📚 IB AQA Maths: High-Frequency Topic Summary | IB AQA 数学：高频考点总结
IB AQA Mathematics covers a broad range of topics, but certain core concepts appear in exams with remarkable regularity. This summary identifies the most frequently assessed areas—from algebraic manipulation to probability distributions—and provides key insights to sharpen your revision. Understanding these high-frequency topics will help you target your study time and build confidence for both calculator and non-calculator papers.
IB AQA 数学涵盖了广泛的知识点，但有一些核心概念在考试中出现的频率非常高。本文总结了最常考的范围——从代数运算到概率分布——并提供关键要点，帮助你更有针对性地复习。熟悉这些高频考点能让你合理分配学习时间，无论是允许使用计算器还是不允许使用计算器的试卷，都能增强信心。
1. Algebraic Manipulation | 代数运算
Fluency in expanding, factorising and simplifying expressions is essential. Common tasks include expanding (a+b)2 = a2 + 2ab + b2 and recognising the difference of squares a2−b2 = (a+b)(a−b).
熟练掌握展开、因式分解和化简表达式至关重要。常见的运算包括展开 (a+b)2 = a2 + 2ab + b2，以及识别平方差 a2−b2 = (a+b)(a−b)。
Rational expressions such as (x2−1)/(x−1) must be simplified, noting restrictions like x ≠ 1. Surds and indices also appear regularly, requiring the laws am × an = am+n and √a × √b = √ab.
有理表达式如 (x2−1)/(x−1) 需要化简，并注明 x ≠ 1 等限制。根式和指数也经常出现，需要用到指数律 am × an = am+n 和 √a × √b = √ab。
Partial fractions and algebraic division are tested in more advanced problems, particularly when integrating rational functions.
部分分式与代数除法会在较复杂的题目中考查，尤其是在对有理函数进行积分时。
2. Functions and Graphs | 函数与图像
Understanding domain, range and inverse functions is frequently examined. For a function f(x), the inverse f−1(x) reflects the graph in the line y=x, and its domain is the range of f(x).
定义域、值域和反函数的概念常常是考试重点。对于函数 f(x)，其反函数 f−1(x) 的图像关于直线 y=x 对称，且反函数的定义域就是原函数的值域。
Transformations of graphs—translations, stretches and reflections— the graph vertically by a units, while y = f(2x) compresses horizontally by a factor of ½.
图像的变换——平移、伸缩和反射——需要熟练应用。例如 y = f(x) + a 使图像垂直平移 a 个单位，而 y = f(2x) 则使图像水平压缩为原来的 ½。
Modulus functions |f(x)| and composite functions gf(x) are common in both equation-solving and sketching tasks.
绝对值函数 |f(x)| 和复合函数 gf(x) 在解方程与画图题中经常出现。
3. Quadratics, Equations and Inequalities | 二次方程与不等式
The quadratic formula is a must-know tool:
x = (−b ± √b2−4ac) / 2a
二次公式是必须掌握的工具：
x = (−b ± √b2−4ac) / 2a
Discriminant Δ = b2 ves two distinct real roots, Δ = 0 gives one repeated root, and Δ < 0 gives no real roots.
判别式 Δ = b2−4ac 决定了实数根的个数：Δ > 0 有两个不等实根，Δ = 0 有一个重根，Δ < 0 无实根。
Solving inequalities often involves sketching the quadratic graph first. When multiplying or dividing by a negative quantity, remember to reverse the inequality sign.
解不等式时常常先画出二次函数草图。当乘以或除以负数时，注意要反向不等号。
4. Trigonometry | 三角学
Exact values for sin, cos and tan at 0°, 30°, 45°, 60°, 90° must be memorised. The two key triangles—45-45-90 and 30-60-90—help derive these values quickly.
必须熟记 0°、30°、45°、60°、90° 的正弦、余弦和正切的精确值。利用两个关键三角形——45-45-90 和 30-60-90——可以快速推导这些值。
Trigonometric identities such as sin2θ + cos2θ ≡ 1 and tanθ ≡ sinθ / cosθ are fundamental for simplifying expressions and solving equations.
三角恒等式如 sin2θ + cos2θ ≡ 1 和 tanθ ≡ sinθ / cosθ 是化简表达式和求解方程的基础。
The sine and cosine rules (a/sinA = b/sinB = c/sinC; a2 = b2+c2−2bc cosA) are essential for non‑right‑angled triangles, as is the area formula ½ab sinC.
正弦定理和余弦定理（a/sinA = b/sinB = c/sinC；a2 = b2+c2−2bc cosA）对于非直角三角形必不可少，面积公式 ½ab sinC 也同样重要。
sively in calculus; remember that π radians = 180° and arc length s = rθ, sector area = ½r2θ.
弧度制在微积分中广泛应用；记住 π 弧度 = 180°，弧长 s = rθ，扇形面积 = ½r2θ。
5. Exponentials and Logarithms | 指数与对数
The exponential function ex and natural logarithm ln x are inverses: ln(ex) = x. The graph y = ex passes through (0,1) with a horizontal asymptote at y=0.
指数函数 ex 与自然对数 ln x 互为反函数：ln(ex) = x。y = ex 的图像经过 (0,1)，水平渐近线为 y=0。
Log laws simplify products and powers: loga(xy) = logax + logay, loga(xn) = n logax, and the change of base formula logba = logca / logcb.
对数运算律可以简化乘积和幂：loga(xy) = logax + logay，loga(xn) = n logax，以及换底公式 logba = logca / logcb。
Exponential growth and decay models y = aekt are common in applied problems. Solving equations often requires taking logs on both sides.
指数增长与衰减模型 y = aekt 在应用题中十分常见。解这类方程通常需要对两边取对数。
6. Differentiation | 微分
Differentiation from first principles uses the limit f'(x) = limh→0 [f(x+h)−f(x)]/h, but the power rule d/dx (xn) = nxn−1 speeds up most calculations.
从第一性原理求导需要用到极限 f'(x) = limh→0 [f(x+h)−f(x)]/h，但实际计算中幂函数求导法则 d/dx (xn) = nxn−1 更常用。
oduct and quotient rules are essential for composite functions. For y = f(g(x)), dy/dx = f'(g(x)) × g'(x). For y = uv, dy/dx = u'v + v'u.
复合函数的求导离不开链式法则、乘积法则和商法则。若 y = f(g(x))，则 dy/dx = f'(g(x)) × g'(x)；若 y = uv，则 dy/dx = u'v + v'u。
Derivatives of ex, ln x, sin x and cos x must be known instantly: d/dx (ex) = ex, d/dx (ln x) = 1/x, d/dx (sin x) = cos x, d/dx (cos x) = −sin x.
必须牢记 ex、ln x、sin x 和 cos x 的导数：d/dx (ex) = ex，d/dx (ln x) = 1/x，d/dx (sin x) = cos x，d/dx (cos x) = −sin x。
Applications include finding gradients of tangents and normals, stationary points, and optimising real-life quantities.
导数的应用包括求切线和法线的斜率、求驻点以及优化实际生活中的量。
7. Integration | 积分
Indefinite integration reverses differentiation: ∫ xn dx = xn+1/(n+1) + C (n ≠ −1). The integral of 1/x is ln|x| + C.
不定积分是微分的逆运算：∫ xn dx = xn+1/(n+1) + C（n ≠ −1）。1/x 的积分是 ln|x| + C。
Definite integrals calculate the area under a curve between two bounds. Always remember to subtract the lower limit value from the upper limit value.
定积分可以计算曲线下方两个界限之间的面积。务必记得用上限值减去下限值。
u dv = uv − ∫ v du) appear regularly. Recognising when to use which technique is crucial.
换元积分法和分部积分法（∫ u dv = uv − ∫ v du）经常出现，关键在于判断使用哪种方法。
olution, and solving differential equations like dy/dx = ky are common higher-mark topics.
曲线之间的面积、旋转体体积以及求解微分方程（如 dy/dx = ky）是常见的高分值考点。
8. Sequences and Series | 数列与级数
Arithmetic sequences have a common difference d: un = a + (n−1)d and sum Sn = n/2 [2a + (n−1)d] or n/2 (a+l).
等差数列的公差为 d：通项 un = a + (n−1)d，求和公式为 Sn = n/2 [2a + (n−1)d] 或 n/2 (a+l)。
Geometric sequences have a common ratio r: un = arn−1. The sum of the first n terms is Sn = a(1−rn)/(1−r) for r ≠ 1.
等比数列的公比为 r：通项 un = arn−1，前 n 项和为 Sn = a(1−rn)/(1−r)，其中 r ≠ 1。
Infinite geometric series converge when |r| < 1, with sum to infinity S∞ = a/(1−r). This is frequently tested with context problems.
当 |r| < 1 时，无穷等比级数收敛，其无穷和为 S∞ = a/(1−r)，这一知识点常在实际应用题中考查。
The binomial expansion (1+x)n = 1 + nx + n(n−1)x2/2! + ... is valid for |x| < 1 and rational n. Be prepared to expand and approximate.
二项展开式 (1+x)n = 1 + nx + n(n−1)x2/2! + ... 在 |x| < 1 且 n 为有理数时成立，要求能够展开并进行近似计算。
9. Vectors | 向量
ectors are expressed in component form or as i, j, k unit vectors. Magnitude is found with |v| = √(x2+y2+z2).
向量可以用分量形式或单位向量 i、j、k 表示。模长计算公式为 |v| = √(x2+y2+z2)。
Scalar (dot) product a·b = |a||b|cosθ helps find angles between vectors and test perpendicularity (a·b = 0).
数量积（点积）a·b = |a||b|cosθ 可用于求向量之间的夹角以及判断垂直关系（a·b = 0）。
ations of lines r = a + t b and planes are tested, along with intersections and distances.
直线的向量方程 r = a + t b 以及平面的方程都是考试内容，还包括交点和距离的计算。
10. Probability and Statistics | 概率与统计
Basic probability rules: P(A∪B) = P(A) + P(B) − P(A∩B). Conditional probability P(A|B) = P(A∩B) / P(B). Tree diagrams and Venn diagrams help visualise multi-stage problems.
基本概率公式：P(A∪B) = P(A) + P(B) − P(A∩B)。条件概率 P(A|B) = P(A∩B) / P(B)。树状图和文氏图可以帮助理清多阶段问题。
Discrete random variables and their expectation E(X) = Σ x P(X=x), variance Var(X) = E(X2) − [E(X)]2 appear frequently. The binomial distribution X ~ B(n,p) with mean np and variance np(1−p) is a major topic.
离散随机变量及其期望 E(X) = Σ x P(X=x)、方差 Var(X) = E(X2) − [E(X)]2 经常出现。二项分布 X ~ B(n,p) 的均值为 np，方差为 np(1−p)，是重要的考点。
The normal distribution X ~ N(μ,σ2 /σ. Inverse normal calculations and approximations (e.g., normal approximation to binomial) are common.
正态分布 X ~ N(μ,σ2) 需要标准化 z = (x−μ)/σ。反查正态分布以及用正态分布近似二项分布等方法是常见题型。
Hypothesis testing involves stating null and alternative hypotheses, calculating a test statistic and interpreting the p-value or critical region.
假设检验需要设立零假设与备择假设，计算检验统计量，并解读 p 值或临界域。
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