OCR数学 纯数 证明 三角 微积分 向量 数列
OCR 的 A Level Mathematics A 代号 H240，三份卷子里的纯数部分加起来占了两百多分。纯数不是单独一张卷，它被拆进 Paper 1、Paper 2 与 Paper 3 的每一节。这篇文章按纯数的十个主题逐块拆开，每块配一个可以自己算一遍的例子。文中的试卷结构、公式册位置与有效数字要求都取 OCR 官方 specification 与样卷。
OCR's A Level Mathematics A (H240) spreads pure mathematics across all three papers instead of examining it in one sitting. The notes below follow the pure topics in the specification and attach one worked example to each, using the paper structure, formula booklet and accuracy rules printed by OCR. Anyone revising from a topic list alone tends to miss how much of every paper is pure.
三卷结构、公式册与三位有效数字
Paper 1 是纯数单节卷，满分 100 分，时长 2 小时，占总成绩三分之一。Paper 2 与 Paper 3 各分两节，纯数约占 50 分，另一节分别是统计与力学。Paper 2 的统计节里会出现预发布数据集（large data set）的题目，OCR 在考前放出一份电子表格给课堂使用。三份卷子需要的公式都印在试题册第 2、3 页，余弦定理这类长公式不用背。非精确答案按 3 位有效数字给，题目另有说明时以题目为准。计算器要遵守 JCQ 规则，能自动因式分解或用符号做微积分的机型不能带进考场。很多学生以为公式册在手就不用背，其实公式册只给结论，选哪一条仍然要练。
Every paper lasts two hours and carries 100 marks, and the formulae are printed on pages 2 and 3 of the question paper, so the memory work shifts from recall to selection. The JCQ calculator rules still exclude machines that factorise or differentiate symbolically, and numerical answers are expected to three significant figures unless a question says otherwise. Many candidates treat the booklet as a safety net and then freeze because they cannot decide which result applies.
证明的三种写法与反例反驳
纯数的第一块内容是证明，spec 要求掌握演绎证明、穷举证明、反证法，以及用反例推翻一个命题。演绎证明从一个为真的前提出发，用代数一路推到结论。穷举要把情形列全，比如证明 n 为整数时 n^2+n 一定是偶数，只要分 n 为偶数与 n 为奇数两种。反证法的经典例子是根号 2 无法写成两个整数之比，设它等于既约分数 p/q，推出 p 与 q 都是偶数，与既约矛盾。反例只需要一个，命题 n^2+n+41 对所有正整数都取质数，在 n=40 时得到 1681，也就是 41 的平方。常见的错误是反证法里假设完忘记回到原命题，或者穷举漏掉一种情形。阅卷时这类题的分数集中在推理链是否闭合，而不是结论是否正确。
Proof questions award most marks for a closed chain of reasoning, so a correct conclusion drawn from an assumption that was never discharged scores poorly. Exhaustion needs every case listed, while one counterexample is enough to destroy a universal claim, as n^2+n+41 shows at n=40. Candidates who write that the result follows obviously are the ones examiners mark down.
多项式除法、因式定理与余数定理
因式定理说 (x-a) 是 f(x) 的因式，当且仅当 f(a)=0。余数定理说 f(x) 除以 (x-a) 的余数就是 f(a)。分解三次式时先试常数项的小因数，正负都要试，因为 (x+2) 对应的是 a 等于负 2。试出一个根之后用多项式除法或比较系数降到二次，再按常规方法分解。常见的错误是把 (x+2) 当成 a 等于 2 代入，符号一错整题都塌。这个工具在后面求曲线与坐标轴交点、化简有理函数时都会反复出现。多做几道含参系数的题，符号敏感度会上来。
The factor theorem and the remainder theorem are the same substitution viewed from two angles, and both fail silently when a sign is dropped. When a cubic is given, trial factors come from the divisors of the constant term with both signs tested. Slips here usually cost the whole question rather than a single mark.
圆的方程与切线垂直半径
圆的方程有两种写法，标准式是 (x-a)^2+(y-b)^2=r^2，一般式要靠配方还原圆心与半径。以 x^2+y^2-4x+6y-12=0 为例，配方后是 (x-2)^2+(y+3)^2=25，圆心在 (2,-3)，半径 5。切线与半径垂直，所以切点处切线斜率等于半径斜率的负倒数。判断直线与圆的位置关系，用圆心到直线的距离与半径比较，相等是相切，小于是相交，大于是相离。很多学生以为看到圆就要配方，题目给的是标准式时直接读圆心更快。这一节的计算量不大，失分多在切线斜率忘取负倒数。
Circle questions turn on two facts: completing the square recovers the centre and radius, and a tangent meets the radius at right angles. Once the centre is known, the distance to the line decides whether it cuts, touches or misses the circle. Reading the centre straight off a standard-form equation saves time better spent on the tangent algebra.
等差等比数列与无穷级数收敛
等差数列的通项是 u_n=a+(n-1)d，前 n 项和可以用首末项平均再乘项数。等比数列的通项是 u_n=ar^(n-1)，求和公式是 a 乘 (1-r^n) 再除以 1-r。公比的绝对值小于 1 时，无穷级数收敛到 a/(1-r)。收敛条件这一句比公式本身更容易丢分，公比等于 1 或绝对值不小于 1 时级数发散，写和就不成立。举例说，首项 3、公比三分之二的无穷级数和是 9。常见的错误是把公比的符号弄错，或者把项数数成 n-1。做题时先把 a 与 r 写在草稿边上，比直接套公式稳。
Geometric series are examined far more often than arithmetic ones at A level because convergence is easy to get wrong. The sum to infinity exists only when the modulus of the common ratio is below one, and an answer quoting a finite sum otherwise loses the method marks. Counting the number of terms incorrectly is the other recurring slip.
二项展开与 |x| 小于 1 的适用范围
n 为正整数时，(a+b)^n 用二项系数展开，通项里 a 的指数从 n 降到 0。A level 还要掌握 n 为有理数时 (1+x)^n 的展开，spec 写明这个展开只在 |x| 小于 1 时有效。以 (1+x)^(1/2) 为例，前三项是 1 加 x/2 减 x^2/8，代 x=0.01 得到的近似精度很高。常见的错误是忽略有效范围，直接代 x=2 去算近似，结果越展开越偏。展开式的项数按题目要求截断，答案里保留 3 位有效数字即可。
Binomial expansion splits into a finite case for positive integer powers and an infinite series for rational powers, and only the second one needs a validity condition. The expansion of (1+x)^n for rational n holds when the modulus of x is below one, which is exactly why substituting a large value gives nonsense. Examiners expect the range to be stated, not assumed.
三角恒等式与 R sin(x+α) 化简
三角恒等式的起点是 sin^2 x + cos^2 x = 1 与 tan x = sin x / cos x。二倍角公式由 cos 2x 的三种写法派生，哪一种好用取决于题目要消掉什么。把 a sin x + b cos x 写成 R sin(x+α) 时，R 等于根号下 a^2+b^2，tan α 等于 b/a，还要按系数符号确定 α 所在象限。以 3 sin x + 4 cos x 为例，R 是 5，最大值 5，最小值负 5。这类化简在解方程与求最值时都会出现。常见的错误是把 α 算成 tan α = a/b，分子分母颠倒了。
Writing a sin x + b cos x as a single sine wave converts a messy expression into one with an obvious maximum and minimum. The amplitude is the square root of the sum of squares, and the phase shift needs both the ratio and the signs of the coefficients. Reversing that ratio is the commonest error in this topic.
弧度制、弧长与扇形面积
弧度制下 π 等于 180 度，弧长公式是 s=rθ，扇形面积公式是 A 等于二分之一的 r^2 乘 θ，θ 必须是弧度。半径 6 厘米、圆心角 1.2 弧度时，弧长 7.2 厘米，面积 21.6 平方厘米。小角近似给出 sin θ 约等于 θ，cos θ 约等于 1 减 θ^2/2，tan θ 约等于 θ，同样只在弧度下成立。常见的错误是把角度直接代进这些公式，答案会差出五十多倍。题目若混用两种单位，先统一成弧度再动笔。
Radians exist so that arc length and sector area collapse into single products, and the small-angle approximations inherit the same unit condition. Substituting degrees into s=rθ or into sin θ close to θ produces answers wrong by a factor of about 57. The formula booklet prints the results without the reminder.
指数与对数在建模题里的位置
e^x 与 ln x 互为反函数，对数律把乘法变成加法，ln(ab) 等于 ln a 加 ln b，ln(a/b) 等于 ln a 减 ln b，ln(a^k) 等于 k ln a。建模题常给 y=ae^(bx) 这样的关系，两边取对数得 ln y 等于 ln a 加 bx，画成直线就能从截距与斜率读出 a 与 b。常见的错误是把 ln(a+b) 拆成两项相加，这条式子本身不成立。反函数那里还要会解 e^x=5 得到 x=ln 5，答案保留三位有效数字。
Logarithms are examined as a modelling tool more often than as pure algebra, and the standard move is to linearise an exponential relationship before reading off constants. Taking logs of y=ae^(bx) gives a straight line whose intercept is ln a and whose gradient is b. Writing ln(a+b) as the sum of two logs is the error examiners see most.
链式法则、隐函数与参数方程求导
复合函数用链式法则，乘积与商用对应公式，隐函数求导时对 y 求导要乘一个 dy/dx。参数方程给出的曲线，导数等于 (dy/dt) 除以 (dx/dt)。以 x^2+y^2=25 为例，两边求导得 2x 加 2y 乘 dy/dx 等于 0，所以 dy/dx 等于负 x/y。二阶导数的符号用来判断驻点是极大还是极小，正为极小，负为极大。常见的错误是隐函数求导漏掉 y 的链式项，把 2y 直接写下来。参数方程求二阶导要再除一次 dx/dt，这一处最容易漏。
Implicit and parametric differentiation are both applications of the chain rule, and forgetting the extra factor is the most expensive mistake in this topic. A second derivative then classifies each stationary point through its sign. Questions asking for the nature of a point expect that sign argument, not only the coordinate.
分部积分与换元法的取舍
换元法适合被积函数里出现导数与函数本身配对的结构，比如 2x/(x^2+1) 的积分直接得 ln(x^2+1) 加 c。分部积分适合乘积，公式是 ∫u dv 等于 uv 减 ∫v du，选 u 的原则是让它求导之后更简单。∫x e^x dx 是典型例子，∫ln x dx 也能用分部积分，把 ln x 当 u、dx 当 dv，结果是 x ln x 减 x 加 c。常见的错误是选错 u 与 dv，越积越复杂，这时换一个顺序通常更快。定积分用分部积分时要记得把上下限带进 uv 那一项。
Substitution suits integrands built from a function and its derivative, while integration by parts suits products and needs u chosen so that differentiation simplifies it. Picking the parts in the wrong order usually produces a harder integral rather than a wrong one, which is why a quick retry pays off. The integral of ln x is the standard test of that judgement.
定积分求面积与梯形法则
定积分求面积时，曲线在 x 轴上方取正值，下方取负值，跨过 x 轴的部分要分段计算。两条曲线之间的面积用上减下，先求交点定出积分上下限。梯形法则把区间分成若干等宽带，公式是带宽一半乘首末两项加中间各项的两倍之和。常见的错误是忘记检查曲线是否在区间内穿过 x 轴，答案因此偏小甚至为零。这个法则给的是近似值，题目通常要求把每条带的纵坐标写出来，过程分比结果分多。
Definite integrals measure signed area, so any region crossing the axis has to be split before the pieces are added. Between two curves the integrand is the upper function minus the lower one, with limits taken from the intersections. The trapezium rule is an approximation and examiners still expect the strip width and the ordinates to be shown.
Newton-Raphson 迭代与符号变化定位根
数值方法先要定位根，条件是 f(a) 与 f(b) 符号相反，并且函数在区间上连续。牛顿拉夫森迭代的公式是 x(n+1) 等于 x(n) 减 f(x(n)) 除以 f'(x(n))，从一个接近根的初值出发，通常几步就收敛。不动点迭代写成 x=g(x)，局部收敛的条件是 g'(x) 的绝对值小于 1。常见的错误是只写符号变化就判定根存在，漏掉连续这个前提。迭代过程要求写出每一步的数值，收敛慢的题目会明确给出迭代次数。
Numerical methods questions reward precision about conditions rather than speed of iteration. A sign change alone does not guarantee a root unless the function is continuous on the interval, and Newton-Raphson needs a starting value close enough to converge. Fixed-point iteration converges locally only when the modulus of g'(x) stays below one.
三维向量与几何证明
三维向量写成 ai+bj+ck 的形式，模长是根号下 a^2+b^2+c^2，方向相同的单位向量用向量除以模长。几何证明里，两条线平行看向量是否成倍数，垂直看点积是否为零。证明三点共线要先证两个向量平行，再说它们共用一个点，缺了后一句结论不完整。常见的错误是只写平行就宣布共线，忽略了共用点这一步。点积与模长的计算过程要写全，一步答案往往只有一分。
Vector geometry reduces to three tests: a scalar multiple for parallel lines, a zero dot product for perpendicular ones, and a shared point for collinearity. Omitting the shared point is the commonest lost mark in proof questions built on vectors. Magnitude and unit vectors are usually worth a mark each on their own.
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