📚 STEP Mathematics Core Topics and Revision Strategies | STEP数学核心考点与备考要点分析
STEP, the Sixth Term Examination Paper, is an admissions mathematics test used by the University of Cambridge and several other universities. It is designed to assess whether candidates can think like a mathematician, not merely recall routine A-level methods.
STEP（第六学期入学考试）是剑桥大学等院校使用的数学录取测试。它考查的是考生能否像数学家一样思考，而不仅仅是回忆 A-Level 常规方法。
1. Understanding the STEP Papers | 理解 STEP 试卷
STEP 2 assumes knowledge from A-level Mathematics, while STEP 3 assumes the additional content in A-level Further Mathematics. Both papers are 3 hours long and contain 12 questions, usually eight in pure mathematics, three in mechanics and three in probability/statistics.
STEP 2 假定考生已掌握 A-Level 数学的内容；STEP 3 则假定考生掌握 A-Level 进阶数学中的额外内容。两份试卷均为 3 小时，共 12 道题，通常包括 8 道纯数题、3 道力学题和 3 道概率统计题。
Candidates are asked to answer six questions. Each question is marked out of 20, and a high grade such as ‘1’ or ‘S’ depends on clear reasoning, correct notation and the ability to recover from partial errors.
考生需要任选 6 题作答。每题满分 20 分，要取得 “1” 或 “S” 这类高分，必须展示清晰的推理、正确的符号，并能在出现部分错误后及时修正。
No calculators are allowed, and all working must be shown. This means arithmetic accuracy and algebraic fluency are central to success.
考试不允许使用计算器，并且必须展示全部过程。这意味着计算准确度和代数熟练度是成功的关键。
2. Pure Mathematics: Algebra, Inequalities and Polynomials | 纯数：代数、不等式与多项式
Algebra dominates every STEP paper. Candidates must factorise expressions, manipulate equations and solve problems quickly and accurately.
代数是每份 STEP 试卷的核心。考生必须快速准确地因式分解、变形表达式并解方程。
For a cubic equation ax³ + bx² + cx + d = 0 with roots α, β and γ, the elementary symmetric identities are essential:
对于三次方程 ax³ + bx² + cx + d = 0，若根为 α、β、γ，下面这些对称恒等式必须熟练：
α + β + γ = −b/a, αβ + βγ + γα = c/a, αβγ = −d/a
These identities reduce problems about polynomial roots to algebra that does not require solving the original equation.
这些恒等式可将关于多项式根的问题转化为不需要解原方程的代数运算。
Inequalities often involve absolute values and rational expressions. For example, |x − 2| < 3 requires a sign-aware approach. Multiplying both sides of a rational inequality by a denominator is dangerous because the sign of the denominator may change.
不等式常涉及绝对值与分式表达式。例如 |x − 2| < 3 需要谨慎的符号讨论。对分式不等式两边直接乘以分母是危险的，因为我们不知道分母的正负。
|x − a| < R ⇔ a − R < x < a + R
You should also know the remainder theorem, the factor theorem and the binomial expansion. In STEP, however, these are starting points rather than the final objective.
你还需要掌握余数定理、因式定理和二项式展开。在 STEP 中，这些只是起点，而不是最终考查对象。
3. Calculus and Differential Equations | 微积分与微分方程
STEP calculus questions are not usually purely mechanical; they often ask you to model a situation with a derivative, estimate a small change, or identify why a proposed solution is false.
STEP 的微积分题通常不是单纯计算，而是要求你用导数建立模型、估计微小变化，或判断某个解答错在哪里。
The product and quotient rules are fundamental:
乘积法则和商法则是最基本的工具：
(uv)′ = u′v + uv′, (u/v)′ = (u′v − uv′)/v²
Integration by parts is used repeatedly in STEP. The formula should be embedded in your memory:
分部积分在 STEP 中反复出现。下面这个公式应当牢记：
∫ u dv = uv − ∫ v du
For first-order differential equations, the separator method is common. Given dy/dx = f(x)g(y), separate and integrate:
对于一阶微分方程，分离变量法很常用。若已知 dy/dx = f(x)g(y)，可分离变量并积分：
∫ 1/g(y) dy = ∫ f(x) dx
Candidates should also understand second-order equations, especially those with constant coefficients. Characteristic equations with real and repeated roots, as well as particular integrals, appear in STEP 3.
考生还应理解二阶常系数线性微分方程。STEP 3 中会出现特征方程实根、重根以及特解的内容。
4. Complex Numbers and Vectors | 复数与向量
STEP 2 and STEP 3 treat complex numbers geometrically. You need to connect the Argand diagram to loci, rotations, inequalities and regions.
STEP 2 与 STEP 3 从几何角度考查复数。你需要将阿尔冈图与轨迹、旋转、不等式和区域联系起来。
De Moivre’s theorem is one of the most powerful results in the complex-number section:
棣莫弗定理是复数部分最有力的工具之一：
(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)
It is used to derive multiple-angle identities, roots of unity and exact trigonometric values. Euler’s notation e^(iθ) = cosθ + i sinθ is also needed.
它可用于推导多倍角公式、单位根和精确三角函数值。欧拉公式 e^(iθ) = cosθ + i sinθ 也是必备知识。
In vectors, the scalar product gives both a geometric and an algebraic approach:
在向量中，点积同时具有几何意义和代数意义：
a · b = |a||b| cos θ, a · b = a₁b₁ + a₂b₂ + a₃b₃
You must also know vector equations of lines and planes, perpendicular distance calculations and when three points are collinear or coplanar.
你还需要掌握直线和平面的向量方程、点到直线或平面的距离计算，以及判断三点共线或四点共面。
5. Proof and Mathematical Reasoning | 证明与数学推理
The biggest difference between STEP and normal school exams is the need to produce independent proofs. Mathematical induction, contradiction and counterexamples all appear regularly.
STEP 与普通考试最大的区别在于，它要求你独立完成证明。数学归纳法、反证法和举反例都会经常出现。
For induction, check a suitable base case, state the induction hypothesis clearly, prove the step from n = k to n = k + 1, and finish with a conclusion.
使用归纳法时，要验证合适的基础情形，清楚写出归纳假设，证明从 n = k 到 n = k + 1 的递推步骤，并给出结论。
Proof by contradiction starts with the opposite statement and derives a logical impossibility. A classic example is proving that √2 is irrational.
反证法先假设要证的结论不成立，然后推出逻辑上不可能的结果。经典的例子是证明 √2 是无理数。
STEP also rewards constructing explicit counterexamples. If a conjecture says ‘all functions are increasing’, one careful graph may be enough to dismiss it.
STEP 也重视构造明确的反例。如果某个猜想声称 “所有函数都递增”，一张仔细绘制的图就足以否定它。
6. Mechanics: Kinematics and Dynamics | 力学：运动学与动力学
Mechanics questions usually form three of the twelve STEP questions. They reward clear diagrams and a consistent sign convention.
力学题通常占 12 题中的 3 题。它们要求画好严谨的受力图，并全程使用统一的符号约定。
The SUVAT equations are valid only for constant acceleration:
SUVAT 公式只适用于匀加速运动：
v = u + at, s = ut + ½at², v² = u² + 2as
Newton’s second law is often used together with energy methods:
牛顿第二定律通常与能量法一起使用：
F = ma, Work done = ∫ F dx
Friction requires careful treatment. At rest, friction can take any value up to μR; when sliding, the magnitude is μR. The direction of friction must oppose relative motion.
摩擦力的处理需要格外小心。静止时，
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