📚 AQA International Mathematics A MA04 Pure Maths 4 Revision | AQA 国际数学 A MA04 纯数学 4 复习
This revision guide targets the high-yield skills assessed in AQA International Mathematics A MA04, using the June 2023 question paper at 07:00 GMT as a reference point. Work through each section alongside past papers to build confidence in Pure Mathematics 4.
本复习指南围绕 AQA 国际数学 A MA04 的重点技能展开，并以 2023 年 6 月 07:00 GMT 试卷为参考。建议结合历年真题逐节练习，以建立纯数学 4 的解题信心。
1. Integration by Partial Fractions | 部分分式积分
Many MA04 integrals begin with a rational expression such as (3x + 5) / ((x + 1)(x - 2)). Split the fraction into A/(x + 1) + B/(x - 2), solve for A and B, then integrate each term to natural logarithms.
MA04 的很多积分从形如 (3x + 5) / ((x + 1)(x - 2)) 的有理式开始。先把分式拆成 A/(x + 1) + B/(x - 2)，求出 A 和 B，再逐项积分得到自然对数。
Remember to write absolute values inside logarithms: ∫ 1/(ax + b) dx = (1/a) ln|ax + b| er fractions should be divided first, or the decomposition must include a polynomial term.
请记住对数内要写绝对值：∫ 1/(ax + b) dx = (1/a) ln|ax + b| + C。如果是有理假分式，应先做多项式除法，或者分解时加上一个多项式项。
2. Integration by Parts | 分部积分法
ula ∫u dv/dx dx = uv - ∫v du/dx dx when the integrand is a product such as x sin x or x² eˣ s when differentiated, usually a logarithm, inverse trig function, or positive power of x.
当被积函数是 x sin x 或 x² eˣ 这样的乘积时，使用公式 ∫u dv/dx dx = uv - ∫v du/dx dx。选择 u 时优先选求导后简化的函数，通常是对数、反三角函数或 x 的正次幂。
For example, with ∫x eˣ dx, take u = x and dv/dx = eˣ, giving x eˣ - eˣ ation by parts is needed, keep the same choice of u throughout and watch for sign changes.
例如，对 ∫x eˣ dx，令 u = x、dv/dx = eˣ，得到 x eˣ - eˣ + C。如果需要多次分部积分，每次都要保持相同的 u 选择，并注意符号变化。
3. Integration by Substitution | 换元积分法
e whole integral in terms of u. For definite integrals, change the limits from x-values to u-values before evaluating, so you do not need to convert back.
出现复合函数时，设 u = g(x)，求出 du/dx，并把整个积分用 u 表示。对于定积分，求值前要先把上下限从 x 值换成 u 值，这样就不必再换回 x。
Watch for hidden substitutions: if f'(x)/f(x) appears, the integral is ln|f(x)| + C. Also recognise patterns such as sin²x cos x, which suggests u = sin x.
注意隐藏的换元：如果出现 f'(x)/f(x)，积分结果就是 ln|f(x)| + C。还要识别 sin²x cos x 这类结构，它提示设 u = sin x。
4. Parametric Equations and Differentiation | 参数方程与求导
dx/dt). This avoids the need to eliminate the parameter, which can be messy or impossible.
对于由 x = f(t)、y = g(t) 给出的曲线，梯度为 dy/dx = (dy/dt) ÷ (dx/dt)。这样就不必消去参数，而消去参数有时很复杂甚至不可能。
dt (dy/dx) ÷ dx/dt. Simplify the first derivative before differentiating again, and express your final answer in terms of t.
二阶导数为 d²y/dx² = d/dt (dy/dx) ÷ dx/dt。再次求导前要先把一阶导数化简，并用 t 表示最终结果。
5. Differential Equations | 微分方程
For separable equations, collect all y-terms with dy and all x-terms with dx, then integrate both sides. Add one constant of integration on one side only, because combining two constants is unnecessary.
对于可分离变量的微分方程，把所有含 y 的项与 dy 放在一边，所有含 x 的项与 dx 放在另一边，然后两边积分。只在一边加一个积分常数，因为两个常数可以合并。
y models often appear as dy/dt = ky, giving y = A eᵏᵗ. Be ready to interpret k in context.
用初始条件求特解。指数增长和衰减模型常以 dy/dt = ky 出现，其通解为 y = A eᵏᵗ。还要能结合题目背景解释 k 的含义。
6. Binomial Expansion for Rational Powers | 有理次幂二项式展开
The expansion (1 + x)ⁿ = 1 + nx + n(n - 1)/2! x² + n(n - 1)(n - 2)/3! x³ + ... is valid for rational n when |x| < 1. Rewrite expressions like 1/√4 - x as (1/2)(1 - x/4)⁻¹/² before expanding.
当 |x| < 1 且 n 为有理数时，(1 + x)ⁿ = 1 + nx + n(n - 1)/2! x² + n(n - 1)(n - 2)/3! x³ + ... 成立。先把 1/√4 - x 写成 (1/2)(1 - x/4)⁻¹/² 再展开。
State the range of validity explicitly, such as |x| < 4 for the example above. Most mark schemes award a mark for this condition, so write it at the end of the expansion.
要明确写出收敛范围，例如上例中为 |x| < 4。多数评分标准会为这个条件给一分，所以要在展开式的末尾写出。
| Original form | Rewrite | Validity |
| 1/(4 + x) | (1/4)(1 + x/4)⁻¹ | |x| < 4 |
| √9 - x | 3(1 - x/9)¹/² | |x| < 9 |
7. Vectors in Three Dimensions | 三维向量
A vector in three dimensions has the form a i + b j + c k. Its magnitude is √a² + b² + c², and the distance between two points is the magnitude of their difference vector.
三维向量形如 a i + b j + c k。它的模为 √a² + b² + c²，两点间的距离就是两点差向量的模。
The scalar product is a·b = |a||b| perpendicular. This is often used to prove a right angle or to find an unknown component.
数量积为 a·b = |a||b| cos θ，其中 θ 是两向量的夹角。若 a·b = 0，则两向量垂直。这一性质常用于证明直角或求未知分量。
8. Vector Equations of Lines and Planes | 直线与平面的向量方程
a is a point on the line and b is the direction vector. To show two lines intersect, set their equations equal and solve for λ and μ, then check the third component is consistent.
直线可写作 r = a + λb，其中 a 是直线上一点，b 是方向向量。要证明两直线相交，令两方程相等并解出 λ 和 μ，然后检查第三个分量是否一致。
n is the normal vector. The angle between a line and a plane is found using cos θ with the normal, then taking 90° - θ. Do not use sin θ with the normal without justification.
平面常以 r·n = d 出现，其中 n 是法向量。直线与平面的夹角可先用 cos θ 求直线与法向量的夹角，再用 90° - θ 得到。不要未经说明就用 sin θ 直接对法向量算。
9. Volumes of Revolution | 旋转体体积
∫ [f(x)]² dx between the given limits. Square the function before integrating, not after.
曲线 y = f(x) 绕 x 轴旋转 360° 时，体积为 V = π ∫ [f(x)]² dx，积分限为给定区间。要先平方再积分，而不是积分后再平方。
/dt dt, with t-limits that correspond to the required x-limits. This form avoids eliminating the parameter and is commonly tested in MA04.
对于参数曲线，使用 V = π ∫ y² dx/dt dt，t 的上下限要与 x 的积分限对应。这种形式避免消去参数，是 MA04 常见考点。
10. Numerical Methods | 数值方法
The Newton-Raphson method uses xₙ₊₁ = xₙ - f(xₙ)/f' ximate a root. A starting value near the root is usually given or identified from a graph.
牛顿-拉弗森法使用 xₙ₊₁ = xₙ - f(xₙ)/f'(xₙ) 逼近根。初始值通常由题目给出或从图像中确定。
ₙ) can converge or diverge. Show each substitution and round to the accuracy requested in the question. State the final root clearly.
形如 xₙ₊₁ = g(xₙ) 的迭代公式可能收敛也可能发散。要写出每一步代入过程，并按题目要求保留精度。最后要清楚地写出求得的根。
11. MA04 Exam Technique | MA04 考试技巧
In the June 2023 MA04 paper, questions often combine two Pure 4 topics, such as parametric differentiation followed by integration by parts. Read the whole question before starting so you can plan the order of your calculations.
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