📚 Year 13 WJEC Mathematics: Core Concept Review | Year 13 WJEC 数学：核心知识点梳理
The Year 13 WJEC Mathematics course builds on AS knowledge, introducing advanced pure topics alongside applied modules. A thorough command of core concepts—from partial fractions to complex numbers—is essential for tackling the final examinations with confidence. This structured overview highlights the fundamental techniques, key formulae, and common pitfalls that you must master.
Year 13 WJEC 数学课程在 AS 阶段的基础上进一步深化，引入纯数高级主题及应用模块。熟练掌握从部分分式到复数的核心概念，是自信应对最终考试的关键。本文系统梳理了必须掌握的基础方法、重要公式和常见易错点。
1. Partial Fractions and Rational Functions | 部分分式与有理函数
When integrating rational expressions, you often need to split them into partial fractions. For distinct linear factors, write (3x+1)/((x-1)(x+2)) ≡ A/(x-1) + B/(x+2). Solve for A and B by equating coefficients or using convenient x-values. Always check for an improper fraction first—if the degree of the numerator is not less than the denominator, perform polynomial long division before decomposition.
在积分有理式时，通常需要将其拆分为部分分式。对于相异的线性因式，可将 (3x+1)/((x-1)(x+2)) 写成 A/(x-1) + B/(x+2) 的形式，通过比较系数或代入合适的 x 值求出 A 和 B。务必先检查是否为假分式——若分子的次数不低于分母，应先进行多项式长除再分解。
For a repeated linear factor, such as (x+3)/(x-2)², use the form A/(x-2) + B/(x-2)². When an irreducible quadratic factor appears, e.g., x²+1, include a term of the form (Ax+B)/(x²+1). The cover-up method works only for linear factors, so be ready to expand and compare coefficients. In WJEC papers, partial fractions frequently appear followed by integration or series expansion.
对于重线性因式如 (x+3)/(x-2)²，应设为 A/(x-2) + B/(x-2)²。若含有不可约二次因式如 x²+1，则需添加 (Ax+B)/(x²+1) 项。遮盖法仅适用于线性因式，因此要熟练进行展开和系数比较。在 WJEC 试卷中，部分分式常与积分或级数展开结合考查。
2. Advanced Trigonometry | 进阶三角学
You must be fluent with sec x ≡ 1/cos x, cosec x ≡ 1/sin x, and cot x ≡ 1/tan x ≡ cos x/sin x. Key identities include 1+tan² x ≡ sec² x and 1+cot² x ≡ x(cot x) = -cosec² x. These enable you to solve equations and tackle integration problems.
你必须熟练运用 sec x ≡ 1/cos x, cosec x ≡ 1/sin x 和 cot x ≡ 1/tan x ≡ cos x/sin x。核心恒等式有 1+tan² x ≡ sec² x 和 1+cot² x ≡ cosec² x。导数公式为 d/dx(sec x) = sec x tan x，d/dx(cosec x) = -cosec x cot x，d/dx(cot x) = -cosec² x。这些知识能帮助你解方程和处理积分题。
eck for undefined values, as sec and cosec have vertical asymptotes. Compound-angle formulas, double-angle identities like cos 2θ = 1 - 2 sin² θ, and the R-formula a cos θ + b sin θ = R cos(θ ± α) are essential tools for simplifying expressions and finding maximum/minimum values.
在指定区间内解三角方程时，要将整式统一表示为单一函数（如 cos 或 tan）。注意定义域的限制，因为 sec 和 cosec 存在垂直渐近线。和角公式、倍角公式（如 cos 2θ = 1 - 2 sin² θ）以及辅助角公式 a cos θ + b sin θ = R cos(θ ± α) 都是化简表达式和寻求最值的重要工具。
3. Exponentials and Logarithms | 指数与对数
The natural exponential function eˣ has the unique property that its derivative is itself: d/dx(eˣ) = eˣ. For y = eᵏˣ, the derivative is y' = k eᵏˣ is defined for x > 0, with derivative d/dx(ln x) = 1/x. The two functions are inverses: eln x = x and ln(eˣ lify products and quotients: ln(ab) = ln a + ln b, ln(a/b) = ln a - ln b, ln(aⁿ) = n ln a.
自然指数函数 eˣ 的独特之处在于其导数为自身：d/dx(eˣ) = eˣ。对于 y = eᵏˣ，有 y' = k eᵏˣ。自然对数 ln x 的定义域为 x > 0，其导数为 d/dx(ln x) = 1/x。二者互为反函数：eln x = x，ln(eˣ) = x。对数求导法可简化乘商运算：ln(ab) = ln a + ln b，ln(a/b) = ln a - ln b，ln(aⁿ) = n ln a。
be population growth (k > 0) and radioactive decay (k < 0). When solving equations like e²ˣ⁺¹ = 5, take natural logs of both sides. Integration standard results include ∫ eˣ dx = eˣ + C and ∫ 1/x dx = ln|x| + C. Don’t forget that integrals of the form ∫ f'(x)/f(x) dx directly yield ln|f(x)| + C, a pattern that frequently appears in WJEC exercises.
形如 P = P₀ eᵏᵗ 的指数模型可用于描述种群增长 (k > 0) 或放射性衰变 (k < 0)。求解 e²ˣ⁺¹ = 5 这类方程时，对方程两边取自然对数即可。积分中需牢记 ∫ eˣ dx = eˣ + C 和 ∫ 1/x dx = ln|x| + C。不要忽视 ∫ f'(x)/f(x) dx = ln|f(x)| + C 这一常用结论，在 WJEC 试题中经常出现。
4. Sequences and Series | 序列与级数
The binomial expansion (1+x)ⁿ = 1 + nx + [n(n-1)/2!] x² + … is valid for any rational exponent n, provided |x| < 1. You must be confident in expanding expressions like (1-2x)⁻¹ or √1+x [2a + (n-1)d]; for geometric series Sₙ = a(1-rⁿ)/(1-r), and the sum to infinity exists for |r| < 1, S∞ = a/(1-r).
二项展开式 (1+x)ⁿ = 1 + nx + [n(n-1)/2!] x² + … 对任意有理指数 n 成立，前提是 |x| < 1。你必须能够展开形如 (1-2x)⁻¹ 或 √1+x 的表达式，并指明其有效性范围。对于等差数列，前 n 项和 Sₙ = n/2 [2a + (n-1)d]；对于等比数列，Sₙ = a(1-rⁿ)/(1-r)，当 |r| < 1 时无穷级数收敛至 S∞ = a/(1-r)。
WJEC also explores Maclaurin series as an extension. For example, eˣ = 1 + x + x²/2! + x³/3! + … , sin x = x - x³/3! + x⁵/5! - … , and cos x = 1 - x²/2! + x⁴/4! - … imating the error. Recognising these series helps with limits and differential equations as well.
WJEC 还涉及麦克劳林级数作为拓展。例如 eˣ = 1 + x + x²/2! + x³/3! + … ，sin x = x - x³/3! + x⁵/5! - … ，cos x = 1 - x²/2! + x⁴/4! - … 。题目可能要求通过函数在 0 处的导数推导出这些级数。另一常见任务是截断级数以近似计算并估计误差。熟悉这些级数也有助于处理极限和微分方程。
5. Differentiation Techniques | 微分技巧
r the chain rule, product rule, and quotient rule with fluency. For y = f(g(x)), dy/dx = f'(g(x)) g'(x). For product: d/dx(uv) = u'v + uv'. Quotient: d/dx(u/v) = (u'v - uv' e both sides with respect to x, remembering to include dy/dx whenever y is differentiated.
进入 Year 13，你需要熟练掌握链式法则、乘积法则和商法则。对于 y = f(g(x))，有 dy/dx = f'(g(x)) g'(x)。乘积法则：d/dx(uv) = u'v + uv'。商法则：d/dx(u/v) = (u'v - uv')/v²。隐函数求导时，等式两边同时对 x 求导，注意每次对 y 求导都要乘上 dy/dx。
with respect to t, then dividing by dx/dt. Applications include finding equations of tangents and normals, solving related rates problems, and optimisation: set dy/dx = 0 and use the second derivative test (d²y/dx² > 0 for minimum, d²y/dx² < 0 for maximum).
对于由参数方程 x = f(t), y = g(t) 定义的函数，使用 dy/dx = (dy/dt)/(dx/dt) 求导。二阶导数 d²y/dx² 则需先对 t 求导 dy/dx，再除以 dx/dt。应用场景包括求切线和法线方程、解相关变化率问题以及最优化：令 dy/dx = 0，利用二阶导数检验（d²y/dx² > 0 时取极小值，d²y/dx² < 0 时取极大值）。
6. Integration Methods | 积分方法
ts. For substitution, look for an inner function whose derivative is also present. For ∫ x √x²+1 et u = x²+1. Integration by parts follows ∫ u dv = uv - ∫ v du; typical uses include ∫ x eˣ ng partial fractions, and handle trigonometric integrals like ∫ sin² x dx by using cos 2x.
在 AS 积分基础上，你将学习换元积分法和分部积分法。换元时需寻找内层函数及其导数的组合，例如 ∫ x √x²+1 dx 可令 u = x²+1。分部积分公式为 ∫ u dv = uv - ∫ v du，常见例题如 ∫ x eˣ dx 或 ∫ x ln x dx。你还要能运用部分分式积分有理函数，以及处理形如 ∫ sin² x dx 的三角积分，这时可利用 cos 2x 降次。
∫ y(t) (dx/dt) dt, with limits changing accordingly. Volumes of revolution about the x-axis are V = π ∫ y² dx; for y-axis revolution, V = π ∫ x² dy. You may also encounter improper integrals, where a limit is infinite or the integrand is unbounded—evaluate by replacing the problem endpoint with a variable and taking the limit.
常用标准结果有 ∫ sec² x dx = tan x + C 和 ∫ cosec x cot x dx = -cosec x + C。若函数由参数方程给出，曲线下方面积按 ∫ y dx = ∫ y(t) (dx/dt) dt 计算，并同步变换积分限。绕 x 轴旋转的体积公式为 V = π ∫ y² dx；绕 y 轴旋转则为 V = π ∫ x² dy。此外，你可能遇到反常积分，即积分限无穷或被积函数无界——此时将奇点替换为变量并取极限进行计算。
7. Differential Equations | 微分方程
emember to include the constant of integration and find particular solutions using initial conditions. Real-world contexts include Newton’s law of cooling, population growth, and chemical reaction rates, where you set up the equation from a verbal description.
一阶可分离变量的微分方程是重点内容。对于 dy/dx = f(x) g(y) 的形式，移项得 ∫ 1/g(y) dy = ∫ f(x) dx。积分后切记添加积分常数，并利用初始条件求出特解。现实情境包括牛顿冷却定律、种群增长及化学反应速率，需根据文字描述建立方程。
times the integration leads to a logarithmic form, and you must be comfortable manipulating exponentials to make y the subject. Practice setting up proportional relationships: “the rate of change of P is proportional to P” translates to dP/dt = kP.
WJEC 还可能考查建模题，要求对解的实际意义进行解释，如当 t → ∞ 时的极限值。有时积分结果为对数形式，你需熟练运用指数运算将 y 表出。要练习建立比例关系：“P 的变化率与 P 成正比”对应微分方程 dP/dt = kP。
8. Numerical Methods | 数值方法
f(a) and f(b) have opposite signs. The iterative formula xₙ₊₁ = g(xₙ) converges if -1 < g' (x). Always show the iterations step by step and present results to the required accuracy.
数值求根从寻找满足 f(a) 与 f(b) 异号的区间开始。迭代公式 xₙ₊₁ = g(xₙ) 在根附近若 -1 < g'(x) < 1 则收敛。WJEC 常要求由 f(x)=0 变形为 x = g(x) 以建立迭代式。务必逐步展示迭代过程，并按题目要求精度给出结果。
The Newton-Raphson method uses xₙ₊₁ = xₙ - f(xₙ)/f'(xₙ). It generally converges faster than simple iteration, but can fail if f' rical integration, the trapezium rule approximates ∫ f(x) dx as h/2 [y₀ + yₙ + 2(y₁ + y₂ + … ips. This topic often appears with a table of values to fill in and final area estimation.
牛顿-拉夫逊方法利用 xₙ₊₁ = xₙ - f(xₙ)/f'(xₙ) 进行迭代。它通常比简单迭代收敛更快，但当 f'(xₙ) 接近零时可能失效。数值积分方面，梯形法则将 ∫ f(x) dx 近似为 h/2 [y₀ + yₙ + 2(y₁ + y₂ + … + yₙ₋₁)]，其中 h = (b-a)/n。需要理解增加分段数如何提高精度。该主题常以表格填空题结合面积估算的形式出现。
9. Vectors in 3D | 三维向量
Vectors in three dimensions extend i, j, k notation. The dot product a·b = |a||b| ine and b is the direction vector. To find the angle between two lines, use the dot product of their direction vectors.
三维向量沿用 i, j, k 的表示法。点积 a·b = |a||b| cos θ 用于计算向量夹角。若 a·b = 0，则两向量垂直。直线的向量方程为 r = a + λ b，其中 a 为直线上一点，b 为方向向量。求两直线夹角时，只需对其方向向量作点积运算。
tion of a line and a plane is found by substituting the line equation into the plane equation. Distances between points, lines, and planes require careful application of vector projection. WJEC questions often blend coordinates with vectors expecting you to move flexibly between forms.
平面可用向量形式 r·n = p 表达，其中 n 为法向量。要得到笛卡尔方程，展开成 ax + by + cz = d 即可。直线与平面的交点可通过将直线方程代入平面方程求解。点、线、面之间的距离计算需灵活运用向量投影。WJEC 题目常混合坐标与向量，要求你在不同形式间自由切换。
10. Complex Numbers | 复数
by i² = -1. A complex number z = a + bi has real part a and imaginary part b. Its modulus |z| = √a²+b² is the angle from the positive real axis. The polar form z = r(cos θ + i sin θ) and Euler's form z = r eiθ are powerful for multiplication and division: multiply moduli, add arguments.
虚数单位 i 定义为 i² = -1。复数 z = a + bi 的实部为 a，虚部为 b。其模 |z| = √a²+b² 表示到原点的距离，辐角 arg(z) 则是从正实轴起算的角度。极坐标形式 z = r(cos θ + i sin θ) 和欧拉形式 z = r eiθ 在乘除运算中极为高效：模相乘，幅角相加。
e n nth roots of unity, zⁿ = 1, are equally spaced points on the unit circle, given by cos(2kπ/n) + i sin(2kπ/n). WJEC also expects you to solve polynomial equations with real coefficients where complex roots occur in conjugate pairs, and to represent loci such as |z - a| = r (a circle) in the Argand diagram.
棣莫弗定理指出，对任意整数 n，[r(cos θ + i sin θ)]ⁿ = rⁿ (cos nθ + i sin nθ)。由此可计算复数的幂和根。方程 zⁿ = 1 的 n 个 n 次单位根均匀分布在单位圆上，可表示为 cos(2kπ/n) + i sin(2kπ/n)。WJEC 还要求求解实系数多项式方程，其复数根以共轭对的形式出现，并在阿尔冈图中
Published by TutorHao | Year 13 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