📚 IB Mathematics: Paper 1 Question Types Analysis | IB 数学：Paper 1 题型解析
Paper 1 in the IB Mathematics: Analysis and Approaches course, at both Standard and Higher Level, is a non‑calculator paper that demands fluent algebraic skills, sharp mental arithmetic, and a deep understanding of fundamental concepts. Short‑response and extended structured questions probe how well you can manipulate expressions, solve equations, reason logically, and set out clear mathematical arguments – all without electronic assistance.
在 IB 数学分析与方法（AA）课程中，标准水平和高水平的 Paper 1 均为不可使用计算器的试卷。它要求扎实的代数技巧、敏锐的心算能力以及对基础概念的深刻理解。试卷中的简答题和结构化拓展题旨在考查学生在无电子辅助的情况下，能否熟练地处理表达式、求解方程、进行逻辑推理并清晰地书写数学论证。
1. Overview of Paper 1 | Paper 1 概述
Paper 1 accounts for 40% of the final grade at SL (80 marks, 90 minutes) and 30% at HL (110 marks, 120 minutes). It includes a mix of compulsory short questions and longer, multi‑part problems. No calculator is allowed, which means examiners expect exact values (such as √2, π, or simplified surds), step‑by‑step working, and analytical reasoning.
Paper 1 在 SL 中占最终成绩的 40%（80 分，90 分钟），在 HL 中占 30%（110 分，120 分钟）。试卷包含必做的短答题和多步长问题。因为禁止使用计算器，评卷者要求考生给出精确值（如 √2、π 或化简后的根式）、展示完整的解题步骤以及分析性推理过程。
2. Algebraic Manipulation | 代数运算
Expanding, factorising, simplifying radicals and handling exponents are fundamental. For example, you might be asked to expand (x + 2)³ without a calculator: (x + 2)(x² + 4x + 4) = x³ + 6x² + 12x + 8. Another common task is to rationalise a denominator such as 1/(√3 – 1) by multiplying by the conjugate, yielding (√3 + 1)/2.
展开、因式分解、化简根式以及处理指数是基本技能。例如，你可能会被要求不用计算器展开 (x + 2)³：(x + 2)(x² + 4x + 4) = x³ + 6x² + 12x + 8。另一个常见任务是有理化分母，比如 1/(√3 – 1)，乘以共轭式得到 (√3 + 1)/2。
Solving exponential and logarithmic equations by hand is also typical. For instance, 2ˣ⁺¹ = 8 can be rewritten as 2ˣ⁺¹ = 2³, so x + 1 = 3 and x = 2. With logarithms, transformations like logₐ(MN) = logₐM + logₐN are tested frequently.
手算求解指数方程和对数方程也是常见题型。例如，2ˣ⁺¹ = 8 可改写为 2ˣ⁺¹ = 2³，因此 x + 1 = 3 且 x = 2。涉及对数时，经常会考查诸如 logₐ(MN) = logₐM + logₐN 的恒等变换。
3. Functions and Equations | 函数与方程
Quadratic functions appear in all forms: f(x) = ax² + bx + c, vertex form a(x – h)² + k, and factorised form. You must be able to find the discriminant Δ = b² – 4ac to determine the nature of roots. When Δ > 0, there are two distinct real roots; if Δ = 0, one repeated root; and if Δ < 0, no real roots.
二次函数以各种形式出现：f(x) = ax² + bx + c、顶点式 a(x – h)² + k 以及因式分解式。你必须会求判别式 Δ = b² – 4ac 以判断根的性质。当 Δ > 0 时有两个不等实根；Δ = 0 时有一个重根；Δ < 0 时无实根。
Function transformations – translations, stretches, reflections – are another key topic. For example, if f(x) = x², then g(x) = f(2x – 1) + 3 represents a horizontal compression by factor ½, a shift right by ½ unit, and a vertical shift up by 3. Inverse functions are equally important: for f(x) = (2x + 3)/(x – 1), swap variables and rearrange to obtain f⁻¹(x) = (x + 3)/(x – 2).
函数变换——平移、伸缩、对称——是另一关键主题。例如，若 f(x) = x²，则 g(x) = f(2x – 1) + 3 表示水平方向压缩至原来的 ½，再向右平移 ½ 个单位，并垂直向上平移 3 个单位。反函数同样重要：对于 f(x) = (2x + 3)/(x – 1)，交换变量并进行整理即可得到 f⁻¹(x) = (x + 3)/(x – 2)。
4. Trigonometry | 三角学
Radian measure replaces degrees in most analytic work. You need to convert fluently: 180° = π rad, so 30° = π/6, 45° = π/4, and so on. Exact values of sine, cosine and tangent for these angles must be memorised:
在大多数分析性题目中，弧度制替代了角度制。你需要熟练转换：180° = π rad，因此 30° = π/6，45° = π/4，等等。必须熟记这些角的正弦、余弦和正切的精确值：
| θ (rad) | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0 | 0 | 1 | 0 |
| π/6 | 1/2 | √3/2 | 1/√3 |
| π/4 | √2/2 | √2/2 | 1 |
| π/3 | √3/2 | 1/2 | √3 |
| π/2 | 1 | 0 | undefined |
Trigonometric equations, such as sin 2x = √3/2 for 0 ≤ x ≤ π, require you to find the general solution within the given interval. Using identities like sin²θ + cos²θ = 1 and the double‑angle formulas is standard. Paper 1 often asks for an exact answer in terms of π.
解三角方程，例如在区间 0 ≤ x ≤ π 内求解 sin 2x = √3/2，需要你在给定区间内找出通解。熟用恒等式如 sin²θ + cos²θ = 1 以及倍角公式是基本要求。Paper 1 通常要求用 π 表示精确答案。
5. Calculus: Differentiation | 微积分：微分
You are expected to know the derivatives of basic functions – polynomials, eˣ, ln x, sin x, cos x, tan x – and to apply the product, quotient and chain rules flawlessly. For instance, differentiate f(x) = eˣ sin x: f'(x) = eˣ sin x + eˣ cos x = eˣ(sin x + cos x).
你需要掌握基本函数（多项式、eˣ、ln x、sin x、cos x、tan x）的导数，并能熟练运用乘法法则、除法法则和链式法则。例如，对 f(x) = eˣ sin x 求导：f'(x) = eˣ sin x + eˣ cos x = eˣ(sin x + cos x)。
Finding equations of tangents and normals is a recurring theme. Given a curve y = x³ – 3x + 1 at x = 1, compute y’ = 3x² – 3, so the gradient at x=1 is 0, giving a horizontal tangent y = -1. Optimisation problems, where you set the first derivative to zero and justify the nature of stationary points using the second derivative, are also common.
求切线和法线方程是反复出现的题型。对于曲线 y = x³ – 3x + 1 在 x = 1 处，计算 y’ = 3x² – 3，因此 x = 1 处斜率为 0，得到水平切线 y = -1。优化问题也很常见：令一阶导数为零，并用二阶导数判断驻点的性质。
6. Calculus: Integration | 微积分：积分
Indefinite integration as the reverse of differentiation is central: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1). Definite integrals give the exact area under a curve. A typical question might ask for the area between y = x² and y = x+2, requiring you to find intersection points at x = -1 and x = 2, then evaluate ∫₋₁² [(x+2) – x²] dx.
作为微分逆运算的不定积分是核心内容：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C（n ≠ -1）。定积分用于求解曲线下的精确面积。一道典型题目可能会要求计算 y = x² 与 y = x+2 之间的面积，这就需要先求出交点 x = -1 和 x = 2，然后计算 ∫₋₁² [(x+2) – x²] dx。
Integration by substitution is tested without a calculator by providing an appropriate substitution. For example, to evaluate ∫ (2x+1)√(x² + x) dx, let u = x² + x, so du = (2x+1) dx, and the integral becomes ∫ √u du = ⅔ u³/² + C = ⅔ (x² + x)³/² + C.
换元积分法在 Paper 1 中会给出合适的代换方式进行考查。例如，计算 ∫ (2x+1)√(x² + x) dx，可设 u = x² + x，则 du = (2x+1) dx，积分化为 ∫ √u du = ⅔ u³/² + C = ⅔ (x² + x)³/² + C。
7. Sequences, Series and the Binomial Theorem | 数列、级数与二项式定理
Arithmetic and geometric sequences appear regularly. You need the nth term formulas aₙ = a₁ + (n-1)d and aₙ = a₁ rⁿ⁻¹, together with the sum formulas Sₙ = n/2 (2a₁ + (n-1)d) and Sₙ = a₁(1 – rⁿ)/(1 – r) for |r| < 1. A problem could combine sequences with logarithms, for instance finding the number of terms in a geometric progression where the last term is given.
等差和等比数列经常出现。你需要掌握通项公式 aₙ = a₁ + (n-1)d 以及 aₙ = a₁ rⁿ⁻¹，还有求和公式 Sₙ = n/2 (2a₁ + (n-1)d) 与当 |r| < 1 时的 Sₙ = a₁(1 - rⁿ)/(1 - r)。题目可能会将数列与对数结合，例如已知等比数列的末项，求项数。
The binomial expansion (a + b)ⁿ = Σ ⁿCᵣ aⁿ⁻ʳ bʳ (r = 0 to n) is used to expand expressions such as (1 + x)⁴ or to find a specific term without full expansion. At HL, you also encounter expansions for rational exponents using the infinite series (1 + x)ⁿ = 1 + nx + n(n-1)x²/2! + … for |x| < 1.
二项式展开 (a + b)ⁿ = Σ ⁿCᵣ aⁿ⁻ʳ bʳ（r 从 0 到 n）用于展开如 (1 + x)⁴ 的表达式，或在不完全展开的情况下求出特定项。在 HL 中，你还会遇到有理指数情形的无穷级数展开：(1 + x)ⁿ = 1 + nx + n(n-1)x²/2! + …，要求 |x| < 1。
8. Probability and Statistics | 概率与统计
Counting principles, permutations and combinations (ⁿPᵣ and ⁿCᵣ) are foundational. A typical question: “How many ways can a committee of 3 be chosen from 5 men and 4 women if it must include at least one woman?” Use complementary counting. Probability calculations often involve tree diagrams and conditional probability P(A|B) = P(A ∩ B)/P(B).
计数原理、排列（ⁿPᵣ）与组合（ⁿCᵣ）是基础。一道典型题目：“从 5 名男士和 4 名女士中选出 3 人组成委员会，至少包含一名女士，共有多少种选法？”可使用补集计数。概率计算常涉及树形图和条件概率 P(A|B) = P(A ∩ B)/P(B)。
Discrete random variables and their expected value E(X) = Σ x P(X = x) are tested without calculators, so the numbers involved are manageable. You may be asked to construct a probability distribution table and verify that the sum of probabilities equals 1.
离散随机变量及其期望值 E(X) = Σ x P(X = x) 会在不使用计算器的情况下考查，因此所涉及的数字都比较容易处理。你可能会被要求列出概率分布表，并验证概率之和为 1。
9. Vectors | 向量
Vectors in two and three dimensions are written as column vectors or in i, j, k notation. You need to compute magnitude |v| = √(x² + y² + z²), scalar (dot) product v · w = x₁x₂ + y₁y₂ + z₁z₂, and the angle between vectors using cos θ = (v · w)/(|v||w|).
二维和三维向量可写成列向量形式或 i、j、k 标记。你需要计算模长 |v| = √(x² + y² + z²)、数量积（点积）v · w = x₁x₂ + y₁y₂ + z₁z₂，并利用 cos θ = (v · w)/(|v||w|) 求出向量夹角。
Vector equations of lines, such as r = a + λb, are examined in the context of intersections and relative positions. For example, find the point of intersection between the line r = (1, 2, 3) + λ(1, -1, 2) and the plane x + 2y – z = 5. Substituting the parametric equations into the Cartesian equation yields λ = 1, giving the point (2, 1, 5).
直线的向量方程，如 r = a + λb，会在交点与相对位置的情境中进行考查。例如，求直线 r = (1, 2, 3) + λ(1, -1, 2) 与平面 x + 2y – z = 5 的交点。将参数方程代入笛卡儿方程，得到 λ = 1，交点为 (2, 1, 5)。
10. Proof and Mathematical Reasoning | 证明与数学推理
Paper 1 frequently includes a proof question, especially at HL. Mathematical induction is a standard tool: prove a statement for n = 1, assume true for n = k, and show it follows for n = k + 1. A classic example is proving that Σ r² = n(n+1)(2n+1)/6 for all positive integers n.
Paper 1 经常包含一道证明题，尤其在 HL 中。数学归纳法是标准工具：证明命题对 n = 1 成立，假设对 n = k 成立，进而证明对 n = k + 1 也成立。一个经典例子是证明对所有正整数 n，有 Σ r² = n(n+1)(2n+1)/6。
Direct proof and proof by contradiction also appear. You might be asked to prove that √2 is irrational: assume √2 = p/q in lowest terms, square both sides to obtain 2 = p²/q², then deduce that both p and q are even, contradicting the assumption of no common factors.
直接证明和反证法也会出现。你可能会被要求证明 √2 是无理数：假设 √2 = p/q 为最简分数，两边平方得到 2 = p²/q²，然后推出 p 和 q 均为偶数，与“无公因数”的假设相矛盾。
11. Exam Techniques and Tips | 应试技巧与建议
Read the entire question before you start writing; sometimes a later part gives a hint. Show all your working – even if your final answer is wrong, method marks can be earned. Use a ruler for graphs and label axes. If you get stuck on one part, move on and return to it later; the paper is designed to be completed within the time.
动笔前先通读整道题，有时后面的小问会给出提示。展示所有解题步骤——哪怕最终答案错误，仍可获得方法分。用尺子画图并标记坐标轴。如果在某一部分卡住，跳过它并稍后返回；试卷是按时间可控来设计的。
Write down exact forms unless the question specifies otherwise. Simplify fractions, rationalise denominators, and leave answers as surds or multiples of π. When solving trigonometric equations, remember to consider all quadrants within the given interval, and sketch a unit circle or graph if necessary.
除非题目特别说明，否则一律使用精确形式。化简分数、有理化分母
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