📚 IB Math Review Set A: Comprehensive Practice & Solutions | IB 数学复习题集A：综合练习与解析
Review Set A is a carefully designed compilation of problems that target the core topics of the IB Mathematics: Analysis and Approaches course. It covers foundational algebraic techniques, function theory, trigonometry, calculus, sequences, and probability. Working through this set will sharpen your problem-solving skills and highlight the connections between different areas of the syllabus.
复习题集A 是一套精心设计的练习汇编，针对 IB 数学分析与方法的各核心主题。它涵盖基础代数技巧、函数理论、三角学、微积分、数列和概率。通过完成这套题目，你将提升解题能力，并深化对各知识模块之间联系的理解。
1. Understanding the Scope of Review Set A | 了解复习集 A 的考查范围
This review set typically draws from Chapters 1 through 8 of the IB Mathematics AA textbook, including algebraic expressions, functions, exponentials and logarithms, trigonometry, and introductory calculus. Each question is designed to test not only procedural fluency but also conceptual understanding.
这套复习题通常选自 IB 数学 AA 教材的第 1 至第 8 章，内容涵盖代数表达式、函数、指数与对数、三角学以及微积分初步。每一道题不仅考查计算熟练度，更检验对概念的深层理解。
You may encounter tasks that require you to transform a function and interpret its behaviour, solve equations analytically and graphically, or apply differentiation to find rates of change and tangents. A systematic approach to each problem will help you identify which skills are being assessed.
你可能会遇到需要变换函数并解释其性质、通过解析和图像方式解方程、或者应用微分求变化率和切线的问题。对每道题采取系统化的处理方式，将帮助你准确识别所要考查的技能。
Spend a few minutes scanning the entire set before diving in. Note the distribution of marks and decide whether to start with the topics you find most comfortable or those that carry the highest weighting.
在动手解题之前，先用几分钟浏览整套题目。留意各题的分数分布，然后决定是从你最有把握的主题入手，还是优先攻克分值最高的题目。
2. Algebraic Manipulation & Simplification | 代数运算与化简
Review Set A often begins with exercises on expanding brackets, factorising quadratics, and simplifying rational expressions. For example, simplify (x² - 5x + 6) ÷ (x - 2) and state any restrictions. Recognising common factor patterns saves time.
复习集 A 通常从展开括号、分解二次式以及化简有理表达式入手。例如，化简 (x² - 5x + 6) ÷ (x - 2) 并说明限制条件。识别常见的因式分解模式能节省大量时间。
Be careful when dealing with negative signs and fractions. A common mistake is to forget to apply the distributive law correctly: (x - 3)² must be expanded as x² - 6x + 9, not x² - 9.
处理负号和分数时要格外小心。一个常见错误是忘记正确应用分配律：(x - 3)² 必须展开为 x² - 6x + 9，而不是 x² - 9。
When simplifying complex fractions, multiply numerator and denominator by the lowest common denominator, cancel only factors—never individual terms—and present your final answer in the simplest factored form.
在化简复分式时，用最小公分母通分分子和分母，只约去因式——绝不要约去单独的项——并将最终答案写成最简因式分解形式。
3. Functions & Their Graphs | 函数及其图像
ermine inverse functions, and sketch graphs after transformations. For instance, if f(x) = √2x + 1, you must state the domain and range before finding the inverse f⁻¹(x).
函数类题目考查你求复合函数 f(g(x))、确定反函数以及通过变换绘制图像的能力。例如，若 f(x) = √2x + 1，你必须先给出定义域和值域，再求出反函数 f⁻¹(x)。
emember that horizontal stretches and shifts act inside the function argument and may feel “backwards”: y = f(3x) compresses the graph horizontally, not stretches it.
变换 y = a f(b(x - h)) + k 经常出现。记住，水平伸缩和平移作用在函数参数内部，并可能显得“反直觉”：y = f(3x) 会水平压缩图像，而不是拉宽。
(2x + 1)/(x - 3), locate the vertical and horizontal asymptotes before plotting the curve.
画图时使用数值表，并标出截距和渐近线等关键点。对于有理函数如 g(x) = (2x + 1)/(x - 3)，先确定垂直和水平渐近线，再绘制曲线。
4. Exponential & Logarithmic Equations | 指数与对数方程
You will likely see problems like: Solve 5ˣ⁺¹ = 125ˣ + log₃(x - 1) = 2. Converting between exponential and logarithmic forms is the foundation.
你很可能会看到类似这样的题目：解 5ˣ⁺¹ = 125ˣ⁻² 和解 log₃(2x) + log₃(x - 1) = 2。熟练转换指数式与对数式是解题的基础。
For the first equation, express both sides with base 5: 125 = 5³, so 5ˣ⁺¹ = (5³)ˣ⁻² = 5³ˣ giving x + 1 = 3x - 6 → x = 3.5. Always check if the base is positive and not equal to 1.
对于第一个方程，将两边都写成以 5 为底的幂：125 = 5³，所以 5ˣ⁺¹ = (5³)ˣ⁻² = 5³ˣ⁻⁶，得到 x + 1 = 3x - 6 → x = 3.5。务必检查底数是否为正且不等于 1。
ntial form. Do not forget to reject any solutions that make the argument of a logarithm non-positive.
对于对数方程，利用 logₐ m + logₐ n = logₐ (mn) 将和式合并为乘积，再转化为指数形式。切勿忘记舍去那些使对数真数非正的解。
5. Trigonometric Functions & Identities | 三角函数与恒等式
x)/(sin x). Radian measure is the default angle unit in IB exams unless otherwise stated.
复习集 A 通常包含在 0 ≤ θ ≤ 2π 内解方程 sin 2θ = 0.5，或证明恒等式 (sin x)/(1 + cos x) = (1 - cos x)/(sin x)。除非另有说明，IB 考试默认使用弧度制。
nce angle is π/6, so 2θ = π/6 + 2kπ or 2θ = 5π/6 + 2kπ. Dividing by 2 yields the θ values.
解三角方程时，先求出主值，再通过加上周期来生成指定区间内的所有解。对于 sin 2θ = 0.5，参考角为 π/6，因此 2θ = π/6 + 2kπ 或 2θ = 5π/6 + 2kπ。除以 2 即得到 θ 的值。
ine. Use the Pythagorean identity sin²x + cos²x = 1 and double-angle formulas to simplify.
证明恒等式时，从较复杂的一边入手，并将所有表达式用正弦和余弦改写。利用勾股恒等式 sin²x + cos²x = 1 以及倍角公式进行化简。
6. Differentiation Rules & Applications | 微分法则与应用
in rule combinations. You might be asked to differentiate f(x) = (2x³ - 5)⁴ or g(x) = x² eˣ.
复习集 A 中的微分类题目涵盖从简单的幂运算法则到乘法、除法和链式法则的组合。你或许会遇到求导 f(x) = (2x³ - 5)⁴ 或 g(x) = x² eˣ 的问题。
For f(x) = (2x³ - 5)⁴, use the chain rule: f'(x) = 4(2x³ - 5)³ × (6x²) = 24x²(2x³ - 5)³. Clearly show the “outside” derivative multiplied by the derivative of the inside.
对于 f(x) = (2x³ - 5)⁴，使用链式法则：f'(x) = 4(2x³ - 5)³ × (6x²) = 24x²(2x³ - 5)³。务必清晰地写出“外部”导数乘以内层导数。
normals, and determining increasing/decreasing intervals. The tangent at x = a has slope f'(a) and passes through (a, f(a)). The normal line is perpendicular, with slope -1/ f'(a).
应用部分包括求切线和法线方程，以及确定函数的增减区间。x = a 处的切线斜率为 f'(a)，并通过点 (a, f(a))。法线与之垂直，斜率为 -1/ f'(a)。
7. Integration & Definite Integrals | 积分与定积分
iation. Typical problems ask you to find ∫ (6x² - 4x + 3) dx or evaluate ∫₀² (3x² - 2) dx.
积分作为微分的逆运算被引入。典型题目包括求 ∫ (6x² - 4x + 3) dx 或计算定积分 ∫₀² (3x² - 2) dx。
Always add a constant of integration “+ C” the definite integral, apply the limits: ∫₀² (3x² - 2) dx = [x³ - 2x]₀² = (8 - 4) - 0 = 4.
不定积分永远要加上积分常数 “+ C”。对于定积分，代入上下限：∫₀² (3x² - 2) dx = [x³ - 2x]₀² = (8 - 4) - 0 = 4。
Area problems require you to find the region bounded by a curve and the x-axis. If the curve goes below the x-axis, split the integral and take absolute values to avoid negative area.
面积问题要求你求出曲线与 x 轴围成的区域面积。如果曲线的一部分在 x 轴下方，需将积分拆分并对负值部分取绝对值，以避免面积抵消。
8. Sequences & Series | 数列与级数
nd d = 3, or the sum of the first 12 terms of a geometric series with r = 0.5 and u₁ = 64.
等差数列和等比数列是常考内容。你可能会被要求求出首项 u₁ = 7、公差 d = 3 的等差数列的第 20 项，或者求首项 u₁ = 64、公比 r = 0.5 的等比数列前 12 项的和。
+ uₙ) or Sₙ = n/2 [2u₁ + (n - 1)d] for arithmetic; Sₙ = u₁ (1 - rⁿ)/(1 - r) for geometric.
相关公式为：等差数列 uₙ = u₁ + (n - 1)d，等比数列 uₙ = u₁ rⁿ⁻¹。求和公式：等差数列 Sₙ = n/2 (u₁ + uₙ) 或 Sₙ = n/2 [2u₁ + (n - 1)d]；等比数列 Sₙ = u₁ (1 - rⁿ)/(1 - r)。
For infinite geometric series where |r| inity is S∞ = u₁ / (1 - r). Ensure the condition is satisfied before applying the formula.
对于 |r| < 1 的无穷等比级数，无穷和为 S∞ = u₁ / (1 - r)。使用公式前必须确认满足收敛条件。
9. Probability & Statistical Distributions | 概率与统计分布
Basic probability combined with binomial distribution problems often closes Review Set A. For instance: A fair die is rolled 5 times. Find the probability of obtaining exactly two sixes.
基础概率结合二项分布的问题通常作为复习集 A 的收尾。例如：一枚公平骰子掷 5 次，求恰好出现两次 6 点的概率。
Let X ~ 36) × (125/216) ≈ 0.1608. Show the binomial coefficient and the calculation steps clearly.
设 X ~ B(5, 1/6)。则 P(X = 2) = C(5,2) × (1/6)² × (5/6)³ = 10 × (1/36) × (125/216) ≈ 0.1608。清晰写出二项式系数和计算步骤。
npq. If you are asked to find the most likely number of successes, compare np and np + p.
二项分布的均值与方差分别为 μ = np 和 σ² = npq。若要求最可能成功次数，可比较 np 与 np + p。
10. Using the GDC Effectively | 高效使用图形计算器
In IB Mathematics, your graphing calculator is a powerful tool for checking solutions, graphing complex functions, and performing numerical integration. However, Review Set A is designed to test your understanding with and without technology.
在 IB 数学中，图形计算器是检验解法、绘制复杂函数图像和进行数值积分的强大工具。但复习集 A 的设计同时考查你在使用和不使用技术辅助时的理解程度。
Always sketch a rough graph by hand before relying on the GDC. Use the calculator to find intersections, zeros, or maxima/minima, but record the key points on paper so the examiner can follow your reasoning.
在依赖 GDC 之前，始终先手绘粗略草图。用计算器找出交点、零点或极值点，但要在纸面上记录关键点，以便考官理解你的推理过程。
Be aware of the limitations: the GDC may display discontinuities incorrectly, and its numerical solver can miss solutions if the initial guess is poorly chosen.
务必注意其局限性：GDC 可能错误显示间断点，其数值求解器也可能因初始猜测不当而遗漏解。
11. Common Mistakes & How to Avoid Them | 常见错误与避免方法
One frequent error is mishandling negative signs inside brackets when expanding. Always write the terms carefully, and if necessary, use extra brackets in your working. Another is forgetting to rationalise denominators or simplify surds when answers require exact form.
一个常见错误是展开括号时对负号处理不当。务必仔细书写各项，必要时在工作步骤中增加额外的括号。另一个错误是当题目要求精确值形式时，忘记分母有理化或简化根式。
In calculus, confusing the derivative of eˣ (which remains eˣ) with the derivative of xⁿ is a typical slip. Also, when integrating, students sometimes forget to divide by the new exponent or the derivative of the inner function in the reverse chain rule.
在微积分中，混淆 eˣ 的导数（仍为 eˣ）与 xⁿ 的导数是一个常见失误。此外，在积分时学生有时会忘记除以新指数，或在逆链式法则中漏除内层函数的导数。
Create a personalised “error log” as you work through Review Set A. Noting down each mistake and the correct method reinforces good habits and reduces the chance of repeating errors in the final exam.
在完成复习集 A 的过程中，建立自己的“错题日志”。记录下每个错误和正确的解法，能强化良好习惯，并降低在最终考试中重蹈覆辙的可能。
12. Exam Strategy & Final Tips | 考试策略与终极提示
When tackling Review Set A under timed conditions, allocate the time proportionally to the marks. If a question is worth 6 marks, spend no more than about 6–7 minutes on it before moving on. Attempt every problem; partial credit is awarded for valid methods even if the final answer is incorrect.
在限时条件下完成复习集 A 时，按分值比例分配时间。如果一道题值 6 分，就投入不超过 6–7 分钟，然后继续前进。尝试每一道题；即使最终答案不正确，有效的解题方法也能获得部分分数。
Show all steps—do not skip algebraic manipulation. Examiners look for logical flow, correct substitution, and clear final answers. Use proper mathematical notation throughout: write limits on integrals, the “dy/dx” operator, and the equal sign correctly.
展示所有步骤——不要省略代数运算。考官注重逻辑流程、正确的代入和清晰的最终答案。全程使用规范的数学符号：正确书写积分的上下限、“dy/dx”算符以及等号。
Finally, after completing Review Set A, review the problems you found challenging and redo them without looking at the solution. This active recall cements the concepts and builds the confidence needed for your IB exam.
最后，在完成复习集 A 之后，回顾你觉得有挑战性的题目，并脱离答案重新做一遍。这种主动回忆能巩固概念，并建立应对 IB 考试所需的自信。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图