📚 A-Level CIE Mathematics: End-of-Term Revision Checklist | A-Level CIE 数学：期末复习提纲
This article provides a structured end-of-term revision checklist for CIE A-Level Mathematics (9709), covering key topics from Pure Mathematics, Mechanics and Statistics. Use it to identify your strengths, pinpoint gaps and focus your final preparation efficiently before the exam.
本文为 CIE A-Level 数学 (9709) 提供一份结构化的期末复习提纲，涵盖纯数、力学和统计的核心主题。借助这份提纲，你可以快速定位自己的强项与薄弱环节，在考前进行有针对性的高效复习。
1. Algebra and Functions | 代数与函数
Master quadratic equations: the standard form ax² + bx + c = 0 solves to x = [-b ± √b² - 4ac nt Δ = b² - 4ac tells you whether roots are real and distinct, real and equal, or complex.
掌握二次方程：标准形式 ax² + bx + c = 0 的解为 x = [-b ± √b² - 4ac] / 2a，判别式 Δ = b² - 4ac 决定了根是相异实根、相等实根还是复数根。
Be able to complete the square to find the vertex of a parabola and to solve quadratic inequalities using sign diagrams or graphical methods. Remember that multiplying by a negative reverses the inequality sign.
熟练使用配方法求抛物线的顶点，并利用符号表或图像法解二次不等式。注意两边同乘负数会反转不等号方向。
s for factorising cubics and higher-degree polynomials. If f(a)=0, then (x-a) is a factor.
多项式除法、因式定理与余数定理是分解三次及高次多项式的核心工具。若 f(a)=0，则 (x-a) 为因式。
e formula logₐb = log_cb / log_ca. Be ready to solve exponential equations by taking logs.
掌握指数与对数法则：logₐ(xy)=logₐx+logₐy，logₐ(x/y)=logₐx-logₐy 以及换底公式 logₐb=log_cb/log_ca。能熟练运用取对数解指数方程。
Modulus functions create piecewise definitions: |x| = x for x≥0 and -x for x<0. Solve equations and inequalities such as |2x-3| = 5 by splitting into cases.
绝对值函数需要分段处理：|x| 在 x≥0 时为 x，x<0 时为 -x。通过分情况讨论求解形如 |2x-3|=5 的方程和不等式。
etches vertically. An inverse function f⁻¹ exists only if f is one-to-one; reflect in y=x.
函数变换：y=f(x)+a 为垂直平移，y=f(x+a) 为水平平移（注意方向相反），y=a f(x) 为垂直伸缩。反函数 f⁻¹ 仅在原函数为一一映射时存在，图像关于 y=x 对称。
2. Coordinate Geometry | 坐标几何
/(x₂ - x₁). Parallel lines share the same gradient; perpendicular lines satisfy m₁m₂ = -1.
直线方程可写为 y=mx+c 或 y-y₁=m(x-x₁)。斜率 m=(y₂-y₁)/(x₂-x₁)。平行线斜率相等，垂直线满足 m₁m₂=-1。
entre (a,b) and radius r. The general form x²+y²+2gx+2fy+c=0 has centre (-g,-f) and radius √g²+f²-c.
圆的几何：标准方程 (x-a)²+(y-b)²=r² 表示圆心 (a,b)、半径 r。一般式 x²+y²+2gx+2fy+c=0 的圆心为 (-g,-f)，半径 √g²+f²-c。
use the discriminant condition Δ = 0 after substituting the line equation into the circle.
半径在切点处与切线垂直。求切线方程时，可将直线方程代入圆的方程，利用判别式 Δ=0 得到切线的条件。
(dx/dt). Locus problems often lead to Cartesian equations after eliminating the parameter.
参数方程 (P3) 通过第三变量描述曲线；求导时使用 dy/dx=(dy/dt)/(dx/dt)。轨迹问题消去参数后常可得到直角坐标方程。
3. Sequences and Series | 数列与级数
(n-1)d] or n/2 (a + l). Geometric progression: uₙ = arⁿ⁻¹, sum Sₙ = a(1-rⁿ)/(1-r) for r≠1.
等差数列：第 n 项 uₙ=a+(n-1)d，前 n 项和 Sₙ=n/2[2a+(n-1)d] 或 n/2(a+l)。等比数列：uₙ=arⁿ⁻¹，当 r≠1 时 Sₙ=a(1-rⁿ)/(1-r)。
The sum to infinity of a convergent geometric series with |r|<1 is S∞ = a/(1-r). Be able to find the range of r for convergence in applied problems.
当 |r|<1 时，收敛等比级数的无穷和为 S∞=a/(1-r)。需要能根据实际情境求出收敛的 r 范围。
Binomial expansion: (1+x)ⁿ = 1 + nx + [n(n-1)/2!]x² + ... for |x| l n. In CIE, you may need to expand compound expressions like (a+bx)ⁿ by factoring out aⁿ.
二项展开式：当 |x|<1 时 (1+x)ⁿ=1+nx+[n(n-1)/2!]x²+...，对于有理数 n 成立。在 CIE 中常需通过提取 aⁿ 展开形如 (a+bx)ⁿ 的式子。
Series expansion of standard functions (P3): know the Maclaurin series for eˣ, sin x, cos x, and ln(1+x). Be ready to derive further expansions by substitution and manipulation.
标准函数的级数展开 (P3)：熟记 eˣ、sin x、cos x 和 ln(1+x) 的麦克劳林级数。能通过代换与恒等变形得到更复杂的展开式。
4. Trigonometry | 三角学
length s = rθ, sector area A = ½ r²θ. Always set calculators to radian mode for calculus.
弧度是自然的角量度：π 弧度 = 180°。弧长 s=rθ，扇形面积 A=½ r²θ。进行微积分运算时务必把计算器调至弧度模式。
sinθ/cosθ. Compound angle formulas: sin(A±B) = sinA cosB ± cosA sinB, cos(A±B) = cosA cosB ∓ sinA sinB.
核心恒等式：sin²θ+cos²θ=1，tanθ=sinθ/cosθ。和差公式：sin(A±B)=sinA cosB±cosA sinB，cos(A±B)=cosA cosB∓ sinA sinB。
sin²θ = 2cos²θ - 1 = 1 - 2sin²θ. These are crucial for integration and solving equations.
倍角公式：sin 2θ=2 sinθ cosθ，cos 2θ=cos²θ-sin²θ=2cos²θ-1=1-2sin²θ。它们在积分和解方程中至关重要。
Solving trigonometric equations: express in a single trig ratio, find basic angles, and use quadrant diagrams or CAST to list all solutions within the given interval.
解三角方程：化为单一三角比，求出基础角，再借助象限图或 CAST 规则列出指定区间内的所有解。
Harmonic form: R sin(θ ± α) or R cos(θ ± α) where R = √a²+b² and tanα = b/a. This simplifies finding the maximum/minimum values of a sinθ + b cosθ.
辅助角形式：R sin(θ±α) 或 R cos(θ±α)，其中 R=√a²+b²，tanα=b/a。利用此形式可以轻松求出 a sinθ+b cosθ 的最大最小值。
5. Differentiation | 微分
Standard derivatives: d/dx (xⁿ) = nxⁿ⁻¹, d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x, d/dx (sin x) = cos x, d/dx (cos x) = -sin x, d/dx (tan x) = sec²x.
标准导数公式：d/dx (xⁿ)=nxⁿ⁻¹，d/dx (eˣ)=eˣ，d/dx (ln x)=1/x，d/dx (sin x)=cos x，d/dx (cos x)=-sin x，d/dx (tan x)=sec²x。
Product rule: d/dx (uv) = u'v + uv'. Quotient rule: d/dx (u/v) = (u'v - uv' dx = dy/du × du/dx. Apply to composite functions, parametric and implicit differentiation.
乘法法则：d/dx (uv)=u'v+uv'。除法法则：d/dx (u/v)=(u'v-uv')/v²。链式法则：dy/dx=dy/du×du/dx。可应用于复合函数、参数方程和隐函数求导。
dy/dx whenever differentiating y. Parametric differentiation uses dy/dx = (dy/dt)/(dx/dt).
隐函数微分 (P3) 将 y 视为 x 的函数，两边同时对 x 求导，每当对 y 求导时补上 dy/dx。参数微分使用 dy/dx=(dy/dt)/(dx/dt)。
d derivative test or first derivative sign change. Maxima, minima and points of inflection all matter.
二阶导数 d²y/dx² 刻画凹凸性。驻点满足 dy/dx=0，可用二阶导数检验或一阶导数符号变化进行分类。极大值、极小值与拐点都是常见考点。
change, and optimisation problems. Set up a model, express the quantity to optimise in one variable and differentiate.
应用：切线与法线 (y-y₁=m(x-x₁))、相关变化率以及最优化问题。建立模型，将待优化的量表示为单一变量的函数，然后求导找最值。
6. Integration | 积分
tegration as reverse differentiation: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C for n≠-1, and ∫ 1/x dx = ln|x| + C. Also know ∫ eˣ dx = eˣ + C, ∫ sin x dx = -cos x + C, ∫ cos x dx = sin x + C.
积分作为微分的逆运算：∫ xⁿ dx = xⁿ⁺¹/(n+1)+C (n≠-1)，∫ 1/x dx=ln|x|+C。同时牢记 ∫ eˣ dx=eˣ+C，∫ sin x dx=-cos x+C，∫ cos x dx=sin x+C。
= linear function, u² = expression under a square root, or trigonometric substitutions for √a²-x² forms.
利用换元积分法简化被积函数。对定积分要一并更换积分上下限。常见换元：令 u 为线性函数，或令 u² 等于根号内的式子，或对 √a²-x² 形式使用三角换元。
garithmic, inverse trig, algebraic, trigonometric, exponential). Often used for products of x and eˣ, sin x, or ln x.
分部积分 (P3)：∫ u dv = uv - ∫ v du。根据 LIATE 规则选择 u（对数、反三角、代数、三角、指数）。常用于 x 与 eˣ、sin x 或 ln x 的乘积。
lue if the curve dips below the axis. The area between two curves is ∫ (upper - lower) dx.
曲线与 x 轴之间的面积由 ∫ y dx 给出，若曲线部分在 x 轴下方则取绝对值。两条曲线间的面积为 ∫ (上曲线 - 下曲线) dx。
s other than axes, adjust the integrand accordingly. Always sketch the region to set up correct limits.
绕 x 轴旋转的体积：V = π ∫ y² dx。若旋转轴非坐标轴，需相应调整被积函数。建议先画草图以正确设定积分限。
+ 2(y₁+...+yₙ₋₁) + yₙ] where h = (b-a)/n. Useful when analytical integration is difficult.
数值积分：梯形法则用 h/2 [y₀+2(y₁+…+yₙ₋₁)+yₙ] 近似 ∫ f(x) dx，其中 h=(b-a)/n。当解析积分困难时此法尤为有用。
7. Differential Equations (P3) | 微分方程 (P3)
dy = ∫ f(x) dx. Integrate both sides and include the constant of integration immediately.
一阶可分离变量的微分方程：将 dy/dx=f(x)g(y) 改写为 ∫ 1/g(y) dy = ∫ f(x) dx，两边积分并立即加上积分常数。
ude exponential growth/decay dy/dx = ky and limited growth where the solution involves e⁻ᵏˣ.
利用已知条件求出特解。常见模型包括指数增长/衰减 dy/dx=ky 以及含有 e⁻ᵏˣ 的受限增长模型。
When variables cannot be separated, an integrating factor may be needed, but CIE P3 mainly focuses on clearly separable types. Always verify your solution by differentiating.
当变量不可分离时可能需要积分因子，但 CIE P3 主要考查明显的分离变量类型。求出解后务必通过求导验证其正确性。
8. Vectors (P3) | 向量 (P3)
A vector in 3D is written as ai + bj + ck or as a column vector. Magnitude: |v| = √a²+b²+c². The dot product a·b = |a||b| cos θ, used to find angles between vectors.
三维向量可写为 ai+bj+ck 或列向量。模长：|v|=√a²+b²+c²。点积 a·b=|a||b| cos θ，用于求两向量夹角。
a point on the line and b is the direction vector. To check if a point lies on a line, see if a consistent λ exists.
直线方程：r=a+λb，其中 a 为直线上一点，b 为方向向量。判断某点是否在直线上，只需检查是否存在一致的 λ 值。
he scalar form ax+by+cz=d. Find n via cross product of two direction vectors in the plane.
平面方程：r·n=a·n，其中 n 为法向量；也可写成标量形式 ax+by+cz=d。平面的法向量可通过平面内两个方向向量的叉积求得。
Intersection problems: line with plane (substitute line into plane), two planes (solve simultaneous equations), or finding perpendicular distance from a point to a line/plane using vector formulae.
求交问题：直线与平面相交（将直线代入平面方程），两平面相交（解联立方程），以及利用向量公式求点到直线或点到平面的垂直距离。
9. Complex Numbers (P3) | 复数 (P3)
multiplication (i² = -1), and division (multiply numerator and denominator by conjugate).
复数 z=x+iy 的实部为 x，虚部为 y。共轭复数为 z*=x-iy。基本运算：加减、乘法（注意 i²=-1）以及除法（分子分母同乘共轭）。
Modulus |z| = √x²+y² nd diagram. Argument arg(z) = θ, measured from the positive real axis, usually in (-π, π].
模 |z|=√x²+y² 表示复平面上点到原点的距离。辐角 arg(z)=θ 从正实轴量起，通常取值在 (-π,π] 区间。
on: multiply moduli, add arguments. Division: divide moduli, subtract arguments. de Moivre's theorem: (cosθ + i sinθ)ⁿ = cos nθ + i sin nθ.
极坐标形式：z=r(cosθ+i sinθ) 或 r cisθ。乘法：模相乘，辐角相加；除法：模相除，辐角相减。棣莫弗定理：(cosθ+i sinθ)ⁿ=cos nθ+i sin nθ。
Use de Moivre to find roots of complex numbers, such as the nth roots of unity. Equate real and imaginary parts to derive trigonometric identities.
利用棣莫弗定理求复数的根，例如 n 次单位根。通过实部与虚部分别相等可推导三角恒等式。
10. Mechanics (M1) – Kinematics and Forces | 力学 (M1) – 运动学与力
and s = vt - ½ at². Choose the equation that avoids the unknown you are not interested in.
匀加速运动的 SUVAT 方程：v=u+at，s=ut+½ at²，s=½ (u+v)t，v²=u²+2as，以及 s=vt-½ at²。选择不含无关未知量的方程进行求解。
Displacement–time and velocity–time graphs: gradient gives velocity/acceleration, area under v–t graph gives displacement. Motion under gravity uses a = g = 9.8 m s⁻² downwards.
位移–时间图与速度–时间图：斜率表示速度/加速度，v–t 图下的面积代表位移。重力作用下的运动取向下加速度 a=g=9.8 m s⁻²。
Newton' ections. Friction F ≤ μR, where R is normal reaction and μ is the coefficient of friction.
牛顿运动定律：合力 F=ma。将力沿垂直方向进行分解。摩擦力 F≤μR，其中 R 为法向反作用力，μ 为摩擦系数。
ach mass separately or treat the whole system. Include tension in strings and consider pulley problems where one mass moves up as the other moves down.
连接体问题：分别对各质点应用 F=ma，或整体考虑系统。计算中
Published by TutorHao | A-Level 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