📚 Edexcel A Level Maths Practice Set 079: Mixed Pure Topics | 爱德思 A Level 数学练习第079套：纯数混合专题
This set brings together exam-style questions and worked strategies from the high-yield pure mathematics content tested in Edexcel A Level Mathematics Paper 1. Each section pairs a concise explanation with a model example so you can revise the method and then apply it under timed conditions.
本练习第079套汇集了爱德思 A Level 数学卷一纯数部分的高频考点，以考试风格例题配合解题策略。每一节都采用简明讲解与典型示例配对，帮助你先复习方法，再在限时条件下独立应用。
1. Algebraic Simplification and Surds | 代数化简与根式
Surds questions in Edexcel papers usually require rationalising a denominator or simplifying an expression such as (2 + √3)(5 − √3). Always look for the difference of two squares before multiplying out brackets.
爱德思试卷中的根式题通常要求有理化分母，或化简形如 (2 + √3)(5 − √3) 的表达式。展开括号前，始终优先寻找平方差结构。
For example, to rationalise 1/(√5 + √3), multiply the numerator and denominator by the conjugate √5 − √3. The denominator becomes 5 − 3 = 2, so the result is ½(√5 − √3). This technique avoids leaving a surd in the denominator.
例如，要对 1/(√5 + √3) 有理化，可将分子分母同乘共轭根式 √5 − √3。分母变为 5 − 3 = 2，因此结果为 ½(√5 − √3)。这一技巧可以避免分母中保留根式。
1/(√5 + √3) × (√5 − √3)/(√5 − √3) = (√5 − √3)/2
2. Quadratic Discriminant and Inequalities | 二次判别式与不等式
The discriminant Δ = b² − 4ac tells you how many real roots the quadratic ax² + bx + c = 0 has. If Δ > 0 there are two distinct real roots; if Δ = 0 there is exactly one repeated root; if Δ < 0 there are no real roots.
判别式 Δ = b² − 4ac 可以判断二次方程 ax² + bx + c = 0 有多少个实根。若 Δ > 0，则有两个不同实根；若 Δ = 0，则有一个重根；若 Δ < 0，则没有实根。
For a quadratic inequality such as x² − 5x + 6 > 0, first solve the equation x² − 5x + 6 = 0 to get x = 2 and x = 3. Then sketch the parabola or use a sign table to identify x < 2 or x > 3 as the solution region.
对于形如 x² − 5x + 6 > 0 的二次不等式，先解方程 x² − 5x + 6 = 0，得到 x = 2 与 x = 3。然后画出抛物线草图或使用符号表，确定解集为 x < 2 或 x > 3。
x² − 5x + 6 > 0 ⇒ (x − 2)(x − 3) > 0 ⇒ x < 2 or x > 3
3. Polynomial Division and Factor Theorem | 多项式除法与因式定理
The factor theorem states that (x − a) is a factor of a polynomial f(x) if and only if f(a) = 0. This is often used with polynomial division to factorise cubic or quartic expressions.
因式定理指出，(x − a) 是多项式 f(x) 的因式，当且仅当 f(a) = 0。这通常与多项式除法一起使用，用于对三次或四次表达式进行因式分解。
When dividing f(x) by (x − a), you can use algebraic long division or compare coefficients. The remainder theorem is equally useful: the remainder when f(x) is divided by (x − a) is exactly f(a).
用 (x − a) 除 f(x) 时，可以使用代数长除法或比较系数法。余式定理同样有用：f(x) 除以 (x − a) 的余式恰好是 f(a)。
4. Graphs and Function Transformations | 函数图像与变换
You need to recognise how the graph of y = f(x) changes under transformations. For example, y = f(x + 2) is a translation 2 units to the left, while y = f(2x) is a horizontal stretch with scale factor 1/2.
你需要识别 y = f(x) 的图像在变换下如何变化。例如，y = f(x + 2) 表示向左平移 2 个单位，而 y = f(2x) 表示水平方向以 1/2 为比例因子进行拉伸。
The modulus graph y = |f(x)| keeps all non-negative parts of f(x) and reflects any negative parts in the x-axis. Solving |x − 3| = 5 means considering both x − 3 = 5 and x − 3 = −5, giving x = 8 or x = −2.
绝对值图像 y = |f(x)| 保留 f(x) 的所有非负部分，并将任何负值部分关于 x 轴反射。求解 |x − 3| = 5 时，需要同时考虑 x − 3 = 5 和 x − 3 = −5，得到 x = 8 或 x = −2。
5. Differentiation Rules and Tangents | 微分法则与切线
The power rule states that if y = xⁿ, then dy/dx = n xⁿ⁻¹. For composite functions use the chain rule: if y = f(u) and u = g(x), then dy/dx = dy/du × du/dx.
幂法则为：若 y = xⁿ，则 dy/dx = n xⁿ⁻¹。对于复合函数使用链式法则：若 y = f(u) 且 u = g(x)，则 dy/dx = dy/du × du/dx。
To find the equation of a tangent at x = a, first find the gradient m = dy/dx at that point, then use y − y₁ = m(x − x₁). A normal line has gradient −1/m.
要求 x = a 处的切线方程，先求出该点的斜率 m = dy/dx，再用 y − y₁ = m(x − x₁) 写出方程。法线的斜率为 −1/m。
d/dx [xⁿ] = n xⁿ⁻¹, and d/dx [f(g(x))] = f ‘(g(x)) g ‘(x)
6. Integration and Area Under Curves | 积分与曲线下面积
Integration reverses differentiation. The basic rule is ∫ xⁿ dx = xⁿ⁺¹/(n+1) + c for n ≠ −1. Definite integrals from a to b are written as ∫ₐᵇ f(x) dx and are evaluated as F(b) − F(a).
积分是微分的逆运算。基本法则为 ∫ xⁿ dx = xⁿ⁺¹/(n+1) + c，其中 n ≠ −1。从 a 到 b 的定积分写作 ∫ₐᵇ f(x) dx，计算结果为 F(b) − F(a)。
The area between a curve and the x-axis from x = a to x = b is found by ∫ₐᵇ f(x) dx when f(x) is positive. If the curve crosses the x-axis, split the integral and take absolute values of each region.
当 f(x) 为正时，曲线与 x 轴之间从 x = a 到 x = b 的面积由 ∫ₐᵇ f(x) dx 给出。如果曲线穿过 x 轴，需要拆分积分，并对每一段区域取绝对值。
7. Trigonometric Equations and Identities | 三角方程与恒等式
Core identities include sin²θ + cos²θ ≡ 1 and tan θ ≡ sin θ / cos θ. These allow you to rewrite equations such as 2cos²θ + cos θ − 1 = 0 as a quadratic in cos θ.
核心恒等式包括 sin²θ + cos²θ ≡ 1 和 tan θ ≡ sin θ / cos θ。这些恒等式可以将 2cos²θ + cos θ − 1 = 0 这样的方程改写为关于 cos θ 的二次方程。
When solving trig equations, always identify all solutions in the required interval. For 2cos²θ + cos θ − 1 = 0, solve for cos θ = 1/2 or cos θ = −1, then use the cosine graph or CAST diagram to list every solution.
解三角方程时，必须找出指定区间内的所有解。对于 2cos²θ + cos θ − 1 = 0，解得 cos θ = 1/2 或 cos θ = −1，然后借助余弦图像或 CAST 图列出全部解。
8. Exponentials and Logarithms | 指数与对数
The natural exponential and logarithm are inverses: eˣ and ln x satisfy ln(eˣ) = x and e^(ln x) = x. To solve an equation such as e²ˣ = 5, take natural logs to get 2x = ln 5.
自然指数与自然对数互为逆运算：eˣ 与 ln x 满足 ln(eˣ) = x 且 e^(ln x) = x。要求解 e²ˣ = 5，可两边取自然对数得到 2x = ln 5。
Use the log laws ln(ab) = ln a + ln b, ln(a/b) = ln a − ln b and ln(aᵏ) = k ln a to condense or expand logarithmic expressions. Exponential growth models are often written as P = P₀e^(kt).
使用对数法则 ln(ab) = ln a + ln b、ln(a/b) = ln a − ln b 和 ln(aᵏ) = k ln a 来合并或展开对数表达式。指数增长模型通常写作 P = P₀e^(kt)。
9. Sequences and Series | 数列与级数
For an arithmetic sequence, the nth term is uₙ = a + (n − 1)d and the sum of the first n terms is Sₙ = n/2 (2a + (n − 1)d). Here a is the first term and d is the common difference.
对于等差数列，第 n 项为 uₙ = a + (n − 1)d，前 n 项和为 Sₙ = n/2 (2a + (n − 1)d)。其中 a 为首项，d 为公差。
For a geometric sequence, the nth term is uₙ = arⁿ⁻¹ and the sum of the first n terms is Sₙ = a(1 − rⁿ)/(1 − r). If |r| < 1, the infinite sum converges to S∞ = a/(1 − r).
对于等比数列，第 n 项为 uₙ = arⁿ⁻¹，前 n 项和为 Sₙ = a(1 − rⁿ)/(1 − r)。若 |r| < 1，则无穷级数收敛于 S∞ = a/(1 − r)。
10. Binomial Expansion | 二项展开
For positive integer n, the binomial expansion is (a + b)ⁿ = Σ nCr aⁿ⁻ʳ bʳ, where nCr = n! / [r!(n − r)!]. In Edexcel papers you often expand (1 + x)ⁿ for small powers or use the first few terms.
对于正整数 n，二项展开式为 (a + b)ⁿ = Σ nCr aⁿ⁻ʳ bʳ，其中 nCr = n! / [r!(n − r)!]。在爱德思考试中，常对 (1 + x)ⁿ 进行小次数展开，或使用前几项。
For rational n, the expansion (1 + x)ⁿ = 1 + nx + n(n−1)/2! x² + … is only valid when |x| < 1. You may be asked to state the range of validity or expand an expression like (1 − 2x)⁻¹.
对于有理数 n，展开式 (1 + x)ⁿ = 1 + nx + n(n−1)/2! x² + … 仅在 |x| < 1 时有效。题目可能要求写出有效范围，或展开 (1 − 2x)⁻¹ 这样的表达式。
11. Vectors in Pure Contexts | 纯数中的向量
In pure mathematics, vectors are used to model position, direction and distance in two or three dimensions. The magnitude of a vector a = xi + yj + zk is |a| = √(x² + y² + z²).
在纯数学中，向量用于表示二维或三维空间中的位置、方向和距离。向量 a = xi + yj + zk 的模为 |a| = √(x² + y² + z²)。
The scalar product a·b = |a||b| cos θ is used to find angles between vectors. If a and b are perpendicular, then a·b = 0. For coordinates, a·b = x₁x₂ + y₁y₂ + z₁z₂.
数量积 a·b = |a||b| cos θ 用于求向量间的夹角。若 a 与 b 垂直，则 a·b = 0。在坐标形式下，a·b = x₁x₂ + y₁y₂ + z₁z₂。
12. Exam Strategy and Common Pitfalls | 考试策略与常见错误
In Edexcel A Level Maths, method marks matter. Always show your working clearly, especially in differentiation, integration and equation solving, so that even a small slip does not lose the entire question.
在爱德思 A Level 数学中，步骤分非常重要。务必清晰写出解题过程，尤其是在微分、积分和方程求解中，这样即使出现小失误也不会丢掉整道题的分数。
Common pitfalls include forgetting the constant of integration +c, missing the validity condition |x| < 1 in binomial expansion, and using degrees instead of radians in calculus. Check exact answers when the question asks for surd or log form.
常见错误包括漏写积分常数 +c、忘记二项展开的有效条件 |x| < 1，以及在微积分中使用角度制而非弧度制。当题目要求根式或对数形式时，应核对是否为精确答案。
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