📚 Edexcel A-Level Maths Mixed Practice: Page 087 | Edexcel A-Level 数学综合练习：第087页
4)-087. It is written for Edexcel A-Level Mathematics and focuses on pure mathematics topics that appear repeatedly in timed exercise sets and mock papers.
本文梳理了像 pdfjoiner_(4)-087 这类综合练习页中最常考查的核心技能。文章面向 Edexcel A-Level 数学，重点覆盖在限时练习和模拟卷中反复出现的纯数学内容。
1. Understanding the Command Words | 理解指令词
Before attempting any mixed page, identify exactly what the examiner is asking. Command words such as 'show that', 'prove', 'find', 'hence', and 'state' determine how much working must be shown and what form the final answer should take.
在开始做任何综合练习页之前，先要准确判断题目要求。像 'show that'、'prove'、'find'、'hence' 和 'state' 这类指令词决定了需要展示多少步骤，以及最终答案应采用什么形式。
A 'show that' question usually gives the answer in the question. You must produce clear intermediate lines leading to that given result without using the answer itself as an assumption.
'show that' 类题目通常会在题干中给出结论。你必须写出清晰的中间步骤，逐步推出该结论，而不能直接把结论当作已知条件来使用。
A 'hence' question depends on the previous part. If you could not complete part (a), you can often still use the printed result to attempt part (b) and gain method marks.
'hence' 类题目依赖前一问的结果。如果你没有完整做出第 (a) 问，通常仍然可以利用题中给出的结果尝试第 (b) 问，从而获得方法分。
2. Algebraic Manipulation and Fractions | 代数运算与分式
Many mixed pages begin with simplifying algebraic fractions. Factorise numerators and denominators first, then cancel common factors. Always state any excluded values where the original denominator is zero.
许多综合练习页会从化简代数分式开始。首先对分子和分母进行因式分解，然后约去公因式。凡是原分母为零的取值，都要明确写出排除条件。
For example, simplify (3x² − 12) ÷ (x − 2). Factor the numerator as 3(x − 2)(x + 2), then cancel the (x − 2) factor.
例如，化简 (3x² − 12) ÷ (x − 2)。先把分子因式分解为 3(x − 2)(x + 2)，然后约去 (x − 2) 这个公因式。
(3x² − 12) ÷ (x − 2) = 3(x + 2), x ≠ 2
When adding or subtracting rational expressions, use a common denominator. Do not multiply only part of the expression, and keep the numerator in factorised form where possible.
在进行分式加减时，要先通分。不要只给部分表达式乘以公分母，并且在可能的情况下保持分子为因式分解形式。
For partial fractions, the denominator type decides the split: linear factors give A/(ax+b), repeated linear factors give A/(ax+b) + B/(ax+b)², and irreducible quadratics give (Ax+B)/(ax²+bx+c).
对于部分分式，分母类型决定了拆分形式：线性因式对应 A/(ax+b)，重复线性因式对应 A/(ax+b) + B/(ax+b)²，不可约二次因式对应 (Ax+B)/(ax²+bx+c)。
3. Functions and Graphs | 函数与图像
inside out, and check the domain of the inner function before applying the outer function.
复合函数写作 fg(x) = f(g(x))。计算时从内层函数开始，并在代入外层函数之前检查内层函数的定义域。
t of f(x). To find it, write y = f(x), rearrange to make x the subject, then swap x and y.
反函数 f⁻¹(x) 可以逆转 f(x) 的作用。求反函数时，先写出 y = f(x)，将 x 表示成 y 的表达式，然后交换 x 与 y。
line y = x. The domain of f⁻¹ is the range of f, and the range of f⁻¹ is the domain of f.
y = f⁻¹(x) 的图像是 y = f(x) 关于直线 y = x 的对称图形。f⁻¹ 的定义域就是 f 的值域，f⁻¹ 的值域就是 f 的定义域。
consider horizontal translation, then stretch, then vertical stretch, then vertical translation, but always follow the exact form given in the question.
函数图像的变换必须按正确顺序进行。对于 y = af(bx + c) + d，通常先考虑水平平移，再考虑水平伸缩，然后是垂直伸缩，最后是垂直平移，但具体顺序要严格按照题目给出的形式处理。
4. Trigonometric Equations and Identities | 三角方程与恒等式
Trigonometric equations require you to find all solutions in a given interval. Use the quadrant rules or the graphs of sin θ, cos θ and tan θ to identify the correct additional solutions.
三角方程需要求出给定区间内的所有解。可以利用象限规则或 sin θ、cos θ、tan θ 的图像来确定正确的补充解。
then sin θ = 1/2. The principal solution is θ = π/6, and the second solution is θ = 5π/6.
例如，若在 0 ≤ θ ≤ 2π 内有 2 sin θ = 1，则 sin θ = 1/2。主解为 θ = π/6，另一个解为 θ = 5π/6。
2 sin θ = 1 ⇒ θ = π/6 or θ = 5π/6, 0 ≤ θ ≤ 2π
olving equations. Replace one trigonometric function by the other to obtain a quadratic in a single function.
sin² θ + cos² θ = 1 和 tan θ = sin θ / cos θ 等恒等式是解三角方程的基础。将一种三角函数用另一种表示，通常可以得到关于单一三角函数的二次方程。
θ + b sin θ = c, express the left side in the form R cos(θ − α) or R sin(θ + α), where R = √a² + b² and α is found from tan α = b/a.
对于 a cos θ + b sin θ = c 型方程，可以将左边写成 R cos(θ − α) 或 R sin(θ + α) 的形式，其中 R = √a² + b²，α 由 tan α = b/a 确定。
5. Differentiation Techniques | 微分技巧
. This rule is central to differentiating powers of brackets, exponentials and logarithms.
当一个函数位于另一个函数内部时，使用链式法则。若 y = f(g(x))，则 dy/dx = f′(g(x)) g′(x)。该法则对括号幂函数、指数函数和对数函数求导都至关重要。
u dv/dx. The quotient rule states that if y = u/v, then dy/dx = (v du/dx − u dv/dx) / v².
乘法法则：若 y = uv，则 dy/dx = v du/dx + u dv/dx。除法法则：若 y = u/v，则 dy/dx = (v du/dx − u dv/dx) / v²。
dy/dx = v du/dx + u dv/dx
dy/dx = (v du/dx − u dv/dx) / v²
A table of standard derivatives helps with mixed questions:
标准导数表有助于应对混合题：
| f(x) | f′(x) |
|---|---|
| xⁿ | n xⁿ⁻¹ |
| sin x | cos x |
| cos x | −sin x |
| tan x | sec² x |
| eˣ | eˣ |
| ln x | 1 / x |
² or a sign change test to classify each point as a local maximum, local minimum or point of inflection.
驻点出现在 dy/dx = 0 的位置。可以使用二阶导数 d²y/dx² 或符号变化法判断每个点是局部极大值、局部极小值还是拐点。
6. Integration and Area | 积分与面积
verse of differentiation. The basic power rule is ∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + C for n ≠ −1.
积分是微分的逆运算。基本幂函数积分为 ∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + C，其中 n ≠ −1。
∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + C, n ≠ −1
is given by ∫ₐᵇ f(x) dx, but areas below the axis are counted as negative by the integral.
对于定积分，先求出原函数，再代入上下限相减。曲线与 x 轴之间从 x = a 到 x = b 的面积由 ∫ₐᵇ f(x) dx 给出，但位于 x 轴下方的面积在积分中计为负值。
∫ₐᵇ f(x) dx = F(b) − F(a)
When finding the total area enclosed by a curve and the x-axis, split the integral where the curve crosses the axis. Take the absolute value of each section before adding.
求曲线与 x 轴围成的总面积时，要在曲线穿过 x 轴的位置拆分积分。对每一段取绝对值后再相加。
the upper curve and lower curve: Area = ∫ₐᵇ [f(x) − g(x)] dx, where f(x) ≥ g(x) on [a, b].
求两条曲线之间的面积时，对上方曲线减下方曲线的差进行积分：面积 = ∫ₐᵇ [f(x) − g(x)] dx，其中在 [a, b] 上 f(x) ≥ g(x)。
7. Exponentials and Logarithms | 指数与对数
Exponential functions of the form y = aˣ have rate of change proportional to their value. The natural exponential eˣ is its own derivative and its own integral.
形如 y = aˣ 的指数函数，其变化率与函数值成正比。自然指数 eˣ 的导数与积分都等于它本身。
laws are logₐ(xy) = logₐ x + logₐ y, logₐ(x/y) = logₐ x − logₐ y, and logₐ(xᵏ) = k logₐ x.
对数是指数的逆运算。核心法则包括 logₐ(xy) = logₐ x + logₐ y，logₐ(x/y) = logₐ x − logₐ y，以及 logₐ(xᵏ) = k logₐ x。
ln eˣ = x and eˡⁿˣ = x
ial amount, k is the growth rate, and t is time. Taking natural logs linearises the equation.
增长与衰减问题常用模型 P = P₀ eᵏᵗ，其中 P₀ 为初始量，k 为增长率，t 为时间。对方程两边取自然对数可以将其线性化。
When solving equations such as 2ˣ = 5, take logs of both sides: ln 2ˣ = ln 5, so x ln 2 = ln 5, giving x = ln 5 / ln 2.
解 2ˣ = 5 这类方程时，两边取对数：ln 2ˣ = ln 5，因此 x ln 2 = ln 5，解得 x = ln 5 / ln 2。
8. Sequences and Series | 数列与级数
e first n terms is Sₙ = ½ n [2a + (n − 1)d] or Sₙ = ½ n (a + l), where l is the last term.
等差数列具有固定公差 d。第 n 项为 uₙ = a + (n − 1)d，前 n 项和为 Sₙ = ½ n [2a + (n − 1)d]，也可以写成 Sₙ = ½ n (a + l)，其中 l 为末项。
Sₙ = ½ n [2a + (n − 1)d]
th term is uₙ = arⁿ⁻¹. The sum of the first n terms is Sₙ = a(1 − rⁿ) / (1 − r) for r ≠ 1.
等比数列具有固定公比 r。第 n 项为 uₙ = arⁿ⁻¹。前 n 项和为 Sₙ = a(1 − rⁿ) / (1 − r)，其中 r ≠ 1。
Sₙ = a(1 − rⁿ) / (1 − r)
An infinite geometric series converges only when |r| < 1. Its sum to infinity is S∞ = a / (1 − r). If |r| ≥ 1, the series diverges and has no finite sum.
无穷等比级数只有在 |r| < 1 时收敛。其无穷项和为 S∞ = a / (1 − r)。如果 |r| ≥ 1，级数发散，不存在有限和。
9. Vectors in Pure Mathematics | 纯数学中的向量
Vectors have magnitude and direction. The magnitude of a vector a = xi + yj + zk is |a| = √x² + y² + z². A unit vector in the direction of a is a / |a|.
向量具有大小和方向。向量 a = xi + yj + zk 的模为 |a| = √x² + y² + z²。沿 a 方向的单位向量为 a / |a|。
|a| = √x² + y² + z²
product of two vectors a and b is a · b = x₁x₂ + y₁y₂ + z₁z₂. It is also given by a · b = |a||b| cos θ, where θ is the angle between the vectors.
两个向量 a 和 b 的数量积为 a · b = x₁x₂ + y₁y₂ + z₁z₂。同时也有 a · b = |a||b| cos θ，其中 θ 是两向量之间的夹角。
a · b = |a||b| cos θ
o, the vectors are perpendicular. This condition is frequently used to find unknown components.
若 a · b = 0 且两个向量均为非零向量，则这两个向量互相垂直。这一条件常用于求解未知分量。
10. Numerical Methods and Iteration | 数值方法与迭代
When an equation cannot be solved exactly, numerical methods give approximate roots. A change of sign between f(a) and f(b) indicates a root lies in the interval [a, b], provided f is continuous.
当方程无法精确求解时，数值方法可以给出近似根。若 f(a) 与 f(b) 异号，并且 f 连续，则在区间 [a, b] 内存在一个根。
t c = (a + b) / 2, then replace either a or b with c so that the sign change is preserved.
二分法不断将区间缩小一半。计算中点 c = (a + b) / 2，然后用 c 替换 a 或 b，使新的区间两端仍然保持异号。
xₙ − f(xₙ) / f′(xₙ). It usually converges quickly, but can fail if f′(xₙ) is close to zero or the starting value is poor.
牛顿-拉夫森迭代公式为 xₙ₊₁ = xₙ − f(xₙ) / f′(xₙ)。它通常收敛很快，但如果 f′(xₙ) 接近零或初值选取不当，则可能失效。
xₙ₊₁ = xₙ − f(xₙ) / f′(xₙ)
Iteration questions often ask for the root correct to two or three decimal places. Continue until successive approximations agree to the required accuracy.
迭代题常要求将根精确到两位或三位小数。继续迭代，直到连续两次近似值在所需精度下保持一致。
11. Proof and Mathematical Argument | 证明与数学论证
Proof by deduction uses known facts and logical steps to reach a conclusion. For example, the sum of an odd and an even number can be shown to be odd algebraically.
演绎证明利用已知事实和逻辑步骤推出结论。例如，可以通过代数方法证明一个奇数与一个偶数之和仍为奇数。
Proof by exhaustion checks every possible case in a finite set. It is useful for small ranges, such as proving a statement for all integers from 1 to 10.
穷举证明需要检查有限集合中的每一种可能情况。它适用于较小范围，例如证明某个命题对 1 到 10 的所有整数都成立。
Proof by contradiction assumes the opposite of the required result, then shows this assumption leads to a logical contradiction. It is often used to prove that √2 is irrational.
反证法先假设与待证结论相反的命题成立，然后证明该假设会导致逻辑矛盾。它常用于证明 √2 是无理数。
In exam answers, always state your conclusion clearly. Use the word 'therefore' or 'hence' to link your working to the final statement.
在考试作答中，一定要清晰写出结论。使用 'therefore' 或 'hence' 等词将你的推导过程与最终结论连接起来。
12. Exam Strategy for Mixed Practice Pages | 混合练习页的应考策略
Start by scanning the whole page and mark the questions you find easiest. Completing these first builds confidence and secures early marks before spending time on harder parts.
先快速浏览整页，标出你认为最简单的题目。先完成这些题目可以建立信心，并在难题上花费时间之前确保拿到基础分。
Show all working even if the answer seems obvious. Edexcel awards method marks for correct intermediate steps, so a small slip in the final answer does not lose all marks.
即使答案看起来很明显，也要展示完整步骤。E
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