。分部积分法：∫ u dv = uv – ∫ v du，用于处理形如 x e^x 或 x sin x 等乘积。

Volumes of revolution: rotating y = f(x) about the x-axis yields V = π ∫_a^b [f(x)]² dx. For rotation about the y-axis, rearrange to x = g(y).

旋转体体积：函数 y = f(x) 绕 x 轴旋转得到 V = π ∫_a^b [f(x)]² dx。绕 y 轴旋转时需转化为 x = g(y)。

7. Vectors & 3D Geometry | 向量与三维几何

A vector has magnitude and direction. In component form, v = x i + y j + z k. The magnitude is |v| = √(x² + y² + z²). A unit vector in the direction of v is v/|v|.

向量有大小和方向。分量形式：v = x i + y j + z k。模长为 |v| = √(x² + y² + z²)。v 方向的单位向量为 v/|v|。

The dot product a·b = |a||b| cosθ = a₁b₁ + a₂b₂ + a₃b₃. It is used to find the angle between vectors and to test perpendicularity (a·b = 0).

点积 a·b = |a||b| cosθ = a₁b₁ + a₂b₂ + a₃b₃。用于求两向量夹角及判定垂直 (a·b = 0)。

The cross product a × b (only in 3D, OCR Further / IB HL) gives a vector perpendicular to both a and b, with magnitude |a||b| sinθ. Its direction follows the right-hand rule.

叉积 a × b（仅限三维，OCR 进阶 / IB HL）给出一个同时垂直于 a 和 b 的向量，大小为 |a||b| sinθ，方向遵循右手定则。

Vector equation of a line: r = a + t d, where a is a point on the line and d is the direction vector. A plane has equation r·n = a·n, where n is the normal vector.

直线的向量方程：r = a + t d，a 为直线上一点，d 为方向向量。平面的方程为 r·n = a·n，其中 n 为法向量。

8. Probability & Distributions | 概率与分布

Probability basics: P(A ∪ B) = P(A) + P(B) – P(A ∩ B). For mutually exclusive events, P(A ∩ B) = 0. For independent events, P(A ∩ B) = P(A)P(B).

概率基础：P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。互斥事件有 P(A ∩ B) = 0。独立事件有 P(A ∩ B) = P(A)P(B)。

Conditional probability: P(A|B) = P(A ∩ B)/P(B). Exam questions often involve tree diagrams or Venn diagrams to visualise problems.

条件概率：P(A|B) = P(A ∩ B)/P(B)。考试常通过树状图或文氏图使问题可视化。

Binomial distribution B(n, p): X ~ B(n, p), P(X = r) = C(n, r) p^r (1-p)^n-r. Mean = np, Variance = np(1-p). Valid when trials are independent and fixed n.

二项分布 B(n, p)：X ~ B(n, p)，P(X = r) = C(n, r) p^r (1-p)^n-r。均值 = np，方差 = np(1-p)。适用条件是各次试验独立且 n 固定。

The normal distribution N(μ, σ²) is symmetrical and bell-shaped. Standardise using Z = (X – μ)/σ ~ N(0,1). For sample means, X̄ ~ N(μ, σ²/n) by the Central Limit Theorem.

正态分布 N(μ, σ²) 是对称的钟形曲线。通过 Z = (X – μ)/σ ~ N(0,1) 进行标准化。对于样本均值，由中心极限定理得 X̄ ~ N(μ, σ²/n)。

9. Statistics & Hypothesis Tests | 统计与假设检验

Descriptive statistics include mean, median, mode, variance, and standard deviation s = √(Σ(x – x̄)²/(n-1)) for a sample. Box plots show median, quartiles, and outliers.

描述性统计包括均值、中位数、众数、

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