📚 High-Scoring Techniques for A-Level Further Maths June 2018 Paper 2 | A-Level进阶数学2018年6月试卷2高分技巧
The June 2018 A-Level Further Mathematics Paper 2 (Core Pure Mathematics 2) is a decisive examination that challenges students with advanced topics such as complex numbers, hyperbolic functions, polar coordinates, differential equations and numerical methods. Excelling not only requires diligent practice but also a strategic approach to revision and answering techniques. This article breaks down high-yield tips and common pitfalls for each major topic area, helping you to maximise your marks and boost confidence before the exam.
2018年6月的A-Level进阶数学试卷二（核心纯数二）是一场关键考试，涵盖复数、双曲函数、极坐标、微分方程与数值方法等高阶课题。要想取得高分，不仅需要大量的练习，还需掌握高效的复习策略与答题技巧。本文逐一剖析各核心考点的提分要点与常见失分点，帮助你在考前最大化得分并增强信心。
1. Complex Numbers & de Moivre’s Theorem | 复数与棣莫弗定理
Always express complex numbers in modulus-argument form r(cosθ + i sinθ) or reⁱθ before applying de Moivre’s theorem. This makes finding powers and nth roots systematic and far less error-prone.
应用棣莫弗定理前，务必将复数写为模-辐角形式 r(cosθ + i sinθ) 或 reⁱθ，这样求幂和求 n 次方根会变得系统化，且不易出错。
When expanding sin(nθ) or cos(nθ) using de Moivre and the binomial theorem, pay close attention to signs and grouping of real and imaginary parts. Clearly separate terms involving i to avoid algebraic slips.
当利用棣莫弗定理和二项式定理展开 sin(nθ) 或 cos(nθ) 时，要格外留意符号以及实部与虚部的分组。清晰分离含 i 的项，防止代数出错。
For loci problems (e.g., |z – a| = k or arg(z – a) = θ), sketch the geometric representation immediately. Interpreting the question geometrically often reveals the answer faster than pure algebraic manipulation.
遇到轨迹问题（如 |z – a| = k 或 arg(z – a) = θ），立即画出几何示意图。通过几何意义解读问题往往比纯粹代数推导更快得出答案。
2. Series & the Method of Differences | 级数与差分法
In method of differences questions, write the first few terms and the last few terms explicitly to help you visualise cancellation. Check whether the series telescopes partially or fully, and then write the sum using the remaining un-cancelled parts.
在差分法问题中，写出前几项与最后几项，直观展示相消情况。观察级数是部分抵消还是全部抵消，然后根据剩余未抵消部分写出级数和。
When evaluating sums like Σ (1/(r(r+1))) or Σ (ln(r+1) – ln r), express the general term as a difference f(r) – f(r+1) or f(r+1) – f(r) and clearly state the amount of terms left at both ends.
求 Σ (1/(r(r+1))) 或 Σ (ln(r+1) – ln r) 等和式时，将通项拆分为 f(r) – f(r+1) 或 f(r+1) – f(r) 的形式，并明确指出首尾各保留多少项。
Many June 2018 Paper 2 candidates lost marks by forgetting to adjust the expression for r starting from 2 or another offset. Always double-check the index range and whether the given expression is valid from the stated lower bound.
2018年6月试卷二中不少考生因忽略下标偏移（如 r 从 2 开始）而失分。务必反复核对求和范围以及给定表达式在起始下标处是否成立。
3. Maclaurin Series & Taylor Expansion | 麦克劳林级数与泰勒展开
Memorise standard Maclaurin series for eˣ, sin x, cos x, ln(1+x), (1+x)ⁿ, and arctan x. Then build composite series by substituting powers of x or using algebraic combinations accurately.
熟记 eˣ, sin x, cos x, ln(1+x), (1+x)ⁿ 和 arctan x 的标准麦克劳林展开式。之后可通过替换 x 的幂次或进行代数组合准确构造复合函数的级数。
When a question asks for the series up to x⁴, stop at that term and avoid unnecessarily carrying high-order terms. Present the final answer in ascending powers of x with simplified coefficients.
当题目要求展开至 x⁴ 为止时，只需求到该项，不要引入多余的高阶项。最终答案按 x 的升幂排列并化简系数。
Always state the range of validity after obtaining a Maclaurin or Taylor series. For ln(1+x) and (1+x)ⁿ, the expansion is valid for |x|<1; adjustments may be needed for composite arguments.
求出麦克劳林或泰勒级数后，一定要注明收敛范围。例如 ln(1+x) 和 (1+x)ⁿ 的展开式仅在 |x|<1 成立；对于复合变量，需相应调整。
4. Hyperbolic Functions: Differentiation & Integration | 双曲函数：微分与积分
Recall the definitions: sinh x = (eˣ – e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2, and analogous tanh x. Know key identities such as cosh²x – sinh²x = 1 and sinh 2x = 2 sinh x cosh x without hesitation.
牢记定义式：sinh x = (eˣ – e⁻ˣ)/2，cosh x = (eˣ + e⁻ˣ)/2 以及相应的 tanh x。熟练掌握 cosh²x – sinh²x = 1、sinh 2x = 2 sinh x cosh x 等核心恒等式。
For differentiation, d/dx(sinh x) = cosh x, d/dx(cosh x) = sinh x, d/dx(tanh x) = sech²x. When integrating, reverse these rules and whenever possible express hyperbolic functions in exponential form to simplify messy integrals.
微分法则：d/dx(sinh x) = cosh x，d/dx(cosh x) = sinh x，d/dx(tanh x) = sech²x。积分时逆向运用这些公式，遇到复杂积分可先将双曲函数转换为指数形式再处理。
Many students confuse the inverse hyperbolic functions’ logarithmic forms. Practice converting arsinh x = ln(x + √(x²+1)) and arcosh x = ln(x + √(x²-1)) quickly, especially before solving differential equations.
不少学生混淆反双曲函数的对数形式。考前要强化 arsinh x = ln(x + √(x²+1))、arcosh x = ln(x + √(x²-1)) 等转化，尤其在解微分方程时用得着。
5. Polar Coordinates: Curves & Areas | 极坐标：曲线与面积
When sketching polar curves r = f(θ), first identify symmetry and key values (θ = 0, π/2, π, etc.). Mark where r = 0 to find tangents at the pole. Use these reference points to draw a smooth loop or spiral.
作极坐标曲线 r = f(θ) 草图时，先判断对称性并计算关键值（θ = 0, π/2, π 等）。标出 r = 0 的点以找到极点处的切线，再利用参考点描出平滑的封闭环或螺旋线。
For area calculations, use A = ½ ∫ r² dθ between the correct limits. Determine the limits from the diagram or by solving r = 0. In the June 2018 paper, many errors came from using the wrong limits or forgetting the ½ factor.
计算面积时，应用 A = ½ ∫ r² dθ，并选择正确的积分限。根据曲线图或解 r = 0 确定上下限。2018年6月试卷中，许多错误源于限值不当或遗漏系数 ½。
When finding the tangent gradient dy/dx, use dy/dθ over dx/dθ and recall x = r cosθ, y = r sinθ. Simplify using the product rule and carefully evaluate at the required angle.
求切线斜率 dy/dx 时，利用 dy/dθ / dx/dθ，并记住 x = r cosθ, y = r sinθ。运用乘积法则化简表达式，并在指定角度处精确求值。
6. First-Order Differential Equations & Integrating Factors | 一阶微分方程与积分因子
For dy/dx + P(x)y = Q(x), the integrating factor is e^(∫P dx). Multiply the entire equation by this factor, then the left side becomes d/dx(y × integrating factor). Integrate both sides and solve for y.
对于 dy/dx + P(x)y = Q(x) 型方程，积分因子为 e^(∫P dx)。将方程两边同乘该因子，左侧即化为 d/dx(y × 积分因子)。再两边积分并解出 y。
Always simplify the integrating factor to the simplest exponential form. For example, if ∫P dx = 2ln x, then e^(∫P dx) = x², not e^(2ln x). This avoids messy algebra when differentiating.
务必将积分因子化简为最简指数形式。例如，若 ∫P dx = 2ln x，则 e^(∫P dx) = x² 而非 e^(2ln x)，这样在求导时运算更简捷。
After integration, include the constant of integration immediately. Boundary conditions given in a problem are used to find the particular solution; always check that your solution satisfies the original differential equation.
积分后立即加上积分常数。题目给出的边界条件用于确定特解；要养成习惯将所得解代回原方程验证其正确性。
7. Second-Order Linear Differential Equations | 二阶线性微分方程
For homogeneous equations ay” + by’ + cy = 0, form the auxiliary equation am² + bm + c = 0. For real distinct roots m₁, m₂, the general solution is y = Ae^(m₁x) + Be^(m₂x). For repeated or complex roots, use the standard formulas without delay.
对于齐次方程 ay” + by’ + cy = 0，构造辅助方程 am² + bm + c = 0。若为不等实根 m₁, m₂，通解为 y = Ae^(m₁x) + Be^(m₂x)。对于重根或复根，套用标准公式即可。
When finding a particular integral for a non-homogeneous term like ke^(px), try y = λe^(px); for polynomials or trigonometric functions, choose a similar form with undetermined coefficients. If the trial form overlaps with the complementary function, multiply by x.
为 k e^(px) 型非齐次项求特解时，可试设 y = λe^(px)；对于多项式或三角函数，采用含未定系数的相似形式。若试解形式与补函数重合，则需乘以 x。
In the 2018 exam, some candidates lost marks by forgetting to apply initial conditions to find A and B. Write down the derivative y’ clearly, substitute values, and solve the resulting simultaneous equations carefully.
2018年考试中，部分考生因忘记利用初始条件求解 A 和 B 而失分。应清晰地写出导数 y’，代值后建立的方程组，并细心求解。
8. Numerical Methods: Euler, Mid-ordinate & Simpson’s Rule | 数值方法：欧拉法、中矩形法与辛普森法则
For Euler’s method y_(n+1) = y_n + h·f(x_n, y_n), use a well-organised table to calculate successive values. Round to the required decimal places at each step as specified in the question to avoid cumulative rounding errors.
运用欧拉法 y_(n+1) = y_n + h·f(x_n, y_n) 时，使用结构清晰的表格计算连续值。每一步严格按题目要求的小数位数取整，以防积累舍入误差。
The mid-ordinate rule: ∫_a^b f(x)dx ≈ h Σ f(x_i) with x_i at midpoints. Remember h = (b – a)/n. Double-check the number of strips and that midpoints are correctly computed before summing.
中矩形法则：∫_a^b f(x)dx ≈ h Σ f(x_i)，其中 x_i 为中点。记住步长 h = (b – a)/n。求和前务必核对条带数与中点的计算是否正确。
Simpson’s rule requires an even number of strips (n even) and uses the pattern 1,4,2,4,…,2,4,1 for ordinates. Always write the multiplier sequence before performing calculations to avoid pattern mistakes.
辛普森法则要求条带数 n 为偶数，纵坐标乘数规律为 1,4,2,4,…,2,4,1。先写下乘数序列再进行计算，可有效避免模式错误。
9. Handling Proofs & Rigour | 处理证明题与严谨性
Proofs by induction frequently appear in series, divisibility, and matrix problems. For the inductive step: “Assume true for n = k, then prove for n = k + 1”. Clearly state where you use the induction hypothesis.
数学归纳法常出现在级数、整除性及矩阵问题中。论证步骤：“假设 n = k 时成立，再推导 n = k + 1 时成立”。要清晰标出使用归纳假设的地方。
When proving de Moivre’s theorem by induction, start from the left-hand side for n = k + 1 and factor out a complex factor (cosθ + i sinθ) before applying the induction hypothesis. Present your work in a logical flow.
用归纳法证明棣莫弗定理时，从 n = k + 1 时的左侧出发，提取因子 (cosθ + i sinθ)，再运用归纳假设。书写过程要保持逻辑清晰。
In any proof, avoid jumping steps. Examiners expect each algebraic manipulation to be justified. Even simple factorisations or trigonometric identities should be referenced or written stepwise.
任何证明题都不可省去关键步骤。考官希望每一步代数变换都有依据。即使简单的因式分解或三角恒等式，也应按步书写或注明依据。
10. Exam Strategy & Time Management | 考试策略与时间管理
Paper 2 consists of a mix of shorter and longer structured questions. Aim to spend roughly one minute per mark. Leave the last 15-20 minutes for checking and tackling any skipped sub-questions.
试卷二包含结构化的短问题与长问题。大致按每分钟答完 1 分题目的节奏分配时间，留出最后 15–20 分钟检查并补做跳过的子问题。
Read through the whole paper during the first few minutes to identify questions where you feel most confident. Attempt those first to secure marks and build momentum, but be careful not to spend too long on any single part.
开考前几分钟通览全卷，识别自己最自信的题目。优先作答这些题以确保拿到分数并建立信心，但要注意不在某个小问上耗时过久。
Present working logically and show all intermediate steps. In “show that” questions, demonstrate every manipulation so the examiner can follow your reasoning; marks are often awarded for method even if a slip occurs.
解答过程要条理分明，展示所有中间步骤。对于“证明……”类题目，把每一步变形都写出来，让考官能跟踪你的推理；即便出现计算失误，方法分依旧可得。
Finally, if you get stuck, move on and mark the question to revisit. A fresh look after solving other problems often reveals the missing insight. Keep calm and maintain steady pace throughout the exam.
最后，如果卡壳先跳过并做好标记，回头再看。解完其它问题后重新审视，常常能获得之前缺失的思路。整场考试保持镇定、匀速作答。
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Cancel reply