📚 IGCSE OCR Maths: Last-Minute Revision Notes | IGCSE OCR 数学：考前冲刺笔记
These last-minute revision notes cover the essential topics, formulas and exam tricks for the IGCSE OCR Mathematics specification. Use them to check your understanding, plug gaps and boost your confidence before the final paper.
这份考前冲刺笔记涵盖了 IGCSE OCR 数学考试的核心主题、公式和应试技巧。用它来查漏补缺，巩固理解，在临考前提升信心。
1. Number Foundations | 数与基础运算
Prime factorisation is the key to highest common factor (HCF) and lowest common multiple (LCM). Express any integer as a product of primes using a factor tree.
质因数分解是求最大公因数（HCF）和最小公倍数（LCM）的关键。用因子树将任意整数写成质数的乘积。
To find the HCF, take only the common prime factors, each raised to the lowest power that appears in both numbers.
求 HCF 时，只取公共的质因数，每个质因数取两数中出现的最低次幂。
To find the LCM, take all prime factors from either number, each raised to the highest power that appears.
求 LCM 时，取所有出现的质因数，每个质因数取最高次幂。
Standard form is written as a × 10ⁿ, where 1 ≤ a < 10 and n is an integer. Remember: positive n for large numbers, negative n for small decimals.
标准形式写为 a × 10ⁿ，其中 1 ≤ a < 10，n 为整数。记住：大数用正指数 n，小数用负指数 n。
Rounding and estimation: round to a given number of significant figures or decimal places before calculating to check if your answer is reasonable.
四舍五入与估算：计算前先按要求保留有效数字或小数位数，以便快速判断答案是否合理。
2. Fractions, Decimals, Percentages and Ratio | 分数、小数、百分数和比
To convert a recurring decimal to a fraction, set up an equation, multiply by a power of 10 and subtract. Example: x = 0.7̅, 10x = 7.7̅ → 9x = 7 → x = 7/9.
将循环小数转化为分数，可以设方程、乘 10 的幂再相减。例如：x = 0.7̅, 10x = 7.7̅ → 9x = 7 → x = 7/9。
Percentage increase or decrease: new value = original × (1 ± r/100). Reverse percentages: divide by the multiplier to find the original amount.
百分比增减：新值 = 原值 × (1 ± r/100)。逆向百分比：用乘数逆运算，除以该乘数求原值。
Ratio problems often require sharing in a given ratio. Add the parts, then divide the total quantity by the sum. Multiply each part by this unit value.
比例问题常按给定比例分配。先将比的各项相加，总量除以总份数，再将每项乘以单位值。
Direct proportion: y = kx. Inverse proportion: y = k/x. Use given data to find the constant k, then apply it to the required value.
正比例：y = kx。反比例：y = k/x。先利用已知数据求常数 k，再代入目标值求解。
3. Algebra Essentials | 代数基础
Expand brackets carefully, multiplying each term inside. For double brackets, use FOIL: (a+b)(c+d) = ac + ad + bc + bd.
展开括号时，确保括号内每一项都相乘。双括号可用 FOIL 法则：(a+b)(c+d) = ac + ad + bc + bd。
Factorising is the reverse process. Always look for a common factor first. For quadratics, find two numbers that multiply to ac and add to b.
因式分解是展开的逆过程。首先检查公因式。对于二次式，寻找两个数，使其乘积等于 ac，和等于 b。
Solving linear equations: isolate the variable by performing the same operation on both sides. With fractions, multiply through by the common denominator.
解一次方程：在等式两边同时进行相同运算以分离变量。含有分母时，两边同乘公分母消去分数。
Inequalities: treat like equations, but reverse the inequality sign when multiplying or dividing by a negative number. Show solution on a number line with open/closed circles.
不等式：解法类似方程，但乘或除负数时，不等号方向要反转。用数轴表示解集，空心圆表示不含等于，实心圆表示包含等于。
4. Quadratics and Simultaneous Equations | 二次方程与联立方程组
Quadratic equations can be solved by factorising, using the quadratic formula x = [−b ± √(b² − 4ac)] / 2a, or completing the square.
二次方程可通过因式分解、求根公式 x = [−b ± √(b² − 4ac)] / 2a 或配方法求解。
The discriminant b² − 4ac tells you the nature of the roots: positive → two distinct real roots, zero → one repeated root, negative → no real roots.
判别式 b² − 4ac 揭示根的性质：大于 0 → 两个不等实根，等于 0 → 一个重根，小于 0 → 无实根。
Simultaneous equations: elimination works well for linear pairs. For one linear and one quadratic, substitute the linear expression into the quadratic and solve.
联立方程组：线性方程组用代入法或消元法。若有一个二次方程，将线性表达式代入二次方程，化简后求解。
Always check your solutions by substituting back into the original equations. Watch for extraneous answers when squaring.
始终将答案代回原方程检验。当涉及平方时，注意可能产生增根。
5. Graphs and Functions | 图与函数
Straight line graphs have the form y = mx + c, where m is the gradient and c is the y‑intercept. Parallel lines have the same gradient.
直线图的方程为 y = mx + c，其中 m 是斜率，c 是截距。平行线的斜率相等。
For a quadratic graph y = ax² + bx + c, the turning point is a minimum if a > 0, or a maximum if a < 0. Plot the vertex and y‑intercept first.
二次函数图像 y = ax² + bx + c 中，若 a > 0 则顶点为最小值，a < 0 则为最大值。先标出顶点和 y 轴截距再描点。
Recognise shapes of cubic, reciprocal and exponential graphs. Sketching guide: consider intercepts, asymptotes and behaviour for large x.
熟悉三次函数、反比例函数和指数函数的图像形状。画草图时注意截距、渐近线和 x 趋向无穷时的变化趋势。
Gradient of a curve at a point is found by drawing a tangent and calculating its slope. This links directly to differentiation.
曲线上某点的斜率可通过作该点的切线并计算其斜率得出。这与微分直接相关。
6. Geometry and Circle Theorems | 几何与圆定理
Angle basics: angles on a straight line sum to 180°, angles around a point sum to 360°, vertically opposite angles are equal.
基本角度：直线上邻角之和为 180°，绕一点的周角之和为 360°，对顶角相等。
Circle theorems you must know: angle at centre is twice angle at circumference; angles in the same segment are equal; opposite angles in a cyclic quadrilateral sum to 180°.
必掌握的圆定理：圆心角是圆周角的两倍；同弧上的圆周角相等；圆内接四边形对角之和为 180°。
Tangent theorems: the angle between a tangent and a radius is 90°; tangents from an external point are equal in length; alternate segment theorem.
切线定理：切线与半径的夹角为 90°；圆外一点引出的两切线长度相等；弦切角定理（交错弧定理）。
For bearings, always measure clockwise from North and give answers as three‑digit numbers. Draw a clear North line at each point.
方位角始终从正北顺时针量取，并用三位数字表示。在每个点都应画出清楚的正北方向线。
7. Mensuration and Compound Measures | 测量与复合量度
Area formulas: triangle: ½ × base × height; parallelogram: base × vertical height; trapezium: ½ (a + b) × h; circle: πr².
面积公式：三角形：½ × 底 × 高；平行四边形：底 × 垂直高；梯形：½ (a + b) × h；圆：πr²。
Volume formulas: prism: area of cross‑section × length; cylinder: πr²h; pyramid: ⅓ × base area × height; cone: ⅓ πr²h; sphere: ⁴/₃ πr³.
体积公式：棱柱：底面积 × 长；圆柱体：πr²h；棱锥：⅓ × 底面积 × 高；圆锥：⅓ πr²h；球：⁴/₃ πr³。
Surface area: add the areas of all faces. For cones, curved surface area = πrl; sphere = 4πr². Always check if units need converting.
表面积：将所有面的面积相加。圆锥侧面积 = πrl；球表面积 = 4πr²。注意单位是否需要换算。
Compound measures: speed = distance / time; density = mass / volume; pressure = force / area. Use the triangle method to rearrange.
复合量度：速度 = 距离 / 时间；密度 = 质量 / 体积；压强 = 力 / 面积。可用三角形法快速变形公式。
8. Trigonometry | 三角学
In right‑angled triangles, use SOH CAH TOA: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
直角三角形中用 SOH CAH TOA：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。
For non‑right triangles, the sine rule: a/sin A = b/sin B = c/sin C. Use it for two angles and a side, or two sides and a non‑included angle (be careful of the ambiguous case).
非直角三角形用正弦定理：a/sin A = b/sin B = c/sin C。适用于已知两角一边或两边及一对角的情况（注意可能有两解）。
Cosine rule: a² = b² + c² − 2bc cos A. Use it for two sides and the included angle, or all three sides. Area of triangle: ½ ab sin C.
余弦定理：a² = b² + c² − 2bc cos A。适用于两边夹角或三边已知。三角形面积公式：½ ab sin C。
In 3D trigonometry, identify the right‑angled triangle within the solid. Draw a neat sketch of that triangle separately. Pythagoras’ theorem is heavily used to find missing lengths first.
立体三角问题中，找出立体内部的直角三角形，单独画出该三角形的草图。通常先用勾股定理求出所需的边长。
9. Vectors and Transformations | 向量与变换
A vector is written as a column (x over y) or as xi + yj. Addition and scalar multiplication are done component‑wise. The magnitude |v| = √(x² + y²).
向量可写成列向量 (x y)ᵀ 或 xi + yj。向量的加减与数乘按分量进行。模长 |v| = √(x² + y²)。
Translations are described by a vector; every point moves the same distance in the same direction. Reflections need a mirror line; give its equation or draw it.
平移用向量描述，每个点按相同方向和距离移动。反射需要镜面线，要写明镜面线的方程或画出。
Rotation: state centre, angle and direction. Enlargement: state centre and scale factor. If negative, the shape is also inverted through the centre.
旋转：说明旋转中心、角度和方向。放大：说明中心及比例因子。比例因子为负时，图形还会倒反。
Describe a single transformation equivalent to a combination of two transformations. Vectors help prove parallel lines and collinearity.
能够用一个单一变换描述两次变换的组合。向量可以用于证明平行线和共线问题。
10. Probability and Statistics | 概率与统计
Probability of an event = number of favourable outcomes / total number of outcomes. Sum of probabilities of all mutually exclusive events is 1.
事件的概率 = 有利结果的数量 / 所有可能结果的总数。所有互斥事件的概率之和为 1。
Tree diagrams: multiply along branches, add across different paths. Independent events: P(A and B) = P(A) × P(B). Conditional probability: P(A|B) = P(A and B)/P(B).
树状图：沿分支相乘，不同路径的结果相加。独立事件：P(A 与 B) = P(A) × P(B)。条件概率：P(A|B) = P(A 与 B)/P(B)。
Statistical measures: mean = sum of values ÷ number of values; median = middle value (or mean of two middles); mode = most frequent; range = max – min.
统计量：平均数 = 总和 ÷ 个数；中位数 = 中间值（或中间两数的平均）；众数 = 出现次数最多的值；极差 = 最大值 − 最小值。
For grouped data, the modal class is the class with highest frequency. Estimate the mean using midpoints. Cumulative frequency graphs give medians and quartiles.
分组数据中，众数所在组为频数最高的组。用组中值估算平均数。累积频数图可读取中位数和四分位数。
Box plots show the five‑number summary: minimum, lower quartile, median, upper quartile, maximum. Compare distributions using these visual summaries.
箱线图展示五数概括：最小值、下四分位数、中位数、上四分位数、最大值。用这些图形可直观比较分布情况。
11. Introduction to Calculus | 微积分基础
Differentiation finds the gradient of a curve. For y = xⁿ, dy/dx = nxⁿ⁻¹. Sum or difference: differentiate term by term.
微分可以求出曲线的斜率。对于 y = xⁿ，dy/dx = nxⁿ⁻¹。和或差的导数等于各项导数的和或差。
Second derivative d²y/dx² tells you the nature of a stationary point: positive → minimum, negative → maximum, zero → possibly an inflection (check sign change).
二阶导数 d²y/dx² 说明驻点的性质：大于 0 → 极小值，小于 0 → 极大值，等于 0 → 可能为拐点（需检查两侧符号变化）。
To find the equation of a tangent at a point, calculate dy/dx for the gradient, then use y − y₁ = m(x − x₁). Normal gradient is the negative reciprocal.
求曲线在某点的切线方程：先计算 dy/dx 得到斜率 m，再用点斜式 y − y₁ = m(x − x₁)。法线的斜率是 −1/m。
Kinematics: if displacement s = f(t), then velocity v = ds/dt, acceleration a = dv/dt. Use differentiation to move from displacement to velocity to acceleration.
运动学：若位移 s = f(t)，则速度 v = ds/dt，加速度 a = dv/dt。通过微分可以从位移求出速度和加速度。
Basic integration is the reverse of differentiation: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + c (n ≠ −1). It finds the area under a curve between limits.
基本积分是微分的逆运算：∫ xⁿ dx = xⁿ⁺¹/(n+1) + c (n ≠ −1)。积分可用于计算曲线下在上下限之间的面积。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导